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

Theorem vmappw 23771
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 14429 . . . 4  |-  ( P  e.  Prime  ->  P  e.  NN )
2 nnnn0 10843 . . . 4  |-  ( K  e.  NN  ->  K  e.  NN0 )
3 nnexpcl 12223 . . . 4  |-  ( ( P  e.  NN  /\  K  e.  NN0 )  -> 
( P ^ K
)  e.  NN )
41, 2, 3syl2an 475 . . 3  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( P ^ K )  e.  NN )
5 eqid 2402 . . . 4  |-  { p  e.  Prime  |  p  ||  ( P ^ K ) }  =  { p  e.  Prime  |  p  ||  ( P ^ K ) }
65vmaval 23768 . . 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 17 . 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 2763 . . . . . 6  |-  { p  e.  Prime  |  p  ||  ( P ^ K ) }  =  { p  |  ( p  e. 
Prime  /\  p  ||  ( P ^ K ) ) }
9 prmdvdsexpb 14465 . . . . . . . . . . . . 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 1204 . . . . . . . . . . 11  |-  ( ( P  e.  Prime  /\  K  e.  NN  /\  p  e. 
Prime )  ->  ( p 
||  ( P ^ K )  ->  p  =  P ) )
12113expa 1197 . . . . . . . . . 10  |-  ( ( ( P  e.  Prime  /\  K  e.  NN )  /\  p  e.  Prime )  ->  ( p  ||  ( P ^ K )  ->  p  =  P ) )
1312expimpd 601 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  (
( p  e.  Prime  /\  p  ||  ( P ^ K ) )  ->  p  =  P ) )
14 simpl 455 . . . . . . . . . . 11  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  P  e.  Prime )
15 prmz 14430 . . . . . . . . . . . 12  |-  ( P  e.  Prime  ->  P  e.  ZZ )
16 iddvdsexp 14216 . . . . . . . . . . . 12  |-  ( ( P  e.  ZZ  /\  K  e.  NN )  ->  P  ||  ( P ^ K ) )
1715, 16sylan 469 . . . . . . . . . . 11  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  P  ||  ( P ^ K
) )
1814, 17jca 530 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( P  e.  Prime  /\  P  ||  ( P ^ K
) ) )
19 eleq1 2474 . . . . . . . . . . 11  |-  ( p  =  P  ->  (
p  e.  Prime  <->  P  e.  Prime ) )
20 breq1 4398 . . . . . . . . . . 11  |-  ( p  =  P  ->  (
p  ||  ( P ^ K )  <->  P  ||  ( P ^ K ) ) )
2119, 20anbi12d 709 . . . . . . . . . 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 190 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  (
( p  e.  Prime  /\  p  ||  ( P ^ K ) )  <-> 
p  =  P ) )
24 elsn 3986 . . . . . . . 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 2540 . . . . . 6  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  { p  |  ( p  e. 
Prime  /\  p  ||  ( P ^ K ) ) }  =  { P } )
278, 26syl5eq 2455 . . . . 5  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  { p  e.  Prime  |  p  ||  ( P ^ K ) }  =  { P } )
2827fveq2d 5853 . . . 4  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( # `
 { p  e. 
Prime  |  p  ||  ( P ^ K ) } )  =  ( # `  { P } ) )
29 hashsng 12486 . . . . 5  |-  ( P  e.  Prime  ->  ( # `  { P } )  =  1 )
3029adantr 463 . . . 4  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( # `
 { P }
)  =  1 )
3128, 30eqtrd 2443 . . 3  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( # `
 { p  e. 
Prime  |  p  ||  ( P ^ K ) } )  =  1 )
3231iftrued 3893 . 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 ) } ) )
3327unieqd 4201 . . . 4  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  U. {
p  e.  Prime  |  p 
||  ( P ^ K ) }  =  U. { P } )
34 unisng 4207 . . . . 5  |-  ( P  e.  Prime  ->  U. { P }  =  P
)
3534adantr 463 . . . 4  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  U. { P }  =  P
)
3633, 35eqtrd 2443 . . 3  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  U. {
p  e.  Prime  |  p 
||  ( P ^ K ) }  =  P )
3736fveq2d 5853 . 2  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( log `  U. { p  e.  Prime  |  p  ||  ( P ^ K ) } )  =  ( log `  P ) )
387, 32, 373eqtrd 2447 1  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  (Λ `  ( P ^ K
) )  =  ( log `  P ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842   {cab 2387   {crab 2758   ifcif 3885   {csn 3972   U.cuni 4191   class class class wbr 4395   ` cfv 5569  (class class class)co 6278   0cc0 9522   1c1 9523   NNcn 10576   NN0cn0 10836   ZZcz 10905   ^cexp 12210   #chash 12452    || cdvds 14195   Primecprime 14426   logclog 23234  Λcvma 23746
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599  ax-pre-sup 9600
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-1st 6784  df-2nd 6785  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-2o 7168  df-oadd 7171  df-er 7348  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-sup 7935  df-card 8352  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-div 10248  df-nn 10577  df-2 10635  df-3 10636  df-n0 10837  df-z 10906  df-uz 11128  df-rp 11266  df-fz 11727  df-fl 11966  df-mod 12035  df-seq 12152  df-exp 12211  df-hash 12453  df-cj 13081  df-re 13082  df-im 13083  df-sqrt 13217  df-abs 13218  df-dvds 14196  df-gcd 14354  df-prm 14427  df-vma 23752
This theorem is referenced by:  vmaprm  23772  vmacl  23773  efvmacl  23775  vmalelog  23861  vmasum  23872  chpval2  23874  rplogsumlem2  24051  rpvmasumlem  24053
  Copyright terms: Public domain W3C validator