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Theorem snmlfval 23913
Description: The function  F from snmlval 23914 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 5866 . . . . 5  |-  ( n  =  N  ->  (
1 ... n )  =  ( 1 ... N
) )
2 rabeq 2782 . . . . 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 15 . . . 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 5529 . . 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 19 . . 3  |-  ( n  =  N  ->  n  =  N )
64, 5oveq12d 5876 . 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 5883 . 2  |-  ( (
# `  { k  e.  ( 1 ... N
)  |  ( |_
`  ( ( A  x.  ( R ^
k ) )  mod 
R ) )  =  B } )  /  N )  e.  _V
96, 7, 8fvmpt 5602 1  |-  ( N  e.  NN  ->  ( F `  N )  =  ( ( # `  { k  e.  ( 1 ... N )  |  ( |_ `  ( ( A  x.  ( R ^ k ) )  mod  R ) )  =  B }
)  /  N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   {crab 2547    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   1c1 8738    x. cmul 8742    / cdiv 9423   NNcn 9746   ...cfz 10782   |_cfl 10924    mod cmo 10973   ^cexp 11104   #chash 11337
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861
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