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Theorem pceu 13898
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 750 . . . 4  |-  ( ( P  e.  Prime  /\  ( N  e.  QQ  /\  N  =/=  0 ) )  ->  N  e.  QQ )
2 elq 10945 . . . 4  |-  ( N  e.  QQ  <->  E. x  e.  ZZ  E. y  e.  NN  N  =  ( x  /  y ) )
31, 2sylib 196 . . 3  |-  ( ( P  e.  Prime  /\  ( N  e.  QQ  /\  N  =/=  0 ) )  ->  E. x  e.  ZZ  E. y  e.  NN  N  =  ( x  / 
y ) )
4 ovex 6107 . . . . . . . . 9  |-  ( S  -  T )  e. 
_V
5 biidd 237 . . . . . . . . 9  |-  ( z  =  ( S  -  T )  ->  ( N  =  ( x  /  y )  <->  N  =  ( x  /  y
) ) )
64, 5ceqsexv 3000 . . . . . . . 8  |-  ( E. z ( z  =  ( S  -  T
)  /\  N  =  ( x  /  y
) )  <->  N  =  ( x  /  y
) )
7 exancom 1640 . . . . . . . 8  |-  ( E. z ( z  =  ( S  -  T
)  /\  N  =  ( x  /  y
) )  <->  E. z
( N  =  ( x  /  y )  /\  z  =  ( S  -  T ) ) )
86, 7bitr3i 251 . . . . . . 7  |-  ( N  =  ( x  / 
y )  <->  E. z
( N  =  ( x  /  y )  /\  z  =  ( S  -  T ) ) )
98rexbii 2732 . . . . . 6  |-  ( E. y  e.  NN  N  =  ( x  / 
y )  <->  E. y  e.  NN  E. z ( N  =  ( x  /  y )  /\  z  =  ( S  -  T ) ) )
10 rexcom4 2984 . . . . . 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 249 . . . . 5  |-  ( E. y  e.  NN  N  =  ( x  / 
y )  <->  E. z E. y  e.  NN  ( N  =  (
x  /  y )  /\  z  =  ( S  -  T ) ) )
1211rexbii 2732 . . . 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 2984 . . . 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 249 . . 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 196 . 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 2435 . . . . . . . . . . 11  |-  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  s } ,  RR ,  <  )  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  s } ,  RR ,  <  )
19 eqid 2435 . . . . . . . . . . 11  |-  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  t } ,  RR ,  <  )  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  t } ,  RR ,  <  )
20 simp11l 1094 . . . . . . . . . . 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 1095 . . . . . . . . . . 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 1014 . . . . . . . . . . 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 1098 . . . . . . . . . . 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 984 . . . . . . . . . . 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 1011 . . . . . . . . . . 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 13897 . . . . . . . . . 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 1099 . . . . . . . . . 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 1012 . . . . . . . . . 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 2477 . . . . . . . . 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 1181 . . . . . . . 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 2839 . . . . . . 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 1181 . . . . . 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 710 . . . . 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 2839 . . . 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 431 . . 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 1687 . 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 2441 . . . . . 6  |-  ( z  =  w  ->  (
z  =  ( S  -  T )  <->  w  =  ( S  -  T
) ) )
3837anbi2d 698 . . . . 5  |-  ( z  =  w  ->  (
( N  =  ( x  /  y )  /\  z  =  ( S  -  T ) )  <->  ( N  =  ( x  /  y
)  /\  w  =  ( S  -  T
) ) ) )
39382rexbidv 2750 . . . 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 6089 . . . . . . . . 9  |-  ( x  =  s  ->  (
x  /  y )  =  ( s  / 
y ) )
4140eqeq2d 2446 . . . . . . . 8  |-  ( x  =  s  ->  ( N  =  ( x  /  y )  <->  N  =  ( s  /  y
) ) )
42 breq2 4286 . . . . . . . . . . . . 13  |-  ( x  =  s  ->  (
( P ^ n
)  ||  x  <->  ( P ^ n )  ||  s ) )
4342rabbidv 2956 . . . . . . . . . . . 12  |-  ( x  =  s  ->  { n  e.  NN0  |  ( P ^ n )  ||  x }  =  {
n  e.  NN0  | 
( P ^ n
)  ||  s }
)
4443supeq1d 7686 . . . . . . . . . . 11  |-  ( x  =  s  ->  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  x } ,  RR ,  <  )  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  s } ,  RR ,  <  ) )
4516, 44syl5eq 2479 . . . . . . . . . 10  |-  ( x  =  s  ->  S  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  s } ,  RR ,  <  ) )
4645oveq1d 6097 . . . . . . . . 9  |-  ( x  =  s  ->  ( S  -  T )  =  ( sup ( { n  e.  NN0  |  ( P ^ n
)  ||  s } ,  RR ,  <  )  -  T ) )
4746eqeq2d 2446 . . . . . . . 8  |-  ( x  =  s  ->  (
w  =  ( S  -  T )  <->  w  =  ( sup ( { n  e.  NN0  |  ( P ^ n )  ||  s } ,  RR ,  <  )  -  T ) ) )
4841, 47anbi12d 705 . . . . . . 7  |-  ( x  =  s  ->  (
( N  =  ( x  /  y )  /\  w  =  ( S  -  T ) )  <->  ( N  =  ( s  /  y
)  /\  w  =  ( sup ( { n  e.  NN0  |  ( P ^ n )  ||  s } ,  RR ,  <  )  -  T ) ) ) )
4948rexbidv 2728 . . . . . 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 6090 . . . . . . . . 9  |-  ( y  =  t  ->  (
s  /  y )  =  ( s  / 
t ) )
5150eqeq2d 2446 . . . . . . . 8  |-  ( y  =  t  ->  ( N  =  ( s  /  y )  <->  N  =  ( s  /  t
) ) )
52 breq2 4286 . . . . . . . . . . . . 13  |-  ( y  =  t  ->  (
( P ^ n
)  ||  y  <->  ( P ^ n )  ||  t ) )
5352rabbidv 2956 . . . . . . . . . . . 12  |-  ( y  =  t  ->  { n  e.  NN0  |  ( P ^ n )  ||  y }  =  {
n  e.  NN0  | 
( P ^ n
)  ||  t }
)
5453supeq1d 7686 . . . . . . . . . . 11  |-  ( y  =  t  ->  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  y } ,  RR ,  <  )  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  t } ,  RR ,  <  ) )
5517, 54syl5eq 2479 . . . . . . . . . 10  |-  ( y  =  t  ->  T  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  t } ,  RR ,  <  ) )
5655oveq2d 6098 . . . . . . . . 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 2446 . . . . . . . 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 705 . . . . . . 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 2940 . . . . . 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 261 . . . . 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 2940 . . . 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 261 . . 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 2316 . 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 659 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 369    /\ w3a 960   A.wal 1362    = wceq 1364   E.wex 1591    e. wcel 1757   E!weu 2256    =/= wne 2598   E.wrex 2708   {crab 2711   class class class wbr 4282  (class class class)co 6082   supcsup 7680   RRcr 9271   0cc0 9272    < clt 9408    - cmin 9585    / cdiv 9983   NNcn 10312   NN0cn0 10569   ZZcz 10636   QQcq 10943   ^cexp 11851    || cdivides 13520   Primecprime 13748
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1671  ax-6 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2416  ax-sep 4403  ax-nul 4411  ax-pow 4460  ax-pr 4521  ax-un 6363  ax-cnex 9328  ax-resscn 9329  ax-1cn 9330  ax-icn 9331  ax-addcl 9332  ax-addrcl 9333  ax-mulcl 9334  ax-mulrcl 9335  ax-mulcom 9336  ax-addass 9337  ax-mulass 9338  ax-distr 9339  ax-i2m1 9340  ax-1ne0 9341  ax-1rid 9342  ax-rnegex 9343  ax-rrecex 9344  ax-cnre 9345  ax-pre-lttri 9346  ax-pre-lttrn 9347  ax-pre-ltadd 9348  ax-pre-mulgt0 9349  ax-pre-sup 9350
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1702  df-eu 2260  df-mo 2261  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2966  df-sbc 3178  df-csb 3279  df-dif 3321  df-un 3323  df-in 3325  df-ss 3332  df-pss 3334  df-nul 3628  df-if 3782  df-pw 3852  df-sn 3868  df-pr 3870  df-tp 3872  df-op 3874  df-uni 4082  df-int 4119  df-iun 4163  df-br 4283  df-opab 4341  df-mpt 4342  df-tr 4376  df-eprel 4621  df-id 4625  df-po 4630  df-so 4631  df-fr 4668  df-we 4670  df-ord 4711  df-on 4712  df-lim 4713  df-suc 4714  df-xp 4835  df-rel 4836  df-cnv 4837  df-co 4838  df-dm 4839  df-rn 4840  df-res 4841  df-ima 4842  df-iota 5371  df-fun 5410  df-fn 5411  df-f 5412  df-f1 5413  df-fo 5414  df-f1o 5415  df-fv 5416  df-riota 6041  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-om 6468  df-1st 6568  df-2nd 6569  df-recs 6820  df-rdg 6854  df-1o 6910  df-2o 6911  df-oadd 6914  df-er 7091  df-en 7301  df-dom 7302  df-sdom 7303  df-fin 7304  df-sup 7681  df-pnf 9410  df-mnf 9411  df-xr 9412  df-ltxr 9413  df-le 9414  df-sub 9587  df-neg 9588  df-div 9984  df-nn 10313  df-2 10370  df-3 10371  df-n0 10570  df-z 10637  df-uz 10852  df-q 10944  df-rp 10982  df-fl 11628  df-mod 11695  df-seq 11793  df-exp 11852  df-cj 12574  df-re 12575  df-im 12576  df-sqr 12710  df-abs 12711  df-dvds 13521  df-gcd 13676  df-prm 13749
This theorem is referenced by:  pczpre  13899  pcdiv  13904
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