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Theorem vmaval 23513
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 10562 . . . . . 6  |-  NN  e.  _V
2 prmnn 14232 . . . . . . 7  |-  ( p  e.  Prime  ->  p  e.  NN )
32ssriv 3503 . . . . . 6  |-  Prime  C_  NN
41, 3ssexi 4601 . . . . 5  |-  Prime  e.  _V
54rabex 4607 . . . 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 4460 . . . . . . . . 9  |-  ( x  =  A  ->  (
p  ||  x  <->  p  ||  A
) )
98rabbidv 3101 . . . . . . . 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 2516 . . . . . . 7  |-  ( x  =  A  ->  { p  e.  Prime  |  p  ||  x }  =  S
)
127, 11sylan9eqr 2520 . . . . . 6  |-  ( ( x  =  A  /\  s  =  { p  e.  Prime  |  p  ||  x } )  ->  s  =  S )
1312fveq2d 5876 . . . . 5  |-  ( ( x  =  A  /\  s  =  { p  e.  Prime  |  p  ||  x } )  ->  ( # `
 s )  =  ( # `  S
) )
1413eqeq1d 2459 . . . 4  |-  ( ( x  =  A  /\  s  =  { p  e.  Prime  |  p  ||  x } )  ->  (
( # `  s )  =  1  <->  ( # `  S
)  =  1 ) )
1512unieqd 4261 . . . . 5  |-  ( ( x  =  A  /\  s  =  { p  e.  Prime  |  p  ||  x } )  ->  U. s  =  U. S )
1615fveq2d 5876 . . . 4  |-  ( ( x  =  A  /\  s  =  { p  e.  Prime  |  p  ||  x } )  ->  ( log `  U. s )  =  ( log `  U. S ) )
1714, 16ifbieq1d 3967 . . 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 3457 . 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 23497 . 2  |- Λ  =  ( x  e.  NN  |->  [_ { p  e.  Prime  |  p  ||  x }  /  s ]_ if ( ( # `  s
)  =  1 ,  ( log `  U. s ) ,  0 ) )
20 fvex 5882 . . 3  |-  ( log `  U. S )  e. 
_V
21 c0ex 9607 . . 3  |-  0  e.  _V
2220, 21ifex 4013 . 2  |-  if ( ( # `  S
)  =  1 ,  ( log `  U. S ) ,  0 )  e.  _V
2318, 19, 22fvmpt 5956 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 1395    e. wcel 1819   {crab 2811   _Vcvv 3109   [_csb 3430   ifcif 3944   U.cuni 4251   class class class wbr 4456   ` cfv 5594   0cc0 9509   1c1 9510   NNcn 10556   #chash 12408    || cdvds 13998   Primecprime 14229   logclog 23068  Λcvma 23491
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-i2m1 9577  ax-1ne0 9578  ax-rrecex 9581  ax-cnre 9582
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-om 6700  df-recs 7060  df-rdg 7094  df-nn 10557  df-prm 14230  df-vma 23497
This theorem is referenced by:  isppw  23514  vmappw  23516
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