Theorem pcprod 14059

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.

Step	Hyp	Ref	Expression
1		pcprod.1	. . . . . 6 |- F = ( n e. NN |-> if ( n e. Prime , ( n ^ ( n pCnt N ) ) , 1 ) )
2		pccl 14018	. . . . . . . . 9 |- ( ( n e. Prime /\ N e. NN ) -> ( n pCnt N ) e. NN0 )
3	2	ancoms 453	. . . . . . . 8 |- ( ( N e. NN /\ n e. Prime ) -> ( n pCnt N ) e. NN0 )
4	3	ralrimiva 2822	. . . . . . 7 |- ( N e. NN -> A. n e. Prime ( n pCnt N ) e. NN0 )
5	4	adantl 466	. . . . . 6 |- ( ( p e. Prime /\ N e. NN ) -> A. n e. Prime ( n pCnt N ) e. NN0 )
6		simpr 461	. . . . . 6 |- ( ( p e. Prime /\ N e. NN ) -> N e. NN )
7		simpl 457	. . . . . 6 |- ( ( p e. Prime /\ N e. NN ) -> p e. Prime )
8		oveq1 6197	. . . . . 6 |- ( n = p -> ( n pCnt N ) = ( p pCnt N ) )
9	1, 5, 6, 7, 8	pcmpt 14056	. . . . 5 |- ( ( p e. Prime /\ N e. NN ) -> ( p pCnt ( seq 1 ( x. , F ) ` N ) ) = if ( p <_ N , ( p pCnt N ) , 0 ) )
10		iftrue 3895	. . . . . . 7 |- ( p <_ N -> if ( p <_ N , ( p pCnt N ) , 0 ) = ( p pCnt N ) )
11	10	adantl 466	. . . . . 6 |- ( ( ( p e. Prime /\ N e. NN ) /\ p <_ N ) -> if ( p <_ N , ( p pCnt N ) , 0 ) = ( p pCnt N ) )
12		iffalse 3897	. . . . . . . 8 |- ( -. p <_ N -> if ( p <_ N , ( p pCnt N ) , 0 ) = 0 )
13	12	adantl 466	. . . . . . 7 |- ( ( ( p e. Prime /\ N e. NN ) /\ -. p <_ N ) -> if ( p <_ N , ( p pCnt N ) , 0 ) = 0 )
14		prmz 13869	. . . . . . . . . 10 |- ( p e. Prime -> p e. ZZ )
15		dvdsle 13680	. . . . . . . . . 10 |- ( ( p e. ZZ /\ N e. NN ) -> ( p || N -> p <_ N ) )
16	14, 15	sylan 471	. . . . . . . . 9 |- ( ( p e. Prime /\ N e. NN ) -> ( p || N -> p <_ N ) )
17	16	con3dimp 441	. . . . . . . 8 |- ( ( ( p e. Prime /\ N e. NN ) /\ -. p <_ N ) -> -. p || N )
18		pceq0 14039	. . . . . . . . 9 |- ( ( p e. Prime /\ N e. NN ) -> ( ( p pCnt N ) = 0 <-> -. p || N ) )
19	18	adantr 465	. . . . . . . 8 |- ( ( ( p e. Prime /\ N e. NN ) /\ -. p <_ N ) -> ( ( p pCnt N ) = 0 <-> -. p || N ) )
20	17, 19	mpbird 232	. . . . . . 7 |- ( ( ( p e. Prime /\ N e. NN ) /\ -. p <_ N ) -> ( p pCnt N ) = 0 )
21	13, 20	eqtr4d 2495	. . . . . 6 |- ( ( ( p e. Prime /\ N e. NN ) /\ -. p <_ N ) -> if ( p <_ N , ( p pCnt N ) , 0 ) = ( p pCnt N ) )
22	11, 21	pm2.61dan 789	. . . . 5 |- ( ( p e. Prime /\ N e. NN ) -> if ( p <_ N , ( p pCnt N ) , 0 ) = ( p pCnt N ) )
23	9, 22	eqtrd 2492	. . . 4 |- ( ( p e. Prime /\ N e. NN ) -> ( p pCnt ( seq 1 ( x. , F ) ` N ) ) = ( p pCnt N ) )
24	23	ancoms 453	. . 3 |- ( ( N e. NN /\ p e. Prime ) -> ( p pCnt ( seq 1 ( x. , F ) ` N ) ) = ( p pCnt N ) )
25	24	ralrimiva 2822	. 2 |- ( N e. NN -> A. p e. Prime ( p pCnt ( seq 1 ( x. , F ) ` N ) ) = ( p pCnt N ) )
26	1, 4	pcmptcl 14055	. . . . . 6 |- ( N e. NN -> ( F : NN --> NN /\ seq 1 ( x. , F ) : NN --> NN ) )
27	26	simprd 463	. . . . 5 |- ( N e. NN -> seq 1 ( x. , F ) : NN --> NN )
28		ffvelrn 5940	. . . . 5 |- ( ( seq 1 ( x. , F ) : NN --> NN /\ N e. NN ) -> ( seq 1 ( x. , F ) ` N ) e. NN )
29	27, 28	mpancom 669	. . . 4 |- ( N e. NN -> ( seq 1 ( x. , F ) ` N ) e. NN )
30	29	nnnn0d 10737	. . 3 |- ( N e. NN -> ( seq 1 ( x. , F ) ` N ) e. NN0 )
31		nnnn0 10687	. . 3 |- ( N e. NN -> N e. NN0 )
32		pc11 14048	. . 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 ) ) )
33	30, 31, 32	syl2anc 661	. 2 |- ( N e. NN -> ( ( seq 1 ( x. , F ) ` N ) = N <-> A. p e. Prime ( p pCnt ( seq 1 ( x. , F ) ` N ) ) = ( p pCnt N ) ) )
34	25, 33	mpbird 232	1 |- ( N e. NN -> ( seq 1 ( x. , F ) ` N ) = N )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1370   e. wcel 1758   A.wral 2795   ifcif 3889   class class class wbr 4390   |-> cmpt 4448   -->wf 5512   ` cfv 5516  (class class class)co 6190   0cc0 9383   1c1 9384   x. cmul 9388   <_ cle 9520   NNcn 10423   NN0cn0 10680   ZZcz 10747   seqcseq 11907   ^cexp 11966   || cdivides 13637   Primecprime 13865   pCnt cpc 14005

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-cnex 9439  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459  ax-pre-mulgt0 9460  ax-pre-sup 9461

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-int 4227  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-om 6577  df-1st 6677  df-2nd 6678  df-recs 6932  df-rdg 6966  df-1o 7020  df-2o 7021  df-oadd 7024  df-er 7201  df-en 7411  df-dom 7412  df-sdom 7413  df-fin 7414  df-sup 7792  df-pnf 9521  df-mnf 9522  df-xr 9523  df-ltxr 9524  df-le 9525  df-sub 9698  df-neg 9699  df-div 10095  df-nn 10424  df-2 10481  df-3 10482  df-n0 10681  df-z 10748  df-uz 10963  df-q 11055  df-rp 11093  df-fz 11539  df-fl 11743  df-mod 11810  df-seq 11908  df-exp 11967  df-cj 12690  df-re 12691  df-im 12692  df-sqr 12826  df-abs 12827  df-dvds 13638  df-gcd 13793  df-prm 13866  df-pc 14006

This theorem is referenced by:  pclogsum  22670