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Theorem pcprod 14501
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 14460 . . . . . . . . 9  |-  ( ( n  e.  Prime  /\  N  e.  NN )  ->  (
n  pCnt  N )  e.  NN0 )
32ancoms 451 . . . . . . . 8  |-  ( ( N  e.  NN  /\  n  e.  Prime )  -> 
( n  pCnt  N
)  e.  NN0 )
43ralrimiva 2868 . . . . . . 7  |-  ( N  e.  NN  ->  A. n  e.  Prime  ( n  pCnt  N )  e.  NN0 )
54adantl 464 . . . . . 6  |-  ( ( p  e.  Prime  /\  N  e.  NN )  ->  A. n  e.  Prime  ( n  pCnt  N )  e.  NN0 )
6 simpr 459 . . . . . 6  |-  ( ( p  e.  Prime  /\  N  e.  NN )  ->  N  e.  NN )
7 simpl 455 . . . . . 6  |-  ( ( p  e.  Prime  /\  N  e.  NN )  ->  p  e.  Prime )
8 oveq1 6277 . . . . . 6  |-  ( n  =  p  ->  (
n  pCnt  N )  =  ( p  pCnt  N ) )
91, 5, 6, 7, 8pcmpt 14498 . . . . 5  |-  ( ( p  e.  Prime  /\  N  e.  NN )  ->  (
p  pCnt  (  seq 1 (  x.  ,  F ) `  N
) )  =  if ( p  <_  N ,  ( p  pCnt  N ) ,  0 ) )
10 iftrue 3935 . . . . . . 7  |-  ( p  <_  N  ->  if ( p  <_  N , 
( p  pCnt  N
) ,  0 )  =  ( p  pCnt  N ) )
1110adantl 464 . . . . . 6  |-  ( ( ( p  e.  Prime  /\  N  e.  NN )  /\  p  <_  N
)  ->  if (
p  <_  N , 
( p  pCnt  N
) ,  0 )  =  ( p  pCnt  N ) )
12 iffalse 3938 . . . . . . . 8  |-  ( -.  p  <_  N  ->  if ( p  <_  N ,  ( p  pCnt  N ) ,  0 )  =  0 )
1312adantl 464 . . . . . . 7  |-  ( ( ( p  e.  Prime  /\  N  e.  NN )  /\  -.  p  <_  N )  ->  if ( p  <_  N , 
( p  pCnt  N
) ,  0 )  =  0 )
14 prmz 14308 . . . . . . . . . 10  |-  ( p  e.  Prime  ->  p  e.  ZZ )
15 dvdsle 14118 . . . . . . . . . 10  |-  ( ( p  e.  ZZ  /\  N  e.  NN )  ->  ( p  ||  N  ->  p  <_  N )
)
1614, 15sylan 469 . . . . . . . . 9  |-  ( ( p  e.  Prime  /\  N  e.  NN )  ->  (
p  ||  N  ->  p  <_  N ) )
1716con3dimp 439 . . . . . . . 8  |-  ( ( ( p  e.  Prime  /\  N  e.  NN )  /\  -.  p  <_  N )  ->  -.  p  ||  N )
18 pceq0 14481 . . . . . . . . 9  |-  ( ( p  e.  Prime  /\  N  e.  NN )  ->  (
( p  pCnt  N
)  =  0  <->  -.  p  ||  N ) )
1918adantr 463 . . . . . . . 8  |-  ( ( ( p  e.  Prime  /\  N  e.  NN )  /\  -.  p  <_  N )  ->  (
( p  pCnt  N
)  =  0  <->  -.  p  ||  N ) )
2017, 19mpbird 232 . . . . . . 7  |-  ( ( ( p  e.  Prime  /\  N  e.  NN )  /\  -.  p  <_  N )  ->  (
p  pCnt  N )  =  0 )
2113, 20eqtr4d 2498 . . . . . 6  |-  ( ( ( p  e.  Prime  /\  N  e.  NN )  /\  -.  p  <_  N )  ->  if ( p  <_  N , 
( p  pCnt  N
) ,  0 )  =  ( p  pCnt  N ) )
2211, 21pm2.61dan 789 . . . . 5  |-  ( ( p  e.  Prime  /\  N  e.  NN )  ->  if ( p  <_  N , 
( p  pCnt  N
) ,  0 )  =  ( p  pCnt  N ) )
239, 22eqtrd 2495 . . . 4  |-  ( ( p  e.  Prime  /\  N  e.  NN )  ->  (
p  pCnt  (  seq 1 (  x.  ,  F ) `  N
) )  =  ( p  pCnt  N )
)
2423ancoms 451 . . 3  |-  ( ( N  e.  NN  /\  p  e.  Prime )  -> 
( p  pCnt  (  seq 1 (  x.  ,  F ) `  N
) )  =  ( p  pCnt  N )
)
2524ralrimiva 2868 . 2  |-  ( N  e.  NN  ->  A. p  e.  Prime  ( p  pCnt  (  seq 1 (  x.  ,  F ) `  N ) )  =  ( p  pCnt  N
) )
261, 4pcmptcl 14497 . . . . . 6  |-  ( N  e.  NN  ->  ( F : NN --> NN  /\  seq 1 (  x.  ,  F ) : NN --> NN ) )
2726simprd 461 . . . . 5  |-  ( N  e.  NN  ->  seq 1 (  x.  ,  F ) : NN --> NN )
28 ffvelrn 6005 . . . . 5  |-  ( (  seq 1 (  x.  ,  F ) : NN --> NN  /\  N  e.  NN )  ->  (  seq 1 (  x.  ,  F ) `  N
)  e.  NN )
2927, 28mpancom 667 . . . 4  |-  ( N  e.  NN  ->  (  seq 1 (  x.  ,  F ) `  N
)  e.  NN )
3029nnnn0d 10848 . . 3  |-  ( N  e.  NN  ->  (  seq 1 (  x.  ,  F ) `  N
)  e.  NN0 )
31 nnnn0 10798 . . 3  |-  ( N  e.  NN  ->  N  e.  NN0 )
32 pc11 14490 . . 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 659 . 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 232 1  |-  ( N  e.  NN  ->  (  seq 1 (  x.  ,  F ) `  N
)  =  N )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823   A.wral 2804   ifcif 3929   class class class wbr 4439    |-> cmpt 4497   -->wf 5566   ` cfv 5570  (class class class)co 6270   0cc0 9481   1c1 9482    x. cmul 9486    <_ cle 9618   NNcn 10531   NN0cn0 10791   ZZcz 10860    seqcseq 12092   ^cexp 12151    || cdvds 14073   Primecprime 14304    pCnt cpc 14447
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-oadd 7126  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-sup 7893  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11083  df-q 11184  df-rp 11222  df-fz 11676  df-fl 11910  df-mod 11979  df-seq 12093  df-exp 12152  df-cj 13017  df-re 13018  df-im 13019  df-sqrt 13153  df-abs 13154  df-dvds 14074  df-gcd 14232  df-prm 14305  df-pc 14448
This theorem is referenced by:  pclogsum  23691
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