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Theorem pcprod 14919
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 14878 . . . . . . . . 9  |-  ( ( n  e.  Prime  /\  N  e.  NN )  ->  (
n  pCnt  N )  e.  NN0 )
32ancoms 460 . . . . . . . 8  |-  ( ( N  e.  NN  /\  n  e.  Prime )  -> 
( n  pCnt  N
)  e.  NN0 )
43ralrimiva 2809 . . . . . . 7  |-  ( N  e.  NN  ->  A. n  e.  Prime  ( n  pCnt  N )  e.  NN0 )
54adantl 473 . . . . . 6  |-  ( ( p  e.  Prime  /\  N  e.  NN )  ->  A. n  e.  Prime  ( n  pCnt  N )  e.  NN0 )
6 simpr 468 . . . . . 6  |-  ( ( p  e.  Prime  /\  N  e.  NN )  ->  N  e.  NN )
7 simpl 464 . . . . . 6  |-  ( ( p  e.  Prime  /\  N  e.  NN )  ->  p  e.  Prime )
8 oveq1 6315 . . . . . 6  |-  ( n  =  p  ->  (
n  pCnt  N )  =  ( p  pCnt  N ) )
91, 5, 6, 7, 8pcmpt 14916 . . . . 5  |-  ( ( p  e.  Prime  /\  N  e.  NN )  ->  (
p  pCnt  (  seq 1 (  x.  ,  F ) `  N
) )  =  if ( p  <_  N ,  ( p  pCnt  N ) ,  0 ) )
10 iftrue 3878 . . . . . . 7  |-  ( p  <_  N  ->  if ( p  <_  N , 
( p  pCnt  N
) ,  0 )  =  ( p  pCnt  N ) )
1110adantl 473 . . . . . 6  |-  ( ( ( p  e.  Prime  /\  N  e.  NN )  /\  p  <_  N
)  ->  if (
p  <_  N , 
( p  pCnt  N
) ,  0 )  =  ( p  pCnt  N ) )
12 iffalse 3881 . . . . . . . 8  |-  ( -.  p  <_  N  ->  if ( p  <_  N ,  ( p  pCnt  N ) ,  0 )  =  0 )
1312adantl 473 . . . . . . 7  |-  ( ( ( p  e.  Prime  /\  N  e.  NN )  /\  -.  p  <_  N )  ->  if ( p  <_  N , 
( p  pCnt  N
) ,  0 )  =  0 )
14 prmz 14705 . . . . . . . . . 10  |-  ( p  e.  Prime  ->  p  e.  ZZ )
15 dvdsle 14427 . . . . . . . . . 10  |-  ( ( p  e.  ZZ  /\  N  e.  NN )  ->  ( p  ||  N  ->  p  <_  N )
)
1614, 15sylan 479 . . . . . . . . 9  |-  ( ( p  e.  Prime  /\  N  e.  NN )  ->  (
p  ||  N  ->  p  <_  N ) )
1716con3dimp 448 . . . . . . . 8  |-  ( ( ( p  e.  Prime  /\  N  e.  NN )  /\  -.  p  <_  N )  ->  -.  p  ||  N )
18 pceq0 14899 . . . . . . . . 9  |-  ( ( p  e.  Prime  /\  N  e.  NN )  ->  (
( p  pCnt  N
)  =  0  <->  -.  p  ||  N ) )
1918adantr 472 . . . . . . . 8  |-  ( ( ( p  e.  Prime  /\  N  e.  NN )  /\  -.  p  <_  N )  ->  (
( p  pCnt  N
)  =  0  <->  -.  p  ||  N ) )
2017, 19mpbird 240 . . . . . . 7  |-  ( ( ( p  e.  Prime  /\  N  e.  NN )  /\  -.  p  <_  N )  ->  (
p  pCnt  N )  =  0 )
2113, 20eqtr4d 2508 . . . . . 6  |-  ( ( ( p  e.  Prime  /\  N  e.  NN )  /\  -.  p  <_  N )  ->  if ( p  <_  N , 
( p  pCnt  N
) ,  0 )  =  ( p  pCnt  N ) )
2211, 21pm2.61dan 808 . . . . 5  |-  ( ( p  e.  Prime  /\  N  e.  NN )  ->  if ( p  <_  N , 
( p  pCnt  N
) ,  0 )  =  ( p  pCnt  N ) )
239, 22eqtrd 2505 . . . 4  |-  ( ( p  e.  Prime  /\  N  e.  NN )  ->  (
p  pCnt  (  seq 1 (  x.  ,  F ) `  N
) )  =  ( p  pCnt  N )
)
2423ancoms 460 . . 3  |-  ( ( N  e.  NN  /\  p  e.  Prime )  -> 
( p  pCnt  (  seq 1 (  x.  ,  F ) `  N
) )  =  ( p  pCnt  N )
)
2524ralrimiva 2809 . 2  |-  ( N  e.  NN  ->  A. p  e.  Prime  ( p  pCnt  (  seq 1 (  x.  ,  F ) `  N ) )  =  ( p  pCnt  N
) )
261, 4pcmptcl 14915 . . . . . 6  |-  ( N  e.  NN  ->  ( F : NN --> NN  /\  seq 1 (  x.  ,  F ) : NN --> NN ) )
2726simprd 470 . . . . 5  |-  ( N  e.  NN  ->  seq 1 (  x.  ,  F ) : NN --> NN )
28 ffvelrn 6035 . . . . 5  |-  ( (  seq 1 (  x.  ,  F ) : NN --> NN  /\  N  e.  NN )  ->  (  seq 1 (  x.  ,  F ) `  N
)  e.  NN )
2927, 28mpancom 682 . . . 4  |-  ( N  e.  NN  ->  (  seq 1 (  x.  ,  F ) `  N
)  e.  NN )
3029nnnn0d 10949 . . 3  |-  ( N  e.  NN  ->  (  seq 1 (  x.  ,  F ) `  N
)  e.  NN0 )
31 nnnn0 10900 . . 3  |-  ( N  e.  NN  ->  N  e.  NN0 )
32 pc11 14908 . . 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 673 . 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 240 1  |-  ( N  e.  NN  ->  (  seq 1 (  x.  ,  F ) `  N
)  =  N )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    /\ wa 376    = wceq 1452    e. wcel 1904   A.wral 2756   ifcif 3872   class class class wbr 4395    |-> cmpt 4454   -->wf 5585   ` cfv 5589  (class class class)co 6308   0cc0 9557   1c1 9558    x. cmul 9562    <_ cle 9694   NNcn 10631   NN0cn0 10893   ZZcz 10961    seqcseq 12251   ^cexp 12310    || cdvds 14382   Primecprime 14701    pCnt cpc 14865
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-sup 7974  df-inf 7975  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-n0 10894  df-z 10962  df-uz 11183  df-q 11288  df-rp 11326  df-fz 11811  df-fl 12061  df-mod 12130  df-seq 12252  df-exp 12311  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-dvds 14383  df-gcd 14548  df-prm 14702  df-pc 14866
This theorem is referenced by:  pclogsum  24222
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