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Theorem dya2iocival 34275
Description: The function 𝐼 returns closed-below open-above dyadic rational intervals covering the real line. This is the same construction as in dyadmbl 25635. (Contributed by Thierry Arnoux, 24-Sep-2017.)
Hypotheses
Ref Expression
sxbrsiga.0 𝐽 = (topGen‘ran (,))
dya2ioc.1 𝐼 = (𝑥 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))))
Assertion
Ref Expression
dya2iocival ((𝑁 ∈ ℤ ∧ 𝑋 ∈ ℤ) → (𝑋𝐼𝑁) = ((𝑋 / (2↑𝑁))[,)((𝑋 + 1) / (2↑𝑁))))
Distinct variable group:   𝑥,𝑛
Allowed substitution hints:   𝐼(𝑥,𝑛)   𝐽(𝑥,𝑛)   𝑁(𝑥,𝑛)   𝑋(𝑥,𝑛)

Proof of Theorem dya2iocival
Dummy variables 𝑚 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq1 7438 . . . 4 (𝑢 = 𝑋 → (𝑢 / (2↑𝑚)) = (𝑋 / (2↑𝑚)))
2 oveq1 7438 . . . . 5 (𝑢 = 𝑋 → (𝑢 + 1) = (𝑋 + 1))
32oveq1d 7446 . . . 4 (𝑢 = 𝑋 → ((𝑢 + 1) / (2↑𝑚)) = ((𝑋 + 1) / (2↑𝑚)))
41, 3oveq12d 7449 . . 3 (𝑢 = 𝑋 → ((𝑢 / (2↑𝑚))[,)((𝑢 + 1) / (2↑𝑚))) = ((𝑋 / (2↑𝑚))[,)((𝑋 + 1) / (2↑𝑚))))
5 oveq2 7439 . . . . 5 (𝑚 = 𝑁 → (2↑𝑚) = (2↑𝑁))
65oveq2d 7447 . . . 4 (𝑚 = 𝑁 → (𝑋 / (2↑𝑚)) = (𝑋 / (2↑𝑁)))
75oveq2d 7447 . . . 4 (𝑚 = 𝑁 → ((𝑋 + 1) / (2↑𝑚)) = ((𝑋 + 1) / (2↑𝑁)))
86, 7oveq12d 7449 . . 3 (𝑚 = 𝑁 → ((𝑋 / (2↑𝑚))[,)((𝑋 + 1) / (2↑𝑚))) = ((𝑋 / (2↑𝑁))[,)((𝑋 + 1) / (2↑𝑁))))
9 dya2ioc.1 . . . 4 𝐼 = (𝑥 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))))
10 oveq1 7438 . . . . . 6 (𝑢 = 𝑥 → (𝑢 / (2↑𝑚)) = (𝑥 / (2↑𝑚)))
11 oveq1 7438 . . . . . . 7 (𝑢 = 𝑥 → (𝑢 + 1) = (𝑥 + 1))
1211oveq1d 7446 . . . . . 6 (𝑢 = 𝑥 → ((𝑢 + 1) / (2↑𝑚)) = ((𝑥 + 1) / (2↑𝑚)))
1310, 12oveq12d 7449 . . . . 5 (𝑢 = 𝑥 → ((𝑢 / (2↑𝑚))[,)((𝑢 + 1) / (2↑𝑚))) = ((𝑥 / (2↑𝑚))[,)((𝑥 + 1) / (2↑𝑚))))
14 oveq2 7439 . . . . . . 7 (𝑚 = 𝑛 → (2↑𝑚) = (2↑𝑛))
1514oveq2d 7447 . . . . . 6 (𝑚 = 𝑛 → (𝑥 / (2↑𝑚)) = (𝑥 / (2↑𝑛)))
1614oveq2d 7447 . . . . . 6 (𝑚 = 𝑛 → ((𝑥 + 1) / (2↑𝑚)) = ((𝑥 + 1) / (2↑𝑛)))
1715, 16oveq12d 7449 . . . . 5 (𝑚 = 𝑛 → ((𝑥 / (2↑𝑚))[,)((𝑥 + 1) / (2↑𝑚))) = ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))))
1813, 17cbvmpov 7528 . . . 4 (𝑢 ∈ ℤ, 𝑚 ∈ ℤ ↦ ((𝑢 / (2↑𝑚))[,)((𝑢 + 1) / (2↑𝑚)))) = (𝑥 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))))
199, 18eqtr4i 2768 . . 3 𝐼 = (𝑢 ∈ ℤ, 𝑚 ∈ ℤ ↦ ((𝑢 / (2↑𝑚))[,)((𝑢 + 1) / (2↑𝑚))))
20 ovex 7464 . . 3 ((𝑋 / (2↑𝑁))[,)((𝑋 + 1) / (2↑𝑁))) ∈ V
214, 8, 19, 20ovmpo 7593 . 2 ((𝑋 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑋𝐼𝑁) = ((𝑋 / (2↑𝑁))[,)((𝑋 + 1) / (2↑𝑁))))
2221ancoms 458 1 ((𝑁 ∈ ℤ ∧ 𝑋 ∈ ℤ) → (𝑋𝐼𝑁) = ((𝑋 / (2↑𝑁))[,)((𝑋 + 1) / (2↑𝑁))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  ran crn 5686  cfv 6561  (class class class)co 7431  cmpo 7433  1c1 11156   + caddc 11158   / cdiv 11920  2c2 12321  cz 12613  (,)cioo 13387  [,)cico 13389  cexp 14102  topGenctg 17482
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436
This theorem is referenced by:  dya2iocress  34276  dya2iocbrsiga  34277  dya2icoseg  34279
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