ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  pceu Unicode version
Theorem pceu 12651
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 9745 . . . 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 9385 . . . . . . . . . . . . . 14 |- 0 e. ZZ
9 zq 9749 . . . . . . . . . . . . . 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 9762 . . . . . . . . . . . . 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 4069 . . . . . . . . . 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 9498 . . . . . . . . . . 11 |- ( ( ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ y e. NN ) /\ N = ( x / y ) ) -> x e. CC )
17 nnz 9393 . . . . . . . . . . . . . 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 9498 . . . . . . . . . . 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 9084 . . . . . . . . . . 11 |- ( ( ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ y e. NN ) /\ N = ( x / y ) ) -> y # 0 )
2316, 20, 22divap0bd 8877 . . . . . . . . . 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 9386 . . . . . . . . . 10 |- ( ( ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ y e. NN ) /\ N = ( x / y ) ) -> 0 e. ZZ )
26 zapne 9449 . . . . . . . . . 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 1842 . . . . . . 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 12486 . . . . . . . . . . . . . . 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 2205 . . . . . . . . . . . . . . 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 12645 . . . . . . . . . . . . . 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 1248 . . . . . . . . . . . 12 |- ( ( ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ y e. NN ) /\ x =/= 0 ) -> S e. NN0 )
4241nn0zd 9495 . . . . . . . . . . 11 |- ( ( ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ y e. NN ) /\ x =/= 0 ) -> S e. ZZ )
43 nnne0 9066 . . . . . . . . . . . . . . . 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 2205 . . . . . . . . . . . . . . . 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 12645 . . . . . . . . . . . . . . 15 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( y e. ZZ /\ y =/= 0 ) ) -> ( T e. NN0 /\ ( P ^ T ) || y ) )
4833, 18, 44, 47syl12anc 1248 . . . . . . . . . . . . . 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 9495 . . . . . . . . . . 11 |- ( ( ( ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) /\ x e. ZZ ) /\ y e. NN ) /\ x =/= 0 ) -> T e. ZZ )
5242, 51zsubcld 9502 . . . . . . . . . 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 2902 . . . . . . . . . 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 1631 . . . . . . . . 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 707 . . . . . 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 2503 . . . . 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 2503 . . . 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 2795 . . . . . 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 2513 . . . . 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 2795 . . . . 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 2205 . . . . . . . . . . 11 |- sup ( { n e. NN0 | ( P ^ n ) || s } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || s } , RR , < )
69 eqid 2205 . . . . . . . . . . 11 |- sup ( { n e. NN0 | ( P ^ n ) || t } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || t } , RR , < )
70 simp11l 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 , < ) ) ) ) -> P e. Prime )
71 simp11r 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 =/= 0 )
72 simp12 1031 . . . . . . . . . . 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 1115 . . . . . . . . . . 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 1001 . . . . . . . . . . 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 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 , < ) ) ) ) -> N = ( s / t ) )
7638, 46, 68, 69, 70, 71, 72, 73, 74, 75pceulem 12650 . . . . . . . . . 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 1116 . . . . . . . . . 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 1029 . . . . . . . . . 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 2248 . . . . . . . . 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 1205 . . . . . . . 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 2630 . . . . . . 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 1205 . . . . . 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 2630 . . . 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 1898 . 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 2212 . . . . . 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 2531 . . . 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 5953 . . . . . . . . 9 |- ( x = s -> ( x / y ) = ( s / y ) )
9190eqeq2d 2217 . . . . . . . 8 |- ( x = s -> ( N = ( x / y ) <-> N = ( s / y ) ) )
92 breq2 4049 . . . . . . . . . . . . 13 |- ( x = s -> ( ( P ^ n ) || x <-> ( P ^ n ) || s ) )
9392rabbidv 2761 . . . . . . . . . . . 12 |- ( x = s -> { n e. NN0 | ( P ^ n ) || x } = { n e. NN0 | ( P ^ n ) || s } )
9493supeq1d 7091 . . . . . . . . . . 11 |- ( x = s -> sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || s } , RR , < ) )
9538, 94eqtrid 2250 . . . . . . . . . 10 |- ( x = s -> S = sup ( { n e. NN0 | ( P ^ n ) || s } , RR , < ) )
9695oveq1d 5961 . . . . . . . . 9 |- ( x = s -> ( S - T ) = ( sup ( { n e. NN0 | ( P ^ n ) || s } , RR , < ) - T ) )
9796eqeq2d 2217 . . . . . . . 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 2507 . . . . . 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 5954 . . . . . . . . 9 |- ( y = t -> ( s / y ) = ( s / t ) )
101100eqeq2d 2217 . . . . . . . 8 |- ( y = t -> ( N = ( s / y ) <-> N = ( s / t ) ) )
102 breq2 4049 . . . . . . . . . . . . 13 |- ( y = t -> ( ( P ^ n ) || y <-> ( P ^ n ) || t ) )
103102rabbidv 2761 . . . . . . . . . . . 12 |- ( y = t -> { n e. NN0 | ( P ^ n ) || y } = { n e. NN0 | ( P ^ n ) || t } )
104103supeq1d 7091 . . . . . . . . . . 11 |- ( y = t -> sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || t } , RR , < ) )
10546, 104eqtrid 2250 . . . . . . . . . 10 |- ( y = t -> T = sup ( { n e. NN0 | ( P ^ n ) || t } , RR , < ) )
106105oveq2d 5962 . . . . . . . . 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 2217 . . . . . . . 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 2743 . . . . . 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 2743 . . . 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 2116 . 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 981   A.wal 1371   = wceq 1373   E.wex 1515   E!weu 2054   e. wcel 2176   =/= wne 2376   E.wrex 2485   {crab 2488   class class class wbr 4045   ` cfv 5272  (class class class)co 5946   supcsup 7086   RRcr 7926   0cc0 7927   < clt 8109   - cmin 8245   # cap 8656   / cdiv 8747   NNcn 9038   2c2 9089   NN0cn0 9297   ZZcz 9374   ZZ>=cuz 9650   QQcq 9742   ^cexp 10685   || cdvds 12131   Primecprime 12462
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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4160  ax-sep 4163  ax-nul 4171  ax-pow 4219  ax-pr 4254  ax-un 4481  ax-setind 4586  ax-iinf 4637  ax-cnex 8018  ax-resscn 8019  ax-1cn 8020  ax-1re 8021  ax-icn 8022  ax-addcl 8023  ax-addrcl 8024  ax-mulcl 8025  ax-mulrcl 8026  ax-addcom 8027  ax-mulcom 8028  ax-addass 8029  ax-mulass 8030  ax-distr 8031  ax-i2m1 8032  ax-0lt1 8033  ax-1rid 8034  ax-0id 8035  ax-rnegex 8036  ax-precex 8037  ax-cnre 8038  ax-pre-ltirr 8039  ax-pre-ltwlin 8040  ax-pre-lttrn 8041  ax-pre-apti 8042  ax-pre-ltadd 8043  ax-pre-mulgt0 8044  ax-pre-mulext 8045  ax-arch 8046  ax-caucvg 8047
This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-if 3572  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4046  df-opab 4107  df-mpt 4108  df-tr 4144  df-id 4341  df-po 4344  df-iso 4345  df-iord 4414  df-on 4416  df-ilim 4417  df-suc 4419  df-iom 4640  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-res 4688  df-ima 4689  df-iota 5233  df-fun 5274  df-fn 5275  df-f 5276  df-f1 5277  df-fo 5278  df-f1o 5279  df-fv 5280  df-isom 5281  df-riota 5901  df-ov 5949  df-oprab 5950  df-mpo 5951  df-1st 6228  df-2nd 6229  df-recs 6393  df-frec 6479  df-1o 6504  df-2o 6505  df-er 6622  df-en 6830  df-sup 7088  df-inf 7089  df-pnf 8111  df-mnf 8112  df-xr 8113  df-ltxr 8114  df-le 8115  df-sub 8247  df-neg 8248  df-reap 8650  df-ap 8657  df-div 8748  df-inn 9039  df-2 9097  df-3 9098  df-4 9099  df-n0 9298  df-z 9375  df-uz 9651  df-q 9743  df-rp 9778  df-fz 10133  df-fzo 10267  df-fl 10415  df-mod 10470  df-seqfrec 10595  df-exp 10686  df-cj 11186  df-re 11187  df-im 11188  df-rsqrt 11342  df-abs 11343  df-dvds 12132  df-gcd 12308  df-prm 12463
This theorem is referenced by:  pcval  12652  pczpre  12653  pcdiv  12658
  Copyright terms: Public domain W3C validator