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Theorem pceu 14796
Description: Uniqueness for the prime power function. (Contributed by Mario Carneiro, 23-Feb-2014.)
Hypotheses
Ref Expression
pcval.1  |-  S  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  x } ,  RR ,  <  )
pcval.2  |-  T  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  y } ,  RR ,  <  )
Assertion
Ref Expression
pceu  |-  ( ( P  e.  Prime  /\  ( N  e.  QQ  /\  N  =/=  0 ) )  ->  E! z E. x  e.  ZZ  E. y  e.  NN  ( N  =  ( x  /  y
)  /\  z  =  ( S  -  T
) ) )
Distinct variable groups:    x, n, y, z, N    P, n, x, y, z    z, S   
z, T
Allowed substitution hints:    S( x, y, n)    T( x, y, n)

Proof of Theorem pceu
Dummy variables  s 
t  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simprl 762 . . . 4  |-  ( ( P  e.  Prime  /\  ( N  e.  QQ  /\  N  =/=  0 ) )  ->  N  e.  QQ )
2 elq 11274 . . . 4  |-  ( N  e.  QQ  <->  E. x  e.  ZZ  E. y  e.  NN  N  =  ( x  /  y ) )
31, 2sylib 199 . . 3  |-  ( ( P  e.  Prime  /\  ( N  e.  QQ  /\  N  =/=  0 ) )  ->  E. x  e.  ZZ  E. y  e.  NN  N  =  ( x  / 
y ) )
4 ovex 6334 . . . . . . . . 9  |-  ( S  -  T )  e. 
_V
5 biidd 240 . . . . . . . . 9  |-  ( z  =  ( S  -  T )  ->  ( N  =  ( x  /  y )  <->  N  =  ( x  /  y
) ) )
64, 5ceqsexv 3118 . . . . . . . 8  |-  ( E. z ( z  =  ( S  -  T
)  /\  N  =  ( x  /  y
) )  <->  N  =  ( x  /  y
) )
7 exancom 1716 . . . . . . . 8  |-  ( E. z ( z  =  ( S  -  T
)  /\  N  =  ( x  /  y
) )  <->  E. z
( N  =  ( x  /  y )  /\  z  =  ( S  -  T ) ) )
86, 7bitr3i 254 . . . . . . 7  |-  ( N  =  ( x  / 
y )  <->  E. z
( N  =  ( x  /  y )  /\  z  =  ( S  -  T ) ) )
98rexbii 2924 . . . . . 6  |-  ( E. y  e.  NN  N  =  ( x  / 
y )  <->  E. y  e.  NN  E. z ( N  =  ( x  /  y )  /\  z  =  ( S  -  T ) ) )
10 rexcom4 3101 . . . . . 6  |-  ( E. y  e.  NN  E. z ( N  =  ( x  /  y
)  /\  z  =  ( S  -  T
) )  <->  E. z E. y  e.  NN  ( N  =  (
x  /  y )  /\  z  =  ( S  -  T ) ) )
119, 10bitri 252 . . . . 5  |-  ( E. y  e.  NN  N  =  ( x  / 
y )  <->  E. z E. y  e.  NN  ( N  =  (
x  /  y )  /\  z  =  ( S  -  T ) ) )
1211rexbii 2924 . . . 4  |-  ( E. x  e.  ZZ  E. y  e.  NN  N  =  ( x  / 
y )  <->  E. x  e.  ZZ  E. z E. y  e.  NN  ( N  =  ( x  /  y )  /\  z  =  ( S  -  T ) ) )
13 rexcom4 3101 . . . 4  |-  ( E. x  e.  ZZ  E. z E. y  e.  NN  ( N  =  (
x  /  y )  /\  z  =  ( S  -  T ) )  <->  E. z E. x  e.  ZZ  E. y  e.  NN  ( N  =  ( x  /  y
)  /\  z  =  ( S  -  T
) ) )
1412, 13bitri 252 . . 3  |-  ( E. x  e.  ZZ  E. y  e.  NN  N  =  ( x  / 
y )  <->  E. z E. x  e.  ZZ  E. y  e.  NN  ( N  =  ( x  /  y )  /\  z  =  ( S  -  T ) ) )
153, 14sylib 199 . 2  |-  ( ( P  e.  Prime  /\  ( N  e.  QQ  /\  N  =/=  0 ) )  ->  E. z E. x  e.  ZZ  E. y  e.  NN  ( N  =  ( x  /  y
)  /\  z  =  ( S  -  T
) ) )
16 pcval.1 . . . . . . . . . . 11  |-  S  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  x } ,  RR ,  <  )
17 pcval.2 . . . . . . . . . . 11  |-  T  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  y } ,  RR ,  <  )
18 eqid 2422 . . . . . . . . . . 11  |-  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  s } ,  RR ,  <  )  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  s } ,  RR ,  <  )
19 eqid 2422 . . . . . . . . . . 11  |-  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  t } ,  RR ,  <  )  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  t } ,  RR ,  <  )
20 simp11l 1116 . . . . . . . . . . 11  |-  ( ( ( ( P  e. 
Prime  /\  N  =/=  0
)  /\  ( x  e.  ZZ  /\  y  e.  NN )  /\  ( N  =  ( x  /  y )  /\  z  =  ( S  -  T ) ) )  /\  ( s  e.  ZZ  /\  t  e.  NN )  /\  ( N  =  ( s  /  t )  /\  w  =  ( sup ( { n  e.  NN0  |  ( P ^ n
)  ||  s } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( P ^ n )  ||  t } ,  RR ,  <  ) ) ) )  ->  P  e.  Prime )
21 simp11r 1117 . . . . . . . . . . 11  |-  ( ( ( ( P  e. 
Prime  /\  N  =/=  0
)  /\  ( x  e.  ZZ  /\  y  e.  NN )  /\  ( N  =  ( x  /  y )  /\  z  =  ( S  -  T ) ) )  /\  ( s  e.  ZZ  /\  t  e.  NN )  /\  ( N  =  ( s  /  t )  /\  w  =  ( sup ( { n  e.  NN0  |  ( P ^ n
)  ||  s } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( P ^ n )  ||  t } ,  RR ,  <  ) ) ) )  ->  N  =/=  0
)
22 simp12 1036 . . . . . . . . . . 11  |-  ( ( ( ( P  e. 
Prime  /\  N  =/=  0
)  /\  ( x  e.  ZZ  /\  y  e.  NN )  /\  ( N  =  ( x  /  y )  /\  z  =  ( S  -  T ) ) )  /\  ( s  e.  ZZ  /\  t  e.  NN )  /\  ( N  =  ( s  /  t )  /\  w  =  ( sup ( { n  e.  NN0  |  ( P ^ n
)  ||  s } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( P ^ n )  ||  t } ,  RR ,  <  ) ) ) )  ->  ( x  e.  ZZ  /\  y  e.  NN ) )
23 simp13l 1120 . . . . . . . . . . 11  |-  ( ( ( ( P  e. 
Prime  /\  N  =/=  0
)  /\  ( x  e.  ZZ  /\  y  e.  NN )  /\  ( N  =  ( x  /  y )  /\  z  =  ( S  -  T ) ) )  /\  ( s  e.  ZZ  /\  t  e.  NN )  /\  ( N  =  ( s  /  t )  /\  w  =  ( sup ( { n  e.  NN0  |  ( P ^ n
)  ||  s } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( P ^ n )  ||  t } ,  RR ,  <  ) ) ) )  ->  N  =  ( x  /  y ) )
24 simp2 1006 . . . . . . . . . . 11  |-  ( ( ( ( P  e. 
Prime  /\  N  =/=  0
)  /\  ( x  e.  ZZ  /\  y  e.  NN )  /\  ( N  =  ( x  /  y )  /\  z  =  ( S  -  T ) ) )  /\  ( s  e.  ZZ  /\  t  e.  NN )  /\  ( N  =  ( s  /  t )  /\  w  =  ( sup ( { n  e.  NN0  |  ( P ^ n
)  ||  s } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( P ^ n )  ||  t } ,  RR ,  <  ) ) ) )  ->  ( s  e.  ZZ  /\  t  e.  NN ) )
25 simp3l 1033 . . . . . . . . . . 11  |-  ( ( ( ( P  e. 
Prime  /\  N  =/=  0
)  /\  ( x  e.  ZZ  /\  y  e.  NN )  /\  ( N  =  ( x  /  y )  /\  z  =  ( S  -  T ) ) )  /\  ( s  e.  ZZ  /\  t  e.  NN )  /\  ( N  =  ( s  /  t )  /\  w  =  ( sup ( { n  e.  NN0  |  ( P ^ n
)  ||  s } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( P ^ n )  ||  t } ,  RR ,  <  ) ) ) )  ->  N  =  ( s  /  t ) )
2616, 17, 18, 19, 20, 21, 22, 23, 24, 25pceulem 14795 . . . . . . . . . 10  |-  ( ( ( ( P  e. 
Prime  /\  N  =/=  0
)  /\  ( x  e.  ZZ  /\  y  e.  NN )  /\  ( N  =  ( x  /  y )  /\  z  =  ( S  -  T ) ) )  /\  ( s  e.  ZZ  /\  t  e.  NN )  /\  ( N  =  ( s  /  t )  /\  w  =  ( sup ( { n  e.  NN0  |  ( P ^ n
)  ||  s } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( P ^ n )  ||  t } ,  RR ,  <  ) ) ) )  ->  ( S  -  T )  =  ( sup ( { n  e.  NN0  |  ( P ^ n )  ||  s } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  t } ,  RR ,  <  )
) )
27 simp13r 1121 . . . . . . . . . 10  |-  ( ( ( ( P  e. 
Prime  /\  N  =/=  0
)  /\  ( x  e.  ZZ  /\  y  e.  NN )  /\  ( N  =  ( x  /  y )  /\  z  =  ( S  -  T ) ) )  /\  ( s  e.  ZZ  /\  t  e.  NN )  /\  ( N  =  ( s  /  t )  /\  w  =  ( sup ( { n  e.  NN0  |  ( P ^ n
)  ||  s } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( P ^ n )  ||  t } ,  RR ,  <  ) ) ) )  ->  z  =  ( S  -  T ) )
28 simp3r 1034 . . . . . . . . . 10  |-  ( ( ( ( P  e. 
Prime  /\  N  =/=  0
)  /\  ( x  e.  ZZ  /\  y  e.  NN )  /\  ( N  =  ( x  /  y )  /\  z  =  ( S  -  T ) ) )  /\  ( s  e.  ZZ  /\  t  e.  NN )  /\  ( N  =  ( s  /  t )  /\  w  =  ( sup ( { n  e.  NN0  |  ( P ^ n
)  ||  s } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( P ^ n )  ||  t } ,  RR ,  <  ) ) ) )  ->  w  =  ( sup ( { n  e.  NN0  |  ( P ^ n )  ||  s } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  t } ,  RR ,  <  )
) )
2926, 27, 283eqtr4d 2473 . . . . . . . . 9  |-  ( ( ( ( P  e. 
Prime  /\  N  =/=  0
)  /\  ( x  e.  ZZ  /\  y  e.  NN )  /\  ( N  =  ( x  /  y )  /\  z  =  ( S  -  T ) ) )  /\  ( s  e.  ZZ  /\  t  e.  NN )  /\  ( N  =  ( s  /  t )  /\  w  =  ( sup ( { n  e.  NN0  |  ( P ^ n
)  ||  s } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( P ^ n )  ||  t } ,  RR ,  <  ) ) ) )  ->  z  =  w )
30293exp 1204 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  N  =/=  0 )  /\  ( x  e.  ZZ  /\  y  e.  NN )  /\  ( N  =  ( x  /  y )  /\  z  =  ( S  -  T ) ) )  ->  ( ( s  e.  ZZ  /\  t  e.  NN )  ->  (
( N  =  ( s  /  t )  /\  w  =  ( sup ( { n  e.  NN0  |  ( P ^ n )  ||  s } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  t } ,  RR ,  <  )
) )  ->  z  =  w ) ) )
3130rexlimdvv 2920 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  N  =/=  0 )  /\  ( x  e.  ZZ  /\  y  e.  NN )  /\  ( N  =  ( x  /  y )  /\  z  =  ( S  -  T ) ) )  ->  ( E. s  e.  ZZ  E. t  e.  NN  ( N  =  ( s  /  t
)  /\  w  =  ( sup ( { n  e.  NN0  |  ( P ^ n )  ||  s } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  t } ,  RR ,  <  )
) )  ->  z  =  w ) )
32313exp 1204 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  =/=  0 )  ->  (
( x  e.  ZZ  /\  y  e.  NN )  ->  ( ( N  =  ( x  / 
y )  /\  z  =  ( S  -  T ) )  -> 
( E. s  e.  ZZ  E. t  e.  NN  ( N  =  ( s  /  t
)  /\  w  =  ( sup ( { n  e.  NN0  |  ( P ^ n )  ||  s } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  t } ,  RR ,  <  )
) )  ->  z  =  w ) ) ) )
3332adantrl 720 . . . . 5  |-  ( ( P  e.  Prime  /\  ( N  e.  QQ  /\  N  =/=  0 ) )  -> 
( ( x  e.  ZZ  /\  y  e.  NN )  ->  (
( N  =  ( x  /  y )  /\  z  =  ( S  -  T ) )  ->  ( E. s  e.  ZZ  E. t  e.  NN  ( N  =  ( s  /  t
)  /\  w  =  ( sup ( { n  e.  NN0  |  ( P ^ n )  ||  s } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  t } ,  RR ,  <  )
) )  ->  z  =  w ) ) ) )
3433rexlimdvv 2920 . . . 4  |-  ( ( P  e.  Prime  /\  ( N  e.  QQ  /\  N  =/=  0 ) )  -> 
( E. x  e.  ZZ  E. y  e.  NN  ( N  =  ( x  /  y
)  /\  z  =  ( S  -  T
) )  ->  ( E. s  e.  ZZ  E. t  e.  NN  ( N  =  ( s  /  t )  /\  w  =  ( sup ( { n  e.  NN0  |  ( P ^ n
)  ||  s } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( P ^ n )  ||  t } ,  RR ,  <  ) ) )  -> 
z  =  w ) ) )
3534impd 432 . . 3  |-  ( ( P  e.  Prime  /\  ( N  e.  QQ  /\  N  =/=  0 ) )  -> 
( ( E. x  e.  ZZ  E. y  e.  NN  ( N  =  ( x  /  y
)  /\  z  =  ( S  -  T
) )  /\  E. s  e.  ZZ  E. t  e.  NN  ( N  =  ( s  /  t
)  /\  w  =  ( sup ( { n  e.  NN0  |  ( P ^ n )  ||  s } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  t } ,  RR ,  <  )
) ) )  -> 
z  =  w ) )
3635alrimivv 1768 . 2  |-  ( ( P  e.  Prime  /\  ( N  e.  QQ  /\  N  =/=  0 ) )  ->  A. z A. w ( ( E. x  e.  ZZ  E. y  e.  NN  ( N  =  ( x  /  y
)  /\  z  =  ( S  -  T
) )  /\  E. s  e.  ZZ  E. t  e.  NN  ( N  =  ( s  /  t
)  /\  w  =  ( sup ( { n  e.  NN0  |  ( P ^ n )  ||  s } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  t } ,  RR ,  <  )
) ) )  -> 
z  =  w ) )
37 eqeq1 2426 . . . . . 6  |-  ( z  =  w  ->  (
z  =  ( S  -  T )  <->  w  =  ( S  -  T
) ) )
3837anbi2d 708 . . . . 5  |-  ( z  =  w  ->  (
( N  =  ( x  /  y )  /\  z  =  ( S  -  T ) )  <->  ( N  =  ( x  /  y
)  /\  w  =  ( S  -  T
) ) ) )
39382rexbidv 2943 . . . 4  |-  ( z  =  w  ->  ( E. x  e.  ZZ  E. y  e.  NN  ( N  =  ( x  /  y )  /\  z  =  ( S  -  T ) )  <->  E. x  e.  ZZ  E. y  e.  NN  ( N  =  ( x  /  y
)  /\  w  =  ( S  -  T
) ) ) )
40 oveq1 6313 . . . . . . . . 9  |-  ( x  =  s  ->  (
x  /  y )  =  ( s  / 
y ) )
4140eqeq2d 2436 . . . . . . . 8  |-  ( x  =  s  ->  ( N  =  ( x  /  y )  <->  N  =  ( s  /  y
) ) )
42 breq2 4427 . . . . . . . . . . . . 13  |-  ( x  =  s  ->  (
( P ^ n
)  ||  x  <->  ( P ^ n )  ||  s ) )
4342rabbidv 3071 . . . . . . . . . . . 12  |-  ( x  =  s  ->  { n  e.  NN0  |  ( P ^ n )  ||  x }  =  {
n  e.  NN0  | 
( P ^ n
)  ||  s }
)
4443supeq1d 7970 . . . . . . . . . . 11  |-  ( x  =  s  ->  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  x } ,  RR ,  <  )  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  s } ,  RR ,  <  ) )
4516, 44syl5eq 2475 . . . . . . . . . 10  |-  ( x  =  s  ->  S  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  s } ,  RR ,  <  ) )
4645oveq1d 6321 . . . . . . . . 9  |-  ( x  =  s  ->  ( S  -  T )  =  ( sup ( { n  e.  NN0  |  ( P ^ n
)  ||  s } ,  RR ,  <  )  -  T ) )
4746eqeq2d 2436 . . . . . . . 8  |-  ( x  =  s  ->  (
w  =  ( S  -  T )  <->  w  =  ( sup ( { n  e.  NN0  |  ( P ^ n )  ||  s } ,  RR ,  <  )  -  T ) ) )
4841, 47anbi12d 715 . . . . . . 7  |-  ( x  =  s  ->  (
( N  =  ( x  /  y )  /\  w  =  ( S  -  T ) )  <->  ( N  =  ( s  /  y
)  /\  w  =  ( sup ( { n  e.  NN0  |  ( P ^ n )  ||  s } ,  RR ,  <  )  -  T ) ) ) )
4948rexbidv 2936 . . . . . 6  |-  ( x  =  s  ->  ( E. y  e.  NN  ( N  =  (
x  /  y )  /\  w  =  ( S  -  T ) )  <->  E. y  e.  NN  ( N  =  (
s  /  y )  /\  w  =  ( sup ( { n  e.  NN0  |  ( P ^ n )  ||  s } ,  RR ,  <  )  -  T ) ) ) )
50 oveq2 6314 . . . . . . . . 9  |-  ( y  =  t  ->  (
s  /  y )  =  ( s  / 
t ) )
5150eqeq2d 2436 . . . . . . . 8  |-  ( y  =  t  ->  ( N  =  ( s  /  y )  <->  N  =  ( s  /  t
) ) )
52 breq2 4427 . . . . . . . . . . . . 13  |-  ( y  =  t  ->  (
( P ^ n
)  ||  y  <->  ( P ^ n )  ||  t ) )
5352rabbidv 3071 . . . . . . . . . . . 12  |-  ( y  =  t  ->  { n  e.  NN0  |  ( P ^ n )  ||  y }  =  {
n  e.  NN0  | 
( P ^ n
)  ||  t }
)
5453supeq1d 7970 . . . . . . . . . . 11  |-  ( y  =  t  ->  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  y } ,  RR ,  <  )  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  t } ,  RR ,  <  ) )
5517, 54syl5eq 2475 . . . . . . . . . 10  |-  ( y  =  t  ->  T  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  t } ,  RR ,  <  ) )
5655oveq2d 6322 . . . . . . . . 9  |-  ( y  =  t  ->  ( sup ( { n  e. 
NN0  |  ( P ^ n )  ||  s } ,  RR ,  <  )  -  T )  =  ( sup ( { n  e.  NN0  |  ( P ^ n
)  ||  s } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( P ^ n )  ||  t } ,  RR ,  <  ) ) )
5756eqeq2d 2436 . . . . . . . 8  |-  ( y  =  t  ->  (
w  =  ( sup ( { n  e. 
NN0  |  ( P ^ n )  ||  s } ,  RR ,  <  )  -  T )  <-> 
w  =  ( sup ( { n  e. 
NN0  |  ( P ^ n )  ||  s } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  t } ,  RR ,  <  )
) ) )
5851, 57anbi12d 715 . . . . . . 7  |-  ( y  =  t  ->  (
( N  =  ( s  /  y )  /\  w  =  ( sup ( { n  e.  NN0  |  ( P ^ n )  ||  s } ,  RR ,  <  )  -  T ) )  <->  ( N  =  ( s  /  t
)  /\  w  =  ( sup ( { n  e.  NN0  |  ( P ^ n )  ||  s } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  t } ,  RR ,  <  )
) ) ) )
5958cbvrexv 3055 . . . . . 6  |-  ( E. y  e.  NN  ( N  =  ( s  /  y )  /\  w  =  ( sup ( { n  e.  NN0  |  ( P ^ n
)  ||  s } ,  RR ,  <  )  -  T ) )  <->  E. t  e.  NN  ( N  =  ( s  /  t
)  /\  w  =  ( sup ( { n  e.  NN0  |  ( P ^ n )  ||  s } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  t } ,  RR ,  <  )
) ) )
6049, 59syl6bb 264 . . . . 5  |-  ( x  =  s  ->  ( E. y  e.  NN  ( N  =  (
x  /  y )  /\  w  =  ( S  -  T ) )  <->  E. t  e.  NN  ( N  =  (
s  /  t )  /\  w  =  ( sup ( { n  e.  NN0  |  ( P ^ n )  ||  s } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  t } ,  RR ,  <  )
) ) ) )
6160cbvrexv 3055 . . . 4  |-  ( E. x  e.  ZZ  E. y  e.  NN  ( N  =  ( x  /  y )  /\  w  =  ( S  -  T ) )  <->  E. s  e.  ZZ  E. t  e.  NN  ( N  =  ( s  /  t
)  /\  w  =  ( sup ( { n  e.  NN0  |  ( P ^ n )  ||  s } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  t } ,  RR ,  <  )
) ) )
6239, 61syl6bb 264 . . 3  |-  ( z  =  w  ->  ( E. x  e.  ZZ  E. y  e.  NN  ( N  =  ( x  /  y )  /\  z  =  ( S  -  T ) )  <->  E. s  e.  ZZ  E. t  e.  NN  ( N  =  ( s  /  t
)  /\  w  =  ( sup ( { n  e.  NN0  |  ( P ^ n )  ||  s } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  t } ,  RR ,  <  )
) ) ) )
6362eu4 2317 . 2  |-  ( E! z E. x  e.  ZZ  E. y  e.  NN  ( N  =  ( x  /  y
)  /\  z  =  ( S  -  T
) )  <->  ( E. z E. x  e.  ZZ  E. y  e.  NN  ( N  =  ( x  /  y )  /\  z  =  ( S  -  T ) )  /\  A. z A. w ( ( E. x  e.  ZZ  E. y  e.  NN  ( N  =  ( x  /  y
)  /\  z  =  ( S  -  T
) )  /\  E. s  e.  ZZ  E. t  e.  NN  ( N  =  ( s  /  t
)  /\  w  =  ( sup ( { n  e.  NN0  |  ( P ^ n )  ||  s } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  t } ,  RR ,  <  )
) ) )  -> 
z  =  w ) ) )
6415, 36, 63sylanbrc 668 1  |-  ( ( P  e.  Prime  /\  ( N  e.  QQ  /\  N  =/=  0 ) )  ->  E! z E. x  e.  ZZ  E. y  e.  NN  ( N  =  ( x  /  y
)  /\  z  =  ( S  -  T
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    /\ w3a 982   A.wal 1435    = wceq 1437   E.wex 1657    e. wcel 1872   E!weu 2269    =/= wne 2614   E.wrex 2772   {crab 2775   class class class wbr 4423  (class class class)co 6306   supcsup 7964   RRcr 9546   0cc0 9547    < clt 9683    - cmin 9868    / cdiv 10277   NNcn 10617   NN0cn0 10877   ZZcz 10945   QQcq 11272   ^cexp 12279    || cdvds 14305   Primecprime 14622
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6598  ax-cnex 9603  ax-resscn 9604  ax-1cn 9605  ax-icn 9606  ax-addcl 9607  ax-addrcl 9608  ax-mulcl 9609  ax-mulrcl 9610  ax-mulcom 9611  ax-addass 9612  ax-mulass 9613  ax-distr 9614  ax-i2m1 9615  ax-1ne0 9616  ax-1rid 9617  ax-rnegex 9618  ax-rrecex 9619  ax-cnre 9620  ax-pre-lttri 9621  ax-pre-lttrn 9622  ax-pre-ltadd 9623  ax-pre-mulgt0 9624  ax-pre-sup 9625
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rmo 2779  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-int 4256  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6268  df-ov 6309  df-oprab 6310  df-mpt2 6311  df-om 6708  df-1st 6808  df-2nd 6809  df-wrecs 7040  df-recs 7102  df-rdg 7140  df-1o 7194  df-2o 7195  df-oadd 7198  df-er 7375  df-en 7582  df-dom 7583  df-sdom 7584  df-fin 7585  df-sup 7966  df-inf 7967  df-pnf 9685  df-mnf 9686  df-xr 9687  df-ltxr 9688  df-le 9689  df-sub 9870  df-neg 9871  df-div 10278  df-nn 10618  df-2 10676  df-3 10677  df-n0 10878  df-z 10946  df-uz 11168  df-q 11273  df-rp 11311  df-fl 12035  df-mod 12104  df-seq 12221  df-exp 12280  df-cj 13163  df-re 13164  df-im 13165  df-sqrt 13299  df-abs 13300  df-dvds 14306  df-gcd 14469  df-prm 14623
This theorem is referenced by:  pczpre  14797  pcdiv  14802
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