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Theorem pceu 12948
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 531 . . . 4 |- ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) -> N e. QQ )
2 elq 9917 . . . 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 533 . . . . . . . . . . . . 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 9551 . . . . . . . . . . . . . 14 |- 0 e. ZZ
9 zq 9921 . . . . . . . . . . . . . 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 9934 . . . . . . . . . . . . 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 4117 . . . . . . . . . 10 |- ( ( ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ y e. NN ) /\ N = ( x / y ) ) -> ( x / y ) # 0 )
15 simpllr 536 . . . . . . . . . . . 12 |- ( ( ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ y e. NN ) /\ N = ( x / y ) ) -> x e. ZZ )
1615zcnd 9664 . . . . . . . . . . 11 |- ( ( ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ y e. NN ) /\ N = ( x / y ) ) -> x e. CC )
17 nnz 9559 . . . . . . . . . . . . . 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 9664 . . . . . . . . . . 11 |- ( ( ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ y e. NN ) /\ N = ( x / y ) ) -> y e. CC )
21 simplr 529 . . . . . . . . . . . 12 |- ( ( ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ y e. NN ) /\ N = ( x / y ) ) -> y e. NN )
2221nnap0d 9248 . . . . . . . . . . 11 |- ( ( ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ y e. NN ) /\ N = ( x / y ) ) -> y # 0 )
2316, 20, 22divap0bd 9041 . . . . . . . . . 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 9552 . . . . . . . . . 10 |- ( ( ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ y e. NN ) /\ N = ( x / y ) ) -> 0 e. ZZ )
26 zapne 9615 . . . . . . . . . 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 1867 . . . . . . 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 12783 . . . . . . . . . . . . . . 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 536 . . . . . . . . . . . . 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 2231 . . . . . . . . . . . . . . 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 12942 . . . . . . . . . . . . . 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 1272 . . . . . . . . . . . 12 |- ( ( ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ y e. NN ) /\ x =/= 0 ) -> S e. NN0 )
4241nn0zd 9661 . . . . . . . . . . 11 |- ( ( ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ y e. NN ) /\ x =/= 0 ) -> S e. ZZ )
43 nnne0 9230 . . . . . . . . . . . . . . . 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 2231 . . . . . . . . . . . . . . . 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 12942 . . . . . . . . . . . . . . 15 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( y e. ZZ /\ y =/= 0 ) ) -> ( T e. NN0 /\ ( P ^ T ) || y ) )
4833, 18, 44, 47syl12anc 1272 . . . . . . . . . . . . . 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 9661 . . . . . . . . . . 11 |- ( ( ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ y e. NN ) /\ x =/= 0 ) -> T e. ZZ )
5242, 51zsubcld 9668 . . . . . . . . . 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 2936 . . . . . . . . . 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 1657 . . . . . . . . 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 713 . . . . . 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 2530 . . . . 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 2530 . . . 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 2827 . . . . . 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 2540 . . . . 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 2827 . . . . 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 2231 . . . . . . . . . . 11 |- sup ( { n e. NN0 | ( P ^ n ) || s } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || s } , RR , < )
69 eqid 2231 . . . . . . . . . . 11 |- sup ( { n e. NN0 | ( P ^ n ) || t } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || t } , RR , < )
70 simp11l 1135 . . . . . . . . . . 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 1136 . . . . . . . . . . 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 1055 . . . . . . . . . . 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 1139 . . . . . . . . . . 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 1025 . . . . . . . . . . 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 1052 . . . . . . . . . . 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 12947 . . . . . . . . . 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 1140 . . . . . . . . . 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 1053 . . . . . . . . . 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 2274 . . . . . . . . 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 1229 . . . . . . . 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 2658 . . . . . . 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 1229 . . . . . 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 2658 . . . 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 1923 . 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 2238 . . . . . 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 2558 . . . 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 6035 . . . . . . . . 9 |- ( x = s -> ( x / y ) = ( s / y ) )
9190eqeq2d 2243 . . . . . . . 8 |- ( x = s -> ( N = ( x / y ) <-> N = ( s / y ) ) )
92 breq2 4097 . . . . . . . . . . . . 13 |- ( x = s -> ( ( P ^ n ) || x <-> ( P ^ n ) || s ) )
9392rabbidv 2792 . . . . . . . . . . . 12 |- ( x = s -> { n e. NN0 | ( P ^ n ) || x } = { n e. NN0 | ( P ^ n ) || s } )
9493supeq1d 7246 . . . . . . . . . . 11 |- ( x = s -> sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || s } , RR , < ) )
9538, 94eqtrid 2276 . . . . . . . . . 10 |- ( x = s -> S = sup ( { n e. NN0 | ( P ^ n ) || s } , RR , < ) )
9695oveq1d 6043 . . . . . . . . 9 |- ( x = s -> ( S - T ) = ( sup ( { n e. NN0 | ( P ^ n ) || s } , RR , < ) - T ) )
9796eqeq2d 2243 . . . . . . . 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 2534 . . . . . 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 6036 . . . . . . . . 9 |- ( y = t -> ( s / y ) = ( s / t ) )
101100eqeq2d 2243 . . . . . . . 8 |- ( y = t -> ( N = ( s / y ) <-> N = ( s / t ) ) )
102 breq2 4097 . . . . . . . . . . . . 13 |- ( y = t -> ( ( P ^ n ) || y <-> ( P ^ n ) || t ) )
103102rabbidv 2792 . . . . . . . . . . . 12 |- ( y = t -> { n e. NN0 | ( P ^ n ) || y } = { n e. NN0 | ( P ^ n ) || t } )
104103supeq1d 7246 . . . . . . . . . . 11 |- ( y = t -> sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || t } , RR , < ) )
10546, 104eqtrid 2276 . . . . . . . . . 10 |- ( y = t -> T = sup ( { n e. NN0 | ( P ^ n ) || t } , RR , < ) )
106105oveq2d 6044 . . . . . . . . 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 2243 . . . . . . . 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 2773 . . . . . 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 2773 . . . 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 2142 . 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 1005   A.wal 1396   = wceq 1398   E.wex 1541   E!weu 2079   e. wcel 2202   =/= wne 2403   E.wrex 2512   {crab 2515   class class class wbr 4093   ` cfv 5333  (class class class)co 6028   supcsup 7241   RRcr 8091   0cc0 8092   < clt 8273   - cmin 8409   # cap 8820   / cdiv 8911   NNcn 9202   2c2 9253   NN0cn0 9461   ZZcz 9540   ZZ>=cuz 9816   QQcq 9914   ^cexp 10863   || cdvds 12428   Primecprime 12759
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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-mulrcl 8191  ax-addcom 8192  ax-mulcom 8193  ax-addass 8194  ax-mulass 8195  ax-distr 8196  ax-i2m1 8197  ax-0lt1 8198  ax-1rid 8199  ax-0id 8200  ax-rnegex 8201  ax-precex 8202  ax-cnre 8203  ax-pre-ltirr 8204  ax-pre-ltwlin 8205  ax-pre-lttrn 8206  ax-pre-apti 8207  ax-pre-ltadd 8208  ax-pre-mulgt0 8209  ax-pre-mulext 8210  ax-arch 8211  ax-caucvg 8212
This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-isom 5342  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-frec 6600  df-1o 6625  df-2o 6626  df-er 6745  df-en 6953  df-sup 7243  df-inf 7244  df-pnf 8275  df-mnf 8276  df-xr 8277  df-ltxr 8278  df-le 8279  df-sub 8411  df-neg 8412  df-reap 8814  df-ap 8821  df-div 8912  df-inn 9203  df-2 9261  df-3 9262  df-4 9263  df-n0 9462  df-z 9541  df-uz 9817  df-q 9915  df-rp 9950  df-fz 10306  df-fzo 10440  df-fl 10593  df-mod 10648  df-seqfrec 10773  df-exp 10864  df-cj 11482  df-re 11483  df-im 11484  df-rsqrt 11638  df-abs 11639  df-dvds 12429  df-gcd 12605  df-prm 12760
This theorem is referenced by:  pcval  12949  pczpre  12950  pcdiv  12955
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