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Theorem 1arithlem4 13286
Description: Lemma for 1arith 13287. (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
StepHypRef 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 )
32ffvelrnda 5862 . . . . . 6  |-  ( (
ph  /\  y  e.  Prime )  ->  ( F `  y )  e.  NN0 )
43ralrimiva 2781 . . . . 5  |-  ( ph  ->  A. y  e.  Prime  ( F `  y )  e.  NN0 )
51, 4pcmptcl 13252 . . . 4  |-  ( ph  ->  ( G : NN --> NN  /\  seq  1 (  x.  ,  G ) : NN --> NN ) )
65simprd 450 . . 3  |-  ( ph  ->  seq  1 (  x.  ,  G ) : NN --> NN )
7 1arithlem4.4 . . 3  |-  ( ph  ->  N  e.  NN )
86, 7ffvelrnd 5863 . 2  |-  ( ph  ->  (  seq  1 (  x.  ,  G ) `
 N )  e.  NN )
9 1arith.1 . . . . . . 7  |-  M  =  ( n  e.  NN  |->  ( p  e.  Prime  |->  ( p  pCnt  n ) ) )
1091arithlem2 13284 . . . . . 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 )
) )
118, 10sylan 458 . . . . 5  |-  ( (
ph  /\  q  e.  Prime )  ->  ( ( M `  (  seq  1 (  x.  ,  G ) `  N
) ) `  q
)  =  ( q 
pCnt  (  seq  1
(  x.  ,  G
) `  N )
) )
124adantr 452 . . . . . 6  |-  ( (
ph  /\  q  e.  Prime )  ->  A. y  e.  Prime  ( F `  y )  e.  NN0 )
137adantr 452 . . . . . 6  |-  ( (
ph  /\  q  e.  Prime )  ->  N  e.  NN )
14 simpr 448 . . . . . 6  |-  ( (
ph  /\  q  e.  Prime )  ->  q  e.  Prime )
15 fveq2 5720 . . . . . 6  |-  ( y  =  q  ->  ( F `  y )  =  ( F `  q ) )
161, 12, 13, 14, 15pcmpt 13253 . . . . 5  |-  ( (
ph  /\  q  e.  Prime )  ->  ( q  pCnt  (  seq  1 (  x.  ,  G ) `
 N ) )  =  if ( q  <_  N ,  ( F `  q ) ,  0 ) )
1713nnred 10007 . . . . . 6  |-  ( (
ph  /\  q  e.  Prime )  ->  N  e.  RR )
18 prmz 13075 . . . . . . . 8  |-  ( q  e.  Prime  ->  q  e.  ZZ )
1918zred 10367 . . . . . . 7  |-  ( q  e.  Prime  ->  q  e.  RR )
2019adantl 453 . . . . . 6  |-  ( (
ph  /\  q  e.  Prime )  ->  q  e.  RR )
21 ifid 3763 . . . . . . 7  |-  if ( q  <_  N , 
( F `  q
) ,  ( F `
 q ) )  =  ( F `  q )
22 1arithlem4.5 . . . . . . . . 9  |-  ( (
ph  /\  ( q  e.  Prime  /\  N  <_  q ) )  ->  ( F `  q )  =  0 )
2322anassrs 630 . . . . . . . 8  |-  ( ( ( ph  /\  q  e.  Prime )  /\  N  <_  q )  ->  ( F `  q )  =  0 )
2423ifeq2d 3746 . . . . . . 7  |-  ( ( ( ph  /\  q  e.  Prime )  /\  N  <_  q )  ->  if ( q  <_  N ,  ( F `  q ) ,  ( F `  q ) )  =  if ( q  <_  N , 
( F `  q
) ,  0 ) )
2521, 24syl5reqr 2482 . . . . . 6  |-  ( ( ( ph  /\  q  e.  Prime )  /\  N  <_  q )  ->  if ( q  <_  N ,  ( F `  q ) ,  0 )  =  ( F `
 q ) )
26 iftrue 3737 . . . . . . 7  |-  ( q  <_  N  ->  if ( q  <_  N ,  ( F `  q ) ,  0 )  =  ( F `
 q ) )
2726adantl 453 . . . . . 6  |-  ( ( ( ph  /\  q  e.  Prime )  /\  q  <_  N )  ->  if ( q  <_  N ,  ( F `  q ) ,  0 )  =  ( F `
 q ) )
2817, 20, 25, 27lecasei 9171 . . . . 5  |-  ( (
ph  /\  q  e.  Prime )  ->  if (
q  <_  N , 
( F `  q
) ,  0 )  =  ( F `  q ) )
2911, 16, 283eqtrrd 2472 . . . 4  |-  ( (
ph  /\  q  e.  Prime )  ->  ( F `  q )  =  ( ( M `  (  seq  1 (  x.  ,  G ) `  N
) ) `  q
) )
3029ralrimiva 2781 . . 3  |-  ( ph  ->  A. q  e.  Prime  ( F `  q )  =  ( ( M `
 (  seq  1
(  x.  ,  G
) `  N )
) `  q )
)
3191arithlem3 13285 . . . . 5  |-  ( (  seq  1 (  x.  ,  G ) `  N )  e.  NN  ->  ( M `  (  seq  1 (  x.  ,  G ) `  N
) ) : Prime --> NN0 )
328, 31syl 16 . . . 4  |-  ( ph  ->  ( M `  (  seq  1 (  x.  ,  G ) `  N
) ) : Prime --> NN0 )
33 ffn 5583 . . . . 5  |-  ( F : Prime --> NN0  ->  F  Fn  Prime )
34 ffn 5583 . . . . 5  |-  ( ( M `  (  seq  1 (  x.  ,  G ) `  N
) ) : Prime --> NN0 
->  ( M `  (  seq  1 (  x.  ,  G ) `  N
) )  Fn  Prime )
35 eqfnfv 5819 . . . . 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
) ) )
3633, 34, 35syl2an 464 . . . 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
) ) )
372, 32, 36syl2anc 643 . . 3  |-  ( ph  ->  ( F  =  ( M `  (  seq  1 (  x.  ,  G ) `  N
) )  <->  A. q  e.  Prime  ( F `  q )  =  ( ( M `  (  seq  1 (  x.  ,  G ) `  N
) ) `  q
) ) )
3830, 37mpbird 224 . 2  |-  ( ph  ->  F  =  ( M `
 (  seq  1
(  x.  ,  G
) `  N )
) )
39 fveq2 5720 . . . 4  |-  ( x  =  (  seq  1
(  x.  ,  G
) `  N )  ->  ( M `  x
)  =  ( M `
 (  seq  1
(  x.  ,  G
) `  N )
) )
4039eqeq2d 2446 . . 3  |-  ( x  =  (  seq  1
(  x.  ,  G
) `  N )  ->  ( F  =  ( M `  x )  <-> 
F  =  ( M `
 (  seq  1
(  x.  ,  G
) `  N )
) ) )
4140rspcev 3044 . 2  |-  ( ( (  seq  1 (  x.  ,  G ) `
 N )  e.  NN  /\  F  =  ( M `  (  seq  1 (  x.  ,  G ) `  N
) ) )  ->  E. x  e.  NN  F  =  ( M `  x ) )
428, 38, 41syl2anc 643 1  |-  ( ph  ->  E. x  e.  NN  F  =  ( M `  x ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697   E.wrex 2698   ifcif 3731   class class class wbr 4204    e. cmpt 4258    Fn wfn 5441   -->wf 5442   ` cfv 5446  (class class class)co 6073   RRcr 8981   0cc0 8982   1c1 8983    x. cmul 8987    <_ cle 9113   NNcn 9992   NN0cn0 10213    seq cseq 11315   ^cexp 11374   Primecprime 13071    pCnt cpc 13202
This theorem is referenced by:  1arith  13287
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-q 10567  df-rp 10605  df-fz 11036  df-fl 11194  df-mod 11243  df-seq 11316  df-exp 11375  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-dvds 12845  df-gcd 12999  df-prm 13072  df-pc 13203
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