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Theorem pcgcd1 12270
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 5859 . . . 4  |-  ( B  =  0  ->  ( A  gcd  B )  =  ( A  gcd  0
) )
21oveq2d 5867 . . 3  |-  ( B  =  0  ->  ( P  pCnt  ( A  gcd  B ) )  =  ( P  pCnt  ( A  gcd  0 ) ) )
3 simp2 993 . . . . . . 7  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  A  e.  ZZ )
4 gcdid0 11924 . . . . . . 7  |-  ( A  e.  ZZ  ->  ( A  gcd  0 )  =  ( abs `  A
) )
53, 4syl 14 . . . . . 6  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  gcd  0 )  =  ( abs `  A
) )
65oveq2d 5867 . . . . 5  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  ( P  pCnt  ( A  gcd  0 ) )  =  ( P  pCnt  ( abs `  A ) ) )
7 zq 9574 . . . . . . 7  |-  ( A  e.  ZZ  ->  A  e.  QQ )
8 pcabs 12268 . . . . . . 7  |-  ( ( P  e.  Prime  /\  A  e.  QQ )  ->  ( P  pCnt  ( abs `  A
) )  =  ( P  pCnt  A )
)
97, 8sylan2 284 . . . . . 6  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  ( P  pCnt  ( abs `  A
) )  =  ( P  pCnt  A )
)
1093adant3 1012 . . . . 5  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  ( P  pCnt  ( abs `  A
) )  =  ( P  pCnt  A )
)
116, 10eqtrd 2203 . . . 4  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  ( P  pCnt  ( A  gcd  0 ) )  =  ( P  pCnt  A
) )
1211adantr 274 . . 3  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( P  pCnt  A
)  <_  ( P  pCnt  B ) )  -> 
( P  pCnt  ( A  gcd  0 ) )  =  ( P  pCnt  A ) )
132, 12sylan9eqr 2225 . 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 ) )
14 simpl1 995 . . . . 5  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  ->  P  e.  Prime )
153adantr 274 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  ->  A  e.  ZZ )
16 simpl3 997 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  ->  B  e.  ZZ )
17 simprr 527 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  ->  B  =/=  0 )
18 simpr 109 . . . . . . . . 9  |-  ( ( A  =  0  /\  B  =  0 )  ->  B  =  0 )
1918necon3ai 2389 . . . . . . . 8  |-  ( B  =/=  0  ->  -.  ( A  =  0  /\  B  =  0
) )
2017, 19syl 14 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  ->  -.  ( A  =  0  /\  B  =  0 ) )
21 gcdn0cl 11906 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  -.  ( A  =  0  /\  B  =  0 ) )  ->  ( A  gcd  B )  e.  NN )
2215, 16, 20, 21syl21anc 1232 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  -> 
( A  gcd  B
)  e.  NN )
2322nnzd 9322 . . . . 5  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  -> 
( A  gcd  B
)  e.  ZZ )
24 gcddvds 11907 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( A  gcd  B )  ||  A  /\  ( A  gcd  B ) 
||  B ) )
2515, 16, 24syl2anc 409 . . . . . 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 ) )
2625simpld 111 . . . . 5  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  -> 
( A  gcd  B
)  ||  A )
27 pcdvdstr 12269 . . . . 5  |-  ( ( P  e.  Prime  /\  (
( A  gcd  B
)  e.  ZZ  /\  A  e.  ZZ  /\  ( A  gcd  B )  ||  A ) )  -> 
( P  pCnt  ( A  gcd  B ) )  <_  ( P  pCnt  A ) )
2814, 23, 15, 26, 27syl13anc 1235 . . . 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 ) )
2915, 7syl 14 . . . . . . . . . 10  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  ->  A  e.  QQ )
30 pcxcl 12254 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  A  e.  QQ )  ->  ( P  pCnt  A )  e. 
RR* )
3114, 29, 30syl2anc 409 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  -> 
( P  pCnt  A
)  e.  RR* )
32 pczcl 12241 . . . . . . . . . . 11  |-  ( ( P  e.  Prime  /\  ( B  e.  ZZ  /\  B  =/=  0 ) )  -> 
( P  pCnt  B
)  e.  NN0 )
3314, 16, 17, 32syl12anc 1231 . . . . . . . . . 10  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  -> 
( P  pCnt  B
)  e.  NN0 )
3433nn0red 9178 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  -> 
( P  pCnt  B
)  e.  RR )
35 pcge0 12255 . . . . . . . . . . 11  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  0  <_  ( P  pCnt  A
) )
3614, 15, 35syl2anc 409 . . . . . . . . . 10  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  -> 
0  <_  ( P  pCnt  A ) )
37 ge0gtmnf 9769 . . . . . . . . . 10  |-  ( ( ( P  pCnt  A
)  e.  RR*  /\  0  <_  ( P  pCnt  A
) )  -> -oo  <  ( P  pCnt  A )
)
3831, 36, 37syl2anc 409 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  -> -oo  <  ( P  pCnt  A ) )
39 simprl 526 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  -> 
( P  pCnt  A
)  <_  ( P  pCnt  B ) )
40 xrre 9766 . . . . . . . . 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 )
4131, 34, 38, 39, 40syl22anc 1234 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  -> 
( P  pCnt  A
)  e.  RR )
42 pnfnre 7950 . . . . . . . . . . . 12  |- +oo  e/  RR
4342neli 2437 . . . . . . . . . . 11  |-  -. +oo  e.  RR
44 pc0 12247 . . . . . . . . . . . . 13  |-  ( P  e.  Prime  ->  ( P 
pCnt  0 )  = +oo )
4514, 44syl 14 . . . . . . . . . . . 12  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  -> 
( P  pCnt  0
)  = +oo )
4645eleq1d 2239 . . . . . . . . . . 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 ) )
4743, 46mtbiri 670 . . . . . . . . . 10  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  ->  -.  ( P  pCnt  0
)  e.  RR )
48 oveq2 5859 . . . . . . . . . . . 12  |-  ( A  =  0  ->  ( P  pCnt  A )  =  ( P  pCnt  0
) )
4948eleq1d 2239 . . . . . . . . . . 11  |-  ( A  =  0  ->  (
( P  pCnt  A
)  e.  RR  <->  ( P  pCnt  0 )  e.  RR ) )
5049notbid 662 . . . . . . . . . 10  |-  ( A  =  0  ->  ( -.  ( P  pCnt  A
)  e.  RR  <->  -.  ( P  pCnt  0 )  e.  RR ) )
5147, 50syl5ibrcom 156 . . . . . . . . 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 ) )
5251necon2ad 2397 . . . . . . . 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 ) )
5341, 52mpd 13 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  ->  A  =/=  0 )
54 pczdvds 12256 . . . . . . 7  |-  ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  A  =/=  0 ) )  -> 
( P ^ ( P  pCnt  A ) ) 
||  A )
5514, 15, 53, 54syl12anc 1231 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  -> 
( P ^ ( P  pCnt  A ) ) 
||  A )
56 pczcl 12241 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  A  =/=  0 ) )  -> 
( P  pCnt  A
)  e.  NN0 )
5714, 15, 53, 56syl12anc 1231 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  -> 
( P  pCnt  A
)  e.  NN0 )
58 pcdvdsb 12262 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  B  e.  ZZ  /\  ( P 
pCnt  A )  e.  NN0 )  ->  ( ( P 
pCnt  A )  <_  ( P  pCnt  B )  <->  ( P ^ ( P  pCnt  A ) )  ||  B
) )
5914, 16, 57, 58syl3anc 1233 . . . . . . 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
) )
6039, 59mpbid 146 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  -> 
( P ^ ( P  pCnt  A ) ) 
||  B )
61 prmnn 12053 . . . . . . . . . 10  |-  ( P  e.  Prime  ->  P  e.  NN )
6214, 61syl 14 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  ->  P  e.  NN )
6362, 57nnexpcld 10620 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  -> 
( P ^ ( P  pCnt  A ) )  e.  NN )
6463nnzd 9322 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  -> 
( P ^ ( P  pCnt  A ) )  e.  ZZ )
65 dvdsgcd 11956 . . . . . . 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 ) ) )
6664, 15, 16, 65syl3anc 1233 . . . . . 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 ) ) )
6755, 60, 66mp2and 431 . . . . 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 ) )
68 pcdvdsb 12262 . . . . . 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 ) ) )
6914, 23, 57, 68syl3anc 1233 . . . . 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 ) ) )
7067, 69mpbird 166 . . . 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
) ) )
7114, 22pccld 12243 . . . . . 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 )
7271nn0red 9178 . . . . 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 )
7372, 41letri3d 8024 . . . 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 ) ) ) ) )
7428, 70, 73mpbir2and 939 . . 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 ) )
7574anassrs 398 . 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 ) )
76 simpl3 997 . . . 4  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( P  pCnt  A
)  <_  ( P  pCnt  B ) )  ->  B  e.  ZZ )
77 0zd 9213 . . . 4  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( P  pCnt  A
)  <_  ( P  pCnt  B ) )  -> 
0  e.  ZZ )
78 zdceq 9276 . . . 4  |-  ( ( B  e.  ZZ  /\  0  e.  ZZ )  -> DECID  B  =  0 )
7976, 77, 78syl2anc 409 . . 3  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( P  pCnt  A
)  <_  ( P  pCnt  B ) )  -> DECID  B  =  0 )
80 dcne 2351 . . 3  |-  (DECID  B  =  0  <->  ( B  =  0  \/  B  =/=  0 ) )
8179, 80sylib 121 . 2  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( P  pCnt  A
)  <_  ( P  pCnt  B ) )  -> 
( B  =  0  \/  B  =/=  0
) )
8213, 75, 81mpjaodan 793 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    /\ wa 103    <-> wb 104    \/ wo 703  DECID wdc 829    /\ w3a 973    = wceq 1348    e. wcel 2141    =/= wne 2340   class class class wbr 3987   ` cfv 5196  (class class class)co 5851   RRcr 7762   0cc0 7763   +oocpnf 7940   -oocmnf 7941   RR*cxr 7942    < clt 7943    <_ cle 7944   NNcn 8867   NN0cn0 9124   ZZcz 9201   QQcq 9567   ^cexp 10464   abscabs 10950    || cdvds 11738    gcd cgcd 11886   Primecprime 12050    pCnt cpc 12227
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-iinf 4570  ax-cnex 7854  ax-resscn 7855  ax-1cn 7856  ax-1re 7857  ax-icn 7858  ax-addcl 7859  ax-addrcl 7860  ax-mulcl 7861  ax-mulrcl 7862  ax-addcom 7863  ax-mulcom 7864  ax-addass 7865  ax-mulass 7866  ax-distr 7867  ax-i2m1 7868  ax-0lt1 7869  ax-1rid 7870  ax-0id 7871  ax-rnegex 7872  ax-precex 7873  ax-cnre 7874  ax-pre-ltirr 7875  ax-pre-ltwlin 7876  ax-pre-lttrn 7877  ax-pre-apti 7878  ax-pre-ltadd 7879  ax-pre-mulgt0 7880  ax-pre-mulext 7881  ax-arch 7882  ax-caucvg 7883
This theorem depends on definitions:  df-bi 116  df-stab 826  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3526  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-tr 4086  df-id 4276  df-po 4279  df-iso 4280  df-iord 4349  df-on 4351  df-ilim 4352  df-suc 4354  df-iom 4573  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-isom 5205  df-riota 5807  df-ov 5854  df-oprab 5855  df-mpo 5856  df-1st 6117  df-2nd 6118  df-recs 6282  df-frec 6368  df-1o 6393  df-2o 6394  df-er 6510  df-en 6716  df-sup 6958  df-inf 6959  df-pnf 7945  df-mnf 7946  df-xr 7947  df-ltxr 7948  df-le 7949  df-sub 8081  df-neg 8082  df-reap 8483  df-ap 8490  df-div 8579  df-inn 8868  df-2 8926  df-3 8927  df-4 8928  df-n0 9125  df-z 9202  df-uz 9477  df-q 9568  df-rp 9600  df-fz 9955  df-fzo 10088  df-fl 10215  df-mod 10268  df-seqfrec 10391  df-exp 10465  df-cj 10795  df-re 10796  df-im 10797  df-rsqrt 10951  df-abs 10952  df-dvds 11739  df-gcd 11887  df-prm 12051  df-pc 12228
This theorem is referenced by:  pcgcd  12271
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