:: INT_1 semantic presentation
Lemma1:
for b1 being set holds
not ( b1 in [:{0},NAT :] \ {[0,0]} & ( for b2 being Element of NAT holds
not b1 = - b2 ) )
Lemma2:
for b1 being set
for b2 being Element of NAT holds
( b1 = - b2 & b2 <> b1 implies b1 in [:{0},NAT :] \ {[0,0]} )
:: deftheorem Def1 defines INT INT_1:def 1 :
for b
1 being
set holds
( b
1 = INT iff for b
2 being
set holds
( b
2 in b
1 iff not for b
3 being
Element of
NAT holds
( not b
2 = b
3 & not b
2 = - b
3 ) ) );
:: deftheorem Def2 defines integer INT_1:def 2 :
theorem Th1: :: INT_1:1
canceled;
theorem Th2: :: INT_1:2
canceled;
theorem Th3: :: INT_1:3
canceled;
theorem Th4: :: INT_1:4
canceled;
theorem Th5: :: INT_1:5
canceled;
theorem Th6: :: INT_1:6
canceled;
theorem Th7: :: INT_1:7
theorem Th8: :: INT_1:8
Lemma7:
for b1 being set holds
( b1 in INT implies b1 in REAL )
by NUMBERS:15;
theorem Th9: :: INT_1:9
canceled;
theorem Th10: :: INT_1:10
canceled;
theorem Th11: :: INT_1:11
canceled;
theorem Th12: :: INT_1:12
canceled;
theorem Th13: :: INT_1:13
canceled;
theorem Th14: :: INT_1:14
canceled;
theorem Th15: :: INT_1:15
canceled;
theorem Th16: :: INT_1:16
theorem Th17: :: INT_1:17
theorem Th18: :: INT_1:18
theorem Th19: :: INT_1:19
Lemma11:
for b1 being Element of NAT holds
( b1 < 1 implies b1 = 0 )
Lemma12:
for b1 being Integer holds
( 0 < b1 implies 1 <= b1 )
theorem Th20: :: INT_1:20
for b
1, b
2 being
Integer holds
( b
1 < b
2 implies b
1 + 1
<= b
2 )
theorem Th21: :: INT_1:21
for b
1 being
Integer holds
( b
1 < 0 implies b
1 <= - 1 )
theorem Th22: :: INT_1:22
for b
1, b
2 being
Integer holds
( b
1 * b
2 = 1 iff ( ( b
1 = 1 & b
2 = 1 ) or ( b
1 = - 1 & b
2 = - 1 ) ) )
theorem Th23: :: INT_1:23
for b
1, b
2 being
Integer holds
( b
1 * b
2 = - 1 iff ( ( b
1 = - 1 & b
2 = 1 ) or ( b
1 = 1 & b
2 = - 1 ) ) )
Lemma16:
for b1, b2 being Integer holds
not ( b1 <= b2 & ( for b3 being Element of NAT holds
not b1 + b3 = b2 ) )
Lemma17:
for b1, b2 being Integer holds
not ( b1 <= b2 & ( for b3 being Element of NAT holds
not b2 - b3 = b1 ) )
Lemma18:
for b1 being real number holds
b1 - 1 < b1
by XREAL_1:148;
:: deftheorem Def3 defines are_congruent_mod INT_1:def 3 :
theorem Th24: :: INT_1:24
canceled;
theorem Th25: :: INT_1:25
canceled;
theorem Th26: :: INT_1:26
canceled;
theorem Th27: :: INT_1:27
canceled;
theorem Th28: :: INT_1:28
canceled;
theorem Th29: :: INT_1:29
canceled;
theorem Th30: :: INT_1:30
canceled;
theorem Th31: :: INT_1:31
canceled;
theorem Th32: :: INT_1:32
theorem Th33: :: INT_1:33
theorem Th34: :: INT_1:34
theorem Th35: :: INT_1:35
theorem Th36: :: INT_1:36
theorem Th37: :: INT_1:37
theorem Th38: :: INT_1:38
theorem Th39: :: INT_1:39
theorem Th40: :: INT_1:40
theorem Th41: :: INT_1:41
theorem Th42: :: INT_1:42
theorem Th43: :: INT_1:43
theorem Th44: :: INT_1:44
theorem Th45: :: INT_1:45
Lemma22:
for b1 being real number holds
not for b2 being Element of NAT holds not b1 < b2
:: deftheorem Def4 defines [\ INT_1:def 4 :
theorem Th46: :: INT_1:46
canceled;
theorem Th47: :: INT_1:47
theorem Th48: :: INT_1:48
theorem Th49: :: INT_1:49
canceled;
theorem Th50: :: INT_1:50
theorem Th51: :: INT_1:51
theorem Th52: :: INT_1:52
:: deftheorem Def5 defines [/ INT_1:def 5 :
theorem Th53: :: INT_1:53
canceled;
theorem Th54: :: INT_1:54
theorem Th55: :: INT_1:55
theorem Th56: :: INT_1:56
canceled;
theorem Th57: :: INT_1:57
theorem Th58: :: INT_1:58
theorem Th59: :: INT_1:59
theorem Th60: :: INT_1:60
theorem Th61: :: INT_1:61
theorem Th62: :: INT_1:62
theorem Th63: :: INT_1:63
theorem Th64: :: INT_1:64
theorem Th65: :: INT_1:65
theorem Th66: :: INT_1:66
:: deftheorem Def6 defines frac INT_1:def 6 :
theorem Th67: :: INT_1:67
canceled;
theorem Th68: :: INT_1:68
theorem Th69: :: INT_1:69
theorem Th70: :: INT_1:70
theorem Th71: :: INT_1:71
theorem Th72: :: INT_1:72
:: deftheorem Def7 defines div INT_1:def 7 :
:: deftheorem Def8 defines mod INT_1:def 8 :
for b
1, b
2 being
Integer holds
( ( b
2 <> 0 implies b
1 mod b
2 = b
1 - ((b1 div b2) * b2) ) & ( not b
2 <> 0 implies b
1 mod b
2 = 0 ) );
:: deftheorem Def9 defines divides INT_1:def 9 :
theorem Th73: :: INT_1:73
canceled;
theorem Th74: :: INT_1:74
theorem Th75: :: INT_1:75
theorem Th76: :: INT_1:76
theorem Th77: :: INT_1:77
theorem Th78: :: INT_1:78
theorem Th79: :: INT_1:79
for b
1, b
2 being
Integer holds
( b
1 < b
2 implies b
1 <= b
2 - 1 )
theorem Th80: :: INT_1:80
theorem Th81: :: INT_1:81
theorem Th82: :: INT_1:82
theorem Th83: :: INT_1:83
for b
1, b
2 being
Integer holds
not ( 0
< b
1 & 1
< b
2 & not b
1 div b
2 < b
1 )