Users' Mathboxes Mathbox for Mario Carneiro < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  snmlfval Unicode version

Theorem snmlfval 23928
Description: The function  F from snmlval 23929 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 5882 . . . . 5  |-  ( n  =  N  ->  (
1 ... n )  =  ( 1 ... N
) )
2 rabeq 2795 . . . . 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 5545 . . 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 5892 . 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 5899 . 2  |-  ( (
# `  { k  e.  ( 1 ... N
)  |  ( |_
`  ( ( A  x.  ( R ^
k ) )  mod 
R ) )  =  B } )  /  N )  e.  _V
96, 7, 8fvmpt 5618 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 1632    e. wcel 1696   {crab 2560    e. cmpt 4093   ` cfv 5271  (class class class)co 5874   1c1 8754    x. cmul 8758    / cdiv 9439   NNcn 9762   ...cfz 10798   |_cfl 10940    mod cmo 10989   ^cexp 11120   #chash 11353
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877
  Copyright terms: Public domain W3C validator