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Theorem pceu 12278
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 9611 . . . 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 9253 . . . . . . . . . . . . . 14 |- 0 e. ZZ
9 zq 9615 . . . . . . . . . . . . . 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 9628 . . . . . . . . . . . . 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 4024 . . . . . . . . . 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 9365 . . . . . . . . . . 11 |- ( ( ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ y e. NN ) /\ N = ( x / y ) ) -> x e. CC )
17 nnz 9261 . . . . . . . . . . . . . 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 9365 . . . . . . . . . . 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 8954 . . . . . . . . . . 11 |- ( ( ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ y e. NN ) /\ N = ( x / y ) ) -> y # 0 )
2316, 20, 22divap0bd 8748 . . . . . . . . . 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 9254 . . . . . . . . . 10 |- ( ( ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ y e. NN ) /\ N = ( x / y ) ) -> 0 e. ZZ )
26 zapne 9316 . . . . . . . . . 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 1819 . . . . . . 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 12114 . . . . . . . . . . . . . . 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 2177 . . . . . . . . . . . . . . 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 12272 . . . . . . . . . . . . . 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 1236 . . . . . . . . . . . 12 |- ( ( ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ y e. NN ) /\ x =/= 0 ) -> S e. NN0 )
4241nn0zd 9362 . . . . . . . . . . 11 |- ( ( ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ y e. NN ) /\ x =/= 0 ) -> S e. ZZ )
43 nnne0 8936 . . . . . . . . . . . . . . . 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 2177 . . . . . . . . . . . . . . . 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 12272 . . . . . . . . . . . . . . 15 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( y e. ZZ /\ y =/= 0 ) ) -> ( T e. NN0 /\ ( P ^ T ) || y ) )
4833, 18, 44, 47syl12anc 1236 . . . . . . . . . . . . . 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 9362 . . . . . . . . . . 11 |- ( ( ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ y e. NN ) /\ x =/= 0 ) -> T e. ZZ )
5242, 51zsubcld 9369 . . . . . . . . . 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 2866 . . . . . . . . . 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 1608 . . . . . . . . 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 705 . . . . . 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 2474 . . . . 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 2474 . . . 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 2760 . . . . . 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 2484 . . . . 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 2760 . . . . 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 2177 . . . . . . . . . . 11 |- sup ( { n e. NN0 | ( P ^ n ) || s } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || s } , RR , < )
69 eqid 2177 . . . . . . . . . . 11 |- sup ( { n e. NN0 | ( P ^ n ) || t } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || t } , RR , < )
70 simp11l 1108 . . . . . . . . . . 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 1109 . . . . . . . . . . 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 1028 . . . . . . . . . . 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 1112 . . . . . . . . . . 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 998 . . . . . . . . . . 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 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 , < ) ) ) ) -> N = ( s / t ) )
7638, 46, 68, 69, 70, 71, 72, 73, 74, 75pceulem 12277 . . . . . . . . . 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 1113 . . . . . . . . . 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 1026 . . . . . . . . . 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 2220 . . . . . . . . 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 1202 . . . . . . . 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 2601 . . . . . . 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 1202 . . . . . 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 2601 . . . 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 1875 . 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 2184 . . . . . 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 2502 . . . 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 5876 . . . . . . . . 9 |- ( x = s -> ( x / y ) = ( s / y ) )
9190eqeq2d 2189 . . . . . . . 8 |- ( x = s -> ( N = ( x / y ) <-> N = ( s / y ) ) )
92 breq2 4004 . . . . . . . . . . . . 13 |- ( x = s -> ( ( P ^ n ) || x <-> ( P ^ n ) || s ) )
9392rabbidv 2726 . . . . . . . . . . . 12 |- ( x = s -> { n e. NN0 | ( P ^ n ) || x } = { n e. NN0 | ( P ^ n ) || s } )
9493supeq1d 6980 . . . . . . . . . . 11 |- ( x = s -> sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || s } , RR , < ) )
9538, 94eqtrid 2222 . . . . . . . . . 10 |- ( x = s -> S = sup ( { n e. NN0 | ( P ^ n ) || s } , RR , < ) )
9695oveq1d 5884 . . . . . . . . 9 |- ( x = s -> ( S - T ) = ( sup ( { n e. NN0 | ( P ^ n ) || s } , RR , < ) - T ) )
9796eqeq2d 2189 . . . . . . . 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 2478 . . . . . 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 5877 . . . . . . . . 9 |- ( y = t -> ( s / y ) = ( s / t ) )
101100eqeq2d 2189 . . . . . . . 8 |- ( y = t -> ( N = ( s / y ) <-> N = ( s / t ) ) )
102 breq2 4004 . . . . . . . . . . . . 13 |- ( y = t -> ( ( P ^ n ) || y <-> ( P ^ n ) || t ) )
103102rabbidv 2726 . . . . . . . . . . . 12 |- ( y = t -> { n e. NN0 | ( P ^ n ) || y } = { n e. NN0 | ( P ^ n ) || t } )
104103supeq1d 6980 . . . . . . . . . . 11 |- ( y = t -> sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || t } , RR , < ) )
10546, 104eqtrid 2222 . . . . . . . . . 10 |- ( y = t -> T = sup ( { n e. NN0 | ( P ^ n ) || t } , RR , < ) )
106105oveq2d 5885 . . . . . . . . 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 2189 . . . . . . . 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 2708 . . . . . 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 2708 . . . 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 2088 . 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 978   A.wal 1351   = wceq 1353   E.wex 1492   E!weu 2026   e. wcel 2148   =/= wne 2347   E.wrex 2456   {crab 2459   class class class wbr 4000   ` cfv 5212  (class class class)co 5869   supcsup 6975   RRcr 7801   0cc0 7802   < clt 7982   - cmin 8118   # cap 8528   / cdiv 8618   NNcn 8908   2c2 8959   NN0cn0 9165   ZZcz 9242   ZZ>=cuz 9517   QQcq 9608   ^cexp 10505   || cdvds 11778   Primecprime 12090
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-mulrcl 7901  ax-addcom 7902  ax-mulcom 7903  ax-addass 7904  ax-mulass 7905  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-1rid 7909  ax-0id 7910  ax-rnegex 7911  ax-precex 7912  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918  ax-pre-mulgt0 7919  ax-pre-mulext 7920  ax-arch 7921  ax-caucvg 7922
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-ilim 4366  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-isom 5221  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-frec 6386  df-1o 6411  df-2o 6412  df-er 6529  df-en 6735  df-sup 6977  df-inf 6978  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-reap 8522  df-ap 8529  df-div 8619  df-inn 8909  df-2 8967  df-3 8968  df-4 8969  df-n0 9166  df-z 9243  df-uz 9518  df-q 9609  df-rp 9641  df-fz 9996  df-fzo 10129  df-fl 10256  df-mod 10309  df-seqfrec 10432  df-exp 10506  df-cj 10835  df-re 10836  df-im 10837  df-rsqrt 10991  df-abs 10992  df-dvds 11779  df-gcd 11927  df-prm 12091
This theorem is referenced by:  pcval  12279  pczpre  12280  pcdiv  12285
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