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Theorem pcdiv 13155
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 957 . . 3  |-  ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  A  =/=  0 )  /\  B  e.  NN )  ->  P  e.  Prime )
2 simp2l 983 . . . 4  |-  ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  A  =/=  0 )  /\  B  e.  NN )  ->  A  e.  ZZ )
3 simp3 959 . . . 4  |-  ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  A  =/=  0 )  /\  B  e.  NN )  ->  B  e.  NN )
4 znq 10512 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( A  /  B
)  e.  QQ )
52, 3, 4syl2anc 643 . . 3  |-  ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  A  =/=  0 )  /\  B  e.  NN )  ->  ( A  /  B )  e.  QQ )
62zcnd 10310 . . . 4  |-  ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  A  =/=  0 )  /\  B  e.  NN )  ->  A  e.  CC )
73nncnd 9950 . . . 4  |-  ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  A  =/=  0 )  /\  B  e.  NN )  ->  B  e.  CC )
8 simp2r 984 . . . 4  |-  ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  A  =/=  0 )  /\  B  e.  NN )  ->  A  =/=  0 )
93nnne0d 9978 . . . 4  |-  ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  A  =/=  0 )  /\  B  e.  NN )  ->  B  =/=  0 )
106, 7, 8, 9divne0d 9740 . . 3  |-  ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  A  =/=  0 )  /\  B  e.  NN )  ->  ( A  /  B )  =/=  0 )
11 eqid 2389 . . . 4  |-  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  x } ,  RR ,  <  )  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  x } ,  RR ,  <  )
12 eqid 2389 . . . 4  |-  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  y } ,  RR ,  <  )  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  y } ,  RR ,  <  )
1311, 12pcval 13147 . . 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 1182 . 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 2389 . . . . . . . 8  |-  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  A } ,  RR ,  <  )  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  A } ,  RR ,  <  )
1615pczpre 13150 . . . . . . 7  |-  ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  A  =/=  0 ) )  -> 
( P  pCnt  A
)  =  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  A } ,  RR ,  <  )
)
17163adant3 977 . . . . . 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 10237 . . . . . . . . 9  |-  ( B  e.  NN  ->  B  e.  ZZ )
19 nnne0 9966 . . . . . . . . 9  |-  ( B  e.  NN  ->  B  =/=  0 )
2018, 19jca 519 . . . . . . . 8  |-  ( B  e.  NN  ->  ( B  e.  ZZ  /\  B  =/=  0 ) )
21 eqid 2389 . . . . . . . . 9  |-  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  B } ,  RR ,  <  )  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  B } ,  RR ,  <  )
2221pczpre 13150 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  ( B  e.  ZZ  /\  B  =/=  0 ) )  -> 
( P  pCnt  B
)  =  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  B } ,  RR ,  <  )
)
2320, 22sylan2 461 . . . . . . 7  |-  ( ( P  e.  Prime  /\  B  e.  NN )  ->  ( P  pCnt  B )  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  B } ,  RR ,  <  ) )
24233adant2 976 . . . . . 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 6040 . . . . 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 2389 . . . . 5  |-  ( A  /  B )  =  ( A  /  B
)
2725, 26jctil 524 . . . 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 6029 . . . . . . 7  |-  ( x  =  A  ->  (
x  /  y )  =  ( A  / 
y ) )
2928eqeq2d 2400 . . . . . 6  |-  ( x  =  A  ->  (
( A  /  B
)  =  ( x  /  y )  <->  ( A  /  B )  =  ( A  /  y ) ) )
30 breq2 4159 . . . . . . . . . 10  |-  ( x  =  A  ->  (
( P ^ n
)  ||  x  <->  ( P ^ n )  ||  A ) )
3130rabbidv 2893 . . . . . . . . 9  |-  ( x  =  A  ->  { n  e.  NN0  |  ( P ^ n )  ||  x }  =  {
n  e.  NN0  | 
( P ^ n
)  ||  A }
)
3231supeq1d 7388 . . . . . . . 8  |-  ( x  =  A  ->  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  x } ,  RR ,  <  )  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  A } ,  RR ,  <  ) )
3332oveq1d 6037 . . . . . . 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 2400 . . . . . 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 692 . . . . 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 6030 . . . . . . 7  |-  ( y  =  B  ->  ( A  /  y )  =  ( A  /  B
) )
3736eqeq2d 2400 . . . . . 6  |-  ( y  =  B  ->  (
( A  /  B
)  =  ( A  /  y )  <->  ( A  /  B )  =  ( A  /  B ) ) )
38 breq2 4159 . . . . . . . . . 10  |-  ( y  =  B  ->  (
( P ^ n
)  ||  y  <->  ( P ^ n )  ||  B ) )
3938rabbidv 2893 . . . . . . . . 9  |-  ( y  =  B  ->  { n  e.  NN0  |  ( P ^ n )  ||  y }  =  {
n  e.  NN0  | 
( P ^ n
)  ||  B }
)
4039supeq1d 7388 . . . . . . . 8  |-  ( y  =  B  ->  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  y } ,  RR ,  <  )  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  B } ,  RR ,  <  ) )
4140oveq2d 6038 . . . . . . 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 2400 . . . . . 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 692 . . . . 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 3005 . . . 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 1184 . . 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 6047 . . . 4  |-  ( ( P  pCnt  A )  -  ( P  pCnt  B ) )  e.  _V
4711, 12pceu 13149 . . . . 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 1182 . . . 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 2395 . . . . . . 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 685 . . . . . 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 2694 . . . . 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 5386 . . . 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 645 . . 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 202 . 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 2421 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 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   E!weu 2240    =/= wne 2552   E.wrex 2652   {crab 2655   _Vcvv 2901   class class class wbr 4155   iotacio 5358  (class class class)co 6022   supcsup 7382   RRcr 8924   0cc0 8925    < clt 9055    - cmin 9225    / cdiv 9611   NNcn 9934   NN0cn0 10155   ZZcz 10216   QQcq 10508   ^cexp 11311    || cdivides 12781   Primecprime 13008    pCnt cpc 13139
This theorem is referenced by:  pcqmul  13156  pcqcl  13159  pcid  13175  pcneg  13176  pc2dvds  13181  pcz  13183  pcaddlem  13186  pcadd  13187  pcmpt2  13191  pcbc  13198  sylow1lem1  15161  chtublem  20864
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 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002  ax-pre-sup 9003
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 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-int 3995  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-riota 6487  df-recs 6571  df-rdg 6606  df-1o 6662  df-2o 6663  df-oadd 6666  df-er 6843  df-en 7048  df-dom 7049  df-sdom 7050  df-fin 7051  df-sup 7383  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-div 9612  df-nn 9935  df-2 9992  df-3 9993  df-n0 10156  df-z 10217  df-uz 10423  df-q 10509  df-rp 10547  df-fl 11131  df-mod 11180  df-seq 11253  df-exp 11312  df-cj 11833  df-re 11834  df-im 11835  df-sqr 11969  df-abs 11970  df-dvds 12782  df-gcd 12936  df-prm 13009  df-pc 13140
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