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Theorem dya2iocival 30644
Description: The function 𝐼 returns closed-below open-above dyadic rational intervals covering the real line. This is the same construction as in dyadmbl 23568. (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 6820 . . . 4 (𝑢 = 𝑋 → (𝑢 / (2↑𝑚)) = (𝑋 / (2↑𝑚)))
2 oveq1 6820 . . . . 5 (𝑢 = 𝑋 → (𝑢 + 1) = (𝑋 + 1))
32oveq1d 6828 . . . 4 (𝑢 = 𝑋 → ((𝑢 + 1) / (2↑𝑚)) = ((𝑋 + 1) / (2↑𝑚)))
41, 3oveq12d 6831 . . 3 (𝑢 = 𝑋 → ((𝑢 / (2↑𝑚))[,)((𝑢 + 1) / (2↑𝑚))) = ((𝑋 / (2↑𝑚))[,)((𝑋 + 1) / (2↑𝑚))))
5 oveq2 6821 . . . . 5 (𝑚 = 𝑁 → (2↑𝑚) = (2↑𝑁))
65oveq2d 6829 . . . 4 (𝑚 = 𝑁 → (𝑋 / (2↑𝑚)) = (𝑋 / (2↑𝑁)))
75oveq2d 6829 . . . 4 (𝑚 = 𝑁 → ((𝑋 + 1) / (2↑𝑚)) = ((𝑋 + 1) / (2↑𝑁)))
86, 7oveq12d 6831 . . 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 6820 . . . . . 6 (𝑢 = 𝑥 → (𝑢 / (2↑𝑚)) = (𝑥 / (2↑𝑚)))
11 oveq1 6820 . . . . . . 7 (𝑢 = 𝑥 → (𝑢 + 1) = (𝑥 + 1))
1211oveq1d 6828 . . . . . 6 (𝑢 = 𝑥 → ((𝑢 + 1) / (2↑𝑚)) = ((𝑥 + 1) / (2↑𝑚)))
1310, 12oveq12d 6831 . . . . 5 (𝑢 = 𝑥 → ((𝑢 / (2↑𝑚))[,)((𝑢 + 1) / (2↑𝑚))) = ((𝑥 / (2↑𝑚))[,)((𝑥 + 1) / (2↑𝑚))))
14 oveq2 6821 . . . . . . 7 (𝑚 = 𝑛 → (2↑𝑚) = (2↑𝑛))
1514oveq2d 6829 . . . . . 6 (𝑚 = 𝑛 → (𝑥 / (2↑𝑚)) = (𝑥 / (2↑𝑛)))
1614oveq2d 6829 . . . . . 6 (𝑚 = 𝑛 → ((𝑥 + 1) / (2↑𝑚)) = ((𝑥 + 1) / (2↑𝑛)))
1715, 16oveq12d 6831 . . . . 5 (𝑚 = 𝑛 → ((𝑥 / (2↑𝑚))[,)((𝑥 + 1) / (2↑𝑚))) = ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))))
1813, 17cbvmpt2v 6900 . . . 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 2785 . . 3 𝐼 = (𝑢 ∈ ℤ, 𝑚 ∈ ℤ ↦ ((𝑢 / (2↑𝑚))[,)((𝑢 + 1) / (2↑𝑚))))
20 ovex 6841 . . 3 ((𝑋 / (2↑𝑁))[,)((𝑋 + 1) / (2↑𝑁))) ∈ V
214, 8, 19, 20ovmpt2 6961 . 2 ((𝑋 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑋𝐼𝑁) = ((𝑋 / (2↑𝑁))[,)((𝑋 + 1) / (2↑𝑁))))
2221ancoms 468 1 ((𝑁 ∈ ℤ ∧ 𝑋 ∈ ℤ) → (𝑋𝐼𝑁) = ((𝑋 / (2↑𝑁))[,)((𝑋 + 1) / (2↑𝑁))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  ran crn 5267  cfv 6049  (class class class)co 6813  cmpt2 6815  1c1 10129   + caddc 10131   / cdiv 10876  2c2 11262  cz 11569  (,)cioo 12368  [,)cico 12370  cexp 13054  topGenctg 16300
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-iota 6012  df-fun 6051  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818
This theorem is referenced by:  dya2iocress  30645  dya2iocbrsiga  30646  dya2icoseg  30648
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