Theorem snmlval 36065

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 7419	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (𝑟 − 1) = (𝑅 − 1))
2	1	oveq2d 7428	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (0...(𝑟 − 1)) = (0...(𝑅 − 1)))
3		oveq1 7419	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 = 𝑅 → (𝑟↑𝑘) = (𝑅↑𝑘))
4	3	oveq2d 7428	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = 𝑅 → (𝑥 · (𝑟↑𝑘)) = (𝑥 · (𝑅↑𝑘)))
5		id 23	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = 𝑅 → 𝑟 = 𝑅)
6	4, 5	oveq12d 7430	. . . . . . . . . . . . . 14 ⊢ (𝑟 = 𝑅 → ((𝑥 · (𝑟↑𝑘)) mod 𝑟) = ((𝑥 · (𝑅↑𝑘)) mod 𝑅))
7	6	fveqeq2d 6885	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑅 → ((⌊‘((𝑥 · (𝑟↑𝑘)) mod 𝑟)) = 𝑏 ↔ (⌊‘((𝑥 · (𝑅↑𝑘)) mod 𝑅)) = 𝑏))
8	7	rabbidv 3420	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑅 → {𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑟↑𝑘)) mod 𝑟)) = 𝑏} = {𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑅↑𝑘)) mod 𝑅)) = 𝑏})
9	8	fveq2d 6881	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑟↑𝑘)) mod 𝑟)) = 𝑏}) = (♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑅↑𝑘)) mod 𝑅)) = 𝑏}))
10	9	oveq1d 7427	. . . . . . . . . 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 5199	. . . . . . . . 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 7420	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (1 / 𝑟) = (1 / 𝑅))
13	11, 12	breq12d 5116	. . . . . . . 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 3335	. . . . . . 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 3420	. . . . . 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 11272	. . . . . . 7 ⊢ ℝ ∈ V
18	17	rabex 5300	. . . . . 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 6985	. . . . 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 2847	. . . 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 7419	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝐴 → (𝑥 · (𝑅↑𝑘)) = (𝐴 · (𝑅↑𝑘)))
22	21	fvoveq1d 7434	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐴 → (⌊‘((𝑥 · (𝑅↑𝑘)) mod 𝑅)) = (⌊‘((𝐴 · (𝑅↑𝑘)) mod 𝑅)))
23	22	eqeq1d 2763	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → ((⌊‘((𝑥 · (𝑅↑𝑘)) mod 𝑅)) = 𝑏 ↔ (⌊‘((𝐴 · (𝑅↑𝑘)) mod 𝑅)) = 𝑏))
24	23	rabbidv 3420	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → {𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑅↑𝑘)) mod 𝑅)) = 𝑏} = {𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝐴 · (𝑅↑𝑘)) mod 𝑅)) = 𝑏})
25	24	fveq2d 6881	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑅↑𝑘)) mod 𝑅)) = 𝑏}) = (♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝐴 · (𝑅↑𝑘)) mod 𝑅)) = 𝑏}))
26	25	oveq1d 7427	. . . . . . . 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 5199	. . . . . . 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 5113	. . . . . 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 3186	. . . . 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 3645	. . . 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 585	. 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 6235	. . . 4 ⊢ dom 𝑆 ⊆ (ℤ≥‘2)
34		elfvdm 6911	. . . 4 ⊢ (𝐴 ∈ (𝑆‘𝑅) → 𝑅 ∈ dom 𝑆)
35	33, 34	sselid 3929	. . 3 ⊢ (𝐴 ∈ (𝑆‘𝑅) → 𝑅 ∈ (ℤ≥‘2))
36	35	pm4.71ri 570	. 2 ⊢ (𝐴 ∈ (𝑆‘𝑅) ↔ (𝑅 ∈ (ℤ≥‘2) ∧ 𝐴 ∈ (𝑆‘𝑅)))
37		3anass 1111	. 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)))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3077  {crab 3413   class class class wbr 5103   ↦ cmpt 5186  dom cdm 5651  ‘cfv 6531  (class class class)co 7412  ℝcr 11180  0cc0 11181  1c1 11182   · cmul 11186   − cmin 11522   / cdiv 11954  ℕcn 12316  2c2 12378  ℤ_≥cuz 12946  ...cfz 13620  ⌊cfl 13910   mod cmo 13989  ↑cexp 14184  ♯chash 14454   ⇝ cli 15631

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391  ax-cnex 11237  ax-resscn 11238

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6487  df-fun 6533  df-fv 6539  df-ov 7415

This theorem is used by:  snmlflim  36066