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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 3473 | . . 3 ⊢ 𝑢 ∈ V | |
3 | vex 3473 | . . 3 ⊢ 𝑣 ∈ V | |
4 | 2, 3 | xpex 7749 | . 2 ⊢ (𝑢 × 𝑣) ∈ V |
5 | 1, 4 | fnmpoi 8068 | 1 ⊢ 𝑅 Fn (ran 𝐼 × ran 𝐼) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 × cxp 5670 ran crn 5673 Fn wfn 6537 ‘cfv 6542 (class class class)co 7414 ∈ cmpo 7416 1c1 11133 + caddc 11135 / cdiv 11895 2c2 12291 ℤcz 12582 (,)cioo 13350 [,)cico 13352 ↑cexp 14052 topGenctg 17412 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-fv 6550 df-oprab 7418 df-mpo 7419 df-1st 7987 df-2nd 7988 |
This theorem is referenced by: dya2iocuni 33893 |
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