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Theorem snmlfval 32474
Description: The function 𝐹 from snmlval 32475 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 · (𝑅𝑘)) mod 𝑅)) = 𝐵}) / 𝑛))
Assertion
Ref Expression
snmlfval (𝑁 ∈ ℕ → (𝐹𝑁) = ((♯‘{𝑘 ∈ (1...𝑁) ∣ (⌊‘((𝐴 · (𝑅𝑘)) 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
StepHypRef Expression
1 oveq2 7153 . . . . 5 (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁))
21rabeqdv 3482 . . . 4 (𝑛 = 𝑁 → {𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝐴 · (𝑅𝑘)) mod 𝑅)) = 𝐵} = {𝑘 ∈ (1...𝑁) ∣ (⌊‘((𝐴 · (𝑅𝑘)) mod 𝑅)) = 𝐵})
32fveq2d 6667 . . 3 (𝑛 = 𝑁 → (♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝐴 · (𝑅𝑘)) mod 𝑅)) = 𝐵}) = (♯‘{𝑘 ∈ (1...𝑁) ∣ (⌊‘((𝐴 · (𝑅𝑘)) mod 𝑅)) = 𝐵}))
4 id 22 . . 3 (𝑛 = 𝑁𝑛 = 𝑁)
53, 4oveq12d 7163 . 2 (𝑛 = 𝑁 → ((♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝐴 · (𝑅𝑘)) mod 𝑅)) = 𝐵}) / 𝑛) = ((♯‘{𝑘 ∈ (1...𝑁) ∣ (⌊‘((𝐴 · (𝑅𝑘)) mod 𝑅)) = 𝐵}) / 𝑁))
6 snmlff.f . 2 𝐹 = (𝑛 ∈ ℕ ↦ ((♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝐴 · (𝑅𝑘)) mod 𝑅)) = 𝐵}) / 𝑛))
7 ovex 7178 . 2 ((♯‘{𝑘 ∈ (1...𝑁) ∣ (⌊‘((𝐴 · (𝑅𝑘)) mod 𝑅)) = 𝐵}) / 𝑁) ∈ V
85, 6, 7fvmpt 6761 1 (𝑁 ∈ ℕ → (𝐹𝑁) = ((♯‘{𝑘 ∈ (1...𝑁) ∣ (⌊‘((𝐴 · (𝑅𝑘)) mod 𝑅)) = 𝐵}) / 𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wcel 2105  {crab 3139  cmpt 5137  cfv 6348  (class class class)co 7145  1c1 10526   · cmul 10530   / cdiv 11285  cn 11626  ...cfz 12880  cfl 13148   mod cmo 13225  cexp 13417  chash 13678
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7148
This theorem is referenced by: (None)
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