:: MATRIX_7 semantic presentation
Lemma1:
for b1, b2 being set holds
( b1 <> b2 implies ( {b1,b2} \ {b2} = {b1} & {b1,b2} \ {b1} = {b2} ) )
by ZFMISC_1:23;
theorem Th1: :: MATRIX_7:1
theorem Th2: :: MATRIX_7:2
Lemma4:
idseq 2 = <*1,2*>
by FINSEQ_2:61;
theorem Th3: :: MATRIX_7:3
theorem Th4: :: MATRIX_7:4
theorem Th5: :: MATRIX_7:5
theorem Th6: :: MATRIX_7:6
theorem Th7: :: MATRIX_7:7
Lemma10:
<*1,2*> <> <*2,1*>
by GROUP_7:2;
theorem Th8: :: MATRIX_7:8
theorem Th9: :: MATRIX_7:9
theorem Th10: :: MATRIX_7:10
theorem Th11: :: MATRIX_7:11
theorem Th12: :: MATRIX_7:12
for b
1 being
Fieldfor b
2 being
Matrix of 2,b
1 holds
Det b
2 = ((b2 * 1,1) * (b2 * 2,2)) - ((b2 * 1,2) * (b2 * 2,1))
theorem Th13: :: MATRIX_7:13
theorem Th14: :: MATRIX_7:14
theorem Th15: :: MATRIX_7:15
:: deftheorem Def1 defines IFIN MATRIX_7:def 1 :
for b
1, b
2, b
3, b
4 being
set holds
( ( b
1 in b
2 implies
IFIN b
1,b
2,b
3,b
4 = b
3 ) & ( not b
1 in b
2 implies
IFIN b
1,b
2,b
3,b
4 = b
4 ) );
theorem Th16: :: MATRIX_7:16
:: deftheorem Def2 defines diagonal MATRIX_7:def 2 :
theorem Th17: :: MATRIX_7:17
theorem Th18: :: MATRIX_7:18
theorem Th19: :: MATRIX_7:19
:: deftheorem Def3 defines @ MATRIX_7:def 3 :
theorem Th20: :: MATRIX_7:20
:: deftheorem Def4 defines " MATRIX_7:def 4 :
theorem Th21: :: MATRIX_7:21
theorem Th22: :: MATRIX_7:22
theorem Th23: :: MATRIX_7:23
theorem Th24: :: MATRIX_7:24
theorem Th25: :: MATRIX_7:25
theorem Th26: :: MATRIX_7:26
theorem Th27: :: MATRIX_7:27
Lemma29:
for b1 being Nat
for b2 being Element of Permutations b1 holds
( b2 is even & b1 >= 1 implies b2 " is even )
Lemma30:
for b1 being Nat
for b2 being Permutation of Seg b1 holds (b2 " ) " = b2
by FUNCT_1:65;
theorem Th28: :: MATRIX_7:28
theorem Th29: :: MATRIX_7:29
theorem Th30: :: MATRIX_7:30
theorem Th31: :: MATRIX_7:31
theorem Th32: :: MATRIX_7:32
theorem Th33: :: MATRIX_7:33
theorem Th34: :: MATRIX_7:34
theorem Th35: :: MATRIX_7:35
theorem Th36: :: MATRIX_7:36
theorem Th37: :: MATRIX_7:37