Theorem snmlfval 35573

Description: The function 𝐹 from snmlval 35574 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 7368	. . . . 5 ⊢ (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁))
2	1	rabeqdv 3408	. . . 4 ⊢ (𝑛 = 𝑁 → {𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝐴 · (𝑅↑𝑘)) mod 𝑅)) = 𝐵} = {𝑘 ∈ (1...𝑁) ∣ (⌊‘((𝐴 · (𝑅↑𝑘)) mod 𝑅)) = 𝐵})
3	2	fveq2d 6835	. . 3 ⊢ (𝑛 = 𝑁 → (♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝐴 · (𝑅↑𝑘)) mod 𝑅)) = 𝐵}) = (♯‘{𝑘 ∈ (1...𝑁) ∣ (⌊‘((𝐴 · (𝑅↑𝑘)) mod 𝑅)) = 𝐵}))
4		id 22	. . 3 ⊢ (𝑛 = 𝑁 → 𝑛 = 𝑁)
5	3, 4	oveq12d 7378	. 2 ⊢ (𝑛 = 𝑁 → ((♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝐴 · (𝑅↑𝑘)) mod 𝑅)) = 𝐵}) / 𝑛) = ((♯‘{𝑘 ∈ (1...𝑁) ∣ (⌊‘((𝐴 · (𝑅↑𝑘)) mod 𝑅)) = 𝐵}) / 𝑁))
6		snmlff.f	. 2 ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ ((♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝐴 · (𝑅↑𝑘)) mod 𝑅)) = 𝐵}) / 𝑛))
7		ovex 7393	. 2 ⊢ ((♯‘{𝑘 ∈ (1...𝑁) ∣ (⌊‘((𝐴 · (𝑅↑𝑘)) mod 𝑅)) = 𝐵}) / 𝑁) ∈ V
8	5, 6, 7	fvmpt 6939	1 ⊢ (𝑁 ∈ ℕ → (𝐹‘𝑁) = ((♯‘{𝑘 ∈ (1...𝑁) ∣ (⌊‘((𝐴 · (𝑅↑𝑘)) mod 𝑅)) = 𝐵}) / 𝑁))

Syntax hints:   → wi 4   = wceq 1548   ∈ wcel 2121  {crab 3393   ↦ cmpt 5156  ‘cfv 6489  (class class class)co 7360  1c1 11034   · cmul 11038   / cdiv 11802  ℕcn 12169  ...cfz 13456  ⌊cfl 13744   mod cmo 13823  ↑cexp 14018  ♯chash 14287

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 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-sep 5221  ax-nul 5231  ax-pr 5365

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-br 5076  df-opab 5138  df-mpt 5157  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-iota 6445  df-fun 6491  df-fv 6497  df-ov 7363

This theorem is referenced by: (None)