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Theorem 1arithlem4 12989
Description: Lemma for 1arith 12990. (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 5679 . . . . . . 7  |-  ( ( F : Prime --> NN0  /\  y  e.  Prime )  -> 
( F `  y
)  e.  NN0 )
42, 3sylan 457 . . . . . 6  |-  ( (
ph  /\  y  e.  Prime )  ->  ( F `  y )  e.  NN0 )
54ralrimiva 2639 . . . . 5  |-  ( ph  ->  A. y  e.  Prime  ( F `  y )  e.  NN0 )
61, 5pcmptcl 12955 . . . 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 5679 . . 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 12987 . . . . . 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 5541 . . . . . 6  |-  ( y  =  q  ->  ( F `  y )  =  ( F `  q ) )
181, 14, 15, 16, 17pcmpt 12956 . . . . 5  |-  ( (
ph  /\  q  e.  Prime )  ->  ( q  pCnt  (  seq  1 (  x.  ,  G ) `
 N ) )  =  if ( q  <_  N ,  ( F `  q ) ,  0 ) )
1915nnred 9777 . . . . . 6  |-  ( (
ph  /\  q  e.  Prime )  ->  N  e.  RR )
20 prmz 12778 . . . . . . . 8  |-  ( q  e.  Prime  ->  q  e.  ZZ )
2120zred 10133 . . . . . . 7  |-  ( q  e.  Prime  ->  q  e.  RR )
2221adantl 452 . . . . . 6  |-  ( (
ph  /\  q  e.  Prime )  ->  q  e.  RR )
23 ifid 3610 . . . . . . 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 3593 . . . . . . 7  |-  ( ( ( ph  /\  q  e.  Prime )  /\  N  <_  q )  ->  if ( q  <_  N ,  ( F `  q ) ,  ( F `  q ) )  =  if ( q  <_  N , 
( F `  q
) ,  0 ) )
2723, 26syl5reqr 2343 . . . . . 6  |-  ( ( ( ph  /\  q  e.  Prime )  /\  N  <_  q )  ->  if ( q  <_  N ,  ( F `  q ) ,  0 )  =  ( F `
 q ) )
28 iftrue 3584 . . . . . . 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 8942 . . . . 5  |-  ( (
ph  /\  q  e.  Prime )  ->  if (
q  <_  N , 
( F `  q
) ,  0 )  =  ( F `  q ) )
3113, 18, 303eqtrrd 2333 . . . 4  |-  ( (
ph  /\  q  e.  Prime )  ->  ( F `  q )  =  ( ( M `  (  seq  1 (  x.  ,  G ) `  N
) ) `  q
) )
3231ralrimiva 2639 . . 3  |-  ( ph  ->  A. q  e.  Prime  ( F `  q )  =  ( ( M `
 (  seq  1
(  x.  ,  G
) `  N )
) `  q )
)
33111arithlem3 12988 . . . . 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 5405 . . . . 5  |-  ( F : Prime --> NN0  ->  F  Fn  Prime )
36 ffn 5405 . . . . 5  |-  ( ( M `  (  seq  1 (  x.  ,  G ) `  N
) ) : Prime --> NN0 
->  ( M `  (  seq  1 (  x.  ,  G ) `  N
) )  Fn  Prime )
37 eqfnfv 5638 . . . . 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 5541 . . . 4  |-  ( x  =  (  seq  1
(  x.  ,  G
) `  N )  ->  ( M `  x
)  =  ( M `
 (  seq  1
(  x.  ,  G
) `  N )
) )
4241eqeq2d 2307 . . 3  |-  ( x  =  (  seq  1
(  x.  ,  G
) `  N )  ->  ( F  =  ( M `  x )  <-> 
F  =  ( M `
 (  seq  1
(  x.  ,  G
) `  N )
) ) )
4342rspcev 2897 . 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 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   ifcif 3578   class class class wbr 4039    e. cmpt 4093    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874   RRcr 8752   0cc0 8753   1c1 8754    x. cmul 8758    <_ cle 8884   NNcn 9762   NN0cn0 9981    seq cseq 11062   ^cexp 11120   Primecprime 12774    pCnt cpc 12905
This theorem is referenced by:  1arith  12990
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-q 10333  df-rp 10371  df-fz 10799  df-fl 10941  df-mod 10990  df-seq 11063  df-exp 11121  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-dvds 12548  df-gcd 12702  df-prm 12775  df-pc 12906
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