![]() |
Mathbox for Mario Carneiro |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > snmlfval | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
snmlff.f | ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ ((♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝐴 · (𝑅↑𝑘)) mod 𝑅)) = 𝐵}) / 𝑛)) |
Ref | Expression |
---|---|
snmlfval | ⊢ (𝑁 ∈ ℕ → (𝐹‘𝑁) = ((♯‘{𝑘 ∈ (1...𝑁) ∣ (⌊‘((𝐴 · (𝑅↑𝑘)) mod 𝑅)) = 𝐵}) / 𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 7404 | . . . . 5 ⊢ (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁)) | |
2 | 1 | rabeqdv 3448 | . . . 4 ⊢ (𝑛 = 𝑁 → {𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝐴 · (𝑅↑𝑘)) mod 𝑅)) = 𝐵} = {𝑘 ∈ (1...𝑁) ∣ (⌊‘((𝐴 · (𝑅↑𝑘)) mod 𝑅)) = 𝐵}) |
3 | 2 | fveq2d 6885 | . . 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 7414 | . 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 7429 | . 2 ⊢ ((♯‘{𝑘 ∈ (1...𝑁) ∣ (⌊‘((𝐴 · (𝑅↑𝑘)) mod 𝑅)) = 𝐵}) / 𝑁) ∈ V | |
8 | 5, 6, 7 | fvmpt 6987 | 1 ⊢ (𝑁 ∈ ℕ → (𝐹‘𝑁) = ((♯‘{𝑘 ∈ (1...𝑁) ∣ (⌊‘((𝐴 · (𝑅↑𝑘)) 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 ℕcn 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) |
Copyright terms: Public domain | W3C validator |