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Theorem pc11 13253
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 6089 . . 3  |-  ( A  =  B  ->  (
p  pCnt  A )  =  ( p  pCnt  B ) )
21ralrimivw 2790 . 2  |-  ( A  =  B  ->  A. p  e.  Prime  ( p  pCnt  A )  =  ( p 
pCnt  B ) )
3 nn0z 10304 . . . 4  |-  ( A  e.  NN0  ->  A  e.  ZZ )
4 nn0z 10304 . . . 4  |-  ( B  e.  NN0  ->  B  e.  ZZ )
5 zq 10580 . . . . . . . . . . 11  |-  ( A  e.  ZZ  ->  A  e.  QQ )
6 pcxcl 13234 . . . . . . . . . . 11  |-  ( ( p  e.  Prime  /\  A  e.  QQ )  ->  (
p  pCnt  A )  e.  RR* )
75, 6sylan2 461 . . . . . . . . . 10  |-  ( ( p  e.  Prime  /\  A  e.  ZZ )  ->  (
p  pCnt  A )  e.  RR* )
8 zq 10580 . . . . . . . . . . 11  |-  ( B  e.  ZZ  ->  B  e.  QQ )
9 pcxcl 13234 . . . . . . . . . . 11  |-  ( ( p  e.  Prime  /\  B  e.  QQ )  ->  (
p  pCnt  B )  e.  RR* )
108, 9sylan2 461 . . . . . . . . . 10  |-  ( ( p  e.  Prime  /\  B  e.  ZZ )  ->  (
p  pCnt  B )  e.  RR* )
117, 10anim12dan 811 . . . . . . . . 9  |-  ( ( p  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ ) )  -> 
( ( p  pCnt  A )  e.  RR*  /\  (
p  pCnt  B )  e.  RR* ) )
12 xrletri3 10745 . . . . . . . . 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 16 . . . . . . . 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 440 . . . . . . 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 2721 . . . . . 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 2838 . . . . . 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 253 . . . . 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 13252 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  ||  B  <->  A. p  e.  Prime  (
p  pCnt  A )  <_  ( p  pCnt  B
) ) )
19 pc2dvds 13252 . . . . . . 7  |-  ( ( B  e.  ZZ  /\  A  e.  ZZ )  ->  ( B  ||  A  <->  A. p  e.  Prime  (
p  pCnt  B )  <_  ( p  pCnt  A
) ) )
2019ancoms 440 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( B  ||  A  <->  A. p  e.  Prime  (
p  pCnt  B )  <_  ( p  pCnt  A
) ) )
2118, 20anbi12d 692 . . . . 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 248 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A. p  e. 
Prime  ( p  pCnt  A
)  =  ( p 
pCnt  B )  <->  ( A  ||  B  /\  B  ||  A ) ) )
233, 4, 22syl2an 464 . . 3  |-  ( ( A  e.  NN0  /\  B  e.  NN0 )  -> 
( A. p  e. 
Prime  ( p  pCnt  A
)  =  ( p 
pCnt  B )  <->  ( A  ||  B  /\  B  ||  A ) ) )
24 dvdseq 12897 . . . 4  |-  ( ( ( A  e.  NN0  /\  B  e.  NN0 )  /\  ( A  ||  B  /\  B  ||  A ) )  ->  A  =  B )
2524ex 424 . . 3  |-  ( ( A  e.  NN0  /\  B  e.  NN0 )  -> 
( ( A  ||  B  /\  B  ||  A
)  ->  A  =  B ) )
2623, 25sylbid 207 . 2  |-  ( ( A  e.  NN0  /\  B  e.  NN0 )  -> 
( A. p  e. 
Prime  ( p  pCnt  A
)  =  ( p 
pCnt  B )  ->  A  =  B ) )
272, 26impbid2 196 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    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705   class class class wbr 4212  (class class class)co 6081   RR*cxr 9119    <_ cle 9121   NN0cn0 10221   ZZcz 10282   QQcq 10574    || cdivides 12852   Primecprime 13079    pCnt cpc 13210
This theorem is referenced by:  pcprod  13264  prmreclem2  13285  1arith  13295  isppw2  20898  sqf11  20922  bposlem3  21070
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-2o 6725  df-oadd 6728  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-n0 10222  df-z 10283  df-uz 10489  df-q 10575  df-rp 10613  df-fz 11044  df-fl 11202  df-mod 11251  df-seq 11324  df-exp 11383  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-dvds 12853  df-gcd 13007  df-prm 13080  df-pc 13211
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