:: MATRIX_9 semantic presentation
Lemma1:
for b1 being Nat holds idseq b1 in Permutations b1
by MATRIX_2:def 11;
theorem Th1: :: MATRIX_9:1
for b
1, b
2 being
set holds
( b
1 in b
2 implies
{b1} in Fin b
2 )
Lemma3:
<*1,2*> <> <*2,1*>
by GROUP_7:2;
Lemma4:
<*2,1*> in Permutations 2
by TARSKI:def 2, MATRIX_7:3;
Lemma5:
for b1 being Nat
for b2, b3 being Element of the carrier of (Group_of_Perm b1)
for b4, b5 being Element of Permutations b1 holds
( b2 = b4 & b3 = b5 implies b2 * b3 = b5 * b4 )
by MATRIX_7:9;
Lemma6:
for b1 being FinSequence-like Function holds
( b1 is one-to-one implies Rev b1 is one-to-one )
theorem Th2: :: MATRIX_9:2
theorem Th3: :: MATRIX_9:3
theorem Th4: :: MATRIX_9:4
theorem Th5: :: MATRIX_9:5
theorem Th6: :: MATRIX_9:6
:: deftheorem Def1 defines PPath_product MATRIX_9:def 1 :
:: deftheorem Def2 defines Per MATRIX_9:def 2 :
theorem Th7: :: MATRIX_9:7
theorem Th8: :: MATRIX_9:8
theorem Th9: :: MATRIX_9:9
theorem Th10: :: MATRIX_9:10
theorem Th11: :: MATRIX_9:11
theorem Th12: :: MATRIX_9:12
theorem Th13: :: MATRIX_9:13
for b
1 being
Fieldfor b
2, b
3, b
4, b
5 being
Element of b
1 holds
Det (b2,b3 ][ b4,b5) = (b2 * b5) - (b3 * b4)
theorem Th14: :: MATRIX_9:14
for b
1 being
Fieldfor b
2, b
3, b
4, b
5 being
Element of b
1 holds
Per (b2,b3 ][ b4,b5) = (b2 * b5) + (b3 * b4)
theorem Th15: :: MATRIX_9:15
theorem Th16: :: MATRIX_9:16
theorem Th17: :: MATRIX_9:17
theorem Th18: :: MATRIX_9:18
theorem Th19: :: MATRIX_9:19
Permutations 3
= {<*1,2,3*>,<*3,2,1*>,<*1,3,2*>,<*2,3,1*>,<*2,1,3*>,<*3,1,2*>}
theorem Th20: :: MATRIX_9:20
for b
1 being
Fieldfor b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9, b
10 being
Element of b
1for b
11 being
Matrix of 3,b
1 holds
( b
11 = <*<*b2,b3,b4*>,<*b5,b6,b7*>,<*b8,b9,b10*>*> implies for b
12 being
Element of
Permutations 3 holds
( b
12 = <*1,2,3*> implies
Path_matrix b
12,b
11 = <*b2,b6,b10*> ) )
theorem Th21: :: MATRIX_9:21
for b
1 being
Fieldfor b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9, b
10 being
Element of b
1for b
11 being
Matrix of 3,b
1 holds
( b
11 = <*<*b2,b3,b4*>,<*b5,b6,b7*>,<*b8,b9,b10*>*> implies for b
12 being
Element of
Permutations 3 holds
( b
12 = <*3,2,1*> implies
Path_matrix b
12,b
11 = <*b4,b6,b8*> ) )
theorem Th22: :: MATRIX_9:22
for b
1 being
Fieldfor b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9, b
10 being
Element of b
1for b
11 being
Matrix of 3,b
1 holds
( b
11 = <*<*b2,b3,b4*>,<*b5,b6,b7*>,<*b8,b9,b10*>*> implies for b
12 being
Element of
Permutations 3 holds
( b
12 = <*1,3,2*> implies
Path_matrix b
12,b
11 = <*b2,b7,b9*> ) )
theorem Th23: :: MATRIX_9:23
for b
1 being
Fieldfor b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9, b
10 being
Element of b
1for b
11 being
Matrix of 3,b
1 holds
( b
11 = <*<*b2,b3,b4*>,<*b5,b6,b7*>,<*b8,b9,b10*>*> implies for b
12 being
Element of
Permutations 3 holds
( b
12 = <*2,3,1*> implies
Path_matrix b
12,b
11 = <*b3,b7,b8*> ) )
theorem Th24: :: MATRIX_9:24
for b
1 being
Fieldfor b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9, b
10 being
Element of b
1for b
11 being
Matrix of 3,b
1 holds
( b
11 = <*<*b2,b3,b4*>,<*b5,b6,b7*>,<*b8,b9,b10*>*> implies for b
12 being
Element of
Permutations 3 holds
( b
12 = <*2,1,3*> implies
Path_matrix b
12,b
11 = <*b3,b5,b10*> ) )
theorem Th25: :: MATRIX_9:25
for b
1 being
Fieldfor b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9, b
10 being
Element of b
1for b
11 being
Matrix of 3,b
1 holds
( b
11 = <*<*b2,b3,b4*>,<*b5,b6,b7*>,<*b8,b9,b10*>*> implies for b
12 being
Element of
Permutations 3 holds
( b
12 = <*3,1,2*> implies
Path_matrix b
12,b
11 = <*b4,b5,b9*> ) )
theorem Th26: :: MATRIX_9:26
theorem Th27: :: MATRIX_9:27
(
<*1,3,2*> in Permutations 3 &
<*2,3,1*> in Permutations 3 &
<*2,1,3*> in Permutations 3 &
<*3,1,2*> in Permutations 3 &
<*1,2,3*> in Permutations 3 &
<*3,2,1*> in Permutations 3 )
Lemma29:
( <*1,2,3*> <> <*3,2,1*> & <*1,2,3*> <> <*1,3,2*> & <*1,2,3*> <> <*2,3,1*> & <*1,2,3*> <> <*2,1,3*> & <*1,2,3*> <> <*3,1,2*> & <*3,2,1*> <> <*1,3,2*> & <*3,2,1*> <> <*2,3,1*> & <*3,2,1*> <> <*2,1,3*> & <*3,2,1*> <> <*3,1,2*> & <*1,3,2*> <> <*2,3,1*> & <*1,3,2*> <> <*2,1,3*> & <*1,3,2*> <> <*3,1,2*> & <*2,3,1*> <> <*2,1,3*> & <*2,3,1*> <> <*3,1,2*> & <*2,1,3*> <> <*3,1,2*> )
by GROUP_7:3;
theorem Th28: :: MATRIX_9:28
theorem Th29: :: MATRIX_9:29
theorem Th30: :: MATRIX_9:30
theorem Th31: :: MATRIX_9:31
theorem Th32: :: MATRIX_9:32
theorem Th33: :: MATRIX_9:33
theorem Th34: :: MATRIX_9:34
theorem Th35: :: MATRIX_9:35
theorem Th36: :: MATRIX_9:36
theorem Th37: :: MATRIX_9:37
(
<*2,1,3*> * <*1,3,2*> = <*2,3,1*> &
<*1,3,2*> * <*2,1,3*> = <*3,1,2*> &
<*2,1,3*> * <*3,2,1*> = <*3,1,2*> &
<*3,2,1*> * <*2,1,3*> = <*2,3,1*> &
<*3,2,1*> * <*3,2,1*> = <*1,2,3*> &
<*2,1,3*> * <*2,1,3*> = <*1,2,3*> &
<*1,3,2*> * <*1,3,2*> = <*1,2,3*> &
<*1,3,2*> * <*2,3,1*> = <*3,2,1*> &
<*2,3,1*> * <*2,3,1*> = <*3,1,2*> &
<*2,3,1*> * <*3,1,2*> = <*1,2,3*> &
<*3,1,2*> * <*2,3,1*> = <*1,2,3*> &
<*3,1,2*> * <*3,1,2*> = <*2,3,1*> &
<*1,3,2*> * <*3,2,1*> = <*2,3,1*> &
<*3,2,1*> * <*1,3,2*> = <*3,1,2*> )
theorem Th38: :: MATRIX_9:38
theorem Th39: :: MATRIX_9:39
Lemma40:
<*1,2,3*> is even Permutation of Seg 3
by MATRIX_2:29, FINSEQ_2:62;
Lemma41:
<*2,3,1*> is even Permutation of Seg 3
Lemma42:
<*3,1,2*> is even Permutation of Seg 3
theorem Th40: :: MATRIX_9:40
Lemma43:
for b1 being Permutation of Seg 3 holds b1 * <*1,2,3*> = b1
Lemma44:
for b1 being Permutation of Seg 3 holds <*1,2,3*> * b1 = b1
theorem Th41: :: MATRIX_9:41
theorem Th42: :: MATRIX_9:42
theorem Th43: :: MATRIX_9:43
theorem Th44: :: MATRIX_9:44
theorem Th45: :: MATRIX_9:45
theorem Th46: :: MATRIX_9:46
for b
1 being
Fieldfor b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9, b
10 being
Element of b
1for b
11 being
Matrix of 3,b
1 holds
( b
11 = <*<*b2,b3,b4*>,<*b5,b6,b7*>,<*b8,b9,b10*>*> implies
Det b
11 = ((((((b2 * b6) * b10) - ((b4 * b6) * b8)) - ((b2 * b7) * b9)) + ((b3 * b7) * b8)) - ((b3 * b5) * b10)) + ((b4 * b5) * b9) )
theorem Th47: :: MATRIX_9:47
for b
1 being
Fieldfor b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9, b
10 being
Element of b
1for b
11 being
Matrix of 3,b
1 holds
( b
11 = <*<*b2,b3,b4*>,<*b5,b6,b7*>,<*b8,b9,b10*>*> implies
Per b
11 = ((((((b2 * b6) * b10) + ((b4 * b6) * b8)) + ((b2 * b7) * b9)) + ((b3 * b7) * b8)) + ((b3 * b5) * b10)) + ((b4 * b5) * b9) )
theorem Th48: :: MATRIX_9:48
theorem Th49: :: MATRIX_9:49
theorem Th50: :: MATRIX_9:50
theorem Th51: :: MATRIX_9:51
theorem Th52: :: MATRIX_9:52
Lemma54:
for b1 being Nat
for b2 being Field
for b3 being Matrix of b1,b2 holds
( ex b4 being Nat st
( b4 in Seg b1 & ( for b5 being Nat holds
( b5 in Seg b1 implies (Col b3,b4) . b5 = 0. b2 ) ) ) implies the add of b2 $$ (FinOmega (Permutations b1)),(PPath_product b3) = 0. b2 )
theorem Th53: :: MATRIX_9:53
theorem Th54: :: MATRIX_9:54
theorem Th55: :: MATRIX_9:55
theorem Th56: :: MATRIX_9:56