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Theorem pcgcd1 12929
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 5866 . . . 4  |-  ( B  =  0  ->  ( A  gcd  B )  =  ( A  gcd  0
) )
21oveq2d 5874 . . 3  |-  ( B  =  0  ->  ( P  pCnt  ( A  gcd  B ) )  =  ( P  pCnt  ( A  gcd  0 ) ) )
32eqeq1d 2291 . 2  |-  ( B  =  0  ->  (
( P  pCnt  ( A  gcd  B ) )  =  ( P  pCnt  A )  <->  ( P  pCnt  ( A  gcd  0 ) )  =  ( P 
pCnt  A ) ) )
4 simpl1 958 . . . . 5  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  ->  P  e.  Prime )
5 simp2 956 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  A  e.  ZZ )
65adantr 451 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  ->  A  e.  ZZ )
7 simpl3 960 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  ->  B  e.  ZZ )
8 simprr 733 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  ->  B  =/=  0 )
9 simpr 447 . . . . . . . . 9  |-  ( ( A  =  0  /\  B  =  0 )  ->  B  =  0 )
109necon3ai 2486 . . . . . . . 8  |-  ( B  =/=  0  ->  -.  ( A  =  0  /\  B  =  0
) )
118, 10syl 15 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  ->  -.  ( A  =  0  /\  B  =  0 ) )
12 gcdn0cl 12693 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  -.  ( A  =  0  /\  B  =  0 ) )  ->  ( A  gcd  B )  e.  NN )
136, 7, 11, 12syl21anc 1181 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  -> 
( A  gcd  B
)  e.  NN )
1413nnzd 10116 . . . . 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 12694 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( A  gcd  B )  ||  A  /\  ( A  gcd  B ) 
||  B ) )
166, 7, 15syl2anc 642 . . . . . 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 445 . . . . 5  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  -> 
( A  gcd  B
)  ||  A )
18 pcdvdstr 12928 . . . . 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 1184 . . . 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 10322 . . . . . . . . . . 11  |-  ( A  e.  ZZ  ->  A  e.  QQ )
216, 20syl 15 . . . . . . . . . 10  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  ->  A  e.  QQ )
22 pcxcl 12913 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  A  e.  QQ )  ->  ( P  pCnt  A )  e. 
RR* )
234, 21, 22syl2anc 642 . . . . . . . . 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 12901 . . . . . . . . . . 11  |-  ( ( P  e.  Prime  /\  ( B  e.  ZZ  /\  B  =/=  0 ) )  -> 
( P  pCnt  B
)  e.  NN0 )
254, 7, 8, 24syl12anc 1180 . . . . . . . . . 10  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  -> 
( P  pCnt  B
)  e.  NN0 )
2625nn0red 10019 . . . . . . . . 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 12914 . . . . . . . . . . 11  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  0  <_  ( P  pCnt  A
) )
284, 6, 27syl2anc 642 . . . . . . . . . 10  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  -> 
0  <_  ( P  pCnt  A ) )
29 ge0gtmnf 10501 . . . . . . . . . 10  |-  ( ( ( P  pCnt  A
)  e.  RR*  /\  0  <_  ( P  pCnt  A
) )  ->  -oo  <  ( P  pCnt  A )
)
3023, 28, 29syl2anc 642 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  ->  -oo  <  ( P  pCnt  A ) )
31 simprl 732 . . . . . . . . 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 10498 . . . . . . . . 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 1183 . . . . . . . 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 8874 . . . . . . . . . . . 12  |-  +oo  e/  RR
35 df-nel 2449 . . . . . . . . . . . 12  |-  (  +oo  e/  RR  <->  -.  +oo  e.  RR )
3634, 35mpbi 199 . . . . . . . . . . 11  |-  -.  +oo  e.  RR
37 pc0 12907 . . . . . . . . . . . . 13  |-  ( P  e.  Prime  ->  ( P 
pCnt  0 )  = 
+oo )
384, 37syl 15 . . . . . . . . . . . 12  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  -> 
( P  pCnt  0
)  =  +oo )
3938eleq1d 2349 . . . . . . . . . . 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 294 . . . . . . . . . 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 5866 . . . . . . . . . . . 12  |-  ( A  =  0  ->  ( P  pCnt  A )  =  ( P  pCnt  0
) )
4241eleq1d 2349 . . . . . . . . . . 11  |-  ( A  =  0  ->  (
( P  pCnt  A
)  e.  RR  <->  ( P  pCnt  0 )  e.  RR ) )
4342notbid 285 . . . . . . . . . 10  |-  ( A  =  0  ->  ( -.  ( P  pCnt  A
)  e.  RR  <->  -.  ( P  pCnt  0 )  e.  RR ) )
4440, 43syl5ibrcom 213 . . . . . . . . 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 2494 . . . . . . . 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 14 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  ->  A  =/=  0 )
47 pczdvds 12915 . . . . . . 7  |-  ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  A  =/=  0 ) )  -> 
( P ^ ( P  pCnt  A ) ) 
||  A )
484, 6, 46, 47syl12anc 1180 . . . . . 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 12901 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  A  =/=  0 ) )  -> 
( P  pCnt  A
)  e.  NN0 )
504, 6, 46, 49syl12anc 1180 . . . . . . . 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 12921 . . . . . . . 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 1182 . . . . . . 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 201 . . . . . 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 12761 . . . . . . . . . 10  |-  ( P  e.  Prime  ->  P  e.  NN )
554, 54syl 15 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  ->  P  e.  NN )
5655, 50nnexpcld 11266 . . . . . . . 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 10116 . . . . . . 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 12722 . . . . . . 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 1182 . . . . . 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 660 . . . . 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 12921 . . . . . 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 1182 . . . . 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 223 . . . 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 12903 . . . . . 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 10019 . . . . 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 8961 . . . 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 888 . . 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 629 . 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 12703 . . . . . 6  |-  ( A  e.  ZZ  ->  ( A  gcd  0 )  =  ( abs `  A
) )
705, 69syl 15 . . . . 5  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  gcd  0 )  =  ( abs `  A
) )
7170oveq2d 5874 . . . 4  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  ( P  pCnt  ( A  gcd  0 ) )  =  ( P  pCnt  ( abs `  A ) ) )
72 pcabs 12927 . . . . . 6  |-  ( ( P  e.  Prime  /\  A  e.  QQ )  ->  ( P  pCnt  ( abs `  A
) )  =  ( P  pCnt  A )
)
7320, 72sylan2 460 . . . . 5  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  ( P  pCnt  ( abs `  A
) )  =  ( P  pCnt  A )
)
74733adant3 975 . . . 4  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  ( P  pCnt  ( abs `  A
) )  =  ( P  pCnt  A )
)
7571, 74eqtrd 2315 . . 3  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  ( P  pCnt  ( A  gcd  0 ) )  =  ( P  pCnt  A
) )
7675adantr 451 . 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 2521 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446    e/ wnel 2447   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   RRcr 8736   0cc0 8737    +oocpnf 8864    -oocmnf 8865   RR*cxr 8866    < clt 8867    <_ cle 8868   NNcn 9746   NN0cn0 9965   ZZcz 10024   QQcq 10316   ^cexp 11104   abscabs 11719    || cdivides 12531    gcd cgcd 12685   Primecprime 12758    pCnt cpc 12889
This theorem is referenced by:  pcgcd  12930
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317  df-rp 10355  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-dvds 12532  df-gcd 12686  df-prm 12759  df-pc 12890
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