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Theorem snmlfval 34252
Description: The function 𝐹 from snmlval 34253 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 7404 . . . . 5 (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁))
21rabeqdv 3448 . . . 4 (𝑛 = 𝑁 → {𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝐴 · (𝑅𝑘)) mod 𝑅)) = 𝐵} = {𝑘 ∈ (1...𝑁) ∣ (⌊‘((𝐴 · (𝑅𝑘)) mod 𝑅)) = 𝐵})
32fveq2d 6885 . . 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 7414 . 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 7429 . 2 ((♯‘{𝑘 ∈ (1...𝑁) ∣ (⌊‘((𝐴 · (𝑅𝑘)) mod 𝑅)) = 𝐵}) / 𝑁) ∈ V
85, 6, 7fvmpt 6987 1 (𝑁 ∈ ℕ → (𝐹𝑁) = ((♯‘{𝑘 ∈ (1...𝑁) ∣ (⌊‘((𝐴 · (𝑅𝑘)) mod 𝑅)) = 𝐵}) / 𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  {crab 3433  cmpt 5227  cfv 6535  (class class class)co 7396  1c1 11098   · cmul 11102   / cdiv 11858  cn 12199  ...cfz 13471  cfl 13742   mod cmo 13821  cexp 14014  chash 14277
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5295  ax-nul 5302  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4905  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-iota 6487  df-fun 6537  df-fv 6543  df-ov 7399
This theorem is referenced by: (None)
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