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Theorem pcdiv 14585
Description: Division property of the prime power function. (Contributed by Mario Carneiro, 1-Mar-2014.)
Assertion
Ref Expression
pcdiv  |-  ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  A  =/=  0 )  /\  B  e.  NN )  ->  ( P  pCnt  ( A  /  B ) )  =  ( ( P  pCnt  A )  -  ( P 
pCnt  B ) ) )

Proof of Theorem pcdiv
Dummy variables  x  n  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp1 997 . . 3  |-  ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  A  =/=  0 )  /\  B  e.  NN )  ->  P  e.  Prime )
2 simp2l 1023 . . . 4  |-  ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  A  =/=  0 )  /\  B  e.  NN )  ->  A  e.  ZZ )
3 simp3 999 . . . 4  |-  ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  A  =/=  0 )  /\  B  e.  NN )  ->  B  e.  NN )
4 znq 11231 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( A  /  B
)  e.  QQ )
52, 3, 4syl2anc 659 . . 3  |-  ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  A  =/=  0 )  /\  B  e.  NN )  ->  ( A  /  B )  e.  QQ )
62zcnd 11009 . . . 4  |-  ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  A  =/=  0 )  /\  B  e.  NN )  ->  A  e.  CC )
73nncnd 10592 . . . 4  |-  ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  A  =/=  0 )  /\  B  e.  NN )  ->  B  e.  CC )
8 simp2r 1024 . . . 4  |-  ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  A  =/=  0 )  /\  B  e.  NN )  ->  A  =/=  0 )
93nnne0d 10621 . . . 4  |-  ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  A  =/=  0 )  /\  B  e.  NN )  ->  B  =/=  0 )
106, 7, 8, 9divne0d 10377 . . 3  |-  ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  A  =/=  0 )  /\  B  e.  NN )  ->  ( A  /  B )  =/=  0 )
11 eqid 2402 . . . 4  |-  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  x } ,  RR ,  <  )  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  x } ,  RR ,  <  )
12 eqid 2402 . . . 4  |-  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  y } ,  RR ,  <  )  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  y } ,  RR ,  <  )
1311, 12pcval 14577 . . 3  |-  ( ( P  e.  Prime  /\  (
( A  /  B
)  e.  QQ  /\  ( A  /  B
)  =/=  0 ) )  ->  ( P  pCnt  ( A  /  B
) )  =  ( iota z E. x  e.  ZZ  E. y  e.  NN  ( ( A  /  B )  =  ( x  /  y
)  /\  z  =  ( sup ( { n  e.  NN0  |  ( P ^ n )  ||  x } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  y } ,  RR ,  <  )
) ) ) )
141, 5, 10, 13syl12anc 1228 . 2  |-  ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  A  =/=  0 )  /\  B  e.  NN )  ->  ( P  pCnt  ( A  /  B ) )  =  ( iota z E. x  e.  ZZ  E. y  e.  NN  (
( A  /  B
)  =  ( x  /  y )  /\  z  =  ( sup ( { n  e.  NN0  |  ( P ^ n
)  ||  x } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( P ^ n )  ||  y } ,  RR ,  <  ) ) ) ) )
15 eqid 2402 . . . . . . . 8  |-  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  A } ,  RR ,  <  )  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  A } ,  RR ,  <  )
1615pczpre 14580 . . . . . . 7  |-  ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  A  =/=  0 ) )  -> 
( P  pCnt  A
)  =  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  A } ,  RR ,  <  )
)
17163adant3 1017 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  A  =/=  0 )  /\  B  e.  NN )  ->  ( P  pCnt  A )  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  A } ,  RR ,  <  ) )
18 nnz 10927 . . . . . . . . 9  |-  ( B  e.  NN  ->  B  e.  ZZ )
19 nnne0 10609 . . . . . . . . 9  |-  ( B  e.  NN  ->  B  =/=  0 )
2018, 19jca 530 . . . . . . . 8  |-  ( B  e.  NN  ->  ( B  e.  ZZ  /\  B  =/=  0 ) )
21 eqid 2402 . . . . . . . . 9  |-  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  B } ,  RR ,  <  )  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  B } ,  RR ,  <  )
2221pczpre 14580 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  ( B  e.  ZZ  /\  B  =/=  0 ) )  -> 
( P  pCnt  B
)  =  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  B } ,  RR ,  <  )
)
2320, 22sylan2 472 . . . . . . 7  |-  ( ( P  e.  Prime  /\  B  e.  NN )  ->  ( P  pCnt  B )  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  B } ,  RR ,  <  ) )
24233adant2 1016 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  A  =/=  0 )  /\  B  e.  NN )  ->  ( P  pCnt  B )  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  B } ,  RR ,  <  ) )
2517, 24oveq12d 6296 . . . . 5  |-  ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  A  =/=  0 )  /\  B  e.  NN )  ->  (
( P  pCnt  A
)  -  ( P 
pCnt  B ) )  =  ( sup ( { n  e.  NN0  | 
( P ^ n
)  ||  A } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( P ^ n )  ||  B } ,  RR ,  <  ) ) )
26 eqid 2402 . . . . 5  |-  ( A  /  B )  =  ( A  /  B
)
2725, 26jctil 535 . . . 4  |-  ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  A  =/=  0 )  /\  B  e.  NN )  ->  (
( A  /  B
)  =  ( A  /  B )  /\  ( ( P  pCnt  A )  -  ( P 
pCnt  B ) )  =  ( sup ( { n  e.  NN0  | 
( P ^ n
)  ||  A } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( P ^ n )  ||  B } ,  RR ,  <  ) ) ) )
28 oveq1 6285 . . . . . . 7  |-  ( x  =  A  ->  (
x  /  y )  =  ( A  / 
y ) )
2928eqeq2d 2416 . . . . . 6  |-  ( x  =  A  ->  (
( A  /  B
)  =  ( x  /  y )  <->  ( A  /  B )  =  ( A  /  y ) ) )
30 breq2 4399 . . . . . . . . . 10  |-  ( x  =  A  ->  (
( P ^ n
)  ||  x  <->  ( P ^ n )  ||  A ) )
3130rabbidv 3051 . . . . . . . . 9  |-  ( x  =  A  ->  { n  e.  NN0  |  ( P ^ n )  ||  x }  =  {
n  e.  NN0  | 
( P ^ n
)  ||  A }
)
3231supeq1d 7939 . . . . . . . 8  |-  ( x  =  A  ->  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  x } ,  RR ,  <  )  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  A } ,  RR ,  <  ) )
3332oveq1d 6293 . . . . . . 7  |-  ( x  =  A  ->  ( sup ( { n  e. 
NN0  |  ( P ^ n )  ||  x } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  y } ,  RR ,  <  )
)  =  ( sup ( { n  e. 
NN0  |  ( P ^ n )  ||  A } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  y } ,  RR ,  <  )
) )
3433eqeq2d 2416 . . . . . 6  |-  ( x  =  A  ->  (
( ( P  pCnt  A )  -  ( P 
pCnt  B ) )  =  ( sup ( { n  e.  NN0  | 
( P ^ n
)  ||  x } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( P ^ n )  ||  y } ,  RR ,  <  ) )  <->  ( ( P  pCnt  A )  -  ( P  pCnt  B ) )  =  ( sup ( { n  e. 
NN0  |  ( P ^ n )  ||  A } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  y } ,  RR ,  <  )
) ) )
3529, 34anbi12d 709 . . . . 5  |-  ( x  =  A  ->  (
( ( A  /  B )  =  ( x  /  y )  /\  ( ( P 
pCnt  A )  -  ( P  pCnt  B ) )  =  ( sup ( { n  e.  NN0  |  ( P ^ n
)  ||  x } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( P ^ n )  ||  y } ,  RR ,  <  ) ) )  <->  ( ( A  /  B )  =  ( A  /  y
)  /\  ( ( P  pCnt  A )  -  ( P  pCnt  B ) )  =  ( sup ( { n  e. 
NN0  |  ( P ^ n )  ||  A } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  y } ,  RR ,  <  )
) ) ) )
36 oveq2 6286 . . . . . . 7  |-  ( y  =  B  ->  ( A  /  y )  =  ( A  /  B
) )
3736eqeq2d 2416 . . . . . 6  |-  ( y  =  B  ->  (
( A  /  B
)  =  ( A  /  y )  <->  ( A  /  B )  =  ( A  /  B ) ) )
38 breq2 4399 . . . . . . . . . 10  |-  ( y  =  B  ->  (
( P ^ n
)  ||  y  <->  ( P ^ n )  ||  B ) )
3938rabbidv 3051 . . . . . . . . 9  |-  ( y  =  B  ->  { n  e.  NN0  |  ( P ^ n )  ||  y }  =  {
n  e.  NN0  | 
( P ^ n
)  ||  B }
)
4039supeq1d 7939 . . . . . . . 8  |-  ( y  =  B  ->  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  y } ,  RR ,  <  )  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  B } ,  RR ,  <  ) )
4140oveq2d 6294 . . . . . . 7  |-  ( y  =  B  ->  ( sup ( { n  e. 
NN0  |  ( P ^ n )  ||  A } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  y } ,  RR ,  <  )
)  =  ( sup ( { n  e. 
NN0  |  ( P ^ n )  ||  A } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  B } ,  RR ,  <  )
) )
4241eqeq2d 2416 . . . . . 6  |-  ( y  =  B  ->  (
( ( P  pCnt  A )  -  ( P 
pCnt  B ) )  =  ( sup ( { n  e.  NN0  | 
( P ^ n
)  ||  A } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( P ^ n )  ||  y } ,  RR ,  <  ) )  <->  ( ( P  pCnt  A )  -  ( P  pCnt  B ) )  =  ( sup ( { n  e. 
NN0  |  ( P ^ n )  ||  A } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  B } ,  RR ,  <  )
) ) )
4337, 42anbi12d 709 . . . . 5  |-  ( y  =  B  ->  (
( ( A  /  B )  =  ( A  /  y )  /\  ( ( P 
pCnt  A )  -  ( P  pCnt  B ) )  =  ( sup ( { n  e.  NN0  |  ( P ^ n
)  ||  A } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( P ^ n )  ||  y } ,  RR ,  <  ) ) )  <->  ( ( A  /  B )  =  ( A  /  B
)  /\  ( ( P  pCnt  A )  -  ( P  pCnt  B ) )  =  ( sup ( { n  e. 
NN0  |  ( P ^ n )  ||  A } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  B } ,  RR ,  <  )
) ) ) )
4435, 43rspc2ev 3171 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  NN  /\  (
( A  /  B
)  =  ( A  /  B )  /\  ( ( P  pCnt  A )  -  ( P 
pCnt  B ) )  =  ( sup ( { n  e.  NN0  | 
( P ^ n
)  ||  A } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( P ^ n )  ||  B } ,  RR ,  <  ) ) ) )  ->  E. x  e.  ZZ  E. y  e.  NN  (
( A  /  B
)  =  ( x  /  y )  /\  ( ( P  pCnt  A )  -  ( P 
pCnt  B ) )  =  ( sup ( { n  e.  NN0  | 
( P ^ n
)  ||  x } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( P ^ n )  ||  y } ,  RR ,  <  ) ) ) )
452, 3, 27, 44syl3anc 1230 . . 3  |-  ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  A  =/=  0 )  /\  B  e.  NN )  ->  E. x  e.  ZZ  E. y  e.  NN  ( ( A  /  B )  =  ( x  /  y
)  /\  ( ( P  pCnt  A )  -  ( P  pCnt  B ) )  =  ( sup ( { n  e. 
NN0  |  ( P ^ n )  ||  x } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  y } ,  RR ,  <  )
) ) )
46 ovex 6306 . . . 4  |-  ( ( P  pCnt  A )  -  ( P  pCnt  B ) )  e.  _V
4711, 12pceu 14579 . . . . 5  |-  ( ( P  e.  Prime  /\  (
( A  /  B
)  e.  QQ  /\  ( A  /  B
)  =/=  0 ) )  ->  E! z E. x  e.  ZZ  E. y  e.  NN  (
( A  /  B
)  =  ( x  /  y )  /\  z  =  ( sup ( { n  e.  NN0  |  ( P ^ n
)  ||  x } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( P ^ n )  ||  y } ,  RR ,  <  ) ) ) )
481, 5, 10, 47syl12anc 1228 . . . 4  |-  ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  A  =/=  0 )  /\  B  e.  NN )  ->  E! z E. x  e.  ZZ  E. y  e.  NN  (
( A  /  B
)  =  ( x  /  y )  /\  z  =  ( sup ( { n  e.  NN0  |  ( P ^ n
)  ||  x } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( P ^ n )  ||  y } ,  RR ,  <  ) ) ) )
49 eqeq1 2406 . . . . . . 7  |-  ( z  =  ( ( P 
pCnt  A )  -  ( P  pCnt  B ) )  ->  ( z  =  ( sup ( { n  e.  NN0  | 
( P ^ n
)  ||  x } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( P ^ n )  ||  y } ,  RR ,  <  ) )  <->  ( ( P  pCnt  A )  -  ( P  pCnt  B ) )  =  ( sup ( { n  e. 
NN0  |  ( P ^ n )  ||  x } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  y } ,  RR ,  <  )
) ) )
5049anbi2d 702 . . . . . 6  |-  ( z  =  ( ( P 
pCnt  A )  -  ( P  pCnt  B ) )  ->  ( ( ( A  /  B )  =  ( x  / 
y )  /\  z  =  ( sup ( { n  e.  NN0  |  ( P ^ n
)  ||  x } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( P ^ n )  ||  y } ,  RR ,  <  ) ) )  <->  ( ( A  /  B )  =  ( x  /  y
)  /\  ( ( P  pCnt  A )  -  ( P  pCnt  B ) )  =  ( sup ( { n  e. 
NN0  |  ( P ^ n )  ||  x } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  y } ,  RR ,  <  )
) ) ) )
51502rexbidv 2925 . . . . 5  |-  ( z  =  ( ( P 
pCnt  A )  -  ( P  pCnt  B ) )  ->  ( E. x  e.  ZZ  E. y  e.  NN  ( ( A  /  B )  =  ( x  /  y
)  /\  z  =  ( sup ( { n  e.  NN0  |  ( P ^ n )  ||  x } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  y } ,  RR ,  <  )
) )  <->  E. x  e.  ZZ  E. y  e.  NN  ( ( A  /  B )  =  ( x  /  y
)  /\  ( ( P  pCnt  A )  -  ( P  pCnt  B ) )  =  ( sup ( { n  e. 
NN0  |  ( P ^ n )  ||  x } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  y } ,  RR ,  <  )
) ) ) )
5251iota2 5559 . . . 4  |-  ( ( ( ( P  pCnt  A )  -  ( P 
pCnt  B ) )  e. 
_V  /\  E! z E. x  e.  ZZ  E. y  e.  NN  (
( A  /  B
)  =  ( x  /  y )  /\  z  =  ( sup ( { n  e.  NN0  |  ( P ^ n
)  ||  x } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( P ^ n )  ||  y } ,  RR ,  <  ) ) ) )  ->  ( E. x  e.  ZZ  E. y  e.  NN  ( ( A  /  B )  =  ( x  /  y
)  /\  ( ( P  pCnt  A )  -  ( P  pCnt  B ) )  =  ( sup ( { n  e. 
NN0  |  ( P ^ n )  ||  x } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  y } ,  RR ,  <  )
) )  <->  ( iota z E. x  e.  ZZ  E. y  e.  NN  (
( A  /  B
)  =  ( x  /  y )  /\  z  =  ( sup ( { n  e.  NN0  |  ( P ^ n
)  ||  x } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( P ^ n )  ||  y } ,  RR ,  <  ) ) ) )  =  ( ( P 
pCnt  A )  -  ( P  pCnt  B ) ) ) )
5346, 48, 52sylancr 661 . . 3  |-  ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  A  =/=  0 )  /\  B  e.  NN )  ->  ( E. x  e.  ZZ  E. y  e.  NN  (
( A  /  B
)  =  ( x  /  y )  /\  ( ( P  pCnt  A )  -  ( P 
pCnt  B ) )  =  ( sup ( { n  e.  NN0  | 
( P ^ n
)  ||  x } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( P ^ n )  ||  y } ,  RR ,  <  ) ) )  <->  ( iota z E. x  e.  ZZ  E. y  e.  NN  (
( A  /  B
)  =  ( x  /  y )  /\  z  =  ( sup ( { n  e.  NN0  |  ( P ^ n
)  ||  x } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( P ^ n )  ||  y } ,  RR ,  <  ) ) ) )  =  ( ( P 
pCnt  A )  -  ( P  pCnt  B ) ) ) )
5445, 53mpbid 210 . 2  |-  ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  A  =/=  0 )  /\  B  e.  NN )  ->  ( iota z E. x  e.  ZZ  E. y  e.  NN  ( ( A  /  B )  =  ( x  /  y
)  /\  z  =  ( sup ( { n  e.  NN0  |  ( P ^ n )  ||  x } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  y } ,  RR ,  <  )
) ) )  =  ( ( P  pCnt  A )  -  ( P 
pCnt  B ) ) )
5514, 54eqtrd 2443 1  |-  ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  A  =/=  0 )  /\  B  e.  NN )  ->  ( P  pCnt  ( A  /  B ) )  =  ( ( P  pCnt  A )  -  ( P 
pCnt  B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842   E!weu 2238    =/= wne 2598   E.wrex 2755   {crab 2758   _Vcvv 3059   class class class wbr 4395   iotacio 5531  (class class class)co 6278   supcsup 7934   RRcr 9521   0cc0 9522    < clt 9658    - cmin 9841    / cdiv 10247   NNcn 10576   NN0cn0 10836   ZZcz 10905   QQcq 11227   ^cexp 12210    || cdvds 14195   Primecprime 14426    pCnt cpc 14569
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599  ax-pre-sup 9600
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-1st 6784  df-2nd 6785  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-2o 7168  df-oadd 7171  df-er 7348  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-sup 7935  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-div 10248  df-nn 10577  df-2 10635  df-3 10636  df-n0 10837  df-z 10906  df-uz 11128  df-q 11228  df-rp 11266  df-fl 11966  df-mod 12035  df-seq 12152  df-exp 12211  df-cj 13081  df-re 13082  df-im 13083  df-sqrt 13217  df-abs 13218  df-dvds 14196  df-gcd 14354  df-prm 14427  df-pc 14570
This theorem is referenced by:  pcqmul  14586  pcqcl  14589  pcid  14605  pcneg  14606  pc2dvds  14611  pcz  14613  pcaddlem  14616  pcadd  14617  pcmpt2  14621  pcbc  14628  sylow1lem1  16942  chtublem  23867
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