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Mirrors > Home > MPE Home > Th. List > dyadval | Structured version Visualization version GIF version |
Description: Value of the dyadic rational function 𝐹. (Contributed by Mario Carneiro, 26-Mar-2015.) |
Ref | Expression |
---|---|
dyadmbl.1 | ⊢ 𝐹 = (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ 〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) |
Ref | Expression |
---|---|
dyadval | ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ0) → (𝐴𝐹𝐵) = 〈(𝐴 / (2↑𝐵)), ((𝐴 + 1) / (2↑𝐵))〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . . 4 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴) | |
2 | oveq2 7409 | . . . 4 ⊢ (𝑦 = 𝐵 → (2↑𝑦) = (2↑𝐵)) | |
3 | 1, 2 | oveqan12d 7420 | . . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 / (2↑𝑦)) = (𝐴 / (2↑𝐵))) |
4 | oveq1 7408 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 + 1) = (𝐴 + 1)) | |
5 | 4, 2 | oveqan12d 7420 | . . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ((𝑥 + 1) / (2↑𝑦)) = ((𝐴 + 1) / (2↑𝐵))) |
6 | 3, 5 | opeq12d 4873 | . 2 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉 = 〈(𝐴 / (2↑𝐵)), ((𝐴 + 1) / (2↑𝐵))〉) |
7 | dyadmbl.1 | . 2 ⊢ 𝐹 = (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ 〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) | |
8 | opex 5454 | . 2 ⊢ 〈(𝐴 / (2↑𝐵)), ((𝐴 + 1) / (2↑𝐵))〉 ∈ V | |
9 | 6, 7, 8 | ovmpoa 7555 | 1 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ0) → (𝐴𝐹𝐵) = 〈(𝐴 / (2↑𝐵)), ((𝐴 + 1) / (2↑𝐵))〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 〈cop 4626 (class class class)co 7401 ∈ cmpo 7403 1c1 11107 + caddc 11109 / cdiv 11868 2c2 12264 ℕ0cn0 12469 ℤcz 12555 ↑cexp 14024 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pr 5417 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-sbc 3770 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-br 5139 df-opab 5201 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-iota 6485 df-fun 6535 df-fv 6541 df-ov 7404 df-oprab 7405 df-mpo 7406 |
This theorem is referenced by: dyadovol 25444 dyadss 25445 dyaddisjlem 25446 dyadmaxlem 25448 opnmbllem 25452 opnmbllem0 37014 |
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