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 3497 | . . 3 ⊢ 𝑢 ∈ V | |
3 | vex 3497 | . . 3 ⊢ 𝑣 ∈ V | |
4 | 2, 3 | xpex 7475 | . 2 ⊢ (𝑢 × 𝑣) ∈ V |
5 | 1, 4 | fnmpoi 7767 | 1 ⊢ 𝑅 Fn (ran 𝐼 × ran 𝐼) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 × cxp 5552 ran crn 5555 Fn wfn 6349 ‘cfv 6354 (class class class)co 7155 ∈ cmpo 7157 1c1 10537 + caddc 10539 / cdiv 11296 2c2 11691 ℤcz 11980 (,)cioo 12737 [,)cico 12739 ↑cexp 13428 topGenctg 16710 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-fv 6362 df-oprab 7159 df-mpo 7160 df-1st 7688 df-2nd 7689 |
This theorem is referenced by: dya2iocuni 31541 |
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