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

Theorem snmlflim 23915
Description: If  A is simply normal, then the function  F of relative density of  B in the digit string converges to  1  /  R, i.e. the set of occurences of  B in the digit string has natural density  1  /  R. (Contributed by Mario Carneiro, 6-Apr-2015.)
Hypotheses
Ref Expression
snml.s  |-  S  =  ( r  e.  (
ZZ>= `  2 )  |->  { x  e.  RR  |  A. b  e.  (
0 ... ( r  - 
1 ) ) ( n  e.  NN  |->  ( ( # `  {
k  e.  ( 1 ... n )  |  ( |_ `  (
( x  x.  (
r ^ k ) )  mod  r ) )  =  b } )  /  n ) )  ~~>  ( 1  / 
r ) } )
snml.f  |-  F  =  ( n  e.  NN  |->  ( ( # `  {
k  e.  ( 1 ... n )  |  ( |_ `  (
( A  x.  ( R ^ k ) )  mod  R ) )  =  B } )  /  n ) )
Assertion
Ref Expression
snmlflim  |-  ( ( A  e.  ( S `
 R )  /\  B  e.  ( 0 ... ( R  - 
1 ) ) )  ->  F  ~~>  ( 1  /  R ) )
Distinct variable groups:    k, b, n, x, A    B, b,
k, n    F, b    r, b, R, k, n, x
Allowed substitution hints:    A( r)    B( x, r)    S( x, k, n, r, b)    F( x, k, n, r)

Proof of Theorem snmlflim
StepHypRef Expression
1 snml.s . . . 4  |-  S  =  ( r  e.  (
ZZ>= `  2 )  |->  { x  e.  RR  |  A. b  e.  (
0 ... ( r  - 
1 ) ) ( n  e.  NN  |->  ( ( # `  {
k  e.  ( 1 ... n )  |  ( |_ `  (
( x  x.  (
r ^ k ) )  mod  r ) )  =  b } )  /  n ) )  ~~>  ( 1  / 
r ) } )
21snmlval 23914 . . 3  |-  ( A  e.  ( S `  R )  <->  ( R  e.  ( ZZ>= `  2 )  /\  A  e.  RR  /\ 
A. b  e.  ( 0 ... ( R  -  1 ) ) ( n  e.  NN  |->  ( ( # `  {
k  e.  ( 1 ... n )  |  ( |_ `  (
( A  x.  ( R ^ k ) )  mod  R ) )  =  b } )  /  n ) )  ~~>  ( 1  /  R
) ) )
32simp3bi 972 . 2  |-  ( A  e.  ( S `  R )  ->  A. b  e.  ( 0 ... ( R  -  1 ) ) ( n  e.  NN  |->  ( ( # `  { k  e.  ( 1 ... n )  |  ( |_ `  ( ( A  x.  ( R ^ k ) )  mod  R ) )  =  b } )  /  n ) )  ~~>  ( 1  /  R ) )
4 eqeq2 2292 . . . . . . . . 9  |-  ( b  =  B  ->  (
( |_ `  (
( A  x.  ( R ^ k ) )  mod  R ) )  =  b  <->  ( |_ `  ( ( A  x.  ( R ^ k ) )  mod  R ) )  =  B ) )
54rabbidv 2780 . . . . . . . 8  |-  ( b  =  B  ->  { k  e.  ( 1 ... n )  |  ( |_ `  ( ( A  x.  ( R ^ k ) )  mod  R ) )  =  b }  =  { k  e.  ( 1 ... n )  |  ( |_ `  ( ( A  x.  ( R ^ k ) )  mod  R ) )  =  B }
)
65fveq2d 5529 . . . . . . 7  |-  ( b  =  B  ->  ( # `
 { k  e.  ( 1 ... n
)  |  ( |_
`  ( ( A  x.  ( R ^
k ) )  mod 
R ) )  =  b } )  =  ( # `  {
k  e.  ( 1 ... n )  |  ( |_ `  (
( A  x.  ( R ^ k ) )  mod  R ) )  =  B } ) )
76oveq1d 5873 . . . . . 6  |-  ( b  =  B  ->  (
( # `  { k  e.  ( 1 ... n )  |  ( |_ `  ( ( A  x.  ( R ^ k ) )  mod  R ) )  =  b } )  /  n )  =  ( ( # `  {
k  e.  ( 1 ... n )  |  ( |_ `  (
( A  x.  ( R ^ k ) )  mod  R ) )  =  B } )  /  n ) )
87mpteq2dv 4107 . . . . 5  |-  ( b  =  B  ->  (
n  e.  NN  |->  ( ( # `  {
k  e.  ( 1 ... n )  |  ( |_ `  (
( A  x.  ( R ^ k ) )  mod  R ) )  =  b } )  /  n ) )  =  ( n  e.  NN  |->  ( ( # `  { k  e.  ( 1 ... n )  |  ( |_ `  ( ( A  x.  ( R ^ k ) )  mod  R ) )  =  B }
)  /  n ) ) )
9 snml.f . . . . 5  |-  F  =  ( n  e.  NN  |->  ( ( # `  {
k  e.  ( 1 ... n )  |  ( |_ `  (
( A  x.  ( R ^ k ) )  mod  R ) )  =  B } )  /  n ) )
108, 9syl6eqr 2333 . . . 4  |-  ( b  =  B  ->  (
n  e.  NN  |->  ( ( # `  {
k  e.  ( 1 ... n )  |  ( |_ `  (
( A  x.  ( R ^ k ) )  mod  R ) )  =  b } )  /  n ) )  =  F )
1110breq1d 4033 . . 3  |-  ( b  =  B  ->  (
( n  e.  NN  |->  ( ( # `  {
k  e.  ( 1 ... n )  |  ( |_ `  (
( A  x.  ( R ^ k ) )  mod  R ) )  =  b } )  /  n ) )  ~~>  ( 1  /  R
)  <->  F  ~~>  ( 1  /  R ) ) )
1211rspccva 2883 . 2  |-  ( ( A. b  e.  ( 0 ... ( R  -  1 ) ) ( n  e.  NN  |->  ( ( # `  {
k  e.  ( 1 ... n )  |  ( |_ `  (
( A  x.  ( R ^ k ) )  mod  R ) )  =  b } )  /  n ) )  ~~>  ( 1  /  R
)  /\  B  e.  ( 0 ... ( R  -  1 ) ) )  ->  F  ~~>  ( 1  /  R
) )
133, 12sylan 457 1  |-  ( ( A  e.  ( S `
 R )  /\  B  e.  ( 0 ... ( R  - 
1 ) ) )  ->  F  ~~>  ( 1  /  R ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   {crab 2547   class class class wbr 4023    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   RRcr 8736   0cc0 8737   1c1 8738    x. cmul 8742    - cmin 9037    / cdiv 9423   NNcn 9746   2c2 9795   ZZ>=cuz 10230   ...cfz 10782   |_cfl 10924    mod cmo 10973   ^cexp 11104   #chash 11337    ~~> cli 11958
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-13 1686  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-pow 4188  ax-pr 4214  ax-cnex 8793  ax-resscn 8794
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-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861
  Copyright terms: Public domain W3C validator