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Theorem vmappw 22582
Description: Value of the von Mangoldt function at a prime power. (Contributed by Mario Carneiro, 7-Apr-2016.)
Assertion
Ref Expression
vmappw  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  (Λ `  ( P ^ K
) )  =  ( log `  P ) )

Proof of Theorem vmappw
Dummy variable  p is distinct from all other variables.
StepHypRef Expression
1 prmnn 13879 . . . 4  |-  ( P  e.  Prime  ->  P  e.  NN )
2 nnnn0 10692 . . . 4  |-  ( K  e.  NN  ->  K  e.  NN0 )
3 nnexpcl 11990 . . . 4  |-  ( ( P  e.  NN  /\  K  e.  NN0 )  -> 
( P ^ K
)  e.  NN )
41, 2, 3syl2an 477 . . 3  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( P ^ K )  e.  NN )
5 eqid 2452 . . . 4  |-  { p  e.  Prime  |  p  ||  ( P ^ K ) }  =  { p  e.  Prime  |  p  ||  ( P ^ K ) }
65vmaval 22579 . . 3  |-  ( ( P ^ K )  e.  NN  ->  (Λ `  ( P ^ K
) )  =  if ( ( # `  {
p  e.  Prime  |  p 
||  ( P ^ K ) } )  =  1 ,  ( log `  U. {
p  e.  Prime  |  p 
||  ( P ^ K ) } ) ,  0 ) )
74, 6syl 16 . 2  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  (Λ `  ( P ^ K
) )  =  if ( ( # `  {
p  e.  Prime  |  p 
||  ( P ^ K ) } )  =  1 ,  ( log `  U. {
p  e.  Prime  |  p 
||  ( P ^ K ) } ) ,  0 ) )
8 df-rab 2805 . . . . . 6  |-  { p  e.  Prime  |  p  ||  ( P ^ K ) }  =  { p  |  ( p  e. 
Prime  /\  p  ||  ( P ^ K ) ) }
9 prmdvdsexpb 13914 . . . . . . . . . . . . 13  |-  ( ( p  e.  Prime  /\  P  e.  Prime  /\  K  e.  NN )  ->  ( p 
||  ( P ^ K )  <->  p  =  P ) )
109biimpd 207 . . . . . . . . . . . 12  |-  ( ( p  e.  Prime  /\  P  e.  Prime  /\  K  e.  NN )  ->  ( p 
||  ( P ^ K )  ->  p  =  P ) )
11103coml 1195 . . . . . . . . . . 11  |-  ( ( P  e.  Prime  /\  K  e.  NN  /\  p  e. 
Prime )  ->  ( p 
||  ( P ^ K )  ->  p  =  P ) )
12113expa 1188 . . . . . . . . . 10  |-  ( ( ( P  e.  Prime  /\  K  e.  NN )  /\  p  e.  Prime )  ->  ( p  ||  ( P ^ K )  ->  p  =  P ) )
1312expimpd 603 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  (
( p  e.  Prime  /\  p  ||  ( P ^ K ) )  ->  p  =  P ) )
14 simpl 457 . . . . . . . . . . 11  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  P  e.  Prime )
15 prmz 13880 . . . . . . . . . . . 12  |-  ( P  e.  Prime  ->  P  e.  ZZ )
16 iddvdsexp 13669 . . . . . . . . . . . 12  |-  ( ( P  e.  ZZ  /\  K  e.  NN )  ->  P  ||  ( P ^ K ) )
1715, 16sylan 471 . . . . . . . . . . 11  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  P  ||  ( P ^ K
) )
1814, 17jca 532 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( P  e.  Prime  /\  P  ||  ( P ^ K
) ) )
19 eleq1 2524 . . . . . . . . . . 11  |-  ( p  =  P  ->  (
p  e.  Prime  <->  P  e.  Prime ) )
20 breq1 4398 . . . . . . . . . . 11  |-  ( p  =  P  ->  (
p  ||  ( P ^ K )  <->  P  ||  ( P ^ K ) ) )
2119, 20anbi12d 710 . . . . . . . . . 10  |-  ( p  =  P  ->  (
( p  e.  Prime  /\  p  ||  ( P ^ K ) )  <-> 
( P  e.  Prime  /\  P  ||  ( P ^ K ) ) ) )
2218, 21syl5ibrcom 222 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  (
p  =  P  -> 
( p  e.  Prime  /\  p  ||  ( P ^ K ) ) ) )
2313, 22impbid 191 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  (
( p  e.  Prime  /\  p  ||  ( P ^ K ) )  <-> 
p  =  P ) )
24 elsn 3994 . . . . . . . 8  |-  ( p  e.  { P }  <->  p  =  P )
2523, 24syl6bbr 263 . . . . . . 7  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  (
( p  e.  Prime  /\  p  ||  ( P ^ K ) )  <-> 
p  e.  { P } ) )
2625abbi1dv 2590 . . . . . 6  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  { p  |  ( p  e. 
Prime  /\  p  ||  ( P ^ K ) ) }  =  { P } )
278, 26syl5eq 2505 . . . . 5  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  { p  e.  Prime  |  p  ||  ( P ^ K ) }  =  { P } )
2827fveq2d 5798 . . . 4  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( # `
 { p  e. 
Prime  |  p  ||  ( P ^ K ) } )  =  ( # `  { P } ) )
29 hashsng 12248 . . . . 5  |-  ( P  e.  Prime  ->  ( # `  { P } )  =  1 )
3029adantr 465 . . . 4  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( # `
 { P }
)  =  1 )
3128, 30eqtrd 2493 . . 3  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( # `
 { p  e. 
Prime  |  p  ||  ( P ^ K ) } )  =  1 )
32 iftrue 3900 . . 3  |-  ( (
# `  { p  e.  Prime  |  p  ||  ( P ^ K ) } )  =  1  ->  if ( (
# `  { p  e.  Prime  |  p  ||  ( P ^ K ) } )  =  1 ,  ( log `  U. { p  e.  Prime  |  p  ||  ( P ^ K ) } ) ,  0 )  =  ( log `  U. { p  e.  Prime  |  p  ||  ( P ^ K ) } ) )
3331, 32syl 16 . 2  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  if ( ( # `  {
p  e.  Prime  |  p 
||  ( P ^ K ) } )  =  1 ,  ( log `  U. {
p  e.  Prime  |  p 
||  ( P ^ K ) } ) ,  0 )  =  ( log `  U. { p  e.  Prime  |  p  ||  ( P ^ K ) } ) )
3427unieqd 4204 . . . 4  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  U. {
p  e.  Prime  |  p 
||  ( P ^ K ) }  =  U. { P } )
35 unisng 4210 . . . . 5  |-  ( P  e.  Prime  ->  U. { P }  =  P
)
3635adantr 465 . . . 4  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  U. { P }  =  P
)
3734, 36eqtrd 2493 . . 3  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  U. {
p  e.  Prime  |  p 
||  ( P ^ K ) }  =  P )
3837fveq2d 5798 . 2  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( log `  U. { p  e.  Prime  |  p  ||  ( P ^ K ) } )  =  ( log `  P ) )
397, 33, 383eqtrd 2497 1  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  (Λ `  ( P ^ K
) )  =  ( log `  P ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   {cab 2437   {crab 2800   ifcif 3894   {csn 3980   U.cuni 4194   class class class wbr 4395   ` cfv 5521  (class class class)co 6195   0cc0 9388   1c1 9389   NNcn 10428   NN0cn0 10685   ZZcz 10752   ^cexp 11977   #chash 12215    || cdivides 13648   Primecprime 13876   logclog 22134  Λcvma 22557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-cnex 9444  ax-resscn 9445  ax-1cn 9446  ax-icn 9447  ax-addcl 9448  ax-addrcl 9449  ax-mulcl 9450  ax-mulrcl 9451  ax-mulcom 9452  ax-addass 9453  ax-mulass 9454  ax-distr 9455  ax-i2m1 9456  ax-1ne0 9457  ax-1rid 9458  ax-rnegex 9459  ax-rrecex 9460  ax-cnre 9461  ax-pre-lttri 9462  ax-pre-lttrn 9463  ax-pre-ltadd 9464  ax-pre-mulgt0 9465  ax-pre-sup 9466
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-reu 2803  df-rmo 2804  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-int 4232  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-om 6582  df-1st 6682  df-2nd 6683  df-recs 6937  df-rdg 6971  df-1o 7025  df-2o 7026  df-oadd 7029  df-er 7206  df-en 7416  df-dom 7417  df-sdom 7418  df-fin 7419  df-sup 7797  df-card 8215  df-pnf 9526  df-mnf 9527  df-xr 9528  df-ltxr 9529  df-le 9530  df-sub 9703  df-neg 9704  df-div 10100  df-nn 10429  df-2 10486  df-3 10487  df-n0 10686  df-z 10753  df-uz 10968  df-rp 11098  df-fz 11550  df-fl 11754  df-mod 11821  df-seq 11919  df-exp 11978  df-hash 12216  df-cj 12701  df-re 12702  df-im 12703  df-sqr 12837  df-abs 12838  df-dvds 13649  df-gcd 13804  df-prm 13877  df-vma 22563
This theorem is referenced by:  vmaprm  22583  vmacl  22584  efvmacl  22586  vmalelog  22672  vmasum  22683  chpval2  22685  rplogsumlem2  22862  rpvmasumlem  22864
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