:: Operations on Subspaces in Vector Space
:: by Wojciech A. Trybulec
::
:: Received July 27, 1990
:: Copyright (c) 1990-2026 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 RLVECT_1, ALGSTR_0, BINOP_1, VECTSP_1, LATTICES, XBOOLE_0,
RLSUB_1, SUBSET_1, ARYTM_3, STRUCT_0, TARSKI, SUPINF_2, ARYTM_1, RLSUB_2,
ZFMISC_1, FUNCT_1, RELAT_1, GROUP_1, FINSEQ_4, MCART_1, EQREL_1, PBOOLE;
notations TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, BINOP_1, RELAT_1, FUNCT_1,
STRUCT_0, ALGSTR_0, LATTICES, RELSET_1, RLVECT_1, GROUP_1, VECTSP_1,
DOMAIN_1, VECTSP_4;
constructors BINOP_1, REALSET1, LATTICES, VECTSP_4;
registrations SUBSET_1, STRUCT_0, LATTICES, VECTSP_1, VECTSP_4, RELAT_1,
XTUPLE_0;
requirements SUBSET, BOOLE;
definitions VECTSP_4, TARSKI, XBOOLE_0;
equalities VECTSP_4, XBOOLE_0, RLVECT_1;
expansions VECTSP_4, TARSKI, XBOOLE_0;
theorems BINOP_1, FUNCT_1, LATTICES, MCART_1, ORDERS_1, RLSUB_2, TARSKI,
VECTSP_1, VECTSP_4, ZFMISC_1, RLVECT_1, RELAT_1, VECTSP_2, XBOOLE_0,
XBOOLE_1, STRUCT_0, XTUPLE_0;
schemes BINOP_1, FUNCT_1, ORDERS_1, RELSET_1, CLASSES1, XFAMILY;
begin
reserve GF for add-associative right_zeroed right_complementable Abelian
associative well-unital distributive non empty doubleLoopStr;
reserve M for Abelian add-associative right_zeroed right_complementable
vector-distributive scalar-distributive scalar-associative scalar-unital
non empty ModuleStr over GF;
reserve W,W1,W2,W3 for Subspace of M;
reserve u,u1,u2,v,v1,v2 for Element of M;
reserve X,Y for set, x,y,y1,y2 for object;
definition
let GF;
let M;
let W1,W2;
func W1 + W2 -> strict Subspace of M means
:Def1:
the carrier of it = {v + u : v in W1 & u in W2};
existence
proof
reconsider V1 = the carrier of W1, V2 = the carrier of W2 as Subset of M
by VECTSP_4:def 2;
set VS = {v + u where u,v is Element of M: v in W1 & u in W2};
VS c= the carrier of M
proof
let x be object;
assume x in VS;
then ex v2,v1 st x = v1 + v2 & v1 in W1 & v2 in W2;
hence thesis;
end;
then reconsider VS as Subset of M;
A1: 0.M = 0.M + 0.M by RLVECT_1:4;
0.M in W1 & 0.M in W2 by VECTSP_4:17;
then
A2: 0.M in VS by A1;
A3: VS = {v + u where v,u: v in V1 & u in V2}
proof
thus VS c= {v + u where v,u: v in V1 & u in V2}
proof
let x be object;
assume x in VS;
then consider u,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 v,u: v in V1 & u in V2};
then consider v,u 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 VECTSP_4:33;
hence thesis by A2,A3,VECTSP_4:6,34;
end;
uniqueness by VECTSP_4:29;
end;
Lm1: W1 + W2 = W2 + W1
proof
set A = {v + u : v in W1 & u in W2};
set B = {v + u : v in W2 & u in W1};
A1: B c= A
proof
let x be object;
assume x in B;
then ex u,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 u,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;
definition
let GF;
let M;
let W1,W2;
func W1 /\ W2 -> strict Subspace of M 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 M;
0.M in W2 by VECTSP_4:17;
then
A1: 0.M in VW2 by STRUCT_0:def 5;
VW1 c= VV & VW2 c= VV by VECTSP_4:def 2;
then VW1 /\ VW2 c= VV /\ VV by XBOOLE_1:27;
then reconsider V1 = VW1, V2 = VW2, V3 = VW1 /\ VW2 as Subset of M by
VECTSP_4:def 2;
V1 is linearly-closed & V2 is linearly-closed by VECTSP_4:33;
then
A2: V3 is linearly-closed by VECTSP_4:7;
0.M in W1 by VECTSP_4:17;
then 0.M in VW1 by STRUCT_0:def 5;
then VW1 /\ VW2 <> {} by A1,XBOOLE_0:def 4;
hence thesis by A2,VECTSP_4:34;
end;
uniqueness by VECTSP_4:29;
commutativity;
end;
theorem Th1:
x in W1 + W2 iff ex v1,v2 st v1 in W1 & v2 in W2 & x = v1 + v2
proof
thus x in W1 + W2 implies ex v1,v2 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 : v in W1 & u in W2} by Def1;
then consider v2,v1 such that
A1: x = v1 + v2 & v1 in W1 & v2 in W2;
take v1,v2;
thus thesis by A1;
end;
given v1,v2 such that
A2: v1 in W1 & v2 in W2 & x = v1 + v2;
x in {v + u : 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:
v in W1 or v in W2 implies v in W1 + W2
proof
assume
A1: v in W1 or v in W2;
now
per cases by A1;
suppose
A2: v in W1;
v = v + 0.M & 0.M in W2 by RLVECT_1:4,VECTSP_4:17;
hence thesis by A2,Th1;
end;
suppose
A3: v in W2;
v = 0.M + v & 0.M in W1 by RLVECT_1:4,VECTSP_4:17;
hence thesis by A3,Th1;
end;
end;
hence thesis;
end;
theorem Th3:
x in W1 /\ W2 iff x in W1 & x in W2
proof
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;
Lm2: the carrier of W1 c= the carrier of W1 + W2
proof
let x be object;
set A = {v + u : v in W1 & u in W2};
assume x in the carrier of W1;
then reconsider v = x as Element of W1;
reconsider v as Element of M by VECTSP_4:10;
A1: v = v + 0.M by RLVECT_1:4;
v in W1 & 0.M in W2 by STRUCT_0:def 5,VECTSP_4:17;
then x in A by A1;
hence thesis by Def1;
end;
Lm3: for W2 being strict Subspace of M holds the carrier of W1 c= the carrier
of W2 implies W1 + W2 = W2
proof
let W2 be strict Subspace of M;
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 : v in W1 & u in W2} by Def1;
then consider u,v such that
A3: x = v + u and
A4: v in W1 and
A5: u in W2;
W1 is Subspace of W2 by A1,VECTSP_4:27;
then v in W2 by A4,VECTSP_4:8;
then v + u in W2 by A5,VECTSP_4:20;
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 VECTSP_4:29;
end;
theorem
for W being strict Subspace of M holds W + W = W by Lm3;
theorem
W1 + W2 = W2 + W1 by Lm1;
theorem Th6:
W1 + (W2 + W3) = (W1 + W2) + W3
proof
set A = {v + u : v in W1 & u in W2};
set B = {v + u : v in W2 & u in W3};
set C = {v + u : v in W1 + W2 & u in W3};
set D = {v + u : 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 u,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 u2,u1 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 u,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 u2,u1 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:
W1 is Subspace of W1 + W2 & W2 is Subspace of W1 + W2
proof
the carrier of W1 c= the carrier of W1 + W2 by Lm2;
hence W1 is Subspace of W1 + W2 by VECTSP_4:27;
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 VECTSP_4:27;
end;
theorem Th8:
for W2 being strict Subspace of M holds W1 is Subspace of W2 iff W1 + W2 = W2
proof
let W2 be strict Subspace of M;
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 VECTSP_4:def 2;
hence thesis by Lm3;
end;
thus thesis by Th7;
end;
theorem Th9:
for W being strict Subspace of M holds (0).M + W = W & W + (0).M = W
proof
let W be strict Subspace of M;
(0).M is Subspace of W by VECTSP_4:39;
then the carrier of (0).M c= the carrier of W by VECTSP_4:def 2;
hence (0).M + W = W by Lm3;
hence thesis by Lm1;
end;
Lm4: for W ex W9 being strict Subspace of M st the carrier of W = the carrier
of W9
proof
let W;
take W9 = W + W;
thus the carrier of W c= the carrier of W9 by Lm2;
let x be object;
assume x in the carrier of W9;
then x in {v + u : v in W & u in W} by Def1;
then ex v2,v1 st x = v1 + v2 & v1 in W & v2 in W;
then x in W by VECTSP_4:20;
hence thesis by STRUCT_0:def 5;
end;
Lm5: for W,W9,W1 being Subspace of M st the carrier of W = the carrier of W9
holds W1 + W = W1 + W9 & W + W1 = W9 + W1
proof
let W,W9,W1 be Subspace of M such that
A1: the carrier of W = the carrier of W9;
A2: now
let v;
set W1W9 = {v1 + v2 where v2,v1: v1 in W1 & v2 in W9};
set W1W = {v1 + v2 where v2,v1: v1 in W1 & v2 in W};
thus v in W1 + W implies v in W1 + W9
proof
assume v in W1 + W;
then v in the carrier of W1 + W by STRUCT_0:def 5;
then v in W1W by Def1;
then consider v2,v1 such that
A3: v = v1 + v2 & v1 in W1 and
A4: v2 in W;
v2 in the carrier of W9 by A1,A4,STRUCT_0:def 5;
then v2 in W9 by STRUCT_0:def 5;
then v in W1W9 by A3;
then v in the carrier of W1 + W9 by Def1;
hence thesis by STRUCT_0:def 5;
end;
assume v in W1 + W9;
then v in the carrier of W1 + W9 by STRUCT_0:def 5;
then v in W1W9 by Def1;
then consider v2,v1 such that
A5: v = v1 + v2 & v1 in W1 and
A6: v2 in W9;
v2 in the carrier of W by A1,A6,STRUCT_0:def 5;
then v2 in W by STRUCT_0:def 5;
then v in W1W by A5;
then v in the carrier of W1 + W by Def1;
hence v in W1 + W by STRUCT_0:def 5;
end;
hence W1 + W = W1 + W9 by VECTSP_4:30;
W1 + W = W + W1 & W1 + W9 = W9 + W1 by Lm1;
hence thesis by A2,VECTSP_4:30;
end;
Lm6: for W being Subspace of M holds W is Subspace of (Omega).M
proof
let W be Subspace of M;
thus the carrier of W c= the carrier of (Omega).M by VECTSP_4:def 2;
thus 0.W = 0.M by VECTSP_4:def 2
.= 0.(Omega).M by VECTSP_4:def 2;
thus thesis by VECTSP_4:def 2;
end;
theorem
(0).M + (Omega).M = the ModuleStr of M & (Omega). M + (0).M = the
ModuleStr of M by Th9;
theorem Th11:
(Omega).M + W = the ModuleStr of M & W + (Omega).M = the ModuleStr of M
proof
consider W9 being strict Subspace of M such that
A1: the carrier of W9 = the carrier of (Omega).M;
A2: the carrier of W c= the carrier of W9 by A1,VECTSP_4:def 2;
A3: W9 is Subspace of (Omega).M by Lm6;
W + (Omega).M = W + W9 by A1,Lm5
.= W9 by A2,Lm3
.= the ModuleStr of M by A1,A3,VECTSP_4:31;
hence thesis by Lm1;
end;
theorem
for M being strict Abelian add-associative right_zeroed
right_complementable vector-distributive scalar-distributive
scalar-associative scalar-unital non empty ModuleStr over GF holds (Omega).M
+ (Omega).M = M by Th11;
theorem
for W being strict Subspace of M holds W /\ W = W
proof
let W be strict Subspace of M;
the carrier of W = (the carrier of W) /\ (the carrier of W);
hence thesis by Def2;
end;
theorem Th14:
W1 /\ (W2 /\ W3) = (W1 /\ W2) /\ W3
proof
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;
Lm7: the carrier of W1 /\ W2 c= the carrier of W1
proof
the carrier of W1 /\ W2 = (the carrier of W1) /\ (the carrier of W2) by Def2;
hence thesis by XBOOLE_1:17;
end;
theorem Th15:
W1 /\ W2 is Subspace of W1 & W1 /\ W2 is Subspace of W2
proof
the carrier of W1 /\ W2 c= the carrier of W1 by Lm7;
hence W1 /\ W2 is Subspace of W1 by VECTSP_4:27;
the carrier of W2 /\ W1 c= the carrier of W2 by Lm7;
hence thesis by VECTSP_4:27;
end;
Lm8: for W,W9,W1 being Subspace of M st the carrier of W = the carrier of W9
holds W1 /\ W = W1 /\ W9 & W /\ W1 = W9 /\ W1
proof
let W,W9,W1 be Subspace of M;
assume the carrier of W = the carrier of W9;
then
A1: the carrier of W1 /\ W = (the carrier of W1) /\ (the carrier of W9) by Def2
.= the carrier of W1 /\ W9 by Def2;
hence W1 /\ W = W1 /\ W9 by VECTSP_4:29;
thus thesis by A1,VECTSP_4:29;
end;
theorem Th16:
(for W1 being strict Subspace of M holds W1 is Subspace of W2
implies W1 /\ W2 = W1) & for W1 st W1 /\ W2 = W1 holds W1 is Subspace of W2
proof
thus for W1 being strict Subspace of M holds W1 is Subspace of W2 implies W1
/\ W2 = W1
proof
let W1 be strict Subspace of M;
assume W1 is Subspace of W2;
then
A1: the carrier of W1 c= the carrier of W2 by VECTSP_4:def 2;
the carrier of W1 /\ W2 = (the carrier of W1) /\ (the carrier of W2)
by Def2;
hence thesis by A1,VECTSP_4:29,XBOOLE_1:28;
end;
thus thesis by Th15;
end;
theorem
W1 is Subspace of W2 implies W1 /\ W3 is Subspace of W2 /\ W3
proof
set A1 = the carrier of W1;
set A2 = the carrier of W2;
set A3 = the carrier of W3;
set A4 = the carrier of W1 /\ W3;
assume W1 is Subspace of W2;
then A1 c= A2 by VECTSP_4:def 2;
then A1 /\ A3 c= A2 /\ A3 by XBOOLE_1:26;
then A4 c= A2 /\ A3 by Def2;
then A4 c= the carrier of W2 /\ W3 by Def2;
hence thesis by VECTSP_4:27;
end;
theorem
W1 is Subspace of W3 implies W1 /\ W2 is Subspace of W3
proof
assume
A1: W1 is Subspace of W3;
W1 /\ W2 is Subspace of W1 by Th15;
hence thesis by A1,VECTSP_4:26;
end;
theorem
W1 is Subspace of W2 & W1 is Subspace of W3 implies W1 is Subspace of
W2 /\ W3
proof
assume
A1: W1 is Subspace of W2 & W1 is Subspace of W3;
now
let v;
assume v in W1;
then v in W2 & v in W3 by A1,VECTSP_4:8;
hence v in W2 /\ W3 by Th3;
end;
hence thesis by VECTSP_4:28;
end;
theorem Th20:
(0).M /\ W = (0).M & W /\ (0).M = (0).M
proof
0.M in W by VECTSP_4:17;
then 0.M in the carrier of W by STRUCT_0:def 5;
then {0.M} c= the carrier of W by ZFMISC_1:31;
then
A1: {0.M} /\ (the carrier of W) = {0.M} by XBOOLE_1:28;
A2: the carrier of (0).M /\ W = (the carrier of (0).M) /\ (the carrier of W)
by Def2
.= {0.M} /\ (the carrier of W) by VECTSP_4:def 3;
hence (0).M /\ W = (0).M by A1,VECTSP_4:def 3;
thus thesis by A2,A1,VECTSP_4:def 3;
end;
theorem Th21:
for W being strict Subspace of M holds (Omega).M /\ W = W & W /\
(Omega).M = W
proof
let W be strict Subspace of M;
A1: the carrier of (Omega).M /\ W = (the carrier of the ModuleStr of M) /\ (
the carrier of W) & the carrier of W c= the carrier of M by Def2,VECTSP_4:def 2
;
hence (Omega).M /\ W = W by VECTSP_4:29,XBOOLE_1:28;
thus thesis by A1,VECTSP_4:29,XBOOLE_1:28;
end;
theorem
for M being strict Abelian add-associative right_zeroed
right_complementable vector-distributive scalar-distributive
scalar-associative scalar-unital non empty ModuleStr over GF holds (Omega).M
/\ (Omega).M = M by Th21;
Lm9: the carrier of W1 /\ W2 c= the carrier of W1 + W2
proof
the carrier of W1 /\ W2 c= the carrier of W1 & the carrier of W1 c= the
carrier of W1 + W2 by Lm2,Lm7;
hence thesis;
end;
theorem
W1 /\ W2 is Subspace of W1 + W2 by Lm9,VECTSP_4:27;
Lm10: the carrier of (W1 /\ W2) + W2 = the carrier of W2
proof
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 v,u: u in W1 /\ W2 & v in W2} by Def1;
then consider v,u 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,VECTSP_4:20;
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 W2 being strict Subspace of M holds (W1 /\ W2) + W2 = W2 by Lm10,
VECTSP_4:29;
Lm11: the carrier of W1 /\ (W1 + W2) = the carrier of W1
proof
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 M by VECTSP_4:def 2;
then reconsider x1 = x as Element of M by A2;
A3: x1 + 0.M = x1 & 0.M in W2 by RLVECT_1:4,VECTSP_4:17;
x in W1 by A2,STRUCT_0:def 5;
then x in {u + v where v,u: 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 W1 being strict Subspace of M holds W1 /\ (W1 + W2) = W1 by Lm11,
VECTSP_4:29;
Lm12: the carrier of (W1 /\ W2) + (W2 /\ W3) c= the carrier of W2 /\ (W1 + W3)
proof
let x be object;
assume x in the carrier of (W1 /\ W2) + (W2 /\ W3);
then x in {u + v where v,u: u in W1 /\ W2 & v in W2 /\ W3} by Def1;
then consider v,u 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,VECTSP_4:20;
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
(W1 /\ W2) + (W2 /\ W3) is Subspace of W2 /\ (W1 + W3) by Lm12,VECTSP_4:27;
Lm13: W1 is Subspace of W2 implies the carrier of W2 /\ (W1 + W3) = the
carrier of (W1 /\ W2) + (W2 /\ W3)
proof
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 v,u: u in W1 & v in W3} by Def1;
then consider v1,u1 such that
A3: x = u1 + v1 and
A4: u1 in W1 and
A5: v1 in W3;
A6: u1 in W2 by A1,A4,VECTSP_4:8;
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,VECTSP_4:23;
then v1 + (u1 - u1) in W2 by RLVECT_1:def 3;
then v1 + 0.M in W2 by VECTSP_1:19;
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 Lm12;
end;
theorem
W1 is Subspace of W2 implies W2 /\ (W1 + W3) = (W1 /\ W2) + (W2 /\ W3)
by Lm13,VECTSP_4:29;
Lm14: 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 x in {u + v where v,u: u in W2 & v in W1 /\ W3} by Def1;
then consider v,u 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 u2,u1: 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 v2,v1: 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
W2 + (W1 /\ W3) is Subspace of (W1 + W2) /\ (W2 + W3) by Lm14,VECTSP_4:27;
Lm15: W1 is Subspace of W2 implies the carrier of W2 + (W1 /\ W3) = the
carrier of (W1 + W2) /\ (W2 + W3)
proof
reconsider V2 = the carrier of W2 as Subset of M by VECTSP_4:def 2;
A1: V2 is linearly-closed by VECTSP_4:33;
assume W1 is Subspace of W2;
then
A2: the carrier of W1 c= the carrier of W2 by VECTSP_4:def 2;
thus the carrier of W2 + (W1 /\ W3) c= the carrier of (W1 + W2) /\ (W2 + W3)
by Lm14;
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 u2,u1: u1 in W1 & u2 in W2} by Def1;
then consider u2,u1 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;
then
A5: u1 + u2 in W2 by STRUCT_0:def 5;
0.M in W1 /\ W3 & (u1 + u2) + 0.M = u1 + u2 by RLVECT_1:4,VECTSP_4:17;
then x in {u + v where v,u: u in W2 & v in W1 /\ W3} by A3,A5;
hence thesis by Def1;
end;
theorem
W1 is Subspace of W2 implies W2 + (W1 /\ W3) = (W1 + W2) /\ (W2 + W3)
by Lm15,VECTSP_4:29;
theorem Th30:
for W1 being strict Subspace of M holds W1 is Subspace of W3
implies W1 + (W2 /\ W3) = (W1 + W2) /\ W3
proof
let W1 be strict Subspace of M;
assume
A1: W1 is Subspace of W3;
hence (W1 + W2) /\ W3 = (W1 /\ W3) + (W3 /\ W2) by Lm13,VECTSP_4:29
.= W1 + (W2 /\ W3) by A1,Th16;
end;
theorem
for W1,W2 being strict Subspace of M holds W1 + W2 = W2 iff W1 /\ W2 = W1
proof
let W1,W2 be strict Subspace of M;
W1 + W2 = W2 iff W1 is Subspace of W2 by Th8;
hence thesis by Th16;
end;
theorem
for W2,W3 being strict Subspace of M holds W1 is Subspace of W2
implies W1 + W3 is Subspace of W2 + W3
proof
let W2,W3 be strict Subspace of M;
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
W1 is Subspace of W2 implies W1 is Subspace of W2 + W3
proof
assume
A1: W1 is Subspace of W2;
W2 is Subspace of W2 + W3 by Th7;
hence thesis by A1,VECTSP_4:26;
end;
theorem
W1 is Subspace of W3 & W2 is Subspace of W3 implies W1 + W2 is Subspace of W3
proof
assume
A1: W1 is Subspace of W3 & W2 is Subspace of W3;
now
let v;
assume v in W1 + W2;
then consider v1,v2 such that
A2: v1 in W1 & v2 in W2 and
A3: v = v1 + v2 by Th1;
v1 in W3 & v2 in W3 by A1,A2,VECTSP_4:8;
hence v in W3 by A3,VECTSP_4:20;
end;
hence thesis by VECTSP_4:28;
end;
theorem
(ex W 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
set VW1 = the carrier of W1;
set VW2 = the carrier of W2;
thus (ex W 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 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,VECTSP_4:27;
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 Element of M by VECTSP_4:10;
reconsider A1 = VW as Subset of M by VECTSP_4:def 2;
A6: A1 is linearly-closed by VECTSP_4:33;
not VW1 c= VW2 by A2,VECTSP_4:27;
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 Element of M by VECTSP_4:10;
A9: now
reconsider A2 = VW2 as Subset of M by VECTSP_4:def 2;
A10: A2 is linearly-closed by VECTSP_4:33;
assume x + y in VW2;
then (x + y) - y in VW2 by A10,VECTSP_4:3;
then x + (y - y) in VW2 by RLVECT_1:def 3;
then x + 0.M in VW2 by VECTSP_1:19;
hence contradiction by A8,RLVECT_1:4;
end;
A11: now
reconsider A2 = VW1 as Subset of M by VECTSP_4:def 2;
A12: A2 is linearly-closed by VECTSP_4:33;
assume x + y in VW1;
then (y + x) - x in VW1 by A12,VECTSP_4:3;
then y + (x - x) in VW1 by RLVECT_1:def 3;
then y + 0.M in VW1 by VECTSP_1:19;
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;
hence thesis by A1,A11,A9,XBOOLE_0:def 3;
end;
A13: now
assume W1 is Subspace of W2;
then VW1 c= VW2 by VECTSP_4:def 2;
then VW1 \/ VW2 = VW2 by XBOOLE_1:12;
hence thesis;
end;
A14: now
assume W2 is Subspace of W1;
then VW2 c= VW1 by VECTSP_4:def 2;
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;
definition
let GF;
let M;
func Subspaces(M) -> set means
:Def3:
for x being object
holds x in it iff ex W being strict Subspace of M st W = x;
existence
proof
defpred R[object, object] means
ex W being strict Subspace of M st $2 = W & $1 = the carrier of W;
defpred P[object] means
ex W being strict Subspace of M st $1 = the carrier of W;
consider B being set such that
A1: for x being set holds x in B iff x in bool the carrier of M & P[x] from
XFAMILY:sch 1;
A2: for x,y1,y2 being object st R[x,y1] & R[x,y2] holds y1 = y2
by VECTSP_4:29;
consider f being Function such that
A3: for x,y being object holds [x,y] in f iff x in B & R[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 M 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 holds y in rng f iff ex W being strict Subspace of M st y = W
proof
let y;
thus y in rng f implies ex W being strict Subspace of M st y = W
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 M st y = W & x = the carrier of W
by A3;
hence thesis;
end;
given W being strict Subspace of M such that
A8: y = W;
reconsider W = y as Subspace of M by A8;
reconsider x = the carrier of W as set;
A9: y is set by TARSKI:1;
the carrier of W c= the carrier of M by VECTSP_4:def 2;
then
A10: x in dom f by A1,A6,A8;
then [x,y] in f by A3,A6,A8;
then y = f.x by A10,FUNCT_1:def 2,A9;
hence thesis by A10,FUNCT_1:def 3;
end;
hence thesis;
end;
uniqueness
proof
let D1,D2 be set;
assume
A11: for x holds x in D1 iff ex W being strict Subspace of M st x= W;
assume
A12: for x holds x in D2 iff ex W being strict Subspace of M st x = W;
now
let x be object;
thus x in D1 implies x in D2
proof
assume x in D1;
then ex W being strict Subspace of M st x= W by A11;
hence thesis by A12;
end;
assume x in D2;
then ex W being strict Subspace of M st x= W by A12;
hence x in D1 by A11;
end;
hence thesis by TARSKI:2;
end;
end;
registration
let GF;
let M;
cluster Subspaces(M) -> non empty;
coherence
proof
set W = the strict Subspace of M;
W in Subspaces(M) by Def3;
hence thesis;
end;
end;
theorem
for M being strict Abelian add-associative right_zeroed
right_complementable vector-distributive scalar-distributive
scalar-associative scalar-unital non empty ModuleStr over GF holds M in
Subspaces(M)
proof
let M be strict Abelian add-associative right_zeroed right_complementable
vector-distributive scalar-distributive scalar-associative scalar-unital
non empty ModuleStr over GF;
ex W9 being strict Subspace of M st the carrier of (Omega).M = the
carrier of W9;
hence thesis by Def3;
end;
definition
let GF;
let M;
let W1,W2;
pred M is_the_direct_sum_of W1,W2 means
the ModuleStr of M = W1 + W2 & W1 /\ W2 = (0).M;
end;
Lm16: W1 + W2 = the ModuleStr of M iff for v being Element of M ex v1,v2 being
Element of M st v1 in W1 & v2 in W2 & v = v1 + v2
proof
thus W1 + W2 = the ModuleStr of M implies for v being Element of M ex v1,v2
being Element of M st v1 in W1 & v2 in W2 & v = v1 + v2
by RLVECT_1:1,Th1;
assume
A1: for v being Element of M ex v1,v2 being Element of M st v1 in W1 &
v2 in W2 & v = v1 + v2;
now
thus W1 + W2 is Subspace of (Omega).M by Lm6;
let u be Element of M;
ex v1,v2 being Element of M st v1 in W1 & v2 in W2 & u = v1 + v2 by A1;
hence u in W1 + W2 by Th1;
end;
hence thesis by VECTSP_4:32;
end;
reserve F for Field;
reserve V for VectSp of F;
reserve W for Subspace of V;
definition
let F,V,W;
mode Linear_Compl of W -> Subspace of V means
:Def5:
V is_the_direct_sum_of it,W;
existence
proof
defpred P[set,set] means ex W1,W2 being strict Subspace of V st $1 = W1 &
$2 = W2 & W1 is Subspace of W2;
defpred P[set] means
ex W1 being strict Subspace of V st $1 = W1 & W /\ W1 = (0).V;
consider X such that
A1: for x being set holds x in X iff x in Subspaces(V) & P[x]
from XFAMILY:sch 1;
W /\ (0).V = (0).V & (0).V in Subspaces(V) by Def3,Th20;
then reconsider X as non empty set by A1;
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 Z[set, set] means [$1,$2] in R;
A3: now
let x,y be Element of X;
assume ( Z[x,y])& Z[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 VECTSP_4:25;
end;
A4: for Y st Y c= X & (for x,y being Element of X st x in Y & y in Y
holds Z[x,y] or Z[y,x]) ex y being Element of X st for x being Element of X st
x in Y holds Z[x,y]
proof
let Y;
assume that
A5: Y c= X and
A6: 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
A7: 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 A7;
end;
suppose
A8: Y <> {};
defpred P[object,object] means
ex W1 being strict Subspace of V st $1 = W1
& $2 = the carrier of W1;
A9: for x being object st x in Y ex y being object st P[x,y]
proof
let x be object;
assume x in Y;
then consider W1 being strict Subspace of V such that
A10: x = W1 and
W /\ W1 = (0).V by A1,A5;
reconsider y = the carrier of W1 as set;
take y;
take W1;
thus thesis by A10;
end;
consider f being Function such that
A11: dom f = Y and
A12: for x being object st x in Y holds P[x, f.x]
from CLASSES1:sch 1(A9);
set Z = union(rng f);
now
let x be object;
assume x in Z;
then consider Y9 being set such that
A13: x in Y9 and
A14: Y9 in rng f by TARSKI:def 4;
consider y being object such that
A15: y in dom f and
A16: f.y = Y9 by A14,FUNCT_1:def 3;
consider W1 being strict Subspace of V such that
y = W1 and
A17: f.y = the carrier of W1 by A11,A12,A15;
the carrier of W1 c= the carrier of V by VECTSP_4:def 2;
hence x in the carrier of V by A13,A16,A17;
end;
then reconsider Z as Subset of V by TARSKI:def 3;
A18: Z is linearly-closed
proof
thus for v1,v2 being Element of V st v1 in Z & v2 in Z holds v1 +
v2 in Z
proof
let v1,v2 be Element of V;
assume that
A19: v1 in Z and
A20: v2 in Z;
consider Y1 being set such that
A21: v1 in Y1 and
A22: Y1 in rng f by A19,TARSKI:def 4;
consider y1 being object such that
A23: y1 in dom f and
A24: f.y1 = Y1 by A22,FUNCT_1:def 3;
consider Y2 being set such that
A25: v2 in Y2 and
A26: Y2 in rng f by A20,TARSKI:def 4;
consider y2 being object such that
A27: y2 in dom f and
A28: f.y2 = Y2 by A26,FUNCT_1:def 3;
consider W1 being strict Subspace of V such that
A29: y1 = W1 and
A30: f.y1 = the carrier of W1 by A11,A12,A23;
consider W2 being strict Subspace of V such that
A31: y2 = W2 and
A32: f.y2 = the carrier of W2 by A11,A12,A27;
reconsider y1,y2 as Element of X by A5,A11,A23,A27;
now
per cases by A6,A11,A23,A27;
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 A29,A31,
VECTSP_4:def 2;
then
A33: v1 in W2 by A21,A24,A30,STRUCT_0:def 5;
v2 in W2 by A25,A28,A32,STRUCT_0:def 5;
then v1 + v2 in W2 by A33,VECTSP_4:20;
then
A34: v1 + v2 in the carrier of W2 by STRUCT_0:def 5;
f.y2 in rng f by A27,FUNCT_1:def 3;
hence thesis by A32,A34,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 A29,A31,
VECTSP_4:def 2;
then
A35: v2 in W1 by A25,A28,A32,STRUCT_0:def 5;
v1 in W1 by A21,A24,A30,STRUCT_0:def 5;
then v1 + v2 in W1 by A35,VECTSP_4:20;
then
A36: v1 + v2 in the carrier of W1 by STRUCT_0:def 5;
f.y1 in rng f by A23,FUNCT_1:def 3;
hence thesis by A30,A36,TARSKI:def 4;
end;
end;
hence thesis;
end;
let a be Element of F, v1 be Element of V;
assume v1 in Z;
then consider Y1 being set such that
A37: v1 in Y1 and
A38: Y1 in rng f by TARSKI:def 4;
consider y1 being object such that
A39: y1 in dom f and
A40: f.y1 = Y1 by A38,FUNCT_1:def 3;
consider W1 being strict Subspace of V such that
y1 = W1 and
A41: f.y1 = the carrier of W1 by A11,A12,A39;
v1 in W1 by A37,A40,A41,STRUCT_0:def 5;
then a * v1 in W1 by VECTSP_4:21;
then
A42: a * v1 in the carrier of W1 by STRUCT_0:def 5;
f.y1 in rng f by A39,FUNCT_1:def 3;
hence thesis by A41,A42,TARSKI:def 4;
end;
set z = the Element of rng f;
A43: rng f <> {} by A8,A11,RELAT_1:42;
then consider z1 being object such that
A44: z1 in dom f and
A45: f.z1 = z by FUNCT_1:def 3;
ex W3 being strict Subspace of V st z1 = W3 & f.z1 = the carrier
of W3 by A11,A12,A44;
then Z <> {} by A43,A45,ORDERS_1:6;
then consider E being strict Subspace of V such that
A46: Z = the carrier of E by A18,VECTSP_4:34;
now
let u be Element of V;
thus u in W /\ E implies u in (0).V
proof
assume
A47: u in W /\ E;
then
A48: u in W by Th3;
u in E by A47,Th3;
then u in Z by A46,STRUCT_0:def 5;
then consider Y1 being set such that
A49: u 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;
A53: ex W2 being strict Subspace of V st y1 = W2 & W /\ W2 =
(0).V by A1,A5,A11,A51;
consider W1 being strict Subspace of V such that
A54: y1 = W1 and
A55: f.y1 = the carrier of W1 by A11,A12,A51;
u in W1 by A49,A52,A55,STRUCT_0:def 5;
hence thesis by A54,A48,A53,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 VECTSP_4:def 3;
then u = 0.V by TARSKI:def 1;
hence u in W /\ E by VECTSP_4:17;
end;
then
A56: W /\ E = (0).V by VECTSP_4:30;
E in Subspaces(V) by Def3;
then reconsider y9 = E as Element of X by A1,A56;
take y = y9;
let x be Element of X;
assume
A57: x in Y;
then consider W1 being strict Subspace of V such that
A58: x = W1 and
A59: f.x = the carrier of W1 by A12;
now
let u be Element of V;
assume u in W1;
then
A60: u in the carrier of W1 by STRUCT_0:def 5;
the carrier of W1 in rng f by A11,A57,A59,FUNCT_1:def 3;
then u in Z by A60,TARSKI:def 4;
hence u in E by A46,STRUCT_0:def 5;
end;
then W1 is Subspace of E by VECTSP_4:28;
hence [x,y] in R by A2,A58;
end;
end;
hence thesis;
end;
A61: now
let x,y,z be Element of X;
assume that
A62: Z[x,y] and
A63: Z[y,z];
consider W1,W2 being strict Subspace of V such that
A64: x = W1 and
A65: y = W2 & W1 is Subspace of W2 by A2,A62;
consider W3,W4 being strict Subspace of V such that
A66: y = W3 and
A67: z = W4 and
A68: W3 is Subspace of W4 by A2,A63;
W1 is Subspace of W4 by A65,A66,A68,VECTSP_4:26;
hence Z[x,z] by A2,A64,A67;
end;
A69: now
let x be Element of X;
consider W1 being strict Subspace of V such that
A70: x = W1 and
W /\ W1 = (0).V by A1;
W1 is Subspace of W1 by VECTSP_4:24;
hence Z[x,x] by A2,A70;
end;
consider x being Element of X such that
A71: for y being Element of X st x <> y holds not Z[x,y] from
ORDERS_1:sch 1 (A69,A3,A61,A4);
consider L being strict Subspace of V such that
A72: x = L and
A73: W /\ L = (0).V by A1;
take L;
thus the ModuleStr of V = L + W
proof
assume not thesis;
then consider v being Element of V such that
A74: for v1,v2 being Element of V holds not v1 in L or not v2 in W
or v <> v1 + v2 by Lm16;
v = 0.V + v & 0.V in W by RLVECT_1:4,VECTSP_4:17;
then
A75: not v in L by A74;
set A = the set of all a * v where a is Element of F ;
A76: 1_F * v in A;
now
let x be object;
assume x in A;
then ex a being Element of F st x = a * v;
hence x in the carrier of V;
end;
then reconsider A as Subset of V by TARSKI:def 3;
A is linearly-closed
proof
thus for v1,v2 being Element of V st v1 in A & v2 in A holds v1 + v2
in A
proof
let v1,v2 be Element of V;
assume v1 in A;
then consider a1 being Element of F such that
A77: v1 = a1 * v;
assume v2 in A;
then consider a2 being Element of F such that
A78: v2 = a2 * v;
v1 + v2 = (a1 + a2) * v by A77,A78,VECTSP_1:def 15;
hence thesis;
end;
let a be Element of F, v1 be Element of V;
assume v1 in A;
then consider a1 being Element of F such that
A79: v1 = a1 * v;
a * v1 = (a * a1) * v by A79,VECTSP_1:def 16;
hence thesis;
end;
then consider Z being strict Subspace of V such that
A80: the carrier of Z = A by A76,VECTSP_4:34;
A81: not v in L + W by A74,Th1;
now
let u be Element of V;
thus u in Z /\ (W + L) implies u in (0).V
proof
assume
A82: u in Z /\ (W + L);
then u in Z by Th3;
then u in A by A80,STRUCT_0:def 5;
then consider a being Element of F such that
A83: u = a * v;
now
u in W + L by A82,Th3;
then a" * (a * v) in W + L by A83,VECTSP_4:21;
then
A84: (a" * a) * v in W + L by VECTSP_1:def 16;
assume a <> 0.F;
then 1_F * v in W + L by A84,VECTSP_1:def 10;
then 1_F * v in L + W by Lm1;
hence contradiction by A81;
end;
then u = 0.V by A83,VECTSP_1:14;
hence thesis by VECTSP_4:17;
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 VECTSP_4:def 3;
then u = 0.V by TARSKI:def 1;
hence u in Z /\ (W + L) by VECTSP_4:17;
end;
then
A85: Z /\ (W + L) = (0).V by VECTSP_4:30;
now
let u be Element of V;
thus u in (Z + L) /\ W implies u in (0).V
proof
assume
A86: u in (Z + L) /\ W;
then u in Z + L by Th3;
then consider v1,v2 being Element of V such that
A87: v1 in Z and
A88: v2 in L and
A89: u = v1 + v2 by Th1;
A90: u in W by A86,Th3;
then
A91: u in W + L by Th2;
v1 = u - v2 & v2 in W + L by A88,A89,Th2,VECTSP_2:2;
then v1 in W + L by A91,VECTSP_4:23;
then v1 in Z /\ (W + L) by A87,Th3;
then v1 in the carrier of (0).V by A85,STRUCT_0:def 5;
then v1 in {0.V} by VECTSP_4:def 3;
then v1 = 0.V by TARSKI:def 1;
then v2 = u by A89,RLVECT_1:4;
hence thesis by A73,A88,A90,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 VECTSP_4:def 3;
then u = 0.V by TARSKI:def 1;
hence u in (Z + L) /\ W by VECTSP_4:17;
end;
then
A92: W /\ (Z + L) = (0).V by VECTSP_4:30;
(Z + L) in Subspaces(V) by Def3;
then reconsider x1 = Z + L as Element of X by A1,A92;
L is Subspace of Z + L by Th7;
then
A93: [x,x1] in R by A2,A72;
v in A by A76;
then v in Z by A80,STRUCT_0:def 5;
then Z + L <> L by A75,Th2;
hence contradiction by A71,A72,A93;
end;
thus thesis by A73;
end;
end;
Lm17: for W1,W2 be Subspace of M holds M is_the_direct_sum_of W1,W2 implies M
is_the_direct_sum_of W2,W1
by Lm1;
reserve W,W1,W2 for Subspace of V;
theorem
V is_the_direct_sum_of W1,W2 implies W2 is Linear_Compl of W1
by Lm17,Def5;
theorem Th38:
for 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,Lm17;
theorem Th39:
for L being Linear_Compl of W holds W + L = the ModuleStr of V &
L + W = the ModuleStr of V
proof
let L be Linear_Compl of W;
V is_the_direct_sum_of W,L by Th38;
hence W + L = the ModuleStr of V;
hence thesis by Lm1;
end;
theorem Th40:
for L being Linear_Compl of W holds W /\ L = (0).V & L /\ W = (0).V
proof
let L be Linear_Compl of W;
A1: V is_the_direct_sum_of W,L by Th38;
hence W /\ L = (0).V;
thus thesis by A1;
end;
reserve W1,W2 for Subspace of M;
theorem
M is_the_direct_sum_of W1,W2 implies M is_the_direct_sum_of W2,W1 by Lm17;
theorem Th42:
M is_the_direct_sum_of (0).M,(Omega).M & M is_the_direct_sum_of
(Omega). M,(0).M
by Th9,Th20;
reserve W for Subspace of V;
theorem
for L being Linear_Compl of W holds W is Linear_Compl of L
proof
let 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 Lm17;
hence thesis by Def5;
end;
theorem
(0).V is Linear_Compl of (Omega).V & (Omega).V is Linear_Compl of (0). V
by Th42,Def5;
reserve W1,W2 for Subspace of M;
reserve u,u1,u2,v for Element of M;
reserve C1 for Coset of W1;
reserve C2 for Coset of W2;
theorem Th45:
C1 meets C2 implies C1 /\ C2 is Coset of W1 /\ W2
proof
set v = the Element of C1 /\ C2;
set C = C1 /\ C2;
assume
A1: C1 /\ C2 <> {};
then reconsider v as Element of M by TARSKI:def 3;
v in C2 by A1,XBOOLE_0:def 4;
then
A2: C2 = v + W2 by VECTSP_4:77;
v in C1 by A1,XBOOLE_0:def 4;
then
A3: C1 = v + W1 by VECTSP_4:77;
C is Coset of W1 /\ W2
proof
take v;
thus C c= v + W1 /\ W2
proof
let x be object;
assume
A4: x in C;
then x in C1 by XBOOLE_0:def 4;
then consider u1 such that
A5: u1 in W1 and
A6: x = v + u1 by A3,VECTSP_4:42;
x in C2 by A4,XBOOLE_0:def 4;
then consider u2 such that
A7: u2 in W2 and
A8: x = v + u2 by A2,VECTSP_4:42;
u1 = u2 by A6,A8,RLVECT_1:8;
then u1 in W1 /\ W2 by A5,A7,Th3;
hence thesis by A6;
end;
let x be object;
assume x in v + (W1 /\ W2);
then consider u such that
A9: u in W1 /\ W2 and
A10: x = v + u by VECTSP_4:42;
u in W2 by A9,Th3;
then
A11: x in {v + u2 : u2 in W2} by A10;
u in W1 by A9,Th3;
then x in {v + u1 : u1 in W1} by A10;
hence thesis by A3,A2,A11,XBOOLE_0:def 4;
end;
hence thesis;
end;
theorem Th46:
M is_the_direct_sum_of W1,W2 iff for C1 being Coset of W1, C2
being Coset of W2 ex v being Element of M st C1 /\ C2 = {v}
proof
set VW1 = the carrier of W1;
set VW2 = the carrier of W2;
A1: W1 + W2 is Subspace of (Omega).M by Lm6;
thus M is_the_direct_sum_of W1,W2 implies for C1 being Coset of W1, C2 being
Coset of W2 ex v being Element of M st C1 /\ C2 = {v}
proof
assume
A2: M is_the_direct_sum_of W1,W2;
then
A3: the ModuleStr of M = W1 + W2;
let C1 be Coset of W1, C2 be Coset of W2;
consider v1 being Element of M such that
A4: C1 = v1 + W1 by VECTSP_4:def 6;
v1 in (Omega).M by RLVECT_1:1;
then consider v11,v12 being Element of M such that
A5: v11 in W1 and
A6: v12 in W2 and
A7: v1 = v11 + v12 by A3,Th1;
consider v2 being Element of M such that
A8: C2 = v2 + W2 by VECTSP_4:def 6;
v2 in (Omega).M by RLVECT_1:1;
then consider v21,v22 being Element of M 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,VECTSP_2:2;
then v21 in C2 by A8,A10,VECTSP_4:62;
then C2 = v21 + W2 by VECTSP_4:77;
then
A14: x in C2 by A6,A13;
v12 = v1 - v11 by A7,VECTSP_2:2;
then v12 in C1 by A4,A5,VECTSP_4:62;
then C1 = v12 + W1 by VECTSP_4:77;
then x in C1 by A9,A13;
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 Th45;
A16: v in {v} by TARSKI:def 1;
W1 /\ W2 = (0).M by A2;
then ex u being Element of M st C = {u} by VECTSP_4:72;
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 Element
of M st C1 /\ C2 = {v};
A18: VW2 is Coset of W2 by VECTSP_4:73;
A19: the carrier of M c= the carrier of W1 + W2
proof
let x be object;
assume x in the carrier of M;
then reconsider u = x as Element of M;
consider C1 being Coset of W1 such that
A20: u in C1 by VECTSP_4:68;
consider v being Element of M such that
A21: C1 /\ VW2 = {v} by A18,A17;
A22: v in {v} by TARSKI:def 1;
then v in C1 by A21,XBOOLE_0:def 4;
then consider v1 being Element of M such that
A23: v1 in W1 and
A24: u - v1 = v by A20,VECTSP_4:79;
v in VW2 by A21,A22,XBOOLE_0:def 4;
then
A25: v in W2 by STRUCT_0:def 5;
u = v1 + v by A24,VECTSP_2:2;
then x in W1 + W2 by A25,A23,Th1;
hence thesis by STRUCT_0:def 5;
end;
VW1 is Coset of W1 by VECTSP_4:73;
then consider v being Element of M such that
A26: VW1 /\ VW2 = {v} by A18,A17;
the carrier of W1 + W2 c= the carrier of M by VECTSP_4:def 2;
then the carrier of M = the carrier of W1 + W2 by A19;
hence the ModuleStr of M = W1 + W2 by A1,VECTSP_4:31;
0.M in W2 by VECTSP_4:17;
then
A27: 0.M in VW2 by STRUCT_0:def 5;
0.M in W1 by VECTSP_4:17;
then 0.M in VW1 by STRUCT_0:def 5;
then
A28: 0.M in {v} by A26,A27,XBOOLE_0:def 4;
the carrier of (0).M = {0.M} by VECTSP_4:def 3
.= VW1 /\ VW2 by A26,A28,TARSKI:def 1
.= the carrier of W1 /\ W2 by Def2;
hence thesis by VECTSP_4:29;
end;
theorem
for M being strict Abelian add-associative right_zeroed
right_complementable vector-distributive scalar-distributive
scalar-associative scalar-unital
non empty ModuleStr over GF, W1,W2 being
Subspace of M holds W1 + W2 = M iff for v being Element of M ex v1,v2 being
Element of M st v1 in W1 & v2 in W2 & v = v1 + v2 by Lm16;
theorem Th48:
for v,v1,v2,u1,u2 being Element of M holds M
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 implies v1 = u1 & v2 = u2
proof
let v,v1,v2,u1,u2 be Element of M;
reconsider C2 = v1 + W2 as Coset of W2 by VECTSP_4:def 6;
reconsider C1 = the carrier of W1 as Coset of W1 by VECTSP_4:73;
A1: v1 in C2 by VECTSP_4:44;
assume M is_the_direct_sum_of W1,W2;
then consider u being Element of M such that
A2: C1 /\ C2 = {u} by Th46;
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,VECTSP_4:23;
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,VECTSP_2:2
.= v1 + (v2 - u2) by RLVECT_1:def 3;
then u1 in C2 by A7;
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
M = W1 + W2 & (ex v st for v1,v2,u1,u2 being Element of M st v = v1 +
v2 & v = u1 + u2 & v1 in W1 & u1 in W1 & v2 in W2 & u2 in W2 holds v1 = u1 & v2
= u2) implies M is_the_direct_sum_of W1,W2
proof
assume
A1: M = W1 + W2;
the carrier of (0).M = {0.M} & (0).M is Subspace of W1 /\ W2 by VECTSP_4:39
,def 3;
then
A2: {0.M} c= the carrier of W1 /\ W2 by VECTSP_4:def 2;
given v such that
A3: for v1,v2,u1,u2 being Element of M 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).M by A1;
then the carrier of W1 /\ W2 <> {0.M} by VECTSP_4:def 3;
then {0.M} 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.M} by XBOOLE_0:6;
A6: x in W1 /\ W2 by A4,STRUCT_0:def 5;
then x in M by VECTSP_4:9;
then reconsider u = x as Element of M by STRUCT_0:def 5;
consider v1,v2 being Element of M such that
A7: v1 in W1 and
A8: v2 in W2 and
A9: v = v1 + v2 by A1,Lm16;
A10: v = v1 + v2 + 0.M by A9,RLVECT_1:4
.= (v1 + v2) + (u - u) by VECTSP_1:19
.= ((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 A6,Th3;
then
A11: v2 - u in W2 by A8,VECTSP_4:23;
x in W1 by A6,Th3;
then v1 + u in W1 by A7,VECTSP_4:20;
then v2 - u = v2 by A3,A7,A8,A9,A10,A11;
then v2 + (- u) = v2 + 0.M by RLVECT_1:4;
then - u = 0.M by RLVECT_1:8;
then
A12: u = - 0.M by RLVECT_1:17;
x <> 0.M by A5,TARSKI:def 1;
hence thesis by A12,RLVECT_1:12;
end;
reserve t1,t2 for Element of [:the carrier of M, the carrier of M:];
definition
let GF,M,v,W1,W2;
assume
A1: M is_the_direct_sum_of W1,W2;
func v |-- (W1,W2) -> Element of [:the carrier of M,the carrier of M:] means
:Def6:
v = it`1 + it`2 & it`1 in W1 & it`2 in W2;
existence
proof
W1 + W2 = the ModuleStr of M by A1;
then consider v1,v2 being Element of M such that
A2: v1 in W1 & v2 in W2 & v = v1 + v2 by Lm16;
take [v1,v2];
thus thesis by A2;
end;
uniqueness
proof
let t1,t2;
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,Th48;
t1 = [t1`1,t1`2] by MCART_1:21;
hence thesis by A3,MCART_1:21;
end;
end;
theorem Th50:
M is_the_direct_sum_of W1,W2 implies (v |-- (W1,W2))`1 = (v |-- (W2,W1))`2
proof
assume
A1: M is_the_direct_sum_of W1,W2;
then
A2: (v |-- (W1,W2))`2 in W2 by Def6;
A3: M is_the_direct_sum_of W2,W1 by A1,Lm17;
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,Th48;
end;
theorem Th51:
M is_the_direct_sum_of W1,W2 implies (v |-- (W1,W2))`2 = (v |-- (W2,W1))`1
proof
assume
A1: M is_the_direct_sum_of W1,W2;
then
A2: (v |-- (W1,W2))`2 in W2 by Def6;
A3: M is_the_direct_sum_of W2,W1 by A1,Lm17;
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,Th48;
end;
reserve W for Subspace of V;
theorem
for L being Linear_Compl of W, v being Element of V, t being Element
of [:the carrier of V,the carrier of V:] holds t`1 + t`2 = v & t`1 in W & t`2
in L implies t = v |-- (W,L)
proof
let L be Linear_Compl of W;
let v be Element of V;
let t be Element of [:the carrier of V,the carrier of V:];
V is_the_direct_sum_of W,L by Th38;
hence thesis by Def6;
end;
theorem
for L being Linear_Compl of W, v being Element of V holds (v |-- (W,L)
)`1 + (v |-- (W,L))`2 = v
proof
let L be Linear_Compl of W;
let v be Element of V;
V is_the_direct_sum_of W,L by Th38;
hence thesis by Def6;
end;
theorem
for L being Linear_Compl of W, v being Element of V holds (v |-- (W,L)
)`1 in W & (v |-- (W,L))`2 in L
proof
let L be Linear_Compl of W;
let v be Element of V;
V is_the_direct_sum_of W,L by Th38;
hence thesis by Def6;
end;
theorem
for L being Linear_Compl of W, v being Element of V holds (v |-- (W,L)
)`1 = (v |-- (L,W))`2
by Th38,Th50;
theorem
for L being Linear_Compl of W, v being Element of V holds (v |-- (W,L)
)`2 = (v |-- (L,W))`1
by Th38,Th51;
reserve A1,A2,B for Element of Subspaces(M),
W1,W2 for Subspace of M;
definition
let GF;
let M;
func SubJoin M -> BinOp of Subspaces M means
:Def7:
for A1,A2,W1,W2 st A1 = W1 & A2 = W2 holds it.(A1,A2) = W1 + W2;
existence
proof
defpred P[set,set,set] means for W1,W2 st $1 = W1 & $2 = W2 holds $3 = W1
+ W2;
A1: for A1,A2 ex B st P[A1,A2,B]
proof
let A1,A2;
consider W1 being strict Subspace of M such that
A2: W1 = A1 by Def3;
consider W2 being strict Subspace of M such that
A3: W2 = A2 by Def3;
reconsider C = W1 + W2 as Element of Subspaces M by Def3;
take C;
thus thesis by A2,A3;
end;
thus ex o being BinOp of Subspaces M st for A1,A2 holds P[A1,A2,o.(A1,A2)]
from BINOP_1:sch 3(A1);
end;
uniqueness
proof
let o1,o2 be BinOp of Subspaces M;
assume
A4: for A1,A2,W1,W2 st A1 = W1 & A2 = W2 holds o1.(A1,A2) = W1 + W2;
assume
A5: for A1,A2,W1,W2 st A1 = W1 & A2 = W2 holds o2.(A1,A2) = W1 + W2;
now
let x,y be set;
assume that
A6: x in Subspaces M and
A7: y in Subspaces M;
reconsider A = x, B = y as Element of Subspaces M by A6,A7;
consider W1 being strict Subspace of M such that
A8: W1 = x by A6,Def3;
consider W2 being strict Subspace of M such that
A9: W2 = y by A7,Def3;
o1.(A,B) = W1 + W2 by A4,A8,A9;
hence o1.(x,y) = o2.(x,y) by A5,A8,A9;
end;
hence thesis by BINOP_1:1;
end;
end;
definition
let GF;
let M;
func SubMeet M -> BinOp of Subspaces M means
:Def8:
for A1,A2,W1,W2 st A1 = W1 & A2 = W2 holds it.(A1,A2) = W1 /\ W2;
existence
proof
defpred P[set,set,set] means for W1,W2 st $1 = W1 & $2 = W2 holds $3 = W1
/\ W2;
A1: for A1,A2 ex B st P[A1,A2,B]
proof
let A1,A2;
consider W1 being strict Subspace of M such that
A2: W1 = A1 by Def3;
consider W2 being strict Subspace of M such that
A3: W2 = A2 by Def3;
reconsider C = W1 /\ W2 as Element of Subspaces M by Def3;
take C;
thus thesis by A2,A3;
end;
thus ex o being BinOp of Subspaces M st for A1,A2 holds P[A1,A2,o.(A1,A2)]
from BINOP_1:sch 3(A1);
end;
uniqueness
proof
let o1,o2 be BinOp of Subspaces M;
assume
A4: for A1,A2,W1,W2 st A1 = W1 & A2 = W2 holds o1.(A1,A2) = W1 /\ W2;
assume
A5: for A1,A2,W1,W2 st A1 = W1 & A2 = W2 holds o2.(A1,A2) = W1 /\ W2;
now
let x,y be set;
assume that
A6: x in Subspaces M and
A7: y in Subspaces M;
reconsider A = x, B = y as Element of Subspaces M by A6,A7;
consider W1 being strict Subspace of M such that
A8: W1 = x by A6,Def3;
consider W2 being strict Subspace of M such that
A9: W2 = y by A7,Def3;
o1.(A,B) = W1 /\ W2 by A4,A8,A9;
hence o1.(x,y) = o2.(x,y) by A5,A8,A9;
end;
hence thesis by BINOP_1:1;
end;
end;
theorem Th57:
LattStr (# Subspaces(M), SubJoin(M), SubMeet(M) #) is Lattice
proof
set S = LattStr (# Subspaces(M), SubJoin(M), SubMeet(M) #);
A1: for A,B being Element of S holds A "/\" B = B "/\" A
proof
let A,B be Element of S;
consider W1 being strict Subspace of M such that
A2: W1 = A by Def3;
consider W2 being strict Subspace of M such that
A3: W2 = B by Def3;
thus A "/\" B = SubMeet(M).(A,B) by LATTICES:def 2
.= W1 /\ W2 by A2,A3,Def8
.= SubMeet(M).(B,A) by A2,A3,Def8
.= B "/\" A by LATTICES:def 2;
end;
A4: for A,B being Element of S holds (A "/\" B) "\/" B = B
proof
let A,B be Element of S;
consider W1 being strict Subspace of M such that
A5: W1 = A by Def3;
consider W2 being strict Subspace of M such that
A6: W2 = B by Def3;
reconsider AB = W1 /\ W2 as Element of S by Def3;
thus (A "/\" B) "\/" B = SubJoin(M).(A "/\" B,B) by LATTICES:def 1
.= SubJoin(M).(SubMeet(M).(A,B),B) by LATTICES:def 2
.= SubJoin(M).(AB,B) by A5,A6,Def8
.= (W1 /\ W2) + W2 by A6,Def7
.= B by A6,Lm10,VECTSP_4:29;
end;
A7: for A,B,C being Element of S holds A "/\" (B "/\" C) = (A "/\" B) "/\" C
proof
let A,B,C be Element of S;
consider W1 being strict Subspace of M such that
A8: W1 = A by Def3;
consider W2 being strict Subspace of M such that
A9: W2 = B by Def3;
consider W3 being strict Subspace of M such that
A10: W3 = C by Def3;
reconsider AB = W1 /\ W2, BC = W2 /\ W3 as Element of S by Def3;
thus A "/\" (B "/\" C) = SubMeet(M).(A,B "/\" C) by LATTICES:def 2
.= SubMeet(M).(A,SubMeet(M).(B,C)) by LATTICES:def 2
.= SubMeet(M).(A,BC) by A9,A10,Def8
.= W1 /\ (W2 /\ W3) by A8,Def8
.= (W1 /\ W2) /\ W3 by Th14
.= SubMeet(M).(AB,C) by A10,Def8
.= SubMeet(M).(SubMeet(M).(A,B),C) by A8,A9,Def8
.= SubMeet(M).(A "/\" B,C) by LATTICES:def 2
.= (A "/\" B) "/\" C by LATTICES:def 2;
end;
A11: for A,B,C being Element of S holds A "\/" (B "\/" C) = (A "\/" B) "\/" C
proof
let A,B,C be Element of S;
consider W1 being strict Subspace of M such that
A12: W1 = A by Def3;
consider W2 being strict Subspace of M such that
A13: W2 = B by Def3;
consider W3 being strict Subspace of M such that
A14: W3 = C by Def3;
reconsider AB = W1 + W2, BC = W2 + W3 as Element of S by Def3;
thus A "\/" (B "\/" C) = SubJoin(M).(A,B "\/" C) by LATTICES:def 1
.= SubJoin(M).(A,SubJoin(M).(B,C)) by LATTICES:def 1
.= SubJoin(M).(A,BC) by A13,A14,Def7
.= W1 + (W2 + W3) by A12,Def7
.= (W1 + W2) + W3 by Th6
.= SubJoin(M).(AB,C) by A14,Def7
.= SubJoin(M).(SubJoin(M).(A,B),C) by A12,A13,Def7
.= SubJoin(M).(A "\/" B,C) by LATTICES:def 1
.= (A "\/" B) "\/" C by LATTICES:def 1;
end;
A15: for A,B being Element of S holds A "/\" (A "\/" B) = A
proof
let A,B be Element of S;
consider W1 being strict Subspace of M such that
A16: W1 = A by Def3;
consider W2 being strict Subspace of M such that
A17: W2 = B by Def3;
reconsider AB = W1 + W2 as Element of S by Def3;
thus A "/\" (A "\/" B) = SubMeet(M).(A,A "\/" B) by LATTICES:def 2
.= SubMeet(M).(A,SubJoin(M).(A,B)) by LATTICES:def 1
.= SubMeet(M).(A,AB) by A16,A17,Def7
.= W1 /\ (W1 + W2) by A16,Def8
.= A by A16,Lm11,VECTSP_4:29;
end;
for A,B being Element of S holds A "\/" B = B "\/" A
proof
let A,B be Element of S;
consider W1 being strict Subspace of M such that
A18: W1 = A by Def3;
consider W2 being strict Subspace of M such that
A19: W2 = B by Def3;
thus A "\/" B = SubJoin(M).(A,B) by LATTICES:def 1
.= W1 + W2 by A18,A19,Def7
.= W2 + W1 by Lm1
.= SubJoin(M).(B,A) by A18,A19,Def7
.= B "\/" A by LATTICES:def 1;
end;
then S is join-commutative join-associative meet-absorbing meet-commutative
meet-associative join-absorbing by A11,A4,A1,A7,A15,LATTICES:def 4,def 5
,def 6,def 7,def 8,def 9;
hence thesis;
end;
theorem Th58:
LattStr (# Subspaces(M), SubJoin(M), SubMeet(M) #) is 0_Lattice
proof
set S = LattStr (# Subspaces(M), SubJoin(M), SubMeet(M) #);
ex C being Element of S st for A being Element of S holds C "/\" A = C &
A "/\" C = C
proof
reconsider C = (0).M as Element of S by Def3;
take C;
let A be Element of S;
consider W being strict Subspace of M such that
A1: W = A by Def3;
thus C "/\" A = SubMeet(M).(C,A) by LATTICES:def 2
.= (0).M /\ W by A1,Def8
.= C by Th20;
thus A "/\" C = SubMeet(M).(A,C) by LATTICES:def 2
.= W /\ (0).M by A1,Def8
.= C by Th20;
end;
hence thesis by Th57,LATTICES:def 13;
end;
theorem Th59:
LattStr (# Subspaces(M), SubJoin(M), SubMeet(M) #) is 1_Lattice
proof
set S = LattStr (# Subspaces(M), SubJoin(M), SubMeet(M) #);
ex C being Element of S st for A being Element of S holds C "\/" A = C &
A "\/" C = C
proof
consider W9 being strict Subspace of M such that
A1: the carrier of W9 = the carrier of (Omega).M;
reconsider C = W9 as Element of S by Def3;
take C;
let A be Element of S;
consider W being strict Subspace of M such that
A2: W = A by Def3;
A3: C is Subspace of (Omega).M by Lm6;
thus C "\/" A = SubJoin(M).(C,A) by LATTICES:def 1
.= W9 + W by A2,Def7
.= (Omega).M + W by A1,Lm5
.= the ModuleStr of M by Th11
.= C by A1,A3,VECTSP_4:31;
thus A "\/" C = SubJoin(M).(A,C) by LATTICES:def 1
.= W + W9 by A2,Def7
.= W + (Omega).M by A1,Lm5
.= the ModuleStr of M by Th11
.= C by A1,A3,VECTSP_4:31;
end;
hence thesis by Th57,LATTICES:def 14;
end;
theorem Th60:
LattStr (# Subspaces(M), SubJoin(M), SubMeet(M) #) is 01_Lattice
proof
LattStr (# Subspaces(M), SubJoin(M), SubMeet(M) #) is lower-bounded
upper-bounded Lattice by Th58,Th59;
hence thesis;
end;
theorem
LattStr (# Subspaces(M), SubJoin(M), SubMeet(M) #) is M_Lattice
proof
set S = LattStr (# Subspaces(M), SubJoin(M), SubMeet(M) #);
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;
assume
A1: A [= C;
consider W1 being strict Subspace of M such that
A2: W1 = A by Def3;
consider W3 being strict Subspace of M such that
A3: W3 = C by Def3;
W1 + W3 = SubJoin(M).(A,C) by A2,A3,Def7
.= A "\/" C by LATTICES:def 1
.= W3 by A1,A3,LATTICES:def 3;
then
A4: W1 is Subspace of W3 by Th8;
consider W2 being strict Subspace of M such that
A5: W2 = B by Def3;
reconsider AB = W1 + W2 as Element of S by Def3;
reconsider BC = W2 /\ W3 as Element of S by Def3;
thus A "\/" (B "/\" C) = SubJoin(M).(A,B "/\" C) by LATTICES:def 1
.= SubJoin(M).(A,SubMeet(M).(B,C)) by LATTICES:def 2
.= SubJoin(M).(A,BC) by A5,A3,Def8
.= W1 + (W2 /\ W3) by A2,Def7
.= (W1 + W2) /\ W3 by A4,Th30
.= SubMeet(M).(AB,C) by A3,Def8
.= SubMeet(M).(SubJoin(M).(A,B),C) by A2,A5,Def7
.= SubMeet(M).(A "\/" B,C) by LATTICES:def 1
.= (A "\/" B) "/\" C by LATTICES:def 2;
end;
hence thesis by Th57,LATTICES:def 12;
end;
theorem
for F being Field, V being VectSp of F holds LattStr (# Subspaces(V),
SubJoin(V), SubMeet(V) #) is C_Lattice
proof
let F be Field, V be VectSp of F;
reconsider S = LattStr (# Subspaces(V), SubJoin(V), SubMeet(V) #) as
01_Lattice by Th60;
reconsider S0 = S as 0_Lattice;
reconsider S1 = S as 1_Lattice;
consider W9 being strict Subspace of V such that
A1: the carrier of W9 = the carrier of (Omega).V;
reconsider I = W9 as Element of S by Def3;
reconsider I1 = I as Element of S1;
reconsider Z = (0).V as Element of S by Def3;
reconsider Z0 = Z as Element of S0;
now
let A be Element of S0;
consider W being strict Subspace of V such that
A2: W = A by Def3;
thus A "/\" Z0 = SubMeet(V).(A,Z0) by LATTICES:def 2
.= W /\ (0).V by A2,Def8
.= Z0 by Th20;
end;
then
A3: Bottom S = Z by RLSUB_2:64;
now
let A be Element of S1;
consider W being strict Subspace of V such that
A4: W = A by Def3;
A5: W9 is Subspace of (Omega).V by Lm6;
thus A "\/" I1 = SubJoin(V).(A,I1) by LATTICES:def 1
.= W + W9 by A4,Def7
.= W + (Omega).V by A1,Lm5
.= the ModuleStr of V by Th11
.= W9 by A1,A5,VECTSP_4:31;
end;
then
A6: Top S = I by RLSUB_2:65;
now
A7: I is Subspace of (Omega).V by Lm6;
let A be Element of S;
consider W being strict Subspace of V such that
A8: W = A by Def3;
set L = the Linear_Compl of W;
consider W99 being strict Subspace of V such that
A9: the carrier of W99 = the carrier of L by Lm4;
reconsider B9 = W99 as Element of S by Def3;
take B = B9;
A10: B "/\" A = SubMeet(V).(B,A) by LATTICES:def 2
.= W99 /\ W by A8,Def8
.= L /\ W by A9,Lm8
.= Bottom S by A3,Th40;
B "\/" A = SubJoin(V).(B,A) by LATTICES:def 1
.= W99 + W by A8,Def7
.= L + W by A9,Lm5
.= the ModuleStr of V by Th39
.= Top S by A1,A6,A7,VECTSP_4:31;
hence B is_a_complement_of A by A10,LATTICES:def 18;
end;
hence thesis by LATTICES:def 19;
end;