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Theorem pc11 12932
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 5866 . . 3  |-  ( A  =  B  ->  (
p  pCnt  A )  =  ( p  pCnt  B ) )
21ralrimivw 2627 . 2  |-  ( A  =  B  ->  A. p  e.  Prime  ( p  pCnt  A )  =  ( p 
pCnt  B ) )
3 nn0z 10046 . . . 4  |-  ( A  e.  NN0  ->  A  e.  ZZ )
4 nn0z 10046 . . . 4  |-  ( B  e.  NN0  ->  B  e.  ZZ )
5 zq 10322 . . . . . . . . . . 11  |-  ( A  e.  ZZ  ->  A  e.  QQ )
6 pcxcl 12913 . . . . . . . . . . 11  |-  ( ( p  e.  Prime  /\  A  e.  QQ )  ->  (
p  pCnt  A )  e.  RR* )
75, 6sylan2 460 . . . . . . . . . 10  |-  ( ( p  e.  Prime  /\  A  e.  ZZ )  ->  (
p  pCnt  A )  e.  RR* )
8 zq 10322 . . . . . . . . . . 11  |-  ( B  e.  ZZ  ->  B  e.  QQ )
9 pcxcl 12913 . . . . . . . . . . 11  |-  ( ( p  e.  Prime  /\  B  e.  QQ )  ->  (
p  pCnt  B )  e.  RR* )
108, 9sylan2 460 . . . . . . . . . 10  |-  ( ( p  e.  Prime  /\  B  e.  ZZ )  ->  (
p  pCnt  B )  e.  RR* )
117, 10anim12dan 810 . . . . . . . . 9  |-  ( ( p  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ ) )  -> 
( ( p  pCnt  A )  e.  RR*  /\  (
p  pCnt  B )  e.  RR* ) )
12 xrletri3 10486 . . . . . . . . 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 15 . . . . . . . 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 439 . . . . . . 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 2559 . . . . . 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 2675 . . . . . 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 252 . . . . 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 12931 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  ||  B  <->  A. p  e.  Prime  (
p  pCnt  A )  <_  ( p  pCnt  B
) ) )
19 pc2dvds 12931 . . . . . . 7  |-  ( ( B  e.  ZZ  /\  A  e.  ZZ )  ->  ( B  ||  A  <->  A. p  e.  Prime  (
p  pCnt  B )  <_  ( p  pCnt  A
) ) )
2019ancoms 439 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( B  ||  A  <->  A. p  e.  Prime  (
p  pCnt  B )  <_  ( p  pCnt  A
) ) )
2118, 20anbi12d 691 . . . . 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 247 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A. p  e. 
Prime  ( p  pCnt  A
)  =  ( p 
pCnt  B )  <->  ( A  ||  B  /\  B  ||  A ) ) )
233, 4, 22syl2an 463 . . 3  |-  ( ( A  e.  NN0  /\  B  e.  NN0 )  -> 
( A. p  e. 
Prime  ( p  pCnt  A
)  =  ( p 
pCnt  B )  <->  ( A  ||  B  /\  B  ||  A ) ) )
24 dvdseq 12576 . . . 4  |-  ( ( ( A  e.  NN0  /\  B  e.  NN0 )  /\  ( A  ||  B  /\  B  ||  A ) )  ->  A  =  B )
2524ex 423 . . 3  |-  ( ( A  e.  NN0  /\  B  e.  NN0 )  -> 
( ( A  ||  B  /\  B  ||  A
)  ->  A  =  B ) )
2623, 25sylbid 206 . 2  |-  ( ( A  e.  NN0  /\  B  e.  NN0 )  -> 
( A. p  e. 
Prime  ( p  pCnt  A
)  =  ( p 
pCnt  B )  ->  A  =  B ) )
272, 26impbid2 195 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 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   class class class wbr 4023  (class class class)co 5858   RR*cxr 8866    <_ cle 8868   NN0cn0 9965   ZZcz 10024   QQcq 10316    || cdivides 12531   Primecprime 12758    pCnt cpc 12889
This theorem is referenced by:  pcprod  12943  prmreclem2  12964  1arith  12974  isppw2  20353  sqf11  20377  bposlem3  20525
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317  df-rp 10355  df-fz 10783  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-dvds 12532  df-gcd 12686  df-prm 12759  df-pc 12890
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