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Theorem pceu 12464
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 529 . . . 4 |- ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) -> N e. QQ )
2 elq 9696 . . . 4 |- ( N e. QQ <-> E. x e. ZZ E. y e. NN N = ( x / y ) )
31, 2sylib 122 . . 3 |- ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) -> E. x e. ZZ E. y e. NN N = ( x / y ) )
4 simpr 110 . . . . . . . . . . 11 |- ( ( ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ y e. NN ) /\ N = ( x / y ) ) -> N = ( x / y ) )
5 simprr 531 . . . . . . . . . . . . 13 |- ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) -> N =/= 0 )
65ad3antrrr 492 . . . . . . . . . . . 12 |- ( ( ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ y e. NN ) /\ N = ( x / y ) ) -> N =/= 0 )
71ad3antrrr 492 . . . . . . . . . . . . 13 |- ( ( ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ y e. NN ) /\ N = ( x / y ) ) -> N e. QQ )
8 0z 9337 . . . . . . . . . . . . . 14 |- 0 e. ZZ
9 zq 9700 . . . . . . . . . . . . . 14 |- ( 0 e. ZZ -> 0 e. QQ )
108, 9mp1i 10 . . . . . . . . . . . . 13 |- ( ( ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ y e. NN ) /\ N = ( x / y ) ) -> 0 e. QQ )
11 qapne 9713 . . . . . . . . . . . . 13 |- ( ( N e. QQ /\ 0 e. QQ ) -> ( N # 0 <-> N =/= 0 ) )
127, 10, 11syl2anc 411 . . . . . . . . . . . 12 |- ( ( ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ y e. NN ) /\ N = ( x / y ) ) -> ( N # 0 <-> N =/= 0 ) )
136, 12mpbird 167 . . . . . . . . . . 11 |- ( ( ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ y e. NN ) /\ N = ( x / y ) ) -> N # 0 )
144, 13eqbrtrrd 4057 . . . . . . . . . 10 |- ( ( ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ y e. NN ) /\ N = ( x / y ) ) -> ( x / y ) # 0 )
15 simpllr 534 . . . . . . . . . . . 12 |- ( ( ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ y e. NN ) /\ N = ( x / y ) ) -> x e. ZZ )
1615zcnd 9449 . . . . . . . . . . 11 |- ( ( ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ y e. NN ) /\ N = ( x / y ) ) -> x e. CC )
17 nnz 9345 . . . . . . . . . . . . . 14 |- ( y e. NN -> y e. ZZ )
1817adantl 277 . . . . . . . . . . . . 13 |- ( ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ y e. NN ) -> y e. ZZ )
1918adantr 276 . . . . . . . . . . . 12 |- ( ( ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ y e. NN ) /\ N = ( x / y ) ) -> y e. ZZ )
2019zcnd 9449 . . . . . . . . . . 11 |- ( ( ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ y e. NN ) /\ N = ( x / y ) ) -> y e. CC )
21 simplr 528 . . . . . . . . . . . 12 |- ( ( ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ y e. NN ) /\ N = ( x / y ) ) -> y e. NN )
2221nnap0d 9036 . . . . . . . . . . 11 |- ( ( ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ y e. NN ) /\ N = ( x / y ) ) -> y # 0 )
2316, 20, 22divap0bd 8829 . . . . . . . . . 10 |- ( ( ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ y e. NN ) /\ N = ( x / y ) ) -> ( x # 0 <-> ( x / y ) # 0 ) )
2414, 23mpbird 167 . . . . . . . . 9 |- ( ( ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ y e. NN ) /\ N = ( x / y ) ) -> x # 0 )
25 0zd 9338 . . . . . . . . . 10 |- ( ( ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ y e. NN ) /\ N = ( x / y ) ) -> 0 e. ZZ )
26 zapne 9400 . . . . . . . . . 10 |- ( ( x e. ZZ /\ 0 e. ZZ ) -> ( x # 0 <-> x =/= 0 ) )
2715, 25, 26syl2anc 411 . . . . . . . . 9 |- ( ( ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ y e. NN ) /\ N = ( x / y ) ) -> ( x # 0 <-> x =/= 0 ) )
2824, 27mpbid 147 . . . . . . . 8 |- ( ( ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ y e. NN ) /\ N = ( x / y ) ) -> x =/= 0 )
2928ex 115 . . . . . . 7 |- ( ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ y e. NN ) -> ( N = ( x / y ) -> x =/= 0 ) )
3029adantrd 279 . . . . . . . 8 |- ( ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ y e. NN ) -> ( ( N = ( x / y ) /\ z = ( S - T ) ) -> x =/= 0 ) )
3130exlimdv 1833 . . . . . . 7 |- ( ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ y e. NN ) -> ( E. z ( N = ( x / y ) /\ z = ( S - T ) ) -> x =/= 0 ) )
32 prmuz2 12299 . . . . . . . . . . . . . . 15 |- ( P e. Prime -> P e. ( ZZ>= ` 2 ) )
3332ad3antrrr 492 . . . . . . . . . . . . . 14 |- ( ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ y e. NN ) -> P e. ( ZZ>= ` 2 ) )
3433adantr 276 . . . . . . . . . . . . 13 |- ( ( ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ y e. NN ) /\ x =/= 0 ) -> P e. ( ZZ>= ` 2 ) )
35 simpllr 534 . . . . . . . . . . . . 13 |- ( ( ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ y e. NN ) /\ x =/= 0 ) -> x e. ZZ )
36 simpr 110 . . . . . . . . . . . . 13 |- ( ( ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ y e. NN ) /\ x =/= 0 ) -> x =/= 0 )
37 eqid 2196 . . . . . . . . . . . . . . 15 |- { n e. NN0 | ( P ^ n ) || x } = { n e. NN0 | ( P ^ n ) || x }
38 pcval.1 . . . . . . . . . . . . . . 15 |- S = sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < )
3937, 38pcprecl 12458 . . . . . . . . . . . . . 14 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( x e. ZZ /\ x =/= 0 ) ) -> ( S e. NN0 /\ ( P ^ S ) || x ) )
4039simpld 112 . . . . . . . . . . . . 13 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( x e. ZZ /\ x =/= 0 ) ) -> S e. NN0 )
4134, 35, 36, 40syl12anc 1247 . . . . . . . . . . . 12 |- ( ( ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ y e. NN ) /\ x =/= 0 ) -> S e. NN0 )
4241nn0zd 9446 . . . . . . . . . . 11 |- ( ( ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ y e. NN ) /\ x =/= 0 ) -> S e. ZZ )
43 nnne0 9018 . . . . . . . . . . . . . . . 16 |- ( y e. NN -> y =/= 0 )
4443adantl 277 . . . . . . . . . . . . . . 15 |- ( ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ y e. NN ) -> y =/= 0 )
45 eqid 2196 . . . . . . . . . . . . . . . 16 |- { n e. NN0 | ( P ^ n ) || y } = { n e. NN0 | ( P ^ n ) || y }
46 pcval.2 . . . . . . . . . . . . . . . 16 |- T = sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < )
4745, 46pcprecl 12458 . . . . . . . . . . . . . . 15 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( y e. ZZ /\ y =/= 0 ) ) -> ( T e. NN0 /\ ( P ^ T ) || y ) )
4833, 18, 44, 47syl12anc 1247 . . . . . . . . . . . . . 14 |- ( ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ y e. NN ) -> ( T e. NN0 /\ ( P ^ T ) || y ) )
4948simpld 112 . . . . . . . . . . . . 13 |- ( ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ y e. NN ) -> T e. NN0 )
5049adantr 276 . . . . . . . . . . . 12 |- ( ( ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ y e. NN ) /\ x =/= 0 ) -> T e. NN0 )
5150nn0zd 9446 . . . . . . . . . . 11 |- ( ( ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ y e. NN ) /\ x =/= 0 ) -> T e. ZZ )
5242, 51zsubcld 9453 . . . . . . . . . 10 |- ( ( ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ y e. NN ) /\ x =/= 0 ) -> ( S - T ) e. ZZ )
53 biidd 172 . . . . . . . . . . 11 |- ( z = ( S - T ) -> ( N = ( x / y ) <-> N = ( x / y ) ) )
5453ceqsexgv 2893 . . . . . . . . . 10 |- ( ( S - T ) e. ZZ -> ( E. z ( z = ( S - T ) /\ N = ( x / y ) ) <-> N = ( x / y ) ) )
5552, 54syl 14 . . . . . . . . 9 |- ( ( ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ y e. NN ) /\ x =/= 0 ) -> ( E. z ( z = ( S - T ) /\ N = ( x / y ) ) <-> N = ( x / y ) ) )
56 exancom 1622 . . . . . . . . 9 |- ( E. z ( z = ( S - T ) /\ N = ( x / y ) ) <-> E. z ( N = ( x / y ) /\ z = ( S - T ) ) )
5755, 56bitr3di 195 . . . . . . . 8 |- ( ( ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ y e. NN ) /\ x =/= 0 ) -> ( N = ( x / y ) <-> E. z ( N = ( x / y ) /\ z = ( S - T ) ) ) )
5857ex 115 . . . . . . 7 |- ( ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ y e. NN ) -> ( x =/= 0 -> ( N = ( x / y ) <-> E. z ( N = ( x / y ) /\ z = ( S - T ) ) ) ) )
5929, 31, 58pm5.21ndd 706 . . . . . 6 |- ( ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ y e. NN ) -> ( N = ( x / y ) <-> E. z ( N = ( x / y ) /\ z = ( S - T ) ) ) )
6059rexbidva 2494 . . . . 5 |- ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) -> ( E. y e. NN N = ( x / y ) <-> E. y e. NN E. z ( N = ( x / y ) /\ z = ( S - T ) ) ) )
6160rexbidva 2494 . . . 4 |- ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) -> ( E. x e. ZZ E. y e. NN N = ( x / y ) <-> E. x e. ZZ E. y e. NN E. z ( N = ( x / y ) /\ z = ( S - T ) ) ) )
62 rexcom4 2786 . . . . . 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 ) ) )
6362rexbii 2504 . . . . 5 |- ( E. x e. ZZ E. y e. NN E. z ( N = ( x / y ) /\ z = ( S - T ) ) <-> E. x e. ZZ E. z E. y e. NN ( N = ( x / y ) /\ z = ( S - T ) ) )
64 rexcom4 2786 . . . . 5 |- ( 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 ) ) )
6563, 64bitri 184 . . . 4 |- ( E. x e. ZZ E. y e. NN E. z ( N = ( x / y ) /\ z = ( S - T ) ) <-> E. z E. x e. ZZ E. y e. NN ( N = ( x / y ) /\ z = ( S - T ) ) )
6661, 65bitrdi 196 . . 3 |- ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) -> ( 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 ) ) ) )
673, 66mpbid 147 . 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 ) ) )
68 eqid 2196 . . . . . . . . . . 11 |- sup ( { n e. NN0 | ( P ^ n ) || s } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || s } , RR , < )
69 eqid 2196 . . . . . . . . . . 11 |- sup ( { n e. NN0 | ( P ^ n ) || t } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || t } , RR , < )
70 simp11l 1110 . . . . . . . . . . 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 )
71 simp11r 1111 . . . . . . . . . . 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 )
72 simp12 1030 . . . . . . . . . . 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 ) )
73 simp13l 1114 . . . . . . . . . . 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 ) )
74 simp2 1000 . . . . . . . . . . 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 ) )
75 simp3l 1027 . . . . . . . . . . 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 ) )
7638, 46, 68, 69, 70, 71, 72, 73, 74, 75pceulem 12463 . . . . . . . . . 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 , < ) ) )
77 simp13r 1115 . . . . . . . . . 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 ) )
78 simp3r 1028 . . . . . . . . . 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 , < ) ) )
7976, 77, 783eqtr4d 2239 . . . . . . . . 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 )
80793exp 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 ) ) )
8180rexlimdvv 2621 . . . . . . 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 ) )
82813exp 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 ) ) ) )
8382adantrl 478 . . . . 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 ) ) ) )
8483rexlimdvv 2621 . . . 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 ) ) )
8584impd 254 . . 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 ) )
8685alrimivv 1889 . 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 ) )
87 eqeq1 2203 . . . . . 6 |- ( z = w -> ( z = ( S - T ) <-> w = ( S - T ) ) )
8887anbi2d 464 . . . . 5 |- ( z = w -> ( ( N = ( x / y ) /\ z = ( S - T ) ) <-> ( N = ( x / y ) /\ w = ( S - T ) ) ) )
89882rexbidv 2522 . . . 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 ) ) ) )
90 oveq1 5929 . . . . . . . . 9 |- ( x = s -> ( x / y ) = ( s / y ) )
9190eqeq2d 2208 . . . . . . . 8 |- ( x = s -> ( N = ( x / y ) <-> N = ( s / y ) ) )
92 breq2 4037 . . . . . . . . . . . . 13 |- ( x = s -> ( ( P ^ n ) || x <-> ( P ^ n ) || s ) )
9392rabbidv 2752 . . . . . . . . . . . 12 |- ( x = s -> { n e. NN0 | ( P ^ n ) || x } = { n e. NN0 | ( P ^ n ) || s } )
9493supeq1d 7053 . . . . . . . . . . 11 |- ( x = s -> sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || s } , RR , < ) )
9538, 94eqtrid 2241 . . . . . . . . . 10 |- ( x = s -> S = sup ( { n e. NN0 | ( P ^ n ) || s } , RR , < ) )
9695oveq1d 5937 . . . . . . . . 9 |- ( x = s -> ( S - T ) = ( sup ( { n e. NN0 | ( P ^ n ) || s } , RR , < ) - T ) )
9796eqeq2d 2208 . . . . . . . 8 |- ( x = s -> ( w = ( S - T ) <-> w = ( sup ( { n e. NN0 | ( P ^ n ) || s } , RR , < ) - T ) ) )
9891, 97anbi12d 473 . . . . . . 7 |- ( x = s -> ( ( N = ( x / y ) /\ w = ( S - T ) ) <-> ( N = ( s / y ) /\ w = ( sup ( { n e. NN0 | ( P ^ n ) || s } , RR , < ) - T ) ) ) )
9998rexbidv 2498 . . . . . 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 ) ) ) )
100 oveq2 5930 . . . . . . . . 9 |- ( y = t -> ( s / y ) = ( s / t ) )
101100eqeq2d 2208 . . . . . . . 8 |- ( y = t -> ( N = ( s / y ) <-> N = ( s / t ) ) )
102 breq2 4037 . . . . . . . . . . . . 13 |- ( y = t -> ( ( P ^ n ) || y <-> ( P ^ n ) || t ) )
103102rabbidv 2752 . . . . . . . . . . . 12 |- ( y = t -> { n e. NN0 | ( P ^ n ) || y } = { n e. NN0 | ( P ^ n ) || t } )
104103supeq1d 7053 . . . . . . . . . . 11 |- ( y = t -> sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || t } , RR , < ) )
10546, 104eqtrid 2241 . . . . . . . . . 10 |- ( y = t -> T = sup ( { n e. NN0 | ( P ^ n ) || t } , RR , < ) )
106105oveq2d 5938 . . . . . . . . 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 , < ) ) )
107106eqeq2d 2208 . . . . . . . 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 , < ) ) ) )
108101, 107anbi12d 473 . . . . . . 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 , < ) ) ) ) )
109108cbvrexvw 2734 . . . . . 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 , < ) ) ) )
11099, 109bitrdi 196 . . . . 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 , < ) ) ) ) )
111110cbvrexvw 2734 . . . 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 , < ) ) ) )
11289, 111bitrdi 196 . . 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 , < ) ) ) ) )
113112eu4 2107 . 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 ) ) )
11467, 86, 113sylanbrc 417 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 104   <-> wb 105   /\ w3a 980   A.wal 1362   = wceq 1364   E.wex 1506   E!weu 2045   e. wcel 2167   =/= wne 2367   E.wrex 2476   {crab 2479   class class class wbr 4033   ` cfv 5258  (class class class)co 5922   supcsup 7048   RRcr 7878   0cc0 7879   < clt 8061   - cmin 8197   # cap 8608   / cdiv 8699   NNcn 8990   2c2 9041   NN0cn0 9249   ZZcz 9326   ZZ>=cuz 9601   QQcq 9693   ^cexp 10630   || cdvds 11952   Primecprime 12275
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-mulrcl 7978  ax-addcom 7979  ax-mulcom 7980  ax-addass 7981  ax-mulass 7982  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-1rid 7986  ax-0id 7987  ax-rnegex 7988  ax-precex 7989  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-apti 7994  ax-pre-ltadd 7995  ax-pre-mulgt0 7996  ax-pre-mulext 7997  ax-arch 7998  ax-caucvg 7999
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-po 4331  df-iso 4332  df-iord 4401  df-on 4403  df-ilim 4404  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-isom 5267  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-frec 6449  df-1o 6474  df-2o 6475  df-er 6592  df-en 6800  df-sup 7050  df-inf 7051  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-reap 8602  df-ap 8609  df-div 8700  df-inn 8991  df-2 9049  df-3 9050  df-4 9051  df-n0 9250  df-z 9327  df-uz 9602  df-q 9694  df-rp 9729  df-fz 10084  df-fzo 10218  df-fl 10360  df-mod 10415  df-seqfrec 10540  df-exp 10631  df-cj 11007  df-re 11008  df-im 11009  df-rsqrt 11163  df-abs 11164  df-dvds 11953  df-gcd 12121  df-prm 12276
This theorem is referenced by:  pcval  12465  pczpre  12466  pcdiv  12471
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