:: RLSUB_1 semantic presentation
begin
definition
let V be RealLinearSpace;
let V1 be Subset of V;
attrV1 is linearly-closed means :Def1: :: RLSUB_1:def 1
( ( for v, u being VECTOR of V st v in V1 & u in V1 holds
v + u in V1 ) & ( for a being Real
for v being VECTOR of V st v in V1 holds
a * v in V1 ) );
end;
:: deftheorem Def1 defines linearly-closed RLSUB_1:def_1_:_
for V being RealLinearSpace
for V1 being Subset of V holds
( V1 is linearly-closed iff ( ( for v, u being VECTOR of V st v in V1 & u in V1 holds
v + u in V1 ) & ( for a being Real
for v being VECTOR of V st v in V1 holds
a * v in V1 ) ) );
theorem Th1: :: RLSUB_1:1
for V being RealLinearSpace
for V1 being Subset of V st V1 <> {} & V1 is linearly-closed holds
0. V in V1
proof
let V be RealLinearSpace; ::_thesis: for V1 being Subset of V st V1 <> {} & V1 is linearly-closed holds
0. V in V1
let V1 be Subset of V; ::_thesis: ( V1 <> {} & V1 is linearly-closed implies 0. V in V1 )
assume that
A1: V1 <> {} and
A2: V1 is linearly-closed ; ::_thesis: 0. V in V1
set x = the Element of V1;
reconsider x = the Element of V1 as Element of V by A1, TARSKI:def_3;
0 * x in V1 by A1, A2, Def1;
hence 0. V in V1 by RLVECT_1:10; ::_thesis: verum
end;
theorem Th2: :: RLSUB_1:2
for V being RealLinearSpace
for V1 being Subset of V st V1 is linearly-closed holds
for v being VECTOR of V st v in V1 holds
- v in V1
proof
let V be RealLinearSpace; ::_thesis: for V1 being Subset of V st V1 is linearly-closed holds
for v being VECTOR of V st v in V1 holds
- v in V1
let V1 be Subset of V; ::_thesis: ( V1 is linearly-closed implies for v being VECTOR of V st v in V1 holds
- v in V1 )
assume A1: V1 is linearly-closed ; ::_thesis: for v being VECTOR of V st v in V1 holds
- v in V1
let v be VECTOR of V; ::_thesis: ( v in V1 implies - v in V1 )
assume v in V1 ; ::_thesis: - v in V1
then (- 1) * v in V1 by A1, Def1;
hence - v in V1 by RLVECT_1:16; ::_thesis: verum
end;
theorem :: RLSUB_1:3
for V being RealLinearSpace
for V1 being Subset of V st V1 is linearly-closed holds
for v, u being VECTOR of V st v in V1 & u in V1 holds
v - u in V1
proof
let V be RealLinearSpace; ::_thesis: for V1 being Subset of V st V1 is linearly-closed holds
for v, u being VECTOR of V st v in V1 & u in V1 holds
v - u in V1
let V1 be Subset of V; ::_thesis: ( V1 is linearly-closed implies for v, u being VECTOR of V st v in V1 & u in V1 holds
v - u in V1 )
assume A1: V1 is linearly-closed ; ::_thesis: for v, u being VECTOR of V st v in V1 & u in V1 holds
v - u in V1
let v, u be VECTOR of V; ::_thesis: ( v in V1 & u in V1 implies v - u in V1 )
assume that
A2: v in V1 and
A3: u in V1 ; ::_thesis: v - u in V1
- u in V1 by A1, A3, Th2;
hence v - u in V1 by A1, A2, Def1; ::_thesis: verum
end;
theorem Th4: :: RLSUB_1:4
for V being RealLinearSpace holds {(0. V)} is linearly-closed
proof
let V be RealLinearSpace; ::_thesis: {(0. V)} is linearly-closed
thus for v, u being VECTOR of V st v in {(0. V)} & u in {(0. V)} holds
v + u in {(0. V)} :: according to RLSUB_1:def_1 ::_thesis: for a being Real
for v being VECTOR of V st v in {(0. V)} holds
a * v in {(0. V)}
proof
let v, u be VECTOR of V; ::_thesis: ( v in {(0. V)} & u in {(0. V)} implies v + u in {(0. V)} )
assume ( v in {(0. V)} & u in {(0. V)} ) ; ::_thesis: v + u in {(0. V)}
then ( v = 0. V & u = 0. V ) by TARSKI:def_1;
then v + u = 0. V by RLVECT_1:4;
hence v + u in {(0. V)} by TARSKI:def_1; ::_thesis: verum
end;
let a be Real; ::_thesis: for v being VECTOR of V st v in {(0. V)} holds
a * v in {(0. V)}
let v be VECTOR of V; ::_thesis: ( v in {(0. V)} implies a * v in {(0. V)} )
assume A1: v in {(0. V)} ; ::_thesis: a * v in {(0. V)}
then v = 0. V by TARSKI:def_1;
hence a * v in {(0. V)} by A1, RLVECT_1:10; ::_thesis: verum
end;
theorem :: RLSUB_1:5
for V being RealLinearSpace
for V1 being Subset of V st the carrier of V = V1 holds
V1 is linearly-closed
proof
let V be RealLinearSpace; ::_thesis: for V1 being Subset of V st the carrier of V = V1 holds
V1 is linearly-closed
let V1 be Subset of V; ::_thesis: ( the carrier of V = V1 implies V1 is linearly-closed )
assume A1: the carrier of V = V1 ; ::_thesis: V1 is linearly-closed
hence for v, u being VECTOR of V st v in V1 & u in V1 holds
v + u in V1 ; :: according to RLSUB_1:def_1 ::_thesis: for a being Real
for v being VECTOR of V st v in V1 holds
a * v in V1
let a be Real; ::_thesis: for v being VECTOR of V st v in V1 holds
a * v in V1
let v be VECTOR of V; ::_thesis: ( v in V1 implies a * v in V1 )
assume v in V1 ; ::_thesis: a * v in V1
thus a * v in V1 by A1; ::_thesis: verum
end;
theorem :: RLSUB_1:6
for V being RealLinearSpace
for V1, V2, V3 being Subset of V st V1 is linearly-closed & V2 is linearly-closed & V3 = { (v + u) where v, u is VECTOR of V : ( v in V1 & u in V2 ) } holds
V3 is linearly-closed
proof
let V be RealLinearSpace; ::_thesis: for V1, V2, V3 being Subset of V st V1 is linearly-closed & V2 is linearly-closed & V3 = { (v + u) where v, u is VECTOR of V : ( v in V1 & u in V2 ) } holds
V3 is linearly-closed
let V1, V2, V3 be Subset of V; ::_thesis: ( V1 is linearly-closed & V2 is linearly-closed & V3 = { (v + u) where v, u is VECTOR of V : ( v in V1 & u in V2 ) } implies V3 is linearly-closed )
assume that
A1: ( V1 is linearly-closed & V2 is linearly-closed ) and
A2: V3 = { (v + u) where v, u is VECTOR of V : ( v in V1 & u in V2 ) } ; ::_thesis: V3 is linearly-closed
thus for v, u being VECTOR of V st v in V3 & u in V3 holds
v + u in V3 :: according to RLSUB_1:def_1 ::_thesis: for a being Real
for v being VECTOR of V st v in V3 holds
a * v in V3
proof
let v, u be VECTOR of V; ::_thesis: ( v in V3 & u in V3 implies v + u in V3 )
assume that
A3: v in V3 and
A4: u in V3 ; ::_thesis: v + u in V3
consider v1, v2 being VECTOR of V such that
A5: v = v1 + v2 and
A6: ( v1 in V1 & v2 in V2 ) by A2, A3;
consider u1, u2 being VECTOR of V 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, Def1;
hence v + u in V3 by A2, A9; ::_thesis: verum
end;
let a be Real; ::_thesis: for v being VECTOR of V st v in V3 holds
a * v in V3
let v be VECTOR of V; ::_thesis: ( v in V3 implies a * v in V3 )
assume v in V3 ; ::_thesis: a * v in V3
then consider v1, v2 being VECTOR of V 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, Def1;
hence a * v in V3 by A2, A12; ::_thesis: verum
end;
theorem :: RLSUB_1:7
for V being RealLinearSpace
for V1, V2 being Subset of V st V1 is linearly-closed & V2 is linearly-closed holds
V1 /\ V2 is linearly-closed
proof
let V be RealLinearSpace; ::_thesis: for V1, V2 being Subset of V st V1 is linearly-closed & V2 is linearly-closed holds
V1 /\ V2 is linearly-closed
let V1, V2 be Subset of V; ::_thesis: ( V1 is linearly-closed & V2 is linearly-closed implies V1 /\ V2 is linearly-closed )
assume that
A1: V1 is linearly-closed and
A2: V2 is linearly-closed ; ::_thesis: V1 /\ V2 is linearly-closed
thus for v, u being VECTOR of V st v in V1 /\ V2 & u in V1 /\ V2 holds
v + u in V1 /\ V2 :: according to RLSUB_1:def_1 ::_thesis: for a being Real
for v being VECTOR of V st v in V1 /\ V2 holds
a * v in V1 /\ V2
proof
let v, u be VECTOR of V; ::_thesis: ( v in V1 /\ V2 & u in V1 /\ V2 implies v + u in V1 /\ V2 )
assume A3: ( v in V1 /\ V2 & u in V1 /\ V2 ) ; ::_thesis: v + u in V1 /\ V2
then ( v in V2 & u in V2 ) by XBOOLE_0:def_4;
then A4: v + u in V2 by A2, Def1;
( v in V1 & u in V1 ) by A3, XBOOLE_0:def_4;
then v + u in V1 by A1, Def1;
hence v + u in V1 /\ V2 by A4, XBOOLE_0:def_4; ::_thesis: verum
end;
let a be Real; ::_thesis: for v being VECTOR of V st v in V1 /\ V2 holds
a * v in V1 /\ V2
let v be VECTOR of V; ::_thesis: ( v in V1 /\ V2 implies a * v in V1 /\ V2 )
assume A5: v in V1 /\ V2 ; ::_thesis: a * v in V1 /\ V2
then v in V2 by XBOOLE_0:def_4;
then A6: a * v in V2 by A2, Def1;
v in V1 by A5, XBOOLE_0:def_4;
then a * v in V1 by A1, Def1;
hence a * v in V1 /\ V2 by A6, XBOOLE_0:def_4; ::_thesis: verum
end;
definition
let V be RealLinearSpace;
mode Subspace of V -> RealLinearSpace means :Def2: :: RLSUB_1:def 2
( 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
ex b1 being RealLinearSpace st
( the carrier of b1 c= the carrier of V & 0. b1 = 0. V & the addF of b1 = the addF of V || the carrier of b1 & the Mult of b1 = the Mult of V | [:REAL, the carrier of b1:] )
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 ex b1 being RealLinearSpace st
( the carrier of b1 c= the carrier of V & 0. b1 = 0. V & the addF of b1 = the addF of V || the carrier of b1 & the Mult of b1 = the Mult of V | [:REAL, the carrier of b1:] ) ; ::_thesis: verum
end;
end;
:: deftheorem Def2 defines Subspace RLSUB_1:def_2_:_
for V, b2 being RealLinearSpace holds
( b2 is Subspace of V iff ( the carrier of b2 c= the carrier of V & 0. b2 = 0. V & the addF of b2 = the addF of V || the carrier of b2 & the Mult of b2 = the Mult of V | [:REAL, the carrier of b2:] ) );
theorem :: RLSUB_1:8
for V being RealLinearSpace
for x being set
for W1, W2 being Subspace of V st x in W1 & W1 is Subspace of W2 holds
x in W2
proof
let V be RealLinearSpace; ::_thesis: for x being set
for W1, W2 being Subspace of V st x in W1 & W1 is Subspace of W2 holds
x in W2
let x be set ; ::_thesis: for W1, W2 being Subspace of V st x in W1 & W1 is Subspace of W2 holds
x in W2
let W1, W2 be Subspace of V; ::_thesis: ( x in W1 & W1 is Subspace of W2 implies x in W2 )
assume ( x in W1 & W1 is Subspace of W2 ) ; ::_thesis: x in W2
then ( x in the carrier of W1 & the carrier of W1 c= the carrier of W2 ) by Def2, STRUCT_0:def_5;
hence x in W2 by STRUCT_0:def_5; ::_thesis: verum
end;
theorem Th9: :: RLSUB_1:9
for V being RealLinearSpace
for x being set
for W being Subspace of V st x in W holds
x in V
proof
let V be RealLinearSpace; ::_thesis: for x being set
for W being Subspace of V st x in W holds
x in V
let x be set ; ::_thesis: for W being Subspace of V st x in W holds
x in V
let W be Subspace of V; ::_thesis: ( x in W implies x in V )
assume x in W ; ::_thesis: x in V
then A1: x in the carrier of W by STRUCT_0:def_5;
the carrier of W c= the carrier of V by Def2;
hence x in V by A1, STRUCT_0:def_5; ::_thesis: verum
end;
theorem Th10: :: RLSUB_1:10
for V being RealLinearSpace
for W being Subspace of V
for w being VECTOR of W holds w is VECTOR of V
proof
let V be RealLinearSpace; ::_thesis: for W being Subspace of V
for w being VECTOR of W holds w is VECTOR of V
let W be Subspace of V; ::_thesis: for w being VECTOR of W holds w is VECTOR of V
let w be VECTOR of W; ::_thesis: w is VECTOR of V
w in V by Th9, RLVECT_1:1;
hence w is VECTOR of V by STRUCT_0:def_5; ::_thesis: verum
end;
theorem :: RLSUB_1:11
for V being RealLinearSpace
for W being Subspace of V holds 0. W = 0. V by Def2;
theorem :: RLSUB_1:12
for V being RealLinearSpace
for W1, W2 being Subspace of V holds 0. W1 = 0. W2
proof
let V be RealLinearSpace; ::_thesis: for W1, W2 being Subspace of V holds 0. W1 = 0. W2
let W1, W2 be Subspace of V; ::_thesis: 0. W1 = 0. W2
thus 0. W1 = 0. V by Def2
.= 0. W2 by Def2 ; ::_thesis: verum
end;
theorem Th13: :: RLSUB_1:13
for V being RealLinearSpace
for v, u being VECTOR of V
for W being Subspace of V
for w1, w2 being VECTOR of W st w1 = v & w2 = u holds
w1 + w2 = v + u
proof
let V be RealLinearSpace; ::_thesis: for v, u being VECTOR of V
for W being Subspace of V
for w1, w2 being VECTOR of W st w1 = v & w2 = u holds
w1 + w2 = v + u
let v, u be VECTOR of V; ::_thesis: for W being Subspace of V
for w1, w2 being VECTOR of W st w1 = v & w2 = u holds
w1 + w2 = v + u
let W be Subspace of V; ::_thesis: for w1, w2 being VECTOR of W st w1 = v & w2 = u holds
w1 + w2 = v + u
let w1, w2 be VECTOR of W; ::_thesis: ( w1 = v & w2 = u implies w1 + w2 = v + u )
assume A1: ( v = w1 & u = w2 ) ; ::_thesis: w1 + w2 = v + u
w1 + w2 = ( the addF of V || the carrier of W) . [w1,w2] by Def2;
hence w1 + w2 = v + u by A1, FUNCT_1:49; ::_thesis: verum
end;
theorem Th14: :: RLSUB_1:14
for V being RealLinearSpace
for v being VECTOR of V
for a being Real
for W being Subspace of V
for w being VECTOR of W st w = v holds
a * w = a * v
proof
let V be RealLinearSpace; ::_thesis: for v being VECTOR of V
for a being Real
for W being Subspace of V
for w being VECTOR of W st w = v holds
a * w = a * v
let v be VECTOR of V; ::_thesis: for a being Real
for W being Subspace of V
for w being VECTOR of W st w = v holds
a * w = a * v
let a be Real; ::_thesis: for W being Subspace of V
for w being VECTOR of W st w = v holds
a * w = a * v
let W be Subspace of V; ::_thesis: for w being VECTOR of W st w = v holds
a * w = a * v
let w be VECTOR of W; ::_thesis: ( w = v implies a * w = a * v )
assume A1: w = v ; ::_thesis: a * w = a * v
a * w = ( the Mult of V | [:REAL, the carrier of W:]) . [a,w] by Def2;
hence a * w = a * v by A1, FUNCT_1:49; ::_thesis: verum
end;
theorem Th15: :: RLSUB_1:15
for V being RealLinearSpace
for v being VECTOR of V
for W being Subspace of V
for w being VECTOR of W st w = v holds
- v = - w
proof
let V be RealLinearSpace; ::_thesis: for v being VECTOR of V
for W being Subspace of V
for w being VECTOR of W st w = v holds
- v = - w
let v be VECTOR of V; ::_thesis: for W being Subspace of V
for w being VECTOR of W st w = v holds
- v = - w
let W be Subspace of V; ::_thesis: for w being VECTOR of W st w = v holds
- v = - w
let w be VECTOR of W; ::_thesis: ( w = v implies - v = - w )
A1: ( - v = (- 1) * v & - w = (- 1) * w ) by RLVECT_1:16;
assume w = v ; ::_thesis: - v = - w
hence - v = - w by A1, Th14; ::_thesis: verum
end;
theorem Th16: :: RLSUB_1:16
for V being RealLinearSpace
for v, u being VECTOR of V
for W being Subspace of V
for w1, w2 being VECTOR of W st w1 = v & w2 = u holds
w1 - w2 = v - u
proof
let V be RealLinearSpace; ::_thesis: for v, u being VECTOR of V
for W being Subspace of V
for w1, w2 being VECTOR of W st w1 = v & w2 = u holds
w1 - w2 = v - u
let v, u be VECTOR of V; ::_thesis: for W being Subspace of V
for w1, w2 being VECTOR of W st w1 = v & w2 = u holds
w1 - w2 = v - u
let W be Subspace of V; ::_thesis: for w1, w2 being VECTOR of W st w1 = v & w2 = u holds
w1 - w2 = v - u
let w1, w2 be VECTOR of W; ::_thesis: ( w1 = v & w2 = u implies w1 - w2 = v - u )
assume that
A1: w1 = v and
A2: w2 = u ; ::_thesis: w1 - w2 = v - u
- w2 = - u by A2, Th15;
hence w1 - w2 = v - u by A1, Th13; ::_thesis: verum
end;
Lm1: for V being RealLinearSpace
for V1 being Subset of V
for W being Subspace of V st the carrier of W = V1 holds
V1 is linearly-closed
proof
let V be RealLinearSpace; ::_thesis: for V1 being Subset of V
for W being Subspace of V st the carrier of W = V1 holds
V1 is linearly-closed
let V1 be Subset of V; ::_thesis: for W being Subspace of V st the carrier of W = V1 holds
V1 is linearly-closed
let W be Subspace of V; ::_thesis: ( the carrier of W = V1 implies V1 is linearly-closed )
set VW = the carrier of W;
reconsider WW = W as RealLinearSpace ;
assume A1: the carrier of W = V1 ; ::_thesis: V1 is linearly-closed
thus for v, u being VECTOR of V st v in V1 & u in V1 holds
v + u in V1 :: according to RLSUB_1:def_1 ::_thesis: for a being Real
for v being VECTOR of V st v in V1 holds
a * v in V1
proof
let v, u be VECTOR of V; ::_thesis: ( v in V1 & u in V1 implies v + u in V1 )
assume ( v in V1 & u in V1 ) ; ::_thesis: v + u in V1
then reconsider vv = v, uu = u as VECTOR of WW by A1;
reconsider vw = vv + uu as Element of the carrier of W ;
vw in V1 by A1;
hence v + u in V1 by Th13; ::_thesis: verum
end;
let a be Real; ::_thesis: for v being VECTOR of V st v in V1 holds
a * v in V1
let v be VECTOR of V; ::_thesis: ( v in V1 implies a * v in V1 )
assume v in V1 ; ::_thesis: a * v in V1
then reconsider vv = v as VECTOR of WW by A1;
reconsider vw = a * vv as Element of the carrier of W ;
vw in V1 by A1;
hence a * v in V1 by Th14; ::_thesis: verum
end;
theorem Th17: :: RLSUB_1:17
for V being RealLinearSpace
for W being Subspace of V holds 0. V in W
proof
let V be RealLinearSpace; ::_thesis: for W being Subspace of V holds 0. V in W
let W be Subspace of V; ::_thesis: 0. V in W
0. W in W by RLVECT_1:1;
hence 0. V in W by Def2; ::_thesis: verum
end;
theorem :: RLSUB_1:18
for V being RealLinearSpace
for W1, W2 being Subspace of V holds 0. W1 in W2
proof
let V be RealLinearSpace; ::_thesis: for W1, W2 being Subspace of V holds 0. W1 in W2
let W1, W2 be Subspace of V; ::_thesis: 0. W1 in W2
0. W1 = 0. V by Def2;
hence 0. W1 in W2 by Th17; ::_thesis: verum
end;
theorem :: RLSUB_1:19
for V being RealLinearSpace
for W being Subspace of V holds 0. W in V by Th9, RLVECT_1:1;
theorem Th20: :: RLSUB_1:20
for V being RealLinearSpace
for u, v being VECTOR of V
for W being Subspace of V st u in W & v in W holds
u + v in W
proof
let V be RealLinearSpace; ::_thesis: for u, v being VECTOR of V
for W being Subspace of V st u in W & v in W holds
u + v in W
let u, v be VECTOR of V; ::_thesis: for W being Subspace of V st u in W & v in W holds
u + v in W
let W be Subspace of V; ::_thesis: ( u in W & v in W implies u + v in W )
reconsider VW = the carrier of W as Subset of V by Def2;
assume ( u in W & v in W ) ; ::_thesis: u + v in W
then A1: ( u in the carrier of W & v in the carrier of W ) by STRUCT_0:def_5;
VW is linearly-closed by Lm1;
then u + v in the carrier of W by A1, Def1;
hence u + v in W by STRUCT_0:def_5; ::_thesis: verum
end;
theorem Th21: :: RLSUB_1:21
for V being RealLinearSpace
for v being VECTOR of V
for a being Real
for W being Subspace of V st v in W holds
a * v in W
proof
let V be RealLinearSpace; ::_thesis: for v being VECTOR of V
for a being Real
for W being Subspace of V st v in W holds
a * v in W
let v be VECTOR of V; ::_thesis: for a being Real
for W being Subspace of V st v in W holds
a * v in W
let a be Real; ::_thesis: for W being Subspace of V st v in W holds
a * v in W
let W be Subspace of V; ::_thesis: ( v in W implies a * v in W )
reconsider VW = the carrier of W as Subset of V by Def2;
assume v in W ; ::_thesis: a * v in W
then A1: v in the carrier of W by STRUCT_0:def_5;
VW is linearly-closed by Lm1;
then a * v in the carrier of W by A1, Def1;
hence a * v in W by STRUCT_0:def_5; ::_thesis: verum
end;
theorem Th22: :: RLSUB_1:22
for V being RealLinearSpace
for v being VECTOR of V
for W being Subspace of V st v in W holds
- v in W
proof
let V be RealLinearSpace; ::_thesis: for v being VECTOR of V
for W being Subspace of V st v in W holds
- v in W
let v be VECTOR of V; ::_thesis: for W being Subspace of V st v in W holds
- v in W
let W be Subspace of V; ::_thesis: ( v in W implies - v in W )
assume v in W ; ::_thesis: - v in W
then (- 1) * v in W by Th21;
hence - v in W by RLVECT_1:16; ::_thesis: verum
end;
theorem Th23: :: RLSUB_1:23
for V being RealLinearSpace
for u, v being VECTOR of V
for W being Subspace of V st u in W & v in W holds
u - v in W
proof
let V be RealLinearSpace; ::_thesis: for u, v being VECTOR of V
for W being Subspace of V st u in W & v in W holds
u - v in W
let u, v be VECTOR of V; ::_thesis: for W being Subspace of V st u in W & v in W holds
u - v in W
let W be Subspace of V; ::_thesis: ( u in W & v in W implies u - v in W )
assume that
A1: u in W and
A2: v in W ; ::_thesis: u - v in W
- v in W by A2, Th22;
hence u - v in W by A1, Th20; ::_thesis: verum
end;
theorem Th24: :: RLSUB_1:24
for V being RealLinearSpace
for V1 being Subset of V
for D being non empty set
for d1 being Element of D
for A being BinOp of D
for M being Function of [:REAL,D:],D st V1 = D & d1 = 0. V & A = the addF of V || V1 & M = the Mult of V | [:REAL,V1:] holds
RLSStruct(# D,d1,A,M #) is Subspace of V
proof
let V be RealLinearSpace; ::_thesis: for V1 being Subset of V
for D being non empty set
for d1 being Element of D
for A being BinOp of D
for M being Function of [:REAL,D:],D st V1 = D & d1 = 0. V & A = the addF of V || V1 & M = the Mult of V | [:REAL,V1:] holds
RLSStruct(# D,d1,A,M #) is Subspace of V
let V1 be Subset of V; ::_thesis: for D being non empty set
for d1 being Element of D
for A being BinOp of D
for M being Function of [:REAL,D:],D st V1 = D & d1 = 0. V & A = the addF of V || V1 & M = the Mult of V | [:REAL,V1:] holds
RLSStruct(# D,d1,A,M #) is Subspace of V
let D be non empty set ; ::_thesis: for d1 being Element of D
for A being BinOp of D
for M being Function of [:REAL,D:],D st V1 = D & d1 = 0. V & A = the addF of V || V1 & M = the Mult of V | [:REAL,V1:] holds
RLSStruct(# D,d1,A,M #) is Subspace of V
let d1 be Element of D; ::_thesis: for A being BinOp of D
for M being Function of [:REAL,D:],D st V1 = D & d1 = 0. V & A = the addF of V || V1 & M = the Mult of V | [:REAL,V1:] holds
RLSStruct(# D,d1,A,M #) is Subspace of V
let A be BinOp of D; ::_thesis: for M being Function of [:REAL,D:],D st V1 = D & d1 = 0. V & A = the addF of V || V1 & M = the Mult of V | [:REAL,V1:] holds
RLSStruct(# D,d1,A,M #) is Subspace of V
let M be Function of [:REAL,D:],D; ::_thesis: ( 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 )
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:] ; ::_thesis: RLSStruct(# D,d1,A,M #) is Subspace of V
set W = RLSStruct(# D,d1,A,M #);
A5: for a being Real
for x being VECTOR of RLSStruct(# D,d1,A,M #) holds a * x = the Mult of V . (a,x)
proof
let a be Real; ::_thesis: for x being VECTOR of RLSStruct(# D,d1,A,M #) holds a * x = the Mult of V . (a,x)
let x be VECTOR of RLSStruct(# D,d1,A,M #); ::_thesis: a * x = the Mult of V . (a,x)
thus a * x = the Mult of V . [a,x] by A1, A4, FUNCT_1:49
.= the Mult of V . (a,x) ; ::_thesis: verum
end;
A6: for x, y being VECTOR of RLSStruct(# D,d1,A,M #) holds x + y = the addF of V . (x,y)
proof
let x, y be VECTOR of RLSStruct(# D,d1,A,M #); ::_thesis: x + y = the addF of V . (x,y)
thus x + y = the addF of V . [x,y] by A1, A3, FUNCT_1:49
.= the addF of V . (x,y) ; ::_thesis: verum
end;
A7: ( RLSStruct(# D,d1,A,M #) is Abelian & RLSStruct(# D,d1,A,M #) is add-associative & RLSStruct(# D,d1,A,M #) is right_zeroed & RLSStruct(# D,d1,A,M #) is right_complementable & RLSStruct(# D,d1,A,M #) is vector-distributive & RLSStruct(# D,d1,A,M #) is scalar-distributive & RLSStruct(# D,d1,A,M #) is scalar-associative & RLSStruct(# D,d1,A,M #) is scalar-unital )
proof
set MV = the Mult of V;
set AV = the addF of V;
thus RLSStruct(# D,d1,A,M #) is Abelian ::_thesis: ( RLSStruct(# D,d1,A,M #) is add-associative & RLSStruct(# D,d1,A,M #) is right_zeroed & RLSStruct(# D,d1,A,M #) is right_complementable & RLSStruct(# D,d1,A,M #) is vector-distributive & RLSStruct(# D,d1,A,M #) is scalar-distributive & RLSStruct(# D,d1,A,M #) is scalar-associative & RLSStruct(# D,d1,A,M #) is scalar-unital )
proof
let x, y be VECTOR of RLSStruct(# D,d1,A,M #); :: according to RLVECT_1:def_2 ::_thesis: x + y = y + x
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 ; ::_thesis: verum
end;
thus RLSStruct(# D,d1,A,M #) is add-associative ::_thesis: ( RLSStruct(# D,d1,A,M #) is right_zeroed & RLSStruct(# D,d1,A,M #) is right_complementable & RLSStruct(# D,d1,A,M #) is vector-distributive & RLSStruct(# D,d1,A,M #) is scalar-distributive & RLSStruct(# D,d1,A,M #) is scalar-associative & RLSStruct(# D,d1,A,M #) is scalar-unital )
proof
let x, y, z be VECTOR of RLSStruct(# D,d1,A,M #); :: according to RLVECT_1:def_3 ::_thesis: (x + y) + z = x + (y + z)
reconsider x1 = x, y1 = y, z1 = z as VECTOR of V by A1, TARSKI:def_3;
thus (x + y) + z = the addF of V . ((x + y),z1) by A6
.= (x1 + y1) + z1 by A6
.= x1 + (y1 + z1) by RLVECT_1:def_3
.= the addF of V . (x1,(y + z)) by A6
.= x + (y + z) by A6 ; ::_thesis: verum
end;
thus RLSStruct(# D,d1,A,M #) is right_zeroed ::_thesis: ( RLSStruct(# D,d1,A,M #) is right_complementable & RLSStruct(# D,d1,A,M #) is vector-distributive & RLSStruct(# D,d1,A,M #) is scalar-distributive & RLSStruct(# D,d1,A,M #) is scalar-associative & RLSStruct(# D,d1,A,M #) is scalar-unital )
proof
let x be VECTOR of RLSStruct(# D,d1,A,M #); :: according to RLVECT_1:def_4 ::_thesis: x + (0. RLSStruct(# D,d1,A,M #)) = x
reconsider y = x as VECTOR of V by A1, TARSKI:def_3;
thus x + (0. RLSStruct(# D,d1,A,M #)) = y + (0. V) by A2, A6
.= x by RLVECT_1:4 ; ::_thesis: verum
end;
thus RLSStruct(# D,d1,A,M #) is right_complementable ::_thesis: ( RLSStruct(# D,d1,A,M #) is vector-distributive & RLSStruct(# D,d1,A,M #) is scalar-distributive & RLSStruct(# D,d1,A,M #) is scalar-associative & RLSStruct(# D,d1,A,M #) is scalar-unital )
proof
let x be VECTOR of RLSStruct(# D,d1,A,M #); :: according to ALGSTR_0:def_16 ::_thesis: x is right_complementable
reconsider x1 = x as VECTOR of V by A1, TARSKI:def_3;
consider v being VECTOR of 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
.= (- 1) * x by A5 ;
then reconsider y = v as VECTOR of RLSStruct(# D,d1,A,M #) ;
take y ; :: according to ALGSTR_0:def_11 ::_thesis: x + y = 0. RLSStruct(# D,d1,A,M #)
thus x + y = 0. RLSStruct(# D,d1,A,M #) by A2, A6, A8; ::_thesis: verum
end;
thus for a being real number
for x, y being VECTOR of RLSStruct(# D,d1,A,M #) holds a * (x + y) = (a * x) + (a * y) :: according to RLVECT_1:def_5 ::_thesis: ( RLSStruct(# D,d1,A,M #) is scalar-distributive & RLSStruct(# D,d1,A,M #) is scalar-associative & RLSStruct(# D,d1,A,M #) is scalar-unital )
proof
let a be real number ; ::_thesis: for x, y being VECTOR of RLSStruct(# D,d1,A,M #) holds a * (x + y) = (a * x) + (a * y)
let x, y be VECTOR of RLSStruct(# D,d1,A,M #); ::_thesis: a * (x + y) = (a * x) + (a * y)
reconsider x1 = x, y1 = y as VECTOR of V by A1, TARSKI:def_3;
reconsider a = a as Real by XREAL_0:def_1;
a * (x + y) = the Mult of V . (a,(x + y)) by A5
.= a * (x1 + y1) by A6
.= (a * x1) + (a * y1) by RLVECT_1:def_5
.= the addF of V . (( the Mult of V . (a,x1)),(a * y)) by A5
.= the addF of V . ((a * x),(a * y)) by A5
.= (a * x) + (a * y) by A6 ;
hence a * (x + y) = (a * x) + (a * y) ; ::_thesis: verum
end;
thus for a, b being real number
for x being VECTOR of RLSStruct(# D,d1,A,M #) holds (a + b) * x = (a * x) + (b * x) :: according to RLVECT_1:def_6 ::_thesis: ( RLSStruct(# D,d1,A,M #) is scalar-associative & RLSStruct(# D,d1,A,M #) is scalar-unital )
proof
let a, b be real number ; ::_thesis: for x being VECTOR of RLSStruct(# D,d1,A,M #) holds (a + b) * x = (a * x) + (b * x)
let x be VECTOR of RLSStruct(# D,d1,A,M #); ::_thesis: (a + b) * x = (a * x) + (b * x)
reconsider y = x as VECTOR of V by A1, TARSKI:def_3;
reconsider a = a, b = b as Real by XREAL_0:def_1;
(a + b) * x = (a + b) * y by A5
.= (a * y) + (b * y) by RLVECT_1:def_6
.= the addF of V . (( the Mult of V . (a,y)),(b * x)) by A5
.= the addF of V . ((a * x),(b * x)) by A5
.= (a * x) + (b * x) by A6 ;
hence (a + b) * x = (a * x) + (b * x) ; ::_thesis: verum
end;
thus for a, b being real number
for x being VECTOR of RLSStruct(# D,d1,A,M #) holds (a * b) * x = a * (b * x) :: according to RLVECT_1:def_7 ::_thesis: RLSStruct(# D,d1,A,M #) is scalar-unital
proof
let a, b be real number ; ::_thesis: for x being VECTOR of RLSStruct(# D,d1,A,M #) holds (a * b) * x = a * (b * x)
let x be VECTOR of RLSStruct(# D,d1,A,M #); ::_thesis: (a * b) * x = a * (b * x)
reconsider y = x as VECTOR of V by A1, TARSKI:def_3;
reconsider a = a, b = b as Real by XREAL_0:def_1;
(a * b) * x = (a * b) * y by A5
.= a * (b * y) by RLVECT_1:def_7
.= the Mult of V . (a,(b * x)) by A5
.= a * (b * x) by A5 ;
hence (a * b) * x = a * (b * x) ; ::_thesis: verum
end;
let x be VECTOR of RLSStruct(# D,d1,A,M #); :: according to RLVECT_1:def_8 ::_thesis: 1 * x = x
reconsider y = x as VECTOR of V by A1, TARSKI:def_3;
thus 1 * x = 1 * y by A5
.= x by RLVECT_1:def_8 ; ::_thesis: verum
end;
0. RLSStruct(# D,d1,A,M #) = 0. V by A2;
hence RLSStruct(# D,d1,A,M #) is Subspace of V by A1, A3, A4, A7, Def2; ::_thesis: verum
end;
theorem Th25: :: RLSUB_1:25
for V being RealLinearSpace holds V is Subspace of V
proof
let V be RealLinearSpace; ::_thesis: V is Subspace of V
thus ( the carrier of V c= the carrier of V & 0. V = 0. V ) ; :: according to RLSUB_1:def_2 ::_thesis: ( 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:] )
thus ( 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; ::_thesis: verum
end;
theorem Th26: :: RLSUB_1:26
for V, X being strict RealLinearSpace st V is Subspace of X & X is Subspace of V holds
V = X
proof
let V, X be strict RealLinearSpace; ::_thesis: ( V is Subspace of X & X is Subspace of V implies V = X )
assume that
A1: V is Subspace of X and
A2: X is Subspace of V ; ::_thesis: V = X
set VX = the carrier of X;
set VV = the carrier of V;
( the carrier of V c= the carrier of X & the carrier of X c= the carrier of V ) by A1, A2, Def2;
then A3: the carrier of V = the carrier of X by XBOOLE_0:def_10;
set AX = the addF of X;
set AV = the addF of V;
( the addF of V = the addF of X || the carrier of V & the addF of X = the addF of V || the carrier of X ) by A1, A2, Def2;
then A4: the addF of V = the addF of X by A3, RELAT_1:72;
set MX = the Mult of X;
set MV = the Mult of V;
A5: the Mult of X = the Mult of V | [:REAL, the carrier of X:] by A2, Def2;
( 0. V = 0. X & the Mult of V = the Mult of X | [:REAL, the carrier of V:] ) by A1, Def2;
hence V = X by A3, A4, A5, RELAT_1:72; ::_thesis: verum
end;
theorem Th27: :: RLSUB_1:27
for V, X, Y being RealLinearSpace st V is Subspace of X & X is Subspace of Y holds
V is Subspace of Y
proof
let V, X, Y be RealLinearSpace; ::_thesis: ( V is Subspace of X & X is Subspace of Y implies V is Subspace of Y )
assume that
A1: V is Subspace of X and
A2: X is Subspace of Y ; ::_thesis: V 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 by XBOOLE_1:1; :: according to RLSUB_1:def_2 ::_thesis: ( 0. V = 0. Y & the addF of V = the addF of Y || the carrier of V & the Mult of V = the Mult of Y | [:REAL, the carrier of V:] )
0. V = 0. X by A1, Def2;
hence 0. V = 0. Y by A2, Def2; ::_thesis: ( the addF of V = the addF of Y || the carrier of V & the Mult of V = the Mult of Y | [:REAL, the carrier of V:] )
thus the addF of V = the addF of Y || the carrier of V ::_thesis: the Mult of V = the Mult of Y | [:REAL, 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;
the carrier of V c= the carrier of X by A1, Def2;
then A3: [: the carrier of V, the carrier of V:] c= [: the carrier of X, the carrier of X:] by ZFMISC_1:96;
the addF of V = the addF of X || the carrier of V by A1, Def2;
then the addF of V = ( the addF of Y || the carrier of X) || the carrier of V by A2, Def2;
hence the addF of V = the addF of Y || the carrier of V by A3, FUNCT_1:51; ::_thesis: verum
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;
the carrier of V c= the carrier of X by A1, Def2;
then A4: [:REAL, the carrier of V:] c= [:REAL, the carrier of X:] by ZFMISC_1:95;
the Mult of V = the Mult of X | [:REAL, the carrier of V:] by A1, Def2;
then the Mult of V = ( the Mult of Y | [:REAL, the carrier of X:]) | [:REAL, the carrier of V:] by A2, Def2;
hence the Mult of V = the Mult of Y | [:REAL, the carrier of V:] by A4, FUNCT_1:51; ::_thesis: verum
end;
theorem Th28: :: RLSUB_1:28
for V being RealLinearSpace
for W1, W2 being Subspace of V st the carrier of W1 c= the carrier of W2 holds
W1 is Subspace of W2
proof
let V be RealLinearSpace; ::_thesis: for W1, W2 being Subspace of V st the carrier of W1 c= the carrier of W2 holds
W1 is Subspace of W2
let W1, W2 be Subspace of V; ::_thesis: ( the carrier of W1 c= the carrier of W2 implies W1 is Subspace of W2 )
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 ; ::_thesis: W1 is Subspace of W2
then A2: [: the carrier of W1, the carrier of W1:] c= [: the carrier of W2, the carrier of W2:] 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; :: according to RLSUB_1:def_2 ::_thesis: ( the addF of W1 = the addF of W2 || the carrier of W1 & the Mult of W1 = the Mult of W2 | [:REAL, the carrier of W1:] )
( the addF of W1 = the addF of V || the carrier of W1 & the addF of W2 = the addF of V || the carrier of W2 ) by Def2;
hence the addF of W1 = the addF of W2 || the carrier of W1 by A2, FUNCT_1:51; ::_thesis: the Mult of W1 = the Mult of W2 | [:REAL, the carrier of W1:]
A3: [:REAL, the carrier of W1:] c= [:REAL, the carrier of W2:] by A1, ZFMISC_1:95;
( the Mult of W1 = the Mult of V | [:REAL, the carrier of W1:] & the Mult of W2 = the Mult of V | [:REAL, the carrier of W2:] ) by Def2;
hence the Mult of W1 = the Mult of W2 | [:REAL, the carrier of W1:] by A3, FUNCT_1:51; ::_thesis: verum
end;
theorem :: RLSUB_1:29
for V being RealLinearSpace
for W1, W2 being Subspace of V st ( for v being VECTOR of V st v in W1 holds
v in W2 ) holds
W1 is Subspace of W2
proof
let V be RealLinearSpace; ::_thesis: for W1, W2 being Subspace of V st ( for v being VECTOR of V st v in W1 holds
v in W2 ) holds
W1 is Subspace of W2
let W1, W2 be Subspace of V; ::_thesis: ( ( for v being VECTOR of V st v in W1 holds
v in W2 ) implies W1 is Subspace of W2 )
assume A1: for v being VECTOR of V st v in W1 holds
v in W2 ; ::_thesis: W1 is Subspace of W2
the carrier of W1 c= the carrier of W2
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in the carrier of W1 or x in the carrier of W2 )
assume A2: x in the carrier of W1 ; ::_thesis: x in the carrier of W2
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, STRUCT_0:def_5;
then v in W2 by A1;
hence x in the carrier of W2 by STRUCT_0:def_5; ::_thesis: verum
end;
hence W1 is Subspace of W2 by Th28; ::_thesis: verum
end;
registration
let V be RealLinearSpace;
cluster non empty left_complementable right_complementable complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() for Subspace of V;
existence
ex b1 being Subspace of V st b1 is strict
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 ex b1 being Subspace of V st b1 is strict ; ::_thesis: verum
end;
end;
theorem Th30: :: RLSUB_1:30
for V being RealLinearSpace
for W1, W2 being strict Subspace of V st the carrier of W1 = the carrier of W2 holds
W1 = W2
proof
let V be RealLinearSpace; ::_thesis: for W1, W2 being strict Subspace of V st the carrier of W1 = the carrier of W2 holds
W1 = W2
let W1, W2 be strict Subspace of V; ::_thesis: ( the carrier of W1 = the carrier of W2 implies W1 = W2 )
assume the carrier of W1 = the carrier of W2 ; ::_thesis: W1 = W2
then ( W1 is Subspace of W2 & W2 is Subspace of W1 ) by Th28;
hence W1 = W2 by Th26; ::_thesis: verum
end;
theorem Th31: :: RLSUB_1:31
for V being RealLinearSpace
for W1, W2 being strict Subspace of V st ( for v being VECTOR of V holds
( v in W1 iff v in W2 ) ) holds
W1 = W2
proof
let V be RealLinearSpace; ::_thesis: for W1, W2 being strict Subspace of V st ( for v being VECTOR of V holds
( v in W1 iff v in W2 ) ) holds
W1 = W2
let W1, W2 be strict Subspace of V; ::_thesis: ( ( for v being VECTOR of V holds
( v in W1 iff v in W2 ) ) implies W1 = W2 )
assume A1: for v being VECTOR of V holds
( v in W1 iff v in W2 ) ; ::_thesis: W1 = W2
for x being set holds
( x in the carrier of W1 iff x in the carrier of W2 )
proof
let x be set ; ::_thesis: ( x in the carrier of W1 iff x in the carrier of W2 )
thus ( x in the carrier of W1 implies x in the carrier of W2 ) ::_thesis: ( x in the carrier of W2 implies x in the carrier of W1 )
proof
assume A2: x in the carrier of W1 ; ::_thesis: x in the carrier of W2
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, STRUCT_0:def_5;
then v in W2 by A1;
hence x in the carrier of W2 by STRUCT_0:def_5; ::_thesis: verum
end;
assume A3: x in the carrier of W2 ; ::_thesis: x in the carrier of W1
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, STRUCT_0:def_5;
then v in W1 by A1;
hence x in the carrier of W1 by STRUCT_0:def_5; ::_thesis: verum
end;
then the carrier of W1 = the carrier of W2 by TARSKI:1;
hence W1 = W2 by Th30; ::_thesis: verum
end;
theorem :: RLSUB_1:32
for V being strict RealLinearSpace
for W being strict Subspace of V st the carrier of W = the carrier of V holds
W = V
proof
let V be strict RealLinearSpace; ::_thesis: for W being strict Subspace of V st the carrier of W = the carrier of V holds
W = V
let W be strict Subspace of V; ::_thesis: ( the carrier of W = the carrier of V implies W = V )
assume A1: the carrier of W = the carrier of V ; ::_thesis: W = V
V is Subspace of V by Th25;
hence W = V by A1, Th30; ::_thesis: verum
end;
theorem :: RLSUB_1:33
for V being strict RealLinearSpace
for W being strict Subspace of V st ( for v being VECTOR of V holds
( v in W iff v in V ) ) holds
W = V
proof
let V be strict RealLinearSpace; ::_thesis: for W being strict Subspace of V st ( for v being VECTOR of V holds
( v in W iff v in V ) ) holds
W = V
let W be strict Subspace of V; ::_thesis: ( ( for v being VECTOR of V holds
( v in W iff v in V ) ) implies W = V )
assume A1: for v being VECTOR of V holds
( v in W iff v in V ) ; ::_thesis: W = V
V is Subspace of V by Th25;
hence W = V by A1, Th31; ::_thesis: verum
end;
theorem :: RLSUB_1:34
for V being RealLinearSpace
for V1 being Subset of V
for W being Subspace of V st the carrier of W = V1 holds
V1 is linearly-closed by Lm1;
theorem Th35: :: RLSUB_1:35
for V being RealLinearSpace
for V1 being Subset of V st V1 <> {} & V1 is linearly-closed holds
ex W being strict Subspace of V st V1 = the carrier of W
proof
let V be RealLinearSpace; ::_thesis: for V1 being Subset of V st V1 <> {} & V1 is linearly-closed holds
ex W being strict Subspace of V st V1 = the carrier of W
let V1 be Subset of V; ::_thesis: ( V1 <> {} & V1 is linearly-closed implies ex W being strict Subspace of V st V1 = the carrier of W )
assume that
A1: V1 <> {} and
A2: V1 is linearly-closed ; ::_thesis: ex W being strict Subspace of V st V1 = the carrier of W
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, the carrier of V:] by FUNCT_2:def_1;
then A3: dom ( the Mult of V | [:REAL,V1:]) = [:REAL, the carrier of V:] /\ [:REAL,V1:] by RELAT_1:61;
[:REAL,V1:] c= [:REAL, the carrier of V:] by ZFMISC_1:95;
then A4: dom ( the Mult of V | [:REAL,V1:]) = [:REAL,D:] by A3, XBOOLE_1:28;
now__::_thesis:_for_y_being_set_holds_
(_(_y_in_D_implies_ex_x_being_set_st_
(_x_in_dom_(_the_Mult_of_V_|_[:REAL,V1:])_&_y_=_(_the_Mult_of_V_|_[:REAL,V1:])_._x_)_)_&_(_ex_x_being_set_st_
(_x_in_dom_(_the_Mult_of_V_|_[:REAL,V1:])_&_y_=_(_the_Mult_of_V_|_[:REAL,V1:])_._x_)_implies_y_in_D_)_)
let y be set ; ::_thesis: ( ( y in D implies ex x being set st
( x in dom ( the Mult of V | [:REAL,V1:]) & y = ( the Mult of V | [:REAL,V1:]) . x ) ) & ( ex x being set st
( x in dom ( the Mult of V | [:REAL,V1:]) & y = ( the Mult of V | [:REAL,V1:]) . x ) implies y in D ) )
thus ( y in D implies ex x being set st
( x in dom ( the Mult of V | [:REAL,V1:]) & y = ( the Mult of V | [:REAL,V1:]) . x ) ) ::_thesis: ( ex x being set st
( x in dom ( the Mult of V | [:REAL,V1:]) & y = ( the Mult of V | [:REAL,V1:]) . x ) implies y in D )
proof
assume A5: y in D ; ::_thesis: ex x being set st
( x in dom ( the Mult of V | [:REAL,V1:]) & y = ( the Mult of V | [:REAL,V1:]) . x )
then reconsider v1 = y as Element of the carrier of V ;
A6: [1,y] in [:REAL,D:] by A5, ZFMISC_1:87;
then ( the Mult of V | [:REAL,V1:]) . [1,y] = 1 * v1 by FUNCT_1:49
.= y by RLVECT_1:def_8 ;
hence ex x being set st
( x in dom ( the Mult of V | [:REAL,V1:]) & y = ( the Mult of V | [:REAL,V1:]) . x ) by A4, A6; ::_thesis: verum
end;
given x being set such that A7: x in dom ( the Mult of V | [:REAL,V1:]) and
A8: y = ( the Mult of V | [:REAL,V1:]) . x ; ::_thesis: y in D
consider x1, x2 being set 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 the carrier of V 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, Def1; ::_thesis: verum
end;
then D = rng ( the Mult of V | [:REAL,V1:]) by FUNCT_1:def_3;
then reconsider M = the Mult of V | [:REAL,V1:] 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 = [: the carrier of V, the carrier of V:] by FUNCT_2:def_1;
then dom ( the addF of V || V1) = [: the carrier of V, the carrier of V:] /\ [:V1,V1:] by RELAT_1:61;
then A12: dom ( the addF of V || V1) = [:D,D:] by XBOOLE_1:28;
now__::_thesis:_for_y_being_set_holds_
(_(_y_in_D_implies_ex_x_being_set_st_
(_x_in_dom_(_the_addF_of_V_||_V1)_&_y_=_(_the_addF_of_V_||_V1)_._x_)_)_&_(_ex_x_being_set_st_
(_x_in_dom_(_the_addF_of_V_||_V1)_&_y_=_(_the_addF_of_V_||_V1)_._x_)_implies_y_in_D_)_)
let y be set ; ::_thesis: ( ( y in D implies ex x being set st
( x in dom ( the addF of V || V1) & y = ( the addF of V || V1) . x ) ) & ( ex x being set st
( x in dom ( the addF of V || V1) & y = ( the addF of V || V1) . x ) implies y in D ) )
thus ( y in D implies ex x being set st
( x in dom ( the addF of V || V1) & y = ( the addF of V || V1) . x ) ) ::_thesis: ( ex x being set st
( x in dom ( the addF of V || V1) & y = ( the addF of V || V1) . x ) implies y in D )
proof
assume A13: y in D ; ::_thesis: ex x being set st
( x in dom ( the addF of V || V1) & y = ( the addF of V || V1) . x )
then reconsider v1 = y, v0 = d1 as Element of the carrier of V ;
A14: [d1,y] in [:D,D:] by A13, ZFMISC_1:87;
then ( the addF of V || V1) . [d1,y] = v0 + v1 by FUNCT_1:49
.= y by RLVECT_1:4 ;
hence ex x being set st
( x in dom ( the addF of V || V1) & y = ( the addF of V || V1) . x ) by A12, A14; ::_thesis: verum
end;
given x being set such that A15: x in dom ( the addF of V || V1) and
A16: y = ( the addF of V || V1) . x ; ::_thesis: y in D
consider x1, x2 being set 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 the carrier of V 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, Def1; ::_thesis: verum
end;
then D = rng ( the addF of V || V1) by FUNCT_1:def_3;
then reconsider A = the addF of V || V1 as Function of [:D,D:],D by A12, FUNCT_2:def_1, RELSET_1:4;
set W = RLSStruct(# D,d1,A,M #);
RLSStruct(# D,d1,A,M #) is Subspace of V by Th24;
hence ex W being strict Subspace of V st V1 = the carrier of W ; ::_thesis: verum
end;
definition
let V be RealLinearSpace;
func (0). V -> strict Subspace of V means :Def3: :: RLSUB_1:def 3
the carrier of it = {(0. V)};
correctness
existence
ex b1 being strict Subspace of V st the carrier of b1 = {(0. V)};
uniqueness
for b1, b2 being strict Subspace of V st the carrier of b1 = {(0. V)} & the carrier of b2 = {(0. V)} holds
b1 = b2;
by Th4, Th30, Th35;
end;
:: deftheorem Def3 defines (0). RLSUB_1:def_3_:_
for V being RealLinearSpace
for b2 being strict Subspace of V holds
( b2 = (0). V iff the carrier of b2 = {(0. V)} );
definition
let V be RealLinearSpace;
func (Omega). V -> strict Subspace of V equals :: RLSUB_1:def 4
RLSStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #);
coherence
RLSStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #) is strict Subspace of V
proof
set W = RLSStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #);
A1: for u, v, w being VECTOR of RLSStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #) holds (u + v) + w = u + (v + w)
proof
let u, v, w be VECTOR of RLSStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #); ::_thesis: (u + v) + w = u + (v + 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) ; ::_thesis: verum
end;
A2: for v being VECTOR of RLSStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #) holds v + (0. RLSStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #)) = v
proof
let v be VECTOR of RLSStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #); ::_thesis: v + (0. RLSStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #)) = v
reconsider v9 = v as VECTOR of V ;
thus v + (0. RLSStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #)) = v9 + (0. V)
.= v by RLVECT_1:4 ; ::_thesis: verum
end;
A3: RLSStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #) is right_complementable
proof
let v be VECTOR of RLSStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #); :: according to ALGSTR_0:def_16 ::_thesis: v is right_complementable
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 RLSStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #) ;
take w ; :: according to ALGSTR_0:def_11 ::_thesis: v + w = 0. RLSStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #)
thus v + w = 0. RLSStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #) by A4; ::_thesis: verum
end;
A5: for a being real number
for v, w being VECTOR of RLSStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #) holds a * (v + w) = (a * v) + (a * w)
proof
let a be real number ; ::_thesis: for v, w being VECTOR of RLSStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #) holds a * (v + w) = (a * v) + (a * w)
let v, w be VECTOR of RLSStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #); ::_thesis: a * (v + w) = (a * v) + (a * 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) ; ::_thesis: verum
end;
A6: for a, b being real number
for v being VECTOR of RLSStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #) holds (a * b) * v = a * (b * v)
proof
let a, b be real number ; ::_thesis: for v being VECTOR of RLSStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #) holds (a * b) * v = a * (b * v)
let v be VECTOR of RLSStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #); ::_thesis: (a * b) * v = a * (b * v)
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) ; ::_thesis: verum
end;
A7: for a, b being real number
for v being VECTOR of RLSStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #) holds (a + b) * v = (a * v) + (b * v)
proof
let a, b be real number ; ::_thesis: for v being VECTOR of RLSStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #) holds (a + b) * v = (a * v) + (b * v)
let v be VECTOR of RLSStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #); ::_thesis: (a + b) * v = (a * v) + (b * v)
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) ; ::_thesis: verum
end;
A8: for a being Real
for v, w being VECTOR of RLSStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #)
for v9, w9 being VECTOR of V st v = v9 & w = w9 holds
( v + w = v9 + w9 & a * v = a * v9 ) ;
A9: for v, w being VECTOR of RLSStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #) holds v + w = w + v
proof
let v, w be VECTOR of RLSStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #); ::_thesis: v + w = w + v
reconsider v9 = v, w9 = w as VECTOR of V ;
thus v + w = w9 + v9 by A8
.= w + v ; ::_thesis: verum
end;
for v being VECTOR of RLSStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #) holds 1 * v = v
proof
let v be VECTOR of RLSStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #); ::_thesis: 1 * v = v
reconsider v9 = v as VECTOR of V ;
thus 1 * v = 1 * v9
.= v by RLVECT_1:def_8 ; ::_thesis: verum
end;
then reconsider W = RLSStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #) as RealLinearSpace by A9, A1, A2, A3, A5, A7, A6, RLVECT_1:def_2, RLVECT_1:def_3, RLVECT_1:def_4, RLVECT_1:def_5, RLVECT_1:def_6, RLVECT_1:def_7, RLVECT_1: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 RLSStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #) is strict Subspace of V by A10, Def2; ::_thesis: verum
end;
end;
:: deftheorem defines (Omega). RLSUB_1:def_4_:_
for V being RealLinearSpace holds (Omega). V = RLSStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #);
theorem Th36: :: RLSUB_1:36
for V being RealLinearSpace
for W being Subspace of V holds (0). W = (0). V
proof
let V be RealLinearSpace; ::_thesis: for W being Subspace of V holds (0). W = (0). V
let W be Subspace of V; ::_thesis: (0). W = (0). V
( 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 (0). W = (0). V by A1, Th30; ::_thesis: verum
end;
theorem Th37: :: RLSUB_1:37
for V being RealLinearSpace
for W1, W2 being Subspace of V holds (0). W1 = (0). W2
proof
let V be RealLinearSpace; ::_thesis: for W1, W2 being Subspace of V holds (0). W1 = (0). W2
let W1, W2 be Subspace of V; ::_thesis: (0). W1 = (0). W2
(0). W1 = (0). V by Th36;
hence (0). W1 = (0). W2 by Th36; ::_thesis: verum
end;
theorem :: RLSUB_1:38
for V being RealLinearSpace
for W being Subspace of V holds (0). W is Subspace of V by Th27;
theorem :: RLSUB_1:39
for V being RealLinearSpace
for W being Subspace of V holds (0). V is Subspace of W
proof
let V be RealLinearSpace; ::_thesis: for W being Subspace of V holds (0). V is Subspace of W
let W be Subspace of V; ::_thesis: (0). V is Subspace of W
the carrier of ((0). V) = {(0. V)} by Def3
.= {(0. W)} by Def2 ;
hence (0). V is Subspace of W by Th28; ::_thesis: verum
end;
theorem :: RLSUB_1:40
for V being RealLinearSpace
for W1, W2 being Subspace of V holds (0). W1 is Subspace of W2
proof
let V be RealLinearSpace; ::_thesis: for W1, W2 being Subspace of V holds (0). W1 is Subspace of W2
let W1, W2 be Subspace of V; ::_thesis: (0). W1 is Subspace of W2
(0). W1 = (0). W2 by Th37;
hence (0). W1 is Subspace of W2 ; ::_thesis: verum
end;
theorem :: RLSUB_1:41
for V being strict RealLinearSpace holds V is Subspace of (Omega). V ;
definition
let V be RealLinearSpace;
let v be VECTOR of V;
let W be Subspace of V;
funcv + W -> Subset of V equals :: RLSUB_1:def 5
{ (v + u) where u is VECTOR of V : u in W } ;
coherence
{ (v + u) where u is VECTOR of V : u in W } is Subset of V
proof
set Y = { (v + u) where u is VECTOR of V : u in W } ;
defpred S1[ set ] means ex u being VECTOR of V st
( $1 = v + u & u in W );
consider X being set such that
A1: for x being set holds
( x in X iff ( x in the carrier of V & S1[x] ) ) from XBOOLE_0:sch_1();
X c= the carrier of V
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in X or x in the carrier of V )
assume x in X ; ::_thesis: x in the carrier of V
hence x in the carrier of V by A1; ::_thesis: verum
end;
then reconsider X = X as Subset of V ;
A2: { (v + u) where u is VECTOR of V : u in W } c= X
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { (v + u) where u is VECTOR of V : u in W } or x in X )
assume x in { (v + u) where u is VECTOR of V : u in W } ; ::_thesis: x in X
then ex u being VECTOR of V st
( x = v + u & u in W ) ;
hence x in X by A1; ::_thesis: verum
end;
X c= { (v + u) where u is VECTOR of V : u in W }
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in X or x in { (v + u) where u is VECTOR of V : u in W } )
assume x in X ; ::_thesis: x in { (v + u) where u is VECTOR of V : u in W }
then ex u being VECTOR of V st
( x = v + u & u in W ) by A1;
hence x in { (v + u) where u is VECTOR of V : u in W } ; ::_thesis: verum
end;
hence { (v + u) where u is VECTOR of V : u in W } is Subset of V by A2, XBOOLE_0:def_10; ::_thesis: verum
end;
end;
:: deftheorem defines + RLSUB_1:def_5_:_
for V being RealLinearSpace
for v being VECTOR of V
for W being Subspace of V holds v + W = { (v + u) where u is VECTOR of V : u in W } ;
Lm2: for V being RealLinearSpace
for W being Subspace of V holds (0. V) + W = the carrier of W
proof
let V be RealLinearSpace; ::_thesis: for W being Subspace of V holds (0. V) + W = the carrier of W
let W be Subspace of V; ::_thesis: (0. V) + W = the carrier of W
set A = { ((0. V) + u) where u is VECTOR of V : u in W } ;
A1: the carrier of W c= { ((0. V) + u) where u is VECTOR of V : u in W }
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in the carrier of W or x in { ((0. V) + u) where u is VECTOR of V : u in W } )
assume x in the carrier of W ; ::_thesis: x in { ((0. V) + u) where u is VECTOR of V : u in W }
then A2: x in W by STRUCT_0:def_5;
then x in V by Th9;
then reconsider y = x as Element of V by STRUCT_0:def_5;
(0. V) + y = x by RLVECT_1:4;
hence x in { ((0. V) + u) where u is VECTOR of V : u in W } by A2; ::_thesis: verum
end;
{ ((0. V) + u) where u is VECTOR of V : u in W } c= the carrier of W
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { ((0. V) + u) where u is VECTOR of V : u in W } or x in the carrier of W )
assume x in { ((0. V) + u) where u is VECTOR of V : u in W } ; ::_thesis: x in the carrier of W
then consider u being VECTOR of V such that
A3: x = (0. V) + u and
A4: u in W ;
x = u by A3, RLVECT_1:4;
hence x in the carrier of W by A4, STRUCT_0:def_5; ::_thesis: verum
end;
hence (0. V) + W = the carrier of W by A1, XBOOLE_0:def_10; ::_thesis: verum
end;
definition
let V be RealLinearSpace;
let W be Subspace of V;
mode Coset of W -> Subset of V means :Def6: :: RLSUB_1:def 6
ex v being VECTOR of V st it = v + W;
existence
ex b1 being Subset of V ex v being VECTOR of V st b1 = v + W
proof
reconsider VW = the carrier of W as Subset of V by Def2;
take VW ; ::_thesis: ex v being VECTOR of V st VW = v + W
take 0. V ; ::_thesis: VW = (0. V) + W
thus VW = (0. V) + W by Lm2; ::_thesis: verum
end;
end;
:: deftheorem Def6 defines Coset RLSUB_1:def_6_:_
for V being RealLinearSpace
for W being Subspace of V
for b3 being Subset of V holds
( b3 is Coset of W iff ex v being VECTOR of V st b3 = v + W );
theorem Th42: :: RLSUB_1:42
for V being RealLinearSpace
for v being VECTOR of V
for W being Subspace of V holds
( 0. V in v + W iff v in W )
proof
let V be RealLinearSpace; ::_thesis: for v being VECTOR of V
for W being Subspace of V holds
( 0. V in v + W iff v in W )
let v be VECTOR of V; ::_thesis: for W being Subspace of V holds
( 0. V in v + W iff v in W )
let W be Subspace of V; ::_thesis: ( 0. V in v + W iff v in W )
thus ( 0. V in v + W implies v in W ) ::_thesis: ( v in W implies 0. V in v + W )
proof
assume 0. V in v + W ; ::_thesis: v in W
then consider u being VECTOR of V such that
A1: 0. V = v + u and
A2: u in W ;
v = - u by A1, RLVECT_1:def_10;
hence v in W by A2, Th22; ::_thesis: verum
end;
assume v in W ; ::_thesis: 0. V in v + W
then A3: - v in W by Th22;
0. V = v - v by RLVECT_1:15
.= v + (- v) ;
hence 0. V in v + W by A3; ::_thesis: verum
end;
theorem Th43: :: RLSUB_1:43
for V being RealLinearSpace
for v being VECTOR of V
for W being Subspace of V holds v in v + W
proof
let V be RealLinearSpace; ::_thesis: for v being VECTOR of V
for W being Subspace of V holds v in v + W
let v be VECTOR of V; ::_thesis: for W being Subspace of V holds v in v + W
let W be Subspace of V; ::_thesis: v in v + W
( v + (0. V) = v & 0. V in W ) by Th17, RLVECT_1:4;
hence v in v + W ; ::_thesis: verum
end;
theorem :: RLSUB_1:44
for V being RealLinearSpace
for W being Subspace of V holds (0. V) + W = the carrier of W by Lm2;
theorem Th45: :: RLSUB_1:45
for V being RealLinearSpace
for v being VECTOR of V holds v + ((0). V) = {v}
proof
let V be RealLinearSpace; ::_thesis: for v being VECTOR of V holds v + ((0). V) = {v}
let v be VECTOR of V; ::_thesis: v + ((0). V) = {v}
thus v + ((0). V) c= {v} :: according to XBOOLE_0:def_10 ::_thesis: {v} c= v + ((0). V)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in v + ((0). V) or x in {v} )
assume x in v + ((0). V) ; ::_thesis: x in {v}
then consider u being VECTOR of V 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, STRUCT_0:def_5;
then u = 0. V by A3, TARSKI:def_1;
then x = v by A1, RLVECT_1:4;
hence x in {v} by TARSKI:def_1; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {v} or x in v + ((0). V) )
assume x in {v} ; ::_thesis: x in v + ((0). V)
then A4: x = v by TARSKI:def_1;
( 0. V in (0). V & v = v + (0. V) ) by Th17, RLVECT_1:4;
hence x in v + ((0). V) by A4; ::_thesis: verum
end;
Lm3: for V being RealLinearSpace
for v being VECTOR of V
for W being Subspace of V holds
( v in W iff v + W = the carrier of W )
proof
let V be RealLinearSpace; ::_thesis: for v being VECTOR of V
for W being Subspace of V holds
( v in W iff v + W = the carrier of W )
let v be VECTOR of V; ::_thesis: for W being Subspace of V holds
( v in W iff v + W = the carrier of W )
let W be Subspace of V; ::_thesis: ( v in W iff v + W = the carrier of W )
( 0. V in W & v + (0. V) = v ) by Th17, RLVECT_1:4;
then A1: v in { (v + u) where u is VECTOR of V : u in W } ;
thus ( v in W implies v + W = the carrier of W ) ::_thesis: ( v + W = the carrier of W implies v in W )
proof
assume A2: v in W ; ::_thesis: v + W = the carrier of W
thus v + W c= the carrier of W :: according to XBOOLE_0:def_10 ::_thesis: the carrier of W c= v + W
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in v + W or x in the carrier of W )
assume x in v + W ; ::_thesis: x in the carrier of W
then consider u being VECTOR of V such that
A3: x = v + u and
A4: u in W ;
v + u in W by A2, A4, Th20;
hence x in the carrier of W by A3, STRUCT_0:def_5; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in the carrier of W or x in v + W )
assume x in the carrier of W ; ::_thesis: x in v + W
then reconsider y = x, z = v as Element of W by A2, STRUCT_0:def_5;
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 by RLVECT_1:4 ;
y - z in W by STRUCT_0:def_5;
then A6: y1 - z1 in W by Th16;
y - z = y1 - z1 by Th16;
then z1 + (y1 - z1) = x by A5, Th13;
hence x in v + W by A6; ::_thesis: verum
end;
assume A7: v + W = the carrier of W ; ::_thesis: v in W
assume not v in W ; ::_thesis: contradiction
hence contradiction by A7, A1, STRUCT_0:def_5; ::_thesis: verum
end;
theorem Th46: :: RLSUB_1:46
for V being RealLinearSpace
for v being VECTOR of V holds v + ((Omega). V) = the carrier of V
proof
let V be RealLinearSpace; ::_thesis: for v being VECTOR of V holds v + ((Omega). V) = the carrier of V
let v be VECTOR of V; ::_thesis: v + ((Omega). V) = the carrier of V
v in (Omega). V by STRUCT_0:def_5;
hence v + ((Omega). V) = the carrier of V by Lm3; ::_thesis: verum
end;
theorem Th47: :: RLSUB_1:47
for V being RealLinearSpace
for v being VECTOR of V
for W being Subspace of V holds
( 0. V in v + W iff v + W = the carrier of W )
proof
let V be RealLinearSpace; ::_thesis: for v being VECTOR of V
for W being Subspace of V holds
( 0. V in v + W iff v + W = the carrier of W )
let v be VECTOR of V; ::_thesis: for W being Subspace of V holds
( 0. V in v + W iff v + W = the carrier of W )
let W be Subspace of V; ::_thesis: ( 0. V in v + W iff v + W = the carrier of W )
( 0. V in v + W iff v in W ) by Th42;
hence ( 0. V in v + W iff v + W = the carrier of W ) by Lm3; ::_thesis: verum
end;
theorem :: RLSUB_1:48
for V being RealLinearSpace
for v being VECTOR of V
for W being Subspace of V holds
( v in W iff v + W = the carrier of W ) by Lm3;
theorem Th49: :: RLSUB_1:49
for V being RealLinearSpace
for v being VECTOR of V
for a being Real
for W being Subspace of V st v in W holds
(a * v) + W = the carrier of W
proof
let V be RealLinearSpace; ::_thesis: for v being VECTOR of V
for a being Real
for W being Subspace of V st v in W holds
(a * v) + W = the carrier of W
let v be VECTOR of V; ::_thesis: for a being Real
for W being Subspace of V st v in W holds
(a * v) + W = the carrier of W
let a be Real; ::_thesis: for W being Subspace of V st v in W holds
(a * v) + W = the carrier of W
let W be Subspace of V; ::_thesis: ( v in W implies (a * v) + W = the carrier of W )
assume A1: v in W ; ::_thesis: (a * v) + W = the carrier of W
thus (a * v) + W c= the carrier of W :: according to XBOOLE_0:def_10 ::_thesis: the carrier of W c= (a * v) + W
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (a * v) + W or x in the carrier of W )
assume x in (a * v) + W ; ::_thesis: x in the carrier of W
then consider u being VECTOR of V 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 x in the carrier of W by A2, STRUCT_0:def_5; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in the carrier of W or x in (a * v) + W )
assume A4: x in the carrier of W ; ::_thesis: x in (a * v) + W
then A5: x in W by STRUCT_0:def_5;
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 by RLVECT_1:4 ;
a * v in W by A1, Th21;
then y - (a * v) in W by A5, Th23;
hence x in (a * v) + W by A6; ::_thesis: verum
end;
theorem Th50: :: RLSUB_1:50
for V being RealLinearSpace
for v being VECTOR of V
for a being Real
for W being Subspace of V st a <> 0 & (a * v) + W = the carrier of W holds
v in W
proof
let V be RealLinearSpace; ::_thesis: for v being VECTOR of V
for a being Real
for W being Subspace of V st a <> 0 & (a * v) + W = the carrier of W holds
v in W
let v be VECTOR of V; ::_thesis: for a being Real
for W being Subspace of V st a <> 0 & (a * v) + W = the carrier of W holds
v in W
let a be Real; ::_thesis: for W being Subspace of V st a <> 0 & (a * v) + W = the carrier of W holds
v in W
let W be Subspace of V; ::_thesis: ( a <> 0 & (a * v) + W = the carrier of W implies v in W )
assume that
A1: a <> 0 and
A2: (a * v) + W = the carrier of W ; ::_thesis: v in W
assume not v in W ; ::_thesis: contradiction
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, RLVECT_1:4;
then a * v in { ((a * v) + u) where u is VECTOR of V : u in W } ;
hence contradiction by A2, A3, STRUCT_0:def_5; ::_thesis: verum
end;
theorem Th51: :: RLSUB_1:51
for V being RealLinearSpace
for v being VECTOR of V
for W being Subspace of V holds
( v in W iff (- v) + W = the carrier of W )
proof
let V be RealLinearSpace; ::_thesis: for v being VECTOR of V
for W being Subspace of V holds
( v in W iff (- v) + W = the carrier of W )
let v be VECTOR of V; ::_thesis: for W being Subspace of V holds
( v in W iff (- v) + W = the carrier of W )
let W be Subspace of V; ::_thesis: ( v in W iff (- v) + W = the carrier of W )
( v in W iff ((- 1) * v) + W = the carrier of W ) by Th49, Th50;
hence ( v in W iff (- v) + W = the carrier of W ) by RLVECT_1:16; ::_thesis: verum
end;
theorem Th52: :: RLSUB_1:52
for V being RealLinearSpace
for u, v being VECTOR of V
for W being Subspace of V holds
( u in W iff v + W = (v + u) + W )
proof
let V be RealLinearSpace; ::_thesis: for u, v being VECTOR of V
for W being Subspace of V holds
( u in W iff v + W = (v + u) + W )
let u, v be VECTOR of V; ::_thesis: for W being Subspace of V holds
( u in W iff v + W = (v + u) + W )
let W be Subspace of V; ::_thesis: ( u in W iff v + W = (v + u) + W )
thus ( u in W implies v + W = (v + u) + W ) ::_thesis: ( v + W = (v + u) + W implies u in W )
proof
assume A1: u in W ; ::_thesis: v + W = (v + u) + W
thus v + W c= (v + u) + W :: according to XBOOLE_0:def_10 ::_thesis: (v + u) + W c= v + W
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in v + W or x in (v + u) + W )
assume x in v + W ; ::_thesis: x in (v + u) + W
then consider v1 being VECTOR of V 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, RLVECT_1:4 ;
v1 - u in W by A1, A3, Th23;
hence x in (v + u) + W by A4; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (v + u) + W or x in v + W )
assume x in (v + u) + W ; ::_thesis: x in v + W
then consider v2 being VECTOR of V 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 x in v + W by A7; ::_thesis: verum
end;
assume A8: v + W = (v + u) + W ; ::_thesis: u in W
( 0. V in W & v + (0. V) = v ) by Th17, RLVECT_1:4;
then v in (v + u) + W by A8;
then consider u1 being VECTOR of V such that
A9: v = (v + u) + u1 and
A10: u1 in W ;
( v = v + (0. V) & v = v + (u + u1) ) by A9, RLVECT_1:4, RLVECT_1:def_3;
then u + u1 = 0. V by RLVECT_1:8;
then u = - u1 by RLVECT_1:def_10;
hence u in W by A10, Th22; ::_thesis: verum
end;
theorem :: RLSUB_1:53
for V being RealLinearSpace
for u, v being VECTOR of V
for W being Subspace of V holds
( u in W iff v + W = (v - u) + W )
proof
let V be RealLinearSpace; ::_thesis: for u, v being VECTOR of V
for W being Subspace of V holds
( u in W iff v + W = (v - u) + W )
let u, v be VECTOR of V; ::_thesis: for W being Subspace of V holds
( u in W iff v + W = (v - u) + W )
let W be Subspace of V; ::_thesis: ( u in W iff v + W = (v - u) + W )
A1: ( - u in W implies u in W )
proof
assume - u in W ; ::_thesis: u in W
then - (- u) in W by Th22;
hence u in W by RLVECT_1:17; ::_thesis: verum
end;
( - u in W iff v + W = (v + (- u)) + W ) by Th52;
hence ( u in W iff v + W = (v - u) + W ) by A1, Th22; ::_thesis: verum
end;
theorem Th54: :: RLSUB_1:54
for V being RealLinearSpace
for v, u being VECTOR of V
for W being Subspace of V holds
( v in u + W iff u + W = v + W )
proof
let V be RealLinearSpace; ::_thesis: for v, u being VECTOR of V
for W being Subspace of V holds
( v in u + W iff u + W = v + W )
let v, u be VECTOR of V; ::_thesis: for W being Subspace of V holds
( v in u + W iff u + W = v + W )
let W be Subspace of V; ::_thesis: ( v in u + W iff u + W = v + W )
thus ( v in u + W implies u + W = v + W ) ::_thesis: ( u + W = v + W implies v in u + W )
proof
assume v in u + W ; ::_thesis: u + W = v + 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 :: according to XBOOLE_0:def_10 ::_thesis: v + W c= u + W
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in u + W or x in v + W )
assume x in u + W ; ::_thesis: x in v + W
then consider v1 being VECTOR of V 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 by RLVECT_1:4 ;
then A5: x = v + (v1 + (- z)) by A3, RLVECT_1:def_3
.= v + (v1 - z) ;
v1 - z in W by A2, A4, Th23;
hence x in v + W by A5; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in v + W or x in u + W )
assume x in v + W ; ::_thesis: x in u + W
then consider v2 being VECTOR of V 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 x in u + W ; ::_thesis: verum
end;
thus ( u + W = v + W implies v in u + W ) by Th43; ::_thesis: verum
end;
theorem Th55: :: RLSUB_1:55
for V being RealLinearSpace
for v being VECTOR of V
for W being Subspace of V holds
( v + W = (- v) + W iff v in W )
proof
let V be RealLinearSpace; ::_thesis: for v being VECTOR of V
for W being Subspace of V holds
( v + W = (- v) + W iff v in W )
let v be VECTOR of V; ::_thesis: for W being Subspace of V holds
( v + W = (- v) + W iff v in W )
let W be Subspace of V; ::_thesis: ( v + W = (- v) + W iff v in W )
thus ( v + W = (- v) + W implies v in W ) ::_thesis: ( v in W implies v + W = (- v) + W )
proof
assume v + W = (- v) + W ; ::_thesis: v in W
then v in (- v) + W by Th43;
then consider u being VECTOR of V such that
A1: v = (- v) + u and
A2: u in W ;
0. V = v - ((- v) + u) by A1, RLVECT_1:15
.= (v - (- v)) - u by RLVECT_1:27
.= (v + v) - u by RLVECT_1:17
.= ((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 = (2 ") * u by RLVECT_1:def_8;
hence v in W by A2, Th21; ::_thesis: verum
end;
assume A3: v in W ; ::_thesis: v + W = (- v) + W
then v + W = the carrier of W by Lm3;
hence v + W = (- v) + W by A3, Th51; ::_thesis: verum
end;
theorem Th56: :: RLSUB_1:56
for V being RealLinearSpace
for u, v1, v2 being VECTOR of V
for W being Subspace of V st u in v1 + W & u in v2 + W holds
v1 + W = v2 + W
proof
let V be RealLinearSpace; ::_thesis: for u, v1, v2 being VECTOR of V
for W being Subspace of V st u in v1 + W & u in v2 + W holds
v1 + W = v2 + W
let u, v1, v2 be VECTOR of V; ::_thesis: for W being Subspace of V st u in v1 + W & u in v2 + W holds
v1 + W = v2 + W
let W be Subspace of V; ::_thesis: ( u in v1 + W & u in v2 + W implies v1 + W = v2 + W )
assume that
A1: u in v1 + W and
A2: u in v2 + W ; ::_thesis: v1 + W = 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 :: according to XBOOLE_0:def_10 ::_thesis: v2 + W c= v1 + W
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in v1 + W or x in v2 + W )
assume x in v1 + W ; ::_thesis: x in v2 + W
then consider u1 being VECTOR of V 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 by RLVECT_1:4 ;
then x = (v2 + (x2 - x1)) + u1 by A5, A7, RLVECT_1:def_3
.= v2 + ((x2 - x1) + u1) by RLVECT_1:def_3 ;
hence x in v2 + W by A9; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in v2 + W or x in v1 + W )
assume x in v2 + W ; ::_thesis: x in v1 + W
then consider u1 being VECTOR of V 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 by RLVECT_1:4 ;
then x = (v1 + (x1 - x2)) + u1 by A3, A10, RLVECT_1:def_3
.= v1 + ((x1 - x2) + u1) by RLVECT_1:def_3 ;
hence x in v1 + W by A12; ::_thesis: verum
end;
theorem :: RLSUB_1:57
for V being RealLinearSpace
for u, v being VECTOR of V
for W being Subspace of V st u in v + W & u in (- v) + W holds
v in W
proof
let V be RealLinearSpace; ::_thesis: for u, v being VECTOR of V
for W being Subspace of V st u in v + W & u in (- v) + W holds
v in W
let u, v be VECTOR of V; ::_thesis: for W being Subspace of V st u in v + W & u in (- v) + W holds
v in W
let W be Subspace of V; ::_thesis: ( u in v + W & u in (- v) + W implies v in W )
assume ( u in v + W & u in (- v) + W ) ; ::_thesis: v in W
then v + W = (- v) + W by Th56;
hence v in W by Th55; ::_thesis: verum
end;
theorem Th58: :: RLSUB_1:58
for V being RealLinearSpace
for v being VECTOR of V
for a being Real
for W being Subspace of V st a <> 1 & a * v in v + W holds
v in W
proof
let V be RealLinearSpace; ::_thesis: for v being VECTOR of V
for a being Real
for W being Subspace of V st a <> 1 & a * v in v + W holds
v in W
let v be VECTOR of V; ::_thesis: for a being Real
for W being Subspace of V st a <> 1 & a * v in v + W holds
v in W
let a be Real; ::_thesis: for W being Subspace of V st a <> 1 & a * v in v + W holds
v in W
let W be Subspace of V; ::_thesis: ( a <> 1 & a * v in v + W implies v in W )
assume that
A1: a <> 1 and
A2: a * v in v + W ; ::_thesis: v in W
A3: a - 1 <> 0 by A1;
consider u being VECTOR of V such that
A4: a * v = v + u and
A5: u in W by A2;
u = u + (0. V) by RLVECT_1:4
.= 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 v in W by A5, Th21; ::_thesis: verum
end;
theorem Th59: :: RLSUB_1:59
for V being RealLinearSpace
for v being VECTOR of V
for a being Real
for W being Subspace of V st v in W holds
a * v in v + W
proof
let V be RealLinearSpace; ::_thesis: for v being VECTOR of V
for a being Real
for W being Subspace of V st v in W holds
a * v in v + W
let v be VECTOR of V; ::_thesis: for a being Real
for W being Subspace of V st v in W holds
a * v in v + W
let a be Real; ::_thesis: for W being Subspace of V st v in W holds
a * v in v + W
let W be Subspace of V; ::_thesis: ( v in W implies a * v in v + W )
assume v in W ; ::_thesis: a * v in v + 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 a * v in v + W by A1; ::_thesis: verum
end;
theorem :: RLSUB_1:60
for V being RealLinearSpace
for v being VECTOR of V
for W being Subspace of V holds
( - v in v + W iff v in W )
proof
let V be RealLinearSpace; ::_thesis: for v being VECTOR of V
for W being Subspace of V holds
( - v in v + W iff v in W )
let v be VECTOR of V; ::_thesis: for W being Subspace of V holds
( - v in v + W iff v in W )
let W be Subspace of V; ::_thesis: ( - v in v + W iff v in W )
(- 1) * v = - v by RLVECT_1:16;
hence ( - v in v + W iff v in W ) by Th58, Th59; ::_thesis: verum
end;
theorem Th61: :: RLSUB_1:61
for V being RealLinearSpace
for u, v being VECTOR of V
for W being Subspace of V holds
( u + v in v + W iff u in W )
proof
let V be RealLinearSpace; ::_thesis: for u, v being VECTOR of V
for W being Subspace of V holds
( u + v in v + W iff u in W )
let u, v be VECTOR of V; ::_thesis: for W being Subspace of V holds
( u + v in v + W iff u in W )
let W be Subspace of V; ::_thesis: ( u + v in v + W iff u in W )
thus ( u + v in v + W implies u in W ) ::_thesis: ( u in W implies u + v in v + W )
proof
assume u + v in v + W ; ::_thesis: u in W
then ex v1 being VECTOR of V st
( u + v = v + v1 & v1 in W ) ;
hence u in W by RLVECT_1:8; ::_thesis: verum
end;
assume u in W ; ::_thesis: u + v in v + W
hence u + v in v + W ; ::_thesis: verum
end;
theorem :: RLSUB_1:62
for V being RealLinearSpace
for v, u being VECTOR of V
for W being Subspace of V holds
( v - u in v + W iff u in W )
proof
let V be RealLinearSpace; ::_thesis: for v, u being VECTOR of V
for W being Subspace of V holds
( v - u in v + W iff u in W )
let v, u be VECTOR of V; ::_thesis: for W being Subspace of V holds
( v - u in v + W iff u in W )
let W be Subspace of V; ::_thesis: ( v - u in v + W iff u in W )
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 ( v - u in v + W iff u in W ) by A1, A2, Th61, RLVECT_1:17; ::_thesis: verum
end;
theorem Th63: :: RLSUB_1:63
for V being RealLinearSpace
for u, v being VECTOR of V
for W being Subspace of V holds
( u in v + W iff ex v1 being VECTOR of V st
( v1 in W & u = v + v1 ) )
proof
let V be RealLinearSpace; ::_thesis: for u, v being VECTOR of V
for W being Subspace of V holds
( u in v + W iff ex v1 being VECTOR of V st
( v1 in W & u = v + v1 ) )
let u, v be VECTOR of V; ::_thesis: for W being Subspace of V holds
( u in v + W iff ex v1 being VECTOR of V st
( v1 in W & u = v + v1 ) )
let W be Subspace of V; ::_thesis: ( u in v + W iff ex v1 being VECTOR of V st
( v1 in W & u = v + v1 ) )
thus ( u in v + W implies ex v1 being VECTOR of V st
( v1 in W & u = v + v1 ) ) ::_thesis: ( ex v1 being VECTOR of V st
( v1 in W & u = v + v1 ) implies u in v + W )
proof
assume u in v + W ; ::_thesis: ex v1 being VECTOR of V st
( v1 in W & u = v + v1 )
then ex v1 being VECTOR of V st
( u = v + v1 & v1 in W ) ;
hence ex v1 being VECTOR of V st
( v1 in W & u = v + v1 ) ; ::_thesis: verum
end;
given v1 being VECTOR of V such that A1: ( v1 in W & u = v + v1 ) ; ::_thesis: u in v + W
thus u in v + W by A1; ::_thesis: verum
end;
theorem :: RLSUB_1:64
for V being RealLinearSpace
for u, v being VECTOR of V
for W being Subspace of V holds
( u in v + W iff ex v1 being VECTOR of V st
( v1 in W & u = v - v1 ) )
proof
let V be RealLinearSpace; ::_thesis: for u, v being VECTOR of V
for W being Subspace of V holds
( u in v + W iff ex v1 being VECTOR of V st
( v1 in W & u = v - v1 ) )
let u, v be VECTOR of V; ::_thesis: for W being Subspace of V holds
( u in v + W iff ex v1 being VECTOR of V st
( v1 in W & u = v - v1 ) )
let W be Subspace of V; ::_thesis: ( u in v + W iff ex v1 being VECTOR of V st
( v1 in W & u = v - v1 ) )
thus ( u in v + W implies ex v1 being VECTOR of V st
( v1 in W & u = v - v1 ) ) ::_thesis: ( ex v1 being VECTOR of V st
( v1 in W & u = v - v1 ) implies u in v + W )
proof
assume u in v + W ; ::_thesis: ex v1 being VECTOR of V st
( v1 in W & u = v - v1 )
then consider v1 being VECTOR of V such that
A1: u = v + v1 and
A2: v1 in W ;
take x = - v1; ::_thesis: ( x in W & u = v - x )
thus x in W by A2, Th22; ::_thesis: u = v - x
thus u = v - x by A1, RLVECT_1:17; ::_thesis: verum
end;
given v1 being VECTOR of V such that A3: v1 in W and
A4: u = v - v1 ; ::_thesis: u in v + W
- v1 in W by A3, Th22;
hence u in v + W by A4; ::_thesis: verum
end;
theorem Th65: :: RLSUB_1:65
for V being RealLinearSpace
for v1, v2 being VECTOR of V
for W being Subspace of V holds
( ex v being VECTOR of V st
( v1 in v + W & v2 in v + W ) iff v1 - v2 in W )
proof
let V be RealLinearSpace; ::_thesis: for v1, v2 being VECTOR of V
for W being Subspace of V holds
( ex v being VECTOR of V st
( v1 in v + W & v2 in v + W ) iff v1 - v2 in W )
let v1, v2 be VECTOR of V; ::_thesis: for W being Subspace of V holds
( ex v being VECTOR of V st
( v1 in v + W & v2 in v + W ) iff v1 - v2 in W )
let W be Subspace of V; ::_thesis: ( ex v being VECTOR of V st
( v1 in v + W & v2 in v + W ) iff v1 - v2 in W )
thus ( ex v being VECTOR of V st
( v1 in v + W & v2 in v + W ) implies v1 - v2 in W ) ::_thesis: ( v1 - v2 in W implies ex v being VECTOR of V st
( v1 in v + W & v2 in v + W ) )
proof
given v being VECTOR of V such that A1: v1 in v + W and
A2: v2 in v + W ; ::_thesis: v1 - v2 in W
consider u2 being VECTOR of V such that
A3: u2 in W and
A4: v2 = v + u2 by A2, Th63;
consider u1 being VECTOR of V 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 by RLVECT_1:4 ;
hence v1 - v2 in W by A5, A3, Th23; ::_thesis: verum
end;
assume v1 - v2 in W ; ::_thesis: ex v being VECTOR of V st
( v1 in v + W & v2 in v + W )
then A7: - (v1 - v2) in W by Th22;
take v1 ; ::_thesis: ( v1 in v1 + W & v2 in v1 + W )
thus v1 in v1 + W by Th43; ::_thesis: v2 in v1 + W
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 by RLVECT_1:4 ;
hence v2 in v1 + W by A7; ::_thesis: verum
end;
theorem Th66: :: RLSUB_1:66
for V being RealLinearSpace
for v, u being VECTOR of V
for W being Subspace of V st v + W = u + W holds
ex v1 being VECTOR of V st
( v1 in W & v + v1 = u )
proof
let V be RealLinearSpace; ::_thesis: for v, u being VECTOR of V
for W being Subspace of V st v + W = u + W holds
ex v1 being VECTOR of V st
( v1 in W & v + v1 = u )
let v, u be VECTOR of V; ::_thesis: for W being Subspace of V st v + W = u + W holds
ex v1 being VECTOR of V st
( v1 in W & v + v1 = u )
let W be Subspace of V; ::_thesis: ( v + W = u + W implies ex v1 being VECTOR of V st
( v1 in W & v + v1 = u ) )
assume v + W = u + W ; ::_thesis: ex v1 being VECTOR of V st
( v1 in W & v + v1 = u )
then v in u + W by Th43;
then consider u1 being VECTOR of V such that
A1: v = u + u1 and
A2: u1 in W ;
take v1 = u - v; ::_thesis: ( v1 in W & v + v1 = u )
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; ::_thesis: v + v1 = u
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 by RLVECT_1:4 ; ::_thesis: verum
end;
theorem Th67: :: RLSUB_1:67
for V being RealLinearSpace
for v, u being VECTOR of V
for W being Subspace of V st v + W = u + W holds
ex v1 being VECTOR of V st
( v1 in W & v - v1 = u )
proof
let V be RealLinearSpace; ::_thesis: for v, u being VECTOR of V
for W being Subspace of V st v + W = u + W holds
ex v1 being VECTOR of V st
( v1 in W & v - v1 = u )
let v, u be VECTOR of V; ::_thesis: for W being Subspace of V st v + W = u + W holds
ex v1 being VECTOR of V st
( v1 in W & v - v1 = u )
let W be Subspace of V; ::_thesis: ( v + W = u + W implies ex v1 being VECTOR of V st
( v1 in W & v - v1 = u ) )
assume v + W = u + W ; ::_thesis: ex v1 being VECTOR of V st
( v1 in W & v - v1 = u )
then u in v + W by Th43;
then consider u1 being VECTOR of V such that
A1: u = v + u1 and
A2: u1 in W ;
take v1 = v - u; ::_thesis: ( v1 in W & v - v1 = 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; ::_thesis: v - v1 = u
thus v - v1 = (v - v) + u by RLVECT_1:29
.= (0. V) + u by RLVECT_1:15
.= u by RLVECT_1:4 ; ::_thesis: verum
end;
theorem Th68: :: RLSUB_1:68
for V being RealLinearSpace
for v being VECTOR of V
for W1, W2 being strict Subspace of V holds
( v + W1 = v + W2 iff W1 = W2 )
proof
let V be RealLinearSpace; ::_thesis: for v being VECTOR of V
for W1, W2 being strict Subspace of V holds
( v + W1 = v + W2 iff W1 = W2 )
let v be VECTOR of V; ::_thesis: for W1, W2 being strict Subspace of V holds
( v + W1 = v + W2 iff W1 = W2 )
let W1, W2 be strict Subspace of V; ::_thesis: ( v + W1 = v + W2 iff W1 = W2 )
thus ( v + W1 = v + W2 implies W1 = W2 ) ::_thesis: ( W1 = W2 implies v + W1 = v + W2 )
proof
assume A1: v + W1 = v + W2 ; ::_thesis: W1 = 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 :: according to XBOOLE_0:def_10 ::_thesis: the carrier of W2 c= the carrier of W1
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in the carrier of W1 or x in the carrier of W2 )
assume A3: x in the carrier of W1 ; ::_thesis: x in the carrier of W2
then reconsider y = x as Element of V by A2;
set z = v + y;
x in W1 by A3, STRUCT_0:def_5;
then v + y in v + W2 by A1;
then consider u being VECTOR of V such that
A4: v + y = v + u and
A5: u in W2 ;
y = u by A4, RLVECT_1:8;
hence x in the carrier of W2 by A5, STRUCT_0:def_5; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in the carrier of W2 or x in the carrier of W1 )
assume A6: x in the carrier of W2 ; ::_thesis: x in the carrier of W1
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, STRUCT_0:def_5;
then v + y in v + W1 by A1;
then consider u being VECTOR of V such that
A7: v + y = v + u and
A8: u in W1 ;
y = u by A7, RLVECT_1:8;
hence x in the carrier of W1 by A8, STRUCT_0:def_5; ::_thesis: verum
end;
hence W1 = W2 by Th30; ::_thesis: verum
end;
thus ( W1 = W2 implies v + W1 = v + W2 ) ; ::_thesis: verum
end;
theorem Th69: :: RLSUB_1:69
for V being RealLinearSpace
for v, u being VECTOR of V
for W1, W2 being strict Subspace of V st v + W1 = u + W2 holds
W1 = W2
proof
let V be RealLinearSpace; ::_thesis: for v, u being VECTOR of V
for W1, W2 being strict Subspace of V st v + W1 = u + W2 holds
W1 = W2
let v, u be VECTOR of V; ::_thesis: for W1, W2 being strict Subspace of V st v + W1 = u + W2 holds
W1 = W2
let W1, W2 be strict Subspace of V; ::_thesis: ( v + W1 = u + W2 implies W1 = W2 )
assume A1: v + W1 = u + W2 ; ::_thesis: W1 = W2
set V2 = the carrier of W2;
set V1 = the carrier of W1;
assume A2: W1 <> W2 ; ::_thesis: contradiction
A3: now__::_thesis:_not_the_carrier_of_W1_\_the_carrier_of_W2_<>_{}
set x = the Element of the carrier of W1 \ the carrier of W2;
assume the carrier of W1 \ the carrier of W2 <> {} ; ::_thesis: contradiction
then the Element of the carrier of W1 \ the carrier of W2 in the carrier of W1 by XBOOLE_0:def_5;
then A4: the Element of the carrier of W1 \ the carrier of W2 in W1 by STRUCT_0:def_5;
then the Element of the carrier of W1 \ the carrier of W2 in V by Th9;
then reconsider x = the Element of the carrier of W1 \ the carrier of W2 as Element of V by STRUCT_0:def_5;
set z = v + x;
v + x in u + W2 by A1, A4;
then consider u1 being VECTOR of V such that
A5: v + x = u + u1 and
A6: u1 in W2 ;
x = (0. V) + x by RLVECT_1:4
.= (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 by RLVECT_1:4 ;
then (u + u1) + W2 = (u + u1) + W1 by A1, A6, A7, Th52;
hence contradiction by A2, Th68; ::_thesis: verum
end;
A8: now__::_thesis:_not_the_carrier_of_W2_\_the_carrier_of_W1_<>_{}
set x = the Element of the carrier of W2 \ the carrier of W1;
assume the carrier of W2 \ the carrier of W1 <> {} ; ::_thesis: contradiction
then the Element of the carrier of W2 \ the carrier of W1 in the carrier of W2 by XBOOLE_0:def_5;
then A9: the Element of the carrier of W2 \ the carrier of W1 in W2 by STRUCT_0:def_5;
then the Element of the carrier of W2 \ the carrier of W1 in V by Th9;
then reconsider x = the Element of the carrier of W2 \ the carrier of W1 as Element of V by STRUCT_0:def_5;
set z = u + x;
u + x in v + W1 by A1, A9;
then consider u1 being VECTOR of V such that
A10: u + x = v + u1 and
A11: u1 in W1 ;
x = (0. V) + x by RLVECT_1:4
.= (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 by RLVECT_1:4 ;
then (v + u1) + W1 = (v + u1) + W2 by A1, A11, A12, Th52;
hence contradiction by A2, Th68; ::_thesis: verum
end;
the carrier of W1 <> the carrier of W2 by A2, Th30;
then ( not the carrier of W1 c= the carrier of W2 or not the carrier of W2 c= the carrier of W1 ) by XBOOLE_0:def_10;
hence contradiction by A3, A8, XBOOLE_1:37; ::_thesis: verum
end;
theorem :: RLSUB_1:70
for V being RealLinearSpace
for W being Subspace of V
for C being Coset of W holds
( C is linearly-closed iff C = the carrier of W )
proof
let V be RealLinearSpace; ::_thesis: for W being Subspace of V
for C being Coset of W holds
( C is linearly-closed iff C = the carrier of W )
let W be Subspace of V; ::_thesis: for C being Coset of W holds
( C is linearly-closed iff C = the carrier of W )
let C be Coset of W; ::_thesis: ( C is linearly-closed iff C = the carrier of W )
thus ( C is linearly-closed implies C = the carrier of W ) ::_thesis: ( C = the carrier of W implies C is linearly-closed )
proof
assume A1: C is linearly-closed ; ::_thesis: C = the carrier of W
consider v being VECTOR of V such that
A2: C = v + W by Def6;
C <> {} by A2, Th43;
then 0. V in v + W by A1, A2, Th1;
hence C = the carrier of W by A2, Th47; ::_thesis: verum
end;
thus ( C = the carrier of W implies C is linearly-closed ) by Lm1; ::_thesis: verum
end;
theorem :: RLSUB_1:71
for V being RealLinearSpace
for W1, W2 being strict Subspace of V
for C1 being Coset of W1
for C2 being Coset of W2 st C1 = C2 holds
W1 = W2
proof
let V be RealLinearSpace; ::_thesis: for W1, W2 being strict Subspace of V
for C1 being Coset of W1
for C2 being Coset of W2 st C1 = C2 holds
W1 = W2
let W1, W2 be strict Subspace of V; ::_thesis: for C1 being Coset of W1
for C2 being Coset of W2 st C1 = C2 holds
W1 = W2
let C1 be Coset of W1; ::_thesis: for C2 being Coset of W2 st C1 = C2 holds
W1 = W2
let C2 be Coset of W2; ::_thesis: ( C1 = C2 implies W1 = W2 )
( ex v1 being VECTOR of V st C1 = v1 + W1 & ex v2 being VECTOR of V st C2 = v2 + W2 ) by Def6;
hence ( C1 = C2 implies W1 = W2 ) by Th69; ::_thesis: verum
end;
theorem :: RLSUB_1:72
for V being RealLinearSpace
for v being VECTOR of V holds {v} is Coset of (0). V
proof
let V be RealLinearSpace; ::_thesis: for v being VECTOR of V holds {v} is Coset of (0). V
let v be VECTOR of V; ::_thesis: {v} is Coset of (0). V
v + ((0). V) = {v} by Th45;
hence {v} is Coset of (0). V by Def6; ::_thesis: verum
end;
theorem :: RLSUB_1:73
for V being RealLinearSpace
for V1 being Subset of V st V1 is Coset of (0). V holds
ex v being VECTOR of V st V1 = {v}
proof
let V be RealLinearSpace; ::_thesis: for V1 being Subset of V st V1 is Coset of (0). V holds
ex v being VECTOR of V st V1 = {v}
let V1 be Subset of V; ::_thesis: ( V1 is Coset of (0). V implies ex v being VECTOR of V st V1 = {v} )
assume V1 is Coset of (0). V ; ::_thesis: ex v being VECTOR of V st V1 = {v}
then consider v being VECTOR of V such that
A1: V1 = v + ((0). V) by Def6;
take v ; ::_thesis: V1 = {v}
thus V1 = {v} by A1, Th45; ::_thesis: verum
end;
theorem :: RLSUB_1:74
for V being RealLinearSpace
for W being Subspace of V holds the carrier of W is Coset of W
proof
let V be RealLinearSpace; ::_thesis: for W being Subspace of V holds the carrier of W is Coset of W
let W be Subspace of V; ::_thesis: the carrier of W is Coset of W
the carrier of W = (0. V) + W by Lm2;
hence the carrier of W is Coset of W by Def6; ::_thesis: verum
end;
theorem :: RLSUB_1:75
for V being RealLinearSpace holds the carrier of V is Coset of (Omega). V
proof
let V be RealLinearSpace; ::_thesis: the carrier of V is Coset of (Omega). V
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 = the VECTOR of V + ((Omega). V) by Th46;
hence the carrier of V is Coset of (Omega). V by Def6; ::_thesis: verum
end;
theorem :: RLSUB_1:76
for V being RealLinearSpace
for V1 being Subset of V st V1 is Coset of (Omega). V holds
V1 = the carrier of V
proof
let V be RealLinearSpace; ::_thesis: for V1 being Subset of V st V1 is Coset of (Omega). V holds
V1 = the carrier of V
let V1 be Subset of V; ::_thesis: ( V1 is Coset of (Omega). V implies V1 = the carrier of V )
assume V1 is Coset of (Omega). V ; ::_thesis: V1 = the carrier of V
then ex v being VECTOR of V st V1 = v + ((Omega). V) by Def6;
hence V1 = the carrier of V by Th46; ::_thesis: verum
end;
theorem :: RLSUB_1:77
for V being RealLinearSpace
for W being Subspace of V
for C being Coset of W holds
( 0. V in C iff C = the carrier of W )
proof
let V be RealLinearSpace; ::_thesis: for W being Subspace of V
for C being Coset of W holds
( 0. V in C iff C = the carrier of W )
let W be Subspace of V; ::_thesis: for C being Coset of W holds
( 0. V in C iff C = the carrier of W )
let C be Coset of W; ::_thesis: ( 0. V in C iff C = the carrier of W )
ex v being VECTOR of V st C = v + W by Def6;
hence ( 0. V in C iff C = the carrier of W ) by Th47; ::_thesis: verum
end;
theorem Th78: :: RLSUB_1:78
for V being RealLinearSpace
for u being VECTOR of V
for W being Subspace of V
for C being Coset of W holds
( u in C iff C = u + W )
proof
let V be RealLinearSpace; ::_thesis: for u being VECTOR of V
for W being Subspace of V
for C being Coset of W holds
( u in C iff C = u + W )
let u be VECTOR of V; ::_thesis: for W being Subspace of V
for C being Coset of W holds
( u in C iff C = u + W )
let W be Subspace of V; ::_thesis: for C being Coset of W holds
( u in C iff C = u + W )
let C be Coset of W; ::_thesis: ( u in C iff C = u + W )
thus ( u in C implies C = u + W ) ::_thesis: ( C = u + W implies u in C )
proof
assume A1: u in C ; ::_thesis: C = u + W
ex v being VECTOR of V st C = v + W by Def6;
hence C = u + W by A1, Th54; ::_thesis: verum
end;
thus ( C = u + W implies u in C ) by Th43; ::_thesis: verum
end;
theorem :: RLSUB_1:79
for V being RealLinearSpace
for u, v being VECTOR of V
for W being Subspace of V
for C being Coset of W st u in C & v in C holds
ex v1 being VECTOR of V st
( v1 in W & u + v1 = v )
proof
let V be RealLinearSpace; ::_thesis: for u, v being VECTOR of V
for W being Subspace of V
for C being Coset of W st u in C & v in C holds
ex v1 being VECTOR of V st
( v1 in W & u + v1 = v )
let u, v be VECTOR of V; ::_thesis: for W being Subspace of V
for C being Coset of W st u in C & v in C holds
ex v1 being VECTOR of V st
( v1 in W & u + v1 = v )
let W be Subspace of V; ::_thesis: for C being Coset of W st u in C & v in C holds
ex v1 being VECTOR of V st
( v1 in W & u + v1 = v )
let C be Coset of W; ::_thesis: ( u in C & v in C implies ex v1 being VECTOR of V st
( v1 in W & u + v1 = v ) )
assume ( u in C & v in C ) ; ::_thesis: ex v1 being VECTOR of V st
( v1 in W & u + v1 = v )
then ( C = u + W & C = v + W ) by Th78;
hence ex v1 being VECTOR of V st
( v1 in W & u + v1 = v ) by Th66; ::_thesis: verum
end;
theorem :: RLSUB_1:80
for V being RealLinearSpace
for u, v being VECTOR of V
for W being Subspace of V
for C being Coset of W st u in C & v in C holds
ex v1 being VECTOR of V st
( v1 in W & u - v1 = v )
proof
let V be RealLinearSpace; ::_thesis: for u, v being VECTOR of V
for W being Subspace of V
for C being Coset of W st u in C & v in C holds
ex v1 being VECTOR of V st
( v1 in W & u - v1 = v )
let u, v be VECTOR of V; ::_thesis: for W being Subspace of V
for C being Coset of W st u in C & v in C holds
ex v1 being VECTOR of V st
( v1 in W & u - v1 = v )
let W be Subspace of V; ::_thesis: for C being Coset of W st u in C & v in C holds
ex v1 being VECTOR of V st
( v1 in W & u - v1 = v )
let C be Coset of W; ::_thesis: ( u in C & v in C implies ex v1 being VECTOR of V st
( v1 in W & u - v1 = v ) )
assume ( u in C & v in C ) ; ::_thesis: ex v1 being VECTOR of V st
( v1 in W & u - v1 = v )
then ( C = u + W & C = v + W ) by Th78;
hence ex v1 being VECTOR of V st
( v1 in W & u - v1 = v ) by Th67; ::_thesis: verum
end;
theorem :: RLSUB_1:81
for V being RealLinearSpace
for v1, v2 being VECTOR of V
for W being Subspace of V holds
( ex C being Coset of W st
( v1 in C & v2 in C ) iff v1 - v2 in W )
proof
let V be RealLinearSpace; ::_thesis: for v1, v2 being VECTOR of V
for W being Subspace of V holds
( ex C being Coset of W st
( v1 in C & v2 in C ) iff v1 - v2 in W )
let v1, v2 be VECTOR of V; ::_thesis: for W being Subspace of V holds
( ex C being Coset of W st
( v1 in C & v2 in C ) iff v1 - v2 in W )
let W be Subspace of V; ::_thesis: ( ex C being Coset of W st
( v1 in C & v2 in C ) iff v1 - v2 in W )
thus ( ex C being Coset of W st
( v1 in C & v2 in C ) implies v1 - v2 in W ) ::_thesis: ( v1 - v2 in W implies ex C being Coset of W st
( v1 in C & v2 in C ) )
proof
given C being Coset of W such that A1: ( v1 in C & v2 in C ) ; ::_thesis: v1 - v2 in W
ex v being VECTOR of V st C = v + W by Def6;
hence v1 - v2 in W by A1, Th65; ::_thesis: verum
end;
assume v1 - v2 in W ; ::_thesis: ex C being Coset of W st
( v1 in C & v2 in C )
then consider v being VECTOR of 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 ; ::_thesis: ( v1 in C & v2 in C )
thus ( v1 in C & v2 in C ) by A2; ::_thesis: verum
end;
theorem :: RLSUB_1:82
for V being RealLinearSpace
for u being VECTOR of V
for W being Subspace of V
for B, C being Coset of W st u in B & u in C holds
B = C
proof
let V be RealLinearSpace; ::_thesis: for u being VECTOR of V
for W being Subspace of V
for B, C being Coset of W st u in B & u in C holds
B = C
let u be VECTOR of V; ::_thesis: for W being Subspace of V
for B, C being Coset of W st u in B & u in C holds
B = C
let W be Subspace of V; ::_thesis: for B, C being Coset of W st u in B & u in C holds
B = C
let B, C be Coset of W; ::_thesis: ( u in B & u in C implies B = C )
assume A1: ( u in B & u in C ) ; ::_thesis: B = C
( ex v1 being VECTOR of V st B = v1 + W & ex v2 being VECTOR of V st C = v2 + W ) by Def6;
hence B = C by A1, Th56; ::_thesis: verum
end;