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| Mirrors > Home > MPE Home > Th. List > Mathboxes > dya2iocival | Structured version Visualization version GIF version | ||
| Description: The function 𝐼 returns closed-below open-above dyadic rational intervals covering the real line. This is the same construction as in dyadmbl 25635. (Contributed by Thierry Arnoux, 24-Sep-2017.) |
| Ref | Expression |
|---|---|
| sxbrsiga.0 | ⊢ 𝐽 = (topGen‘ran (,)) |
| dya2ioc.1 | ⊢ 𝐼 = (𝑥 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛)))) |
| Ref | Expression |
|---|---|
| dya2iocival | ⊢ ((𝑁 ∈ ℤ ∧ 𝑋 ∈ ℤ) → (𝑋𝐼𝑁) = ((𝑋 / (2↑𝑁))[,)((𝑋 + 1) / (2↑𝑁)))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | oveq1 7438 | . . . 4 ⊢ (𝑢 = 𝑋 → (𝑢 / (2↑𝑚)) = (𝑋 / (2↑𝑚))) | |
| 2 | oveq1 7438 | . . . . 5 ⊢ (𝑢 = 𝑋 → (𝑢 + 1) = (𝑋 + 1)) | |
| 3 | 2 | oveq1d 7446 | . . . 4 ⊢ (𝑢 = 𝑋 → ((𝑢 + 1) / (2↑𝑚)) = ((𝑋 + 1) / (2↑𝑚))) |
| 4 | 1, 3 | oveq12d 7449 | . . 3 ⊢ (𝑢 = 𝑋 → ((𝑢 / (2↑𝑚))[,)((𝑢 + 1) / (2↑𝑚))) = ((𝑋 / (2↑𝑚))[,)((𝑋 + 1) / (2↑𝑚)))) |
| 5 | oveq2 7439 | . . . . 5 ⊢ (𝑚 = 𝑁 → (2↑𝑚) = (2↑𝑁)) | |
| 6 | 5 | oveq2d 7447 | . . . 4 ⊢ (𝑚 = 𝑁 → (𝑋 / (2↑𝑚)) = (𝑋 / (2↑𝑁))) |
| 7 | 5 | oveq2d 7447 | . . . 4 ⊢ (𝑚 = 𝑁 → ((𝑋 + 1) / (2↑𝑚)) = ((𝑋 + 1) / (2↑𝑁))) |
| 8 | 6, 7 | oveq12d 7449 | . . 3 ⊢ (𝑚 = 𝑁 → ((𝑋 / (2↑𝑚))[,)((𝑋 + 1) / (2↑𝑚))) = ((𝑋 / (2↑𝑁))[,)((𝑋 + 1) / (2↑𝑁)))) |
| 9 | dya2ioc.1 | . . . 4 ⊢ 𝐼 = (𝑥 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛)))) | |
| 10 | oveq1 7438 | . . . . . 6 ⊢ (𝑢 = 𝑥 → (𝑢 / (2↑𝑚)) = (𝑥 / (2↑𝑚))) | |
| 11 | oveq1 7438 | . . . . . . 7 ⊢ (𝑢 = 𝑥 → (𝑢 + 1) = (𝑥 + 1)) | |
| 12 | 11 | oveq1d 7446 | . . . . . 6 ⊢ (𝑢 = 𝑥 → ((𝑢 + 1) / (2↑𝑚)) = ((𝑥 + 1) / (2↑𝑚))) |
| 13 | 10, 12 | oveq12d 7449 | . . . . 5 ⊢ (𝑢 = 𝑥 → ((𝑢 / (2↑𝑚))[,)((𝑢 + 1) / (2↑𝑚))) = ((𝑥 / (2↑𝑚))[,)((𝑥 + 1) / (2↑𝑚)))) |
| 14 | oveq2 7439 | . . . . . . 7 ⊢ (𝑚 = 𝑛 → (2↑𝑚) = (2↑𝑛)) | |
| 15 | 14 | oveq2d 7447 | . . . . . 6 ⊢ (𝑚 = 𝑛 → (𝑥 / (2↑𝑚)) = (𝑥 / (2↑𝑛))) |
| 16 | 14 | oveq2d 7447 | . . . . . 6 ⊢ (𝑚 = 𝑛 → ((𝑥 + 1) / (2↑𝑚)) = ((𝑥 + 1) / (2↑𝑛))) |
| 17 | 15, 16 | oveq12d 7449 | . . . . 5 ⊢ (𝑚 = 𝑛 → ((𝑥 / (2↑𝑚))[,)((𝑥 + 1) / (2↑𝑚))) = ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛)))) |
| 18 | 13, 17 | cbvmpov 7528 | . . . 4 ⊢ (𝑢 ∈ ℤ, 𝑚 ∈ ℤ ↦ ((𝑢 / (2↑𝑚))[,)((𝑢 + 1) / (2↑𝑚)))) = (𝑥 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛)))) |
| 19 | 9, 18 | eqtr4i 2768 | . . 3 ⊢ 𝐼 = (𝑢 ∈ ℤ, 𝑚 ∈ ℤ ↦ ((𝑢 / (2↑𝑚))[,)((𝑢 + 1) / (2↑𝑚)))) |
| 20 | ovex 7464 | . . 3 ⊢ ((𝑋 / (2↑𝑁))[,)((𝑋 + 1) / (2↑𝑁))) ∈ V | |
| 21 | 4, 8, 19, 20 | ovmpo 7593 | . 2 ⊢ ((𝑋 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑋𝐼𝑁) = ((𝑋 / (2↑𝑁))[,)((𝑋 + 1) / (2↑𝑁)))) |
| 22 | 21 | ancoms 458 | 1 ⊢ ((𝑁 ∈ ℤ ∧ 𝑋 ∈ ℤ) → (𝑋𝐼𝑁) = ((𝑋 / (2↑𝑁))[,)((𝑋 + 1) / (2↑𝑁)))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ran crn 5686 ‘cfv 6561 (class class class)co 7431 ∈ cmpo 7433 1c1 11156 + caddc 11158 / cdiv 11920 2c2 12321 ℤcz 12613 (,)cioo 13387 [,)cico 13389 ↑cexp 14102 topGenctg 17482 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 |
| This theorem is referenced by: dya2iocress 34276 dya2iocbrsiga 34277 dya2icoseg 34279 |
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