Theorem dya2iocival 34238

Description: The function 𝐼 returns closed-below open-above dyadic rational intervals covering the real line. This is the same construction as in dyadmbl 25654. (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.

Step	Hyp	Ref	Expression
1		oveq1 7455	. . . 4 ⊢ (𝑢 = 𝑋 → (𝑢 / (2↑𝑚)) = (𝑋 / (2↑𝑚)))
2		oveq1 7455	. . . . 5 ⊢ (𝑢 = 𝑋 → (𝑢 + 1) = (𝑋 + 1))
3	2	oveq1d 7463	. . . 4 ⊢ (𝑢 = 𝑋 → ((𝑢 + 1) / (2↑𝑚)) = ((𝑋 + 1) / (2↑𝑚)))
4	1, 3	oveq12d 7466	. . 3 ⊢ (𝑢 = 𝑋 → ((𝑢 / (2↑𝑚))[,)((𝑢 + 1) / (2↑𝑚))) = ((𝑋 / (2↑𝑚))[,)((𝑋 + 1) / (2↑𝑚))))
5		oveq2 7456	. . . . 5 ⊢ (𝑚 = 𝑁 → (2↑𝑚) = (2↑𝑁))
6	5	oveq2d 7464	. . . 4 ⊢ (𝑚 = 𝑁 → (𝑋 / (2↑𝑚)) = (𝑋 / (2↑𝑁)))
7	5	oveq2d 7464	. . . 4 ⊢ (𝑚 = 𝑁 → ((𝑋 + 1) / (2↑𝑚)) = ((𝑋 + 1) / (2↑𝑁)))
8	6, 7	oveq12d 7466	. . 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 7455	. . . . . 6 ⊢ (𝑢 = 𝑥 → (𝑢 / (2↑𝑚)) = (𝑥 / (2↑𝑚)))
11		oveq1 7455	. . . . . . 7 ⊢ (𝑢 = 𝑥 → (𝑢 + 1) = (𝑥 + 1))
12	11	oveq1d 7463	. . . . . 6 ⊢ (𝑢 = 𝑥 → ((𝑢 + 1) / (2↑𝑚)) = ((𝑥 + 1) / (2↑𝑚)))
13	10, 12	oveq12d 7466	. . . . 5 ⊢ (𝑢 = 𝑥 → ((𝑢 / (2↑𝑚))[,)((𝑢 + 1) / (2↑𝑚))) = ((𝑥 / (2↑𝑚))[,)((𝑥 + 1) / (2↑𝑚))))
14		oveq2 7456	. . . . . . 7 ⊢ (𝑚 = 𝑛 → (2↑𝑚) = (2↑𝑛))
15	14	oveq2d 7464	. . . . . 6 ⊢ (𝑚 = 𝑛 → (𝑥 / (2↑𝑚)) = (𝑥 / (2↑𝑛)))
16	14	oveq2d 7464	. . . . . 6 ⊢ (𝑚 = 𝑛 → ((𝑥 + 1) / (2↑𝑚)) = ((𝑥 + 1) / (2↑𝑛)))
17	15, 16	oveq12d 7466	. . . . 5 ⊢ (𝑚 = 𝑛 → ((𝑥 / (2↑𝑚))[,)((𝑥 + 1) / (2↑𝑚))) = ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))))
18	13, 17	cbvmpov 7545	. . . 4 ⊢ (𝑢 ∈ ℤ, 𝑚 ∈ ℤ ↦ ((𝑢 / (2↑𝑚))[,)((𝑢 + 1) / (2↑𝑚)))) = (𝑥 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))))
19	9, 18	eqtr4i 2771	. . 3 ⊢ 𝐼 = (𝑢 ∈ ℤ, 𝑚 ∈ ℤ ↦ ((𝑢 / (2↑𝑚))[,)((𝑢 + 1) / (2↑𝑚))))
20		ovex 7481	. . 3 ⊢ ((𝑋 / (2↑𝑁))[,)((𝑋 + 1) / (2↑𝑁))) ∈ V
21	4, 8, 19, 20	ovmpo 7610	. 2 ⊢ ((𝑋 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑋𝐼𝑁) = ((𝑋 / (2↑𝑁))[,)((𝑋 + 1) / (2↑𝑁))))
22	21	ancoms 458	1 ⊢ ((𝑁 ∈ ℤ ∧ 𝑋 ∈ ℤ) → (𝑋𝐼𝑁) = ((𝑋 / (2↑𝑁))[,)((𝑋 + 1) / (2↑𝑁))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ran crn 5701  ‘cfv 6573  (class class class)co 7448   ∈ cmpo 7450  1c1 11185   + caddc 11187   / cdiv 11947  2c2 12348  ℤcz 12639  (,)cioo 13407  [,)cico 13409  ↑cexp 14112  topGenctg 17497

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-iota 6525  df-fun 6575  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453

This theorem is referenced by:  dya2iocress  34239  dya2iocbrsiga  34240  dya2icoseg  34242