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Theorem vmaval 23253
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 10554 . . . . . 6  |-  NN  e.  _V
2 prmnn 14096 . . . . . . 7  |-  ( p  e.  Prime  ->  p  e.  NN )
32ssriv 3513 . . . . . 6  |-  Prime  C_  NN
41, 3ssexi 4598 . . . . 5  |-  Prime  e.  _V
54rabex 4604 . . . 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 4457 . . . . . . . . 9  |-  ( x  =  A  ->  (
p  ||  x  <->  p  ||  A
) )
98rabbidv 3110 . . . . . . . 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 2526 . . . . . . 7  |-  ( x  =  A  ->  { p  e.  Prime  |  p  ||  x }  =  S
)
127, 11sylan9eqr 2530 . . . . . 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 2469 . . . 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 3968 . . 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 3467 . 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 23237 . 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 9602 . . 3  |-  0  e.  _V
2220, 21ifex 4014 . 2  |-  if ( ( # `  S
)  =  1 ,  ( log `  U. S ) ,  0 )  e.  _V
2318, 19, 22fvmpt 5957 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 1379    e. wcel 1767   {crab 2821   _Vcvv 3118   [_csb 3440   ifcif 3945   U.cuni 4251   class class class wbr 4453   ` cfv 5594   0cc0 9504   1c1 9505   NNcn 10548   #chash 12385    || cdivides 13864   Primecprime 14093   logclog 22808  Λcvma 23231
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-i2m1 9572  ax-1ne0 9573  ax-rrecex 9576  ax-cnre 9577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  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 6298  df-om 6696  df-recs 7054  df-rdg 7088  df-nn 10549  df-prm 14094  df-vma 23237
This theorem is referenced by:  isppw  23254  vmappw  23256
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