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 7476 | . 2 ⊢ (𝑢 × 𝑣) ∈ V |
5 | 1, 4 | fnmpoi 7768 | 1 ⊢ 𝑅 Fn (ran 𝐼 × ran 𝐼) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 × cxp 5553 ran crn 5556 Fn wfn 6350 ‘cfv 6355 (class class class)co 7156 ∈ cmpo 7158 1c1 10538 + caddc 10540 / cdiv 11297 2c2 11693 ℤcz 11982 (,)cioo 12739 [,)cico 12741 ↑cexp 13430 topGenctg 16711 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-fv 6363 df-oprab 7160 df-mpo 7161 df-1st 7689 df-2nd 7690 |
This theorem is referenced by: dya2iocuni 31541 |
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