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Theorem pceu 13175
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 733 . . . 4  |-  ( ( P  e.  Prime  /\  ( N  e.  QQ  /\  N  =/=  0 ) )  ->  N  e.  QQ )
2 elq 10532 . . . 4  |-  ( N  e.  QQ  <->  E. x  e.  ZZ  E. y  e.  NN  N  =  ( x  /  y ) )
31, 2sylib 189 . . 3  |-  ( ( P  e.  Prime  /\  ( N  e.  QQ  /\  N  =/=  0 ) )  ->  E. x  e.  ZZ  E. y  e.  NN  N  =  ( x  / 
y ) )
4 ovex 6065 . . . . . . . . 9  |-  ( S  -  T )  e. 
_V
5 biidd 229 . . . . . . . . 9  |-  ( z  =  ( S  -  T )  ->  ( N  =  ( x  /  y )  <->  N  =  ( x  /  y
) ) )
64, 5ceqsexv 2951 . . . . . . . 8  |-  ( E. z ( z  =  ( S  -  T
)  /\  N  =  ( x  /  y
) )  <->  N  =  ( x  /  y
) )
7 exancom 1593 . . . . . . . 8  |-  ( E. z ( z  =  ( S  -  T
)  /\  N  =  ( x  /  y
) )  <->  E. z
( N  =  ( x  /  y )  /\  z  =  ( S  -  T ) ) )
86, 7bitr3i 243 . . . . . . 7  |-  ( N  =  ( x  / 
y )  <->  E. z
( N  =  ( x  /  y )  /\  z  =  ( S  -  T ) ) )
98rexbii 2691 . . . . . 6  |-  ( E. y  e.  NN  N  =  ( x  / 
y )  <->  E. y  e.  NN  E. z ( N  =  ( x  /  y )  /\  z  =  ( S  -  T ) ) )
10 rexcom4 2935 . . . . . 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 241 . . . . 5  |-  ( E. y  e.  NN  N  =  ( x  / 
y )  <->  E. z E. y  e.  NN  ( N  =  (
x  /  y )  /\  z  =  ( S  -  T ) ) )
1211rexbii 2691 . . . 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 2935 . . . 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 241 . . 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 189 . 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 2404 . . . . . . . . . . 11  |-  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  s } ,  RR ,  <  )  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  s } ,  RR ,  <  )
19 eqid 2404 . . . . . . . . . . 11  |-  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  t } ,  RR ,  <  )  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  t } ,  RR ,  <  )
20 simp11l 1068 . . . . . . . . . . 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 1069 . . . . . . . . . . 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 988 . . . . . . . . . . 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 1072 . . . . . . . . . . 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 958 . . . . . . . . . . 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 985 . . . . . . . . . . 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 13174 . . . . . . . . . 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 1073 . . . . . . . . . 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 986 . . . . . . . . . 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 2446 . . . . . . . . 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 1152 . . . . . . . 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 2796 . . . . . . 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 1152 . . . . . 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 697 . . . . 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 2796 . . . 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 ) ) )
3534imp3a 421 . . 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 1639 . 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 2410 . . . . . 6  |-  ( z  =  w  ->  (
z  =  ( S  -  T )  <->  w  =  ( S  -  T
) ) )
3837anbi2d 685 . . . . 5  |-  ( z  =  w  ->  (
( N  =  ( x  /  y )  /\  z  =  ( S  -  T ) )  <->  ( N  =  ( x  /  y
)  /\  w  =  ( S  -  T
) ) ) )
39382rexbidv 2709 . . . 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 6047 . . . . . . . . 9  |-  ( x  =  s  ->  (
x  /  y )  =  ( s  / 
y ) )
4140eqeq2d 2415 . . . . . . . 8  |-  ( x  =  s  ->  ( N  =  ( x  /  y )  <->  N  =  ( s  /  y
) ) )
42 breq2 4176 . . . . . . . . . . . . 13  |-  ( x  =  s  ->  (
( P ^ n
)  ||  x  <->  ( P ^ n )  ||  s ) )
4342rabbidv 2908 . . . . . . . . . . . 12  |-  ( x  =  s  ->  { n  e.  NN0  |  ( P ^ n )  ||  x }  =  {
n  e.  NN0  | 
( P ^ n
)  ||  s }
)
4443supeq1d 7409 . . . . . . . . . . 11  |-  ( x  =  s  ->  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  x } ,  RR ,  <  )  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  s } ,  RR ,  <  ) )
4516, 44syl5eq 2448 . . . . . . . . . 10  |-  ( x  =  s  ->  S  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  s } ,  RR ,  <  ) )
4645oveq1d 6055 . . . . . . . . 9  |-  ( x  =  s  ->  ( S  -  T )  =  ( sup ( { n  e.  NN0  |  ( P ^ n
)  ||  s } ,  RR ,  <  )  -  T ) )
4746eqeq2d 2415 . . . . . . . 8  |-  ( x  =  s  ->  (
w  =  ( S  -  T )  <->  w  =  ( sup ( { n  e.  NN0  |  ( P ^ n )  ||  s } ,  RR ,  <  )  -  T ) ) )
4841, 47anbi12d 692 . . . . . . 7  |-  ( x  =  s  ->  (
( N  =  ( x  /  y )  /\  w  =  ( S  -  T ) )  <->  ( N  =  ( s  /  y
)  /\  w  =  ( sup ( { n  e.  NN0  |  ( P ^ n )  ||  s } ,  RR ,  <  )  -  T ) ) ) )
4948rexbidv 2687 . . . . . 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 6048 . . . . . . . . 9  |-  ( y  =  t  ->  (
s  /  y )  =  ( s  / 
t ) )
5150eqeq2d 2415 . . . . . . . 8  |-  ( y  =  t  ->  ( N  =  ( s  /  y )  <->  N  =  ( s  /  t
) ) )
52 breq2 4176 . . . . . . . . . . . . 13  |-  ( y  =  t  ->  (
( P ^ n
)  ||  y  <->  ( P ^ n )  ||  t ) )
5352rabbidv 2908 . . . . . . . . . . . 12  |-  ( y  =  t  ->  { n  e.  NN0  |  ( P ^ n )  ||  y }  =  {
n  e.  NN0  | 
( P ^ n
)  ||  t }
)
5453supeq1d 7409 . . . . . . . . . . 11  |-  ( y  =  t  ->  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  y } ,  RR ,  <  )  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  t } ,  RR ,  <  ) )
5517, 54syl5eq 2448 . . . . . . . . . 10  |-  ( y  =  t  ->  T  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  t } ,  RR ,  <  ) )
5655oveq2d 6056 . . . . . . . . 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 2415 . . . . . . . 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 692 . . . . . . 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 2893 . . . . . 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 253 . . . . 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 2893 . . . 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 253 . . 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 2293 . 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 646 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 set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936   A.wal 1546   E.wex 1547    = wceq 1649    e. wcel 1721   E!weu 2254    =/= wne 2567   E.wrex 2667   {crab 2670   class class class wbr 4172  (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
This theorem is referenced by:  pczpre  13176  pcdiv  13181
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
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