Theorem snmlval 32960

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 ∈ (ℤ≥‘2) ↦ {𝑥 ∈ ℝ ∣ ∀𝑏 ∈ (0...(𝑟 − 1))(𝑛 ∈ ℕ ↦ ((♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑟↑𝑘)) mod 𝑟)) = 𝑏}) / 𝑛)) ⇝ (1 / 𝑟)})

Assertion

Ref	Expression
snmlval	⊢ (𝐴 ∈ (𝑆‘𝑅) ↔ (𝑅 ∈ (ℤ≥‘2) ∧ 𝐴 ∈ ℝ ∧ ∀𝑏 ∈ (0...(𝑅 − 1))(𝑛 ∈ ℕ ↦ ((♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝐴 · (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 7198	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (𝑟 − 1) = (𝑅 − 1))
2	1	oveq2d 7207	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (0...(𝑟 − 1)) = (0...(𝑅 − 1)))
3		oveq1 7198	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 = 𝑅 → (𝑟↑𝑘) = (𝑅↑𝑘))
4	3	oveq2d 7207	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = 𝑅 → (𝑥 · (𝑟↑𝑘)) = (𝑥 · (𝑅↑𝑘)))
5		id 22	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = 𝑅 → 𝑟 = 𝑅)
6	4, 5	oveq12d 7209	. . . . . . . . . . . . . 14 ⊢ (𝑟 = 𝑅 → ((𝑥 · (𝑟↑𝑘)) mod 𝑟) = ((𝑥 · (𝑅↑𝑘)) mod 𝑅))
7	6	fveqeq2d 6703	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑅 → ((⌊‘((𝑥 · (𝑟↑𝑘)) mod 𝑟)) = 𝑏 ↔ (⌊‘((𝑥 · (𝑅↑𝑘)) mod 𝑅)) = 𝑏))
8	7	rabbidv 3380	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑅 → {𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑟↑𝑘)) mod 𝑟)) = 𝑏} = {𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑅↑𝑘)) mod 𝑅)) = 𝑏})
9	8	fveq2d 6699	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑟↑𝑘)) mod 𝑟)) = 𝑏}) = (♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑅↑𝑘)) mod 𝑅)) = 𝑏}))
10	9	oveq1d 7206	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → ((♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑟↑𝑘)) mod 𝑟)) = 𝑏}) / 𝑛) = ((♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑅↑𝑘)) mod 𝑅)) = 𝑏}) / 𝑛))
11	10	mpteq2dv 5136	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (𝑛 ∈ ℕ ↦ ((♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑟↑𝑘)) mod 𝑟)) = 𝑏}) / 𝑛)) = (𝑛 ∈ ℕ ↦ ((♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑅↑𝑘)) mod 𝑅)) = 𝑏}) / 𝑛)))
12		oveq2 7199	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (1 / 𝑟) = (1 / 𝑅))
13	11, 12	breq12d 5052	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → ((𝑛 ∈ ℕ ↦ ((♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑟↑𝑘)) mod 𝑟)) = 𝑏}) / 𝑛)) ⇝ (1 / 𝑟) ↔ (𝑛 ∈ ℕ ↦ ((♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑅↑𝑘)) mod 𝑅)) = 𝑏}) / 𝑛)) ⇝ (1 / 𝑅)))
14	2, 13	raleqbidv 3303	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (∀𝑏 ∈ (0...(𝑟 − 1))(𝑛 ∈ ℕ ↦ ((♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑟↑𝑘)) mod 𝑟)) = 𝑏}) / 𝑛)) ⇝ (1 / 𝑟) ↔ ∀𝑏 ∈ (0...(𝑅 − 1))(𝑛 ∈ ℕ ↦ ((♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑅↑𝑘)) mod 𝑅)) = 𝑏}) / 𝑛)) ⇝ (1 / 𝑅)))
15	14	rabbidv 3380	. . . . . 6 ⊢ (𝑟 = 𝑅 → {𝑥 ∈ ℝ ∣ ∀𝑏 ∈ (0...(𝑟 − 1))(𝑛 ∈ ℕ ↦ ((♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑟↑𝑘)) mod 𝑟)) = 𝑏}) / 𝑛)) ⇝ (1 / 𝑟)} = {𝑥 ∈ ℝ ∣ ∀𝑏 ∈ (0...(𝑅 − 1))(𝑛 ∈ ℕ ↦ ((♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑅↑𝑘)) mod 𝑅)) = 𝑏}) / 𝑛)) ⇝ (1 / 𝑅)})
16		snml.s	. . . . . 6 ⊢ 𝑆 = (𝑟 ∈ (ℤ≥‘2) ↦ {𝑥 ∈ ℝ ∣ ∀𝑏 ∈ (0...(𝑟 − 1))(𝑛 ∈ ℕ ↦ ((♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑟↑𝑘)) mod 𝑟)) = 𝑏}) / 𝑛)) ⇝ (1 / 𝑟)})
17		reex 10785	. . . . . . 7 ⊢ ℝ ∈ V
18	17	rabex 5210	. . . . . 6 ⊢ {𝑥 ∈ ℝ ∣ ∀𝑏 ∈ (0...(𝑅 − 1))(𝑛 ∈ ℕ ↦ ((♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑅↑𝑘)) mod 𝑅)) = 𝑏}) / 𝑛)) ⇝ (1 / 𝑅)} ∈ V
19	15, 16, 18	fvmpt 6796	. . . . 5 ⊢ (𝑅 ∈ (ℤ≥‘2) → (𝑆‘𝑅) = {𝑥 ∈ ℝ ∣ ∀𝑏 ∈ (0...(𝑅 − 1))(𝑛 ∈ ℕ ↦ ((♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑅↑𝑘)) mod 𝑅)) = 𝑏}) / 𝑛)) ⇝ (1 / 𝑅)})
20	19	eleq2d 2816	. . . 4 ⊢ (𝑅 ∈ (ℤ≥‘2) → (𝐴 ∈ (𝑆‘𝑅) ↔ 𝐴 ∈ {𝑥 ∈ ℝ ∣ ∀𝑏 ∈ (0...(𝑅 − 1))(𝑛 ∈ ℕ ↦ ((♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑅↑𝑘)) mod 𝑅)) = 𝑏}) / 𝑛)) ⇝ (1 / 𝑅)}))
21		oveq1 7198	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝐴 → (𝑥 · (𝑅↑𝑘)) = (𝐴 · (𝑅↑𝑘)))
22	21	fvoveq1d 7213	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐴 → (⌊‘((𝑥 · (𝑅↑𝑘)) mod 𝑅)) = (⌊‘((𝐴 · (𝑅↑𝑘)) mod 𝑅)))
23	22	eqeq1d 2738	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → ((⌊‘((𝑥 · (𝑅↑𝑘)) mod 𝑅)) = 𝑏 ↔ (⌊‘((𝐴 · (𝑅↑𝑘)) mod 𝑅)) = 𝑏))
24	23	rabbidv 3380	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → {𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑅↑𝑘)) mod 𝑅)) = 𝑏} = {𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝐴 · (𝑅↑𝑘)) mod 𝑅)) = 𝑏})
25	24	fveq2d 6699	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑅↑𝑘)) mod 𝑅)) = 𝑏}) = (♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝐴 · (𝑅↑𝑘)) mod 𝑅)) = 𝑏}))
26	25	oveq1d 7206	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → ((♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑅↑𝑘)) mod 𝑅)) = 𝑏}) / 𝑛) = ((♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝐴 · (𝑅↑𝑘)) mod 𝑅)) = 𝑏}) / 𝑛))
27	26	mpteq2dv 5136	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑛 ∈ ℕ ↦ ((♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑅↑𝑘)) mod 𝑅)) = 𝑏}) / 𝑛)) = (𝑛 ∈ ℕ ↦ ((♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝐴 · (𝑅↑𝑘)) mod 𝑅)) = 𝑏}) / 𝑛)))
28	27	breq1d 5049	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝑛 ∈ ℕ ↦ ((♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑅↑𝑘)) mod 𝑅)) = 𝑏}) / 𝑛)) ⇝ (1 / 𝑅) ↔ (𝑛 ∈ ℕ ↦ ((♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝐴 · (𝑅↑𝑘)) mod 𝑅)) = 𝑏}) / 𝑛)) ⇝ (1 / 𝑅)))
29	28	ralbidv 3108	. . . . 5 ⊢ (𝑥 = 𝐴 → (∀𝑏 ∈ (0...(𝑅 − 1))(𝑛 ∈ ℕ ↦ ((♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑅↑𝑘)) mod 𝑅)) = 𝑏}) / 𝑛)) ⇝ (1 / 𝑅) ↔ ∀𝑏 ∈ (0...(𝑅 − 1))(𝑛 ∈ ℕ ↦ ((♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝐴 · (𝑅↑𝑘)) mod 𝑅)) = 𝑏}) / 𝑛)) ⇝ (1 / 𝑅)))
30	29	elrab 3591	. . . 4 ⊢ (𝐴 ∈ {𝑥 ∈ ℝ ∣ ∀𝑏 ∈ (0...(𝑅 − 1))(𝑛 ∈ ℕ ↦ ((♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑅↑𝑘)) mod 𝑅)) = 𝑏}) / 𝑛)) ⇝ (1 / 𝑅)} ↔ (𝐴 ∈ ℝ ∧ ∀𝑏 ∈ (0...(𝑅 − 1))(𝑛 ∈ ℕ ↦ ((♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝐴 · (𝑅↑𝑘)) mod 𝑅)) = 𝑏}) / 𝑛)) ⇝ (1 / 𝑅)))
31	20, 30	bitrdi 290	. . 3 ⊢ (𝑅 ∈ (ℤ≥‘2) → (𝐴 ∈ (𝑆‘𝑅) ↔ (𝐴 ∈ ℝ ∧ ∀𝑏 ∈ (0...(𝑅 − 1))(𝑛 ∈ ℕ ↦ ((♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝐴 · (𝑅↑𝑘)) mod 𝑅)) = 𝑏}) / 𝑛)) ⇝ (1 / 𝑅))))
32	31	pm5.32i 578	. 2 ⊢ ((𝑅 ∈ (ℤ≥‘2) ∧ 𝐴 ∈ (𝑆‘𝑅)) ↔ (𝑅 ∈ (ℤ≥‘2) ∧ (𝐴 ∈ ℝ ∧ ∀𝑏 ∈ (0...(𝑅 − 1))(𝑛 ∈ ℕ ↦ ((♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝐴 · (𝑅↑𝑘)) mod 𝑅)) = 𝑏}) / 𝑛)) ⇝ (1 / 𝑅))))
33	16	dmmptss 6084	. . . 4 ⊢ dom 𝑆 ⊆ (ℤ≥‘2)
34		elfvdm 6727	. . . 4 ⊢ (𝐴 ∈ (𝑆‘𝑅) → 𝑅 ∈ dom 𝑆)
35	33, 34	sseldi 3885	. . 3 ⊢ (𝐴 ∈ (𝑆‘𝑅) → 𝑅 ∈ (ℤ≥‘2))
36	35	pm4.71ri 564	. 2 ⊢ (𝐴 ∈ (𝑆‘𝑅) ↔ (𝑅 ∈ (ℤ≥‘2) ∧ 𝐴 ∈ (𝑆‘𝑅)))
37		3anass 1097	. 2 ⊢ ((𝑅 ∈ (ℤ≥‘2) ∧ 𝐴 ∈ ℝ ∧ ∀𝑏 ∈ (0...(𝑅 − 1))(𝑛 ∈ ℕ ↦ ((♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝐴 · (𝑅↑𝑘)) mod 𝑅)) = 𝑏}) / 𝑛)) ⇝ (1 / 𝑅)) ↔ (𝑅 ∈ (ℤ≥‘2) ∧ (𝐴 ∈ ℝ ∧ ∀𝑏 ∈ (0...(𝑅 − 1))(𝑛 ∈ ℕ ↦ ((♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝐴 · (𝑅↑𝑘)) mod 𝑅)) = 𝑏}) / 𝑛)) ⇝ (1 / 𝑅))))
38	32, 36, 37	3bitr4i 306	1 ⊢ (𝐴 ∈ (𝑆‘𝑅) ↔ (𝑅 ∈ (ℤ≥‘2) ∧ 𝐴 ∈ ℝ ∧ ∀𝑏 ∈ (0...(𝑅 − 1))(𝑛 ∈ ℕ ↦ ((♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝐴 · (𝑅↑𝑘)) mod 𝑅)) = 𝑏}) / 𝑛)) ⇝ (1 / 𝑅)))

Syntax hints:   ↔ wb 209   ∧ wa 399   ∧ w3a 1089   = wceq 1543   ∈ wcel 2112  ∀wral 3051  {crab 3055   class class class wbr 5039   ↦ cmpt 5120  dom cdm 5536  ‘cfv 6358  (class class class)co 7191  ℝcr 10693  0cc0 10694  1c1 10695   · cmul 10699   − cmin 11027   / cdiv 11454  ℕcn 11795  2c2 11850  ℤ_≥cuz 12403  ...cfz 13060  ⌊cfl 13330   mod cmo 13407  ↑cexp 13600  ♯chash 13861   ⇝ cli 15010

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-sep 5177  ax-nul 5184  ax-pr 5307  ax-cnex 10750  ax-resscn 10751

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ne 2933  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3400  df-sbc 3684  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-if 4426  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4806  df-br 5040  df-opab 5102  df-mpt 5121  df-id 5440  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-iota 6316  df-fun 6360  df-fv 6366  df-ov 7194

This theorem is referenced by:  snmlflim  32961