Theorem snmlflim 31314

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

Step	Hyp	Ref	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 ) } )
2	1	snmlval 31313	. . 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 ) ) )
3	2	simp3bi 1078	. 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 2633	. . . . . . . . 9 |- ( b = B -> ( ( |_ ` ( ( A x. ( R ^ k ) ) mod R ) ) = b <-> ( |_ ` ( ( A x. ( R ^ k ) ) mod R ) ) = B ) )
5	4	rabbidv 3189	. . . . . . . 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 } )
6	5	fveq2d 6195	. . . . . . 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 } ) )
7	6	oveq1d 6665	. . . . . 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 ) )
8	7	mpteq2dv 4745	. . . . 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 ) )
10	8, 9	syl6eqr 2674	. . . 4 |- ( b = B -> ( n e. NN |-> ( ( # ` { k e. ( 1 ... n ) | ( |_ ` ( ( A x. ( R ^ k ) ) mod R ) ) = b } ) / n ) ) = F )
11	10	breq1d 4663	. . 3 |- ( b = B -> ( ( n e. NN |-> ( ( # ` { k e. ( 1 ... n ) | ( |_ ` ( ( A x. ( R ^ k ) ) mod R ) ) = b } ) / n ) ) ~~> ( 1 / R ) <-> F ~~> ( 1 / R ) ) )
12	11	rspccva 3308	. 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 ) )
13	3, 12	sylan 488	1 |- ( ( A e. ( S ` R ) /\ B e. ( 0 ... ( R - 1 ) ) ) -> F ~~> ( 1 / R ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   class class class wbr 4653   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   RRcr 9935   0cc0 9936   1c1 9937   x. cmul 9941   - cmin 10266   / cdiv 10684   NNcn 11020   2c2 11070   ZZ>=cuz 11687   ...cfz 12326   |_cfl 12591   mod cmo 12668   ^cexp 12860   #chash 13117   ~~> cli 14215

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-cnex 9992  ax-resscn 9993

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653

This theorem is referenced by: (None)