Theorem 1arithlem4 15630

Description: Lemma for 1arith 15631. (Contributed by Mario Carneiro, 30-May-2014.)

Hypotheses

Ref	Expression
1arith.1	|- M = ( n e. NN |-> ( p e. Prime |-> ( p pCnt n ) ) )
1arithlem4.2	|- G = ( y e. NN |-> if ( y e. Prime , ( y ^ ( F ` y ) ) , 1 ) )
1arithlem4.3	|- ( ph -> F : Prime --> NN0 )
1arithlem4.4	|- ( ph -> N e. NN )
1arithlem4.5	|- ( ( ph /\ ( q e. Prime /\ N <_ q ) ) -> ( F ` q ) = 0 )

Assertion

Ref	Expression
1arithlem4	|- ( ph -> E. x e. NN F = ( M ` x ) )

Distinct variable groups:   n, p, q, x, y   F, q, x, y   M, q, x, y   ph, q, y   n, G, p, q, x   n, N, p, q, x

Allowed substitution hints:   ph( x, n, p)   F( n, p)   G( y)   M( n, p)   N( y)

Proof of Theorem 1arithlem4

Step	Hyp	Ref	Expression
1		1arithlem4.2	. . . . 5 |- G = ( y e. NN |-> if ( y e. Prime , ( y ^ ( F ` y ) ) , 1 ) )
2		1arithlem4.3	. . . . . . 7 |- ( ph -> F : Prime --> NN0 )
3	2	ffvelrnda 6359	. . . . . 6 |- ( ( ph /\ y e. Prime ) -> ( F ` y ) e. NN0 )
4	3	ralrimiva 2966	. . . . 5 |- ( ph -> A. y e. Prime ( F ` y ) e. NN0 )
5	1, 4	pcmptcl 15595	. . . 4 |- ( ph -> ( G : NN --> NN /\ seq 1 ( x. , G ) : NN --> NN ) )
6	5	simprd 479	. . 3 |- ( ph -> seq 1 ( x. , G ) : NN --> NN )
7		1arithlem4.4	. . 3 |- ( ph -> N e. NN )
8	6, 7	ffvelrnd 6360	. 2 |- ( ph -> ( seq 1 ( x. , G ) ` N ) e. NN )
9		1arith.1	. . . . . . 7 |- M = ( n e. NN |-> ( p e. Prime |-> ( p pCnt n ) ) )
10	9	1arithlem2 15628	. . . . . 6 |- ( ( ( seq 1 ( x. , G ) ` N ) e. NN /\ q e. Prime ) -> ( ( M ` ( seq 1 ( x. , G ) ` N ) ) ` q ) = ( q pCnt ( seq 1 ( x. , G ) ` N ) ) )
11	8, 10	sylan 488	. . . . 5 |- ( ( ph /\ q e. Prime ) -> ( ( M ` ( seq 1 ( x. , G ) ` N ) ) ` q ) = ( q pCnt ( seq 1 ( x. , G ) ` N ) ) )
12	4	adantr 481	. . . . . 6 |- ( ( ph /\ q e. Prime ) -> A. y e. Prime ( F ` y ) e. NN0 )
13	7	adantr 481	. . . . . 6 |- ( ( ph /\ q e. Prime ) -> N e. NN )
14		simpr 477	. . . . . 6 |- ( ( ph /\ q e. Prime ) -> q e. Prime )
15		fveq2 6191	. . . . . 6 |- ( y = q -> ( F ` y ) = ( F ` q ) )
16	1, 12, 13, 14, 15	pcmpt 15596	. . . . 5 |- ( ( ph /\ q e. Prime ) -> ( q pCnt ( seq 1 ( x. , G ) ` N ) ) = if ( q <_ N , ( F ` q ) , 0 ) )
17	13	nnred 11035	. . . . . 6 |- ( ( ph /\ q e. Prime ) -> N e. RR )
18		prmz 15389	. . . . . . . 8 |- ( q e. Prime -> q e. ZZ )
19	18	zred 11482	. . . . . . 7 |- ( q e. Prime -> q e. RR )
20	19	adantl 482	. . . . . 6 |- ( ( ph /\ q e. Prime ) -> q e. RR )
21		ifid 4125	. . . . . . 7 |- if ( q <_ N , ( F ` q ) , ( F ` q ) ) = ( F ` q )
22		1arithlem4.5	. . . . . . . . 9 |- ( ( ph /\ ( q e. Prime /\ N <_ q ) ) -> ( F ` q ) = 0 )
23	22	anassrs 680	. . . . . . . 8 |- ( ( ( ph /\ q e. Prime ) /\ N <_ q ) -> ( F ` q ) = 0 )
24	23	ifeq2d 4105	. . . . . . 7 |- ( ( ( ph /\ q e. Prime ) /\ N <_ q ) -> if ( q <_ N , ( F ` q ) , ( F ` q ) ) = if ( q <_ N , ( F ` q ) , 0 ) )
25	21, 24	syl5reqr 2671	. . . . . 6 |- ( ( ( ph /\ q e. Prime ) /\ N <_ q ) -> if ( q <_ N , ( F ` q ) , 0 ) = ( F ` q ) )
26		iftrue 4092	. . . . . . 7 |- ( q <_ N -> if ( q <_ N , ( F ` q ) , 0 ) = ( F ` q ) )
27	26	adantl 482	. . . . . 6 |- ( ( ( ph /\ q e. Prime ) /\ q <_ N ) -> if ( q <_ N , ( F ` q ) , 0 ) = ( F ` q ) )
28	17, 20, 25, 27	lecasei 10143	. . . . 5 |- ( ( ph /\ q e. Prime ) -> if ( q <_ N , ( F ` q ) , 0 ) = ( F ` q ) )
29	11, 16, 28	3eqtrrd 2661	. . . 4 |- ( ( ph /\ q e. Prime ) -> ( F ` q ) = ( ( M ` ( seq 1 ( x. , G ) ` N ) ) ` q ) )
30	29	ralrimiva 2966	. . 3 |- ( ph -> A. q e. Prime ( F ` q ) = ( ( M ` ( seq 1 ( x. , G ) ` N ) ) ` q ) )
31	9	1arithlem3 15629	. . . . 5 |- ( ( seq 1 ( x. , G ) ` N ) e. NN -> ( M ` ( seq 1 ( x. , G ) ` N ) ) : Prime --> NN0 )
32	8, 31	syl 17	. . . 4 |- ( ph -> ( M ` ( seq 1 ( x. , G ) ` N ) ) : Prime --> NN0 )
33		ffn 6045	. . . . 5 |- ( F : Prime --> NN0 -> F Fn Prime )
34		ffn 6045	. . . . 5 |- ( ( M ` ( seq 1 ( x. , G ) ` N ) ) : Prime --> NN0 -> ( M ` ( seq 1 ( x. , G ) ` N ) ) Fn Prime )
35		eqfnfv 6311	. . . . 5 |- ( ( F Fn Prime /\ ( M ` ( seq 1 ( x. , G ) ` N ) ) Fn Prime ) -> ( F = ( M ` ( seq 1 ( x. , G ) ` N ) ) <-> A. q e. Prime ( F ` q ) = ( ( M ` ( seq 1 ( x. , G ) ` N ) ) ` q ) ) )
36	33, 34, 35	syl2an 494	. . . 4 |- ( ( F : Prime --> NN0 /\ ( M ` ( seq 1 ( x. , G ) ` N ) ) : Prime --> NN0 ) -> ( F = ( M ` ( seq 1 ( x. , G ) ` N ) ) <-> A. q e. Prime ( F ` q ) = ( ( M ` ( seq 1 ( x. , G ) ` N ) ) ` q ) ) )
37	2, 32, 36	syl2anc 693	. . 3 |- ( ph -> ( F = ( M ` ( seq 1 ( x. , G ) ` N ) ) <-> A. q e. Prime ( F ` q ) = ( ( M ` ( seq 1 ( x. , G ) ` N ) ) ` q ) ) )
38	30, 37	mpbird 247	. 2 |- ( ph -> F = ( M ` ( seq 1 ( x. , G ) ` N ) ) )
39		fveq2 6191	. . . 4 |- ( x = ( seq 1 ( x. , G ) ` N ) -> ( M ` x ) = ( M ` ( seq 1 ( x. , G ) ` N ) ) )
40	39	eqeq2d 2632	. . 3 |- ( x = ( seq 1 ( x. , G ) ` N ) -> ( F = ( M ` x ) <-> F = ( M ` ( seq 1 ( x. , G ) ` N ) ) ) )
41	40	rspcev 3309	. 2 |- ( ( ( seq 1 ( x. , G ) ` N ) e. NN /\ F = ( M ` ( seq 1 ( x. , G ) ` N ) ) ) -> E. x e. NN F = ( M ` x ) )
42	8, 38, 41	syl2anc 693	1 |- ( ph -> E. x e. NN F = ( M ` x ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   ifcif 4086   class class class wbr 4653   |-> cmpt 4729   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   RRcr 9935   0cc0 9936   1c1 9937   x. cmul 9941   <_ cle 10075   NNcn 11020   NN0cn0 11292   seqcseq 12801   ^cexp 12860   Primecprime 15385   pCnt cpc 15541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-fz 12327  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-dvds 14984  df-gcd 15217  df-prm 15386  df-pc 15542

This theorem is referenced by:  1arith  15631