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Theorem 1arithlem4 12973
Description: Lemma for 1arith 12974. (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 )
3 ffvelrn 5663 . . . . . . 7  |-  ( ( F : Prime --> NN0  /\  y  e.  Prime )  -> 
( F `  y
)  e.  NN0 )
42, 3sylan 457 . . . . . 6  |-  ( (
ph  /\  y  e.  Prime )  ->  ( F `  y )  e.  NN0 )
54ralrimiva 2626 . . . . 5  |-  ( ph  ->  A. y  e.  Prime  ( F `  y )  e.  NN0 )
61, 5pcmptcl 12939 . . . 4  |-  ( ph  ->  ( G : NN --> NN  /\  seq  1 (  x.  ,  G ) : NN --> NN ) )
76simprd 449 . . 3  |-  ( ph  ->  seq  1 (  x.  ,  G ) : NN --> NN )
8 1arithlem4.4 . . 3  |-  ( ph  ->  N  e.  NN )
9 ffvelrn 5663 . . 3  |-  ( (  seq  1 (  x.  ,  G ) : NN --> NN  /\  N  e.  NN )  ->  (  seq  1 (  x.  ,  G ) `  N
)  e.  NN )
107, 8, 9syl2anc 642 . 2  |-  ( ph  ->  (  seq  1 (  x.  ,  G ) `
 N )  e.  NN )
11 1arith.1 . . . . . . 7  |-  M  =  ( n  e.  NN  |->  ( p  e.  Prime  |->  ( p  pCnt  n ) ) )
12111arithlem2 12971 . . . . . 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 )
) )
1310, 12sylan 457 . . . . 5  |-  ( (
ph  /\  q  e.  Prime )  ->  ( ( M `  (  seq  1 (  x.  ,  G ) `  N
) ) `  q
)  =  ( q 
pCnt  (  seq  1
(  x.  ,  G
) `  N )
) )
145adantr 451 . . . . . 6  |-  ( (
ph  /\  q  e.  Prime )  ->  A. y  e.  Prime  ( F `  y )  e.  NN0 )
158adantr 451 . . . . . 6  |-  ( (
ph  /\  q  e.  Prime )  ->  N  e.  NN )
16 simpr 447 . . . . . 6  |-  ( (
ph  /\  q  e.  Prime )  ->  q  e.  Prime )
17 fveq2 5525 . . . . . 6  |-  ( y  =  q  ->  ( F `  y )  =  ( F `  q ) )
181, 14, 15, 16, 17pcmpt 12940 . . . . 5  |-  ( (
ph  /\  q  e.  Prime )  ->  ( q  pCnt  (  seq  1 (  x.  ,  G ) `
 N ) )  =  if ( q  <_  N ,  ( F `  q ) ,  0 ) )
1915nnred 9761 . . . . . 6  |-  ( (
ph  /\  q  e.  Prime )  ->  N  e.  RR )
20 prmz 12762 . . . . . . . 8  |-  ( q  e.  Prime  ->  q  e.  ZZ )
2120zred 10117 . . . . . . 7  |-  ( q  e.  Prime  ->  q  e.  RR )
2221adantl 452 . . . . . 6  |-  ( (
ph  /\  q  e.  Prime )  ->  q  e.  RR )
23 ifid 3597 . . . . . . 7  |-  if ( q  <_  N , 
( F `  q
) ,  ( F `
 q ) )  =  ( F `  q )
24 1arithlem4.5 . . . . . . . . 9  |-  ( (
ph  /\  ( q  e.  Prime  /\  N  <_  q ) )  ->  ( F `  q )  =  0 )
2524anassrs 629 . . . . . . . 8  |-  ( ( ( ph  /\  q  e.  Prime )  /\  N  <_  q )  ->  ( F `  q )  =  0 )
2625ifeq2d 3580 . . . . . . 7  |-  ( ( ( ph  /\  q  e.  Prime )  /\  N  <_  q )  ->  if ( q  <_  N ,  ( F `  q ) ,  ( F `  q ) )  =  if ( q  <_  N , 
( F `  q
) ,  0 ) )
2723, 26syl5reqr 2330 . . . . . 6  |-  ( ( ( ph  /\  q  e.  Prime )  /\  N  <_  q )  ->  if ( q  <_  N ,  ( F `  q ) ,  0 )  =  ( F `
 q ) )
28 iftrue 3571 . . . . . . 7  |-  ( q  <_  N  ->  if ( q  <_  N ,  ( F `  q ) ,  0 )  =  ( F `
 q ) )
2928adantl 452 . . . . . 6  |-  ( ( ( ph  /\  q  e.  Prime )  /\  q  <_  N )  ->  if ( q  <_  N ,  ( F `  q ) ,  0 )  =  ( F `
 q ) )
3019, 22, 27, 29lecasei 8926 . . . . 5  |-  ( (
ph  /\  q  e.  Prime )  ->  if (
q  <_  N , 
( F `  q
) ,  0 )  =  ( F `  q ) )
3113, 18, 303eqtrrd 2320 . . . 4  |-  ( (
ph  /\  q  e.  Prime )  ->  ( F `  q )  =  ( ( M `  (  seq  1 (  x.  ,  G ) `  N
) ) `  q
) )
3231ralrimiva 2626 . . 3  |-  ( ph  ->  A. q  e.  Prime  ( F `  q )  =  ( ( M `
 (  seq  1
(  x.  ,  G
) `  N )
) `  q )
)
33111arithlem3 12972 . . . . 5  |-  ( (  seq  1 (  x.  ,  G ) `  N )  e.  NN  ->  ( M `  (  seq  1 (  x.  ,  G ) `  N
) ) : Prime --> NN0 )
3410, 33syl 15 . . . 4  |-  ( ph  ->  ( M `  (  seq  1 (  x.  ,  G ) `  N
) ) : Prime --> NN0 )
35 ffn 5389 . . . . 5  |-  ( F : Prime --> NN0  ->  F  Fn  Prime )
36 ffn 5389 . . . . 5  |-  ( ( M `  (  seq  1 (  x.  ,  G ) `  N
) ) : Prime --> NN0 
->  ( M `  (  seq  1 (  x.  ,  G ) `  N
) )  Fn  Prime )
37 eqfnfv 5622 . . . . 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
) ) )
3835, 36, 37syl2an 463 . . . 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
) ) )
392, 34, 38syl2anc 642 . . 3  |-  ( ph  ->  ( F  =  ( M `  (  seq  1 (  x.  ,  G ) `  N
) )  <->  A. q  e.  Prime  ( F `  q )  =  ( ( M `  (  seq  1 (  x.  ,  G ) `  N
) ) `  q
) ) )
4032, 39mpbird 223 . 2  |-  ( ph  ->  F  =  ( M `
 (  seq  1
(  x.  ,  G
) `  N )
) )
41 fveq2 5525 . . . 4  |-  ( x  =  (  seq  1
(  x.  ,  G
) `  N )  ->  ( M `  x
)  =  ( M `
 (  seq  1
(  x.  ,  G
) `  N )
) )
4241eqeq2d 2294 . . 3  |-  ( x  =  (  seq  1
(  x.  ,  G
) `  N )  ->  ( F  =  ( M `  x )  <-> 
F  =  ( M `
 (  seq  1
(  x.  ,  G
) `  N )
) ) )
4342rspcev 2884 . 2  |-  ( ( (  seq  1 (  x.  ,  G ) `
 N )  e.  NN  /\  F  =  ( M `  (  seq  1 (  x.  ,  G ) `  N
) ) )  ->  E. x  e.  NN  F  =  ( M `  x ) )
4410, 40, 43syl2anc 642 1  |-  ( ph  ->  E. x  e.  NN  F  =  ( M `  x ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   ifcif 3565   class class class wbr 4023    e. cmpt 4077    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   RRcr 8736   0cc0 8737   1c1 8738    x. cmul 8742    <_ cle 8868   NNcn 9746   NN0cn0 9965    seq cseq 11046   ^cexp 11104   Primecprime 12758    pCnt cpc 12889
This theorem is referenced by:  1arith  12974
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317  df-rp 10355  df-fz 10783  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-dvds 12532  df-gcd 12686  df-prm 12759  df-pc 12890
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