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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 3481 | . . 3 ⊢ 𝑢 ∈ V | |
3 | vex 3481 | . . 3 ⊢ 𝑣 ∈ V | |
4 | 2, 3 | xpex 7771 | . 2 ⊢ (𝑢 × 𝑣) ∈ V |
5 | 1, 4 | fnmpoi 8093 | 1 ⊢ 𝑅 Fn (ran 𝐼 × ran 𝐼) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1536 × cxp 5686 ran crn 5689 Fn wfn 6557 ‘cfv 6562 (class class class)co 7430 ∈ cmpo 7432 1c1 11153 + caddc 11155 / cdiv 11917 2c2 12318 ℤcz 12610 (,)cioo 13383 [,)cico 13385 ↑cexp 14098 topGenctg 17483 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-fv 6570 df-oprab 7434 df-mpo 7435 df-1st 8012 df-2nd 8013 |
This theorem is referenced by: dya2iocuni 34264 |
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