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Theorem snmlflim 28974
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 occurrences 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 28973 . . 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 1013 . 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 2472 . . . . . . . . 9  |-  ( b  =  B  ->  (
( |_ `  (
( A  x.  ( R ^ k ) )  mod  R ) )  =  b  <->  ( |_ `  ( ( A  x.  ( R ^ k ) )  mod  R ) )  =  B ) )
54rabbidv 3101 . . . . . . . 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 5876 . . . . . . 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 6311 . . . . . 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 4544 . . . . 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 2516 . . . 4  |-  ( b  =  B  ->  (
n  e.  NN  |->  ( ( # `  {
k  e.  ( 1 ... n )  |  ( |_ `  (
( A  x.  ( R ^ k ) )  mod  R ) )  =  b } )  /  n ) )  =  F )
1110breq1d 4466 . . 3  |-  ( b  =  B  ->  (
( n  e.  NN  |->  ( ( # `  {
k  e.  ( 1 ... n )  |  ( |_ `  (
( A  x.  ( R ^ k ) )  mod  R ) )  =  b } )  /  n ) )  ~~>  ( 1  /  R
)  <->  F  ~~>  ( 1  /  R ) ) )
1211rspccva 3209 . 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 471 1  |-  ( ( A  e.  ( S `
 R )  /\  B  e.  ( 0 ... ( R  - 
1 ) ) )  ->  F  ~~>  ( 1  /  R ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819   A.wral 2807   {crab 2811   class class class wbr 4456    |-> cmpt 4515   ` cfv 5594  (class class class)co 6296   RRcr 9508   0cc0 9509   1c1 9510    x. cmul 9514    - cmin 9824    / cdiv 10227   NNcn 10556   2c2 10606   ZZ>=cuz 11106   ...cfz 11697   |_cfl 11930    mod cmo 11999   ^cexp 12169   #chash 12408    ~~> cli 13319
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-cnex 9565  ax-resscn 9566
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fv 5602  df-ov 6299
This theorem is referenced by: (None)
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