| Mathbox for Thierry Arnoux |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > dya2iocrfn | Structured version Visualization version GIF version | ||
| Description: The function returning dyadic square covering for a given size has domain (ran 𝐼 × ran 𝐼). (Contributed by Thierry Arnoux, 19-Sep-2017.) |
| Ref | Expression |
|---|---|
| sxbrsiga.0 | ⊢ 𝐽 = (topGen‘ran (,)) |
| dya2ioc.1 | ⊢ 𝐼 = (𝑥 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛)))) |
| dya2ioc.2 | ⊢ 𝑅 = (𝑢 ∈ ran 𝐼, 𝑣 ∈ ran 𝐼 ↦ (𝑢 × 𝑣)) |
| Ref | Expression |
|---|---|
| dya2iocrfn | ⊢ 𝑅 Fn (ran 𝐼 × ran 𝐼) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dya2ioc.2 | . 2 ⊢ 𝑅 = (𝑢 ∈ ran 𝐼, 𝑣 ∈ ran 𝐼 ↦ (𝑢 × 𝑣)) | |
| 2 | vex 3484 | . . 3 ⊢ 𝑢 ∈ V | |
| 3 | vex 3484 | . . 3 ⊢ 𝑣 ∈ V | |
| 4 | 2, 3 | xpex 7773 | . 2 ⊢ (𝑢 × 𝑣) ∈ V |
| 5 | 1, 4 | fnmpoi 8095 | 1 ⊢ 𝑅 Fn (ran 𝐼 × ran 𝐼) |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 × cxp 5683 ran crn 5686 Fn wfn 6556 ‘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-pow 5365 ax-pr 5432 ax-un 7755 |
| 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-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 |
| This theorem is referenced by: dya2iocuni 34285 |
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