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Theorem snmlflim 27220
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 27219 . . 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 1005 . 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 2451 . . . . . . . . 9  |-  ( b  =  B  ->  (
( |_ `  (
( A  x.  ( R ^ k ) )  mod  R ) )  =  b  <->  ( |_ `  ( ( A  x.  ( R ^ k ) )  mod  R ) )  =  B ) )
54rabbidv 2963 . . . . . . . 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 5694 . . . . . . 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 6105 . . . . . 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 4378 . . . . 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 2492 . . . 4  |-  ( b  =  B  ->  (
n  e.  NN  |->  ( ( # `  {
k  e.  ( 1 ... n )  |  ( |_ `  (
( A  x.  ( R ^ k ) )  mod  R ) )  =  b } )  /  n ) )  =  F )
1110breq1d 4301 . . 3  |-  ( b  =  B  ->  (
( n  e.  NN  |->  ( ( # `  {
k  e.  ( 1 ... n )  |  ( |_ `  (
( A  x.  ( R ^ k ) )  mod  R ) )  =  b } )  /  n ) )  ~~>  ( 1  /  R
)  <->  F  ~~>  ( 1  /  R ) ) )
1211rspccva 3071 . 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 1369    e. wcel 1756   A.wral 2714   {crab 2718   class class class wbr 4291    e. cmpt 4349   ` cfv 5417  (class class class)co 6090   RRcr 9280   0cc0 9281   1c1 9282    x. cmul 9286    - cmin 9594    / cdiv 9992   NNcn 10321   2c2 10370   ZZ>=cuz 10860   ...cfz 11436   |_cfl 11639    mod cmo 11707   ^cexp 11864   #chash 12102    ~~> cli 12961
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-cnex 9337  ax-resscn 9338
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-rab 2723  df-v 2973  df-sbc 3186  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-br 4292  df-opab 4350  df-mpt 4351  df-id 4635  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fv 5425  df-ov 6093
This theorem is referenced by: (None)
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