Theorem snmlval 34608

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 7418	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (𝑟 − 1) = (𝑅 − 1))
2	1	oveq2d 7427	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (0...(𝑟 − 1)) = (0...(𝑅 − 1)))
3		oveq1 7418	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 = 𝑅 → (𝑟↑𝑘) = (𝑅↑𝑘))
4	3	oveq2d 7427	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = 𝑅 → (𝑥 · (𝑟↑𝑘)) = (𝑥 · (𝑅↑𝑘)))
5		id 22	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = 𝑅 → 𝑟 = 𝑅)
6	4, 5	oveq12d 7429	. . . . . . . . . . . . . 14 ⊢ (𝑟 = 𝑅 → ((𝑥 · (𝑟↑𝑘)) mod 𝑟) = ((𝑥 · (𝑅↑𝑘)) mod 𝑅))
7	6	fveqeq2d 6899	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑅 → ((⌊‘((𝑥 · (𝑟↑𝑘)) mod 𝑟)) = 𝑏 ↔ (⌊‘((𝑥 · (𝑅↑𝑘)) mod 𝑅)) = 𝑏))
8	7	rabbidv 3440	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑅 → {𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑟↑𝑘)) mod 𝑟)) = 𝑏} = {𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑅↑𝑘)) mod 𝑅)) = 𝑏})
9	8	fveq2d 6895	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑟↑𝑘)) mod 𝑟)) = 𝑏}) = (♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑅↑𝑘)) mod 𝑅)) = 𝑏}))
10	9	oveq1d 7426	. . . . . . . . . 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 5250	. . . . . . . . 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 7419	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (1 / 𝑟) = (1 / 𝑅))
13	11, 12	breq12d 5161	. . . . . . . 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 3342	. . . . . . 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 3440	. . . . . 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 11203	. . . . . . 7 ⊢ ℝ ∈ V
18	17	rabex 5332	. . . . . 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 6998	. . . . 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 2819	. . . 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 7418	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝐴 → (𝑥 · (𝑅↑𝑘)) = (𝐴 · (𝑅↑𝑘)))
22	21	fvoveq1d 7433	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐴 → (⌊‘((𝑥 · (𝑅↑𝑘)) mod 𝑅)) = (⌊‘((𝐴 · (𝑅↑𝑘)) mod 𝑅)))
23	22	eqeq1d 2734	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → ((⌊‘((𝑥 · (𝑅↑𝑘)) mod 𝑅)) = 𝑏 ↔ (⌊‘((𝐴 · (𝑅↑𝑘)) mod 𝑅)) = 𝑏))
24	23	rabbidv 3440	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → {𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑅↑𝑘)) mod 𝑅)) = 𝑏} = {𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝐴 · (𝑅↑𝑘)) mod 𝑅)) = 𝑏})
25	24	fveq2d 6895	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑅↑𝑘)) mod 𝑅)) = 𝑏}) = (♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝐴 · (𝑅↑𝑘)) mod 𝑅)) = 𝑏}))
26	25	oveq1d 7426	. . . . . . . 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 5250	. . . . . . 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 5158	. . . . . 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 3177	. . . . 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 3683	. . . 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 286	. . 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 575	. 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 6240	. . . 4 ⊢ dom 𝑆 ⊆ (ℤ≥‘2)
34		elfvdm 6928	. . . 4 ⊢ (𝐴 ∈ (𝑆‘𝑅) → 𝑅 ∈ dom 𝑆)
35	33, 34	sselid 3980	. . 3 ⊢ (𝐴 ∈ (𝑆‘𝑅) → 𝑅 ∈ (ℤ≥‘2))
36	35	pm4.71ri 561	. 2 ⊢ (𝐴 ∈ (𝑆‘𝑅) ↔ (𝑅 ∈ (ℤ≥‘2) ∧ 𝐴 ∈ (𝑆‘𝑅)))
37		3anass 1095	. 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 302	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 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∀wral 3061  {crab 3432   class class class wbr 5148   ↦ cmpt 5231  dom cdm 5676  ‘cfv 6543  (class class class)co 7411  ℝcr 11111  0cc0 11112  1c1 11113   · cmul 11117   − cmin 11448   / cdiv 11875  ℕcn 12216  2c2 12271  ℤ_≥cuz 12826  ...cfz 13488  ⌊cfl 13759   mod cmo 13838  ↑cexp 14031  ♯chash 14294   ⇝ cli 15432

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-cnex 11168  ax-resscn 11169

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7414

This theorem is referenced by:  snmlflim  34609