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Theorem pc11 12239
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 5844 . . 3 |- ( A = B -> ( p pCnt A ) = ( p pCnt B ) )
21ralrimivw 2538 . 2 |- ( A = B -> A. p e. Prime ( p pCnt A ) = ( p pCnt B ) )
3 nn0z 9202 . . . 4 |- ( A e. NN0 -> A e. ZZ )
4 nn0z 9202 . . . 4 |- ( B e. NN0 -> B e. ZZ )
5 zq 9555 . . . . . . . . . . 11 |- ( A e. ZZ -> A e. QQ )
6 pcxcl 12220 . . . . . . . . . . 11 |- ( ( p e. Prime /\ A e. QQ ) -> ( p pCnt A ) e. RR* )
75, 6sylan2 284 . . . . . . . . . 10 |- ( ( p e. Prime /\ A e. ZZ ) -> ( p pCnt A ) e. RR* )
8 zq 9555 . . . . . . . . . . 11 |- ( B e. ZZ -> B e. QQ )
9 pcxcl 12220 . . . . . . . . . . 11 |- ( ( p e. Prime /\ B e. QQ ) -> ( p pCnt B ) e. RR* )
108, 9sylan2 284 . . . . . . . . . 10 |- ( ( p e. Prime /\ B e. ZZ ) -> ( p pCnt B ) e. RR* )
117, 10anim12dan 590 . . . . . . . . 9 |- ( ( p e. Prime /\ ( A e. ZZ /\ B e. ZZ ) ) -> ( ( p pCnt A ) e. RR* /\ ( p pCnt B ) e. RR* ) )
12 xrletri3 9731 . . . . . . . . 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 266 . . . . . . 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 2460 . . . . . 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 2590 . . . . . 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 195 . . . . 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 12238 . . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( A || B <-> A. p e. Prime ( p pCnt A ) <_ ( p pCnt B ) ) )
19 pc2dvds 12238 . . . . . . 7 |- ( ( B e. ZZ /\ A e. ZZ ) -> ( B || A <-> A. p e. Prime ( p pCnt B ) <_ ( p pCnt A ) ) )
2019ancoms 266 . . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( B || A <-> A. p e. Prime ( p pCnt B ) <_ ( p pCnt A ) ) )
2118, 20anbi12d 465 . . . . 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 190 . . . 4 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( A. p e. Prime ( p pCnt A ) = ( p pCnt B ) <-> ( A || B /\ B || A ) ) )
233, 4, 22syl2an 287 . . 3 |- ( ( A e. NN0 /\ B e. NN0 ) -> ( A. p e. Prime ( p pCnt A ) = ( p pCnt B ) <-> ( A || B /\ B || A ) ) )
24 dvdseq 11771 . . . 4 |- ( ( ( A e. NN0 /\ B e. NN0 ) /\ ( A || B /\ B || A ) ) -> A = B )
2524ex 114 . . 3 |- ( ( A e. NN0 /\ B e. NN0 ) -> ( ( A || B /\ B || A ) -> A = B ) )
2623, 25sylbid 149 . 2 |- ( ( A e. NN0 /\ B e. NN0 ) -> ( A. p e. Prime ( p pCnt A ) = ( p pCnt B ) -> A = B ) )
272, 26impbid2 142 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
Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1342   e. wcel 2135   A.wral 2442   class class class wbr 3976  (class class class)co 5836   RR*cxr 7923   <_ cle 7925   NN0cn0 9105   ZZcz 9182   QQcq 9548   || cdvds 11713   Primecprime 12018   pCnt cpc 12193
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 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-coll 4091  ax-sep 4094  ax-nul 4102  ax-pow 4147  ax-pr 4181  ax-un 4405  ax-setind 4508  ax-iinf 4559  ax-cnex 7835  ax-resscn 7836  ax-1cn 7837  ax-1re 7838  ax-icn 7839  ax-addcl 7840  ax-addrcl 7841  ax-mulcl 7842  ax-mulrcl 7843  ax-addcom 7844  ax-mulcom 7845  ax-addass 7846  ax-mulass 7847  ax-distr 7848  ax-i2m1 7849  ax-0lt1 7850  ax-1rid 7851  ax-0id 7852  ax-rnegex 7853  ax-precex 7854  ax-cnre 7855  ax-pre-ltirr 7856  ax-pre-ltwlin 7857  ax-pre-lttrn 7858  ax-pre-apti 7859  ax-pre-ltadd 7860  ax-pre-mulgt0 7861  ax-pre-mulext 7862  ax-arch 7863  ax-caucvg 7864
This theorem depends on definitions:  df-bi 116  df-stab 821  df-dc 825  df-3or 968  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-nel 2430  df-ral 2447  df-rex 2448  df-reu 2449  df-rmo 2450  df-rab 2451  df-v 2723  df-sbc 2947  df-csb 3041  df-dif 3113  df-un 3115  df-in 3117  df-ss 3124  df-nul 3405  df-if 3516  df-pw 3555  df-sn 3576  df-pr 3577  df-op 3579  df-uni 3784  df-int 3819  df-iun 3862  df-br 3977  df-opab 4038  df-mpt 4039  df-tr 4075  df-id 4265  df-po 4268  df-iso 4269  df-iord 4338  df-on 4340  df-ilim 4341  df-suc 4343  df-iom 4562  df-xp 4604  df-rel 4605  df-cnv 4606  df-co 4607  df-dm 4608  df-rn 4609  df-res 4610  df-ima 4611  df-iota 5147  df-fun 5184  df-fn 5185  df-f 5186  df-f1 5187  df-fo 5188  df-f1o 5189  df-fv 5190  df-isom 5191  df-riota 5792  df-ov 5839  df-oprab 5840  df-mpo 5841  df-1st 6100  df-2nd 6101  df-recs 6264  df-frec 6350  df-1o 6375  df-2o 6376  df-er 6492  df-en 6698  df-sup 6940  df-inf 6941  df-pnf 7926  df-mnf 7927  df-xr 7928  df-ltxr 7929  df-le 7930  df-sub 8062  df-neg 8063  df-reap 8464  df-ap 8471  df-div 8560  df-inn 8849  df-2 8907  df-3 8908  df-4 8909  df-n0 9106  df-xnn0 9169  df-z 9183  df-uz 9458  df-q 9549  df-rp 9581  df-fz 9936  df-fzo 10068  df-fl 10195  df-mod 10248  df-seqfrec 10371  df-exp 10445  df-cj 10770  df-re 10771  df-im 10772  df-rsqrt 10926  df-abs 10927  df-dvds 11714  df-gcd 11861  df-prm 12019  df-pc 12194
This theorem is referenced by:  pcprod  12253
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