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

Theorem vmappw 20852
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 13037 . . . 4  |-  ( P  e.  Prime  ->  P  e.  NN )
2 nnnn0 10184 . . . 4  |-  ( K  e.  NN  ->  K  e.  NN0 )
3 nnexpcl 11349 . . . 4  |-  ( ( P  e.  NN  /\  K  e.  NN0 )  -> 
( P ^ K
)  e.  NN )
41, 2, 3syl2an 464 . . 3  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( P ^ K )  e.  NN )
5 eqid 2404 . . . 4  |-  { p  e.  Prime  |  p  ||  ( P ^ K ) }  =  { p  e.  Prime  |  p  ||  ( P ^ K ) }
65vmaval 20849 . . 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 2675 . . . . . 6  |-  { p  e.  Prime  |  p  ||  ( P ^ K ) }  =  { p  |  ( p  e. 
Prime  /\  p  ||  ( P ^ K ) ) }
9 prmdvdsexpb 13070 . . . . . . . . . . . . 13  |-  ( ( p  e.  Prime  /\  P  e.  Prime  /\  K  e.  NN )  ->  ( p 
||  ( P ^ K )  <->  p  =  P ) )
109biimpd 199 . . . . . . . . . . . 12  |-  ( ( p  e.  Prime  /\  P  e.  Prime  /\  K  e.  NN )  ->  ( p 
||  ( P ^ K )  ->  p  =  P ) )
11103coml 1160 . . . . . . . . . . 11  |-  ( ( P  e.  Prime  /\  K  e.  NN  /\  p  e. 
Prime )  ->  ( p 
||  ( P ^ K )  ->  p  =  P ) )
12113expa 1153 . . . . . . . . . 10  |-  ( ( ( P  e.  Prime  /\  K  e.  NN )  /\  p  e.  Prime )  ->  ( p  ||  ( P ^ K )  ->  p  =  P ) )
1312expimpd 587 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  (
( p  e.  Prime  /\  p  ||  ( P ^ K ) )  ->  p  =  P ) )
14 simpl 444 . . . . . . . . . . 11  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  P  e.  Prime )
15 prmz 13038 . . . . . . . . . . . 12  |-  ( P  e.  Prime  ->  P  e.  ZZ )
16 iddvdsexp 12828 . . . . . . . . . . . 12  |-  ( ( P  e.  ZZ  /\  K  e.  NN )  ->  P  ||  ( P ^ K ) )
1715, 16sylan 458 . . . . . . . . . . 11  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  P  ||  ( P ^ K
) )
1814, 17jca 519 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( P  e.  Prime  /\  P  ||  ( P ^ K
) ) )
19 eleq1 2464 . . . . . . . . . . 11  |-  ( p  =  P  ->  (
p  e.  Prime  <->  P  e.  Prime ) )
20 breq1 4175 . . . . . . . . . . 11  |-  ( p  =  P  ->  (
p  ||  ( P ^ K )  <->  P  ||  ( P ^ K ) ) )
2119, 20anbi12d 692 . . . . . . . . . 10  |-  ( p  =  P  ->  (
( p  e.  Prime  /\  p  ||  ( P ^ K ) )  <-> 
( P  e.  Prime  /\  P  ||  ( P ^ K ) ) ) )
2218, 21syl5ibrcom 214 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  (
p  =  P  -> 
( p  e.  Prime  /\  p  ||  ( P ^ K ) ) ) )
2313, 22impbid 184 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  (
( p  e.  Prime  /\  p  ||  ( P ^ K ) )  <-> 
p  =  P ) )
24 elsn 3789 . . . . . . . 8  |-  ( p  e.  { P }  <->  p  =  P )
2523, 24syl6bbr 255 . . . . . . 7  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  (
( p  e.  Prime  /\  p  ||  ( P ^ K ) )  <-> 
p  e.  { P } ) )
2625abbi1dv 2520 . . . . . 6  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  { p  |  ( p  e. 
Prime  /\  p  ||  ( P ^ K ) ) }  =  { P } )
278, 26syl5eq 2448 . . . . 5  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  { p  e.  Prime  |  p  ||  ( P ^ K ) }  =  { P } )
2827fveq2d 5691 . . . 4  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( # `
 { p  e. 
Prime  |  p  ||  ( P ^ K ) } )  =  ( # `  { P } ) )
29 hashsng 11602 . . . . 5  |-  ( P  e.  Prime  ->  ( # `  { P } )  =  1 )
3029adantr 452 . . . 4  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( # `
 { P }
)  =  1 )
3128, 30eqtrd 2436 . . 3  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( # `
 { p  e. 
Prime  |  p  ||  ( P ^ K ) } )  =  1 )
32 iftrue 3705 . . 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 3986 . . . 4  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  U. {
p  e.  Prime  |  p 
||  ( P ^ K ) }  =  U. { P } )
35 unisng 3992 . . . . 5  |-  ( P  e.  Prime  ->  U. { P }  =  P
)
3635adantr 452 . . . 4  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  U. { P }  =  P
)
3734, 36eqtrd 2436 . . 3  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  U. {
p  e.  Prime  |  p 
||  ( P ^ K ) }  =  P )
3837fveq2d 5691 . 2  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  ( log `  U. { p  e.  Prime  |  p  ||  ( P ^ K ) } )  =  ( log `  P ) )
397, 33, 383eqtrd 2440 1  |-  ( ( P  e.  Prime  /\  K  e.  NN )  ->  (Λ `  ( P ^ K
) )  =  ( log `  P ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   {cab 2390   {crab 2670   ifcif 3699   {csn 3774   U.cuni 3975   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   0cc0 8946   1c1 8947   NNcn 9956   NN0cn0 10177   ZZcz 10238   ^cexp 11337   #chash 11573    || cdivides 12807   Primecprime 13034   logclog 20405  Λcvma 20827
This theorem is referenced by:  vmaprm  20853  vmacl  20854  efvmacl  20856  vmalelog  20942  vmasum  20953  chpval2  20955  rplogsumlem2  21132  rpvmasumlem  21134
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-fz 11000  df-fl 11157  df-mod 11206  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-dvds 12808  df-gcd 12962  df-prm 13035  df-vma 20833
  Copyright terms: Public domain W3C validator