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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 3470 | . . 3 ⊢ 𝑢 ∈ V | |
3 | vex 3470 | . . 3 ⊢ 𝑣 ∈ V | |
4 | 2, 3 | xpex 7714 | . 2 ⊢ (𝑢 × 𝑣) ∈ V |
5 | 1, 4 | fnmpoi 8029 | 1 ⊢ 𝑅 Fn (ran 𝐼 × ran 𝐼) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 × cxp 5658 ran crn 5661 Fn wfn 6518 ‘cfv 6523 (class class class)co 7384 ∈ cmpo 7386 1c1 11083 + caddc 11085 / cdiv 11843 2c2 12239 ℤcz 12530 (,)cioo 13296 [,)cico 13298 ↑cexp 13999 topGenctg 17355 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5283 ax-nul 5290 ax-pow 5347 ax-pr 5411 ax-un 7699 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ral 3061 df-rex 3070 df-rab 3426 df-v 3468 df-sbc 3765 df-csb 3881 df-dif 3938 df-un 3940 df-in 3942 df-ss 3952 df-nul 4310 df-if 4514 df-pw 4589 df-sn 4614 df-pr 4616 df-op 4620 df-uni 4893 df-iun 4983 df-br 5133 df-opab 5195 df-mpt 5216 df-id 5558 df-xp 5666 df-rel 5667 df-cnv 5668 df-co 5669 df-dm 5670 df-rn 5671 df-res 5672 df-ima 5673 df-iota 6475 df-fun 6525 df-fn 6526 df-f 6527 df-fv 6531 df-oprab 7388 df-mpo 7389 df-1st 7948 df-2nd 7949 |
This theorem is referenced by: dya2iocuni 33013 |
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