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Theorem snmlflim 23930
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 23929 . . 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 2305 . . . . . . . . 9  |-  ( b  =  B  ->  (
( |_ `  (
( A  x.  ( R ^ k ) )  mod  R ) )  =  b  <->  ( |_ `  ( ( A  x.  ( R ^ k ) )  mod  R ) )  =  B ) )
54rabbidv 2793 . . . . . . . 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 5545 . . . . . . 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 5889 . . . . . 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 4123 . . . . 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 2346 . . . 4  |-  ( b  =  B  ->  (
n  e.  NN  |->  ( ( # `  {
k  e.  ( 1 ... n )  |  ( |_ `  (
( A  x.  ( R ^ k ) )  mod  R ) )  =  b } )  /  n ) )  =  F )
1110breq1d 4049 . . 3  |-  ( b  =  B  ->  (
( n  e.  NN  |->  ( ( # `  {
k  e.  ( 1 ... n )  |  ( |_ `  (
( A  x.  ( R ^ k ) )  mod  R ) )  =  b } )  /  n ) )  ~~>  ( 1  /  R
)  <->  F  ~~>  ( 1  /  R ) ) )
1211rspccva 2896 . 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 1632    e. wcel 1696   A.wral 2556   {crab 2560   class class class wbr 4039    e. cmpt 4093   ` cfv 5271  (class class class)co 5874   RRcr 8752   0cc0 8753   1c1 8754    x. cmul 8758    - cmin 9053    / cdiv 9439   NNcn 9762   2c2 9811   ZZ>=cuz 10246   ...cfz 10798   |_cfl 10940    mod cmo 10989   ^cexp 11120   #chash 11353    ~~> cli 11974
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-13 1698  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-pow 4204  ax-pr 4230  ax-cnex 8809  ax-resscn 8810
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-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877
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