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Theorem pcprod 13256
Description: The product of the primes taken to their respective powers reconstructs the original number. (Contributed by Mario Carneiro, 12-Mar-2014.)
Hypothesis
Ref Expression
pcprod.1  |-  F  =  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( n ^ (
n  pCnt  N )
) ,  1 ) )
Assertion
Ref Expression
pcprod  |-  ( N  e.  NN  ->  (  seq  1 (  x.  ,  F ) `  N
)  =  N )
Distinct variable group:    n, N
Allowed substitution hint:    F( n)

Proof of Theorem pcprod
Dummy variable  p is distinct from all other variables.
StepHypRef Expression
1 pcprod.1 . . . . . 6  |-  F  =  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( n ^ (
n  pCnt  N )
) ,  1 ) )
2 pccl 13215 . . . . . . . . 9  |-  ( ( n  e.  Prime  /\  N  e.  NN )  ->  (
n  pCnt  N )  e.  NN0 )
32ancoms 440 . . . . . . . 8  |-  ( ( N  e.  NN  /\  n  e.  Prime )  -> 
( n  pCnt  N
)  e.  NN0 )
43ralrimiva 2781 . . . . . . 7  |-  ( N  e.  NN  ->  A. n  e.  Prime  ( n  pCnt  N )  e.  NN0 )
54adantl 453 . . . . . 6  |-  ( ( p  e.  Prime  /\  N  e.  NN )  ->  A. n  e.  Prime  ( n  pCnt  N )  e.  NN0 )
6 simpr 448 . . . . . 6  |-  ( ( p  e.  Prime  /\  N  e.  NN )  ->  N  e.  NN )
7 simpl 444 . . . . . 6  |-  ( ( p  e.  Prime  /\  N  e.  NN )  ->  p  e.  Prime )
8 oveq1 6080 . . . . . 6  |-  ( n  =  p  ->  (
n  pCnt  N )  =  ( p  pCnt  N ) )
91, 5, 6, 7, 8pcmpt 13253 . . . . 5  |-  ( ( p  e.  Prime  /\  N  e.  NN )  ->  (
p  pCnt  (  seq  1 (  x.  ,  F ) `  N
) )  =  if ( p  <_  N ,  ( p  pCnt  N ) ,  0 ) )
10 iftrue 3737 . . . . . . 7  |-  ( p  <_  N  ->  if ( p  <_  N , 
( p  pCnt  N
) ,  0 )  =  ( p  pCnt  N ) )
1110adantl 453 . . . . . 6  |-  ( ( ( p  e.  Prime  /\  N  e.  NN )  /\  p  <_  N
)  ->  if (
p  <_  N , 
( p  pCnt  N
) ,  0 )  =  ( p  pCnt  N ) )
12 iffalse 3738 . . . . . . . 8  |-  ( -.  p  <_  N  ->  if ( p  <_  N ,  ( p  pCnt  N ) ,  0 )  =  0 )
1312adantl 453 . . . . . . 7  |-  ( ( ( p  e.  Prime  /\  N  e.  NN )  /\  -.  p  <_  N )  ->  if ( p  <_  N , 
( p  pCnt  N
) ,  0 )  =  0 )
14 prmz 13075 . . . . . . . . . 10  |-  ( p  e.  Prime  ->  p  e.  ZZ )
15 dvdsle 12887 . . . . . . . . . 10  |-  ( ( p  e.  ZZ  /\  N  e.  NN )  ->  ( p  ||  N  ->  p  <_  N )
)
1614, 15sylan 458 . . . . . . . . 9  |-  ( ( p  e.  Prime  /\  N  e.  NN )  ->  (
p  ||  N  ->  p  <_  N ) )
1716con3and 429 . . . . . . . 8  |-  ( ( ( p  e.  Prime  /\  N  e.  NN )  /\  -.  p  <_  N )  ->  -.  p  ||  N )
18 pceq0 13236 . . . . . . . . 9  |-  ( ( p  e.  Prime  /\  N  e.  NN )  ->  (
( p  pCnt  N
)  =  0  <->  -.  p  ||  N ) )
1918adantr 452 . . . . . . . 8  |-  ( ( ( p  e.  Prime  /\  N  e.  NN )  /\  -.  p  <_  N )  ->  (
( p  pCnt  N
)  =  0  <->  -.  p  ||  N ) )
2017, 19mpbird 224 . . . . . . 7  |-  ( ( ( p  e.  Prime  /\  N  e.  NN )  /\  -.  p  <_  N )  ->  (
p  pCnt  N )  =  0 )
2113, 20eqtr4d 2470 . . . . . 6  |-  ( ( ( p  e.  Prime  /\  N  e.  NN )  /\  -.  p  <_  N )  ->  if ( p  <_  N , 
( p  pCnt  N
) ,  0 )  =  ( p  pCnt  N ) )
2211, 21pm2.61dan 767 . . . . 5  |-  ( ( p  e.  Prime  /\  N  e.  NN )  ->  if ( p  <_  N , 
( p  pCnt  N
) ,  0 )  =  ( p  pCnt  N ) )
239, 22eqtrd 2467 . . . 4  |-  ( ( p  e.  Prime  /\  N  e.  NN )  ->  (
p  pCnt  (  seq  1 (  x.  ,  F ) `  N
) )  =  ( p  pCnt  N )
)
2423ancoms 440 . . 3  |-  ( ( N  e.  NN  /\  p  e.  Prime )  -> 
( p  pCnt  (  seq  1 (  x.  ,  F ) `  N
) )  =  ( p  pCnt  N )
)
2524ralrimiva 2781 . 2  |-  ( N  e.  NN  ->  A. p  e.  Prime  ( p  pCnt  (  seq  1 (  x.  ,  F ) `  N ) )  =  ( p  pCnt  N
) )
261, 4pcmptcl 13252 . . . . . 6  |-  ( N  e.  NN  ->  ( F : NN --> NN  /\  seq  1 (  x.  ,  F ) : NN --> NN ) )
2726simprd 450 . . . . 5  |-  ( N  e.  NN  ->  seq  1 (  x.  ,  F ) : NN --> NN )
28 ffvelrn 5860 . . . . 5  |-  ( (  seq  1 (  x.  ,  F ) : NN --> NN  /\  N  e.  NN )  ->  (  seq  1 (  x.  ,  F ) `  N
)  e.  NN )
2927, 28mpancom 651 . . . 4  |-  ( N  e.  NN  ->  (  seq  1 (  x.  ,  F ) `  N
)  e.  NN )
3029nnnn0d 10266 . . 3  |-  ( N  e.  NN  ->  (  seq  1 (  x.  ,  F ) `  N
)  e.  NN0 )
31 nnnn0 10220 . . 3  |-  ( N  e.  NN  ->  N  e.  NN0 )
32 pc11 13245 . . 3  |-  ( ( (  seq  1 (  x.  ,  F ) `
 N )  e. 
NN0  /\  N  e.  NN0 )  ->  ( (  seq  1 (  x.  ,  F ) `  N
)  =  N  <->  A. p  e.  Prime  ( p  pCnt  (  seq  1 (  x.  ,  F ) `  N ) )  =  ( p  pCnt  N
) ) )
3330, 31, 32syl2anc 643 . 2  |-  ( N  e.  NN  ->  (
(  seq  1 (  x.  ,  F ) `
 N )  =  N  <->  A. p  e.  Prime  ( p  pCnt  (  seq  1 (  x.  ,  F ) `  N
) )  =  ( p  pCnt  N )
) )
3425, 33mpbird 224 1  |-  ( N  e.  NN  ->  (  seq  1 (  x.  ,  F ) `  N
)  =  N )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697   ifcif 3731   class class class wbr 4204    e. cmpt 4258   -->wf 5442   ` cfv 5446  (class class class)co 6073   0cc0 8982   1c1 8983    x. cmul 8987    <_ cle 9113   NNcn 9992   NN0cn0 10213   ZZcz 10274    seq cseq 11315   ^cexp 11374    || cdivides 12844   Primecprime 13071    pCnt cpc 13202
This theorem is referenced by:  pclogsum  20991
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 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-q 10567  df-rp 10605  df-fz 11036  df-fl 11194  df-mod 11243  df-seq 11316  df-exp 11375  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-dvds 12845  df-gcd 12999  df-prm 13072  df-pc 13203
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