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Theorem pceu 12186
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 521 . . . 4 |- ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) -> N e. QQ )
2 elq 9538 . . . 4 |- ( N e. QQ <-> E. x e. ZZ E. y e. NN N = ( x / y ) )
31, 2sylib 121 . . 3 |- ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) -> E. x e. ZZ E. y e. NN N = ( x / y ) )
4 simpr 109 . . . . . . . . . . 11 |- ( ( ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ y e. NN ) /\ N = ( x / y ) ) -> N = ( x / y ) )
5 simprr 522 . . . . . . . . . . . . 13 |- ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) -> N =/= 0 )
65ad3antrrr 484 . . . . . . . . . . . 12 |- ( ( ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ y e. NN ) /\ N = ( x / y ) ) -> N =/= 0 )
71ad3antrrr 484 . . . . . . . . . . . . 13 |- ( ( ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ y e. NN ) /\ N = ( x / y ) ) -> N e. QQ )
8 0z 9184 . . . . . . . . . . . . . 14 |- 0 e. ZZ
9 zq 9542 . . . . . . . . . . . . . 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 9555 . . . . . . . . . . . . 13 |- ( ( N e. QQ /\ 0 e. QQ ) -> ( N # 0 <-> N =/= 0 ) )
127, 10, 11syl2anc 409 . . . . . . . . . . . 12 |- ( ( ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ y e. NN ) /\ N = ( x / y ) ) -> ( N # 0 <-> N =/= 0 ) )
136, 12mpbird 166 . . . . . . . . . . 11 |- ( ( ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ y e. NN ) /\ N = ( x / y ) ) -> N # 0 )
144, 13eqbrtrrd 3991 . . . . . . . . . 10 |- ( ( ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ y e. NN ) /\ N = ( x / y ) ) -> ( x / y ) # 0 )
15 simpllr 524 . . . . . . . . . . . 12 |- ( ( ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ y e. NN ) /\ N = ( x / y ) ) -> x e. ZZ )
1615zcnd 9293 . . . . . . . . . . 11 |- ( ( ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ y e. NN ) /\ N = ( x / y ) ) -> x e. CC )
17 nnz 9192 . . . . . . . . . . . . . 14 |- ( y e. NN -> y e. ZZ )
1817adantl 275 . . . . . . . . . . . . 13 |- ( ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ y e. NN ) -> y e. ZZ )
1918adantr 274 . . . . . . . . . . . 12 |- ( ( ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ y e. NN ) /\ N = ( x / y ) ) -> y e. ZZ )
2019zcnd 9293 . . . . . . . . . . 11 |- ( ( ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ y e. NN ) /\ N = ( x / y ) ) -> y e. CC )
21 simplr 520 . . . . . . . . . . . 12 |- ( ( ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ y e. NN ) /\ N = ( x / y ) ) -> y e. NN )
2221nnap0d 8885 . . . . . . . . . . 11 |- ( ( ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ y e. NN ) /\ N = ( x / y ) ) -> y # 0 )
2316, 20, 22divap0bd 8680 . . . . . . . . . 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 166 . . . . . . . . 9 |- ( ( ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ y e. NN ) /\ N = ( x / y ) ) -> x # 0 )
25 0zd 9185 . . . . . . . . . 10 |- ( ( ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ y e. NN ) /\ N = ( x / y ) ) -> 0 e. ZZ )
26 zapne 9244 . . . . . . . . . 10 |- ( ( x e. ZZ /\ 0 e. ZZ ) -> ( x # 0 <-> x =/= 0 ) )
2715, 25, 26syl2anc 409 . . . . . . . . 9 |- ( ( ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ y e. NN ) /\ N = ( x / y ) ) -> ( x # 0 <-> x =/= 0 ) )
2824, 27mpbid 146 . . . . . . . 8 |- ( ( ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ y e. NN ) /\ N = ( x / y ) ) -> x =/= 0 )
2928ex 114 . . . . . . 7 |- ( ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ y e. NN ) -> ( N = ( x / y ) -> x =/= 0 ) )
3029adantrd 277 . . . . . . . 8 |- ( ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ y e. NN ) -> ( ( N = ( x / y ) /\ z = ( S - T ) ) -> x =/= 0 ) )
3130exlimdv 1799 . . . . . . 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 12024 . . . . . . . . . . . . . . 15 |- ( P e. Prime -> P e. ( ZZ>= ` 2 ) )
3332ad3antrrr 484 . . . . . . . . . . . . . 14 |- ( ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ y e. NN ) -> P e. ( ZZ>= ` 2 ) )
3433adantr 274 . . . . . . . . . . . . 13 |- ( ( ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ y e. NN ) /\ x =/= 0 ) -> P e. ( ZZ>= ` 2 ) )
35 simpllr 524 . . . . . . . . . . . . 13 |- ( ( ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ y e. NN ) /\ x =/= 0 ) -> x e. ZZ )
36 simpr 109 . . . . . . . . . . . . 13 |- ( ( ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ y e. NN ) /\ x =/= 0 ) -> x =/= 0 )
37 eqid 2157 . . . . . . . . . . . . . . 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 12180 . . . . . . . . . . . . . 14 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( x e. ZZ /\ x =/= 0 ) ) -> ( S e. NN0 /\ ( P ^ S ) || x ) )
4039simpld 111 . . . . . . . . . . . . 13 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( x e. ZZ /\ x =/= 0 ) ) -> S e. NN0 )
4134, 35, 36, 40syl12anc 1218 . . . . . . . . . . . 12 |- ( ( ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ y e. NN ) /\ x =/= 0 ) -> S e. NN0 )
4241nn0zd 9290 . . . . . . . . . . 11 |- ( ( ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ y e. NN ) /\ x =/= 0 ) -> S e. ZZ )
43 nnne0 8867 . . . . . . . . . . . . . . . 16 |- ( y e. NN -> y =/= 0 )
4443adantl 275 . . . . . . . . . . . . . . 15 |- ( ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ y e. NN ) -> y =/= 0 )
45 eqid 2157 . . . . . . . . . . . . . . . 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 12180 . . . . . . . . . . . . . . 15 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( y e. ZZ /\ y =/= 0 ) ) -> ( T e. NN0 /\ ( P ^ T ) || y ) )
4833, 18, 44, 47syl12anc 1218 . . . . . . . . . . . . . 14 |- ( ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ y e. NN ) -> ( T e. NN0 /\ ( P ^ T ) || y ) )
4948simpld 111 . . . . . . . . . . . . 13 |- ( ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ y e. NN ) -> T e. NN0 )
5049adantr 274 . . . . . . . . . . . 12 |- ( ( ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ y e. NN ) /\ x =/= 0 ) -> T e. NN0 )
5150nn0zd 9290 . . . . . . . . . . 11 |- ( ( ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ y e. NN ) /\ x =/= 0 ) -> T e. ZZ )
5242, 51zsubcld 9297 . . . . . . . . . 10 |- ( ( ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ y e. NN ) /\ x =/= 0 ) -> ( S - T ) e. ZZ )
53 biidd 171 . . . . . . . . . . 11 |- ( z = ( S - T ) -> ( N = ( x / y ) <-> N = ( x / y ) ) )
5453ceqsexgv 2841 . . . . . . . . . 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 1588 . . . . . . . . 9 |- ( E. z ( z = ( S - T ) /\ N = ( x / y ) ) <-> E. z ( N = ( x / y ) /\ z = ( S - T ) ) )
5755, 56bitr3di 194 . . . . . . . 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 114 . . . . . . 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 695 . . . . . 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 2454 . . . . 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 2454 . . . 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 2735 . . . . . 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 2464 . . . . 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 2735 . . . . 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 183 . . . 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 195 . . 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 146 . 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 2157 . . . . . . . . . . 11 |- sup ( { n e. NN0 | ( P ^ n ) || s } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || s } , RR , < )
69 eqid 2157 . . . . . . . . . . 11 |- sup ( { n e. NN0 | ( P ^ n ) || t } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || t } , RR , < )
70 simp11l 1093 . . . . . . . . . . 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 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 , < ) ) ) ) -> N =/= 0 )
72 simp12 1013 . . . . . . . . . . 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 1097 . . . . . . . . . . 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 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 , < ) ) ) ) -> ( s e. ZZ /\ t e. NN ) )
75 simp3l 1010 . . . . . . . . . . 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 12185 . . . . . . . . . 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 1098 . . . . . . . . . 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 1011 . . . . . . . . . 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 2200 . . . . . . . . 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 1184 . . . . . . . 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 2581 . . . . . . 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 1184 . . . . . 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 470 . . . . 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 2581 . . . 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 252 . . 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 1855 . 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 2164 . . . . . 6 |- ( z = w -> ( z = ( S - T ) <-> w = ( S - T ) ) )
8887anbi2d 460 . . . . 5 |- ( z = w -> ( ( N = ( x / y ) /\ z = ( S - T ) ) <-> ( N = ( x / y ) /\ w = ( S - T ) ) ) )
89882rexbidv 2482 . . . 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 5834 . . . . . . . . 9 |- ( x = s -> ( x / y ) = ( s / y ) )
9190eqeq2d 2169 . . . . . . . 8 |- ( x = s -> ( N = ( x / y ) <-> N = ( s / y ) ) )
92 breq2 3971 . . . . . . . . . . . . 13 |- ( x = s -> ( ( P ^ n ) || x <-> ( P ^ n ) || s ) )
9392rabbidv 2701 . . . . . . . . . . . 12 |- ( x = s -> { n e. NN0 | ( P ^ n ) || x } = { n e. NN0 | ( P ^ n ) || s } )
9493supeq1d 6934 . . . . . . . . . . 11 |- ( x = s -> sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || s } , RR , < ) )
9538, 94syl5eq 2202 . . . . . . . . . 10 |- ( x = s -> S = sup ( { n e. NN0 | ( P ^ n ) || s } , RR , < ) )
9695oveq1d 5842 . . . . . . . . 9 |- ( x = s -> ( S - T ) = ( sup ( { n e. NN0 | ( P ^ n ) || s } , RR , < ) - T ) )
9796eqeq2d 2169 . . . . . . . 8 |- ( x = s -> ( w = ( S - T ) <-> w = ( sup ( { n e. NN0 | ( P ^ n ) || s } , RR , < ) - T ) ) )
9891, 97anbi12d 465 . . . . . . 7 |- ( x = s -> ( ( N = ( x / y ) /\ w = ( S - T ) ) <-> ( N = ( s / y ) /\ w = ( sup ( { n e. NN0 | ( P ^ n ) || s } , RR , < ) - T ) ) ) )
9998rexbidv 2458 . . . . . 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 5835 . . . . . . . . 9 |- ( y = t -> ( s / y ) = ( s / t ) )
101100eqeq2d 2169 . . . . . . . 8 |- ( y = t -> ( N = ( s / y ) <-> N = ( s / t ) ) )
102 breq2 3971 . . . . . . . . . . . . 13 |- ( y = t -> ( ( P ^ n ) || y <-> ( P ^ n ) || t ) )
103102rabbidv 2701 . . . . . . . . . . . 12 |- ( y = t -> { n e. NN0 | ( P ^ n ) || y } = { n e. NN0 | ( P ^ n ) || t } )
104103supeq1d 6934 . . . . . . . . . . 11 |- ( y = t -> sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || t } , RR , < ) )
10546, 104syl5eq 2202 . . . . . . . . . 10 |- ( y = t -> T = sup ( { n e. NN0 | ( P ^ n ) || t } , RR , < ) )
106105oveq2d 5843 . . . . . . . . 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 2169 . . . . . . . 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 465 . . . . . . 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 2685 . . . . . 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 195 . . . . 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 2685 . . . 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 195 . . 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 2068 . 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 414 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 103   <-> wb 104   /\ w3a 963   A.wal 1333   = wceq 1335   E.wex 1472   E!weu 2006   e. wcel 2128   =/= wne 2327   E.wrex 2436   {crab 2439   class class class wbr 3967   ` cfv 5173  (class class class)co 5827   supcsup 6929   RRcr 7734   0cc0 7735   < clt 7915   - cmin 8051   # cap 8461   / cdiv 8550   NNcn 8839   2c2 8890   NN0cn0 9096   ZZcz 9173   ZZ>=cuz 9445   QQcq 9535   ^cexp 10428   || cdvds 11695   Primecprime 12000
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4082  ax-sep 4085  ax-nul 4093  ax-pow 4138  ax-pr 4172  ax-un 4396  ax-setind 4499  ax-iinf 4550  ax-cnex 7826  ax-resscn 7827  ax-1cn 7828  ax-1re 7829  ax-icn 7830  ax-addcl 7831  ax-addrcl 7832  ax-mulcl 7833  ax-mulrcl 7834  ax-addcom 7835  ax-mulcom 7836  ax-addass 7837  ax-mulass 7838  ax-distr 7839  ax-i2m1 7840  ax-0lt1 7841  ax-1rid 7842  ax-0id 7843  ax-rnegex 7844  ax-precex 7845  ax-cnre 7846  ax-pre-ltirr 7847  ax-pre-ltwlin 7848  ax-pre-lttrn 7849  ax-pre-apti 7850  ax-pre-ltadd 7851  ax-pre-mulgt0 7852  ax-pre-mulext 7853  ax-arch 7854  ax-caucvg 7855
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3396  df-if 3507  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-int 3810  df-iun 3853  df-br 3968  df-opab 4029  df-mpt 4030  df-tr 4066  df-id 4256  df-po 4259  df-iso 4260  df-iord 4329  df-on 4331  df-ilim 4332  df-suc 4334  df-iom 4553  df-xp 4595  df-rel 4596  df-cnv 4597  df-co 4598  df-dm 4599  df-rn 4600  df-res 4601  df-ima 4602  df-iota 5138  df-fun 5175  df-fn 5176  df-f 5177  df-f1 5178  df-fo 5179  df-f1o 5180  df-fv 5181  df-isom 5182  df-riota 5783  df-ov 5830  df-oprab 5831  df-mpo 5832  df-1st 6091  df-2nd 6092  df-recs 6255  df-frec 6341  df-1o 6366  df-2o 6367  df-er 6483  df-en 6689  df-sup 6931  df-inf 6932  df-pnf 7917  df-mnf 7918  df-xr 7919  df-ltxr 7920  df-le 7921  df-sub 8053  df-neg 8054  df-reap 8455  df-ap 8462  df-div 8551  df-inn 8840  df-2 8898  df-3 8899  df-4 8900  df-n0 9097  df-z 9174  df-uz 9446  df-q 9536  df-rp 9568  df-fz 9920  df-fzo 10052  df-fl 10179  df-mod 10232  df-seqfrec 10355  df-exp 10429  df-cj 10754  df-re 10755  df-im 10756  df-rsqrt 10910  df-abs 10911  df-dvds 11696  df-gcd 11843  df-prm 12001
This theorem is referenced by:  pcval  12187  pczpre  12188  pcdiv  12193
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