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Theorem pc11 12769
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 5975 . . 3 |- ( A = B -> ( p pCnt A ) = ( p pCnt B ) )
21ralrimivw 2582 . 2 |- ( A = B -> A. p e. Prime ( p pCnt A ) = ( p pCnt B ) )
3 nn0z 9427 . . . 4 |- ( A e. NN0 -> A e. ZZ )
4 nn0z 9427 . . . 4 |- ( B e. NN0 -> B e. ZZ )
5 zq 9782 . . . . . . . . . . 11 |- ( A e. ZZ -> A e. QQ )
6 pcxcl 12749 . . . . . . . . . . 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 9782 . . . . . . . . . . 11 |- ( B e. ZZ -> B e. QQ )
9 pcxcl 12749 . . . . . . . . . . 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 600 . . . . . . . . 9 |- ( ( p e. Prime /\ ( A e. ZZ /\ B e. ZZ ) ) -> ( ( p pCnt A ) e. RR* /\ ( p pCnt B ) e. RR* ) )
12 xrletri3 9961 . . . . . . . . 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 2504 . . . . . 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 2634 . . . . . 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 12768 . . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( A || B <-> A. p e. Prime ( p pCnt A ) <_ ( p pCnt B ) ) )
19 pc2dvds 12768 . . . . . . 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 473 . . . . 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 12274 . . . 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
Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   e. wcel 2178   A.wral 2486   class class class wbr 4059  (class class class)co 5967   RR*cxr 8141   <_ cle 8143   NN0cn0 9330   ZZcz 9407   QQcq 9775   || cdvds 12213   Primecprime 12544   pCnt cpc 12722
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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-mulrcl 8059  ax-addcom 8060  ax-mulcom 8061  ax-addass 8062  ax-mulass 8063  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-1rid 8067  ax-0id 8068  ax-rnegex 8069  ax-precex 8070  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-apti 8075  ax-pre-ltadd 8076  ax-pre-mulgt0 8077  ax-pre-mulext 8078  ax-arch 8079  ax-caucvg 8080
This theorem depends on definitions:  df-bi 117  df-stab 833  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-if 3580  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-id 4358  df-po 4361  df-iso 4362  df-iord 4431  df-on 4433  df-ilim 4434  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-isom 5299  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-recs 6414  df-frec 6500  df-1o 6525  df-2o 6526  df-er 6643  df-en 6851  df-sup 7112  df-inf 7113  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-sub 8280  df-neg 8281  df-reap 8683  df-ap 8690  df-div 8781  df-inn 9072  df-2 9130  df-3 9131  df-4 9132  df-n0 9331  df-xnn0 9394  df-z 9408  df-uz 9684  df-q 9776  df-rp 9811  df-fz 10166  df-fzo 10300  df-fl 10450  df-mod 10505  df-seqfrec 10630  df-exp 10721  df-cj 11268  df-re 11269  df-im 11270  df-rsqrt 11424  df-abs 11425  df-dvds 12214  df-gcd 12390  df-prm 12545  df-pc 12723
This theorem is referenced by:  pcprod  12784  1arith  12805
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