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Theorem 1arithlem4 13249
Description: Lemma for 1arith 13250. (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 5829 . . . . . 6  |-  ( (
ph  /\  y  e.  Prime )  ->  ( F `  y )  e.  NN0 )
43ralrimiva 2749 . . . . 5  |-  ( ph  ->  A. y  e.  Prime  ( F `  y )  e.  NN0 )
51, 4pcmptcl 13215 . . . 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 5830 . 2  |-  ( ph  ->  (  seq  1 (  x.  ,  G ) `
 N )  e.  NN )
9 1arith.1 . . . . . . 7  |-  M  =  ( n  e.  NN  |->  ( p  e.  Prime  |->  ( p  pCnt  n ) ) )
1091arithlem2 13247 . . . . . 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 5687 . . . . . 6  |-  ( y  =  q  ->  ( F `  y )  =  ( F `  q ) )
161, 12, 13, 14, 15pcmpt 13216 . . . . 5  |-  ( (
ph  /\  q  e.  Prime )  ->  ( q  pCnt  (  seq  1 (  x.  ,  G ) `
 N ) )  =  if ( q  <_  N ,  ( F `  q ) ,  0 ) )
1713nnred 9971 . . . . . 6  |-  ( (
ph  /\  q  e.  Prime )  ->  N  e.  RR )
18 prmz 13038 . . . . . . . 8  |-  ( q  e.  Prime  ->  q  e.  ZZ )
1918zred 10331 . . . . . . 7  |-  ( q  e.  Prime  ->  q  e.  RR )
2019adantl 453 . . . . . 6  |-  ( (
ph  /\  q  e.  Prime )  ->  q  e.  RR )
21 ifid 3731 . . . . . . 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 3714 . . . . . . 7  |-  ( ( ( ph  /\  q  e.  Prime )  /\  N  <_  q )  ->  if ( q  <_  N ,  ( F `  q ) ,  ( F `  q ) )  =  if ( q  <_  N , 
( F `  q
) ,  0 ) )
2521, 24syl5reqr 2451 . . . . . 6  |-  ( ( ( ph  /\  q  e.  Prime )  /\  N  <_  q )  ->  if ( q  <_  N ,  ( F `  q ) ,  0 )  =  ( F `
 q ) )
26 iftrue 3705 . . . . . . 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 9135 . . . . 5  |-  ( (
ph  /\  q  e.  Prime )  ->  if (
q  <_  N , 
( F `  q
) ,  0 )  =  ( F `  q ) )
2911, 16, 283eqtrrd 2441 . . . 4  |-  ( (
ph  /\  q  e.  Prime )  ->  ( F `  q )  =  ( ( M `  (  seq  1 (  x.  ,  G ) `  N
) ) `  q
) )
3029ralrimiva 2749 . . 3  |-  ( ph  ->  A. q  e.  Prime  ( F `  q )  =  ( ( M `
 (  seq  1
(  x.  ,  G
) `  N )
) `  q )
)
3191arithlem3 13248 . . . . 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 5550 . . . . 5  |-  ( F : Prime --> NN0  ->  F  Fn  Prime )
34 ffn 5550 . . . . 5  |-  ( ( M `  (  seq  1 (  x.  ,  G ) `  N
) ) : Prime --> NN0 
->  ( M `  (  seq  1 (  x.  ,  G ) `  N
) )  Fn  Prime )
35 eqfnfv 5786 . . . . 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 5687 . . . 4  |-  ( x  =  (  seq  1
(  x.  ,  G
) `  N )  ->  ( M `  x
)  =  ( M `
 (  seq  1
(  x.  ,  G
) `  N )
) )
4039eqeq2d 2415 . . 3  |-  ( x  =  (  seq  1
(  x.  ,  G
) `  N )  ->  ( F  =  ( M `  x )  <-> 
F  =  ( M `
 (  seq  1
(  x.  ,  G
) `  N )
) ) )
4140rspcev 3012 . 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 1649    e. wcel 1721   A.wral 2666   E.wrex 2667   ifcif 3699   class class class wbr 4172    e. cmpt 4226    Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6040   RRcr 8945   0cc0 8946   1c1 8947    x. cmul 8951    <_ cle 9077   NNcn 9956   NN0cn0 10177    seq cseq 11278   ^cexp 11337   Primecprime 13034    pCnt cpc 13165
This theorem is referenced by:  1arith  13250
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-q 10531  df-rp 10569  df-fz 11000  df-fl 11157  df-mod 11206  df-seq 11279  df-exp 11338  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-dvds 12808  df-gcd 12962  df-prm 13035  df-pc 13166
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