:: MATRIX_7 semantic presentation
Lemma34:
for x1, x2 being set st x1 <> x2 holds
( {x1,x2} \ {x2} = {x1} & {x1,x2} \ {x1} = {x2} )
by ZFMISC_1:23;
theorem Th1: :: MATRIX_7:1
theorem Th2: :: MATRIX_7:2
Lemma73:
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
Lemma83:
<*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
theorem Th13: :: MATRIX_7:13
theorem Th14: :: MATRIX_7:14
theorem Th15: :: MATRIX_7:15
:: deftheorem Def1 defines IFIN MATRIX_7:def 1 :
for
x,
y,
a,
b being
set holds
( (
x in y implies
IFIN x,
y,
a,
b = a ) & ( not
x in y implies
IFIN x,
y,
a,
b = b ) );
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
Lemma275:
for n being Element of NAT
for IT being Element of Permutations n st IT is even & n >= 1 holds
IT " is even
Lemma283:
for n being Element of NAT
for IT being Permutation of Seg n holds (IT " ) " = IT
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