MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pcdiv Unicode version

Theorem pcdiv 13181
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 10534 . . . 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 10332 . . . 4  |-  ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  A  =/=  0 )  /\  B  e.  NN )  ->  A  e.  CC )
73nncnd 9972 . . . 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 10000 . . . 4  |-  ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  A  =/=  0 )  /\  B  e.  NN )  ->  B  =/=  0 )
106, 7, 8, 9divne0d 9762 . . 3  |-  ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  A  =/=  0 )  /\  B  e.  NN )  ->  ( A  /  B )  =/=  0 )
11 eqid 2404 . . . 4  |-  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  x } ,  RR ,  <  )  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  x } ,  RR ,  <  )
12 eqid 2404 . . . 4  |-  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  y } ,  RR ,  <  )  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  y } ,  RR ,  <  )
1311, 12pcval 13173 . . 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 2404 . . . . . . . 8  |-  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  A } ,  RR ,  <  )  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  A } ,  RR ,  <  )
1615pczpre 13176 . . . . . . 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 10259 . . . . . . . . 9  |-  ( B  e.  NN  ->  B  e.  ZZ )
19 nnne0 9988 . . . . . . . . 9  |-  ( B  e.  NN  ->  B  =/=  0 )
2018, 19jca 519 . . . . . . . 8  |-  ( B  e.  NN  ->  ( B  e.  ZZ  /\  B  =/=  0 ) )
21 eqid 2404 . . . . . . . . 9  |-  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  B } ,  RR ,  <  )  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  B } ,  RR ,  <  )
2221pczpre 13176 . . . . . . . 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 6058 . . . . 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 2404 . . . . 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 6047 . . . . . . 7  |-  ( x  =  A  ->  (
x  /  y )  =  ( A  / 
y ) )
2928eqeq2d 2415 . . . . . 6  |-  ( x  =  A  ->  (
( A  /  B
)  =  ( x  /  y )  <->  ( A  /  B )  =  ( A  /  y ) ) )
30 breq2 4176 . . . . . . . . . 10  |-  ( x  =  A  ->  (
( P ^ n
)  ||  x  <->  ( P ^ n )  ||  A ) )
3130rabbidv 2908 . . . . . . . . 9  |-  ( x  =  A  ->  { n  e.  NN0  |  ( P ^ n )  ||  x }  =  {
n  e.  NN0  | 
( P ^ n
)  ||  A }
)
3231supeq1d 7409 . . . . . . . 8  |-  ( x  =  A  ->  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  x } ,  RR ,  <  )  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  A } ,  RR ,  <  ) )
3332oveq1d 6055 . . . . . . 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 2415 . . . . . 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 6048 . . . . . . 7  |-  ( y  =  B  ->  ( A  /  y )  =  ( A  /  B
) )
3736eqeq2d 2415 . . . . . 6  |-  ( y  =  B  ->  (
( A  /  B
)  =  ( A  /  y )  <->  ( A  /  B )  =  ( A  /  B ) ) )
38 breq2 4176 . . . . . . . . . 10  |-  ( y  =  B  ->  (
( P ^ n
)  ||  y  <->  ( P ^ n )  ||  B ) )
3938rabbidv 2908 . . . . . . . . 9  |-  ( y  =  B  ->  { n  e.  NN0  |  ( P ^ n )  ||  y }  =  {
n  e.  NN0  | 
( P ^ n
)  ||  B }
)
4039supeq1d 7409 . . . . . . . 8  |-  ( y  =  B  ->  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  y } ,  RR ,  <  )  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  B } ,  RR ,  <  ) )
4140oveq2d 6056 . . . . . . 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 2415 . . . . . 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 3020 . . . 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 6065 . . . 4  |-  ( ( P  pCnt  A )  -  ( P  pCnt  B ) )  e.  _V
4711, 12pceu 13175 . . . . 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 2410 . . . . . . 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 2709 . . . . 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 5403 . . . 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 2436 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 1721   E!weu 2254    =/= wne 2567   E.wrex 2667   {crab 2670   _Vcvv 2916   class class class wbr 4172   iotacio 5375  (class class class)co 6040   supcsup 7403   RRcr 8945   0cc0 8946    < clt 9076    - cmin 9247    / cdiv 9633   NNcn 9956   NN0cn0 10177   ZZcz 10238   QQcq 10530   ^cexp 11337    || cdivides 12807   Primecprime 13034    pCnt cpc 13165
This theorem is referenced by:  pcqmul  13182  pcqcl  13185  pcid  13201  pcneg  13202  pc2dvds  13207  pcz  13209  pcaddlem  13212  pcadd  13213  pcmpt2  13217  pcbc  13224  sylow1lem1  15187  chtublem  20948
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-q 10531  df-rp 10569  df-fl 11157  df-mod 11206  df-seq 11279  df-exp 11338  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-dvds 12808  df-gcd 12962  df-prm 13035  df-pc 13166
  Copyright terms: Public domain W3C validator