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Theorem pcdiv 12921
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 955 . . 3  |-  ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  A  =/=  0 )  /\  B  e.  NN )  ->  P  e.  Prime )
2 simp2l 981 . . . 4  |-  ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  A  =/=  0 )  /\  B  e.  NN )  ->  A  e.  ZZ )
3 simp3 957 . . . 4  |-  ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  A  =/=  0 )  /\  B  e.  NN )  ->  B  e.  NN )
4 znq 10336 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( A  /  B
)  e.  QQ )
52, 3, 4syl2anc 642 . . 3  |-  ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  A  =/=  0 )  /\  B  e.  NN )  ->  ( A  /  B )  e.  QQ )
62zcnd 10134 . . . 4  |-  ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  A  =/=  0 )  /\  B  e.  NN )  ->  A  e.  CC )
73nncnd 9778 . . . 4  |-  ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  A  =/=  0 )  /\  B  e.  NN )  ->  B  e.  CC )
8 simp2r 982 . . . 4  |-  ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  A  =/=  0 )  /\  B  e.  NN )  ->  A  =/=  0 )
93nnne0d 9806 . . . 4  |-  ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  A  =/=  0 )  /\  B  e.  NN )  ->  B  =/=  0 )
106, 7, 8, 9divne0d 9568 . . 3  |-  ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  A  =/=  0 )  /\  B  e.  NN )  ->  ( A  /  B )  =/=  0 )
11 eqid 2296 . . . 4  |-  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  x } ,  RR ,  <  )  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  x } ,  RR ,  <  )
12 eqid 2296 . . . 4  |-  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  y } ,  RR ,  <  )  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  y } ,  RR ,  <  )
1311, 12pcval 12913 . . 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 1180 . 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 2296 . . . . . . . 8  |-  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  A } ,  RR ,  <  )  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  A } ,  RR ,  <  )
1615pczpre 12916 . . . . . . 7  |-  ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  A  =/=  0 ) )  -> 
( P  pCnt  A
)  =  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  A } ,  RR ,  <  )
)
17163adant3 975 . . . . . 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 10061 . . . . . . . . 9  |-  ( B  e.  NN  ->  B  e.  ZZ )
19 nnne0 9794 . . . . . . . . 9  |-  ( B  e.  NN  ->  B  =/=  0 )
2018, 19jca 518 . . . . . . . 8  |-  ( B  e.  NN  ->  ( B  e.  ZZ  /\  B  =/=  0 ) )
21 eqid 2296 . . . . . . . . 9  |-  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  B } ,  RR ,  <  )  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  B } ,  RR ,  <  )
2221pczpre 12916 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  ( B  e.  ZZ  /\  B  =/=  0 ) )  -> 
( P  pCnt  B
)  =  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  B } ,  RR ,  <  )
)
2320, 22sylan2 460 . . . . . . 7  |-  ( ( P  e.  Prime  /\  B  e.  NN )  ->  ( P  pCnt  B )  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  B } ,  RR ,  <  ) )
24233adant2 974 . . . . . 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 5892 . . . . 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 2296 . . . . 5  |-  ( A  /  B )  =  ( A  /  B
)
2725, 26jctil 523 . . . 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 5881 . . . . . . 7  |-  ( x  =  A  ->  (
x  /  y )  =  ( A  / 
y ) )
2928eqeq2d 2307 . . . . . 6  |-  ( x  =  A  ->  (
( A  /  B
)  =  ( x  /  y )  <->  ( A  /  B )  =  ( A  /  y ) ) )
30 breq2 4043 . . . . . . . . . 10  |-  ( x  =  A  ->  (
( P ^ n
)  ||  x  <->  ( P ^ n )  ||  A ) )
3130rabbidv 2793 . . . . . . . . 9  |-  ( x  =  A  ->  { n  e.  NN0  |  ( P ^ n )  ||  x }  =  {
n  e.  NN0  | 
( P ^ n
)  ||  A }
)
3231supeq1d 7215 . . . . . . . 8  |-  ( x  =  A  ->  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  x } ,  RR ,  <  )  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  A } ,  RR ,  <  ) )
3332oveq1d 5889 . . . . . . 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 2307 . . . . . 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 691 . . . . 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 5882 . . . . . . 7  |-  ( y  =  B  ->  ( A  /  y )  =  ( A  /  B
) )
3736eqeq2d 2307 . . . . . 6  |-  ( y  =  B  ->  (
( A  /  B
)  =  ( A  /  y )  <->  ( A  /  B )  =  ( A  /  B ) ) )
38 breq2 4043 . . . . . . . . . 10  |-  ( y  =  B  ->  (
( P ^ n
)  ||  y  <->  ( P ^ n )  ||  B ) )
3938rabbidv 2793 . . . . . . . . 9  |-  ( y  =  B  ->  { n  e.  NN0  |  ( P ^ n )  ||  y }  =  {
n  e.  NN0  | 
( P ^ n
)  ||  B }
)
4039supeq1d 7215 . . . . . . . 8  |-  ( y  =  B  ->  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  y } ,  RR ,  <  )  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  B } ,  RR ,  <  ) )
4140oveq2d 5890 . . . . . . 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 2307 . . . . . 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 691 . . . . 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 2905 . . . 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 1182 . . 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 5899 . . . 4  |-  ( ( P  pCnt  A )  -  ( P  pCnt  B ) )  e.  _V
4711, 12pceu 12915 . . . . 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 1180 . . . 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 2302 . . . . . . 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 684 . . . . . 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 2599 . . . . 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 5261 . . . 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 644 . . 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 201 . 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 2328 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 set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   E!weu 2156    =/= wne 2459   E.wrex 2557   {crab 2560   _Vcvv 2801   class class class wbr 4039   iotacio 5233  (class class class)co 5874   supcsup 7209   RRcr 8752   0cc0 8753    < clt 8883    - cmin 9053    / cdiv 9439   NNcn 9762   NN0cn0 9981   ZZcz 10040   QQcq 10332   ^cexp 11120    || cdivides 12547   Primecprime 12774    pCnt cpc 12905
This theorem is referenced by:  pcqmul  12922  pcqcl  12925  pcid  12941  pcneg  12942  pc2dvds  12947  pcz  12949  pcaddlem  12952  pcadd  12953  pcmpt2  12957  pcbc  12964  sylow1lem1  14925  chtublem  20466
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-q 10333  df-rp 10371  df-fl 10941  df-mod 10990  df-seq 11063  df-exp 11121  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-dvds 12548  df-gcd 12702  df-prm 12775  df-pc 12906
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