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Theorem pc11 15584
Description: The prime count function, viewed as a function from NN to ( NN ^m Prime ), is one-to-one. (Contributed by Mario Carneiro, 23-Feb-2014.)
Assertion
Ref Expression
pc11 |- ( ( A e. NN0 /\ B e. NN0 ) -> ( A = B <-> A. p e. Prime ( p pCnt A ) = ( p pCnt B ) ) )
Distinct variable groups:   A, p   B, p
Proof of Theorem pc11
StepHypRef Expression
1 oveq2 6658 . . 3 |- ( A = B -> ( p pCnt A ) = ( p pCnt B ) )
21ralrimivw 2967 . 2 |- ( A = B -> A. p e. Prime ( p pCnt A ) = ( p pCnt B ) )
3 nn0z 11400 . . . 4 |- ( A e. NN0 -> A e. ZZ )
4 nn0z 11400 . . . 4 |- ( B e. NN0 -> B e. ZZ )
5 zq 11794 . . . . . . . . . . 11 |- ( A e. ZZ -> A e. QQ )
6 pcxcl 15565 . . . . . . . . . . 11 |- ( ( p e. Prime /\ A e. QQ ) -> ( p pCnt A ) e. RR* )
75, 6sylan2 491 . . . . . . . . . 10 |- ( ( p e. Prime /\ A e. ZZ ) -> ( p pCnt A ) e. RR* )
8 zq 11794 . . . . . . . . . . 11 |- ( B e. ZZ -> B e. QQ )
9 pcxcl 15565 . . . . . . . . . . 11 |- ( ( p e. Prime /\ B e. QQ ) -> ( p pCnt B ) e. RR* )
108, 9sylan2 491 . . . . . . . . . 10 |- ( ( p e. Prime /\ B e. ZZ ) -> ( p pCnt B ) e. RR* )
117, 10anim12dan 882 . . . . . . . . 9 |- ( ( p e. Prime /\ ( A e. ZZ /\ B e. ZZ ) ) -> ( ( p pCnt A ) e. RR* /\ ( p pCnt B ) e. RR* ) )
12 xrletri3 11985 . . . . . . . . 9 |- ( ( ( p pCnt A ) e. RR* /\ ( p pCnt B ) e. RR* ) -> ( ( p pCnt A ) = ( p pCnt B ) <-> ( ( p pCnt A ) <_ ( p pCnt B ) /\ ( p pCnt B ) <_ ( p pCnt A ) ) ) )
1311, 12syl 17 . . . . . . . 8 |- ( ( p e. Prime /\ ( A e. ZZ /\ B e. ZZ ) ) -> ( ( p pCnt A ) = ( p pCnt B ) <-> ( ( p pCnt A ) <_ ( p pCnt B ) /\ ( p pCnt B ) <_ ( p pCnt A ) ) ) )
1413ancoms 469 . . . . . . 7 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ p e. Prime ) -> ( ( p pCnt A ) = ( p pCnt B ) <-> ( ( p pCnt A ) <_ ( p pCnt B ) /\ ( p pCnt B ) <_ ( p pCnt A ) ) ) )
1514ralbidva 2985 . . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( A. p e. Prime ( p pCnt A ) = ( p pCnt B ) <-> A. p e. Prime ( ( p pCnt A ) <_ ( p pCnt B ) /\ ( p pCnt B ) <_ ( p pCnt A ) ) ) )
16 r19.26 3064 . . . . . 6 |- ( A. p e. Prime ( ( p pCnt A ) <_ ( p pCnt B ) /\ ( p pCnt B ) <_ ( p pCnt A ) ) <-> ( A. p e. Prime ( p pCnt A ) <_ ( p pCnt B ) /\ A. p e. Prime ( p pCnt B ) <_ ( p pCnt A ) ) )
1715, 16syl6bb 276 . . . . 5 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( A. p e. Prime ( p pCnt A ) = ( p pCnt B ) <-> ( A. p e. Prime ( p pCnt A ) <_ ( p pCnt B ) /\ A. p e. Prime ( p pCnt B ) <_ ( p pCnt A ) ) ) )
18 pc2dvds 15583 . . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( A || B <-> A. p e. Prime ( p pCnt A ) <_ ( p pCnt B ) ) )
19 pc2dvds 15583 . . . . . . 7 |- ( ( B e. ZZ /\ A e. ZZ ) -> ( B || A <-> A. p e. Prime ( p pCnt B ) <_ ( p pCnt A ) ) )
2019ancoms 469 . . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( B || A <-> A. p e. Prime ( p pCnt B ) <_ ( p pCnt A ) ) )
2118, 20anbi12d 747 . . . . 5 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( A || B /\ B || A ) <-> ( A. p e. Prime ( p pCnt A ) <_ ( p pCnt B ) /\ A. p e. Prime ( p pCnt B ) <_ ( p pCnt A ) ) ) )
2217, 21bitr4d 271 . . . 4 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( A. p e. Prime ( p pCnt A ) = ( p pCnt B ) <-> ( A || B /\ B || A ) ) )
233, 4, 22syl2an 494 . . 3 |- ( ( A e. NN0 /\ B e. NN0 ) -> ( A. p e. Prime ( p pCnt A ) = ( p pCnt B ) <-> ( A || B /\ B || A ) ) )
24 dvdseq 15036 . . . 4 |- ( ( ( A e. NN0 /\ B e. NN0 ) /\ ( A || B /\ B || A ) ) -> A = B )
2524ex 450 . . 3 |- ( ( A e. NN0 /\ B e. NN0 ) -> ( ( A || B /\ B || A ) -> A = B ) )
2623, 25sylbid 230 . 2 |- ( ( A e. NN0 /\ B e. NN0 ) -> ( A. p e. Prime ( p pCnt A ) = ( p pCnt B ) -> A = B ) )
272, 26impbid2 216 1 |- ( ( A e. NN0 /\ B e. NN0 ) -> ( A = B <-> A. p e. Prime ( p pCnt A ) = ( p pCnt B ) ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   class class class wbr 4653  (class class class)co 6650   RR*cxr 10073   <_ cle 10075   NN0cn0 11292   ZZcz 11377   QQcq 11788   || cdvds 14983   Primecprime 15385   pCnt cpc 15541
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-fz 12327  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-dvds 14984  df-gcd 15217  df-prm 15386  df-pc 15542
This theorem is referenced by:  pcprod  15599  prmreclem2  15621  1arith  15631  isppw2  24841  sqf11  24865  bposlem3  25011
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