:: Operations on Subspaces in Real Unitary Space
:: by Noboru Endou , Takashi Mitsuishi and Yasunari Shidama
::
:: Received October 9, 2002
:: Copyright (c) 2002-2026 Association of Mizar Users
:: (Stowarzyszenie Uzytkownikow Mizara, Bialystok, Poland).
:: This code can be distributed under the GNU General Public Licence
:: version 3.0 or later, or the Creative Commons Attribution-ShareAlike
:: License version 3.0 or later, subject to the binding interpretation
:: detailed in file COPYING.interpretation.
:: See COPYING.GPL and COPYING.CC-BY-SA for the full text of these
:: licenses, or see http://www.gnu.org/licenses/gpl.html and
:: http://creativecommons.org/licenses/by-sa/3.0/.
environ
vocabularies BHSP_1, RLSUB_1, ARYTM_3, STRUCT_0, RLVECT_1, SUBSET_1, TARSKI,
SUPINF_2, XBOOLE_0, ARYTM_1, RLSUB_2, ZFMISC_1, FUNCT_1, RELAT_1, REAL_1,
CARD_1, FINSEQ_4, MCART_1, BINOP_1, LATTICES, EQREL_1, PBOOLE;
notations TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, RELAT_1, ORDINAL1, XCMPLX_0,
XREAL_0, STRUCT_0, FUNCT_1, NUMBERS, BINOP_1, LATTICES, RELSET_1, REAL_1,
DOMAIN_1, RLVECT_1, RLSUB_1, BHSP_1, RUSUB_1;
constructors BINOP_1, REAL_1, MEMBERED, REALSET1, LATTICES, RLSUB_1, RUSUB_1;
registrations SUBSET_1, MEMBERED, STRUCT_0, LATTICES, BHSP_1, RUSUB_1,
RELAT_1, XREAL_0, XTUPLE_0;
requirements NUMERALS, SUBSET, BOOLE;
definitions TARSKI, XBOOLE_0;
equalities XBOOLE_0, RLVECT_1;
expansions TARSKI, XBOOLE_0;
theorems RLVECT_1, FUNCT_1, TARSKI, ZFMISC_1, XBOOLE_0, RELAT_1, RLSUB_1,
XBOOLE_1, RUSUB_1, RLSUB_2, MCART_1, BINOP_1, LATTICES, XCMPLX_0,
ORDERS_1, STRUCT_0, XTUPLE_0;
schemes XBOOLE_0, FUNCT_1, RELSET_1, ORDERS_1, BINOP_1, CLASSES1;
begin :: Definitions of sum and intersection of subspaces.
definition
let V be RealUnitarySpace, W1,W2 be Subspace of V;
func W1 + W2 -> strict Subspace of V means
:Def1:
the carrier of it = {v + u where v,u is VECTOR of V: v in W1 & u in W2};
existence
proof
reconsider V1 = the carrier of W1, V2 = the carrier of W2 as Subset of V
by RUSUB_1:def 1;
set VS = {v + u where v,u is VECTOR of V: v in W1 & u in W2};
VS c= the carrier of V
proof
let x be object;
assume x in VS;
then ex v1,v2 being VECTOR of V st x = v1 + v2 & v1 in W1 & v2 in W2;
hence thesis;
end;
then reconsider VS as Subset of V;
A1: 0.V = 0.V + 0.V by RLVECT_1:4;
0.V in W1 & 0.V in W2 by RUSUB_1:11;
then
A2: 0.V in VS by A1;
A3: VS = {v + u where u,v is VECTOR of V: v in V1 & u in V2}
proof
thus VS c= {v + u where u,v is VECTOR of V: v in V1 & u in V2}
proof
let x be object;
assume x in VS;
then consider v,u being VECTOR of V such that
A4: x = v + u and
A5: v in W1 & u in W2;
v in V1 & u in V2 by A5,STRUCT_0:def 5;
hence thesis by A4;
end;
let x be object;
assume x in {v + u where u,v is VECTOR of V: v in V1 & u in V2};
then consider u,v being VECTOR of V such that
A6: x = v + u and
A7: v in V1 & u in V2;
v in W1 & u in W2 by A7,STRUCT_0:def 5;
hence thesis by A6;
end;
V1 is linearly-closed & V2 is linearly-closed by RUSUB_1:28;
hence thesis by A2,A3,RLSUB_1:6,RUSUB_1:29;
end;
uniqueness by RUSUB_1:24;
end;
definition
let V be RealUnitarySpace, W1,W2 be Subspace of V;
func W1 /\ W2 -> strict Subspace of V means
:Def2:
the carrier of it = (the carrier of W1) /\ (the carrier of W2);
existence
proof
set VW2 = the carrier of W2;
set VW1 = the carrier of W1;
set VV = the carrier of V;
0.V in W2 by RUSUB_1:11;
then
A1: 0.V in VW2 by STRUCT_0:def 5;
VW1 c= VV & VW2 c= VV by RUSUB_1:def 1;
then VW1 /\ VW2 c= VV /\ VV by XBOOLE_1:27;
then reconsider V1 = VW1, V2 = VW2, V3 = VW1 /\ VW2 as Subset of V by
RUSUB_1:def 1;
V1 is linearly-closed & V2 is linearly-closed by RUSUB_1:28;
then
A2: V3 is linearly-closed by RLSUB_1:7;
0.V in W1 by RUSUB_1:11;
then 0.V in VW1 by STRUCT_0:def 5;
then VW1 /\ VW2 <> {} by A1,XBOOLE_0:def 4;
hence thesis by A2,RUSUB_1:29;
end;
uniqueness by RUSUB_1:24;
end;
begin :: Theorems of sum and intersection of subspaces.
theorem Th1:
for V being RealUnitarySpace, W1,W2 being Subspace of V,
x being object
holds x in W1 + W2 iff ex v1,v2 being VECTOR of V st v1 in W1 & v2 in W2 &
x = v1 + v2
proof
let V be RealUnitarySpace;
let W1,W2 be Subspace of V;
let x be object;
thus x in W1 + W2 implies ex v1,v2 being VECTOR of V st v1 in W1 & v2 in W2
& x = v1 + v2
proof
assume x in W1 + W2;
then x in the carrier of W1 + W2 by STRUCT_0:def 5;
then x in {v + u where v,u is VECTOR of V : v in W1 & u in W2} by Def1;
then consider v1,v2 being VECTOR of V such that
A1: x = v1 + v2 & v1 in W1 & v2 in W2;
take v1,v2;
thus thesis by A1;
end;
given v1,v2 be VECTOR of V such that
A2: v1 in W1 & v2 in W2 & x = v1 + v2;
x in {v + u where v,u is VECTOR of V : v in W1 & u in W2} by A2;
then x in the carrier of W1 + W2 by Def1;
hence thesis by STRUCT_0:def 5;
end;
theorem Th2:
for V being RealUnitarySpace, W1,W2 being Subspace of V, v being
VECTOR of V st v in W1 or v in W2 holds v in W1 + W2
proof
let V be RealUnitarySpace;
let W1,W2 be Subspace of V;
let v be VECTOR of V;
assume
A1: v in W1 or v in W2;
now
per cases by A1;
suppose
A2: v in W1;
v = v + 0.V & 0.V in W2 by RLVECT_1:4,RUSUB_1:11;
hence thesis by A2,Th1;
end;
suppose
A3: v in W2;
v = 0.V + v & 0.V in W1 by RLVECT_1:4,RUSUB_1:11;
hence thesis by A3,Th1;
end;
end;
hence thesis;
end;
theorem Th3:
for V being RealUnitarySpace, W1,W2 being Subspace of V, x being
object holds x in W1 /\ W2 iff x in W1 & x in W2
proof
let V be RealUnitarySpace;
let W1,W2 be Subspace of V;
let x be object;
x in W1 /\ W2 iff x in the carrier of W1 /\ W2 by STRUCT_0:def 5;
then x in W1 /\ W2 iff x in (the carrier of W1) /\ (the carrier of W2) by
Def2;
then x in W1 /\ W2 iff x in the carrier of W1 & x in the carrier of W2 by
XBOOLE_0:def 4;
hence thesis by STRUCT_0:def 5;
end;
Lm1: for V being RealUnitarySpace, W1,W2 being Subspace of V holds W1 + W2 =
W2 + W1
proof
let V be RealUnitarySpace;
let W1,W2 be Subspace of V;
set A = {v + u where v,u is VECTOR of V : v in W1 & u in W2};
set B = {v + u where v,u is VECTOR of V : v in W2 & u in W1};
A1: B c= A
proof
let x be object;
assume x in B;
then ex v,u being VECTOR of V st x = v + u & v in W2 & u in W1;
hence thesis;
end;
A2: the carrier of W1 + W2 = A by Def1;
A c= B
proof
let x be object;
assume x in A;
then ex v,u being VECTOR of V st x = v + u & v in W1 & u in W2;
hence thesis;
end;
then A = B by A1;
hence thesis by A2,Def1;
end;
Lm2: for V being RealUnitarySpace, W1,W2 being Subspace of V holds the carrier
of W1 c= the carrier of W1 + W2
proof
let V be RealUnitarySpace;
let W1,W2 be Subspace of V;
let x be object;
set A = {v + u where v,u is VECTOR of V : v in W1 & u in W2};
assume x in the carrier of W1;
then reconsider v = x as Element of W1;
reconsider v as VECTOR of V by RUSUB_1:3;
A1: v = v + 0.V by RLVECT_1:4;
v in W1 & 0.V in W2 by RUSUB_1:11,STRUCT_0:def 5;
then x in A by A1;
hence thesis by Def1;
end;
Lm3: for V being RealUnitarySpace, W1 being Subspace of V, W2 being strict
Subspace of V st the carrier of W1 c= the carrier of W2 holds W1 + W2 = W2
proof
let V be RealUnitarySpace;
let W1 be Subspace of V;
let W2 be strict Subspace of V;
assume
A1: the carrier of W1 c= the carrier of W2;
A2: the carrier of W1 + W2 c= the carrier of W2
proof
let x be object;
assume x in the carrier of W1 + W2;
then x in {v + u where v,u is VECTOR of V : v in W1 & u in W2} by Def1;
then consider v,u being VECTOR of V such that
A3: x = v + u and
A4: v in W1 and
A5: u in W2;
W1 is Subspace of W2 by A1,RUSUB_1:22;
then v in W2 by A4,RUSUB_1:1;
then v + u in W2 by A5,RUSUB_1:14;
hence thesis by A3,STRUCT_0:def 5;
end;
W1 + W2 = W2 + W1 by Lm1;
then the carrier of W2 c= the carrier of W1 + W2 by Lm2;
then the carrier of W1 + W2 = the carrier of W2 by A2;
hence thesis by RUSUB_1:24;
end;
theorem
for V being RealUnitarySpace, W being strict Subspace of V holds W + W
= W by Lm3;
theorem
for V being RealUnitarySpace, W1,W2 being Subspace of V holds W1 + W2
= W2 + W1 by Lm1;
theorem Th6:
for V being RealUnitarySpace, W1,W2,W3 being Subspace of V holds
W1 + (W2 + W3) = (W1 + W2) + W3
proof
let V be RealUnitarySpace;
let W1,W2,W3 be Subspace of V;
set A = {v + u where v,u is VECTOR of V : v in W1 & u in W2};
set B = {v + u where v,u is VECTOR of V : v in W2 & u in W3};
set C = {v + u where v,u is VECTOR of V : v in W1 + W2 & u in W3};
set D = {v + u where v,u is VECTOR of V : v in W1 & u in W2 + W3};
A1: the carrier of W1 + (W2 + W3) = D by Def1;
A2: C c= D
proof
let x be object;
assume x in C;
then consider v,u being VECTOR of V such that
A3: x = v + u and
A4: v in W1 + W2 and
A5: u in W3;
v in the carrier of W1 + W2 by A4,STRUCT_0:def 5;
then v in A by Def1;
then consider u1,u2 being VECTOR of V such that
A6: v = u1 + u2 and
A7: u1 in W1 and
A8: u2 in W2;
u2 + u in B by A5,A8;
then u2 + u in the carrier of W2 + W3 by Def1;
then
A9: u2 + u in W2 + W3 by STRUCT_0:def 5;
v + u =u1 + (u2 + u) by A6,RLVECT_1:def 3;
hence thesis by A3,A7,A9;
end;
D c= C
proof
let x be object;
assume x in D;
then consider v,u being VECTOR of V such that
A10: x = v + u and
A11: v in W1 and
A12: u in W2 + W3;
u in the carrier of W2 + W3 by A12,STRUCT_0:def 5;
then u in B by Def1;
then consider u1,u2 being VECTOR of V such that
A13: u = u1 + u2 and
A14: u1 in W2 and
A15: u2 in W3;
v + u1 in A by A11,A14;
then v + u1 in the carrier of W1 + W2 by Def1;
then
A16: v + u1 in W1 + W2 by STRUCT_0:def 5;
v + u = (v + u1) + u2 by A13,RLVECT_1:def 3;
hence thesis by A10,A15,A16;
end;
then D = C by A2;
hence thesis by A1,Def1;
end;
theorem Th7:
for V being RealUnitarySpace, W1,W2 being Subspace of V holds W1
is Subspace of W1 + W2 & W2 is Subspace of W1 + W2
proof
let V be RealUnitarySpace;
let W1,W2 be Subspace of V;
the carrier of W1 c= the carrier of W1 + W2 by Lm2;
hence W1 is Subspace of W1 + W2 by RUSUB_1:22;
the carrier of W2 c= the carrier of W2 + W1 by Lm2;
then the carrier of W2 c= the carrier of W1 + W2 by Lm1;
hence thesis by RUSUB_1:22;
end;
theorem Th8:
for V being RealUnitarySpace, W1 being Subspace of V, W2 being
strict Subspace of V holds W1 is Subspace of W2 iff W1 + W2 = W2
proof
let V be RealUnitarySpace;
let W1 be Subspace of V;
let W2 be strict Subspace of V;
thus W1 is Subspace of W2 implies W1 + W2 = W2
proof
assume W1 is Subspace of W2;
then the carrier of W1 c= the carrier of W2 by RUSUB_1:def 1;
hence thesis by Lm3;
end;
thus thesis by Th7;
end;
theorem Th9:
for V being RealUnitarySpace, W being strict Subspace of V holds
(0).V + W = W & W + (0).V = W
proof
let V be RealUnitarySpace;
let W be strict Subspace of V;
(0).V is Subspace of W by RUSUB_1:33;
then the carrier of (0).V c= the carrier of W by RUSUB_1:def 1;
hence (0).V + W = W by Lm3;
hence thesis by Lm1;
end;
theorem Th10:
for V being RealUnitarySpace holds (0).V + (Omega).V = the
UNITSTR of V & (Omega).V + (0).V = the UNITSTR of V
proof
let V be RealUnitarySpace;
thus (0).V + (Omega).V = (Omega).V by Th9
.= the UNITSTR of V by RUSUB_1:def 3;
hence thesis by Lm1;
end;
theorem Th11:
for V being RealUnitarySpace, W being Subspace of V holds
(Omega).V + W = the UNITSTR of V & W + (Omega).V = the UNITSTR of V
proof
let V be RealUnitarySpace;
let W be Subspace of V;
the carrier of V = the carrier of the UNITSTR of V & the carrier of W c=
the carrier of V by RUSUB_1:def 1;
then the carrier of W c= the carrier of (Omega).V by RUSUB_1:def 3;
then W + (Omega).V = (Omega).V by Lm3
.= the UNITSTR of V by RUSUB_1:def 3;
hence thesis by Lm1;
end;
theorem
for V being strict RealUnitarySpace holds (Omega).V + (Omega).V = V by Th11;
theorem
for V being RealUnitarySpace, W being strict Subspace of V holds W /\ W = W
proof
let V be RealUnitarySpace;
let W be strict Subspace of V;
the carrier of W = (the carrier of W) /\ (the carrier of W);
hence thesis by Def2;
end;
theorem Th14:
for V being RealUnitarySpace, W1,W2 being Subspace of V holds W1
/\ W2 = W2 /\ W1
proof
let V be RealUnitarySpace;
let W1,W2 be Subspace of V;
the carrier of W1 /\ W2 = (the carrier of W2) /\ (the carrier of W1) by Def2;
hence thesis by Def2;
end;
theorem Th15:
for V being RealUnitarySpace, W1,W2,W3 being Subspace of V holds
W1 /\ (W2 /\ W3) = (W1 /\ W2) /\ W3
proof
let V be RealUnitarySpace;
let W1,W2,W3 be Subspace of V;
set V1 = the carrier of W1;
set V2 = the carrier of W2;
set V3 = the carrier of W3;
the carrier of W1 /\ (W2 /\ W3) = V1 /\ (the carrier of W2 /\ W3) by Def2
.= V1 /\ (V2 /\ V3) by Def2
.= (V1 /\ V2) /\ V3 by XBOOLE_1:16
.= (the carrier of W1 /\ W2) /\ V3 by Def2;
hence thesis by Def2;
end;
Lm4: for V being RealUnitarySpace, W1,W2 being Subspace of V holds the carrier
of W1 /\ W2 c= the carrier of W1
proof
let V be RealUnitarySpace;
let W1,W2 be Subspace of V;
the carrier of W1 /\ W2 = (the carrier of W1) /\ (the carrier of W2) by Def2;
hence thesis by XBOOLE_1:17;
end;
theorem Th16:
for V being RealUnitarySpace, W1,W2 being Subspace of V holds W1
/\ W2 is Subspace of W1 & W1 /\ W2 is Subspace of W2
proof
let V be RealUnitarySpace;
let W1,W2 be Subspace of V;
the carrier of W1 /\ W2 c= the carrier of W1 by Lm4;
hence W1 /\ W2 is Subspace of W1 by RUSUB_1:22;
the carrier of W2 /\ W1 c= the carrier of W2 by Lm4;
then the carrier of W1 /\ W2 c= the carrier of W2 by Th14;
hence thesis by RUSUB_1:22;
end;
theorem Th17:
for V being RealUnitarySpace, W2 being Subspace of V, W1 being
strict Subspace of V holds W1 is Subspace of W2 iff W1 /\ W2 = W1
proof
let V be RealUnitarySpace;
let W2 be Subspace of V;
let W1 be strict Subspace of V;
thus W1 is Subspace of W2 implies W1 /\ W2 = W1
proof
assume W1 is Subspace of W2;
then
A1: the carrier of W1 c= the carrier of W2 by RUSUB_1:def 1;
the carrier of W1 /\ W2 = (the carrier of W1) /\ (the carrier of W2)
by Def2;
hence thesis by A1,RUSUB_1:24,XBOOLE_1:28;
end;
thus thesis by Th16;
end;
theorem Th18:
for V being RealUnitarySpace, W being Subspace of V holds (0).V
/\ W = (0).V & W /\ (0).V = (0).V
proof
let V be RealUnitarySpace;
let W be Subspace of V;
0.V in W by RUSUB_1:11;
then 0.V in the carrier of W by STRUCT_0:def 5;
then {0.V} c= the carrier of W by ZFMISC_1:31;
then
A1: {0.V} /\ (the carrier of W) = {0.V} by XBOOLE_1:28;
the carrier of (0).V /\ W = (the carrier of (0).V) /\ (the carrier of W)
by Def2
.= {0.V} /\ (the carrier of W) by RUSUB_1:def 2;
hence (0).V /\ W = (0).V by A1,RUSUB_1:def 2;
hence thesis by Th14;
end;
theorem
for V being RealUnitarySpace holds (0).V /\ (Omega).V = (0).V &
(Omega).V /\ (0).V = (0).V by Th18;
theorem Th20:
for V being RealUnitarySpace, W being strict Subspace of V holds
(Omega).V /\ W = W & W /\ (Omega).V = W
proof
let V be RealUnitarySpace;
let W be strict Subspace of V;
(Omega).V = the UNITSTR of V by RUSUB_1:def 3;
then
A1: the carrier of (Omega).V /\ W = (the carrier of V) /\ (the carrier of W
) by Def2;
the carrier of W c= the carrier of V by RUSUB_1:def 1;
hence (Omega).V /\ W = W by A1,RUSUB_1:24,XBOOLE_1:28;
hence thesis by Th14;
end;
theorem
for V being strict RealUnitarySpace holds (Omega).V /\ (Omega).V = V
proof
let V be strict RealUnitarySpace;
thus (Omega).V /\ (Omega).V = (Omega).V by Th20
.= V by RUSUB_1:def 3;
end;
Lm5: for V being RealUnitarySpace, W1,W2 being Subspace of V holds the carrier
of W1 /\ W2 c= the carrier of W1 + W2
proof
let V be RealUnitarySpace;
let W1,W2 be Subspace of V;
the carrier of W1 /\ W2 c= the carrier of W1 & the carrier of W1 c= the
carrier of W1 + W2 by Lm2,Lm4;
hence thesis;
end;
theorem
for V being RealUnitarySpace, W1,W2 being Subspace of V holds W1 /\ W2
is Subspace of W1 + W2 by Lm5,RUSUB_1:22;
Lm6: for V being RealUnitarySpace, W1,W2 being Subspace of V holds the carrier
of (W1 /\ W2) + W2 = the carrier of W2
proof
let V be RealUnitarySpace;
let W1,W2 be Subspace of V;
thus the carrier of (W1 /\ W2) + W2 c= the carrier of W2
proof
let x be object;
assume x in the carrier of (W1 /\ W2) + W2;
then x in {u + v where u,v is VECTOR of V : u in W1 /\ W2 & v in W2} by
Def1;
then consider u,v being VECTOR of V such that
A1: x = u + v and
A2: u in W1 /\ W2 and
A3: v in W2;
u in W2 by A2,Th3;
then u + v in W2 by A3,RUSUB_1:14;
hence thesis by A1,STRUCT_0:def 5;
end;
let x be object;
the carrier of W2 c= the carrier of W2 + (W1 /\ W2) by Lm2;
then
A4: the carrier of W2 c= the carrier of (W1 /\ W2) + W2 by Lm1;
assume x in the carrier of W2;
hence thesis by A4;
end;
theorem
for V being RealUnitarySpace, W1 being Subspace of V, W2 being strict
Subspace of V holds (W1 /\ W2) + W2 = W2 by Lm6,RUSUB_1:24;
Lm7: for V being RealUnitarySpace, W1,W2 being Subspace of V holds the carrier
of W1 /\ (W1 + W2) = the carrier of W1
proof
let V be RealUnitarySpace;
let W1,W2 be Subspace of V;
thus the carrier of W1 /\ (W1 + W2) c= the carrier of W1
proof
let x be object;
assume
A1: x in the carrier of W1 /\ (W1 + W2);
the carrier of W1 /\ (W1 + W2) = (the carrier of W1) /\ (the carrier
of W1 + W2) by Def2;
hence thesis by A1,XBOOLE_0:def 4;
end;
let x be object;
assume
A2: x in the carrier of W1;
the carrier of W1 c= the carrier of V by RUSUB_1:def 1;
then reconsider x1 = x as Element of V by A2;
A3: x1 + 0.V = x1 & 0.V in W2 by RLVECT_1:4,RUSUB_1:11;
x in W1 by A2,STRUCT_0:def 5;
then x in {u + v where u,v is VECTOR of V : u in W1 & v in W2} by A3;
then x in the carrier of W1 + W2 by Def1;
then x in (the carrier of W1) /\ (the carrier of W1 + W2) by A2,
XBOOLE_0:def 4;
hence thesis by Def2;
end;
theorem
for V being RealUnitarySpace, W1 being Subspace of V, W2 being strict
Subspace of V holds W2 /\ (W2 + W1) = W2 by Lm7,RUSUB_1:24;
Lm8: for V being RealUnitarySpace, W1,W2,W3 being Subspace of V holds the
carrier of (W1 /\ W2) + (W2 /\ W3) c= the carrier of W2 /\ (W1 + W3)
proof
let V be RealUnitarySpace;
let W1,W2,W3 be Subspace of V;
let x be object;
assume x in the carrier of (W1 /\ W2) + (W2 /\ W3);
then
x in {u + v where u,v is VECTOR of V: u in W1 /\ W2 & v in W2 /\ W3} by Def1;
then consider u,v being VECTOR of V such that
A1: x = u + v and
A2: u in W1 /\ W2 & v in W2 /\ W3;
u in W2 & v in W2 by A2,Th3;
then
A3: x in W2 by A1,RUSUB_1:14;
u in W1 & v in W3 by A2,Th3;
then x in W1 + W3 by A1,Th1;
then x in W2 /\ (W1 + W3) by A3,Th3;
hence thesis by STRUCT_0:def 5;
end;
theorem
for V being RealUnitarySpace, W1,W2,W3 being Subspace of V holds (W1
/\ W2) + (W2 /\ W3) is Subspace of W2 /\ (W1 + W3) by Lm8,RUSUB_1:22;
Lm9: for V being RealUnitarySpace, W1,W2,W3 being Subspace of V st W1 is
Subspace of W2 holds the carrier of W2 /\ (W1 + W3) = the carrier of (W1 /\ W2)
+ (W2 /\ W3)
proof
let V be RealUnitarySpace;
let W1,W2,W3 be Subspace of V;
assume
A1: W1 is Subspace of W2;
thus the carrier of W2 /\ (W1 + W3) c= the carrier of (W1 /\ W2) + (W2 /\ W3
)
proof
let x be object;
assume x in the carrier of W2 /\ (W1 + W3);
then
A2: x in (the carrier of W2) /\ (the carrier of W1 + W3) by Def2;
then x in the carrier of W1 + W3 by XBOOLE_0:def 4;
then x in {u + v where u,v is VECTOR of V : u in W1 & v in W3} by Def1;
then consider u1,v1 being VECTOR of V such that
A3: x = u1 + v1 and
A4: u1 in W1 and
A5: v1 in W3;
A6: u1 in W2 by A1,A4,RUSUB_1:1;
x in the carrier of W2 by A2,XBOOLE_0:def 4;
then u1 + v1 in W2 by A3,STRUCT_0:def 5;
then (v1 + u1) - u1 in W2 by A6,RUSUB_1:17;
then v1 + (u1 - u1) in W2 by RLVECT_1:def 3;
then v1 + 0.V in W2 by RLVECT_1:15;
then v1 in W2 by RLVECT_1:4;
then
A7: v1 in W2 /\ W3 by A5,Th3;
u1 in W1 /\ W2 by A4,A6,Th3;
then x in (W1 /\ W2) + (W2 /\ W3) by A3,A7,Th1;
hence thesis by STRUCT_0:def 5;
end;
thus thesis by Lm8;
end;
theorem
for V being RealUnitarySpace, W1,W2,W3 being Subspace of V st W1 is
Subspace of W2 holds W2 /\ (W1 + W3) = (W1 /\ W2) + (W2 /\ W3) by Lm9,
RUSUB_1:24;
Lm10: for V being RealUnitarySpace, W1,W2,W3 being Subspace of V holds the
carrier of W2 + (W1 /\ W3) c= the carrier of (W1 + W2) /\ (W2 + W3)
proof
let V be RealUnitarySpace;
let W1,W2,W3 be Subspace of V;
let x be object;
assume x in the carrier of W2 + (W1 /\ W3);
then x in {u + v where u,v is VECTOR of V: u in W2 & v in W1 /\ W3} by Def1;
then consider u,v being VECTOR of V such that
A1: x = u + v & u in W2 and
A2: v in W1 /\ W3;
v in W3 by A2,Th3;
then x in {u1 + u2 where u1,u2 is VECTOR of V : u1 in W2 & u2 in W3} by A1;
then
A3: x in the carrier of W2 + W3 by Def1;
v in W1 by A2,Th3;
then x in {v1 + v2 where v1,v2 is VECTOR of V : v1 in W1 & v2 in W2} by A1;
then x in the carrier of W1 + W2 by Def1;
then x in (the carrier of W1 + W2) /\ (the carrier of W2 + W3) by A3,
XBOOLE_0:def 4;
hence thesis by Def2;
end;
theorem
for V being RealUnitarySpace, W1,W2,W3 being Subspace of V holds W2 +
(W1 /\ W3) is Subspace of (W1 + W2) /\ (W2 + W3) by Lm10,RUSUB_1:22;
Lm11: for V being RealUnitarySpace, W1,W2,W3 being Subspace of V st W1 is
Subspace of W2 holds the carrier of W2 + (W1 /\ W3) = the carrier of (W1 + W2)
/\ (W2 + W3)
proof
let V be RealUnitarySpace;
let W1,W2,W3 be Subspace of V;
reconsider V2 = the carrier of W2 as Subset of V by RUSUB_1:def 1;
A1: V2 is linearly-closed by RUSUB_1:28;
assume W1 is Subspace of W2;
then
A2: the carrier of W1 c= the carrier of W2 by RUSUB_1:def 1;
thus the carrier of W2 + (W1 /\ W3) c= the carrier of (W1 + W2) /\ (W2 + W3
) by Lm10;
let x be object;
assume x in the carrier of (W1 + W2) /\ (W2 + W3);
then x in (the carrier of W1 + W2) /\ (the carrier of W2 + W3) by Def2;
then x in the carrier of W1 + W2 by XBOOLE_0:def 4;
then x in {u1 + u2 where u1,u2 is VECTOR of V : u1 in W1 & u2 in W2} by Def1;
then consider u1,u2 being VECTOR of V such that
A3: x = u1 + u2 and
A4: u1 in W1 & u2 in W2;
u1 in the carrier of W1 & u2 in the carrier of W2 by A4,STRUCT_0:def 5;
then u1 + u2 in V2 by A2,A1,RLSUB_1:def 1;
then
A5: u1 + u2 in W2 by STRUCT_0:def 5;
0.V in W1 /\ W3 & (u1 + u2) + 0.V = u1 + u2 by RLVECT_1:4,RUSUB_1:11;
then
x in {u + v where u,v is VECTOR of V : u in W2 & v in W1 /\ W3} by A3,A5;
hence thesis by Def1;
end;
theorem
for V being RealUnitarySpace, W1,W2,W3 being Subspace of V st W1 is
Subspace of W2 holds W2 + (W1 /\ W3) = (W1 + W2) /\ (W2 + W3) by Lm11,
RUSUB_1:24;
theorem Th29:
for V being RealUnitarySpace, W1,W2,W3 being Subspace of V st W1
is strict Subspace of W3 holds W1 + (W2 /\ W3) = (W1 + W2) /\ W3
proof
let V be RealUnitarySpace;
let W1,W2,W3 be Subspace of V;
assume
A1: W1 is strict Subspace of W3;
thus (W1 + W2) /\ W3 = W3 /\ (W1 + W2) by Th14
.= (W1 /\ W3) + (W3 /\ W2) by A1,Lm9,RUSUB_1:24
.= W1 + (W3 /\ W2) by A1,Th17
.= W1 + (W2 /\ W3) by Th14;
end;
theorem
for V being RealUnitarySpace, W1,W2 being strict Subspace of V holds
W1 + W2 = W2 iff W1 /\ W2 = W1
proof
let V be RealUnitarySpace;
let W1,W2 be strict Subspace of V;
W1 + W2 = W2 iff W1 is Subspace of W2 by Th8;
hence thesis by Th17;
end;
theorem
for V being RealUnitarySpace, W1 being Subspace of V, W2,W3 being
strict Subspace of V holds W1 is Subspace of W2 implies W1 + W3 is Subspace of
W2 + W3
proof
let V be RealUnitarySpace;
let W1 be Subspace of V;
let W2,W3 be strict Subspace of V;
assume
A1: W1 is Subspace of W2;
(W1 + W3) + (W2 + W3) = (W1 + W3) + (W3 + W2) by Lm1
.= ((W1 + W3) + W3) + W2 by Th6
.= (W1 + (W3 + W3)) + W2 by Th6
.= (W1 + W3) + W2 by Lm3
.= W1 + (W3 + W2) by Th6
.= W1 + (W2 + W3) by Lm1
.= (W1 + W2) + W3 by Th6
.= W2 + W3 by A1,Th8;
hence thesis by Th8;
end;
theorem
for V being RealUnitarySpace, W1,W2 being Subspace of V holds (ex W
being Subspace of V st the carrier of W = (the carrier of W1) \/ (the carrier
of W2)) iff W1 is Subspace of W2 or W2 is Subspace of W1
proof
let V be RealUnitarySpace;
let W1,W2 be Subspace of V;
set VW1 = the carrier of W1;
set VW2 = the carrier of W2;
thus (ex W being Subspace of V st the carrier of W = (the carrier of W1) \/
(the carrier of W2)) implies W1 is Subspace of W2 or W2 is Subspace of W1
proof
given W be Subspace of V such that
A1: the carrier of W = (the carrier of W1) \/ (the carrier of W2);
set VW = the carrier of W;
assume that
A2: not W1 is Subspace of W2 and
A3: not W2 is Subspace of W1;
not VW2 c= VW1 by A3,RUSUB_1:22;
then consider y being object such that
A4: y in VW2 and
A5: not y in VW1;
reconsider y as Element of VW2 by A4;
reconsider y as VECTOR of V by RUSUB_1:3;
reconsider A1 = VW as Subset of V by RUSUB_1:def 1;
A6: A1 is linearly-closed by RUSUB_1:28;
not VW1 c= VW2 by A2,RUSUB_1:22;
then consider x being object such that
A7: x in VW1 and
A8: not x in VW2;
reconsider x as Element of VW1 by A7;
reconsider x as VECTOR of V by RUSUB_1:3;
A9: now
reconsider A2 = VW2 as Subset of V by RUSUB_1:def 1;
A10: A2 is linearly-closed by RUSUB_1:28;
assume x + y in VW2;
then (x + y) - y in VW2 by A10,RLSUB_1:3;
then x + (y - y) in VW2 by RLVECT_1:def 3;
then x + 0.V in VW2 by RLVECT_1:15;
hence contradiction by A8,RLVECT_1:4;
end;
A11: now
reconsider A2 = VW1 as Subset of V by RUSUB_1:def 1;
A12: A2 is linearly-closed by RUSUB_1:28;
assume x + y in VW1;
then (y + x) - x in VW1 by A12,RLSUB_1:3;
then y + (x - x) in VW1 by RLVECT_1:def 3;
then y + 0.V in VW1 by RLVECT_1:15;
hence contradiction by A5,RLVECT_1:4;
end;
x in VW & y in VW by A1,XBOOLE_0:def 3;
then x + y in VW by A6,RLSUB_1:def 1;
hence thesis by A1,A11,A9,XBOOLE_0:def 3;
end;
A13: now
assume W1 is Subspace of W2;
then VW1 c= VW2 by RUSUB_1:def 1;
then VW1 \/ VW2 = VW2 by XBOOLE_1:12;
hence thesis;
end;
A14: now
assume W2 is Subspace of W1;
then VW2 c= VW1 by RUSUB_1:def 1;
then VW1 \/ VW2 = VW1 by XBOOLE_1:12;
hence thesis;
end;
assume W1 is Subspace of W2 or W2 is Subspace of W1;
hence thesis by A13,A14;
end;
begin :: Introduction of a set of subspaces of real unitary space.
definition
let V be RealUnitarySpace;
func Subspaces(V) -> set means
:Def3:
for x being object holds x in it iff x is strict Subspace of V;
existence
proof
defpred P[object,object] means
ex W being strict Subspace of V st $2 = W & $1 = the carrier of W;
defpred P[object] means
ex W being strict Subspace of V st $1 = the carrier of W;
consider B being set such that
A1: for x being object holds x in B iff x in bool(the carrier of V) & P[x
] from XBOOLE_0:sch 1;
A2: for x,y1,y2 being object st P[x,y1] & P[x,y2] holds y1 = y2 by RUSUB_1:24;
consider f being Function such that
A3: for x,y being object holds [x,y] in f iff x in B & P[x,y] from
FUNCT_1:sch 1 (A2);
for x being object holds x in B iff ex y being object st [x,y] in f
proof
let x be object;
thus x in B implies ex y being object st [x,y] in f
proof
assume
A4: x in B;
then consider W being strict Subspace of V such that
A5: x = the carrier of W by A1;
take W;
thus thesis by A3,A4,A5;
end;
thus thesis by A3;
end;
then
A6: B = dom f by XTUPLE_0:def 12;
for y being object holds y in rng f iff y is strict Subspace of V
proof
let y be object;
thus y in rng f implies y is strict Subspace of V
proof
assume y in rng f;
then consider x being object such that
A7: x in dom f & y = f.x by FUNCT_1:def 3;
[x,y] in f by A7,FUNCT_1:def 2;
then ex W being strict Subspace of V st y = W & x = the carrier of W
by A3;
hence thesis;
end;
assume y is strict Subspace of V;
then reconsider W = y as strict Subspace of V;
reconsider x = the carrier of W as set;
the carrier of W c= the carrier of V by RUSUB_1:def 1;
then
A8: x in dom f by A1,A6;
A9: y is set by TARSKI:1;
[x,y] in f by A3,A6,A8;
then y = f.x by A8,FUNCT_1:def 2,A9;
hence thesis by A8,FUNCT_1:def 3;
end;
hence thesis;
end;
uniqueness
proof
let D1,D2 be set;
assume
A10: for x being object holds x in D1 iff x is strict Subspace of V;
assume
A11: for x being object holds x in D2 iff x is strict Subspace of V;
now
let x be object;
thus x in D1 implies x in D2
proof
assume x in D1;
then x is strict Subspace of V by A10;
hence thesis by A11;
end;
assume x in D2;
then x is strict Subspace of V by A11;
hence x in D1 by A10;
end;
hence thesis by TARSKI:2;
end;
end;
registration
let V be RealUnitarySpace;
cluster Subspaces(V) -> non empty;
coherence
proof
set x = the strict Subspace of V;
x in Subspaces(V) by Def3;
hence thesis;
end;
end;
theorem
for V being strict RealUnitarySpace holds V in Subspaces(V)
proof
let V be strict RealUnitarySpace;
(Omega).V in Subspaces(V) by Def3;
hence thesis by RUSUB_1:def 3;
end;
begin :: Definition of the direct sum and linear complement of subspace
definition
let V be RealUnitarySpace;
let W1,W2 be Subspace of V;
pred V is_the_direct_sum_of W1,W2 means
the UNITSTR of V = W1 + W2 & W1 /\ W2 = (0).V;
end;
Lm12: for V being RealUnitarySpace, W being strict Subspace of V st (for v
being VECTOR of V holds v in W) holds W = the UNITSTR of V
proof
let V be RealUnitarySpace, W be strict Subspace of V;
assume
A1: for v being VECTOR of V holds v in W;
now
let v be VECTOR of V;
thus v in W implies v in (Omega).V
proof
assume v in W;
v in the UNITSTR of V by RLVECT_1:1;
hence thesis by RUSUB_1:def 3;
end;
assume v in (Omega).V;
thus v in W by A1;
end;
then W = (Omega).V by RUSUB_1:25;
hence thesis by RUSUB_1:def 3;
end;
Lm13: for V being RealUnitarySpace, W1,W2 being Subspace of V holds W1 + W2 =
the UNITSTR of V iff for v being VECTOR of V ex v1,v2 being VECTOR of V st v1
in W1 & v2 in W2 & v = v1 + v2
proof
let V be RealUnitarySpace, W1,W2 be Subspace of V;
thus W1 + W2 = the UNITSTR of V implies for v being VECTOR of V ex v1,v2
being VECTOR of V st v1 in W1 & v2 in W2 & v = v1 + v2
by RLVECT_1:1,Th1;
assume
A1: for v being VECTOR of V ex v1,v2 being VECTOR of V st v1 in W1 & v2
in W2 & v = v1 + v2;
now
let u be VECTOR of V;
ex v1,v2 being VECTOR of V st v1 in W1 & v2 in W2 & u = v1 + v2 by A1;
hence u in W1 + W2 by Th1;
end;
hence thesis by Lm12;
end;
Lm14: for V being RealUnitarySpace, W being Subspace of V ex C being strict
Subspace of V st V is_the_direct_sum_of C,W
proof
let V be RealUnitarySpace;
let W be Subspace of V;
defpred P[object] means
ex W1 being strict Subspace of V st $1 = W1 & W /\ W1 = (0).V;
consider X being set such that
A1: for x being object holds x in X iff x in Subspaces(V) & P[x] from
XBOOLE_0:sch 1;
W /\ (0).V = (0).V & (0).V in Subspaces(V) by Def3,Th18;
then reconsider X as non empty set by A1;
defpred P[object,object] means
ex W1,W2 being strict Subspace of V st $1 = W1 & $2
= W2 & W1 is Subspace of W2;
consider R being Relation of X such that
A2: for x,y being Element of X holds [x,y] in R iff P[x,y] from RELSET_1
:sch 2;
defpred Q[set,set] means [$1,$2] in R;
A3: now
let x,y,z be Element of X;
assume that
A4: Q[x,y] and
A5: Q[y,z];
consider W1,W2 being strict Subspace of V such that
A6: x = W1 and
A7: y = W2 & W1 is Subspace of W2 by A2,A4;
consider W3,W4 being strict Subspace of V such that
A8: y = W3 and
A9: z = W4 and
A10: W3 is Subspace of W4 by A2,A5;
W1 is strict Subspace of W4 by A7,A8,A10,RUSUB_1:21;
hence Q[x,z] by A2,A6,A9;
end;
A11: for Y being set st Y c= X & (for x,y being Element of X st x in Y & y
in Y holds Q[x,y] or Q[y,x]) ex y being Element of X st for x being Element of
X st x in Y holds Q[x,y]
proof
let Y be set;
assume that
A12: Y c= X and
A13: for x,y being Element of X st x in Y & y in Y holds [x,y] in R or
[y,x] in R;
now
per cases;
suppose
A14: Y = {};
set y = the Element of X;
take y9 = y;
let x be Element of X;
assume x in Y;
hence [x,y9] in R by A14;
end;
suppose
A15: Y <> {};
defpred P[object,object] means
ex W1 being strict Subspace of V st $1 = W1 & $2 = the carrier of W1;
A16: for x being object st x in Y ex y be object st P[x,y]
proof
let x be object;
assume x in Y;
then consider W1 being strict Subspace of V such that
A17: x = W1 and
W /\ W1 = (0).V by A1,A12;
reconsider y = the carrier of W1 as set;
take y;
take W1;
thus thesis by A17;
end;
consider f being Function such that
A18: dom f = Y and
A19: for x being object st x in Y holds P[x,f.x] from CLASSES1:sch 1(
A16);
set Z = union(rng f);
A20: now
let x be object;
assume x in Z;
then consider Y9 being set such that
A21: x in Y9 and
A22: Y9 in rng f by TARSKI:def 4;
consider y being object such that
A23: y in dom f and
A24: f.y = Y9 by A22,FUNCT_1:def 3;
consider W1 being strict Subspace of V such that
y = W1 and
A25: f.y = the carrier of W1 by A18,A19,A23;
the carrier of W1 c= the carrier of V by RUSUB_1:def 1;
hence x in the carrier of V by A21,A24,A25;
end;
set z = the Element of rng f;
A26: rng f <> {} by A15,A18,RELAT_1:42;
then consider z1 being object such that
A27: z1 in dom f and
A28: f.z1 = z by FUNCT_1:def 3;
reconsider Z as Subset of V by A20,TARSKI:def 3;
ex W3 being strict Subspace of V st z1 = W3 & f.z1 = the carrier
of W3 by A18,A19,A27;
then
A29: Z <> {} by A26,A28,ORDERS_1:6;
A30: for v1,v2 being VECTOR of V st v1 in Z & v2 in Z holds v1 + v2 in Z
proof
let v1,v2 be VECTOR of V;
assume that
A31: v1 in Z and
A32: v2 in Z;
consider Y1 being set such that
A33: v1 in Y1 and
A34: Y1 in rng f by A31,TARSKI:def 4;
consider y1 being object such that
A35: y1 in dom f and
A36: f.y1 = Y1 by A34,FUNCT_1:def 3;
consider Y2 being set such that
A37: v2 in Y2 and
A38: Y2 in rng f by A32,TARSKI:def 4;
consider y2 being object such that
A39: y2 in dom f and
A40: f.y2 = Y2 by A38,FUNCT_1:def 3;
consider W1 being strict Subspace of V such that
A41: y1 = W1 and
A42: f.y1 = the carrier of W1 by A18,A19,A35;
consider W2 being strict Subspace of V such that
A43: y2 = W2 and
A44: f.y2 = the carrier of W2 by A18,A19,A39;
reconsider y1,y2 as Element of X by A12,A18,A35,A39;
now
per cases by A13,A18,A35,A39;
suppose
[y1,y2] in R;
then ex W3,W4 being strict Subspace of V st y1 = W3 & y2 = W4 &
W3 is Subspace of W4 by A2;
then the carrier of W1 c= the carrier of W2 by A41,A43,
RUSUB_1:def 1;
then
A45: v1 in W2 by A33,A36,A42,STRUCT_0:def 5;
v2 in W2 by A37,A40,A44,STRUCT_0:def 5;
then v1 + v2 in W2 by A45,RUSUB_1:14;
then
A46: v1 + v2 in the carrier of W2 by STRUCT_0:def 5;
f.y2 in rng f by A39,FUNCT_1:def 3;
hence thesis by A44,A46,TARSKI:def 4;
end;
suppose
[y2,y1] in R;
then ex W3,W4 being strict Subspace of V st y2 = W3 & y1 = W4 &
W3 is Subspace of W4 by A2;
then the carrier of W2 c= the carrier of W1 by A41,A43,
RUSUB_1:def 1;
then
A47: v2 in W1 by A37,A40,A44,STRUCT_0:def 5;
v1 in W1 by A33,A36,A42,STRUCT_0:def 5;
then v1 + v2 in W1 by A47,RUSUB_1:14;
then
A48: v1 + v2 in the carrier of W1 by STRUCT_0:def 5;
f.y1 in rng f by A35,FUNCT_1:def 3;
hence thesis by A42,A48,TARSKI:def 4;
end;
end;
hence thesis;
end;
for a being Real, v1 being VECTOR of V
st v1 in Z holds a * v1 in Z
proof
let a be Real;
let v1 being VECTOR of V;
assume v1 in Z;
then consider Y1 being set such that
A49: v1 in Y1 and
A50: Y1 in rng f by TARSKI:def 4;
consider y1 being object such that
A51: y1 in dom f and
A52: f.y1 = Y1 by A50,FUNCT_1:def 3;
consider W1 being strict Subspace of V such that
y1 = W1 and
A53: f.y1 = the carrier of W1 by A18,A19,A51;
v1 in W1 by A49,A52,A53,STRUCT_0:def 5;
then a * v1 in W1 by RUSUB_1:15;
then
A54: a * v1 in the carrier of W1 by STRUCT_0:def 5;
f.y1 in rng f by A51,FUNCT_1:def 3;
hence thesis by A53,A54,TARSKI:def 4;
end;
then Z is linearly-closed by A30,RLSUB_1:def 1;
then consider E being strict Subspace of V such that
A55: Z = the carrier of E by A29,RUSUB_1:29;
now
let u be VECTOR of V;
thus u in W /\ E implies u in (0).V
proof
assume
A56: u in W /\ E;
then
A57: u in W by Th3;
u in E by A56,Th3;
then u in Z by A55,STRUCT_0:def 5;
then consider Y1 being set such that
A58: u in Y1 and
A59: Y1 in rng f by TARSKI:def 4;
consider y1 being object such that
A60: y1 in dom f and
A61: f.y1 = Y1 by A59,FUNCT_1:def 3;
A62: ex W2 being strict Subspace of V st y1 = W2 & W /\ W2 = (0).
V by A1,A12,A18,A60;
consider W1 being strict Subspace of V such that
A63: y1 = W1 and
A64: f.y1 = the carrier of W1 by A18,A19,A60;
u in W1 by A58,A61,A64,STRUCT_0:def 5;
hence thesis by A63,A57,A62,Th3;
end;
assume u in (0).V;
then u in the carrier of (0).V by STRUCT_0:def 5;
then u in {0.V} by RUSUB_1:def 2;
then u = 0.V by TARSKI:def 1;
hence u in W /\ E by RUSUB_1:11;
end;
then
A65: W /\ E = (0).V by RUSUB_1:25;
E in Subspaces(V) by Def3;
then reconsider y9 = E as Element of X by A1,A65;
take y = y9;
let x be Element of X;
assume
A66: x in Y;
then consider W1 being strict Subspace of V such that
A67: x = W1 and
A68: f.x = the carrier of W1 by A19;
now
let u be VECTOR of V;
assume u in W1;
then
A69: u in the carrier of W1 by STRUCT_0:def 5;
the carrier of W1 in rng f by A18,A66,A68,FUNCT_1:def 3;
then u in Z by A69,TARSKI:def 4;
hence u in E by A55,STRUCT_0:def 5;
end;
then W1 is strict Subspace of E by RUSUB_1:23;
hence [x,y] in R by A2,A67;
end;
end;
hence thesis;
end;
A70: now
let x be Element of X;
x in Subspaces(V) by A1;
hence x is strict Subspace of V by Def3;
end;
A71: now
let x be Element of X;
reconsider W = x as strict Subspace of V by A70;
W is Subspace of W by RUSUB_1:19;
hence Q[x,x] by A2;
end;
A72: now
let x,y be Element of X;
assume ( Q[x,y])& Q[y,x];
then ( ex W1,W2 being strict Subspace of V st x = W1 & y = W2 & W1 is
Subspace of W2 )& ex W3,W4 being strict Subspace of V st y = W3 & x = W4 & W3
is Subspace of W4 by A2;
hence x = y by RUSUB_1:20;
end;
consider x being Element of X such that
A73: for y being Element of X st x <> y holds not Q[x,y] from ORDERS_1:
sch 1 (A71,A72,A3,A11);
consider L being strict Subspace of V such that
A74: x = L and
A75: W /\ L = (0).V by A1;
take L;
thus the UNITSTR of V = L + W
proof
assume not thesis;
then consider v being VECTOR of V such that
A76: for v1,v2 being VECTOR of V holds not v1 in L or not v2 in W or
v <> v1 + v2 by Lm13;
v = 0.V + v & 0.V in W by RLVECT_1:4,RUSUB_1:11;
then
A77: not v in L by A76;
set A = the set of all a * v where a is Real;
A78: 1 * v in A;
now
let x be object;
assume x in A;
then ex a being Real st x = a * v;
hence x in the carrier of V;
end;
then reconsider A as Subset of V by TARSKI:def 3;
A79: for v1,v2 being VECTOR of V st v1 in A & v2 in A holds v1 + v2 in A
proof
let v1,v2 be VECTOR of V;
assume v1 in A;
then consider a1 being Real such that
A80: v1 = a1 * v;
assume v2 in A;
then consider a2 being Real such that
A81: v2 = a2 * v;
v1 + v2 = (a1 + a2) * v by A80,A81,RLVECT_1:def 6;
hence thesis;
end;
for a being Real, v1 being VECTOR of V
st v1 in A holds a * v1 in A
proof
let a be Real;
let v1 be VECTOR of V;
assume v1 in A;
then consider a1 being Real such that
A82: v1 = a1 * v;
a * v1 = (a * a1) * v by A82,RLVECT_1:def 7;
hence thesis;
end;
then A is linearly-closed by A79,RLSUB_1:def 1;
then consider Z being strict Subspace of V such that
A83: the carrier of Z = A by A78,RUSUB_1:29;
A84: not v in L + W by A76,Th1;
now
let u be VECTOR of V;
thus u in Z /\ (W + L) implies u in (0).V
proof
assume
A85: u in Z /\ (W + L);
then u in Z by Th3;
then u in A by A83,STRUCT_0:def 5;
then consider a be Real such that
A86: u = a * v;
now
reconsider aa=a as Real;
u in W + L by A85,Th3;
then aa" * (aa * v) in W + L by A86,RUSUB_1:15;
then
A87: (a" * a) * v in W + L by RLVECT_1:def 7;
assume a <> 0;
then 1 * v in W + L by A87,XCMPLX_0:def 7;
then 1 * v in L + W by Lm1;
hence contradiction by A84,RLVECT_1:def 8;
end;
then u = 0.V by A86,RLVECT_1:10;
hence thesis by RUSUB_1:11;
end;
assume u in (0).V;
then u in the carrier of (0).V by STRUCT_0:def 5;
then u in {0.V} by RUSUB_1:def 2;
then u = 0.V by TARSKI:def 1;
hence u in Z /\ (W + L) by RUSUB_1:11;
end;
then
A88: Z /\ (W + L) = (0).V by RUSUB_1:25;
now
let u be VECTOR of V;
thus u in (Z + L) /\ W implies u in (0).V
proof
assume
A89: u in (Z + L) /\ W;
then u in Z + L by Th3;
then consider v1,v2 being VECTOR of V such that
A90: v1 in Z and
A91: v2 in L and
A92: u = v1 + v2 by Th1;
A93: u in W by A89,Th3;
then
A94: u in W + L by Th2;
v1 = u - v2 & v2 in W + L by A91,A92,Th2,RLSUB_2:61;
then v1 in W + L by A94,RUSUB_1:17;
then v1 in Z /\ (W + L) by A90,Th3;
then v1 in the carrier of (0).V by A88,STRUCT_0:def 5;
then v1 in {0.V} by RUSUB_1:def 2;
then v1 = 0.V by TARSKI:def 1;
then v2 = u by A92,RLVECT_1:4;
hence thesis by A75,A91,A93,Th3;
end;
assume u in (0).V;
then u in the carrier of (0).V by STRUCT_0:def 5;
then u in {0.V} by RUSUB_1:def 2;
then u = 0.V by TARSKI:def 1;
hence u in (Z + L) /\ W by RUSUB_1:11;
end;
then (Z + L) /\ W = (0).V by RUSUB_1:25;
then
A95: W /\ (Z + L) = (0).V by Th14;
(Z + L) in Subspaces(V) by Def3;
then reconsider x1 = Z + L as Element of X by A1,A95;
L is Subspace of Z + L by Th7;
then
A96: [x,x1] in R by A2,A74;
v in A by A78,RLVECT_1:def 8;
then v in Z by A83,STRUCT_0:def 5;
then Z + L <> L by A77,Th2;
hence contradiction by A73,A74,A96;
end;
thus thesis by A75,Th14;
end;
definition
let V be RealUnitarySpace;
let W be Subspace of V;
mode Linear_Compl of W -> Subspace of V means
:Def5:
V is_the_direct_sum_of it,W;
existence
proof
ex C being strict Subspace of V st V is_the_direct_sum_of C,W by Lm14;
hence thesis;
end;
end;
registration
let V be RealUnitarySpace;
let W be Subspace of V;
cluster strict for Linear_Compl of W;
existence
proof
consider C being strict Subspace of V such that
A1: V is_the_direct_sum_of C,W by Lm14;
C is Linear_Compl of W by A1,Def5;
hence thesis;
end;
end;
Lm15: for V being RealUnitarySpace, W1,W2 being Subspace of V st V
is_the_direct_sum_of W1,W2 holds V is_the_direct_sum_of W2,W1
by Th14,Lm1;
theorem
for V being RealUnitarySpace, W1,W2 being Subspace of V holds V
is_the_direct_sum_of W1,W2 implies W2 is Linear_Compl of W1
by Lm15,Def5;
theorem Th35:
for V being RealUnitarySpace, W being Subspace of V, L being
Linear_Compl of W holds V is_the_direct_sum_of L,W & V is_the_direct_sum_of W,L
by Def5,Lm15;
begin
:: Theorems concerning the direct sum, linear complement
:: and coset of a subspace
theorem Th36:
for V being RealUnitarySpace, W being Subspace of V, L being
Linear_Compl of W holds W + L = the UNITSTR of V & L + W = the UNITSTR of V
proof
let V be RealUnitarySpace, W be Subspace of V, L be Linear_Compl of W;
V is_the_direct_sum_of W,L by Th35;
hence W + L = the UNITSTR of V;
hence thesis by Lm1;
end;
theorem Th37:
for V being RealUnitarySpace, W being Subspace of V, L being
Linear_Compl of W holds W /\ L = (0).V & L /\ W = (0).V
proof
let V be RealUnitarySpace, W be Subspace of V, L be Linear_Compl of W;
V is_the_direct_sum_of W,L by Th35;
hence W /\ L = (0).V;
hence thesis by Th14;
end;
theorem
for V being RealUnitarySpace, W1,W2 being Subspace of V st V
is_the_direct_sum_of W1,W2 holds V is_the_direct_sum_of W2,W1 by Lm15;
theorem Th39:
for V being RealUnitarySpace holds V is_the_direct_sum_of (0).V,
(Omega).V & V is_the_direct_sum_of (Omega).V,(0).V
by Th10,Th18;
theorem
for V being RealUnitarySpace, W being Subspace of V, L being
Linear_Compl of W holds W is Linear_Compl of L
proof
let V be RealUnitarySpace, W be Subspace of V, L be Linear_Compl of W;
V is_the_direct_sum_of L,W by Def5;
then V is_the_direct_sum_of W,L by Lm15;
hence thesis by Def5;
end;
theorem
for V being RealUnitarySpace holds (0).V is Linear_Compl of (Omega).V
& (Omega).V is Linear_Compl of (0).V
by Th39,Def5;
Lm16: for V being RealUnitarySpace, W being Subspace of V, v being VECTOR of V
, x being set holds x in v + W iff ex u being VECTOR of V st u in W & x = v + u
proof
let V be RealUnitarySpace;
let W be Subspace of V;
let v be VECTOR of V;
let x be set;
thus x in v + W implies ex u being VECTOR of V st u in W & x = v + u
proof
assume x in v + W;
then x in {v + u where u is VECTOR of V : u in W} by RUSUB_1:def 4;
then consider u being VECTOR of V such that
A1: x = v + u & u in W;
take u;
thus thesis by A1;
end;
given u be VECTOR of V such that
A2: u in W & x = v + u;
x in {v + v1 where v1 is VECTOR of V : v1 in W} by A2;
hence thesis by RUSUB_1:def 4;
end;
theorem Th42:
for V being RealUnitarySpace, W1,W2 being Subspace of V, C1
being Coset of W1, C2 being Coset of W2 st C1 meets C2 holds C1 /\ C2 is Coset
of W1 /\ W2
proof
let V be RealUnitarySpace;
let W1,W2 be Subspace of V;
let C1 be Coset of W1;
let C2 be Coset of W2;
set v = the Element of C1 /\ C2;
set C = C1 /\ C2;
assume
A1: C1 /\ C2 <> {};
then reconsider v as Element of V by TARSKI:def 3;
v in C1 by A1,XBOOLE_0:def 4;
then
A2: C1 = v + W1 by RUSUB_1:72;
v in C2 by A1,XBOOLE_0:def 4;
then
A3: C2 = v + W2 by RUSUB_1:72;
reconsider v as VECTOR of V;
A4: v + W1 /\ W2 c= C
proof
let x be object;
assume x in v + (W1 /\ W2);
then consider u being VECTOR of V such that
A5: u in W1 /\ W2 and
A6: x = v + u by Lm16;
u in W2 by A5,Th3;
then x in {v + u2 where u2 is VECTOR of V: u2 in W2} by A6;
then
A7: x in C2 by A3,RUSUB_1:def 4;
u in W1 by A5,Th3;
then x in {v + u1 where u1 is VECTOR of V: u1 in W1} by A6;
then x in C1 by A2,RUSUB_1:def 4;
hence thesis by A7,XBOOLE_0:def 4;
end;
C c= v + W1 /\ W2
proof
let x be object;
assume
A8: x in C;
then x in C1 by XBOOLE_0:def 4;
then consider u1 being VECTOR of V such that
A9: u1 in W1 and
A10: x = v + u1 by A2,Lm16;
x in C2 by A8,XBOOLE_0:def 4;
then consider u2 being VECTOR of V such that
A11: u2 in W2 and
A12: x = v + u2 by A3,Lm16;
u1 = u2 by A10,A12,RLVECT_1:8;
then u1 in W1 /\ W2 by A9,A11,Th3;
then x in {v + u where u is VECTOR of V : u in W1 /\ W2} by A10;
hence thesis by RUSUB_1:def 4;
end;
then C = v + W1 /\ W2 by A4;
hence thesis by RUSUB_1:def 5;
end;
Lm17: for V being RealUnitarySpace, W being Subspace of V, v being VECTOR of V
holds ex C being Coset of W st v in C
proof
let V be RealUnitarySpace;
let W be Subspace of V;
let v be VECTOR of V;
reconsider C = v + W as Coset of W by RUSUB_1:def 5;
take C;
thus thesis by RUSUB_1:37;
end;
theorem Th43:
for V being RealUnitarySpace, W1,W2 being Subspace of V holds V
is_the_direct_sum_of W1,W2 iff for C1 being Coset of W1, C2 being Coset of W2
ex v being VECTOR of V st C1 /\ C2 = {v}
proof
let V be RealUnitarySpace, W1,W2 be Subspace of V;
set VW1 = the carrier of W1;
set VW2 = the carrier of W2;
0.V in W2 by RUSUB_1:11;
then
A1: 0.V in VW2 by STRUCT_0:def 5;
thus V is_the_direct_sum_of W1,W2 implies for C1 being Coset of W1, C2 being
Coset of W2 ex v being VECTOR of V st C1 /\ C2 = {v}
proof
assume
A2: V is_the_direct_sum_of W1,W2;
then
A3: the UNITSTR of V = W1 + W2;
let C1 be Coset of W1, C2 be Coset of W2;
consider v1 being VECTOR of V such that
A4: C1 = v1 + W1 by RUSUB_1:def 5;
v1 in the UNITSTR of V by RLVECT_1:1;
then consider v11,v12 being VECTOR of V such that
A5: v11 in W1 and
A6: v12 in W2 and
A7: v1 = v11 + v12 by A3,Th1;
consider v2 being VECTOR of V such that
A8: C2 = v2 + W2 by RUSUB_1:def 5;
v2 in the UNITSTR of V by RLVECT_1:1;
then consider v21,v22 being VECTOR of V such that
A9: v21 in W1 and
A10: v22 in W2 and
A11: v2 = v21 + v22 by A3,Th1;
take v = v12 + v21;
{v} = C1 /\ C2
proof
thus
A12: {v} c= C1 /\ C2
proof
let x be object;
assume x in {v};
then
A13: x = v by TARSKI:def 1;
v21 = v2 - v22 by A11,RLSUB_2:61;
then v21 in C2 by A8,A10,RUSUB_1:58;
then C2 = v21 + W2 by RUSUB_1:72;
then
A14: x in C2 by A6,A13,RUSUB_1:57;
v12 = v1 - v11 by A7,RLSUB_2:61;
then v12 in C1 by A4,A5,RUSUB_1:58;
then C1 = v12 + W1 by RUSUB_1:72;
then x in C1 by A9,A13,RUSUB_1:57;
hence thesis by A14,XBOOLE_0:def 4;
end;
let x be object;
assume
A15: x in C1 /\ C2;
then C1 meets C2;
then reconsider C = C1 /\ C2 as Coset of W1 /\ W2 by Th42;
A16: v in {v} by TARSKI:def 1;
W1 /\ W2 = (0).V by A2;
then ex u being VECTOR of V st C = {u} by RUSUB_1:67;
hence thesis by A12,A15,A16,TARSKI:def 1;
end;
hence thesis;
end;
assume
A17: for C1 being Coset of W1, C2 being Coset of W2 ex v being VECTOR of
V st C1 /\ C2 = {v};
A18: VW2 is Coset of W2 by RUSUB_1:68;
now
let u be VECTOR of V;
consider C1 being Coset of W1 such that
A19: u in C1 by Lm17;
consider v being VECTOR of V such that
A20: C1 /\ VW2 = {v} by A18,A17;
A21: v in {v} by TARSKI:def 1;
then v in C1 by A20,XBOOLE_0:def 4;
then consider v1 being VECTOR of V such that
A22: v1 in W1 and
A23: u - v1 = v by A19,RUSUB_1:74;
v in VW2 by A20,A21,XBOOLE_0:def 4;
then
A24: v in W2 by STRUCT_0:def 5;
u = v1 + v by A23,RLSUB_2:61;
hence u in W1 + W2 by A24,A22,Th1;
end;
hence the UNITSTR of V = W1 + W2 by Lm12;
VW1 is Coset of W1 by RUSUB_1:68;
then consider v being VECTOR of V such that
A25: VW1 /\ VW2 = {v} by A18,A17;
0.V in W1 by RUSUB_1:11;
then 0.V in VW1 by STRUCT_0:def 5;
then
A26: 0.V in {v} by A25,A1,XBOOLE_0:def 4;
the carrier of (0).V = {0.V} by RUSUB_1:def 2
.= VW1 /\ VW2 by A25,A26,TARSKI:def 1
.= the carrier of W1 /\ W2 by Def2;
hence thesis by RUSUB_1:24;
end;
begin :: Decomposition of a vector of real unitary space
theorem
for V being RealUnitarySpace, W1,W2 being Subspace of V holds W1 + W2
= the UNITSTR of V iff for v being VECTOR of V ex v1,v2 being VECTOR of V st v1
in W1 & v2 in W2 & v = v1 + v2 by Lm13;
theorem Th45:
for V being RealUnitarySpace, W1,W2 being Subspace of V, v,v1,v2
,u1,u2 being VECTOR of V st V is_the_direct_sum_of W1,W2 & v = v1 + v2 & v = u1
+ u2 & v1 in W1 & u1 in W1 & v2 in W2 & u2 in W2 holds v1 = u1 & v2 = u2
proof
let V be RealUnitarySpace;
let W1,W2 be Subspace of V;
let v,v1,v2,u1,u2 be VECTOR of V;
reconsider C2 = v1 + W2 as Coset of W2 by RUSUB_1:def 5;
reconsider C1 = the carrier of W1 as Coset of W1 by RUSUB_1:68;
A1: v1 in C2 by RUSUB_1:37;
assume V is_the_direct_sum_of W1,W2;
then consider u being VECTOR of V such that
A2: C1 /\ C2 = {u} by Th43;
assume that
A3: v = v1 + v2 & v = u1 + u2 and
A4: v1 in W1 and
A5: u1 in W1 and
A6: v2 in W2 & u2 in W2;
A7: v2 - u2 in W2 by A6,RUSUB_1:17;
v1 in C1 by A4,STRUCT_0:def 5;
then v1 in C1 /\ C2 by A1,XBOOLE_0:def 4;
then
A8: v1 = u by A2,TARSKI:def 1;
A9: u1 in C1 by A5,STRUCT_0:def 5;
u1 = (v1 + v2) - u2 by A3,RLSUB_2:61
.= v1 + (v2 - u2) by RLVECT_1:def 3;
then u1 in C2 by A7,Lm16;
then
A10: u1 in C1 /\ C2 by A9,XBOOLE_0:def 4;
hence v1 = u1 by A2,A8,TARSKI:def 1;
u1 = u by A10,A2,TARSKI:def 1;
hence thesis by A3,A8,RLVECT_1:8;
end;
theorem
for V being RealUnitarySpace, W1,W2 being Subspace of V st V = W1 + W2
& (ex v being VECTOR of V st for v1,v2,u1,u2 being VECTOR of V st v = v1 + v2 &
v = u1 + u2 & v1 in W1 & u1 in W1 & v2 in W2 & u2 in W2 holds v1 = u1 & v2 = u2
) holds V is_the_direct_sum_of W1,W2
proof
let V be RealUnitarySpace;
let W1,W2 be Subspace of V;
assume
A1: V = W1 + W2;
the carrier of (0).V = {0.V} & (0).V is Subspace of W1 /\ W2 by RUSUB_1:33
,def 2;
then
A2: {0.V} c= the carrier of W1 /\ W2 by RUSUB_1:def 1;
given v be VECTOR of V such that
A3: for v1,v2,u1,u2 being VECTOR of V st v = v1 + v2 & v = u1 + u2 & v1
in W1 & u1 in W1 & v2 in W2 & u2 in W2 holds v1 = u1 & v2 = u2;
assume not thesis;
then W1 /\ W2 <> (0).V by A1;
then the carrier of W1 /\ W2 <> {0.V} by RUSUB_1:def 2;
then {0.V} c< the carrier of W1 /\ W2 by A2;
then consider x being object such that
A4: x in the carrier of W1 /\ W2 and
A5: not x in {0.V} by XBOOLE_0:6;
A6: x <> 0.V by A5,TARSKI:def 1;
A7: x in W1 /\ W2 by A4,STRUCT_0:def 5;
then x in V by RUSUB_1:2;
then reconsider u = x as VECTOR of V by STRUCT_0:def 5;
consider v1,v2 being VECTOR of V such that
A8: v1 in W1 and
A9: v2 in W2 and
A10: v = v1 + v2 by A1,Lm13;
A11: v = v1 + v2 + 0.V by A10,RLVECT_1:4
.= (v1 + v2) + (u - u) by RLVECT_1:15
.= ((v1 + v2) + u) - u by RLVECT_1:def 3
.= ((v1 + u) + v2) - u by RLVECT_1:def 3
.= (v1 + u) + (v2 - u) by RLVECT_1:def 3;
x in W2 by A7,Th3;
then
A12: v2 - u in W2 by A9,RUSUB_1:17;
x in W1 by A7,Th3;
then v1 + u in W1 by A8,RUSUB_1:14;
then v2 - u = v2 by A3,A8,A9,A10,A11,A12
.= v2 - 0.V by RLVECT_1:13;
hence thesis by A6,RLVECT_1:23;
end;
definition
let V be RealUnitarySpace;
let v be VECTOR of V;
let W1,W2 be Subspace of V;
assume
A1: V is_the_direct_sum_of W1,W2;
func v |-- (W1,W2) -> Element of [:the carrier of V, the carrier of V:]
means
:Def6:
v = it`1 + it`2 & it`1 in W1 & it`2 in W2;
existence
proof
W1 + W2 = the UNITSTR of V by A1;
then consider v1,v2 being VECTOR of V such that
A2: v1 in W1 & v2 in W2 & v = v1 + v2 by Lm13;
take [v1,v2];
thus thesis by A2;
end;
uniqueness
proof
let t1,t2 be Element of [:the carrier of V, the carrier of V:];
assume v = t1`1 + t1`2 & t1`1 in W1 & t1`2 in W2 & v = t2`1 + t2`2 & t2`1
in W1 & t2`2 in W2;
then
A3: t1`1 = t2`1 & t1`2 = t2`2 by A1,Th45;
t1 = [t1`1,t1`2] by MCART_1:21;
hence thesis by A3,MCART_1:21;
end;
end;
theorem Th47:
for V being RealUnitarySpace, v being VECTOR of V, W1,W2 being
Subspace of V st V is_the_direct_sum_of W1,W2 holds (v |-- (W1,W2))`1 = (v |--
(W2,W1))`2
proof
let V be RealUnitarySpace;
let v be VECTOR of V;
let W1,W2 be Subspace of V;
assume
A1: V is_the_direct_sum_of W1,W2;
then
A2: (v |-- (W1,W2))`2 in W2 by Def6;
A3: V is_the_direct_sum_of W2,W1 by A1,Lm15;
then
A4: v = (v |-- (W2,W1))`2 + (v |-- (W2,W1))`1 & (v |-- (W2,W1))`1 in W2 by Def6
;
A5: (v |-- (W2,W1))`2 in W1 by A3,Def6;
v = (v |-- (W1,W2))`1 + (v |-- (W1,W2))`2 & (v |-- (W1,W2))`1 in W1 by A1
,Def6;
hence thesis by A1,A2,A4,A5,Th45;
end;
theorem Th48:
for V being RealUnitarySpace, v being VECTOR of V, W1,W2 being
Subspace of V st V is_the_direct_sum_of W1,W2 holds (v |-- (W1,W2))`2 = (v |--
(W2,W1))`1
proof
let V be RealUnitarySpace;
let v be VECTOR of V;
let W1,W2 be Subspace of V;
assume
A1: V is_the_direct_sum_of W1,W2;
then
A2: (v |-- (W1,W2))`2 in W2 by Def6;
A3: V is_the_direct_sum_of W2,W1 by A1,Lm15;
then
A4: v = (v |-- (W2,W1))`2 + (v |-- (W2,W1))`1 & (v |-- (W2,W1))`1 in W2 by Def6
;
A5: (v |-- (W2,W1))`2 in W1 by A3,Def6;
v = (v |-- (W1,W2))`1 + (v |-- (W1,W2))`2 & (v |-- (W1,W2))`1 in W1 by A1
,Def6;
hence thesis by A1,A2,A4,A5,Th45;
end;
theorem
for V being RealUnitarySpace, W being Subspace of V, L being
Linear_Compl of W, v being VECTOR of V, t being Element of [:the carrier of V,
the carrier of V:] st t`1 + t`2 = v & t`1 in W & t`2 in L holds t = v |-- (W,L)
proof
let V be RealUnitarySpace, W be Subspace of V, L be Linear_Compl of W;
V is_the_direct_sum_of W,L by Th35;
hence thesis by Def6;
end;
theorem
for V being RealUnitarySpace, W being Subspace of V, L being
Linear_Compl of W, v being VECTOR of V holds (v |-- (W,L))`1 + (v |-- (W,L))`2
= v
proof
let V be RealUnitarySpace, W be Subspace of V, L be Linear_Compl of W;
V is_the_direct_sum_of W,L by Th35;
hence thesis by Def6;
end;
theorem
for V being RealUnitarySpace, W being Subspace of V, L being
Linear_Compl of W, v being VECTOR of V holds (v |-- (W,L))`1 in W & (v |-- (W,L
))`2 in L
proof
let V be RealUnitarySpace, W be Subspace of V, L be Linear_Compl of W;
V is_the_direct_sum_of W,L by Th35;
hence thesis by Def6;
end;
theorem
for V being RealUnitarySpace, W being Subspace of V, L being
Linear_Compl of W, v being VECTOR of V holds (v |-- (W,L))`1 = (v |-- (L,W))`2
by Th35,Th47;
theorem
for V being RealUnitarySpace, W being Subspace of V, L being
Linear_Compl of W, v being VECTOR of V holds (v |-- (W,L))`2 = (v |-- (L,W))`1
by Th35,Th48;
begin :: Introduction of operations on set of subspaces
definition
let V be RealUnitarySpace;
func SubJoin(V) -> BinOp of Subspaces(V) means
:Def7:
for A1,A2 being
Element of Subspaces(V), W1,W2 being Subspace of V st A1 = W1 & A2 = W2 holds
it.(A1,A2) = W1 + W2;
existence
proof
defpred P[Element of Subspaces(V),Element of Subspaces(V),Element of
Subspaces(V)] means for W1,W2 being Subspace of V st $1 = W1 & $2 = W2 holds $3
= W1 + W2;
A1: for A1,A2 being Element of Subspaces(V) ex B being Element of
Subspaces(V) st P[A1,A2,B]
proof
let A1,A2 be Element of Subspaces(V);
reconsider W1 = A1, W2 = A2 as Subspace of V by Def3;
reconsider C = W1 + W2 as Element of Subspaces(V) by Def3;
take C;
thus thesis;
end;
ex o being BinOp of Subspaces(V) st for a,b being Element of Subspaces(
V) holds P[a,b,o.(a,b)] from BINOP_1:sch 3(A1 );
hence thesis;
end;
uniqueness
proof
let o1,o2 be BinOp of Subspaces(V);
assume
A2: for A1,A2 being Element of Subspaces(V), W1,W2 being Subspace of V
st A1 = W1 & A2 = W2 holds o1.(A1,A2) = W1 + W2;
assume
A3: for A1,A2 being Element of Subspaces(V), W1,W2 being Subspace of V
st A1 = W1 & A2 = W2 holds o2.(A1,A2) = W1 + W2;
now
let x,y be set;
assume
A4: x in Subspaces(V) & y in Subspaces(V);
then reconsider A = x, B = y as Element of Subspaces(V);
reconsider W1 = x, W2 = y as Subspace of V by A4,Def3;
o1.(A,B) = W1 + W2 by A2;
hence o1.(x,y) = o2.(x,y) by A3;
end;
hence thesis by BINOP_1:1;
end;
end;
definition
let V be RealUnitarySpace;
func SubMeet(V) -> BinOp of Subspaces(V) means
:Def8:
for A1,A2 being
Element of Subspaces(V), W1,W2 being Subspace of V st A1 = W1 & A2 = W2 holds
it.(A1,A2) = W1 /\ W2;
existence
proof
defpred P[Element of Subspaces(V),Element of Subspaces(V),Element of
Subspaces(V)] means for W1,W2 being Subspace of V st $1 = W1 & $2 = W2 holds $3
= W1 /\ W2;
A1: for A1,A2 being Element of Subspaces(V) ex B being Element of
Subspaces(V) st P[A1,A2,B]
proof
let A1,A2 be Element of Subspaces(V);
reconsider W1 = A1, W2 = A2 as Subspace of V by Def3;
reconsider C = W1 /\ W2 as Element of Subspaces(V) by Def3;
take C;
thus thesis;
end;
ex o being BinOp of Subspaces(V) st for a,b being Element of Subspaces(
V) holds P[a,b,o.(a,b)] from BINOP_1:sch 3(A1 );
hence thesis;
end;
uniqueness
proof
let o1,o2 be BinOp of Subspaces(V);
assume
A2: for A1,A2 being Element of Subspaces(V), W1,W2 being Subspace of V
st A1 = W1 & A2 = W2 holds o1.(A1,A2) = W1 /\ W2;
assume
A3: for A1,A2 being Element of Subspaces(V), W1,W2 being Subspace of V
st A1 = W1 & A2 = W2 holds o2.(A1,A2) = W1 /\ W2;
now
let x,y be set;
assume
A4: x in Subspaces(V) & y in Subspaces(V);
then reconsider A = x, B = y as Element of Subspaces(V);
reconsider W1 = x, W2 = y as Subspace of V by A4,Def3;
o1.(A,B) = W1 /\ W2 by A2;
hence o1.(x,y) = o2.(x,y) by A3;
end;
hence thesis by BINOP_1:1;
end;
end;
begin :: Theorems of functions SubJoin, SubMeet
theorem Th54:
for V being RealUnitarySpace holds LattStr (# Subspaces(V),
SubJoin(V), SubMeet(V) #) is Lattice
proof
let V be RealUnitarySpace;
set S = LattStr (# Subspaces(V), SubJoin(V), SubMeet(V) #);
A1: for A,B being Element of S holds A "/\" B = B "/\" A
proof
let A,B be Element of S;
reconsider W1 = A, W2 = B as Subspace of V by Def3;
thus A "/\" B = SubMeet(V).(A,B) by LATTICES:def 2
.= W1 /\ W2 by Def8
.= W2 /\ W1 by Th14
.= SubMeet(V).(B,A) by Def8
.= B "/\" A by LATTICES:def 2;
end;
A2: for A,B being Element of S holds (A "/\" B) "\/" B = B
proof
let A,B be Element of S;
reconsider W1 = A, W2 = B as strict Subspace of V by Def3;
reconsider AB = W1 /\ W2 as Element of S by Def3;
thus (A "/\" B) "\/" B = SubJoin(V).(A "/\" B,B) by LATTICES:def 1
.= SubJoin(V).(SubMeet(V).(A,B),B) by LATTICES:def 2
.= SubJoin(V).(AB,B) by Def8
.= (W1 /\ W2) + W2 by Def7
.= B by Lm6,RUSUB_1:24;
end;
A3: for A,B,C being Element of S holds A "\/" (B "\/" C) = (A "\/" B) "\/" C
proof
let A,B,C be Element of S;
reconsider W1 = A, W2 = B, W3 = C as Subspace of V by Def3;
reconsider AB = W1 + W2, BC = W2 + W3 as Element of S by Def3;
thus A "\/" (B "\/" C) = SubJoin(V).(A,B "\/" C) by LATTICES:def 1
.= SubJoin(V).(A,SubJoin(V).(B,C)) by LATTICES:def 1
.= SubJoin(V).(A,BC) by Def7
.= W1 + (W2 + W3) by Def7
.= (W1 + W2) + W3 by Th6
.= SubJoin(V).(AB,C) by Def7
.= SubJoin(V).(SubJoin(V).(A,B),C) by Def7
.= SubJoin(V).(A "\/" B,C) by LATTICES:def 1
.= (A "\/" B) "\/" C by LATTICES:def 1;
end;
A4: for A,B being Element of S holds A "/\" (A "\/" B) = A
proof
let A,B be Element of S;
reconsider W1 = A, W2 = B as strict Subspace of V by Def3;
reconsider AB = W1 + W2 as Element of S by Def3;
thus A "/\" (A "\/" B) = SubMeet(V).(A,A "\/" B) by LATTICES:def 2
.= SubMeet(V).(A,SubJoin(V).(A,B)) by LATTICES:def 1
.= SubMeet(V).(A,AB) by Def7
.= W1 /\ (W1 + W2) by Def8
.= A by Lm7,RUSUB_1:24;
end;
A5: for A,B,C being Element of S holds A "/\" (B "/\" C) = (A "/\" B) "/\" C
proof
let A,B,C be Element of S;
reconsider W1 = A, W2 = B, W3 = C as Subspace of V by Def3;
reconsider AB = W1 /\ W2, BC = W2 /\ W3 as Element of S by Def3;
thus A "/\" (B "/\" C) = SubMeet(V).(A,B "/\" C) by LATTICES:def 2
.= SubMeet(V).(A,SubMeet(V).(B,C)) by LATTICES:def 2
.= SubMeet(V).(A,BC) by Def8
.= W1 /\ (W2 /\ W3) by Def8
.= (W1 /\ W2) /\ W3 by Th15
.= SubMeet(V).(AB,C) by Def8
.= SubMeet(V).(SubMeet(V).(A,B),C) by Def8
.= SubMeet(V).(A "/\" B,C) by LATTICES:def 2
.= (A "/\" B) "/\" C by LATTICES:def 2;
end;
for A,B being Element of S holds A "\/" B = B "\/" A
proof
let A,B be Element of S;
reconsider W1 = A, W2 = B as Subspace of V by Def3;
thus A "\/" B = SubJoin(V).(A,B) by LATTICES:def 1
.= W1 + W2 by Def7
.= W2 + W1 by Lm1
.= SubJoin(V).(B,A) by Def7
.= B "\/" A by LATTICES:def 1;
end;
then S is join-commutative join-associative meet-absorbing meet-commutative
meet-associative join-absorbing by A3,A2,A1,A5,A4,LATTICES:def 4,def 5,def 6
,def 7,def 8,def 9;
hence thesis;
end;
registration
let V be RealUnitarySpace;
cluster LattStr (# Subspaces(V), SubJoin(V), SubMeet(V) #) -> Lattice-like;
coherence by Th54;
end;
theorem Th55:
for V being RealUnitarySpace holds LattStr (# Subspaces(V),
SubJoin(V), SubMeet(V) #) is lower-bounded
proof
let V be RealUnitarySpace;
set S = LattStr (# Subspaces(V), SubJoin(V), SubMeet(V) #);
ex C being Element of S st for A being Element of S holds C "/\" A = C &
A "/\" C = C
proof
reconsider C = (0).V as Element of S by Def3;
take C;
let A be Element of S;
reconsider W = A as Subspace of V by Def3;
thus C "/\" A = SubMeet(V).(C,A) by LATTICES:def 2
.= (0).V /\ W by Def8
.= C by Th18;
hence thesis;
end;
hence thesis by LATTICES:def 13;
end;
theorem Th56:
for V being RealUnitarySpace holds LattStr (# Subspaces(V),
SubJoin(V), SubMeet(V) #) is upper-bounded
proof
let V be RealUnitarySpace;
set S = LattStr (# Subspaces(V), SubJoin(V), SubMeet(V) #);
ex C being Element of S st for A being Element of S holds C "\/" A = C &
A "\/" C = C
proof
reconsider C = (Omega).V as Element of S by Def3;
take C;
let A be Element of S;
reconsider W = A as Subspace of V by Def3;
thus C "\/" A = SubJoin(V).(C,A) by LATTICES:def 1
.= (Omega).V + W by Def7
.= the UNITSTR of V by Th11
.= C by RUSUB_1:def 3;
hence thesis;
end;
hence thesis by LATTICES:def 14;
end;
theorem Th57:
for V being RealUnitarySpace holds LattStr (# Subspaces(V),
SubJoin(V), SubMeet(V) #) is 01_Lattice
proof
let V be RealUnitarySpace;
LattStr (# Subspaces(V), SubJoin(V), SubMeet(V) #) is lower-bounded
upper-bounded Lattice by Th55,Th56;
hence thesis;
end;
theorem Th58:
for V being RealUnitarySpace holds LattStr (# Subspaces(V),
SubJoin(V), SubMeet(V) #) is modular
proof
let V be RealUnitarySpace;
set S = LattStr (# Subspaces(V), SubJoin(V), SubMeet(V) #);
for A,B,C being Element of S st A [= C holds A "\/" (B "/\" C) = (A "\/"
B) "/\" C
proof
let A,B,C be Element of S;
reconsider W1 = A, W2 = B, W3 = C as strict Subspace of V by Def3;
assume
A1: A [= C;
reconsider AB = W1 + W2 as Element of S by Def3;
reconsider BC = W2 /\ W3 as Element of S by Def3;
W1 + W3 = SubJoin(V).(A,C) by Def7
.= A "\/" C by LATTICES:def 1
.= W3 by A1,LATTICES:def 3;
then
A2: W1 is Subspace of W3 by Th8;
thus A "\/" (B "/\" C) = SubJoin(V).(A,B "/\" C) by LATTICES:def 1
.= SubJoin(V).(A,SubMeet(V).(B,C)) by LATTICES:def 2
.= SubJoin(V).(A,BC) by Def8
.= W1 + (W2 /\ W3) by Def7
.= (W1 + W2) /\ W3 by A2,Th29
.= SubMeet(V).(AB,C) by Def8
.= SubMeet(V).(SubJoin(V).(A,B),C) by Def7
.= SubMeet(V).(A "\/" B,C) by LATTICES:def 1
.= (A "\/" B) "/\" C by LATTICES:def 2;
end;
hence thesis by LATTICES:def 12;
end;
theorem Th59:
for V being RealUnitarySpace holds LattStr (# Subspaces(V),
SubJoin(V), SubMeet(V) #) is complemented
proof
let V be RealUnitarySpace;
reconsider S = LattStr (# Subspaces(V), SubJoin(V), SubMeet(V) #) as
01_Lattice by Th57;
reconsider S0 = S as 0_Lattice;
reconsider S1 = S as 1_Lattice;
reconsider Z = (0).V, I = (Omega).V as Element of S by Def3;
reconsider Z0 = Z as Element of S0;
reconsider I1 = I as Element of S1;
now
let A be Element of S0;
reconsider W = A as Subspace of V by Def3;
thus A "/\" Z0 = SubMeet(V).(A,Z0) by LATTICES:def 2
.= W /\ (0).V by Def8
.= Z0 by Th18;
end;
then
A1: Bottom S = Z by RLSUB_2:64;
now
let A be Element of S1;
reconsider W = A as Subspace of V by Def3;
thus A "\/" I1 = SubJoin(V).(A,I1) by LATTICES:def 1
.= W + (Omega).V by Def7
.= the UNITSTR of V by Th11
.= (Omega).V by RUSUB_1:def 3;
end;
then
A2: Top S = I by RLSUB_2:65;
now
let A be Element of S;
reconsider W = A as Subspace of V by Def3;
set L = the strict Linear_Compl of W;
reconsider B9 = L as Element of S by Def3;
take B = B9;
A3: B "/\" A = SubMeet(V).(B,A) by LATTICES:def 2
.= L /\ W by Def8
.= Bottom S by A1,Th37;
B "\/" A = SubJoin(V).(B,A) by LATTICES:def 1
.= L + W by Def7
.= the UNITSTR of V by Th36
.= Top S by A2,RUSUB_1:def 3;
hence B is_a_complement_of A by A3,LATTICES:def 18;
end;
hence thesis by LATTICES:def 19;
end;
registration
let V be RealUnitarySpace;
cluster LattStr (# Subspaces(V), SubJoin(V), SubMeet(V) #) -> lower-bounded
upper-bounded modular complemented;
coherence by Th55,Th56,Th58,Th59;
end;
theorem
for V being RealUnitarySpace, W1,W2,W3 being strict Subspace of V
holds W1 is Subspace of W2 implies W1 /\ W3 is Subspace of W2 /\ W3
proof
let V be RealUnitarySpace, W1,W2,W3 be strict Subspace of V;
set S = LattStr (# Subspaces(V), SubJoin(V), SubMeet(V) #);
reconsider A = W1, B = W2, C = W3, AC = W1 /\ W3, BC = W2 /\ W3 as Element
of S by Def3;
assume
A1: W1 is Subspace of W2;
A "\/" B = SubJoin(V).(A,B) by LATTICES:def 1
.= W1 + W2 by Def7
.= B by A1,Th8;
then A [= B by LATTICES:def 3;
then A "/\" C [= B "/\" C by LATTICES:9;
then
A2: (A "/\" C) "\/" (B "/\" C) = (B "/\" C) by LATTICES:def 3;
A3: B "/\" C = SubMeet(V).(B,C) by LATTICES:def 2
.= W2 /\ W3 by Def8;
(A "/\" C) "\/" (B "/\" C) = SubJoin(V).(A "/\" C,B "/\" C) by LATTICES:def 1
.= SubJoin(V).(SubMeet(V).(A,C),B "/\" C) by LATTICES:def 2
.= SubJoin(V).(SubMeet(V).(A,C),SubMeet(V).(B,C)) by LATTICES:def 2
.= SubJoin(V).(SubMeet(V).(A,C),BC) by Def8
.= SubJoin(V).(AC,BC) by Def8
.= (W1 /\ W3) + (W2 /\ W3) by Def7;
hence thesis by A2,A3,Th8;
end;
begin :: Auxiliary theorems in real unitary space
theorem
for V being RealUnitarySpace, W being strict Subspace of V holds (for
v being VECTOR of V holds v in W) implies W = the UNITSTR of V by Lm12;
theorem
for V being RealUnitarySpace, W being Subspace of V, v being VECTOR of
V holds ex C being Coset of W st v in C by Lm17;
theorem
for V being RealUnitarySpace, W being Subspace of V, v being VECTOR of
V, x being set holds x in v + W iff ex u being VECTOR of V st u in W & x = v +
u by Lm16;