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Theorem pceu 12899
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 732 . . . 4  |-  ( ( P  e.  Prime  /\  ( N  e.  QQ  /\  N  =/=  0 ) )  ->  N  e.  QQ )
2 elq 10318 . . . 4  |-  ( N  e.  QQ  <->  E. x  e.  ZZ  E. y  e.  NN  N  =  ( x  /  y ) )
31, 2sylib 188 . . 3  |-  ( ( P  e.  Prime  /\  ( N  e.  QQ  /\  N  =/=  0 ) )  ->  E. x  e.  ZZ  E. y  e.  NN  N  =  ( x  / 
y ) )
4 ovex 5883 . . . . . . . . 9  |-  ( S  -  T )  e. 
_V
5 biidd 228 . . . . . . . . 9  |-  ( z  =  ( S  -  T )  ->  ( N  =  ( x  /  y )  <->  N  =  ( x  /  y
) ) )
64, 5ceqsexv 2823 . . . . . . . 8  |-  ( E. z ( z  =  ( S  -  T
)  /\  N  =  ( x  /  y
) )  <->  N  =  ( x  /  y
) )
7 exancom 1573 . . . . . . . 8  |-  ( E. z ( z  =  ( S  -  T
)  /\  N  =  ( x  /  y
) )  <->  E. z
( N  =  ( x  /  y )  /\  z  =  ( S  -  T ) ) )
86, 7bitr3i 242 . . . . . . 7  |-  ( N  =  ( x  / 
y )  <->  E. z
( N  =  ( x  /  y )  /\  z  =  ( S  -  T ) ) )
98rexbii 2568 . . . . . 6  |-  ( E. y  e.  NN  N  =  ( x  / 
y )  <->  E. y  e.  NN  E. z ( N  =  ( x  /  y )  /\  z  =  ( S  -  T ) ) )
10 rexcom4 2807 . . . . . 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 240 . . . . 5  |-  ( E. y  e.  NN  N  =  ( x  / 
y )  <->  E. z E. y  e.  NN  ( N  =  (
x  /  y )  /\  z  =  ( S  -  T ) ) )
1211rexbii 2568 . . . 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 2807 . . . 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 240 . . 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 188 . 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 2283 . . . . . . . . . . 11  |-  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  s } ,  RR ,  <  )  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  s } ,  RR ,  <  )
19 eqid 2283 . . . . . . . . . . 11  |-  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  t } ,  RR ,  <  )  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  t } ,  RR ,  <  )
20 simp11l 1066 . . . . . . . . . . 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 1067 . . . . . . . . . . 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 986 . . . . . . . . . . 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 1070 . . . . . . . . . . 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 956 . . . . . . . . . . 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 983 . . . . . . . . . . 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 12898 . . . . . . . . . 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 1071 . . . . . . . . . 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 984 . . . . . . . . . 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 2325 . . . . . . . . 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 1150 . . . . . . . 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 2673 . . . . . . 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 1150 . . . . . 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 696 . . . . 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 2673 . . . 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 420 . . 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 1618 . 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 2289 . . . . . 6  |-  ( z  =  w  ->  (
z  =  ( S  -  T )  <->  w  =  ( S  -  T
) ) )
3837anbi2d 684 . . . . 5  |-  ( z  =  w  ->  (
( N  =  ( x  /  y )  /\  z  =  ( S  -  T ) )  <->  ( N  =  ( x  /  y
)  /\  w  =  ( S  -  T
) ) ) )
39382rexbidv 2586 . . . 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 5865 . . . . . . . . 9  |-  ( x  =  s  ->  (
x  /  y )  =  ( s  / 
y ) )
4140eqeq2d 2294 . . . . . . . 8  |-  ( x  =  s  ->  ( N  =  ( x  /  y )  <->  N  =  ( s  /  y
) ) )
42 breq2 4027 . . . . . . . . . . . . 13  |-  ( x  =  s  ->  (
( P ^ n
)  ||  x  <->  ( P ^ n )  ||  s ) )
4342rabbidv 2780 . . . . . . . . . . . 12  |-  ( x  =  s  ->  { n  e.  NN0  |  ( P ^ n )  ||  x }  =  {
n  e.  NN0  | 
( P ^ n
)  ||  s }
)
4443supeq1d 7199 . . . . . . . . . . 11  |-  ( x  =  s  ->  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  x } ,  RR ,  <  )  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  s } ,  RR ,  <  ) )
4516, 44syl5eq 2327 . . . . . . . . . 10  |-  ( x  =  s  ->  S  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  s } ,  RR ,  <  ) )
4645oveq1d 5873 . . . . . . . . 9  |-  ( x  =  s  ->  ( S  -  T )  =  ( sup ( { n  e.  NN0  |  ( P ^ n
)  ||  s } ,  RR ,  <  )  -  T ) )
4746eqeq2d 2294 . . . . . . . 8  |-  ( x  =  s  ->  (
w  =  ( S  -  T )  <->  w  =  ( sup ( { n  e.  NN0  |  ( P ^ n )  ||  s } ,  RR ,  <  )  -  T ) ) )
4841, 47anbi12d 691 . . . . . . 7  |-  ( x  =  s  ->  (
( N  =  ( x  /  y )  /\  w  =  ( S  -  T ) )  <->  ( N  =  ( s  /  y
)  /\  w  =  ( sup ( { n  e.  NN0  |  ( P ^ n )  ||  s } ,  RR ,  <  )  -  T ) ) ) )
4948rexbidv 2564 . . . . . 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 5866 . . . . . . . . 9  |-  ( y  =  t  ->  (
s  /  y )  =  ( s  / 
t ) )
5150eqeq2d 2294 . . . . . . . 8  |-  ( y  =  t  ->  ( N  =  ( s  /  y )  <->  N  =  ( s  /  t
) ) )
52 breq2 4027 . . . . . . . . . . . . 13  |-  ( y  =  t  ->  (
( P ^ n
)  ||  y  <->  ( P ^ n )  ||  t ) )
5352rabbidv 2780 . . . . . . . . . . . 12  |-  ( y  =  t  ->  { n  e.  NN0  |  ( P ^ n )  ||  y }  =  {
n  e.  NN0  | 
( P ^ n
)  ||  t }
)
5453supeq1d 7199 . . . . . . . . . . 11  |-  ( y  =  t  ->  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  y } ,  RR ,  <  )  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  t } ,  RR ,  <  ) )
5517, 54syl5eq 2327 . . . . . . . . . 10  |-  ( y  =  t  ->  T  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  t } ,  RR ,  <  ) )
5655oveq2d 5874 . . . . . . . . 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 2294 . . . . . . . 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 691 . . . . . . 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 2765 . . . . . 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 252 . . . . 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 2765 . . . 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 252 . . 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 2182 . 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 645 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 358    /\ w3a 934   A.wal 1527   E.wex 1528    = wceq 1623    e. wcel 1684   E!weu 2143    =/= wne 2446   E.wrex 2544   {crab 2547   class class class wbr 4023  (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
This theorem is referenced by:  pczpre  12900  pcdiv  12905
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
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