:: MATRLIN semantic presentation
theorem Th1: :: MATRLIN:1
canceled;
theorem Th2: :: MATRLIN:2
canceled;
theorem Th3: :: MATRLIN:3
theorem Th4: :: MATRLIN:4
theorem Th5: :: MATRLIN:5
theorem Th6: :: MATRLIN:6
theorem Th7: :: MATRLIN:7
theorem Th8: :: MATRLIN:8
canceled;
:: deftheorem Def1 defines FinSequence-yielding MATRLIN:def 1 :
:: deftheorem Def2 defines ^^ MATRLIN:def 2 :
theorem Th9: :: MATRLIN:9
theorem Th10: :: MATRLIN:10
theorem Th11: :: MATRLIN:11
theorem Th12: :: MATRLIN:12
:: deftheorem Def3 defines finite-dimensional MATRLIN:def 3 :
:: deftheorem Def4 defines OrdBasis MATRLIN:def 4 :
Lemma13:
for b1, b2 being FinSequence holds
( len b1 = len b2 & ( for b3 being Nat holds
( b3 in dom b1 implies b1 . b3 = b2 . b3 ) ) implies b1 = b2 )
:: deftheorem Def5 defines + MATRLIN:def 5 :
:: deftheorem Def6 defines * MATRLIN:def 6 :
theorem Th13: :: MATRLIN:13
theorem Th14: :: MATRLIN:14
theorem Th15: :: MATRLIN:15
:: deftheorem Def7 defines lmlt MATRLIN:def 7 :
theorem Th16: :: MATRLIN:16
:: deftheorem Def8 defines Sum MATRLIN:def 8 :
theorem Th17: :: MATRLIN:17
theorem Th18: :: MATRLIN:18
theorem Th19: :: MATRLIN:19
theorem Th20: :: MATRLIN:20
theorem Th21: :: MATRLIN:21
theorem Th22: :: MATRLIN:22
theorem Th23: :: MATRLIN:23
theorem Th24: :: MATRLIN:24
theorem Th25: :: MATRLIN:25
theorem Th26: :: MATRLIN:26
definition
let c
1 be non
empty set ;
let c
2, c
3, c
4 be
Nat;
let c
5 be
Matrix of c
2,c
4,c
1;
let c
6 be
Matrix of c
3,c
4,c
1;
redefine func ^ as c
5 ^ c
6 -> Matrix of a
2 + a
3,a
4,a
1;
coherence
c5 ^ c6 is Matrix of c2 + c3,c4,c1
end;
theorem Th27: :: MATRLIN:27
theorem Th28: :: MATRLIN:28
theorem Th29: :: MATRLIN:29
for b
1, b
2, b
3, b
4, b
5 being
Natfor b
6 being non
empty set for b
7 being
Matrix of b
1,b
2,b
6for b
8 being
Matrix of b
3,b
2,b
6 holds
( b
4 in dom b
8 & b
5 = (len b7) + b
4 implies
Line (b7 ^ b8),b
5 = Line b
8,b
4 )
theorem Th30: :: MATRLIN:30
theorem Th31: :: MATRLIN:31
theorem Th32: :: MATRLIN:32
theorem Th33: :: MATRLIN:33
theorem Th34: :: MATRLIN:34
theorem Th35: :: MATRLIN:35
theorem Th36: :: MATRLIN:36
canceled;
theorem Th37: :: MATRLIN:37
theorem Th38: :: MATRLIN:38
for b
1, b
2 being
Natfor b
3 being
Fieldfor b
4 being
finite-dimensional VectSp of b
3for b
5 being
Matrix of b
1,b
2,the
carrier of b
3 holds
( b
1 > 0 & b
2 > 0 implies for b
6, b
7 being
FinSequence of the
carrier of b
3 holds
(
len b
6 = b
1 &
len b
7 = b
2 & ( for b
8 being
Nat holds
( b
8 in dom b
7 implies b
7 /. b
8 = Sum (mlt b6,(Col b5,b8)) ) ) implies for b
8, b
9 being
FinSequence of the
carrier of b
4 holds
(
len b
8 = b
2 &
len b
9 = b
1 & ( for b
10 being
Nat holds
( b
10 in dom b
9 implies b
9 /. b
10 = Sum (lmlt (Line b5,b10),b8) ) ) implies
Sum (lmlt b6,b9) = Sum (lmlt b7,b8) ) ) )
:: deftheorem Def9 defines |-- MATRLIN:def 9 :
Lemma41:
for b1 being Field
for b2 being finite-dimensional VectSp of b1
for b3 being OrdBasis of b2
for b4 being Element of b2 holds dom (b4 |-- b3) = dom b3
theorem Th39: :: MATRLIN:39
theorem Th40: :: MATRLIN:40
theorem Th41: :: MATRLIN:41
Lemma45:
for b1 being FinSequence
for b2 being set holds
not ( b2 in dom b1 & not len b1 > 0 )
theorem Th42: :: MATRLIN:42
:: deftheorem Def10 defines AutMt MATRLIN:def 10 :
Lemma48:
for b1 being Field
for b2, b3 being finite-dimensional VectSp of b1
for b4 being Function of b2,b3
for b5 being OrdBasis of b2
for b6 being OrdBasis of b3 holds dom (AutMt b4,b5,b6) = dom b5
theorem Th43: :: MATRLIN:43
theorem Th44: :: MATRLIN:44
theorem Th45: :: MATRLIN:45
theorem Th46: :: MATRLIN:46
for b
1 being
Fieldfor b
2, b
3, b
4 being
finite-dimensional VectSp of b
1for b
5 being
Function of b
2,b
3for b
6 being
Function of b
3,b
4for b
7 being
OrdBasis of b
2for b
8 being
OrdBasis of b
3for b
9 being
OrdBasis of b
4 holds
( b
5 is
linear & b
6 is
linear &
len b
7 > 0 &
len b
8 > 0 &
len b
9 > 0 implies
AutMt (b6 * b5),b
7,b
9 = (AutMt b5,b7,b8) * (AutMt b6,b8,b9) )
theorem Th47: :: MATRLIN:47
theorem Th48: :: MATRLIN:48