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Theorem pc11 14265
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 6293 . . 3  |-  ( A  =  B  ->  (
p  pCnt  A )  =  ( p  pCnt  B ) )
21ralrimivw 2879 . 2  |-  ( A  =  B  ->  A. p  e.  Prime  ( p  pCnt  A )  =  ( p 
pCnt  B ) )
3 nn0z 10888 . . . 4  |-  ( A  e.  NN0  ->  A  e.  ZZ )
4 nn0z 10888 . . . 4  |-  ( B  e.  NN0  ->  B  e.  ZZ )
5 zq 11189 . . . . . . . . . . 11  |-  ( A  e.  ZZ  ->  A  e.  QQ )
6 pcxcl 14246 . . . . . . . . . . 11  |-  ( ( p  e.  Prime  /\  A  e.  QQ )  ->  (
p  pCnt  A )  e.  RR* )
75, 6sylan2 474 . . . . . . . . . 10  |-  ( ( p  e.  Prime  /\  A  e.  ZZ )  ->  (
p  pCnt  A )  e.  RR* )
8 zq 11189 . . . . . . . . . . 11  |-  ( B  e.  ZZ  ->  B  e.  QQ )
9 pcxcl 14246 . . . . . . . . . . 11  |-  ( ( p  e.  Prime  /\  B  e.  QQ )  ->  (
p  pCnt  B )  e.  RR* )
108, 9sylan2 474 . . . . . . . . . 10  |-  ( ( p  e.  Prime  /\  B  e.  ZZ )  ->  (
p  pCnt  B )  e.  RR* )
117, 10anim12dan 835 . . . . . . . . 9  |-  ( ( p  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ ) )  -> 
( ( p  pCnt  A )  e.  RR*  /\  (
p  pCnt  B )  e.  RR* ) )
12 xrletri3 11359 . . . . . . . . 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 453 . . . . . . 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 2900 . . . . . 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 2989 . . . . . 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 261 . . . . 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 14264 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  ||  B  <->  A. p  e.  Prime  (
p  pCnt  A )  <_  ( p  pCnt  B
) ) )
19 pc2dvds 14264 . . . . . . 7  |-  ( ( B  e.  ZZ  /\  A  e.  ZZ )  ->  ( B  ||  A  <->  A. p  e.  Prime  (
p  pCnt  B )  <_  ( p  pCnt  A
) ) )
2019ancoms 453 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( B  ||  A  <->  A. p  e.  Prime  (
p  pCnt  B )  <_  ( p  pCnt  A
) ) )
2118, 20anbi12d 710 . . . . 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 256 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A. p  e. 
Prime  ( p  pCnt  A
)  =  ( p 
pCnt  B )  <->  ( A  ||  B  /\  B  ||  A ) ) )
233, 4, 22syl2an 477 . . 3  |-  ( ( A  e.  NN0  /\  B  e.  NN0 )  -> 
( A. p  e. 
Prime  ( p  pCnt  A
)  =  ( p 
pCnt  B )  <->  ( A  ||  B  /\  B  ||  A ) ) )
24 dvdseq 13895 . . . 4  |-  ( ( ( A  e.  NN0  /\  B  e.  NN0 )  /\  ( A  ||  B  /\  B  ||  A ) )  ->  A  =  B )
2524ex 434 . . 3  |-  ( ( A  e.  NN0  /\  B  e.  NN0 )  -> 
( ( A  ||  B  /\  B  ||  A
)  ->  A  =  B ) )
2623, 25sylbid 215 . 2  |-  ( ( A  e.  NN0  /\  B  e.  NN0 )  -> 
( A. p  e. 
Prime  ( p  pCnt  A
)  =  ( p 
pCnt  B )  ->  A  =  B ) )
272, 26impbid2 204 1  |-  ( ( A  e.  NN0  /\  B  e.  NN0 )  -> 
( A  =  B  <->  A. p  e.  Prime  ( p  pCnt  A )  =  ( p  pCnt  B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814   class class class wbr 4447  (class class class)co 6285   RR*cxr 9628    <_ cle 9630   NN0cn0 10796   ZZcz 10865   QQcq 11183    || cdivides 13850   Primecprime 14079    pCnt cpc 14222
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570  ax-pre-sup 9571
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7043  df-rdg 7077  df-1o 7131  df-2o 7132  df-oadd 7135  df-er 7312  df-en 7518  df-dom 7519  df-sdom 7520  df-fin 7521  df-sup 7902  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-div 10208  df-nn 10538  df-2 10595  df-3 10596  df-n0 10797  df-z 10866  df-uz 11084  df-q 11184  df-rp 11222  df-fz 11674  df-fl 11898  df-mod 11966  df-seq 12077  df-exp 12136  df-cj 12898  df-re 12899  df-im 12900  df-sqrt 13034  df-abs 13035  df-dvds 13851  df-gcd 14007  df-prm 14080  df-pc 14223
This theorem is referenced by:  pcprod  14276  prmreclem2  14297  1arith  14307  isppw2  23214  sqf11  23238  bposlem3  23386
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