Theorem 1arithlem4 12292

Description: Lemma for 1arith 12293. (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 5619	. . . . . 6 |- ( ( ph /\ y e. Prime ) -> ( F ` y ) e. NN0 )
4	3	ralrimiva 2538	. . . . 5 |- ( ph -> A. y e. Prime ( F ` y ) e. NN0 )
5	1, 4	pcmptcl 12268	. . . 4 |- ( ph -> ( G : NN --> NN /\ seq 1 ( x. , G ) : NN --> NN ) )
6	5	simprd 113	. . 3 |- ( ph -> seq 1 ( x. , G ) : NN --> NN )
7		1arithlem4.4	. . 3 |- ( ph -> N e. NN )
8	6, 7	ffvelrnd 5620	. 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 12290	. . . . . 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 281	. . . . 5 |- ( ( ph /\ q e. Prime ) -> ( ( M ` ( seq 1 ( x. , G ) ` N ) ) ` q ) = ( q pCnt ( seq 1 ( x. , G ) ` N ) ) )
12	4	adantr 274	. . . . . 6 |- ( ( ph /\ q e. Prime ) -> A. y e. Prime ( F ` y ) e. NN0 )
13	7	adantr 274	. . . . . 6 |- ( ( ph /\ q e. Prime ) -> N e. NN )
14		simpr 109	. . . . . 6 |- ( ( ph /\ q e. Prime ) -> q e. Prime )
15		fveq2 5485	. . . . . 6 |- ( y = q -> ( F ` y ) = ( F ` q ) )
16	1, 12, 13, 14, 15	pcmpt 12269	. . . . 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 398	. . . . . . . 8 |- ( ( ( ph /\ q e. Prime ) /\ N <_ q ) -> ( F ` q ) = 0 )
19	18	ifeq2d 3537	. . . . . . 7 |- ( ( ( ph /\ q e. Prime ) /\ N <_ q ) -> if ( q <_ N , ( F ` q ) , ( F ` q ) ) = if ( q <_ N , ( F ` q ) , 0 ) )
20		prmz 12039	. . . . . . . . . . 11 |- ( q e. Prime -> q e. ZZ )
21	20	adantl 275	. . . . . . . . . 10 |- ( ( ph /\ q e. Prime ) -> q e. ZZ )
22	13	nnzd 9308	. . . . . . . . . 10 |- ( ( ph /\ q e. Prime ) -> N e. ZZ )
23		zdcle 9263	. . . . . . . . . 10 |- ( ( q e. ZZ /\ N e. ZZ ) -> DECID q <_ N )
24	21, 22, 23	syl2anc 409	. . . . . . . . 9 |- ( ( ph /\ q e. Prime ) -> DECID q <_ N )
25		ifiddc 3552	. . . . . . . . 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 274	. . . . . . 7 |- ( ( ( ph /\ q e. Prime ) /\ N <_ q ) -> if ( q <_ N , ( F ` q ) , ( F ` q ) ) = ( F ` q ) )
28	19, 27	eqtr3d 2200	. . . . . 6 |- ( ( ( ph /\ q e. Prime ) /\ N <_ q ) -> if ( q <_ N , ( F ` q ) , 0 ) = ( F ` q ) )
29		iftrue 3524	. . . . . . 7 |- ( q <_ N -> if ( q <_ N , ( F ` q ) , 0 ) = ( F ` q ) )
30	29	adantl 275	. . . . . 6 |- ( ( ( ph /\ q e. Prime ) /\ q <_ N ) -> if ( q <_ N , ( F ` q ) , 0 ) = ( F ` q ) )
31		zletric 9231	. . . . . . 7 |- ( ( N e. ZZ /\ q e. ZZ ) -> ( N <_ q \/ q <_ N ) )
32	22, 21, 31	syl2anc 409	. . . . . 6 |- ( ( ph /\ q e. Prime ) -> ( N <_ q \/ q <_ N ) )
33	28, 30, 32	mpjaodan 788	. . . . 5 |- ( ( ph /\ q e. Prime ) -> if ( q <_ N , ( F ` q ) , 0 ) = ( F ` q ) )
34	11, 16, 33	3eqtrrd 2203	. . . 4 |- ( ( ph /\ q e. Prime ) -> ( F ` q ) = ( ( M ` ( seq 1 ( x. , G ) ` N ) ) ` q ) )
35	34	ralrimiva 2538	. . 3 |- ( ph -> A. q e. Prime ( F ` q ) = ( ( M ` ( seq 1 ( x. , G ) ` N ) ) ` q ) )
36	9	1arithlem3 12291	. . . . 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 5336	. . . . 5 |- ( F : Prime --> NN0 -> F Fn Prime )
39		ffn 5336	. . . . 5 |- ( ( M ` ( seq 1 ( x. , G ) ` N ) ) : Prime --> NN0 -> ( M ` ( seq 1 ( x. , G ) ` N ) ) Fn Prime )
40		eqfnfv 5582	. . . . 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 287	. . . 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 409	. . 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 166	. 2 |- ( ph -> F = ( M ` ( seq 1 ( x. , G ) ` N ) ) )
44		fveq2 5485	. . 3 |- ( x = ( seq 1 ( x. , G ) ` N ) -> ( M ` x ) = ( M ` ( seq 1 ( x. , G ) ` N ) ) )
45	44	rspceeqv 2847	. 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 409	1 |- ( ph -> E. x e. NN F = ( M ` x ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 698  DECID wdc 824   = wceq 1343   e. wcel 2136   A.wral 2443   E.wrex 2444   ifcif 3519   class class class wbr 3981   |-> cmpt 4042   Fn wfn 5182   -->wf 5183   ` cfv 5187  (class class class)co 5841   0cc0 7749   1c1 7750   x. cmul 7754   <_ cle 7930   NNcn 8853   NN0cn0 9110   ZZcz 9187   seqcseq 10376   ^cexp 10450   Primecprime 12035   pCnt cpc 12212

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4096  ax-sep 4099  ax-nul 4107  ax-pow 4152  ax-pr 4186  ax-un 4410  ax-setind 4513  ax-iinf 4564  ax-cnex 7840  ax-resscn 7841  ax-1cn 7842  ax-1re 7843  ax-icn 7844  ax-addcl 7845  ax-addrcl 7846  ax-mulcl 7847  ax-mulrcl 7848  ax-addcom 7849  ax-mulcom 7850  ax-addass 7851  ax-mulass 7852  ax-distr 7853  ax-i2m1 7854  ax-0lt1 7855  ax-1rid 7856  ax-0id 7857  ax-rnegex 7858  ax-precex 7859  ax-cnre 7860  ax-pre-ltirr 7861  ax-pre-ltwlin 7862  ax-pre-lttrn 7863  ax-pre-apti 7864  ax-pre-ltadd 7865  ax-pre-mulgt0 7866  ax-pre-mulext 7867  ax-arch 7868  ax-caucvg 7869

This theorem depends on definitions:  df-bi 116  df-stab 821  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2296  df-ne 2336  df-nel 2431  df-ral 2448  df-rex 2449  df-reu 2450  df-rmo 2451  df-rab 2452  df-v 2727  df-sbc 2951  df-csb 3045  df-dif 3117  df-un 3119  df-in 3121  df-ss 3128  df-nul 3409  df-if 3520  df-pw 3560  df-sn 3581  df-pr 3582  df-op 3584  df-uni 3789  df-int 3824  df-iun 3867  df-br 3982  df-opab 4043  df-mpt 4044  df-tr 4080  df-id 4270  df-po 4273  df-iso 4274  df-iord 4343  df-on 4345  df-ilim 4346  df-suc 4348  df-iom 4567  df-xp 4609  df-rel 4610  df-cnv 4611  df-co 4612  df-dm 4613  df-rn 4614  df-res 4615  df-ima 4616  df-iota 5152  df-fun 5189  df-fn 5190  df-f 5191  df-f1 5192  df-fo 5193  df-f1o 5194  df-fv 5195  df-isom 5196  df-riota 5797  df-ov 5844  df-oprab 5845  df-mpo 5846  df-1st 6105  df-2nd 6106  df-recs 6269  df-frec 6355  df-1o 6380  df-2o 6381  df-er 6497  df-en 6703  df-fin 6705  df-sup 6945  df-inf 6946  df-pnf 7931  df-mnf 7932  df-xr 7933  df-ltxr 7934  df-le 7935  df-sub 8067  df-neg 8068  df-reap 8469  df-ap 8476  df-div 8565  df-inn 8854  df-2 8912  df-3 8913  df-4 8914  df-n0 9111  df-z 9188  df-uz 9463  df-q 9554  df-rp 9586  df-fz 9941  df-fzo 10074  df-fl 10201  df-mod 10254  df-seqfrec 10377  df-exp 10451  df-cj 10780  df-re 10781  df-im 10782  df-rsqrt 10936  df-abs 10937  df-dvds 11724  df-gcd 11872  df-prm 12036  df-pc 12213

This theorem is referenced by:  1arith  12293