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Mirrors > Home > MPE Home > Th. List > Mathboxes > snmlfval | Structured version Visualization version GIF version |
Description: The function 𝐹 from snmlval 31313 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 6658 | . . . . 5 ⊢ (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁)) | |
2 | rabeq 3192 | . . . . 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 6195 | . . 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 6668 | . 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 6678 | . 2 ⊢ ((#‘{𝑘 ∈ (1...𝑁) ∣ (⌊‘((𝐴 · (𝑅↑𝑘)) mod 𝑅)) = 𝐵}) / 𝑁) ∈ V | |
9 | 6, 7, 8 | fvmpt 6282 | 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 1483 ∈ wcel 1990 {crab 2916 ↦ cmpt 4729 ‘cfv 5888 (class class class)co 6650 1c1 9937 · cmul 9941 / cdiv 10684 ℕcn 11020 ...cfz 12326 ⌊cfl 12591 mod cmo 12668 ↑cexp 12860 #chash 13117 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-ov 6653 |
This theorem is referenced by: (None) |
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