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Theorem pc11 13110
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 6093 . . 3 |- ( A = B -> ( p pCnt A ) = ( p pCnt B ) )
21ralrimivw 2624 . 2 |- ( A = B -> A. p e. Prime ( p pCnt A ) = ( p pCnt B ) )
3 nn0z 9664 . . . 4 |- ( A e. NN0 -> A e. ZZ )
4 nn0z 9664 . . . 4 |- ( B e. NN0 -> B e. ZZ )
5 zq 10026 . . . . . . . . . . 11 |- ( A e. ZZ -> A e. QQ )
6 pcxcl 13090 . . . . . . . . . . 11 |- ( ( p e. Prime /\ A e. QQ ) -> ( p pCnt A ) e. RR* )
75, 6sylan2 286 . . . . . . . . . 10 |- ( ( p e. Prime /\ A e. ZZ ) -> ( p pCnt A ) e. RR* )
8 zq 10026 . . . . . . . . . . 11 |- ( B e. ZZ -> B e. QQ )
9 pcxcl 13090 . . . . . . . . . . 11 |- ( ( p e. Prime /\ B e. QQ ) -> ( p pCnt B ) e. RR* )
108, 9sylan2 286 . . . . . . . . . 10 |- ( ( p e. Prime /\ B e. ZZ ) -> ( p pCnt B ) e. RR* )
117, 10anim12dan 608 . . . . . . . . 9 |- ( ( p e. Prime /\ ( A e. ZZ /\ B e. ZZ ) ) -> ( ( p pCnt A ) e. RR* /\ ( p pCnt B ) e. RR* ) )
12 xrletri3 10206 . . . . . . . . 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 14 . . . . . . . 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 268 . . . . . . 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 2546 . . . . . 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 2677 . . . . . 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, 16bitrdi 196 . . . . 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 13109 . . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( A || B <-> A. p e. Prime ( p pCnt A ) <_ ( p pCnt B ) ) )
19 pc2dvds 13109 . . . . . . 7 |- ( ( B e. ZZ /\ A e. ZZ ) -> ( B || A <-> A. p e. Prime ( p pCnt B ) <_ ( p pCnt A ) ) )
2019ancoms 268 . . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( B || A <-> A. p e. Prime ( p pCnt B ) <_ ( p pCnt A ) ) )
2118, 20anbi12d 477 . . . . 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 191 . . . 4 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( A. p e. Prime ( p pCnt A ) = ( p pCnt B ) <-> ( A || B /\ B || A ) ) )
233, 4, 22syl2an 289 . . 3 |- ( ( A e. NN0 /\ B e. NN0 ) -> ( A. p e. Prime ( p pCnt A ) = ( p pCnt B ) <-> ( A || B /\ B || A ) ) )
24 dvdseq 12615 . . . 4 |- ( ( ( A e. NN0 /\ B e. NN0 ) /\ ( A || B /\ B || A ) ) -> A = B )
2524ex 115 . . 3 |- ( ( A e. NN0 /\ B e. NN0 ) -> ( ( A || B /\ B || A ) -> A = B ) )
2623, 25sylbid 150 . 2 |- ( ( A e. NN0 /\ B e. NN0 ) -> ( A. p e. Prime ( p pCnt A ) = ( p pCnt B ) -> A = B ) )
272, 26impbid2 143 1 |- ( ( A e. NN0 /\ B e. NN0 ) -> ( A = B <-> A. p e. Prime ( p pCnt A ) = ( p pCnt B ) ) )
Colors of variables:    wff set class
This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   A.wral 2528   class class class wbr 4130  (class class class)co 6085   RR*cxr 8359   <_ cle 8361   NN0cn0 9563   ZZcz 9644   QQcq 10019   || cdvds 12554   Primecprime 12885   pCnt cpc 13063
This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298  ax-caucvg 8299
This proof depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-isom 5386  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-1o 6687  df-2o 6688  df-er 6807  df-en 7023  df-sup 7324  df-inf 7325  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-div 9003  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-n0 9564  df-xnn0 9631  df-z 9645  df-uz 9922  df-q 10020  df-rp 10055  df-fz 10412  df-fzo 10550  df-fl 10705  df-mod 10760  df-seqfrec 10885  df-exp 10976  df-cj 11607  df-re 11608  df-im 11609  df-rsqrt 11764  df-abs 11765  df-dvds 12555  df-gcd 12731  df-prm 12886  df-pc 13064
This theorem is used by:  pcprod  13125  1arith  13146
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