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.

Step	Hyp	Ref	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 )
3	2	ancoms 451	. . . . . . . 8 |- ( ( N e. NN /\ n e. Prime ) -> ( n pCnt N ) e. NN0 )
4	3	ralrimiva 2868	. . . . . . 7 |- ( N e. NN -> A. n e. Prime ( n pCnt N ) e. NN0 )
5	4	adantl 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 ) )
9	1, 5, 6, 7, 8	pcmpt 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 ) )
11	10	adantl 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 )
13	12	adantl 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 ) )
16	14, 15	sylan 469	. . . . . . . . 9 |- ( ( p e. Prime /\ N e. NN ) -> ( p || N -> p <_ N ) )
17	16	con3dimp 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 ) )
19	18	adantr 463	. . . . . . . 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 2498	. . . . . 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 2495	. . . 4 |- ( ( p e. Prime /\ N e. NN ) -> ( p pCnt ( seq 1 ( x. , F ) ` N ) ) = ( p pCnt N ) )
24	23	ancoms 451	. . 3 |- ( ( N e. NN /\ p e. Prime ) -> ( p pCnt ( seq 1 ( x. , F ) ` N ) ) = ( p pCnt N ) )
25	24	ralrimiva 2868	. 2 |- ( N e. NN -> A. p e. Prime ( p pCnt ( seq 1 ( x. , F ) ` N ) ) = ( p pCnt N ) )
26	1, 4	pcmptcl 14497	. . . . . 6 |- ( N e. NN -> ( F : NN --> NN /\ seq 1 ( x. , F ) : NN --> NN ) )
27	26	simprd 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 )
29	27, 28	mpancom 667	. . . 4 |- ( N e. NN -> ( seq 1 ( x. , F ) ` N ) e. NN )
30	29	nnnn0d 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 ) ) )
33	30, 31, 32	syl2anc 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 ) ) )
34	25, 33	mpbird 232	1 |- ( N e. NN -> ( seq 1 ( x. , F ) ` N ) = N )

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