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Theorem vmappw 22397
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 13762 . . . 4  |-  ( P  e.  Prime  ->  P  e.  NN )
2 nnnn0 10582 . . . 4  |-  ( K  e.  NN  ->  K  e.  NN0 )
3 nnexpcl 11874 . . . 4  |-  ( ( P  e.  NN  /\  K  e.  NN0 )  -> 
( P ^ K
)  e.  NN )
41, 2, 3syl2an 474 . . 3  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( P ^ K )  e.  NN )
5 eqid 2441 . . . 4  |-  { p  e.  Prime  |  p  ||  ( P ^ K ) }  =  { p  e.  Prime  |  p  ||  ( P ^ K ) }
65vmaval 22394 . . 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 2722 . . . . . 6  |-  { p  e.  Prime  |  p  ||  ( P ^ K ) }  =  { p  |  ( p  e. 
Prime  /\  p  ||  ( P ^ K ) ) }
9 prmdvdsexpb 13797 . . . . . . . . . . . . 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 1189 . . . . . . . . . . 11  |-  ( ( P  e.  Prime  /\  K  e.  NN  /\  p  e. 
Prime )  ->  ( p 
||  ( P ^ K )  ->  p  =  P ) )
12113expa 1182 . . . . . . . . . 10  |-  ( ( ( P  e.  Prime  /\  K  e.  NN )  /\  p  e.  Prime )  ->  ( p  ||  ( P ^ K )  ->  p  =  P ) )
1312expimpd 600 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  (
( p  e.  Prime  /\  p  ||  ( P ^ K ) )  ->  p  =  P ) )
14 simpl 454 . . . . . . . . . . 11  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  P  e.  Prime )
15 prmz 13763 . . . . . . . . . . . 12  |-  ( P  e.  Prime  ->  P  e.  ZZ )
16 iddvdsexp 13552 . . . . . . . . . . . 12  |-  ( ( P  e.  ZZ  /\  K  e.  NN )  ->  P  ||  ( P ^ K ) )
1715, 16sylan 468 . . . . . . . . . . 11  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  P  ||  ( P ^ K
) )
1814, 17jca 529 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( P  e.  Prime  /\  P  ||  ( P ^ K
) ) )
19 eleq1 2501 . . . . . . . . . . 11  |-  ( p  =  P  ->  (
p  e.  Prime  <->  P  e.  Prime ) )
20 breq1 4292 . . . . . . . . . . 11  |-  ( p  =  P  ->  (
p  ||  ( P ^ K )  <->  P  ||  ( P ^ K ) ) )
2119, 20anbi12d 705 . . . . . . . . . 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 3888 . . . . . . . 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 2557 . . . . . 6  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  { p  |  ( p  e. 
Prime  /\  p  ||  ( P ^ K ) ) }  =  { P } )
278, 26syl5eq 2485 . . . . 5  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  { p  e.  Prime  |  p  ||  ( P ^ K ) }  =  { P } )
2827fveq2d 5692 . . . 4  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( # `
 { p  e. 
Prime  |  p  ||  ( P ^ K ) } )  =  ( # `  { P } ) )
29 hashsng 12132 . . . . 5  |-  ( P  e.  Prime  ->  ( # `  { P } )  =  1 )
3029adantr 462 . . . 4  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( # `
 { P }
)  =  1 )
3128, 30eqtrd 2473 . . 3  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( # `
 { p  e. 
Prime  |  p  ||  ( P ^ K ) } )  =  1 )
32 iftrue 3794 . . 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 4098 . . . 4  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  U. {
p  e.  Prime  |  p 
||  ( P ^ K ) }  =  U. { P } )
35 unisng 4104 . . . . 5  |-  ( P  e.  Prime  ->  U. { P }  =  P
)
3635adantr 462 . . . 4  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  U. { P }  =  P
)
3734, 36eqtrd 2473 . . 3  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  U. {
p  e.  Prime  |  p 
||  ( P ^ K ) }  =  P )
3837fveq2d 5692 . 2  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( log `  U. { p  e.  Prime  |  p  ||  ( P ^ K ) } )  =  ( log `  P ) )
397, 33, 383eqtrd 2477 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 960    = wceq 1364    e. wcel 1761   {cab 2427   {crab 2717   ifcif 3788   {csn 3874   U.cuni 4088   class class class wbr 4289   ` cfv 5415  (class class class)co 6090   0cc0 9278   1c1 9279   NNcn 10318   NN0cn0 10575   ZZcz 10642   ^cexp 11861   #chash 12099    || cdivides 13531   Primecprime 13759   logclog 21949  Λcvma 22372
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-2o 6917  df-oadd 6920  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-sup 7687  df-card 8105  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-n0 10576  df-z 10643  df-uz 10858  df-rp 10988  df-fz 11434  df-fl 11638  df-mod 11705  df-seq 11803  df-exp 11862  df-hash 12100  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-dvds 13532  df-gcd 13687  df-prm 13760  df-vma 22378
This theorem is referenced by:  vmaprm  22398  vmacl  22399  efvmacl  22401  vmalelog  22487  vmasum  22498  chpval2  22500  rplogsumlem2  22677  rpvmasumlem  22679
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