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Theorem 1arithlem4 14299
Description: Lemma for 1arith 14300. (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 6019 . . . . . 6  |-  ( (
ph  /\  y  e.  Prime )  ->  ( F `  y )  e.  NN0 )
43ralrimiva 2878 . . . . 5  |-  ( ph  ->  A. y  e.  Prime  ( F `  y )  e.  NN0 )
51, 4pcmptcl 14265 . . . 4  |-  ( ph  ->  ( G : NN --> NN  /\  seq 1 (  x.  ,  G ) : NN --> NN ) )
65simprd 463 . . 3  |-  ( ph  ->  seq 1 (  x.  ,  G ) : NN --> NN )
7 1arithlem4.4 . . 3  |-  ( ph  ->  N  e.  NN )
86, 7ffvelrnd 6020 . 2  |-  ( ph  ->  (  seq 1 (  x.  ,  G ) `
 N )  e.  NN )
9 1arith.1 . . . . . . 7  |-  M  =  ( n  e.  NN  |->  ( p  e.  Prime  |->  ( p  pCnt  n ) ) )
1091arithlem2 14297 . . . . . 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 471 . . . . 5  |-  ( (
ph  /\  q  e.  Prime )  ->  ( ( M `  (  seq 1 (  x.  ,  G ) `  N
) ) `  q
)  =  ( q 
pCnt  (  seq 1
(  x.  ,  G
) `  N )
) )
124adantr 465 . . . . . 6  |-  ( (
ph  /\  q  e.  Prime )  ->  A. y  e.  Prime  ( F `  y )  e.  NN0 )
137adantr 465 . . . . . 6  |-  ( (
ph  /\  q  e.  Prime )  ->  N  e.  NN )
14 simpr 461 . . . . . 6  |-  ( (
ph  /\  q  e.  Prime )  ->  q  e.  Prime )
15 fveq2 5864 . . . . . 6  |-  ( y  =  q  ->  ( F `  y )  =  ( F `  q ) )
161, 12, 13, 14, 15pcmpt 14266 . . . . 5  |-  ( (
ph  /\  q  e.  Prime )  ->  ( q  pCnt  (  seq 1 (  x.  ,  G ) `
 N ) )  =  if ( q  <_  N ,  ( F `  q ) ,  0 ) )
1713nnred 10547 . . . . . 6  |-  ( (
ph  /\  q  e.  Prime )  ->  N  e.  RR )
18 prmz 14076 . . . . . . . 8  |-  ( q  e.  Prime  ->  q  e.  ZZ )
1918zred 10962 . . . . . . 7  |-  ( q  e.  Prime  ->  q  e.  RR )
2019adantl 466 . . . . . 6  |-  ( (
ph  /\  q  e.  Prime )  ->  q  e.  RR )
21 ifid 3976 . . . . . . 7  |-  if ( q  <_  N , 
( F `  q
) ,  ( F `
 q ) )  =  ( F `  q )
22 1arithlem4.5 . . . . . . . . 9  |-  ( (
ph  /\  ( q  e.  Prime  /\  N  <_  q ) )  ->  ( F `  q )  =  0 )
2322anassrs 648 . . . . . . . 8  |-  ( ( ( ph  /\  q  e.  Prime )  /\  N  <_  q )  ->  ( F `  q )  =  0 )
2423ifeq2d 3958 . . . . . . 7  |-  ( ( ( ph  /\  q  e.  Prime )  /\  N  <_  q )  ->  if ( q  <_  N ,  ( F `  q ) ,  ( F `  q ) )  =  if ( q  <_  N , 
( F `  q
) ,  0 ) )
2521, 24syl5reqr 2523 . . . . . 6  |-  ( ( ( ph  /\  q  e.  Prime )  /\  N  <_  q )  ->  if ( q  <_  N ,  ( F `  q ) ,  0 )  =  ( F `
 q ) )
26 iftrue 3945 . . . . . . 7  |-  ( q  <_  N  ->  if ( q  <_  N ,  ( F `  q ) ,  0 )  =  ( F `
 q ) )
2726adantl 466 . . . . . 6  |-  ( ( ( ph  /\  q  e.  Prime )  /\  q  <_  N )  ->  if ( q  <_  N ,  ( F `  q ) ,  0 )  =  ( F `
 q ) )
2817, 20, 25, 27lecasei 9686 . . . . 5  |-  ( (
ph  /\  q  e.  Prime )  ->  if (
q  <_  N , 
( F `  q
) ,  0 )  =  ( F `  q ) )
2911, 16, 283eqtrrd 2513 . . . 4  |-  ( (
ph  /\  q  e.  Prime )  ->  ( F `  q )  =  ( ( M `  (  seq 1 (  x.  ,  G ) `  N
) ) `  q
) )
3029ralrimiva 2878 . . 3  |-  ( ph  ->  A. q  e.  Prime  ( F `  q )  =  ( ( M `
 (  seq 1
(  x.  ,  G
) `  N )
) `  q )
)
3191arithlem3 14298 . . . . 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 5729 . . . . 5  |-  ( F : Prime --> NN0  ->  F  Fn  Prime )
34 ffn 5729 . . . . 5  |-  ( ( M `  (  seq 1 (  x.  ,  G ) `  N
) ) : Prime --> NN0 
->  ( M `  (  seq 1 (  x.  ,  G ) `  N
) )  Fn  Prime )
35 eqfnfv 5973 . . . . 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 477 . . . 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 661 . . 3  |-  ( ph  ->  ( F  =  ( M `  (  seq 1 (  x.  ,  G ) `  N
) )  <->  A. q  e.  Prime  ( F `  q )  =  ( ( M `  (  seq 1 (  x.  ,  G ) `  N
) ) `  q
) ) )
3830, 37mpbird 232 . 2  |-  ( ph  ->  F  =  ( M `
 (  seq 1
(  x.  ,  G
) `  N )
) )
39 fveq2 5864 . . . 4  |-  ( x  =  (  seq 1
(  x.  ,  G
) `  N )  ->  ( M `  x
)  =  ( M `
 (  seq 1
(  x.  ,  G
) `  N )
) )
4039eqeq2d 2481 . . 3  |-  ( x  =  (  seq 1
(  x.  ,  G
) `  N )  ->  ( F  =  ( M `  x )  <-> 
F  =  ( M `
 (  seq 1
(  x.  ,  G
) `  N )
) ) )
4140rspcev 3214 . 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 661 1  |-  ( ph  ->  E. x  e.  NN  F  =  ( M `  x ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814   E.wrex 2815   ifcif 3939   class class class wbr 4447    |-> cmpt 4505    Fn wfn 5581   -->wf 5582   ` cfv 5586  (class class class)co 6282   RRcr 9487   0cc0 9488   1c1 9489    x. cmul 9493    <_ cle 9625   NNcn 10532   NN0cn0 10791    seqcseq 12071   ^cexp 12130   Primecprime 14072    pCnt cpc 14215
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-2o 7128  df-oadd 7131  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-sup 7897  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11079  df-q 11179  df-rp 11217  df-fz 11669  df-fl 11893  df-mod 11961  df-seq 12072  df-exp 12131  df-cj 12891  df-re 12892  df-im 12893  df-sqrt 13027  df-abs 13028  df-dvds 13844  df-gcd 14000  df-prm 14073  df-pc 14216
This theorem is referenced by:  1arith  14300
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