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Theorem 1arithlem4 14543
Description: Lemma for 1arith 14544. (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 5963 . . . . . 6  |-  ( (
ph  /\  y  e.  Prime )  ->  ( F `  y )  e.  NN0 )
43ralrimiva 2815 . . . . 5  |-  ( ph  ->  A. y  e.  Prime  ( F `  y )  e.  NN0 )
51, 4pcmptcl 14509 . . . 4  |-  ( ph  ->  ( G : NN --> NN  /\  seq 1 (  x.  ,  G ) : NN --> NN ) )
65simprd 461 . . 3  |-  ( ph  ->  seq 1 (  x.  ,  G ) : NN --> NN )
7 1arithlem4.4 . . 3  |-  ( ph  ->  N  e.  NN )
86, 7ffvelrnd 5964 . 2  |-  ( ph  ->  (  seq 1 (  x.  ,  G ) `
 N )  e.  NN )
9 1arith.1 . . . . . . 7  |-  M  =  ( n  e.  NN  |->  ( p  e.  Prime  |->  ( p  pCnt  n ) ) )
1091arithlem2 14541 . . . . . 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 469 . . . . 5  |-  ( (
ph  /\  q  e.  Prime )  ->  ( ( M `  (  seq 1 (  x.  ,  G ) `  N
) ) `  q
)  =  ( q 
pCnt  (  seq 1
(  x.  ,  G
) `  N )
) )
124adantr 463 . . . . . 6  |-  ( (
ph  /\  q  e.  Prime )  ->  A. y  e.  Prime  ( F `  y )  e.  NN0 )
137adantr 463 . . . . . 6  |-  ( (
ph  /\  q  e.  Prime )  ->  N  e.  NN )
14 simpr 459 . . . . . 6  |-  ( (
ph  /\  q  e.  Prime )  ->  q  e.  Prime )
15 fveq2 5803 . . . . . 6  |-  ( y  =  q  ->  ( F `  y )  =  ( F `  q ) )
161, 12, 13, 14, 15pcmpt 14510 . . . . 5  |-  ( (
ph  /\  q  e.  Prime )  ->  ( q  pCnt  (  seq 1 (  x.  ,  G ) `
 N ) )  =  if ( q  <_  N ,  ( F `  q ) ,  0 ) )
1713nnred 10509 . . . . . 6  |-  ( (
ph  /\  q  e.  Prime )  ->  N  e.  RR )
18 prmz 14320 . . . . . . . 8  |-  ( q  e.  Prime  ->  q  e.  ZZ )
1918zred 10926 . . . . . . 7  |-  ( q  e.  Prime  ->  q  e.  RR )
2019adantl 464 . . . . . 6  |-  ( (
ph  /\  q  e.  Prime )  ->  q  e.  RR )
21 ifid 3919 . . . . . . 7  |-  if ( q  <_  N , 
( F `  q
) ,  ( F `
 q ) )  =  ( F `  q )
22 1arithlem4.5 . . . . . . . . 9  |-  ( (
ph  /\  ( q  e.  Prime  /\  N  <_  q ) )  ->  ( F `  q )  =  0 )
2322anassrs 646 . . . . . . . 8  |-  ( ( ( ph  /\  q  e.  Prime )  /\  N  <_  q )  ->  ( F `  q )  =  0 )
2423ifeq2d 3901 . . . . . . 7  |-  ( ( ( ph  /\  q  e.  Prime )  /\  N  <_  q )  ->  if ( q  <_  N ,  ( F `  q ) ,  ( F `  q ) )  =  if ( q  <_  N , 
( F `  q
) ,  0 ) )
2521, 24syl5reqr 2456 . . . . . 6  |-  ( ( ( ph  /\  q  e.  Prime )  /\  N  <_  q )  ->  if ( q  <_  N ,  ( F `  q ) ,  0 )  =  ( F `
 q ) )
26 iftrue 3888 . . . . . . 7  |-  ( q  <_  N  ->  if ( q  <_  N ,  ( F `  q ) ,  0 )  =  ( F `
 q ) )
2726adantl 464 . . . . . 6  |-  ( ( ( ph  /\  q  e.  Prime )  /\  q  <_  N )  ->  if ( q  <_  N ,  ( F `  q ) ,  0 )  =  ( F `
 q ) )
2817, 20, 25, 27lecasei 9640 . . . . 5  |-  ( (
ph  /\  q  e.  Prime )  ->  if (
q  <_  N , 
( F `  q
) ,  0 )  =  ( F `  q ) )
2911, 16, 283eqtrrd 2446 . . . 4  |-  ( (
ph  /\  q  e.  Prime )  ->  ( F `  q )  =  ( ( M `  (  seq 1 (  x.  ,  G ) `  N
) ) `  q
) )
3029ralrimiva 2815 . . 3  |-  ( ph  ->  A. q  e.  Prime  ( F `  q )  =  ( ( M `
 (  seq 1
(  x.  ,  G
) `  N )
) `  q )
)
3191arithlem3 14542 . . . . 5  |-  ( (  seq 1 (  x.  ,  G ) `  N )  e.  NN  ->  ( M `  (  seq 1 (  x.  ,  G ) `  N
) ) : Prime --> NN0 )
328, 31syl 17 . . . 4  |-  ( ph  ->  ( M `  (  seq 1 (  x.  ,  G ) `  N
) ) : Prime --> NN0 )
33 ffn 5668 . . . . 5  |-  ( F : Prime --> NN0  ->  F  Fn  Prime )
34 ffn 5668 . . . . 5  |-  ( ( M `  (  seq 1 (  x.  ,  G ) `  N
) ) : Prime --> NN0 
->  ( M `  (  seq 1 (  x.  ,  G ) `  N
) )  Fn  Prime )
35 eqfnfv 5913 . . . . 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 475 . . . 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 659 . . 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 5803 . . . 4  |-  ( x  =  (  seq 1
(  x.  ,  G
) `  N )  ->  ( M `  x
)  =  ( M `
 (  seq 1
(  x.  ,  G
) `  N )
) )
4039eqeq2d 2414 . . 3  |-  ( x  =  (  seq 1
(  x.  ,  G
) `  N )  ->  ( F  =  ( M `  x )  <-> 
F  =  ( M `
 (  seq 1
(  x.  ,  G
) `  N )
) ) )
4140rspcev 3157 . 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 659 1  |-  ( ph  ->  E. x  e.  NN  F  =  ( M `  x ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1403    e. wcel 1840   A.wral 2751   E.wrex 2752   ifcif 3882   class class class wbr 4392    |-> cmpt 4450    Fn wfn 5518   -->wf 5519   ` cfv 5523  (class class class)co 6232   RRcr 9439   0cc0 9440   1c1 9441    x. cmul 9445    <_ cle 9577   NNcn 10494   NN0cn0 10754    seqcseq 12059   ^cexp 12118   Primecprime 14316    pCnt cpc 14459
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528  ax-cnex 9496  ax-resscn 9497  ax-1cn 9498  ax-icn 9499  ax-addcl 9500  ax-addrcl 9501  ax-mulcl 9502  ax-mulrcl 9503  ax-mulcom 9504  ax-addass 9505  ax-mulass 9506  ax-distr 9507  ax-i2m1 9508  ax-1ne0 9509  ax-1rid 9510  ax-rnegex 9511  ax-rrecex 9512  ax-cnre 9513  ax-pre-lttri 9514  ax-pre-lttrn 9515  ax-pre-ltadd 9516  ax-pre-mulgt0 9517  ax-pre-sup 9518
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 973  df-3an 974  df-tru 1406  df-fal 1409  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-nel 2599  df-ral 2756  df-rex 2757  df-reu 2758  df-rmo 2759  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-pss 3427  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-tp 3974  df-op 3976  df-uni 4189  df-int 4225  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4487  df-eprel 4731  df-id 4735  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-riota 6194  df-ov 6235  df-oprab 6236  df-mpt2 6237  df-om 6637  df-1st 6736  df-2nd 6737  df-recs 6997  df-rdg 7031  df-1o 7085  df-2o 7086  df-oadd 7089  df-er 7266  df-en 7473  df-dom 7474  df-sdom 7475  df-fin 7476  df-sup 7853  df-pnf 9578  df-mnf 9579  df-xr 9580  df-ltxr 9581  df-le 9582  df-sub 9761  df-neg 9762  df-div 10166  df-nn 10495  df-2 10553  df-3 10554  df-n0 10755  df-z 10824  df-uz 11044  df-q 11144  df-rp 11182  df-fz 11642  df-fl 11877  df-mod 11946  df-seq 12060  df-exp 12119  df-cj 12986  df-re 12987  df-im 12988  df-sqrt 13122  df-abs 13123  df-dvds 14086  df-gcd 14244  df-prm 14317  df-pc 14460
This theorem is referenced by:  1arith  14544
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