MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  1arithlem4 Structured version   Unicode version

Theorem 1arithlem4 13992
Description: Lemma for 1arith 13993. (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 5848 . . . . . 6  |-  ( (
ph  /\  y  e.  Prime )  ->  ( F `  y )  e.  NN0 )
43ralrimiva 2804 . . . . 5  |-  ( ph  ->  A. y  e.  Prime  ( F `  y )  e.  NN0 )
51, 4pcmptcl 13958 . . . 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 5849 . 2  |-  ( ph  ->  (  seq 1 (  x.  ,  G ) `
 N )  e.  NN )
9 1arith.1 . . . . . . 7  |-  M  =  ( n  e.  NN  |->  ( p  e.  Prime  |->  ( p  pCnt  n ) ) )
1091arithlem2 13990 . . . . . 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 5696 . . . . . 6  |-  ( y  =  q  ->  ( F `  y )  =  ( F `  q ) )
161, 12, 13, 14, 15pcmpt 13959 . . . . 5  |-  ( (
ph  /\  q  e.  Prime )  ->  ( q  pCnt  (  seq 1 (  x.  ,  G ) `
 N ) )  =  if ( q  <_  N ,  ( F `  q ) ,  0 ) )
1713nnred 10342 . . . . . 6  |-  ( (
ph  /\  q  e.  Prime )  ->  N  e.  RR )
18 prmz 13772 . . . . . . . 8  |-  ( q  e.  Prime  ->  q  e.  ZZ )
1918zred 10752 . . . . . . 7  |-  ( q  e.  Prime  ->  q  e.  RR )
2019adantl 466 . . . . . 6  |-  ( (
ph  /\  q  e.  Prime )  ->  q  e.  RR )
21 ifid 3831 . . . . . . 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 3813 . . . . . . 7  |-  ( ( ( ph  /\  q  e.  Prime )  /\  N  <_  q )  ->  if ( q  <_  N ,  ( F `  q ) ,  ( F `  q ) )  =  if ( q  <_  N , 
( F `  q
) ,  0 ) )
2521, 24syl5reqr 2490 . . . . . 6  |-  ( ( ( ph  /\  q  e.  Prime )  /\  N  <_  q )  ->  if ( q  <_  N ,  ( F `  q ) ,  0 )  =  ( F `
 q ) )
26 iftrue 3802 . . . . . . 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 9485 . . . . 5  |-  ( (
ph  /\  q  e.  Prime )  ->  if (
q  <_  N , 
( F `  q
) ,  0 )  =  ( F `  q ) )
2911, 16, 283eqtrrd 2480 . . . 4  |-  ( (
ph  /\  q  e.  Prime )  ->  ( F `  q )  =  ( ( M `  (  seq 1 (  x.  ,  G ) `  N
) ) `  q
) )
3029ralrimiva 2804 . . 3  |-  ( ph  ->  A. q  e.  Prime  ( F `  q )  =  ( ( M `
 (  seq 1
(  x.  ,  G
) `  N )
) `  q )
)
3191arithlem3 13991 . . . . 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 5564 . . . . 5  |-  ( F : Prime --> NN0  ->  F  Fn  Prime )
34 ffn 5564 . . . . 5  |-  ( ( M `  (  seq 1 (  x.  ,  G ) `  N
) ) : Prime --> NN0 
->  ( M `  (  seq 1 (  x.  ,  G ) `  N
) )  Fn  Prime )
35 eqfnfv 5802 . . . . 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 5696 . . . 4  |-  ( x  =  (  seq 1
(  x.  ,  G
) `  N )  ->  ( M `  x
)  =  ( M `
 (  seq 1
(  x.  ,  G
) `  N )
) )
4039eqeq2d 2454 . . 3  |-  ( x  =  (  seq 1
(  x.  ,  G
) `  N )  ->  ( F  =  ( M `  x )  <-> 
F  =  ( M `
 (  seq 1
(  x.  ,  G
) `  N )
) ) )
4140rspcev 3078 . 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 1369    e. wcel 1756   A.wral 2720   E.wrex 2721   ifcif 3796   class class class wbr 4297    e. cmpt 4355    Fn wfn 5418   -->wf 5419   ` cfv 5423  (class class class)co 6096   RRcr 9286   0cc0 9287   1c1 9288    x. cmul 9292    <_ cle 9424   NNcn 10327   NN0cn0 10584    seqcseq 11811   ^cexp 11870   Primecprime 13768    pCnt cpc 13908
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364  ax-pre-sup 9365
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-2o 6926  df-oadd 6929  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-sup 7696  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-nn 10328  df-2 10385  df-3 10386  df-n0 10585  df-z 10652  df-uz 10867  df-q 10959  df-rp 10997  df-fz 11443  df-fl 11647  df-mod 11714  df-seq 11812  df-exp 11871  df-cj 12593  df-re 12594  df-im 12595  df-sqr 12729  df-abs 12730  df-dvds 13541  df-gcd 13696  df-prm 13769  df-pc 13909
This theorem is referenced by:  1arith  13993
  Copyright terms: Public domain W3C validator