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Theorem snmlfval 24970
Description: The function  F from snmlval 24971 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 6048 . . . . 5  |-  ( n  =  N  ->  (
1 ... n )  =  ( 1 ... N
) )
2 rabeq 2910 . . . . 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 5691 . . 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 20 . . 3  |-  ( n  =  N  ->  n  =  N )
64, 5oveq12d 6058 . 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 6065 . 2  |-  ( (
# `  { k  e.  ( 1 ... N
)  |  ( |_
`  ( ( A  x.  ( R ^
k ) )  mod 
R ) )  =  B } )  /  N )  e.  _V
96, 7, 8fvmpt 5765 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 1649    e. wcel 1721   {crab 2670    e. cmpt 4226   ` cfv 5413  (class class class)co 6040   1c1 8947    x. cmul 8951    / cdiv 9633   NNcn 9956   ...cfz 10999   |_cfl 11156    mod cmo 11205   ^cexp 11337   #chash 11573
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-iota 5377  df-fun 5415  df-fv 5421  df-ov 6043
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