Theorem 1arithlem4 12366

Description: Lemma for 1arith 12367. (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 5653	. . . . . 6 |- ( ( ph /\ y e. Prime ) -> ( F ` y ) e. NN0 )
4	3	ralrimiva 2550	. . . . 5 |- ( ph -> A. y e. Prime ( F ` y ) e. NN0 )
5	1, 4	pcmptcl 12342	. . . 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 5654	. 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 12364	. . . . . 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 5517	. . . . . 6 |- ( y = q -> ( F ` y ) = ( F ` q ) )
16	1, 12, 13, 14, 15	pcmpt 12343	. . . . 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 3554	. . . . . . 7 |- ( ( ( ph /\ q e. Prime ) /\ N <_ q ) -> if ( q <_ N , ( F ` q ) , ( F ` q ) ) = if ( q <_ N , ( F ` q ) , 0 ) )
20		prmz 12113	. . . . . . . . . . 11 |- ( q e. Prime -> q e. ZZ )
21	20	adantl 277	. . . . . . . . . 10 |- ( ( ph /\ q e. Prime ) -> q e. ZZ )
22	13	nnzd 9376	. . . . . . . . . 10 |- ( ( ph /\ q e. Prime ) -> N e. ZZ )
23		zdcle 9331	. . . . . . . . . 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 3570	. . . . . . . . 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 2212	. . . . . 6 |- ( ( ( ph /\ q e. Prime ) /\ N <_ q ) -> if ( q <_ N , ( F ` q ) , 0 ) = ( F ` q ) )
29		iftrue 3541	. . . . . . 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 9299	. . . . . . 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 798	. . . . 5 |- ( ( ph /\ q e. Prime ) -> if ( q <_ N , ( F ` q ) , 0 ) = ( F ` q ) )
34	11, 16, 33	3eqtrrd 2215	. . . 4 |- ( ( ph /\ q e. Prime ) -> ( F ` q ) = ( ( M ` ( seq 1 ( x. , G ) ` N ) ) ` q ) )
35	34	ralrimiva 2550	. . 3 |- ( ph -> A. q e. Prime ( F ` q ) = ( ( M ` ( seq 1 ( x. , G ) ` N ) ) ` q ) )
36	9	1arithlem3 12365	. . . . 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 5367	. . . . 5 |- ( F : Prime --> NN0 -> F Fn Prime )
39		ffn 5367	. . . . 5 |- ( ( M ` ( seq 1 ( x. , G ) ` N ) ) : Prime --> NN0 -> ( M ` ( seq 1 ( x. , G ) ` N ) ) Fn Prime )
40		eqfnfv 5615	. . . . 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 5517	. . 3 |- ( x = ( seq 1 ( x. , G ) ` N ) -> ( M ` x ) = ( M ` ( seq 1 ( x. , G ) ` N ) ) )
45	44	rspceeqv 2861	. 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 708  DECID wdc 834   = wceq 1353   e. wcel 2148   A.wral 2455   E.wrex 2456   ifcif 3536   class class class wbr 4005   |-> cmpt 4066   Fn wfn 5213   -->wf 5214   ` cfv 5218  (class class class)co 5877   0cc0 7813   1c1 7814   x. cmul 7818   <_ cle 7995   NNcn 8921   NN0cn0 9178   ZZcz 9255   seqcseq 10447   ^cexp 10521   Primecprime 12109   pCnt cpc 12286

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-mulrcl 7912  ax-addcom 7913  ax-mulcom 7914  ax-addass 7915  ax-mulass 7916  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-1rid 7920  ax-0id 7921  ax-rnegex 7922  ax-precex 7923  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927  ax-pre-apti 7928  ax-pre-ltadd 7929  ax-pre-mulgt0 7930  ax-pre-mulext 7931  ax-arch 7932  ax-caucvg 7933

This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-if 3537  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-ilim 4371  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-isom 5227  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-frec 6394  df-1o 6419  df-2o 6420  df-er 6537  df-en 6743  df-fin 6745  df-sup 6985  df-inf 6986  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-sub 8132  df-neg 8133  df-reap 8534  df-ap 8541  df-div 8632  df-inn 8922  df-2 8980  df-3 8981  df-4 8982  df-n0 9179  df-z 9256  df-uz 9531  df-q 9622  df-rp 9656  df-fz 10011  df-fzo 10145  df-fl 10272  df-mod 10325  df-seqfrec 10448  df-exp 10522  df-cj 10853  df-re 10854  df-im 10855  df-rsqrt 11009  df-abs 11010  df-dvds 11797  df-gcd 11946  df-prm 12110  df-pc 12287

This theorem is referenced by:  1arith  12367