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Theorem pcgcd1 13170
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 6021 . . . 4  |-  ( B  =  0  ->  ( A  gcd  B )  =  ( A  gcd  0
) )
21oveq2d 6029 . . 3  |-  ( B  =  0  ->  ( P  pCnt  ( A  gcd  B ) )  =  ( P  pCnt  ( A  gcd  0 ) ) )
32eqeq1d 2388 . 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 2583 . . . . . . . 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 12934 . . . . . . 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 10299 . . . . 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 12935 . . . . . . 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 13169 . . . . 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 10505 . . . . . . . . . . 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 13154 . . . . . . . . . 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 13142 . . . . . . . . . . 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 10200 . . . . . . . . 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 13155 . . . . . . . . . . 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 10685 . . . . . . . . . 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 10682 . . . . . . . . 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 9053 . . . . . . . . . . . 12  |-  +oo  e/  RR
35 df-nel 2546 . . . . . . . . . . . 12  |-  (  +oo  e/  RR  <->  -.  +oo  e.  RR )
3634, 35mpbi 200 . . . . . . . . . . 11  |-  -.  +oo  e.  RR
37 pc0 13148 . . . . . . . . . . . . 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 2446 . . . . . . . . . . 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 6021 . . . . . . . . . . . 12  |-  ( A  =  0  ->  ( P  pCnt  A )  =  ( P  pCnt  0
) )
4241eleq1d 2446 . . . . . . . . . . 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 2591 . . . . . . . 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 13156 . . . . . . 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 13142 . . . . . . . . 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 13162 . . . . . . . 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 13002 . . . . . . . . . 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 11464 . . . . . . . 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 10299 . . . . . . 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 12963 . . . . . . 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 13162 . . . . . 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 13144 . . . . . 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 10200 . . . . 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 9140 . . . 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 12944 . . . . . 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 6029 . . . 4  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  ( P  pCnt  ( A  gcd  0 ) )  =  ( P  pCnt  ( abs `  A ) ) )
72 pcabs 13168 . . . . . 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 2412 . . 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 2618 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 1649    e. wcel 1717    =/= wne 2543    e/ wnel 2544   class class class wbr 4146   ` cfv 5387  (class class class)co 6013   RRcr 8915   0cc0 8916    +oocpnf 9043    -oocmnf 9044   RR*cxr 9045    < clt 9046    <_ cle 9047   NNcn 9925   NN0cn0 10146   ZZcz 10207   QQcq 10499   ^cexp 11302   abscabs 11959    || cdivides 12772    gcd cgcd 12926   Primecprime 12999    pCnt cpc 13130
This theorem is referenced by:  pcgcd  13171
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993  ax-pre-sup 8994
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-int 3986  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-riota 6478  df-recs 6562  df-rdg 6597  df-1o 6653  df-2o 6654  df-oadd 6657  df-er 6834  df-en 7039  df-dom 7040  df-sdom 7041  df-fin 7042  df-sup 7374  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-div 9603  df-nn 9926  df-2 9983  df-3 9984  df-n0 10147  df-z 10208  df-uz 10414  df-q 10500  df-rp 10538  df-fl 11122  df-mod 11171  df-seq 11244  df-exp 11303  df-cj 11824  df-re 11825  df-im 11826  df-sqr 11960  df-abs 11961  df-dvds 12773  df-gcd 12927  df-prm 13000  df-pc 13131
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