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Theorem pcdiv 12905
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 10320 . . . 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 10118 . . . 4  |-  ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  A  =/=  0 )  /\  B  e.  NN )  ->  A  e.  CC )
73nncnd 9762 . . . 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 9790 . . . 4  |-  ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  A  =/=  0 )  /\  B  e.  NN )  ->  B  =/=  0 )
106, 7, 8, 9divne0d 9552 . . 3  |-  ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  A  =/=  0 )  /\  B  e.  NN )  ->  ( A  /  B )  =/=  0 )
11 eqid 2283 . . . 4  |-  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  x } ,  RR ,  <  )  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  x } ,  RR ,  <  )
12 eqid 2283 . . . 4  |-  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  y } ,  RR ,  <  )  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  y } ,  RR ,  <  )
1311, 12pcval 12897 . . 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 2283 . . . . . . . 8  |-  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  A } ,  RR ,  <  )  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  A } ,  RR ,  <  )
1615pczpre 12900 . . . . . . 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 10045 . . . . . . . . 9  |-  ( B  e.  NN  ->  B  e.  ZZ )
19 nnne0 9778 . . . . . . . . 9  |-  ( B  e.  NN  ->  B  =/=  0 )
2018, 19jca 518 . . . . . . . 8  |-  ( B  e.  NN  ->  ( B  e.  ZZ  /\  B  =/=  0 ) )
21 eqid 2283 . . . . . . . . 9  |-  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  B } ,  RR ,  <  )  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  B } ,  RR ,  <  )
2221pczpre 12900 . . . . . . . 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 5876 . . . . 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 2283 . . . . 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 5865 . . . . . . 7  |-  ( x  =  A  ->  (
x  /  y )  =  ( A  / 
y ) )
2928eqeq2d 2294 . . . . . 6  |-  ( x  =  A  ->  (
( A  /  B
)  =  ( x  /  y )  <->  ( A  /  B )  =  ( A  /  y ) ) )
30 breq2 4027 . . . . . . . . . 10  |-  ( x  =  A  ->  (
( P ^ n
)  ||  x  <->  ( P ^ n )  ||  A ) )
3130rabbidv 2780 . . . . . . . . 9  |-  ( x  =  A  ->  { n  e.  NN0  |  ( P ^ n )  ||  x }  =  {
n  e.  NN0  | 
( P ^ n
)  ||  A }
)
3231supeq1d 7199 . . . . . . . 8  |-  ( x  =  A  ->  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  x } ,  RR ,  <  )  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  A } ,  RR ,  <  ) )
3332oveq1d 5873 . . . . . . 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 2294 . . . . . 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 5866 . . . . . . 7  |-  ( y  =  B  ->  ( A  /  y )  =  ( A  /  B
) )
3736eqeq2d 2294 . . . . . 6  |-  ( y  =  B  ->  (
( A  /  B
)  =  ( A  /  y )  <->  ( A  /  B )  =  ( A  /  B ) ) )
38 breq2 4027 . . . . . . . . . 10  |-  ( y  =  B  ->  (
( P ^ n
)  ||  y  <->  ( P ^ n )  ||  B ) )
3938rabbidv 2780 . . . . . . . . 9  |-  ( y  =  B  ->  { n  e.  NN0  |  ( P ^ n )  ||  y }  =  {
n  e.  NN0  | 
( P ^ n
)  ||  B }
)
4039supeq1d 7199 . . . . . . . 8  |-  ( y  =  B  ->  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  y } ,  RR ,  <  )  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  B } ,  RR ,  <  ) )
4140oveq2d 5874 . . . . . . 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 2294 . . . . . 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 2892 . . . 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 5883 . . . 4  |-  ( ( P  pCnt  A )  -  ( P  pCnt  B ) )  e.  _V
4711, 12pceu 12899 . . . . 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 2289 . . . . . . 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 2586 . . . . 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 5245 . . . 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 2315 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 1623    e. wcel 1684   E!weu 2143    =/= wne 2446   E.wrex 2544   {crab 2547   _Vcvv 2788   class class class wbr 4023   iotacio 5217  (class class class)co 5858   supcsup 7193   RRcr 8736   0cc0 8737    < clt 8867    - cmin 9037    / cdiv 9423   NNcn 9746   NN0cn0 9965   ZZcz 10024   QQcq 10316   ^cexp 11104    || cdivides 12531   Primecprime 12758    pCnt cpc 12889
This theorem is referenced by:  pcqmul  12906  pcqcl  12909  pcid  12925  pcneg  12926  pc2dvds  12931  pcz  12933  pcaddlem  12936  pcadd  12937  pcmpt2  12941  pcbc  12948  sylow1lem1  14909  chtublem  20450
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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