Theorem snmlfval 30566

Description: The function 𝐹 from snmlval 30567 maps 𝑁 to the relative density of 𝐵 in the first 𝑁 digits of the digit string of 𝐴 in base 𝑅. (Contributed by Mario Carneiro, 6-Apr-2015.)

Hypothesis

Ref	Expression
snmlff.f	⊢ 𝐹 = (𝑛 ∈ ℕ ↦ ((#‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝐴 · (𝑅↑𝑘)) mod 𝑅)) = 𝐵}) / 𝑛))

Assertion

Ref	Expression
snmlfval	⊢ (𝑁 ∈ ℕ → (𝐹‘𝑁) = ((#‘{𝑘 ∈ (1...𝑁) ∣ (⌊‘((𝐴 · (𝑅↑𝑘)) mod 𝑅)) = 𝐵}) / 𝑁))

Distinct variable groups:   𝐴,𝑛   𝐵,𝑛   𝑘,𝑛,𝑁   𝑅,𝑛

Allowed substitution hints:   𝐴(𝑘)   𝐵(𝑘)   𝑅(𝑘)   𝐹(𝑘,𝑛)

Proof of Theorem snmlfval

Step	Hyp	Ref	Expression
1		oveq2 6557	. . . . 5 ⊢ (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁))
2		rabeq 3166	. . . . 5 ⊢ ((1...𝑛) = (1...𝑁) → {𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝐴 · (𝑅↑𝑘)) mod 𝑅)) = 𝐵} = {𝑘 ∈ (1...𝑁) ∣ (⌊‘((𝐴 · (𝑅↑𝑘)) mod 𝑅)) = 𝐵})
3	1, 2	syl 17	. . . 4 ⊢ (𝑛 = 𝑁 → {𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝐴 · (𝑅↑𝑘)) mod 𝑅)) = 𝐵} = {𝑘 ∈ (1...𝑁) ∣ (⌊‘((𝐴 · (𝑅↑𝑘)) mod 𝑅)) = 𝐵})
4	3	fveq2d 6107	. . 3 ⊢ (𝑛 = 𝑁 → (#‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝐴 · (𝑅↑𝑘)) mod 𝑅)) = 𝐵}) = (#‘{𝑘 ∈ (1...𝑁) ∣ (⌊‘((𝐴 · (𝑅↑𝑘)) mod 𝑅)) = 𝐵}))
5		id 22	. . 3 ⊢ (𝑛 = 𝑁 → 𝑛 = 𝑁)
6	4, 5	oveq12d 6567	. 2 ⊢ (𝑛 = 𝑁 → ((#‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝐴 · (𝑅↑𝑘)) mod 𝑅)) = 𝐵}) / 𝑛) = ((#‘{𝑘 ∈ (1...𝑁) ∣ (⌊‘((𝐴 · (𝑅↑𝑘)) mod 𝑅)) = 𝐵}) / 𝑁))
7		snmlff.f	. 2 ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ ((#‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝐴 · (𝑅↑𝑘)) mod 𝑅)) = 𝐵}) / 𝑛))
8		ovex 6577	. 2 ⊢ ((#‘{𝑘 ∈ (1...𝑁) ∣ (⌊‘((𝐴 · (𝑅↑𝑘)) mod 𝑅)) = 𝐵}) / 𝑁) ∈ V
9	6, 7, 8	fvmpt 6191	1 ⊢ (𝑁 ∈ ℕ → (𝐹‘𝑁) = ((#‘{𝑘 ∈ (1...𝑁) ∣ (⌊‘((𝐴 · (𝑅↑𝑘)) mod 𝑅)) = 𝐵}) / 𝑁))

Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  {crab 2900   ↦ cmpt 4643  ‘cfv 5804  (class class class)co 6549  1c1 9816   · cmul 9820   / cdiv 10563  ℕcn 10897  ...cfz 12197  ⌊cfl 12453   mod cmo 12530  ↑cexp 12722  #chash 12979

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-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

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-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-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552

This theorem is referenced by: (None)