:: ARYTM_3 semantic presentation
Lemma1:
1 in omega
;
:: deftheorem Def1 defines one ARYTM_3:def 1 :
theorem Th1: :: ARYTM_3:1
theorem Th2: :: ARYTM_3:2
theorem Th3: :: ARYTM_3:3
theorem Th4: :: ARYTM_3:4
:: deftheorem Def2 defines are_relative_prime ARYTM_3:def 2 :
theorem Th5: :: ARYTM_3:5
theorem Th6: :: ARYTM_3:6
theorem Th7: :: ARYTM_3:7
defpred S1[ set ] means ex b1 being Ordinal st
( b1 c= a1 & a1 in omega & a1 <> {} & ( for b2, b3, b4 being natural Ordinal holds
not ( b3,b4 are_relative_prime & a1 = b2 *^ b3 & b1 = b2 *^ b4 ) ) );
theorem Th8: :: ARYTM_3:8
:: deftheorem Def3 defines divides ARYTM_3:def 3 :
theorem Th9: :: ARYTM_3:9
theorem Th10: :: ARYTM_3:10
theorem Th11: :: ARYTM_3:11
theorem Th12: :: ARYTM_3:12
canceled;
theorem Th13: :: ARYTM_3:13
theorem Th14: :: ARYTM_3:14
theorem Th15: :: ARYTM_3:15
theorem Th16: :: ARYTM_3:16
Lemma18:
1 = succ 0
;
:: deftheorem Def4 defines lcm ARYTM_3:def 4 :
theorem Th17: :: ARYTM_3:17
theorem Th18: :: ARYTM_3:18
:: deftheorem Def5 defines hcf ARYTM_3:def 5 :
theorem Th19: :: ARYTM_3:19
theorem Th20: :: ARYTM_3:20
theorem Th21: :: ARYTM_3:21
theorem Th22: :: ARYTM_3:22
theorem Th23: :: ARYTM_3:23
theorem Th24: :: ARYTM_3:24
theorem Th25: :: ARYTM_3:25
:: deftheorem Def6 defines RED ARYTM_3:def 6 :
theorem Th26: :: ARYTM_3:26
theorem Th27: :: ARYTM_3:27
theorem Th28: :: ARYTM_3:28
theorem Th29: :: ARYTM_3:29
theorem Th30: :: ARYTM_3:30
theorem Th31: :: ARYTM_3:31
theorem Th32: :: ARYTM_3:32
theorem Th33: :: ARYTM_3:33
set c1 = { [b1,b2] where B is Element of omega , B is Element of omega : ( b1,b2 are_relative_prime & b2 <> {} ) } ;
( 1 <> {} & 1,1 are_relative_prime )
by Th6;
then
[1,1] in { [b1,b2] where B is Element of omega , B is Element of omega : ( b1,b2 are_relative_prime & b2 <> {} ) }
;
then reconsider c2 = { [b1,b2] where B is Element of omega , B is Element of omega : ( b1,b2 are_relative_prime & b2 <> {} ) } as non empty set ;
Lemma38:
for b1, b2 being natural Ordinal holds
( [b1,b2] in c2 implies ( b1,b2 are_relative_prime & b2 <> {} ) )
:: deftheorem Def7 defines RAT+ ARYTM_3:def 7 :
theorem Th34: :: ARYTM_3:34
theorem Th35: :: ARYTM_3:35
theorem Th36: :: ARYTM_3:36
theorem Th37: :: ARYTM_3:37
theorem Th38: :: ARYTM_3:38
theorem Th39: :: ARYTM_3:39
definition
let c
3 be
Element of
RAT+ ;
func numerator c
1 -> Element of
omega means :
Def8:
:: ARYTM_3:def 8
a
2 = a
1 if a
1 in omega otherwise ex b
1 being
natural Ordinal st a
1 = [a2,b1];
existence
( not ( c3 in omega & ( for b1 being Element of omega holds
not b1 = c3 ) ) & not ( not c3 in omega & ( for b1 being Element of omega
for b2 being natural Ordinal holds
not c3 = [b1,b2] ) ) )
correctness
consistency
for b1 being Element of omega holds
verum;
uniqueness
for b1, b2 being Element of omega holds
( ( c3 in omega & b1 = c3 & b2 = c3 implies b1 = b2 ) & ( not c3 in omega & ex b3 being natural Ordinal st c3 = [b1,b3] & ex b3 being natural Ordinal st c3 = [b2,b3] implies b1 = b2 ) );
by ZFMISC_1:33;
func denominator c
1 -> Element of
omega means :
Def9:
:: ARYTM_3:def 9
a
2 = 1
if a
1 in omega otherwise ex b
1 being
natural Ordinal st a
1 = [b1,a2];
existence
( not ( c3 in omega & ( for b1 being Element of omega holds
not b1 = 1 ) ) & not ( not c3 in omega & ( for b1 being Element of omega
for b2 being natural Ordinal holds
not c3 = [b2,b1] ) ) )
correctness
consistency
for b1 being Element of omega holds
verum;
uniqueness
for b1, b2 being Element of omega holds
( ( c3 in omega & b1 = 1 & b2 = 1 implies b1 = b2 ) & ( not c3 in omega & ex b3 being natural Ordinal st c3 = [b3,b1] & ex b3 being natural Ordinal st c3 = [b3,b2] implies b1 = b2 ) );
by ZFMISC_1:33;
end;
:: deftheorem Def8 defines numerator ARYTM_3:def 8 :
:: deftheorem Def9 defines denominator ARYTM_3:def 9 :
theorem Th40: :: ARYTM_3:40
theorem Th41: :: ARYTM_3:41
theorem Th42: :: ARYTM_3:42
theorem Th43: :: ARYTM_3:43
theorem Th44: :: ARYTM_3:44
definition
let c
3, c
4 be
natural Ordinal;
func c
1 / c
2 -> Element of
RAT+ equals :
Def10:
:: ARYTM_3:def 10
{} if a
2 = {} RED a
1,a
2 if RED a
2,a
1 = 1
otherwise [(RED a1,a2),(RED a2,a1)];
coherence
( ( c4 = {} implies {} is Element of RAT+ ) & ( RED c4,c3 = 1 implies RED c3,c4 is Element of RAT+ ) & ( not c4 = {} & not RED c4,c3 = 1 implies [(RED c3,c4),(RED c4,c3)] is Element of RAT+ ) )
consistency
for b1 being Element of RAT+ holds
( c4 = {} & RED c4,c3 = 1 implies ( b1 = {} iff b1 = RED c3,c4 ) )
by Th31;
end;
:: deftheorem Def10 defines / ARYTM_3:def 10 :
theorem Th45: :: ARYTM_3:45
theorem Th46: :: ARYTM_3:46
theorem Th47: :: ARYTM_3:47
theorem Th48: :: ARYTM_3:48
theorem Th49: :: ARYTM_3:49
theorem Th50: :: ARYTM_3:50
theorem Th51: :: ARYTM_3:51
:: deftheorem Def11 defines + ARYTM_3:def 11 :
:: deftheorem Def12 defines *' ARYTM_3:def 12 :
theorem Th52: :: ARYTM_3:52
theorem Th53: :: ARYTM_3:53
theorem Th54: :: ARYTM_3:54
theorem Th55: :: ARYTM_3:55
theorem Th56: :: ARYTM_3:56
theorem Th57: :: ARYTM_3:57
theorem Th58: :: ARYTM_3:58
theorem Th59: :: ARYTM_3:59
theorem Th60: :: ARYTM_3:60
theorem Th61: :: ARYTM_3:61
theorem Th62: :: ARYTM_3:62
theorem Th63: :: ARYTM_3:63
theorem Th64: :: ARYTM_3:64
theorem Th65: :: ARYTM_3:65
theorem Th66: :: ARYTM_3:66
:: deftheorem Def13 defines <=' ARYTM_3:def 13 :
theorem Th67: :: ARYTM_3:67
canceled;
theorem Th68: :: ARYTM_3:68
theorem Th69: :: ARYTM_3:69
for b
1, b
2, b
3 being
Element of
RAT+ holds
( b
1 + b
2 = b
3 + b
2 implies b
1 = b
3 )
theorem Th70: :: ARYTM_3:70
theorem Th71: :: ARYTM_3:71
theorem Th72: :: ARYTM_3:72
theorem Th73: :: ARYTM_3:73
theorem Th74: :: ARYTM_3:74
theorem Th75: :: ARYTM_3:75
theorem Th76: :: ARYTM_3:76
for b
1, b
2, b
3 being
Element of
RAT+ holds
not ( ( ( b
1 < b
2 & b
2 <=' b
3 ) or ( b
1 <=' b
2 & b
2 < b
3 ) ) & not b
1 < b
3 )
by Th74;
theorem Th77: :: ARYTM_3:77
for b
1, b
2, b
3 being
Element of
RAT+ holds
not ( b
1 < b
2 & b
2 < b
3 & not b
1 < b
3 )
by Th74;
theorem Th78: :: ARYTM_3:78
theorem Th79: :: ARYTM_3:79
theorem Th80: :: ARYTM_3:80
theorem Th81: :: ARYTM_3:81
theorem Th82: :: ARYTM_3:82
theorem Th83: :: ARYTM_3:83
theorem Th84: :: ARYTM_3:84
canceled;
theorem Th85: :: ARYTM_3:85
theorem Th86: :: ARYTM_3:86
theorem Th87: :: ARYTM_3:87
theorem Th88: :: ARYTM_3:88
theorem Th89: :: ARYTM_3:89
for b
1, b
2, b
3, b
4 being
Element of
RAT+ holds
not ( b
1 + b
2 = b
3 + b
4 & not b
1 <=' b
3 & not b
2 <=' b
4 )
theorem Th90: :: ARYTM_3:90
theorem Th91: :: ARYTM_3:91
theorem Th92: :: ARYTM_3:92
theorem Th93: :: ARYTM_3:93
theorem Th94: :: ARYTM_3:94
theorem Th95: :: ARYTM_3:95
theorem Th96: :: ARYTM_3:96
theorem Th97: :: ARYTM_3:97
theorem Th98: :: ARYTM_3:98
theorem Th99: :: ARYTM_3:99
theorem Th100: :: ARYTM_3:100
theorem Th101: :: ARYTM_3:101
theorem Th102: :: ARYTM_3:102
theorem Th103: :: ARYTM_3:103