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Theorem pcgcd1 13242
Description: The prime count of a GCD is the minimum of the prime counts of the arguments. (Contributed by Mario Carneiro, 3-Oct-2014.)
Assertion
Ref Expression
pcgcd1  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( P  pCnt  A
)  <_  ( P  pCnt  B ) )  -> 
( P  pCnt  ( A  gcd  B ) )  =  ( P  pCnt  A ) )

Proof of Theorem pcgcd1
StepHypRef Expression
1 oveq2 6081 . . . 4  |-  ( B  =  0  ->  ( A  gcd  B )  =  ( A  gcd  0
) )
21oveq2d 6089 . . 3  |-  ( B  =  0  ->  ( P  pCnt  ( A  gcd  B ) )  =  ( P  pCnt  ( A  gcd  0 ) ) )
32eqeq1d 2443 . 2  |-  ( B  =  0  ->  (
( P  pCnt  ( A  gcd  B ) )  =  ( P  pCnt  A )  <->  ( P  pCnt  ( A  gcd  0 ) )  =  ( P 
pCnt  A ) ) )
4 simpl1 960 . . . . 5  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  ->  P  e.  Prime )
5 simp2 958 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  A  e.  ZZ )
65adantr 452 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  ->  A  e.  ZZ )
7 simpl3 962 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  ->  B  e.  ZZ )
8 simprr 734 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  ->  B  =/=  0 )
9 simpr 448 . . . . . . . . 9  |-  ( ( A  =  0  /\  B  =  0 )  ->  B  =  0 )
109necon3ai 2638 . . . . . . . 8  |-  ( B  =/=  0  ->  -.  ( A  =  0  /\  B  =  0
) )
118, 10syl 16 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  ->  -.  ( A  =  0  /\  B  =  0 ) )
12 gcdn0cl 13006 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  -.  ( A  =  0  /\  B  =  0 ) )  ->  ( A  gcd  B )  e.  NN )
136, 7, 11, 12syl21anc 1183 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  -> 
( A  gcd  B
)  e.  NN )
1413nnzd 10366 . . . . 5  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  -> 
( A  gcd  B
)  e.  ZZ )
15 gcddvds 13007 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( A  gcd  B )  ||  A  /\  ( A  gcd  B ) 
||  B ) )
166, 7, 15syl2anc 643 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  -> 
( ( A  gcd  B )  ||  A  /\  ( A  gcd  B ) 
||  B ) )
1716simpld 446 . . . . 5  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  -> 
( A  gcd  B
)  ||  A )
18 pcdvdstr 13241 . . . . 5  |-  ( ( P  e.  Prime  /\  (
( A  gcd  B
)  e.  ZZ  /\  A  e.  ZZ  /\  ( A  gcd  B )  ||  A ) )  -> 
( P  pCnt  ( A  gcd  B ) )  <_  ( P  pCnt  A ) )
194, 14, 6, 17, 18syl13anc 1186 . . . 4  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  -> 
( P  pCnt  ( A  gcd  B ) )  <_  ( P  pCnt  A ) )
20 zq 10572 . . . . . . . . . . 11  |-  ( A  e.  ZZ  ->  A  e.  QQ )
216, 20syl 16 . . . . . . . . . 10  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  ->  A  e.  QQ )
22 pcxcl 13226 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  A  e.  QQ )  ->  ( P  pCnt  A )  e. 
RR* )
234, 21, 22syl2anc 643 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  -> 
( P  pCnt  A
)  e.  RR* )
24 pczcl 13214 . . . . . . . . . . 11  |-  ( ( P  e.  Prime  /\  ( B  e.  ZZ  /\  B  =/=  0 ) )  -> 
( P  pCnt  B
)  e.  NN0 )
254, 7, 8, 24syl12anc 1182 . . . . . . . . . 10  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  -> 
( P  pCnt  B
)  e.  NN0 )
2625nn0red 10267 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  -> 
( P  pCnt  B
)  e.  RR )
27 pcge0 13227 . . . . . . . . . . 11  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  0  <_  ( P  pCnt  A
) )
284, 6, 27syl2anc 643 . . . . . . . . . 10  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  -> 
0  <_  ( P  pCnt  A ) )
29 ge0gtmnf 10752 . . . . . . . . . 10  |-  ( ( ( P  pCnt  A
)  e.  RR*  /\  0  <_  ( P  pCnt  A
) )  ->  -oo  <  ( P  pCnt  A )
)
3023, 28, 29syl2anc 643 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  ->  -oo  <  ( P  pCnt  A ) )
31 simprl 733 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  -> 
( P  pCnt  A
)  <_  ( P  pCnt  B ) )
32 xrre 10749 . . . . . . . . 9  |-  ( ( ( ( P  pCnt  A )  e.  RR*  /\  ( P  pCnt  B )  e.  RR )  /\  (  -oo  <  ( P  pCnt  A )  /\  ( P 
pCnt  A )  <_  ( P  pCnt  B ) ) )  ->  ( P  pCnt  A )  e.  RR )
3323, 26, 30, 31, 32syl22anc 1185 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  -> 
( P  pCnt  A
)  e.  RR )
34 pnfnre 9119 . . . . . . . . . . . 12  |-  +oo  e/  RR
35 df-nel 2601 . . . . . . . . . . . 12  |-  (  +oo  e/  RR  <->  -.  +oo  e.  RR )
3634, 35mpbi 200 . . . . . . . . . . 11  |-  -.  +oo  e.  RR
37 pc0 13220 . . . . . . . . . . . . 13  |-  ( P  e.  Prime  ->  ( P 
pCnt  0 )  = 
+oo )
384, 37syl 16 . . . . . . . . . . . 12  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  -> 
( P  pCnt  0
)  =  +oo )
3938eleq1d 2501 . . . . . . . . . . 11  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  -> 
( ( P  pCnt  0 )  e.  RR  <->  +oo 
e.  RR ) )
4036, 39mtbiri 295 . . . . . . . . . 10  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  ->  -.  ( P  pCnt  0
)  e.  RR )
41 oveq2 6081 . . . . . . . . . . . 12  |-  ( A  =  0  ->  ( P  pCnt  A )  =  ( P  pCnt  0
) )
4241eleq1d 2501 . . . . . . . . . . 11  |-  ( A  =  0  ->  (
( P  pCnt  A
)  e.  RR  <->  ( P  pCnt  0 )  e.  RR ) )
4342notbid 286 . . . . . . . . . 10  |-  ( A  =  0  ->  ( -.  ( P  pCnt  A
)  e.  RR  <->  -.  ( P  pCnt  0 )  e.  RR ) )
4440, 43syl5ibrcom 214 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  -> 
( A  =  0  ->  -.  ( P  pCnt  A )  e.  RR ) )
4544necon2ad 2646 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  -> 
( ( P  pCnt  A )  e.  RR  ->  A  =/=  0 ) )
4633, 45mpd 15 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  ->  A  =/=  0 )
47 pczdvds 13228 . . . . . . 7  |-  ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  A  =/=  0 ) )  -> 
( P ^ ( P  pCnt  A ) ) 
||  A )
484, 6, 46, 47syl12anc 1182 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  -> 
( P ^ ( P  pCnt  A ) ) 
||  A )
49 pczcl 13214 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  A  =/=  0 ) )  -> 
( P  pCnt  A
)  e.  NN0 )
504, 6, 46, 49syl12anc 1182 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  -> 
( P  pCnt  A
)  e.  NN0 )
51 pcdvdsb 13234 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  B  e.  ZZ  /\  ( P 
pCnt  A )  e.  NN0 )  ->  ( ( P 
pCnt  A )  <_  ( P  pCnt  B )  <->  ( P ^ ( P  pCnt  A ) )  ||  B
) )
524, 7, 50, 51syl3anc 1184 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  -> 
( ( P  pCnt  A )  <_  ( P  pCnt  B )  <->  ( P ^ ( P  pCnt  A ) )  ||  B
) )
5331, 52mpbid 202 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  -> 
( P ^ ( P  pCnt  A ) ) 
||  B )
54 prmnn 13074 . . . . . . . . . 10  |-  ( P  e.  Prime  ->  P  e.  NN )
554, 54syl 16 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  ->  P  e.  NN )
5655, 50nnexpcld 11536 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  -> 
( P ^ ( P  pCnt  A ) )  e.  NN )
5756nnzd 10366 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  -> 
( P ^ ( P  pCnt  A ) )  e.  ZZ )
58 dvdsgcd 13035 . . . . . . 7  |-  ( ( ( P ^ ( P  pCnt  A ) )  e.  ZZ  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  (
( ( P ^
( P  pCnt  A
) )  ||  A  /\  ( P ^ ( P  pCnt  A ) ) 
||  B )  -> 
( P ^ ( P  pCnt  A ) ) 
||  ( A  gcd  B ) ) )
5957, 6, 7, 58syl3anc 1184 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  -> 
( ( ( P ^ ( P  pCnt  A ) )  ||  A  /\  ( P ^ ( P  pCnt  A ) ) 
||  B )  -> 
( P ^ ( P  pCnt  A ) ) 
||  ( A  gcd  B ) ) )
6048, 53, 59mp2and 661 . . . . 5  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  -> 
( P ^ ( P  pCnt  A ) ) 
||  ( A  gcd  B ) )
61 pcdvdsb 13234 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( A  gcd  B )  e.  ZZ  /\  ( P 
pCnt  A )  e.  NN0 )  ->  ( ( P 
pCnt  A )  <_  ( P  pCnt  ( A  gcd  B ) )  <->  ( P ^ ( P  pCnt  A ) )  ||  ( A  gcd  B ) ) )
624, 14, 50, 61syl3anc 1184 . . . . 5  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  -> 
( ( P  pCnt  A )  <_  ( P  pCnt  ( A  gcd  B
) )  <->  ( P ^ ( P  pCnt  A ) )  ||  ( A  gcd  B ) ) )
6360, 62mpbird 224 . . . 4  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  -> 
( P  pCnt  A
)  <_  ( P  pCnt  ( A  gcd  B
) ) )
644, 13pccld 13216 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  -> 
( P  pCnt  ( A  gcd  B ) )  e.  NN0 )
6564nn0red 10267 . . . . 5  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  -> 
( P  pCnt  ( A  gcd  B ) )  e.  RR )
6665, 33letri3d 9207 . . . 4  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  -> 
( ( P  pCnt  ( A  gcd  B ) )  =  ( P 
pCnt  A )  <->  ( ( P  pCnt  ( A  gcd  B ) )  <_  ( P  pCnt  A )  /\  ( P  pCnt  A )  <_  ( P  pCnt  ( A  gcd  B ) ) ) ) )
6719, 63, 66mpbir2and 889 . . 3  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  -> 
( P  pCnt  ( A  gcd  B ) )  =  ( P  pCnt  A ) )
6867anassrs 630 . 2  |-  ( ( ( ( P  e. 
Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( P  pCnt  A )  <_  ( P  pCnt  B ) )  /\  B  =/=  0 )  -> 
( P  pCnt  ( A  gcd  B ) )  =  ( P  pCnt  A ) )
69 gcdid0 13016 . . . . . 6  |-  ( A  e.  ZZ  ->  ( A  gcd  0 )  =  ( abs `  A
) )
705, 69syl 16 . . . . 5  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  gcd  0 )  =  ( abs `  A
) )
7170oveq2d 6089 . . . 4  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  ( P  pCnt  ( A  gcd  0 ) )  =  ( P  pCnt  ( abs `  A ) ) )
72 pcabs 13240 . . . . . 6  |-  ( ( P  e.  Prime  /\  A  e.  QQ )  ->  ( P  pCnt  ( abs `  A
) )  =  ( P  pCnt  A )
)
7320, 72sylan2 461 . . . . 5  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  ( P  pCnt  ( abs `  A
) )  =  ( P  pCnt  A )
)
74733adant3 977 . . . 4  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  ( P  pCnt  ( abs `  A
) )  =  ( P  pCnt  A )
)
7571, 74eqtrd 2467 . . 3  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  ( P  pCnt  ( A  gcd  0 ) )  =  ( P  pCnt  A
) )
7675adantr 452 . 2  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( P  pCnt  A
)  <_  ( P  pCnt  B ) )  -> 
( P  pCnt  ( A  gcd  0 ) )  =  ( P  pCnt  A ) )
773, 68, 76pm2.61ne 2673 1  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( P  pCnt  A
)  <_  ( P  pCnt  B ) )  -> 
( P  pCnt  ( A  gcd  B ) )  =  ( P  pCnt  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598    e/ wnel 2599   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   RRcr 8981   0cc0 8982    +oocpnf 9109    -oocmnf 9110   RR*cxr 9111    < clt 9112    <_ cle 9113   NNcn 9992   NN0cn0 10213   ZZcz 10274   QQcq 10566   ^cexp 11374   abscabs 12031    || cdivides 12844    gcd cgcd 12998   Primecprime 13071    pCnt cpc 13202
This theorem is referenced by:  pcgcd  13243
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-q 10567  df-rp 10605  df-fl 11194  df-mod 11243  df-seq 11316  df-exp 11375  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-dvds 12845  df-gcd 12999  df-prm 13072  df-pc 13203
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