Theorem snmlflim 30568

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 ∈ (ℤ≥‘2) ↦ {𝑥 ∈ ℝ ∣ ∀𝑏 ∈ (0...(𝑟 − 1))(𝑛 ∈ ℕ ↦ ((#‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑟↑𝑘)) mod 𝑟)) = 𝑏}) / 𝑛)) ⇝ (1 / 𝑟)})
snml.f	⊢ 𝐹 = (𝑛 ∈ ℕ ↦ ((#‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝐴 · (𝑅↑𝑘)) mod 𝑅)) = 𝐵}) / 𝑛))

Assertion

Ref	Expression
snmlflim	⊢ ((𝐴 ∈ (𝑆‘𝑅) ∧ 𝐵 ∈ (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 ∈ (ℤ≥‘2) ↦ {𝑥 ∈ ℝ ∣ ∀𝑏 ∈ (0...(𝑟 − 1))(𝑛 ∈ ℕ ↦ ((#‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑟↑𝑘)) mod 𝑟)) = 𝑏}) / 𝑛)) ⇝ (1 / 𝑟)})
2	1	snmlval 30567	. . 3 ⊢ (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)))
3	2	simp3bi 1071	. 2 ⊢ (𝐴 ∈ (𝑆‘𝑅) → ∀𝑏 ∈ (0...(𝑅 − 1))(𝑛 ∈ ℕ ↦ ((#‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝐴 · (𝑅↑𝑘)) mod 𝑅)) = 𝑏}) / 𝑛)) ⇝ (1 / 𝑅))
4		eqeq2 2621	. . . . . . . . 9 ⊢ (𝑏 = 𝐵 → ((⌊‘((𝐴 · (𝑅↑𝑘)) mod 𝑅)) = 𝑏 ↔ (⌊‘((𝐴 · (𝑅↑𝑘)) mod 𝑅)) = 𝐵))
5	4	rabbidv 3164	. . . . . . . 8 ⊢ (𝑏 = 𝐵 → {𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝐴 · (𝑅↑𝑘)) mod 𝑅)) = 𝑏} = {𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝐴 · (𝑅↑𝑘)) mod 𝑅)) = 𝐵})
6	5	fveq2d 6107	. . . . . . 7 ⊢ (𝑏 = 𝐵 → (#‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝐴 · (𝑅↑𝑘)) mod 𝑅)) = 𝑏}) = (#‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝐴 · (𝑅↑𝑘)) mod 𝑅)) = 𝐵}))
7	6	oveq1d 6564	. . . . . 6 ⊢ (𝑏 = 𝐵 → ((#‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝐴 · (𝑅↑𝑘)) mod 𝑅)) = 𝑏}) / 𝑛) = ((#‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝐴 · (𝑅↑𝑘)) mod 𝑅)) = 𝐵}) / 𝑛))
8	7	mpteq2dv 4673	. . . . 5 ⊢ (𝑏 = 𝐵 → (𝑛 ∈ ℕ ↦ ((#‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝐴 · (𝑅↑𝑘)) mod 𝑅)) = 𝑏}) / 𝑛)) = (𝑛 ∈ ℕ ↦ ((#‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝐴 · (𝑅↑𝑘)) mod 𝑅)) = 𝐵}) / 𝑛)))
9		snml.f	. . . . 5 ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ ((#‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝐴 · (𝑅↑𝑘)) mod 𝑅)) = 𝐵}) / 𝑛))
10	8, 9	syl6eqr 2662	. . . 4 ⊢ (𝑏 = 𝐵 → (𝑛 ∈ ℕ ↦ ((#‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝐴 · (𝑅↑𝑘)) mod 𝑅)) = 𝑏}) / 𝑛)) = 𝐹)
11	10	breq1d 4593	. . 3 ⊢ (𝑏 = 𝐵 → ((𝑛 ∈ ℕ ↦ ((#‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝐴 · (𝑅↑𝑘)) mod 𝑅)) = 𝑏}) / 𝑛)) ⇝ (1 / 𝑅) ↔ 𝐹 ⇝ (1 / 𝑅)))
12	11	rspccva 3281	. 2 ⊢ ((∀𝑏 ∈ (0...(𝑅 − 1))(𝑛 ∈ ℕ ↦ ((#‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝐴 · (𝑅↑𝑘)) mod 𝑅)) = 𝑏}) / 𝑛)) ⇝ (1 / 𝑅) ∧ 𝐵 ∈ (0...(𝑅 − 1))) → 𝐹 ⇝ (1 / 𝑅))
13	3, 12	sylan 487	1 ⊢ ((𝐴 ∈ (𝑆‘𝑅) ∧ 𝐵 ∈ (0...(𝑅 − 1))) → 𝐹 ⇝ (1 / 𝑅))

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900   class class class wbr 4583   ↦ cmpt 4643  ‘cfv 5804  (class class class)co 6549  ℝcr 9814  0cc0 9815  1c1 9816   · cmul 9820   − cmin 10145   / cdiv 10563  ℕcn 10897  2c2 10947  ℤ_≥cuz 11563  ...cfz 12197  ⌊cfl 12453   mod cmo 12530  ↑cexp 12722  #chash 12979   ⇝ cli 14063

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-cnex 9871  ax-resscn 9872

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552

This theorem is referenced by: (None)