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Theorem vmaval 22451
Description: Value of the von Mangoldt function. (Contributed by Mario Carneiro, 7-Apr-2016.)
Hypothesis
Ref Expression
vmaval.1  |-  S  =  { p  e.  Prime  |  p  ||  A }
Assertion
Ref Expression
vmaval  |-  ( A  e.  NN  ->  (Λ `  A )  =  if ( ( # `  S
)  =  1 ,  ( log `  U. S ) ,  0 ) )
Distinct variable group:    A, p
Allowed substitution hint:    S( p)

Proof of Theorem vmaval
Dummy variables  s  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nnex 10328 . . . . . 6  |-  NN  e.  _V
2 prmnn 13766 . . . . . . 7  |-  ( p  e.  Prime  ->  p  e.  NN )
32ssriv 3360 . . . . . 6  |-  Prime  C_  NN
41, 3ssexi 4437 . . . . 5  |-  Prime  e.  _V
54rabex 4443 . . . 4  |-  { p  e.  Prime  |  p  ||  x }  e.  _V
65a1i 11 . . 3  |-  ( x  =  A  ->  { p  e.  Prime  |  p  ||  x }  e.  _V )
7 id 22 . . . . . . 7  |-  ( s  =  { p  e. 
Prime  |  p  ||  x }  ->  s  =  {
p  e.  Prime  |  p 
||  x } )
8 breq2 4296 . . . . . . . . 9  |-  ( x  =  A  ->  (
p  ||  x  <->  p  ||  A
) )
98rabbidv 2964 . . . . . . . 8  |-  ( x  =  A  ->  { p  e.  Prime  |  p  ||  x }  =  {
p  e.  Prime  |  p 
||  A } )
10 vmaval.1 . . . . . . . 8  |-  S  =  { p  e.  Prime  |  p  ||  A }
119, 10syl6eqr 2493 . . . . . . 7  |-  ( x  =  A  ->  { p  e.  Prime  |  p  ||  x }  =  S
)
127, 11sylan9eqr 2497 . . . . . 6  |-  ( ( x  =  A  /\  s  =  { p  e.  Prime  |  p  ||  x } )  ->  s  =  S )
1312fveq2d 5695 . . . . 5  |-  ( ( x  =  A  /\  s  =  { p  e.  Prime  |  p  ||  x } )  ->  ( # `
 s )  =  ( # `  S
) )
1413eqeq1d 2451 . . . 4  |-  ( ( x  =  A  /\  s  =  { p  e.  Prime  |  p  ||  x } )  ->  (
( # `  s )  =  1  <->  ( # `  S
)  =  1 ) )
1512unieqd 4101 . . . . 5  |-  ( ( x  =  A  /\  s  =  { p  e.  Prime  |  p  ||  x } )  ->  U. s  =  U. S )
1615fveq2d 5695 . . . 4  |-  ( ( x  =  A  /\  s  =  { p  e.  Prime  |  p  ||  x } )  ->  ( log `  U. s )  =  ( log `  U. S ) )
1714, 16ifbieq1d 3812 . . 3  |-  ( ( x  =  A  /\  s  =  { p  e.  Prime  |  p  ||  x } )  ->  if ( ( # `  s
)  =  1 ,  ( log `  U. s ) ,  0 )  =  if ( ( # `  S
)  =  1 ,  ( log `  U. S ) ,  0 ) )
186, 17csbied 3314 . 2  |-  ( x  =  A  ->  [_ {
p  e.  Prime  |  p 
||  x }  / 
s ]_ if ( (
# `  s )  =  1 ,  ( log `  U. s
) ,  0 )  =  if ( (
# `  S )  =  1 ,  ( log `  U. S
) ,  0 ) )
19 df-vma 22435 . 2  |- Λ  =  ( x  e.  NN  |->  [_ { p  e.  Prime  |  p  ||  x }  /  s ]_ if ( ( # `  s
)  =  1 ,  ( log `  U. s ) ,  0 ) )
20 fvex 5701 . . 3  |-  ( log `  U. S )  e. 
_V
21 c0ex 9380 . . 3  |-  0  e.  _V
2220, 21ifex 3858 . 2  |-  if ( ( # `  S
)  =  1 ,  ( log `  U. S ) ,  0 )  e.  _V
2318, 19, 22fvmpt 5774 1  |-  ( A  e.  NN  ->  (Λ `  A )  =  if ( ( # `  S
)  =  1 ,  ( log `  U. S ) ,  0 ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   {crab 2719   _Vcvv 2972   [_csb 3288   ifcif 3791   U.cuni 4091   class class class wbr 4292   ` cfv 5418   0cc0 9282   1c1 9283   NNcn 10322   #chash 12103    || cdivides 13535   Primecprime 13763   logclog 22006  Λcvma 22429
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-i2m1 9350  ax-1ne0 9351  ax-rrecex 9354  ax-cnre 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-ov 6094  df-om 6477  df-recs 6832  df-rdg 6866  df-nn 10323  df-prm 13764  df-vma 22435
This theorem is referenced by:  isppw  22452  vmappw  22454
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