:: LMOD_7 semantic presentation
scheme :: LMOD_7:sch 3
s3{ F
1()
-> non
empty set , F
2(
Element of F
1(),
Element of F
1(),
Element of F
1())
-> set } :
for b
1, b
2 being
TriOp of F
1() holds
( ( for b
3, b
4, b
5 being
Element of F
1() holds b
1 . b
3,b
4,b
5 = F
2(b
3,b
4,b
5) ) & ( for b
3, b
4, b
5 being
Element of F
1() holds b
2 . b
3,b
4,b
5 = F
2(b
3,b
4,b
5) ) implies b
1 = b
2 )
scheme :: LMOD_7:sch 4
s4{ F
1()
-> non
empty set , F
2(
Element of F
1(),
Element of F
1(),
Element of F
1(),
Element of F
1())
-> set } :
for b
1, b
2 being
QuaOp of F
1() holds
( ( for b
3, b
4, b
5, b
6 being
Element of F
1() holds b
1 . b
3,b
4,b
5,b
6 = F
2(b
3,b
4,b
5,b
6) ) & ( for b
3, b
4, b
5, b
6 being
Element of F
1() holds b
2 . b
3,b
4,b
5,b
6 = F
2(b
3,b
4,b
5,b
6) ) implies b
1 = b
2 )
scheme :: LMOD_7:sch 10
s10{ F
1()
-> non
empty set , F
2()
-> non
empty set , F
3()
-> set , F
4()
-> Element of F
1(), F
5(
set )
-> set , P
1[
set ,
set ], P
2[
set ,
set ] } :
( F
4()
in F
5(F
3()) iff for b
1 being
Element of F
2() holds
( b
1 in F
3() implies P
1[F
4(),b
1] ) )
provided
E1:
F
5(F
3())
= { b1 where B is Element of F1() : P2[b1,F3()] }
and
E2:
( P
2[F
4(),F
3()] iff for b
1 being
Element of F
2() holds
( b
1 in F
3() implies P
1[F
4(),b
1] ) )
Lemma1:
for b1 being AbGroup
for b2, b3, b4 being Element of b1 holds
( - (b2 - b3) = (- b2) - (- b3) & (b2 - b3) + b4 = (b2 + b4) - b3 )
theorem Th1: :: LMOD_7:1
theorem Th2: :: LMOD_7:2
theorem Th3: :: LMOD_7:3
theorem Th4: :: LMOD_7:4
theorem Th5: :: LMOD_7:5
theorem Th6: :: LMOD_7:6
:: deftheorem Def1 defines SUBMODULE_DOMAIN LMOD_7:def 1 :
:: deftheorem Def2 defines LINE LMOD_7:def 2 :
:: deftheorem Def3 defines LINE_DOMAIN LMOD_7:def 3 :
:: deftheorem Def4 defines lines LMOD_7:def 4 :
:: deftheorem Def5 defines HIPERPLANE LMOD_7:def 5 :
:: deftheorem Def6 defines HIPERPLANE_DOMAIN LMOD_7:def 6 :
:: deftheorem Def7 defines hiperplanes LMOD_7:def 7 :
:: deftheorem Def8 defines Sum LMOD_7:def 8 :
:: deftheorem Def9 defines /\ LMOD_7:def 9 :
theorem Th7: :: LMOD_7:7
canceled;
theorem Th8: :: LMOD_7:8
canceled;
theorem Th9: :: LMOD_7:9
canceled;
theorem Th10: :: LMOD_7:10
canceled;
theorem Th11: :: LMOD_7:11
canceled;
theorem Th12: :: LMOD_7:12
canceled;
theorem Th13: :: LMOD_7:13
canceled;
theorem Th14: :: LMOD_7:14
theorem Th15: :: LMOD_7:15
:: deftheorem Def10 defines + LMOD_7:def 10 :
for b
1 being
Ringfor b
2 being
LeftMod of b
1for b
3, b
4, b
5 being
Subset of b
2 holds
( b
5 = b
3 + b
4 iff for b
6 being
set holds
( b
6 in b
5 iff ex b
7, b
8 being
Vector of b
2 st
( b
7 in b
3 & b
8 in b
4 & b
6 = b
7 + b
8 ) ) );
:: deftheorem Def11 defines Vector LMOD_7:def 11 :
theorem Th16: :: LMOD_7:16
theorem Th17: :: LMOD_7:17
theorem Th18: :: LMOD_7:18
:: deftheorem Def12 defines .. LMOD_7:def 12 :
for b
1 being
Ringfor b
2 being
LeftMod of b
1for b
3 being
Subspace of b
2for b
4 being
set holds
( b
4 = b
2 .. b
3 iff for b
5 being
set holds
( b
5 in b
4 iff ex b
6 being
Vector of b
2 st b
5 = b
6 + b
3 ) );
:: deftheorem Def13 defines .. LMOD_7:def 13 :
theorem Th19: :: LMOD_7:19
theorem Th20: :: LMOD_7:20
:: deftheorem Def14 defines - LMOD_7:def 14 :
:: deftheorem Def15 defines + LMOD_7:def 15 :
for b
1 being
Ringfor b
2 being
LeftMod of b
1for b
3 being
Subspace of b
2for b
4, b
5, b
6 being
Element of b
2 .. b
3 holds
( b
6 = b
4 + b
5 iff for b
7, b
8 being
Vector of b
2 holds
( b
4 = b
7 .. b
3 & b
5 = b
8 .. b
3 implies b
6 = (b7 + b8) .. b
3 ) );
definition
let c
1 be
Ring;
let c
2 be
LeftMod of c
1;
let c
3 be
Subspace of c
2;
deffunc H
1(
Element of c
2 .. c
3)
-> Element of c
2 .. c
3 =
- a
1;
func COMPL c
3 -> UnOp of a
2 .. a
3 means :: LMOD_7:def 16
for b
1 being
Element of a
2 .. a
3 holds a
4 . b
1 = - b
1;
existence
ex b1 being UnOp of c2 .. c3 st
for b2 being Element of c2 .. c3 holds b1 . b2 = - b2
uniqueness
for b1, b2 being UnOp of c2 .. c3 holds
( ( for b3 being Element of c2 .. c3 holds b1 . b3 = - b3 ) & ( for b3 being Element of c2 .. c3 holds b2 . b3 = - b3 ) implies b1 = b2 )
deffunc H
2(
Element of c
2 .. c
3,
Element of c
2 .. c
3)
-> Element of c
2 .. c
3 = a
1 + a
2;
func ADD c
3 -> BinOp of a
2 .. a
3 means :
Def17:
:: LMOD_7:def 17
for b
1, b
2 being
Element of a
2 .. a
3 holds a
4 . b
1,b
2 = b
1 + b
2;
existence
ex b1 being BinOp of c2 .. c3 st
for b2, b3 being Element of c2 .. c3 holds b1 . b2,b3 = b2 + b3
uniqueness
for b1, b2 being BinOp of c2 .. c3 holds
( ( for b3, b4 being Element of c2 .. c3 holds b1 . b3,b4 = b3 + b4 ) & ( for b3, b4 being Element of c2 .. c3 holds b2 . b3,b4 = b3 + b4 ) implies b1 = b2 )
end;
:: deftheorem Def16 defines COMPL LMOD_7:def 16 :
:: deftheorem Def17 defines ADD LMOD_7:def 17 :
:: deftheorem Def18 defines . LMOD_7:def 18 :
theorem Th21: :: LMOD_7:21
:: deftheorem Def19 defines . LMOD_7:def 19 :
theorem Th22: :: LMOD_7:22
theorem Th23: :: LMOD_7:23
theorem Th24: :: LMOD_7:24
:: deftheorem Def20 defines * LMOD_7:def 20 :
for b
1 being
Ringfor b
2 being
LeftMod of b
1for b
3 being
Subspace of b
2for b
4 being
Scalar of b
1for b
5, b
6 being
Element of
(b2 . b3) holds
( b
6 = b
4 * b
5 iff for b
7 being
Vector of b
2 holds
( b
5 = b
7 . b
3 implies b
6 = (b4 * b7) . b
3 ) );
definition
let c
1 be
Ring;
let c
2 be
LeftMod of c
1;
let c
3 be
Subspace of c
2;
func LMULT c
3 -> Function of
[:the carrier of a1,the carrier of (a2 . a3):],the
carrier of
(a2 . a3) means :
Def21:
:: LMOD_7:def 21
for b
1 being
Scalar of a
1for b
2 being
Element of
(a2 . a3) holds a
4 . b
1,b
2 = b
1 * b
2;
existence
ex b1 being Function of [:the carrier of c1,the carrier of (c2 . c3):],the carrier of (c2 . c3) st
for b2 being Scalar of c1
for b3 being Element of (c2 . c3) holds b1 . b2,b3 = b2 * b3
uniqueness
for b1, b2 being Function of [:the carrier of c1,the carrier of (c2 . c3):],the carrier of (c2 . c3) holds
( ( for b3 being Scalar of c1
for b4 being Element of (c2 . c3) holds b1 . b3,b4 = b3 * b4 ) & ( for b3 being Scalar of c1
for b4 being Element of (c2 . c3) holds b2 . b3,b4 = b3 * b4 ) implies b1 = b2 )
end;
:: deftheorem Def21 defines LMULT LMOD_7:def 21 :
:: deftheorem Def22 defines / LMOD_7:def 22 :
theorem Th25: :: LMOD_7:25
canceled;
theorem Th26: :: LMOD_7:26
theorem Th27: :: LMOD_7:27
:: deftheorem Def23 defines / LMOD_7:def 23 :
theorem Th28: :: LMOD_7:28
theorem Th29: :: LMOD_7:29
theorem Th30: :: LMOD_7:30
Lemma25:
for b1 being Ring
for b2 being LeftMod of b1
for b3 being Subspace of b2 holds
( b2 / b3 is Abelian & b2 / b3 is add-associative & b2 / b3 is right_zeroed & b2 / b3 is right_complementable )
theorem Th31: :: LMOD_7:31