Theorem snmlval 31313

Description: The property " A is simply normal in base R". A number is simply normal if each digit 0 <_ b < R occurs in the base- R digit string of A with frequency 1 / R (which is consistent with the expectation in an infinite random string of numbers selected from 0 ... R - 1). (Contributed by Mario Carneiro, 6-Apr-2015.)

Hypothesis

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 ) } )

Assertion

Ref	Expression
snmlval	|- ( 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 ) ) )

Distinct variable groups:   k, b, n, x, A   r, b, R, k, n, x

Allowed substitution hints:   A( r)   S( x, k, n, r, b)

Proof of Theorem snmlval

Step	Hyp	Ref	Expression
1		oveq1 6657	. . . . . . . . 9 |- ( r = R -> ( r - 1 ) = ( R - 1 ) )
2	1	oveq2d 6666	. . . . . . . 8 |- ( r = R -> ( 0 ... ( r - 1 ) ) = ( 0 ... ( R - 1 ) ) )
3		oveq1 6657	. . . . . . . . . . . . . . . . 17 |- ( r = R -> ( r ^ k ) = ( R ^ k ) )
4	3	oveq2d 6666	. . . . . . . . . . . . . . . 16 |- ( r = R -> ( x x. ( r ^ k ) ) = ( x x. ( R ^ k ) ) )
5		id 22	. . . . . . . . . . . . . . . 16 |- ( r = R -> r = R )
6	4, 5	oveq12d 6668	. . . . . . . . . . . . . . 15 |- ( r = R -> ( ( x x. ( r ^ k ) ) mod r ) = ( ( x x. ( R ^ k ) ) mod R ) )
7	6	fveq2d 6195	. . . . . . . . . . . . . 14 |- ( r = R -> ( |_ ` ( ( x x. ( r ^ k ) ) mod r ) ) = ( |_ ` ( ( x x. ( R ^ k ) ) mod R ) ) )
8	7	eqeq1d 2624	. . . . . . . . . . . . 13 |- ( r = R -> ( ( |_ ` ( ( x x. ( r ^ k ) ) mod r ) ) = b <-> ( |_ ` ( ( x x. ( R ^ k ) ) mod R ) ) = b ) )
9	8	rabbidv 3189	. . . . . . . . . . . 12 |- ( r = R -> { k e. ( 1 ... n ) | ( |_ ` ( ( x x. ( r ^ k ) ) mod r ) ) = b } = { k e. ( 1 ... n ) | ( |_ ` ( ( x x. ( R ^ k ) ) mod R ) ) = b } )
10	9	fveq2d 6195	. . . . . . . . . . 11 |- ( r = R -> ( # ` { k e. ( 1 ... n ) | ( |_ ` ( ( x x. ( r ^ k ) ) mod r ) ) = b } ) = ( # ` { k e. ( 1 ... n ) | ( |_ ` ( ( x x. ( R ^ k ) ) mod R ) ) = b } ) )
11	10	oveq1d 6665	. . . . . . . . . 10 |- ( r = R -> ( ( # ` { k e. ( 1 ... n ) | ( |_ ` ( ( x x. ( r ^ k ) ) mod r ) ) = b } ) / n ) = ( ( # ` { k e. ( 1 ... n ) | ( |_ ` ( ( x x. ( R ^ k ) ) mod R ) ) = b } ) / n ) )
12	11	mpteq2dv 4745	. . . . . . . . 9 |- ( r = R -> ( n e. NN |-> ( ( # ` { k e. ( 1 ... n ) | ( |_ ` ( ( x x. ( r ^ k ) ) mod r ) ) = b } ) / n ) ) = ( n e. NN |-> ( ( # ` { k e. ( 1 ... n ) | ( |_ ` ( ( x x. ( R ^ k ) ) mod R ) ) = b } ) / n ) ) )
13		oveq2 6658	. . . . . . . . 9 |- ( r = R -> ( 1 / r ) = ( 1 / R ) )
14	12, 13	breq12d 4666	. . . . . . . 8 |- ( r = R -> ( ( n e. NN |-> ( ( # ` { k e. ( 1 ... n ) | ( |_ ` ( ( x x. ( r ^ k ) ) mod r ) ) = b } ) / n ) ) ~~> ( 1 / r ) <-> ( n e. NN |-> ( ( # ` { k e. ( 1 ... n ) | ( |_ ` ( ( x x. ( R ^ k ) ) mod R ) ) = b } ) / n ) ) ~~> ( 1 / R ) ) )
15	2, 14	raleqbidv 3152	. . . . . . 7 |- ( r = R -> ( A. b e. ( 0 ... ( r - 1 ) ) ( n e. NN |-> ( ( # ` { k e. ( 1 ... n ) | ( |_ ` ( ( x x. ( r ^ k ) ) mod r ) ) = b } ) / n ) ) ~~> ( 1 / r ) <-> A. b e. ( 0 ... ( R - 1 ) ) ( n e. NN |-> ( ( # ` { k e. ( 1 ... n ) | ( |_ ` ( ( x x. ( R ^ k ) ) mod R ) ) = b } ) / n ) ) ~~> ( 1 / R ) ) )
16	15	rabbidv 3189	. . . . . 6 |- ( r = R -> { 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 ) } = { 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 ) } )
17		snml.s	. . . . . 6 |- 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 ) } )
18		reex 10027	. . . . . . 7 |- RR e. _V
19	18	rabex 4813	. . . . . 6 |- { 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 ) } e. _V
20	16, 17, 19	fvmpt 6282	. . . . 5 |- ( R e. ( ZZ>= ` 2 ) -> ( S ` R ) = { 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 ) } )
21	20	eleq2d 2687	. . . 4 |- ( R e. ( ZZ>= ` 2 ) -> ( A e. ( S ` R ) <-> A e. { 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 ) } ) )
22		oveq1 6657	. . . . . . . . . . . . . 14 |- ( x = A -> ( x x. ( R ^ k ) ) = ( A x. ( R ^ k ) ) )
23	22	oveq1d 6665	. . . . . . . . . . . . 13 |- ( x = A -> ( ( x x. ( R ^ k ) ) mod R ) = ( ( A x. ( R ^ k ) ) mod R ) )
24	23	fveq2d 6195	. . . . . . . . . . . 12 |- ( x = A -> ( |_ ` ( ( x x. ( R ^ k ) ) mod R ) ) = ( |_ ` ( ( A x. ( R ^ k ) ) mod R ) ) )
25	24	eqeq1d 2624	. . . . . . . . . . 11 |- ( x = A -> ( ( |_ ` ( ( x x. ( R ^ k ) ) mod R ) ) = b <-> ( |_ ` ( ( A x. ( R ^ k ) ) mod R ) ) = b ) )
26	25	rabbidv 3189	. . . . . . . . . 10 |- ( x = A -> { k e. ( 1 ... n ) | ( |_ ` ( ( x x. ( R ^ k ) ) mod R ) ) = b } = { k e. ( 1 ... n ) | ( |_ ` ( ( A x. ( R ^ k ) ) mod R ) ) = b } )
27	26	fveq2d 6195	. . . . . . . . 9 |- ( x = A -> ( # ` { k e. ( 1 ... n ) | ( |_ ` ( ( x x. ( R ^ k ) ) mod R ) ) = b } ) = ( # ` { k e. ( 1 ... n ) | ( |_ ` ( ( A x. ( R ^ k ) ) mod R ) ) = b } ) )
28	27	oveq1d 6665	. . . . . . . 8 |- ( x = A -> ( ( # ` { k e. ( 1 ... n ) | ( |_ ` ( ( x x. ( R ^ k ) ) mod R ) ) = b } ) / n ) = ( ( # ` { k e. ( 1 ... n ) | ( |_ ` ( ( A x. ( R ^ k ) ) mod R ) ) = b } ) / n ) )
29	28	mpteq2dv 4745	. . . . . . 7 |- ( x = A -> ( n e. NN |-> ( ( # ` { k e. ( 1 ... n ) | ( |_ ` ( ( x x. ( R ^ k ) ) mod R ) ) = b } ) / n ) ) = ( n e. NN |-> ( ( # ` { k e. ( 1 ... n ) | ( |_ ` ( ( A x. ( R ^ k ) ) mod R ) ) = b } ) / n ) ) )
30	29	breq1d 4663	. . . . . 6 |- ( x = A -> ( ( n e. NN |-> ( ( # ` { k e. ( 1 ... n ) | ( |_ ` ( ( x x. ( R ^ k ) ) mod R ) ) = b } ) / n ) ) ~~> ( 1 / R ) <-> ( n e. NN |-> ( ( # ` { k e. ( 1 ... n ) | ( |_ ` ( ( A x. ( R ^ k ) ) mod R ) ) = b } ) / n ) ) ~~> ( 1 / R ) ) )
31	30	ralbidv 2986	. . . . 5 |- ( x = A -> ( A. b e. ( 0 ... ( R - 1 ) ) ( n e. NN |-> ( ( # ` { k e. ( 1 ... n ) | ( |_ ` ( ( x x. ( R ^ k ) ) mod R ) ) = b } ) / n ) ) ~~> ( 1 / R ) <-> A. b e. ( 0 ... ( R - 1 ) ) ( n e. NN |-> ( ( # ` { k e. ( 1 ... n ) | ( |_ ` ( ( A x. ( R ^ k ) ) mod R ) ) = b } ) / n ) ) ~~> ( 1 / R ) ) )
32	31	elrab 3363	. . . 4 |- ( A e. { 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 ) } <-> ( 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 ) ) )
33	21, 32	syl6bb 276	. . 3 |- ( R e. ( ZZ>= ` 2 ) -> ( A e. ( S ` R ) <-> ( 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 ) ) ) )
34	33	pm5.32i 669	. 2 |- ( ( R e. ( ZZ>= ` 2 ) /\ 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 ) ) ) )
35	17	dmmptss 5631	. . . 4 |- dom S C_ ( ZZ>= ` 2 )
36		elfvdm 6220	. . . 4 |- ( A e. ( S ` R ) -> R e. dom S )
37	35, 36	sseldi 3601	. . 3 |- ( A e. ( S ` R ) -> R e. ( ZZ>= ` 2 ) )
38	37	pm4.71ri 665	. 2 |- ( A e. ( S ` R ) <-> ( R e. ( ZZ>= ` 2 ) /\ A e. ( S ` R ) ) )
39		3anass 1042	. 2 |- ( ( 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 ) ) <-> ( 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 ) ) ) )
40	34, 38, 39	3bitr4i 292	1 |- ( 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 ) ) )

Syntax hints:   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   class class class wbr 4653   |-> cmpt 4729   dom cdm 5114   ` 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:  snmlflim  31314