Theorem 1arithlem4 12560

Description: Lemma for 1arith 12561. (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	ffvelcdmda 5700	. . . . . 6 |- ( ( ph /\ y e. Prime ) -> ( F ` y ) e. NN0 )
4	3	ralrimiva 2570	. . . . 5 |- ( ph -> A. y e. Prime ( F ` y ) e. NN0 )
5	1, 4	pcmptcl 12536	. . . 4 |- ( ph -> ( G : NN --> NN /\ seq 1 ( x. , G ) : NN --> NN ) )
6	5	simprd 114	. . 3 |- ( ph -> seq 1 ( x. , G ) : NN --> NN )
7		1arithlem4.4	. . 3 |- ( ph -> N e. NN )
8	6, 7	ffvelcdmd 5701	. 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 12558	. . . . . 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 283	. . . . 5 |- ( ( ph /\ q e. Prime ) -> ( ( M ` ( seq 1 ( x. , G ) ` N ) ) ` q ) = ( q pCnt ( seq 1 ( x. , G ) ` N ) ) )
12	4	adantr 276	. . . . . 6 |- ( ( ph /\ q e. Prime ) -> A. y e. Prime ( F ` y ) e. NN0 )
13	7	adantr 276	. . . . . 6 |- ( ( ph /\ q e. Prime ) -> N e. NN )
14		simpr 110	. . . . . 6 |- ( ( ph /\ q e. Prime ) -> q e. Prime )
15		fveq2 5561	. . . . . 6 |- ( y = q -> ( F ` y ) = ( F ` q ) )
16	1, 12, 13, 14, 15	pcmpt 12537	. . . . 5 |- ( ( ph /\ q e. Prime ) -> ( q pCnt ( seq 1 ( x. , G ) ` N ) ) = if ( q <_ N , ( F ` q ) , 0 ) )
17		1arithlem4.5	. . . . . . . . 9 |- ( ( ph /\ ( q e. Prime /\ N <_ q ) ) -> ( F ` q ) = 0 )
18	17	anassrs 400	. . . . . . . 8 |- ( ( ( ph /\ q e. Prime ) /\ N <_ q ) -> ( F ` q ) = 0 )
19	18	ifeq2d 3580	. . . . . . 7 |- ( ( ( ph /\ q e. Prime ) /\ N <_ q ) -> if ( q <_ N , ( F ` q ) , ( F ` q ) ) = if ( q <_ N , ( F ` q ) , 0 ) )
20		prmz 12304	. . . . . . . . . . 11 |- ( q e. Prime -> q e. ZZ )
21	20	adantl 277	. . . . . . . . . 10 |- ( ( ph /\ q e. Prime ) -> q e. ZZ )
22	13	nnzd 9464	. . . . . . . . . 10 |- ( ( ph /\ q e. Prime ) -> N e. ZZ )
23		zdcle 9419	. . . . . . . . . 10 |- ( ( q e. ZZ /\ N e. ZZ ) -> DECID q <_ N )
24	21, 22, 23	syl2anc 411	. . . . . . . . 9 |- ( ( ph /\ q e. Prime ) -> DECID q <_ N )
25		ifiddc 3596	. . . . . . . . 9 |- (DECID q <_ N -> if ( q <_ N , ( F ` q ) , ( F ` q ) ) = ( F ` q ) )
26	24, 25	syl 14	. . . . . . . 8 |- ( ( ph /\ q e. Prime ) -> if ( q <_ N , ( F ` q ) , ( F ` q ) ) = ( F ` q ) )
27	26	adantr 276	. . . . . . 7 |- ( ( ( ph /\ q e. Prime ) /\ N <_ q ) -> if ( q <_ N , ( F ` q ) , ( F ` q ) ) = ( F ` q ) )
28	19, 27	eqtr3d 2231	. . . . . 6 |- ( ( ( ph /\ q e. Prime ) /\ N <_ q ) -> if ( q <_ N , ( F ` q ) , 0 ) = ( F ` q ) )
29		iftrue 3567	. . . . . . 7 |- ( q <_ N -> if ( q <_ N , ( F ` q ) , 0 ) = ( F ` q ) )
30	29	adantl 277	. . . . . 6 |- ( ( ( ph /\ q e. Prime ) /\ q <_ N ) -> if ( q <_ N , ( F ` q ) , 0 ) = ( F ` q ) )
31		zletric 9387	. . . . . . 7 |- ( ( N e. ZZ /\ q e. ZZ ) -> ( N <_ q \/ q <_ N ) )
32	22, 21, 31	syl2anc 411	. . . . . 6 |- ( ( ph /\ q e. Prime ) -> ( N <_ q \/ q <_ N ) )
33	28, 30, 32	mpjaodan 799	. . . . 5 |- ( ( ph /\ q e. Prime ) -> if ( q <_ N , ( F ` q ) , 0 ) = ( F ` q ) )
34	11, 16, 33	3eqtrrd 2234	. . . 4 |- ( ( ph /\ q e. Prime ) -> ( F ` q ) = ( ( M ` ( seq 1 ( x. , G ) ` N ) ) ` q ) )
35	34	ralrimiva 2570	. . 3 |- ( ph -> A. q e. Prime ( F ` q ) = ( ( M ` ( seq 1 ( x. , G ) ` N ) ) ` q ) )
36	9	1arithlem3 12559	. . . . 5 |- ( ( seq 1 ( x. , G ) ` N ) e. NN -> ( M ` ( seq 1 ( x. , G ) ` N ) ) : Prime --> NN0 )
37	8, 36	syl 14	. . . 4 |- ( ph -> ( M ` ( seq 1 ( x. , G ) ` N ) ) : Prime --> NN0 )
38		ffn 5410	. . . . 5 |- ( F : Prime --> NN0 -> F Fn Prime )
39		ffn 5410	. . . . 5 |- ( ( M ` ( seq 1 ( x. , G ) ` N ) ) : Prime --> NN0 -> ( M ` ( seq 1 ( x. , G ) ` N ) ) Fn Prime )
40		eqfnfv 5662	. . . . 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 ) ) )
41	38, 39, 40	syl2an 289	. . . 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 ) ) )
42	2, 37, 41	syl2anc 411	. . 3 |- ( ph -> ( F = ( M ` ( seq 1 ( x. , G ) ` N ) ) <-> A. q e. Prime ( F ` q ) = ( ( M ` ( seq 1 ( x. , G ) ` N ) ) ` q ) ) )
43	35, 42	mpbird 167	. 2 |- ( ph -> F = ( M ` ( seq 1 ( x. , G ) ` N ) ) )
44		fveq2 5561	. . 3 |- ( x = ( seq 1 ( x. , G ) ` N ) -> ( M ` x ) = ( M ` ( seq 1 ( x. , G ) ` N ) ) )
45	44	rspceeqv 2886	. 2 |- ( ( ( seq 1 ( x. , G ) ` N ) e. NN /\ F = ( M ` ( seq 1 ( x. , G ) ` N ) ) ) -> E. x e. NN F = ( M ` x ) )
46	8, 43, 45	syl2anc 411	1 |- ( ph -> E. x e. NN F = ( M ` x ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709  DECID wdc 835   = wceq 1364   e. wcel 2167   A.wral 2475   E.wrex 2476   ifcif 3562   class class class wbr 4034   |-> cmpt 4095   Fn wfn 5254   -->wf 5255   ` cfv 5259  (class class class)co 5925   0cc0 7896   1c1 7897   x. cmul 7901   <_ cle 8079   NNcn 9007   NN0cn0 9266   ZZcz 9343   seqcseq 10556   ^cexp 10647   Primecprime 12300   pCnt cpc 12478

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-mulrcl 7995  ax-addcom 7996  ax-mulcom 7997  ax-addass 7998  ax-mulass 7999  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-1rid 8003  ax-0id 8004  ax-rnegex 8005  ax-precex 8006  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-apti 8011  ax-pre-ltadd 8012  ax-pre-mulgt0 8013  ax-pre-mulext 8014  ax-arch 8015  ax-caucvg 8016

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-isom 5268  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-frec 6458  df-1o 6483  df-2o 6484  df-er 6601  df-en 6809  df-fin 6811  df-sup 7059  df-inf 7060  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-reap 8619  df-ap 8626  df-div 8717  df-inn 9008  df-2 9066  df-3 9067  df-4 9068  df-n0 9267  df-z 9344  df-uz 9619  df-q 9711  df-rp 9746  df-fz 10101  df-fzo 10235  df-fl 10377  df-mod 10432  df-seqfrec 10557  df-exp 10648  df-cj 11024  df-re 11025  df-im 11026  df-rsqrt 11180  df-abs 11181  df-dvds 11970  df-gcd 12146  df-prm 12301  df-pc 12479

This theorem is referenced by:  1arith  12561