:: Subspaces and Cosets of Subspaces in Real Linear Space
:: by Wojciech A. Trybulec
::
:: Received July 24, 1989
:: 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, REAL_1, SUBSET_1, ARYTM_3, RELAT_1, XBOOLE_0, SUPINF_2,
CARD_1, ARYTM_1, STRUCT_0, TARSKI, ALGSTR_0, REALSET1, ZFMISC_1, NUMBERS,
FUNCT_1, BINOP_1, RLSUB_1;
notations TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, ORDINAL1, NUMBERS, XCMPLX_0,
XREAL_0, REAL_1, MCART_1, RELAT_1, FUNCT_1, FUNCT_2, BINOP_1, REALSET1,
DOMAIN_1, STRUCT_0, ALGSTR_0, RLVECT_1;
constructors PARTFUN1, BINOP_1, REAL_1, NAT_1, REALSET1, RLVECT_1, RELSET_1,
NUMBERS;
registrations XBOOLE_0, SUBSET_1, FUNCT_1, RELSET_1, NUMBERS, REALSET1,
STRUCT_0, RLVECT_1, ORDINAL1, ALGSTR_0, XREAL_0;
requirements NUMERALS, BOOLE, SUBSET, ARITHM;
definitions RLVECT_1, TARSKI, XBOOLE_0, ALGSTR_0;
equalities RLVECT_1, REALSET1, BINOP_1, STRUCT_0, ALGSTR_0;
expansions TARSKI, XBOOLE_0, STRUCT_0;
theorems FUNCT_1, FUNCT_2, RLVECT_1, TARSKI, ZFMISC_1, RELAT_1, RELSET_1,
XBOOLE_0, XBOOLE_1, XCMPLX_0, STRUCT_0, ALGSTR_0, XREAL_0;
schemes XBOOLE_0;
begin
reserve V,X,Y for RealLinearSpace;
reserve u,u1,u2,v,v1,v2 for VECTOR of V;
reserve a for Real;
reserve V1,V2,V3 for Subset of V;
reserve x for object;
::
:: Introduction of predicate linearly closed subsets of the carrier.
::
definition
let V;
let V1;
attr V1 is linearly-closed means
(for v,u st v in V1 & u in V1 holds
v + u in V1) & for a,v st v in V1 holds a * v in V1;
end;
theorem Th1:
V1 <> {} & V1 is linearly-closed implies 0.V in V1
proof
assume that
A1: V1 <> {} and
A2: V1 is linearly-closed;
set x = the Element of V1;
reconsider x as Element of V by A1,TARSKI:def 3;
0 * x in V1 by A1,A2;
hence thesis by RLVECT_1:10;
end;
reconsider jj=1 as Element of REAL by XREAL_0:def 1;
theorem Th2:
V1 is linearly-closed implies for v st v in V1 holds - v in V1
proof
assume
A1: V1 is linearly-closed;
let v;
assume v in V1;
then (- jj) * v in V1 by A1;
hence thesis by RLVECT_1:16;
end;
theorem
V1 is linearly-closed implies for v,u st v in V1 & u in V1 holds v - u in V1
proof
assume
A1: V1 is linearly-closed;
let v,u;
assume that
A2: v in V1 and
A3: u in V1;
- u in V1 by A1,A3,Th2;
hence thesis by A1,A2;
end;
theorem Th4:
{0.V} is linearly-closed
proof
thus for v,u st v in {0.V} & u in {0.V} holds v + u in {0.V}
proof
let v,u;
assume v in {0.V} & u in {0.V};
then v = 0.V & u = 0.V by TARSKI:def 1;
then v + u = 0.V;
hence thesis by TARSKI:def 1;
end;
let a,v;
assume
A1: v in {0.V};
then v = 0.V by TARSKI:def 1;
hence thesis by A1;
end;
theorem
the carrier of V = V1 implies V1 is linearly-closed;
theorem
V1 is linearly-closed & V2 is linearly-closed & V3 = {v + u : v in V1
& u in V2} implies V3 is linearly-closed
proof
assume that
A1: V1 is linearly-closed & V2 is linearly-closed and
A2: V3 = {v + u : v in V1 & u in V2};
thus for v,u st v in V3 & u in V3 holds v + u in V3
proof
let v,u;
assume that
A3: v in V3 and
A4: u in V3;
consider v2,v1 such that
A5: v = v1 + v2 and
A6: v1 in V1 & v2 in V2 by A2,A3;
consider u2,u1 such that
A7: u = u1 + u2 and
A8: u1 in V1 & u2 in V2 by A2,A4;
A9: v + u = ((v1 + v2) + u1) + u2 by A5,A7,RLVECT_1:def 3
.= ((v1 + u1) + v2) + u2 by RLVECT_1:def 3
.= (v1 + u1) + (v2 + u2) by RLVECT_1:def 3;
v1 + u1 in V1 & v2 + u2 in V2 by A1,A6,A8;
hence thesis by A2,A9;
end;
let a,v;
assume v in V3;
then consider v2,v1 such that
A10: v = v1 + v2 and
A11: v1 in V1 & v2 in V2 by A2;
A12: a * v = a * v1 + a * v2 by A10,RLVECT_1:def 5;
a * v1 in V1 & a * v2 in V2 by A1,A11;
hence thesis by A2,A12;
end;
theorem
V1 is linearly-closed & V2 is linearly-closed implies V1 /\ V2 is
linearly-closed
proof
assume that
A1: V1 is linearly-closed and
A2: V2 is linearly-closed;
thus for v,u st v in V1 /\ V2 & u in V1 /\ V2 holds v + u in V1 /\ V2
proof
let v,u;
assume
A3: v in V1 /\ V2 & u in V1 /\ V2;
then v in V2 & u in V2 by XBOOLE_0:def 4;
then
A4: v + u in V2 by A2;
v in V1 & u in V1 by A3,XBOOLE_0:def 4;
then v + u in V1 by A1;
hence thesis by A4,XBOOLE_0:def 4;
end;
let a,v;
assume
A5: v in V1 /\ V2;
then v in V2 by XBOOLE_0:def 4;
then
A6: a * v in V2 by A2;
v in V1 by A5,XBOOLE_0:def 4;
then a * v in V1 by A1;
hence thesis by A6,XBOOLE_0:def 4;
end;
definition
let V;
mode Subspace of V -> RealLinearSpace means
:Def2:
the carrier of it c= the
carrier of V & 0.it = 0.V & the addF of it = (the addF of V)||the carrier of it
& the Mult of it = (the Mult of V) | [:REAL, the carrier of it:];
existence
proof
the addF of V = (the addF of V)||the carrier of V & the Mult of V = (
the Mult of V) | [:REAL, the carrier of V:] by RELSET_1:19;
hence thesis;
end;
end;
reserve W,W1,W2 for Subspace of V;
reserve w,w1,w2 for VECTOR of W;
::
:: Axioms of the subspaces of real linear spaces.
::
theorem
x in W1 & W1 is Subspace of W2 implies x in W2
proof
assume x in W1 & W1 is Subspace of W2;
then x in the carrier of W1 & the carrier of W1 c= the carrier of W2 by Def2;
hence thesis;
end;
theorem Th9:
x in W implies x in V
proof
assume x in W;
then
A1: x in the carrier of W;
the carrier of W c= the carrier of V by Def2;
hence thesis by A1;
end;
theorem Th10:
w is VECTOR of V
proof
w in V by Th9,RLVECT_1:1;
hence thesis;
end;
theorem
0.W = 0.V by Def2;
theorem
0.W1 = 0.W2
proof
thus 0.W1 = 0.V by Def2
.= 0.W2 by Def2;
end;
theorem Th13:
w1 = v & w2 = u implies w1 + w2 = v + u
proof
assume
A1: v = w1 & u = w2;
w1 + w2 = ((the addF of V)||the carrier of W).[w1,w2] by Def2;
hence thesis by A1,FUNCT_1:49;
end;
theorem Th14:
w = v implies a * w = a * v
proof
assume
A1: w = v;
reconsider aa=a as Element of REAL by XREAL_0:def 1;
aa * w = ((the Mult of V) | [:REAL, the carrier of W:]).[aa,w] by Def2;
then aa * w = aa * v by A1,FUNCT_1:49;
hence thesis;
end;
theorem Th15:
w = v implies - v = - w
proof
A1: - v = (- jj) * v & - w = (- jj) * w by RLVECT_1:16;
assume w = v;
hence thesis by A1,Th14;
end;
theorem Th16:
w1 = v & w2 = u implies w1 - w2 = v - u
proof
assume that
A1: w1 = v and
A2: w2 = u;
- w2 = - u by A2,Th15;
hence thesis by A1,Th13;
end;
Lm1: the carrier of W = V1 implies V1 is linearly-closed
proof
set VW = the carrier of W;
reconsider WW = W as RealLinearSpace;
assume
A1: the carrier of W = V1;
thus for v,u st v in V1 & u in V1 holds v + u in V1
proof
let v,u;
assume v in V1 & u in V1;
then reconsider vv = v, uu = u as VECTOR of WW by A1;
reconsider vw = vv + uu as Element of VW;
vw in V1 by A1;
hence thesis by Th13;
end;
let a,v;
assume v in V1;
then reconsider vv = v as VECTOR of WW by A1;
reconsider vw = a * vv as Element of VW;
vw in V1 by A1;
hence thesis by Th14;
end;
theorem Th17:
0.V in W
proof
0.W in W;
hence thesis by Def2;
end;
theorem
0.W1 in W2
proof
0.W1 = 0.V by Def2;
hence thesis by Th17;
end;
theorem
0.W in V by Th9,RLVECT_1:1;
theorem Th20:
u in W & v in W implies u + v in W
proof
reconsider VW = the carrier of W as Subset of V by Def2;
assume u in W & v in W;
then
A1: u in the carrier of W & v in the carrier of W;
VW is linearly-closed by Lm1;
then u + v in the carrier of W by A1;
hence thesis;
end;
theorem Th21:
v in W implies a * v in W
proof
reconsider VW = the carrier of W as Subset of V by Def2;
assume v in W;
then
A1: v in the carrier of W;
VW is linearly-closed by Lm1;
then a * v in the carrier of W by A1;
hence thesis;
end;
theorem Th22:
v in W implies - v in W
proof
assume v in W;
then (- jj) * v in W by Th21;
hence thesis by RLVECT_1:16;
end;
theorem Th23:
u in W & v in W implies u - v in W
proof
assume that
A1: u in W and
A2: v in W;
- v in W by A2,Th22;
hence thesis by A1,Th20;
end;
reserve D for non empty set;
reserve d1 for Element of D;
reserve A for BinOp of D;
reserve M for Function of [:REAL,D:],D;
theorem Th24:
V1 = D & d1 = 0.V & A = (the addF of V)||V1 & M = (the Mult of V
) | [:REAL,V1:] implies RLSStruct (# D,d1,A,M #) is Subspace of V
proof
assume that
A1: V1 = D and
A2: d1 = 0.V and
A3: A = (the addF of V)||V1 and
A4: M = (the Mult of V) | [:REAL,V1:];
set W = RLSStruct (# D,d1,A,M #);
A5: for a for x being VECTOR of W holds a * x = (the Mult of V).(a,x)
proof
let a;
let x be VECTOR of W;
reconsider aa=a as Element of REAL by XREAL_0:def 1;
thus a * x = (the Mult of V).[aa,x] by A1,A4,FUNCT_1:49
.= (the Mult of V).(a,x);
end;
A6: for x,y being VECTOR of W holds x + y = (the addF of V).(x,y)
proof
let x,y be VECTOR of W;
thus x + y = (the addF of V).[x,y] by A1,A3,FUNCT_1:49
.= (the addF of V).(x,y);
end;
A7: W is Abelian add-associative right_zeroed right_complementable
vector-distributive scalar-distributive scalar-associative scalar-unital
proof
set MV = the Mult of V;
set AV = the addF of V;
thus W is Abelian
proof
let x,y be VECTOR of W;
reconsider x1 = x, y1 = y as VECTOR of V by A1,TARSKI:def 3;
thus x + y = x1 + y1 by A6
.= y1 + x1
.= y + x by A6;
end;
thus W is add-associative
proof
let x,y,z be VECTOR of W;
reconsider x1 = x, y1 = y, z1 = z as VECTOR of V by A1,TARSKI:def 3;
thus (x + y) + z = AV.(x + y,z1) by A6
.= (x1 + y1) + z1 by A6
.= x1 + (y1 + z1) by RLVECT_1:def 3
.= AV.(x1,y + z) by A6
.= x + (y + z) by A6;
end;
thus W is right_zeroed
proof
let x be VECTOR of W;
reconsider y = x as VECTOR of V by A1,TARSKI:def 3;
thus x + 0.W = y + 0.V by A2,A6
.= x;
end;
thus W is right_complementable
proof
let x be VECTOR of W;
reconsider x1 = x as VECTOR of V by A1,TARSKI:def 3;
consider v such that
A8: x1 + v = 0.V by ALGSTR_0:def 11;
v = - x1 by A8,RLVECT_1:def 10
.= (- 1) * x1 by RLVECT_1:16
.= (- jj) * x by A5;
then reconsider y = v as VECTOR of W;
take y;
thus thesis by A2,A6,A8;
end;
thus for a being Real for x,y being VECTOR of W holds a * (x + y) =
a * x + a * y
proof
let a be Real;
let x,y be VECTOR of W;
reconsider x1 = x, y1 = y as VECTOR of V by A1,TARSKI:def 3;
reconsider a as Real;
a * (x + y) = MV.(a,x + y) by A5
.= a * (x1 + y1) by A6
.= a * x1 + a * y1 by RLVECT_1:def 5
.= AV.(MV.(a,x1),a * y) by A5
.= AV.(a * x, a * y) by A5
.= a * x + a * y by A6;
hence thesis;
end;
thus for a,b being Real for x being VECTOR of W holds (a + b) * x =
a * x + b * x
proof
let a,b be Real;
let x be VECTOR of W;
reconsider y = x as VECTOR of V by A1,TARSKI:def 3;
reconsider a,b as Real;
(a + b) * x = (a + b) * y by A5
.= a * y + b * y by RLVECT_1:def 6
.= AV.(MV.(a,y),b * x) by A5
.= AV.(a * x,b * x) by A5
.= a * x + b * x by A6;
hence thesis;
end;
thus for a,b being Real for x being VECTOR of W holds (a * b) * x =
a * (b * x)
proof
let a,b be Real;
let x be VECTOR of W;
reconsider y = x as VECTOR of V by A1,TARSKI:def 3;
reconsider a,b as Real;
(a * b) * x = (a * b) * y by A5
.= a * (b * y) by RLVECT_1:def 7
.= MV.(a,b * x) by A5
.= a * (b * x) by A5;
hence thesis;
end;
let x be VECTOR of W;
reconsider y = x as VECTOR of V by A1,TARSKI:def 3;
thus 1 * x = jj * y by A5
.= x by RLVECT_1:def 8;
end;
0.W = 0.V by A2;
hence thesis by A1,A3,A4,A7,Def2;
end;
theorem Th25:
V is Subspace of V
proof
thus the carrier of V c= the carrier of V & 0.V = 0.V;
thus thesis by RELSET_1:19;
end;
theorem Th26:
for V,X being strict RealLinearSpace holds V is Subspace of X &
X is Subspace of V implies V = X
proof
let V,X be strict RealLinearSpace;
assume that
A1: V is Subspace of X and
A2: X is Subspace of V;
set VX = the carrier of X;
set VV = the carrier of V;
VV c= VX & VX c= VV by A1,A2,Def2;
then
A3: VV = VX;
set AX = the addF of X;
set AV = the addF of V;
AV = AX||VV & AX = AV||VX by A1,A2,Def2;
then
A4: AV = AX by A3,RELAT_1:72;
set MX = the Mult of X;
set MV = the Mult of V;
A5: MX = MV | [:REAL,VX:] by A2,Def2;
0.V = 0.X & MV = MX | [:REAL,VV:] by A1,Def2;
hence thesis by A3,A4,A5,RELAT_1:72;
end;
theorem Th27:
V is Subspace of X & X is Subspace of Y implies V is Subspace of Y
proof
assume that
A1: V is Subspace of X and
A2: X is Subspace of Y;
the carrier of V c= the carrier of X & the carrier of X c= the carrier
of Y by A1,A2,Def2;
hence the carrier of V c= the carrier of Y;
0.V = 0.X by A1,Def2;
hence 0.V = 0.Y by A2,Def2;
thus the addF of V = (the addF of Y)||the carrier of V
proof
set AY = the addF of Y;
set VX = the carrier of X;
set AX = the addF of X;
set VV = the carrier of V;
set AV = the addF of V;
VV c= VX by A1,Def2;
then
A3: [:VV,VV:] c= [:VX,VX:] by ZFMISC_1:96;
AV = AX||VV by A1,Def2;
then AV = AY||VX||VV by A2,Def2;
hence thesis by A3,FUNCT_1:51;
end;
set MY = the Mult of Y;
set MX = the Mult of X;
set MV = the Mult of V;
set VX = the carrier of X;
set VV = the carrier of V;
VV c= VX by A1,Def2;
then
A4: [:REAL,VV:] c= [:REAL,VX:] by ZFMISC_1:95;
MV = MX | [:REAL,VV:] by A1,Def2;
then MV = (MY | [:REAL,VX:]) | [:REAL,VV:] by A2,Def2;
hence thesis by A4,FUNCT_1:51;
end;
theorem Th28:
the carrier of W1 c= the carrier of W2 implies W1 is Subspace of W2
proof
set VW1 = the carrier of W1;
set VW2 = the carrier of W2;
set AV = the addF of V;
set MV = the Mult of V;
assume
A1: the carrier of W1 c= the carrier of W2;
then
A2: [:VW1,VW1:] c= [:VW2,VW2:] by ZFMISC_1:96;
0.W1 = 0.V by Def2;
hence the carrier of W1 c= the carrier of W2 & 0.W1 = 0.W2 by A1,Def2;
the addF of W1 = AV||VW1 & the addF of W2 = AV||VW2 by Def2;
hence the addF of W1 = (the addF of W2)||the carrier of W1 by A2,FUNCT_1:51;
A3: [:REAL,VW1:] c= [:REAL,VW2:] by A1,ZFMISC_1:95;
the Mult of W1 = MV | [:REAL,VW1:] & the Mult of W2 = MV | [:REAL,VW2 :]
by Def2;
hence thesis by A3,FUNCT_1:51;
end;
theorem
(for v st v in W1 holds v in W2) implies W1 is Subspace of W2
proof
assume
A1: for v st v in W1 holds v in W2;
the carrier of W1 c= the carrier of W2
proof
let x be object;
assume
A2: x in the carrier of W1;
the carrier of W1 c= the carrier of V by Def2;
then reconsider v = x as VECTOR of V by A2;
v in W1 by A2;
then v in W2 by A1;
hence thesis;
end;
hence thesis by Th28;
end;
registration
let V;
cluster strict for Subspace of V;
existence
proof
the carrier of V is Subset of V iff the carrier of V c= the carrier of V;
then reconsider V1 = the carrier of V as Subset of V;
the addF of V = (the addF of V)||V1 & the Mult of V = (the Mult of V)
| [: REAL,V1:] by RELSET_1:19;
then RLSStruct(#the carrier of V,0.V,the addF of V,the Mult of V #) is
Subspace of V by Th24;
hence thesis;
end;
end;
theorem Th30:
for W1,W2 being strict Subspace of V holds the carrier of W1 =
the carrier of W2 implies W1 = W2
proof
let W1,W2 be strict Subspace of V;
assume the carrier of W1 = the carrier of W2;
then W1 is Subspace of W2 & W2 is Subspace of W1 by Th28;
hence thesis by Th26;
end;
theorem Th31:
for W1,W2 being strict Subspace of V holds (for v holds v in W1
iff v in W2) implies W1 = W2
proof
let W1,W2 be strict Subspace of V;
assume
A1: for v holds v in W1 iff v in W2;
for x being object holds x in the carrier of W1 iff x in the carrier of W2
proof let x be object;
thus x in the carrier of W1 implies x in the carrier of W2
proof
assume
A2: x in the carrier of W1;
the carrier of W1 c= the carrier of V by Def2;
then reconsider v = x as VECTOR of V by A2;
v in W1 by A2;
then v in W2 by A1;
hence thesis;
end;
assume
A3: x in the carrier of W2;
the carrier of W2 c= the carrier of V by Def2;
then reconsider v = x as VECTOR of V by A3;
v in W2 by A3;
then v in W1 by A1;
hence thesis;
end;
then the carrier of W1 = the carrier of W2 by TARSKI:2;
hence thesis by Th30;
end;
theorem
for V being strict RealLinearSpace, W being strict Subspace of V holds
the carrier of W = the carrier of V implies W = V
proof
let V be strict RealLinearSpace, W be strict Subspace of V;
assume
A1: the carrier of W = the carrier of V;
V is Subspace of V by Th25;
hence thesis by A1,Th30;
end;
theorem
for V being strict RealLinearSpace, W being strict Subspace of V holds
(for v being VECTOR of V holds v in W iff v in V) implies W = V
proof
let V be strict RealLinearSpace, W be strict Subspace of V;
assume
A1: for v being VECTOR of V holds v in W iff v in V;
V is Subspace of V by Th25;
hence thesis by A1,Th31;
end;
theorem
the carrier of W = V1 implies V1 is linearly-closed by Lm1;
theorem Th35:
V1 <> {} & V1 is linearly-closed implies ex W being strict
Subspace of V st V1 = the carrier of W
proof
assume that
A1: V1 <> {} and
A2: V1 is linearly-closed;
reconsider D = V1 as non empty set by A1;
set M = (the Mult of V) | [:REAL,V1:];
set VV = the carrier of V;
dom(the Mult of V) = [:REAL,VV:] by FUNCT_2:def 1;
then
A3: dom M = [:REAL,VV:] /\ [:REAL,V1:] by RELAT_1:61;
[:REAL,V1:] c= [:REAL,VV:] by ZFMISC_1:95;
then
A4: dom M = [:REAL,D:] by A3,XBOOLE_1:28;
now
let y be object;
thus y in D implies ex x being object st x in dom M & y = M.x
proof
assume
A5: y in D;
then reconsider v1 = y as Element of VV;
A6: [jj,y] in [:REAL,D:] by A5,ZFMISC_1:87;
then M.[1,y] = 1 * v1 by FUNCT_1:49
.= y by RLVECT_1:def 8;
hence thesis by A4,A6;
end;
given x being object such that
A7: x in dom M and
A8: y = M.x;
consider x1,x2 being object such that
A9: x1 in REAL and
A10: x2 in D and
A11: x = [x1,x2] by A4,A7,ZFMISC_1:def 2;
reconsider xx1 = x1 as Real by A9;
reconsider v2 = x2 as Element of VV by A10;
[x1,x2] in [:REAL,V1:] by A9,A10,ZFMISC_1:87;
then y = xx1 * v2 by A8,A11,FUNCT_1:49;
hence y in D by A2,A10;
end;
then D = rng M by FUNCT_1:def 3;
then reconsider M as Function of [:REAL,D:],D by A4,FUNCT_2:def 1,RELSET_1:4;
set A = (the addF of V)||V1;
reconsider d1 = 0.V as Element of D by A2,Th1;
dom(the addF of V) = [:VV,VV:] by FUNCT_2:def 1;
then dom A = [:VV,VV:] /\ [:V1,V1:] by RELAT_1:61;
then
A12: dom A = [:D,D:] by XBOOLE_1:28;
now
let y be object;
thus y in D implies ex x being object st x in dom A & y = A.x
proof
assume
A13: y in D;
then reconsider v1 = y, v0 = d1 as Element of VV;
A14: [d1,y] in [:D,D:] by A13,ZFMISC_1:87;
then A.[d1,y] = v0 + v1 by FUNCT_1:49
.= y;
hence thesis by A12,A14;
end;
given x being object such that
A15: x in dom A and
A16: y = A.x;
consider x1,x2 being object such that
A17: x1 in D & x2 in D and
A18: x = [x1,x2] by A12,A15,ZFMISC_1:def 2;
reconsider v1 = x1, v2 = x2 as Element of VV by A17;
[x1,x2] in [:V1,V1:] by A17,ZFMISC_1:87;
then y = v1 + v2 by A16,A18,FUNCT_1:49;
hence y in D by A2,A17;
end;
then D = rng A by FUNCT_1:def 3;
then reconsider A as Function of [:D,D:],D by A12,FUNCT_2:def 1,RELSET_1:4;
set W = RLSStruct (# D,d1,A,M #);
W is Subspace of V by Th24;
hence thesis;
end;
::
:: Definition of zero subspace and improper subspace of real linear space.
::
definition
let V;
func (0).V -> strict Subspace of V means
:Def3:
the carrier of it = {0.V};
correctness by Th4,Th30,Th35;
end;
definition
let V;
func (Omega).V -> strict Subspace of V equals
the RLSStruct of V;
coherence
proof
set W = the RLSStruct of V;
A1: for u,v,w being VECTOR of W holds (u + v) + w = u + (v + w)
proof
let u,v,w be VECTOR of W;
reconsider u9=u,v9=v,w9=w as VECTOR of V;
thus (u + v) + w = (u9 + v9) + w9 .= u9 + (v9 + w9) by RLVECT_1:def 3
.= u + (v + w);
end;
A2: for v being VECTOR of W holds v + 0.W = v
proof
let v be VECTOR of W;
reconsider v9=v as VECTOR of V;
thus v + 0.W = v9 + 0.V .= v;
end;
A3: W is right_complementable
proof
let v be VECTOR of W;
reconsider v9=v as VECTOR of V;
consider w9 being VECTOR of V such that
A4: v9 + w9 = 0.V by ALGSTR_0:def 11;
reconsider w=w9 as VECTOR of W;
take w;
thus thesis by A4;
end;
A5: for a being Real for v,w being VECTOR of W holds a * (v + w) =
a * v + a * w
proof
let a be Real;
let v,w be VECTOR of W;
reconsider v9=v,w9=w as VECTOR of V;
thus a * (v + w) = a * (v9 + w9) .= a * v9 + a * w9 by RLVECT_1:def 5
.= a * v + a * w;
end;
A6: for a,b being Real for v being VECTOR of W holds (a * b) * v =
a * (b * v)
proof
let a,b be Real;
let v be VECTOR of W;
reconsider v9=v as VECTOR of V;
thus (a * b) * v = (a * b) * v9 .= a * (b * v9) by RLVECT_1:def 7
.= a * (b * v);
end;
A7: for a,b being Real for v being VECTOR of W holds (a + b) * v =
a * v + b * v
proof
let a,b be Real;
let v be VECTOR of W;
reconsider v9=v as VECTOR of V;
thus (a + b) * v = (a + b) * v9 .= a * v9 + b * v9 by RLVECT_1:def 6
.= a * v + b * v;
end;
A8: for a for v,w be VECTOR of W, v9,w9 be VECTOR of V st v=v9 & w=w9
holds v+w = v9+w9 & a*v = a*v9;
A9: for v,w being VECTOR of W holds v + w = w + v
proof
let v,w be VECTOR of W;
reconsider v9=v,w9=w as VECTOR of V;
thus v + w = w9 + v9 by A8
.= w + v;
end;
for v being VECTOR of W holds 1 * v = v
proof
let v be VECTOR of W;
reconsider v9=v as VECTOR of V;
thus 1 * v = 1 * v9 .= v by RLVECT_1:def 8;
end;
then reconsider W as RealLinearSpace by A9,A1,A2,A3,A5,A7,A6,RLVECT_1:def 2
,def 3,def 4,def 5,def 6,def 7,def 8;
A10: the Mult of W = (the Mult of V) | [:REAL, the carrier of W:] by
RELSET_1:19;
0.W = 0.V & the addF of W = (the addF of V)||the carrier of W by
RELSET_1:19;
hence thesis by A10,Def2;
end;
end;
::
:: Definitional theorems of zero subspace and improper subspace.
::
theorem Th36:
(0).W = (0).V
proof
the carrier of (0).W = {0.W} & the carrier of (0).V = {0.V} by Def3;
then
A1: the carrier of (0).W = the carrier of (0).V by Def2;
(0).W is Subspace of V by Th27;
hence thesis by A1,Th30;
end;
theorem Th37:
(0).W1 = (0).W2
proof
(0).W1 = (0).V by Th36;
hence thesis by Th36;
end;
theorem
(0).W is Subspace of V by Th27;
theorem
(0).V is Subspace of W
proof
the carrier of (0).V = {0.V} by Def3
.= {0.W} by Def2;
hence thesis by Th28;
end;
theorem
(0).W1 is Subspace of W2
proof
(0).W1 = (0).W2 by Th37;
hence thesis;
end;
theorem
for V being strict RealLinearSpace holds V is Subspace of (Omega).V;
::
:: Introduction of the cosets of subspace.
::
definition
let V;
let v,W;
func v + W -> Subset of V equals
{v + u : u in W};
coherence
proof
set Y = {v + u : u in W};
defpred P[object] means ex u st $1 = v + u & u in W;
consider X being set such that
A1: for x being object holds x in X iff x in the carrier of V & P[x] from
XBOOLE_0:sch 1;
X c= the carrier of V
by A1;
then reconsider X as Subset of V;
A2: Y c= X
proof
let x be object;
assume x in Y;
then ex u st x = v + u & u in W;
hence thesis by A1;
end;
X c= Y
proof
let x be object;
assume x in X;
then ex u st x = v + u & u in W by A1;
hence thesis;
end;
hence thesis by A2,XBOOLE_0:def 10;
end;
end;
Lm2: 0.V + W = the carrier of W
proof
set A = {0.V + u : u in W};
A1: the carrier of W c= A
proof
let x be object;
assume x in the carrier of W;
then
A2: x in W;
then x in V by Th9;
then reconsider y = x as Element of V;
0.V + y = x;
hence thesis by A2;
end;
A c= the carrier of W
proof
let x be object;
assume x in A;
then consider u such that
A3: x = 0.V + u and
A4: u in W;
x = u by A3;
hence thesis by A4;
end;
hence thesis by A1;
end;
definition
let V;
let W;
mode Coset of W -> Subset of V means
:Def6:
ex v st it = v + W;
existence
proof
reconsider VW = the carrier of W as Subset of V by Def2;
take VW;
take 0.V;
thus thesis by Lm2;
end;
end;
reserve B,C for Coset of W;
::
:: Definitional theorems of the cosets.
::
theorem Th42:
0.V in v + W iff v in W
proof
thus 0.V in v + W implies v in W
proof
assume 0.V in v + W;
then consider u such that
A1: 0.V = v + u and
A2: u in W;
v = - u by A1,RLVECT_1:def 10;
hence thesis by A2,Th22;
end;
assume v in W;
then
A3: - v in W by Th22;
0.V = v - v by RLVECT_1:15
.= v + (- v);
hence thesis by A3;
end;
theorem Th43:
v in v + W
proof
v + 0.V = v & 0.V in W by Th17;
hence thesis;
end;
theorem
0.V + W = the carrier of W by Lm2;
theorem Th45:
v + (0).V = {v}
proof
thus v + (0).V c= {v}
proof
let x be object;
assume x in v + (0).V;
then consider u such that
A1: x = v + u and
A2: u in (0).V;
A3: the carrier of (0).V = {0.V} by Def3;
u in the carrier of (0).V by A2;
then u = 0.V by A3,TARSKI:def 1;
then x = v by A1;
hence thesis by TARSKI:def 1;
end;
let x be object;
assume x in {v};
then
A4: x = v by TARSKI:def 1;
0.V in (0).V & v = v + 0.V by Th17;
hence thesis by A4;
end;
Lm3: v in W iff v + W = the carrier of W
proof
0.V in W & v + 0.V = v by Th17;
then
A1: v in {v + u : u in W};
thus v in W implies v + W = the carrier of W
proof
assume
A2: v in W;
thus v + W c= the carrier of W
proof
let x be object;
assume x in v + W;
then consider u such that
A3: x = v + u and
A4: u in W;
v + u in W by A2,A4,Th20;
hence thesis by A3;
end;
let x be object;
assume x in the carrier of W;
then reconsider y = x, z = v as Element of W by A2;
reconsider y1 = y, z1 = z as VECTOR of V by Th10;
A5: z + (y - z) = (y + z) - z by RLVECT_1:def 3
.= y + (z - z) by RLVECT_1:def 3
.= y + 0.W by RLVECT_1:15
.= x;
y - z in W;
then
A6: y1 - z1 in W by Th16;
y - z = y1 - z1 by Th16;
then z1 + (y1 - z1) = x by A5,Th13;
hence thesis by A6;
end;
assume
A7: v + W = the carrier of W;
assume not v in W;
hence thesis by A7,A1;
end;
theorem Th46:
v + (Omega).V = the carrier of V
by STRUCT_0:def 5,Lm3;
theorem Th47:
0.V in v + W iff v + W = the carrier of W
by Th42,Lm3;
theorem
v in W iff v + W = the carrier of W by Lm3;
theorem Th49:
v in W implies (a * v) + W = the carrier of W
proof
assume
A1: v in W;
thus (a * v) + W c= the carrier of W
proof
let x be object;
assume x in (a * v) + W;
then consider u such that
A2: x = a * v + u and
A3: u in W;
a * v in W by A1,Th21;
then a * v + u in W by A3,Th20;
hence thesis by A2;
end;
let x be object;
assume
A4: x in the carrier of W;
then
A5: x in W;
the carrier of W c= the carrier of V by Def2;
then reconsider y = x as Element of V by A4;
A6: a * v + (y - a * v) = (y + a * v) - a * v by RLVECT_1:def 3
.= y + (a * v - a * v) by RLVECT_1:def 3
.= y + 0.V by RLVECT_1:15
.= x;
a * v in W by A1,Th21;
then y - a * v in W by A5,Th23;
hence thesis by A6;
end;
theorem Th50:
a <> 0 & (a * v) + W = the carrier of W implies v in W
proof
assume that
A1: a <> 0 and
A2: (a * v) + W = the carrier of W;
assume not v in W;
then not 1 * v in W by RLVECT_1:def 8;
then not (a" * a) * v in W by A1,XCMPLX_0:def 7;
then not a" * (a * v) in W by RLVECT_1:def 7;
then
A3: not a * v in W by Th21;
0.V in W & a * v + 0.V = a * v by Th17;
then a * v in {a * v + u : u in W};
hence contradiction by A2,A3;
end;
theorem Th51:
v in W iff - v + W = the carrier of W
proof
v in W iff ((- jj) * v) + W = the carrier of W by Th49,Th50;
hence thesis by RLVECT_1:16;
end;
theorem Th52:
u in W iff v + W = (v + u) + W
proof
thus u in W implies v + W = (v + u) + W
proof
assume
A1: u in W;
thus v + W c= (v + u) + W
proof
let x be object;
assume x in v + W;
then consider v1 such that
A2: x = v + v1 and
A3: v1 in W;
A4: (v + u) + (v1 - u) = v + (u + (v1 - u)) by RLVECT_1:def 3
.= v + ((v1 + u) - u) by RLVECT_1:def 3
.= v + (v1 + (u - u)) by RLVECT_1:def 3
.= v + (v1 + 0.V) by RLVECT_1:15
.= x by A2;
v1 - u in W by A1,A3,Th23;
hence thesis by A4;
end;
let x be object;
assume x in (v + u) + W;
then consider v2 such that
A5: x = (v + u) + v2 and
A6: v2 in W;
A7: x = v + (u + v2) by A5,RLVECT_1:def 3;
u + v2 in W by A1,A6,Th20;
hence thesis by A7;
end;
assume
A8: v + W = (v + u) + W;
0.V in W & v + 0.V = v by Th17;
then v in (v + u) + W by A8;
then consider u1 such that
A9: v = (v + u) + u1 and
A10: u1 in W;
v = v + 0.V & v = v + (u + u1) by A9,RLVECT_1:def 3;
then u + u1 = 0.V by RLVECT_1:8;
then u = - u1 by RLVECT_1:def 10;
hence thesis by A10,Th22;
end;
theorem
u in W iff v + W = (v - u) + W
proof
A1: - u in W implies u in W
proof
assume - u in W;
then - (- u) in W by Th22;
hence thesis;
end;
- u in W iff v + W = (v + (- u)) + W by Th52;
hence thesis by A1,Th22;
end;
theorem Th54:
v in u + W iff u + W = v + W
proof
thus v in u + W implies u + W = v + W
proof
assume v in u + W;
then consider z being VECTOR of V such that
A1: v = u + z and
A2: z in W;
thus u + W c= v + W
proof
let x be object;
assume x in u + W;
then consider v1 such that
A3: x = u + v1 and
A4: v1 in W;
v - z = u + (z - z) by A1,RLVECT_1:def 3
.= u + 0.V by RLVECT_1:15
.= u;
then
A5: x = v + (v1 + (- z)) by A3,RLVECT_1:def 3
.= v + (v1 - z);
v1 - z in W by A2,A4,Th23;
hence thesis by A5;
end;
let x be object;
assume x in v + W;
then consider v2 such that
A6: x = v + v2 & v2 in W;
z + v2 in W & x = u + (z + v2) by A1,A2,A6,Th20,RLVECT_1:def 3;
hence thesis;
end;
thus thesis by Th43;
end;
theorem Th55:
v + W = (- v) + W iff v in W
proof
thus v + W = (- v) + W implies v in W
proof
assume v + W = (- v) + W;
then v in (- v) + W by Th43;
then consider u such that
A1: v = - v + u and
A2: u in W;
reconsider dwa=2 as Real;
0.V = v - (- v + u) by A1,RLVECT_1:15
.= (v - (- v)) - u by RLVECT_1:27
.= (v + v) - u
.= (1 * v + v) - u by RLVECT_1:def 8
.= (1 * v + 1 * v) - u by RLVECT_1:def 8
.= ((1 + 1) * v) - u by RLVECT_1:def 6
.= 2 * v - u;
then 2" * (2 * v) = 2" * u by RLVECT_1:21;
then (2" * 2) * v = 2" * u by RLVECT_1:def 7;
then v = dwa" * u by RLVECT_1:def 8;
hence thesis by A2,Th21;
end;
assume
A3: v in W;
then v + W = the carrier of W by Lm3;
hence thesis by A3,Th51;
end;
theorem Th56:
u in v1 + W & u in v2 + W implies v1 + W = v2 + W
proof
assume that
A1: u in v1 + W and
A2: u in v2 + W;
consider x1 being VECTOR of V such that
A3: u = v1 + x1 and
A4: x1 in W by A1;
consider x2 being VECTOR of V such that
A5: u = v2 + x2 and
A6: x2 in W by A2;
thus v1 + W c= v2 + W
proof
let x be object;
assume x in v1 + W;
then consider u1 such that
A7: x = v1 + u1 and
A8: u1 in W;
x2 - x1 in W by A4,A6,Th23;
then
A9: (x2 - x1) + u1 in W by A8,Th20;
u - x1 = v1 + (x1 - x1) by A3,RLVECT_1:def 3
.= v1 + 0.V by RLVECT_1:15
.= v1;
then x = (v2 + (x2 - x1)) + u1 by A5,A7,RLVECT_1:def 3
.= v2 + ((x2 - x1) + u1) by RLVECT_1:def 3;
hence thesis by A9;
end;
let x be object;
assume x in v2 + W;
then consider u1 such that
A10: x = v2 + u1 and
A11: u1 in W;
x1 - x2 in W by A4,A6,Th23;
then
A12: (x1 - x2) + u1 in W by A11,Th20;
u - x2 = v2 + (x2 - x2) by A5,RLVECT_1:def 3
.= v2 + 0.V by RLVECT_1:15
.= v2;
then x = (v1 + (x1 - x2)) + u1 by A3,A10,RLVECT_1:def 3
.= v1 + ((x1 - x2) + u1) by RLVECT_1:def 3;
hence thesis by A12;
end;
theorem
u in v + W & u in (- v) + W implies v in W
by Th56,Th55;
theorem Th58:
a <> 1 & a * v in v + W implies v in W
proof
assume that
A1: a <> 1 and
A2: a * v in v + W;
A3: a - 1 <> 0 by A1;
consider u such that
A4: a * v = v + u and
A5: u in W by A2;
u = u + 0.V
.= u + (v - v) by RLVECT_1:15
.= a * v - v by A4,RLVECT_1:def 3
.= a * v - 1 * v by RLVECT_1:def 8
.= (a - 1) * v by RLVECT_1:35;
then (a - 1)" * u = ((a - 1)" * (a - 1)) * v by RLVECT_1:def 7;
then 1 * v = (a - 1)" * u by A3,XCMPLX_0:def 7;
then v = (a - 1)" * u by RLVECT_1:def 8;
hence thesis by A5,Th21;
end;
theorem Th59:
v in W implies a * v in v + W
proof
assume v in W;
then
A1: (a - 1) * v in W by Th21;
a * v = ((a - 1) + 1) * v .= (a - 1) * v + 1 * v by RLVECT_1:def 6
.= v + (a - 1) * v by RLVECT_1:def 8;
hence thesis by A1;
end;
theorem
- v in v + W iff v in W
proof
(- jj) * v = - v by RLVECT_1:16;
hence thesis by Th58,Th59;
end;
theorem Th61:
u + v in v + W iff u in W
proof
thus u + v in v + W implies u in W
proof
assume u + v in v + W;
then ex v1 st u + v = v + v1 & v1 in W;
hence thesis by RLVECT_1:8;
end;
assume u in W;
hence thesis;
end;
theorem
v - u in v + W iff u in W
proof
A1: v - u = (- u) + v;
A2: - u in W implies - (- u) in W by Th22;
u in W implies - u in W by Th22;
hence thesis by A1,A2,Th61;
end;
theorem Th63:
u in v + W iff ex v1 st v1 in W & u = v + v1
proof
thus u in v + W implies ex v1 st v1 in W & u = v + v1
proof
assume u in v + W;
then ex v1 st u = v + v1 & v1 in W;
hence thesis;
end;
given v1 such that
A1: v1 in W & u = v + v1;
thus thesis by A1;
end;
theorem
u in v + W iff ex v1 st v1 in W & u = v - v1
proof
thus u in v + W implies ex v1 st v1 in W & u = v - v1
proof
assume u in v + W;
then consider v1 such that
A1: u = v + v1 and
A2: v1 in W;
take x = - v1;
thus x in W by A2,Th22;
thus thesis by A1;
end;
given v1 such that
A3: v1 in W and
A4: u = v - v1;
- v1 in W by A3,Th22;
hence thesis by A4;
end;
theorem Th65:
(ex v st v1 in v + W & v2 in v + W) iff v1 - v2 in W
proof
thus (ex v st v1 in v + W & v2 in v + W) implies v1 - v2 in W
proof
given v such that
A1: v1 in v + W and
A2: v2 in v + W;
consider u2 such that
A3: u2 in W and
A4: v2 = v + u2 by A2,Th63;
consider u1 such that
A5: u1 in W and
A6: v1 = v + u1 by A1,Th63;
v1 - v2 = (u1 + v) + ((- v) - u2) by A6,A4,RLVECT_1:30
.= ((u1 + v) + (- v)) - u2 by RLVECT_1:def 3
.= (u1 + (v + (- v))) - u2 by RLVECT_1:def 3
.= (u1 + 0.V) - u2 by RLVECT_1:5
.= u1 - u2;
hence thesis by A5,A3,Th23;
end;
assume v1 - v2 in W;
then
A7: - (v1 - v2) in W by Th22;
take v1;
thus v1 in v1 + W by Th43;
v1 + (- (v1 - v2)) = v1 + ((- v1) + v2) by RLVECT_1:33
.= (v1 + (- v1)) + v2 by RLVECT_1:def 3
.= 0.V + v2 by RLVECT_1:5
.= v2;
hence thesis by A7;
end;
theorem Th66:
v + W = u + W implies ex v1 st v1 in W & v + v1 = u
proof
assume v + W = u + W;
then v in u + W by Th43;
then consider u1 such that
A1: v = u + u1 and
A2: u1 in W;
take v1 = u - v;
0.V = (u + u1) - v by A1,RLVECT_1:15
.= u1 + (u - v) by RLVECT_1:def 3;
then v1 = - u1 by RLVECT_1:def 10;
hence v1 in W by A2,Th22;
thus v + v1 = (u + v) - v by RLVECT_1:def 3
.= u + (v - v) by RLVECT_1:def 3
.= u + 0.V by RLVECT_1:15
.= u;
end;
theorem Th67:
v + W = u + W implies ex v1 st v1 in W & v - v1 = u
proof
assume v + W = u + W;
then u in v + W by Th43;
then consider u1 such that
A1: u = v + u1 and
A2: u1 in W;
take v1 = v - u;
0.V = (v + u1) - u by A1,RLVECT_1:15
.= u1 + (v - u) by RLVECT_1:def 3;
then v1 = - u1 by RLVECT_1:def 10;
hence v1 in W by A2,Th22;
thus v - v1 = (v - v) + u by RLVECT_1:29
.= 0.V + u by RLVECT_1:15
.= u;
end;
theorem Th68:
for W1,W2 being strict Subspace of V holds v + W1 = v + W2 iff W1 = W2
proof
let W1,W2 be strict Subspace of V;
thus v + W1 = v + W2 implies W1 = W2
proof
assume
A1: v + W1 = v + W2;
the carrier of W1 = the carrier of W2
proof
A2: the carrier of W1 c= the carrier of V by Def2;
thus the carrier of W1 c= the carrier of W2
proof
let x be object;
assume
A3: x in the carrier of W1;
then reconsider y = x as Element of V by A2;
set z = v + y;
x in W1 by A3;
then z in v + W2 by A1;
then consider u such that
A4: z = v + u and
A5: u in W2;
y = u by A4,RLVECT_1:8;
hence thesis by A5;
end;
let x be object;
assume
A6: x in the carrier of W2;
the carrier of W2 c= the carrier of V by Def2;
then reconsider y = x as Element of V by A6;
set z = v + y;
x in W2 by A6;
then z in v + W1 by A1;
then consider u such that
A7: z = v + u and
A8: u in W1;
y = u by A7,RLVECT_1:8;
hence thesis by A8;
end;
hence thesis by Th30;
end;
thus thesis;
end;
theorem Th69:
for W1,W2 being strict Subspace of V holds v + W1 = u + W2 implies W1 = W2
proof
let W1,W2 be strict Subspace of V;
assume
A1: v + W1 = u + W2;
set V2 = the carrier of W2;
set V1 = the carrier of W1;
assume
A2: W1 <> W2;
A3: now
set x = the Element of V1 \ V2;
assume V1 \ V2 <> {};
then x in V1 by XBOOLE_0:def 5;
then
A4: x in W1;
then x in V by Th9;
then reconsider x as Element of V;
set z = v + x;
z in u + W2 by A1,A4;
then consider u1 such that
A5: z = u + u1 and
A6: u1 in W2;
x = 0.V + x
.= v - v + x by RLVECT_1:15
.= - v + (u + u1) by A5,RLVECT_1:def 3;
then
A7: (v + (- v + (u + u1))) + W1 = v + W1 by A4,Th52;
v + (- v + (u + u1)) = (v - v) + (u + u1) by RLVECT_1:def 3
.= 0.V + (u + u1) by RLVECT_1:15
.= u + u1;
then (u + u1) + W2 = (u + u1) + W1 by A1,A6,A7,Th52;
hence thesis by A2,Th68;
end;
A8: now
set x = the Element of V2 \ V1;
assume V2 \ V1 <> {};
then x in V2 by XBOOLE_0:def 5;
then
A9: x in W2;
then x in V by Th9;
then reconsider x as Element of V;
set z = u + x;
z in v + W1 by A1,A9;
then consider u1 such that
A10: z = v + u1 and
A11: u1 in W1;
x = 0.V + x
.= u - u + x by RLVECT_1:15
.= - u + (v + u1) by A10,RLVECT_1:def 3;
then
A12: (u + (- u + (v + u1))) + W2 = u + W2 by A9,Th52;
u + (- u + (v + u1)) = (u - u) + (v + u1) by RLVECT_1:def 3
.= 0.V + (v + u1) by RLVECT_1:15
.= v + u1;
then (v + u1) + W1 = (v + u1) + W2 by A1,A11,A12,Th52;
hence thesis by A2,Th68;
end;
V1 <> V2 by A2,Th30;
then not V1 c= V2 or not V2 c= V1;
hence thesis by A3,A8,XBOOLE_1:37;
end;
::
:: Theorems concerning cosets of subspace
:: regarded as subsets of the carrier.
::
theorem
C is linearly-closed iff C = the carrier of W
proof
thus C is linearly-closed implies C = the carrier of W
proof
assume
A1: C is linearly-closed;
consider v such that
A2: C = v + W by Def6;
C <> {} by A2,Th43;
then 0.V in v + W by A1,A2,Th1;
hence thesis by A2,Th47;
end;
thus thesis by Lm1;
end;
theorem
for W1,W2 being strict Subspace of V, C1 being Coset of W1, C2 being
Coset of W2 holds C1 = C2 implies W1 = W2
proof
let W1,W2 be strict Subspace of V, C1 be Coset of W1, C2 be Coset of W2;
( ex v1 st C1 = v1 + W1)& ex v2 st C2 = v2 + W2 by Def6;
hence thesis by Th69;
end;
theorem
{v} is Coset of (0).V
proof
v + (0).V = {v} by Th45;
hence thesis by Def6;
end;
theorem
V1 is Coset of (0).V implies ex v st V1 = {v}
proof
assume V1 is Coset of (0).V;
then consider v such that
A1: V1 = v + (0).V by Def6;
take v;
thus thesis by A1,Th45;
end;
theorem
the carrier of W is Coset of W
proof
the carrier of W = 0.V + W by Lm2;
hence thesis by Def6;
end;
theorem
the carrier of V is Coset of (Omega).V
proof
set v = the VECTOR of V;
the carrier of V is Subset of V iff the carrier of V c= the carrier of V;
then reconsider A = the carrier of V as Subset of V;
A = v + (Omega).V by Th46;
hence thesis by Def6;
end;
theorem
V1 is Coset of (Omega).V implies V1 = the carrier of V
proof
assume V1 is Coset of (Omega).V;
then ex v st V1 = v + (Omega).V by Def6;
hence thesis by Th46;
end;
theorem
0.V in C iff C = the carrier of W
proof
ex v st C = v + W by Def6;
hence thesis by Th47;
end;
theorem Th78:
u in C iff C = u + W
proof
thus u in C implies C = u + W
proof
assume
A1: u in C;
ex v st C = v + W by Def6;
hence thesis by A1,Th54;
end;
thus thesis by Th43;
end;
theorem
u in C & v in C implies ex v1 st v1 in W & u + v1 = v
proof
assume u in C & v in C;
then C = u + W & C = v + W by Th78;
hence thesis by Th66;
end;
theorem
u in C & v in C implies ex v1 st v1 in W & u - v1 = v
proof
assume u in C & v in C;
then C = u + W & C = v + W by Th78;
hence thesis by Th67;
end;
theorem
(ex C st v1 in C & v2 in C) iff v1 - v2 in W
proof
thus (ex C st v1 in C & v2 in C) implies v1 - v2 in W
proof
given C such that
A1: v1 in C & v2 in C;
ex v st C = v + W by Def6;
hence thesis by A1,Th65;
end;
assume v1 - v2 in W;
then consider v such that
A2: v1 in v + W & v2 in v + W by Th65;
reconsider C = v + W as Coset of W by Def6;
take C;
thus thesis by A2;
end;
theorem
u in B & u in C implies B = C
proof
assume
A1: u in B & u in C;
( ex v1 st B = v1 + W)& ex v2 st C = v2 + W by Def6;
hence thesis by A1,Th56;
end;