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Theorem snmlfval 27149
Description: The function  F from snmlval 27150 maps  N to the relative density of  B in the first  N digits of the digit string of  A in base  R. (Contributed by Mario Carneiro, 6-Apr-2015.)
Hypothesis
Ref Expression
snmlff.f  |-  F  =  ( n  e.  NN  |->  ( ( # `  {
k  e.  ( 1 ... n )  |  ( |_ `  (
( A  x.  ( R ^ k ) )  mod  R ) )  =  B } )  /  n ) )
Assertion
Ref Expression
snmlfval  |-  ( N  e.  NN  ->  ( F `  N )  =  ( ( # `  { k  e.  ( 1 ... N )  |  ( |_ `  ( ( A  x.  ( R ^ k ) )  mod  R ) )  =  B }
)  /  N ) )
Distinct variable groups:    A, n    B, n    k, n, N    R, n
Allowed substitution hints:    A( k)    B( k)    R( k)    F( k, n)

Proof of Theorem snmlfval
StepHypRef Expression
1 oveq2 6098 . . . . 5  |-  ( n  =  N  ->  (
1 ... n )  =  ( 1 ... N
) )
2 rabeq 2964 . . . . 5  |-  ( ( 1 ... n )  =  ( 1 ... N )  ->  { k  e.  ( 1 ... n )  |  ( |_ `  ( ( A  x.  ( R ^ k ) )  mod  R ) )  =  B }  =  { k  e.  ( 1 ... N )  |  ( |_ `  ( ( A  x.  ( R ^ k ) )  mod  R ) )  =  B }
)
31, 2syl 16 . . . 4  |-  ( n  =  N  ->  { k  e.  ( 1 ... n )  |  ( |_ `  ( ( A  x.  ( R ^ k ) )  mod  R ) )  =  B }  =  { k  e.  ( 1 ... N )  |  ( |_ `  ( ( A  x.  ( R ^ k ) )  mod  R ) )  =  B }
)
43fveq2d 5692 . . 3  |-  ( n  =  N  ->  ( # `
 { k  e.  ( 1 ... n
)  |  ( |_
`  ( ( A  x.  ( R ^
k ) )  mod 
R ) )  =  B } )  =  ( # `  {
k  e.  ( 1 ... N )  |  ( |_ `  (
( A  x.  ( R ^ k ) )  mod  R ) )  =  B } ) )
5 id 22 . . 3  |-  ( n  =  N  ->  n  =  N )
64, 5oveq12d 6108 . 2  |-  ( n  =  N  ->  (
( # `  { k  e.  ( 1 ... n )  |  ( |_ `  ( ( A  x.  ( R ^ k ) )  mod  R ) )  =  B } )  /  n )  =  ( ( # `  {
k  e.  ( 1 ... N )  |  ( |_ `  (
( A  x.  ( R ^ k ) )  mod  R ) )  =  B } )  /  N ) )
7 snmlff.f . 2  |-  F  =  ( n  e.  NN  |->  ( ( # `  {
k  e.  ( 1 ... n )  |  ( |_ `  (
( A  x.  ( R ^ k ) )  mod  R ) )  =  B } )  /  n ) )
8 ovex 6115 . 2  |-  ( (
# `  { k  e.  ( 1 ... N
)  |  ( |_
`  ( ( A  x.  ( R ^
k ) )  mod 
R ) )  =  B } )  /  N )  e.  _V
96, 7, 8fvmpt 5771 1  |-  ( N  e.  NN  ->  ( F `  N )  =  ( ( # `  { k  e.  ( 1 ... N )  |  ( |_ `  ( ( A  x.  ( R ^ k ) )  mod  R ) )  =  B }
)  /  N ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1364    e. wcel 1761   {crab 2717    e. cmpt 4347   ` cfv 5415  (class class class)co 6090   1c1 9279    x. cmul 9283    / cdiv 9989   NNcn 10318   ...cfz 11433   |_cfl 11636    mod cmo 11704   ^cexp 11861   #chash 12099
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pr 4528
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3184  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-iota 5378  df-fun 5417  df-fv 5423  df-ov 6093
This theorem is referenced by: (None)
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