Theorem 1arithlem4 12929

Description: Lemma for 1arith 12930. (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 5778	. . . . . 6 |- ( ( ph /\ y e. Prime ) -> ( F ` y ) e. NN0 )
4	3	ralrimiva 2603	. . . . 5 |- ( ph -> A. y e. Prime ( F ` y ) e. NN0 )
5	1, 4	pcmptcl 12905	. . . 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 5779	. 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 12927	. . . . . 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 5635	. . . . . 6 |- ( y = q -> ( F ` y ) = ( F ` q ) )
16	1, 12, 13, 14, 15	pcmpt 12906	. . . . 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 3622	. . . . . . 7 |- ( ( ( ph /\ q e. Prime ) /\ N <_ q ) -> if ( q <_ N , ( F ` q ) , ( F ` q ) ) = if ( q <_ N , ( F ` q ) , 0 ) )
20		prmz 12673	. . . . . . . . . . 11 |- ( q e. Prime -> q e. ZZ )
21	20	adantl 277	. . . . . . . . . 10 |- ( ( ph /\ q e. Prime ) -> q e. ZZ )
22	13	nnzd 9591	. . . . . . . . . 10 |- ( ( ph /\ q e. Prime ) -> N e. ZZ )
23		zdcle 9546	. . . . . . . . . 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 3639	. . . . . . . . 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 2264	. . . . . 6 |- ( ( ( ph /\ q e. Prime ) /\ N <_ q ) -> if ( q <_ N , ( F ` q ) , 0 ) = ( F ` q ) )
29		iftrue 3608	. . . . . . 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 9513	. . . . . . 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 803	. . . . 5 |- ( ( ph /\ q e. Prime ) -> if ( q <_ N , ( F ` q ) , 0 ) = ( F ` q ) )
34	11, 16, 33	3eqtrrd 2267	. . . 4 |- ( ( ph /\ q e. Prime ) -> ( F ` q ) = ( ( M ` ( seq 1 ( x. , G ) ` N ) ) ` q ) )
35	34	ralrimiva 2603	. . 3 |- ( ph -> A. q e. Prime ( F ` q ) = ( ( M ` ( seq 1 ( x. , G ) ` N ) ) ` q ) )
36	9	1arithlem3 12928	. . . . 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 5479	. . . . 5 |- ( F : Prime --> NN0 -> F Fn Prime )
39		ffn 5479	. . . . 5 |- ( ( M ` ( seq 1 ( x. , G ) ` N ) ) : Prime --> NN0 -> ( M ` ( seq 1 ( x. , G ) ` N ) ) Fn Prime )
40		eqfnfv 5740	. . . . 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 5635	. . 3 |- ( x = ( seq 1 ( x. , G ) ` N ) -> ( M ` x ) = ( M ` ( seq 1 ( x. , G ) ` N ) ) )
45	44	rspceeqv 2926	. 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 713  DECID wdc 839   = wceq 1395   e. wcel 2200   A.wral 2508   E.wrex 2509   ifcif 3603   class class class wbr 4086   |-> cmpt 4148   Fn wfn 5319   -->wf 5320   ` cfv 5324  (class class class)co 6013   0cc0 8022   1c1 8023   x. cmul 8027   <_ cle 8205   NNcn 9133   NN0cn0 9392   ZZcz 9469   seqcseq 10699   ^cexp 10790   Primecprime 12669   pCnt cpc 12847

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-mulrcl 8121  ax-addcom 8122  ax-mulcom 8123  ax-addass 8124  ax-mulass 8125  ax-distr 8126  ax-i2m1 8127  ax-0lt1 8128  ax-1rid 8129  ax-0id 8130  ax-rnegex 8131  ax-precex 8132  ax-cnre 8133  ax-pre-ltirr 8134  ax-pre-ltwlin 8135  ax-pre-lttrn 8136  ax-pre-apti 8137  ax-pre-ltadd 8138  ax-pre-mulgt0 8139  ax-pre-mulext 8140  ax-arch 8141  ax-caucvg 8142

This theorem depends on definitions:  df-bi 117  df-stab 836  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-po 4391  df-iso 4392  df-iord 4461  df-on 4463  df-ilim 4464  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-isom 5333  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-frec 6552  df-1o 6577  df-2o 6578  df-er 6697  df-en 6905  df-fin 6907  df-sup 7174  df-inf 7175  df-pnf 8206  df-mnf 8207  df-xr 8208  df-ltxr 8209  df-le 8210  df-sub 8342  df-neg 8343  df-reap 8745  df-ap 8752  df-div 8843  df-inn 9134  df-2 9192  df-3 9193  df-4 9194  df-n0 9393  df-z 9470  df-uz 9746  df-q 9844  df-rp 9879  df-fz 10234  df-fzo 10368  df-fl 10520  df-mod 10575  df-seqfrec 10700  df-exp 10791  df-cj 11393  df-re 11394  df-im 11395  df-rsqrt 11549  df-abs 11550  df-dvds 12339  df-gcd 12515  df-prm 12670  df-pc 12848

This theorem is referenced by:  1arith  12930