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Theorem dya2iocival 33816
Description: The function 𝐼 returns closed-below open-above dyadic rational intervals covering the real line. This is the same construction as in dyadmbl 25503. (Contributed by Thierry Arnoux, 24-Sep-2017.)
Hypotheses
Ref Expression
sxbrsiga.0 𝐽 = (topGenβ€˜ran (,))
dya2ioc.1 𝐼 = (π‘₯ ∈ β„€, 𝑛 ∈ β„€ ↦ ((π‘₯ / (2↑𝑛))[,)((π‘₯ + 1) / (2↑𝑛))))
Assertion
Ref Expression
dya2iocival ((𝑁 ∈ β„€ ∧ 𝑋 ∈ β„€) β†’ (𝑋𝐼𝑁) = ((𝑋 / (2↑𝑁))[,)((𝑋 + 1) / (2↑𝑁))))
Distinct variable group:   π‘₯,𝑛
Allowed substitution hints:   𝐼(π‘₯,𝑛)   𝐽(π‘₯,𝑛)   𝑁(π‘₯,𝑛)   𝑋(π‘₯,𝑛)

Proof of Theorem dya2iocival
Dummy variables π‘š 𝑒 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq1 7421 . . . 4 (𝑒 = 𝑋 β†’ (𝑒 / (2β†‘π‘š)) = (𝑋 / (2β†‘π‘š)))
2 oveq1 7421 . . . . 5 (𝑒 = 𝑋 β†’ (𝑒 + 1) = (𝑋 + 1))
32oveq1d 7429 . . . 4 (𝑒 = 𝑋 β†’ ((𝑒 + 1) / (2β†‘π‘š)) = ((𝑋 + 1) / (2β†‘π‘š)))
41, 3oveq12d 7432 . . 3 (𝑒 = 𝑋 β†’ ((𝑒 / (2β†‘π‘š))[,)((𝑒 + 1) / (2β†‘π‘š))) = ((𝑋 / (2β†‘π‘š))[,)((𝑋 + 1) / (2β†‘π‘š))))
5 oveq2 7422 . . . . 5 (π‘š = 𝑁 β†’ (2β†‘π‘š) = (2↑𝑁))
65oveq2d 7430 . . . 4 (π‘š = 𝑁 β†’ (𝑋 / (2β†‘π‘š)) = (𝑋 / (2↑𝑁)))
75oveq2d 7430 . . . 4 (π‘š = 𝑁 β†’ ((𝑋 + 1) / (2β†‘π‘š)) = ((𝑋 + 1) / (2↑𝑁)))
86, 7oveq12d 7432 . . 3 (π‘š = 𝑁 β†’ ((𝑋 / (2β†‘π‘š))[,)((𝑋 + 1) / (2β†‘π‘š))) = ((𝑋 / (2↑𝑁))[,)((𝑋 + 1) / (2↑𝑁))))
9 dya2ioc.1 . . . 4 𝐼 = (π‘₯ ∈ β„€, 𝑛 ∈ β„€ ↦ ((π‘₯ / (2↑𝑛))[,)((π‘₯ + 1) / (2↑𝑛))))
10 oveq1 7421 . . . . . 6 (𝑒 = π‘₯ β†’ (𝑒 / (2β†‘π‘š)) = (π‘₯ / (2β†‘π‘š)))
11 oveq1 7421 . . . . . . 7 (𝑒 = π‘₯ β†’ (𝑒 + 1) = (π‘₯ + 1))
1211oveq1d 7429 . . . . . 6 (𝑒 = π‘₯ β†’ ((𝑒 + 1) / (2β†‘π‘š)) = ((π‘₯ + 1) / (2β†‘π‘š)))
1310, 12oveq12d 7432 . . . . 5 (𝑒 = π‘₯ β†’ ((𝑒 / (2β†‘π‘š))[,)((𝑒 + 1) / (2β†‘π‘š))) = ((π‘₯ / (2β†‘π‘š))[,)((π‘₯ + 1) / (2β†‘π‘š))))
14 oveq2 7422 . . . . . . 7 (π‘š = 𝑛 β†’ (2β†‘π‘š) = (2↑𝑛))
1514oveq2d 7430 . . . . . 6 (π‘š = 𝑛 β†’ (π‘₯ / (2β†‘π‘š)) = (π‘₯ / (2↑𝑛)))
1614oveq2d 7430 . . . . . 6 (π‘š = 𝑛 β†’ ((π‘₯ + 1) / (2β†‘π‘š)) = ((π‘₯ + 1) / (2↑𝑛)))
1715, 16oveq12d 7432 . . . . 5 (π‘š = 𝑛 β†’ ((π‘₯ / (2β†‘π‘š))[,)((π‘₯ + 1) / (2β†‘π‘š))) = ((π‘₯ / (2↑𝑛))[,)((π‘₯ + 1) / (2↑𝑛))))
1813, 17cbvmpov 7509 . . . 4 (𝑒 ∈ β„€, π‘š ∈ β„€ ↦ ((𝑒 / (2β†‘π‘š))[,)((𝑒 + 1) / (2β†‘π‘š)))) = (π‘₯ ∈ β„€, 𝑛 ∈ β„€ ↦ ((π‘₯ / (2↑𝑛))[,)((π‘₯ + 1) / (2↑𝑛))))
199, 18eqtr4i 2758 . . 3 𝐼 = (𝑒 ∈ β„€, π‘š ∈ β„€ ↦ ((𝑒 / (2β†‘π‘š))[,)((𝑒 + 1) / (2β†‘π‘š))))
20 ovex 7447 . . 3 ((𝑋 / (2↑𝑁))[,)((𝑋 + 1) / (2↑𝑁))) ∈ V
214, 8, 19, 20ovmpo 7573 . 2 ((𝑋 ∈ β„€ ∧ 𝑁 ∈ β„€) β†’ (𝑋𝐼𝑁) = ((𝑋 / (2↑𝑁))[,)((𝑋 + 1) / (2↑𝑁))))
2221ancoms 458 1 ((𝑁 ∈ β„€ ∧ 𝑋 ∈ β„€) β†’ (𝑋𝐼𝑁) = ((𝑋 / (2↑𝑁))[,)((𝑋 + 1) / (2↑𝑁))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1534   ∈ wcel 2099  ran crn 5673  β€˜cfv 6542  (class class class)co 7414   ∈ cmpo 7416  1c1 11125   + caddc 11127   / cdiv 11887  2c2 12283  β„€cz 12574  (,)cioo 13342  [,)cico 13344  β†‘cexp 14044  topGenctg 17404
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-pr 5423
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-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-sbc 3775  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-iota 6494  df-fun 6544  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419
This theorem is referenced by:  dya2iocress  33817  dya2iocbrsiga  33818  dya2icoseg  33820
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