:: BILINEAR semantic presentation
:: deftheorem Def1 BILINEAR:def 1 :
canceled;
:: deftheorem Def2 defines NulForm BILINEAR:def 2 :
definition
let c
1 be non
empty LoopStr ;
let c
2, c
3 be non
empty VectSpStr of c
1;
let c
4, c
5 be
Form of c
2,c
3;
func c
4 + c
5 -> Form of a
2,a
3 means :
Def3:
:: BILINEAR:def 3
for b
1 being
Vector of a
2for b
2 being
Vector of a
3 holds a
6 . b
1,b
2 = (a4 . b1,b2) + (a5 . b1,b2);
existence
ex b1 being Form of c2,c3 st
for b2 being Vector of c2
for b3 being Vector of c3 holds b1 . b2,b3 = (c4 . b2,b3) + (c5 . b2,b3)
uniqueness
for b1, b2 being Form of c2,c3 holds
( ( for b3 being Vector of c2
for b4 being Vector of c3 holds b1 . b3,b4 = (c4 . b3,b4) + (c5 . b3,b4) ) & ( for b3 being Vector of c2
for b4 being Vector of c3 holds b2 . b3,b4 = (c4 . b3,b4) + (c5 . b3,b4) ) implies b1 = b2 )
end;
:: deftheorem Def3 defines + BILINEAR:def 3 :
definition
let c
1 be non
empty HGrStr ;
let c
2, c
3 be non
empty VectSpStr of c
1;
let c
4 be
Form of c
2,c
3;
let c
5 be
Element of c
1;
func c
5 * c
4 -> Form of a
2,a
3 means :
Def4:
:: BILINEAR:def 4
for b
1 being
Vector of a
2for b
2 being
Vector of a
3 holds a
6 . b
1,b
2 = a
5 * (a4 . b1,b2);
existence
ex b1 being Form of c2,c3 st
for b2 being Vector of c2
for b3 being Vector of c3 holds b1 . b2,b3 = c5 * (c4 . b2,b3)
uniqueness
for b1, b2 being Form of c2,c3 holds
( ( for b3 being Vector of c2
for b4 being Vector of c3 holds b1 . b3,b4 = c5 * (c4 . b3,b4) ) & ( for b3 being Vector of c2
for b4 being Vector of c3 holds b2 . b3,b4 = c5 * (c4 . b3,b4) ) implies b1 = b2 )
end;
:: deftheorem Def4 defines * BILINEAR:def 4 :
definition
let c
1 be non
empty LoopStr ;
let c
2, c
3 be non
empty VectSpStr of c
1;
let c
4 be
Form of c
2,c
3;
func - c
4 -> Form of a
2,a
3 means :
Def5:
:: BILINEAR:def 5
for b
1 being
Vector of a
2for b
2 being
Vector of a
3 holds a
5 . b
1,b
2 = - (a4 . b1,b2);
existence
ex b1 being Form of c2,c3 st
for b2 being Vector of c2
for b3 being Vector of c3 holds b1 . b2,b3 = - (c4 . b2,b3)
uniqueness
for b1, b2 being Form of c2,c3 holds
( ( for b3 being Vector of c2
for b4 being Vector of c3 holds b1 . b3,b4 = - (c4 . b3,b4) ) & ( for b3 being Vector of c2
for b4 being Vector of c3 holds b2 . b3,b4 = - (c4 . b3,b4) ) implies b1 = b2 )
end;
:: deftheorem Def5 defines - BILINEAR:def 5 :
:: deftheorem Def6 defines - BILINEAR:def 6 :
:: deftheorem Def7 defines - BILINEAR:def 7 :
definition
let c
1 be non
empty LoopStr ;
let c
2, c
3 be non
empty VectSpStr of c
1;
let c
4, c
5 be
Form of c
2,c
3;
redefine func - as c
4 - c
5 -> Form of a
2,a
3 means :
Def8:
:: BILINEAR:def 8
for b
1 being
Vector of a
2for b
2 being
Vector of a
3 holds a
6 . b
1,b
2 = (a4 . b1,b2) - (a5 . b1,b2);
coherence
c4 - c5 is Form of c2,c3
;
compatibility
for b1 being Form of c2,c3 holds
( b1 = c4 - c5 iff for b2 being Vector of c2
for b3 being Vector of c3 holds b1 . b2,b3 = (c4 . b2,b3) - (c5 . b2,b3) )
end;
:: deftheorem Def8 defines - BILINEAR:def 8 :
theorem Th1: :: BILINEAR:1
theorem Th2: :: BILINEAR:2
theorem Th3: :: BILINEAR:3
theorem Th4: :: BILINEAR:4
theorem Th5: :: BILINEAR:5
theorem Th6: :: BILINEAR:6
theorem Th7: :: BILINEAR:7
theorem Th8: :: BILINEAR:8
E7:
now
let c
1 be non
empty 1-sorted ;
let c
2, c
3 be non
empty VectSpStr of c
1;
let c
4 be
Form of c
2,c
3;
let c
5 be
Element of the
carrier of c
2;
let c
6 be
Element of c
3;
E8:
dom c
4 = [:the carrier of c2,the carrier of c3:]
by FUNCT_2:def 1;
then consider c
7 being
Function such that E9:
(
(curry c4) . c
5 = c
7 &
dom c
7 = the
carrier of c
3 &
rng c
7 c= rng c
4 )
and
for b
1 being
set holds
( b
1 in the
carrier of c
3 implies c
7 . b
1 = c
4 . c
5,b
1 )
by FUNCT_5:36;
rng c
7 c= the
carrier of c
1
by E9, XBOOLE_1:1;
then reconsider c
8 = c
7 as
Function of the
carrier of c
3,the
carrier of c
1 by E9, FUNCT_2:4;
c
8 = (curry c4) . c
5
by E9;
hence
(curry c4) . c
5 is
Functional of c
3
;
consider c
9 being
Function such that E10:
(
(curry' c4) . c
6 = c
9 &
dom c
9 = the
carrier of c
2 &
rng c
9 c= rng c
4 )
and
for b
1 being
set holds
( b
1 in the
carrier of c
2 implies c
9 . b
1 = c
4 . b
1,c
6 )
by E8, FUNCT_5:39;
rng c
9 c= the
carrier of c
1
by E10, XBOOLE_1:1;
then reconsider c
10 = c
9 as
Function of the
carrier of c
2,the
carrier of c
1 by E10, FUNCT_2:4;
c
10 = (curry' c4) . c
6
by E10;
hence
(curry' c4) . c
6 is
Functional of c
2
;
end;
:: deftheorem Def9 defines FunctionalFAF BILINEAR:def 9 :
:: deftheorem Def10 defines FunctionalSAF BILINEAR:def 10 :
theorem Th9: :: BILINEAR:9
theorem Th10: :: BILINEAR:10
theorem Th11: :: BILINEAR:11
theorem Th12: :: BILINEAR:12
theorem Th13: :: BILINEAR:13
theorem Th14: :: BILINEAR:14
theorem Th15: :: BILINEAR:15
theorem Th16: :: BILINEAR:16
theorem Th17: :: BILINEAR:17
theorem Th18: :: BILINEAR:18
theorem Th19: :: BILINEAR:19
theorem Th20: :: BILINEAR:20
definition
let c
1 be non
empty HGrStr ;
let c
2, c
3 be non
empty VectSpStr of c
1;
let c
4 be
Functional of c
2;
let c
5 be
Functional of c
3;
func FormFunctional c
4,c
5 -> Form of a
2,a
3 means :
Def11:
:: BILINEAR:def 11
for b
1 being
Vector of a
2for b
2 being
Vector of a
3 holds a
6 . b
1,b
2 = (a4 . b1) * (a5 . b2);
existence
ex b1 being Form of c2,c3 st
for b2 being Vector of c2
for b3 being Vector of c3 holds b1 . b2,b3 = (c4 . b2) * (c5 . b3)
uniqueness
for b1, b2 being Form of c2,c3 holds
( ( for b3 being Vector of c2
for b4 being Vector of c3 holds b1 . b3,b4 = (c4 . b3) * (c5 . b4) ) & ( for b3 being Vector of c2
for b4 being Vector of c3 holds b2 . b3,b4 = (c4 . b3) * (c5 . b4) ) implies b1 = b2 )
end;
:: deftheorem Def11 defines FormFunctional BILINEAR:def 11 :
theorem Th21: :: BILINEAR:21
theorem Th22: :: BILINEAR:22
theorem Th23: :: BILINEAR:23
theorem Th24: :: BILINEAR:24
theorem Th25: :: BILINEAR:25
theorem Th26: :: BILINEAR:26
:: deftheorem Def12 defines additiveFAF BILINEAR:def 12 :
:: deftheorem Def13 defines additiveSAF BILINEAR:def 13 :
:: deftheorem Def14 defines homogeneousFAF BILINEAR:def 14 :
:: deftheorem Def15 defines homogeneousSAF BILINEAR:def 15 :
theorem Th27: :: BILINEAR:27
theorem Th28: :: BILINEAR:28
theorem Th29: :: BILINEAR:29
theorem Th30: :: BILINEAR:30
theorem Th31: :: BILINEAR:31
theorem Th32: :: BILINEAR:32
theorem Th33: :: BILINEAR:33
theorem Th34: :: BILINEAR:34
theorem Th35: :: BILINEAR:35
theorem Th36: :: BILINEAR:36
theorem Th37: :: BILINEAR:37
theorem Th38: :: BILINEAR:38
theorem Th39: :: BILINEAR:39
for b
1 being non
empty add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr for b
2, b
3 being non
empty add-associative right_zeroed right_complementable VectSp-like VectSpStr of b
1for b
4, b
5 being
Vector of b
2for b
6, b
7 being
Vector of b
3for b
8, b
9 being
Element of b
1for b
10 being
bilinear-Form of b
2,b
3 holds b
10 . (b4 + (b8 * b5)),
(b6 + (b9 * b7)) = ((b10 . b4,b6) + (b9 * (b10 . b4,b7))) + ((b8 * (b10 . b5,b6)) + (b8 * (b9 * (b10 . b5,b7))))
theorem Th40: :: BILINEAR:40
for b
1 being non
empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr for b
2, b
3 being
VectSp of b
1for b
4, b
5 being
Vector of b
2for b
6, b
7 being
Vector of b
3for b
8, b
9 being
Element of b
1for b
10 being
bilinear-Form of b
2,b
3 holds b
10 . (b4 - (b8 * b5)),
(b6 - (b9 * b7)) = ((b10 . b4,b6) - (b9 * (b10 . b4,b7))) - ((b8 * (b10 . b5,b6)) - (b8 * (b9 * (b10 . b5,b7))))
theorem Th41: :: BILINEAR:41
:: deftheorem Def16 defines leftker BILINEAR:def 16 :
:: deftheorem Def17 defines rightker BILINEAR:def 17 :
:: deftheorem Def18 defines diagker BILINEAR:def 18 :
theorem Th42: :: BILINEAR:42
theorem Th43: :: BILINEAR:43
theorem Th44: :: BILINEAR:44
:: deftheorem Def19 defines LKer BILINEAR:def 19 :
:: deftheorem Def20 defines RKer BILINEAR:def 20 :
definition
let c
1 be non
empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr ;
let c
2 be
VectSp of c
1;
let c
3 be non
empty VectSpStr of c
1;
let c
4 be
additiveSAF homogeneousSAF Form of c
2,c
3;
func LQForm c
4 -> additiveSAF homogeneousSAF Form of
(VectQuot a2,(LKer a4)),a
3 means :
Def21:
:: BILINEAR:def 21
for b
1 being
Vector of
(VectQuot a2,(LKer a4))for b
2 being
Vector of a
3for b
3 being
Vector of a
2 holds
( b
1 = b
3 + (LKer a4) implies a
5 . b
1,b
2 = a
4 . b
3,b
2 );
existence
ex b1 being additiveSAF homogeneousSAF Form of (VectQuot c2,(LKer c4)),c3 st
for b2 being Vector of (VectQuot c2,(LKer c4))
for b3 being Vector of c3
for b4 being Vector of c2 holds
( b2 = b4 + (LKer c4) implies b1 . b2,b3 = c4 . b4,b3 )
uniqueness
for b1, b2 being additiveSAF homogeneousSAF Form of (VectQuot c2,(LKer c4)),c3 holds
( ( for b3 being Vector of (VectQuot c2,(LKer c4))
for b4 being Vector of c3
for b5 being Vector of c2 holds
( b3 = b5 + (LKer c4) implies b1 . b3,b4 = c4 . b5,b4 ) ) & ( for b3 being Vector of (VectQuot c2,(LKer c4))
for b4 being Vector of c3
for b5 being Vector of c2 holds
( b3 = b5 + (LKer c4) implies b2 . b3,b4 = c4 . b5,b4 ) ) implies b1 = b2 )
end;
:: deftheorem Def21 defines LQForm BILINEAR:def 21 :
definition
let c
1 be non
empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr ;
let c
2 be non
empty VectSpStr of c
1;
let c
3 be
VectSp of c
1;
let c
4 be
additiveFAF homogeneousFAF Form of c
2,c
3;
func RQForm c
4 -> additiveFAF homogeneousFAF Form of a
2,
(VectQuot a3,(RKer a4)) means :
Def22:
:: BILINEAR:def 22
for b
1 being
Vector of
(VectQuot a3,(RKer a4))for b
2 being
Vector of a
2for b
3 being
Vector of a
3 holds
( b
1 = b
3 + (RKer a4) implies a
5 . b
2,b
1 = a
4 . b
2,b
3 );
existence
ex b1 being additiveFAF homogeneousFAF Form of c2,(VectQuot c3,(RKer c4)) st
for b2 being Vector of (VectQuot c3,(RKer c4))
for b3 being Vector of c2
for b4 being Vector of c3 holds
( b2 = b4 + (RKer c4) implies b1 . b3,b2 = c4 . b3,b4 )
uniqueness
for b1, b2 being additiveFAF homogeneousFAF Form of c2,(VectQuot c3,(RKer c4)) holds
( ( for b3 being Vector of (VectQuot c3,(RKer c4))
for b4 being Vector of c2
for b5 being Vector of c3 holds
( b3 = b5 + (RKer c4) implies b1 . b4,b3 = c4 . b4,b5 ) ) & ( for b3 being Vector of (VectQuot c3,(RKer c4))
for b4 being Vector of c2
for b5 being Vector of c3 holds
( b3 = b5 + (RKer c4) implies b2 . b4,b3 = c4 . b4,b5 ) ) implies b1 = b2 )
end;
:: deftheorem Def22 defines RQForm BILINEAR:def 22 :
definition
let c
1 be non
empty Abelian add-associative right_zeroed right_complementable associative distributive left_unital doubleLoopStr ;
let c
2, c
3 be
VectSp of c
1;
let c
4 be
bilinear-Form of c
2,c
3;
func QForm c
4 -> bilinear-Form of
(VectQuot a2,(LKer a4)),
(VectQuot a3,(RKer a4)) means :
Def23:
:: BILINEAR:def 23
for b
1 being
Vector of
(VectQuot a2,(LKer a4))for b
2 being
Vector of
(VectQuot a3,(RKer a4))for b
3 being
Vector of a
2for b
4 being
Vector of a
3 holds
( b
1 = b
3 + (LKer a4) & b
2 = b
4 + (RKer a4) implies a
5 . b
1,b
2 = a
4 . b
3,b
4 );
existence
ex b1 being bilinear-Form of (VectQuot c2,(LKer c4)),(VectQuot c3,(RKer c4)) st
for b2 being Vector of (VectQuot c2,(LKer c4))
for b3 being Vector of (VectQuot c3,(RKer c4))
for b4 being Vector of c2
for b5 being Vector of c3 holds
( b2 = b4 + (LKer c4) & b3 = b5 + (RKer c4) implies b1 . b2,b3 = c4 . b4,b5 )
uniqueness
for b1, b2 being bilinear-Form of (VectQuot c2,(LKer c4)),(VectQuot c3,(RKer c4)) holds
( ( for b3 being Vector of (VectQuot c2,(LKer c4))
for b4 being Vector of (VectQuot c3,(RKer c4))
for b5 being Vector of c2
for b6 being Vector of c3 holds
( b3 = b5 + (LKer c4) & b4 = b6 + (RKer c4) implies b1 . b3,b4 = c4 . b5,b6 ) ) & ( for b3 being Vector of (VectQuot c2,(LKer c4))
for b4 being Vector of (VectQuot c3,(RKer c4))
for b5 being Vector of c2
for b6 being Vector of c3 holds
( b3 = b5 + (LKer c4) & b4 = b6 + (RKer c4) implies b2 . b3,b4 = c4 . b5,b6 ) ) implies b1 = b2 )
end;
:: deftheorem Def23 defines QForm BILINEAR:def 23 :
theorem Th45: :: BILINEAR:45
theorem Th46: :: BILINEAR:46
theorem Th47: :: BILINEAR:47
theorem Th48: :: BILINEAR:48
theorem Th49: :: BILINEAR:49
theorem Th50: :: BILINEAR:50
theorem Th51: :: BILINEAR:51
theorem Th52: :: BILINEAR:52
theorem Th53: :: BILINEAR:53
theorem Th54: :: BILINEAR:54
theorem Th55: :: BILINEAR:55
theorem Th56: :: BILINEAR:56
theorem Th57: :: BILINEAR:57
:: deftheorem Def24 defines degenerated-on-left BILINEAR:def 24 :
:: deftheorem Def25 defines degenerated-on-right BILINEAR:def 25 :
:: deftheorem Def26 defines symmetric BILINEAR:def 26 :
:: deftheorem Def27 defines alternating BILINEAR:def 27 :
theorem Th58: :: BILINEAR:58
theorem Th59: :: BILINEAR:59
:: deftheorem Def28 defines alternating BILINEAR:def 28 :
theorem Th60: :: BILINEAR:60