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Theorem vmaval 24119
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 10637 . . . . . 6  |-  NN  e.  _V
2 prmnn 14704 . . . . . . 7  |-  ( p  e.  Prime  ->  p  e.  NN )
32ssriv 3422 . . . . . 6  |-  Prime  C_  NN
41, 3ssexi 4541 . . . . 5  |-  Prime  e.  _V
54rabex 4550 . . . 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 4399 . . . . . . . . 9  |-  ( x  =  A  ->  (
p  ||  x  <->  p  ||  A
) )
98rabbidv 3022 . . . . . . . 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 2523 . . . . . . 7  |-  ( x  =  A  ->  { p  e.  Prime  |  p  ||  x }  =  S
)
127, 11sylan9eqr 2527 . . . . . 6  |-  ( ( x  =  A  /\  s  =  { p  e.  Prime  |  p  ||  x } )  ->  s  =  S )
1312fveq2d 5883 . . . . 5  |-  ( ( x  =  A  /\  s  =  { p  e.  Prime  |  p  ||  x } )  ->  ( # `
 s )  =  ( # `  S
) )
1413eqeq1d 2473 . . . 4  |-  ( ( x  =  A  /\  s  =  { p  e.  Prime  |  p  ||  x } )  ->  (
( # `  s )  =  1  <->  ( # `  S
)  =  1 ) )
1512unieqd 4200 . . . . 5  |-  ( ( x  =  A  /\  s  =  { p  e.  Prime  |  p  ||  x } )  ->  U. s  =  U. S )
1615fveq2d 5883 . . . 4  |-  ( ( x  =  A  /\  s  =  { p  e.  Prime  |  p  ||  x } )  ->  ( log `  U. s )  =  ( log `  U. S ) )
1714, 16ifbieq1d 3895 . . 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 3376 . 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 24103 . 2  |- Λ  =  ( x  e.  NN  |->  [_ { p  e.  Prime  |  p  ||  x }  /  s ]_ if ( ( # `  s
)  =  1 ,  ( log `  U. s ) ,  0 ) )
20 fvex 5889 . . 3  |-  ( log `  U. S )  e. 
_V
21 c0ex 9655 . . 3  |-  0  e.  _V
2220, 21ifex 3940 . 2  |-  if ( ( # `  S
)  =  1 ,  ( log `  U. S ) ,  0 )  e.  _V
2318, 19, 22fvmpt 5963 1  |-  ( A  e.  NN  ->  (Λ `  A )  =  if ( ( # `  S
)  =  1 ,  ( log `  U. S ) ,  0 ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376    = wceq 1452    e. wcel 1904   {crab 2760   _Vcvv 3031   [_csb 3349   ifcif 3872   U.cuni 4190   class class class wbr 4395   ` cfv 5589   0cc0 9557   1c1 9558   NNcn 10631   #chash 12553    || cdvds 14382   Primecprime 14701   logclog 23583  Λcvma 24097
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-i2m1 9625  ax-1ne0 9626  ax-rrecex 9629  ax-cnre 9630
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-ov 6311  df-om 6712  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-nn 10632  df-prm 14702  df-vma 24103
This theorem is referenced by:  isppw  24120  vmappw  24122
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