Theorem dya2iocival 34784

Description: The function 𝐼 returns closed-below open-above dyadic rational intervals covering the real line. This is the same construction as in dyadmbl 25828. (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 7420	. . . 4 ⊢ (𝑢 = 𝑋 → (𝑢 / (2↑𝑚)) = (𝑋 / (2↑𝑚)))
2		oveq1 7420	. . . . 5 ⊢ (𝑢 = 𝑋 → (𝑢 + 1) = (𝑋 + 1))
3	2	oveq1d 7428	. . . 4 ⊢ (𝑢 = 𝑋 → ((𝑢 + 1) / (2↑𝑚)) = ((𝑋 + 1) / (2↑𝑚)))
4	1, 3	oveq12d 7431	. . 3 ⊢ (𝑢 = 𝑋 → ((𝑢 / (2↑𝑚))[,)((𝑢 + 1) / (2↑𝑚))) = ((𝑋 / (2↑𝑚))[,)((𝑋 + 1) / (2↑𝑚))))
5		oveq2 7421	. . . . 5 ⊢ (𝑚 = 𝑁 → (2↑𝑚) = (2↑𝑁))
6	5	oveq2d 7429	. . . 4 ⊢ (𝑚 = 𝑁 → (𝑋 / (2↑𝑚)) = (𝑋 / (2↑𝑁)))
7	5	oveq2d 7429	. . . 4 ⊢ (𝑚 = 𝑁 → ((𝑋 + 1) / (2↑𝑚)) = ((𝑋 + 1) / (2↑𝑁)))
8	6, 7	oveq12d 7431	. . 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 7420	. . . . . 6 ⊢ (𝑢 = 𝑥 → (𝑢 / (2↑𝑚)) = (𝑥 / (2↑𝑚)))
11		oveq1 7420	. . . . . . 7 ⊢ (𝑢 = 𝑥 → (𝑢 + 1) = (𝑥 + 1))
12	11	oveq1d 7428	. . . . . 6 ⊢ (𝑢 = 𝑥 → ((𝑢 + 1) / (2↑𝑚)) = ((𝑥 + 1) / (2↑𝑚)))
13	10, 12	oveq12d 7431	. . . . 5 ⊢ (𝑢 = 𝑥 → ((𝑢 / (2↑𝑚))[,)((𝑢 + 1) / (2↑𝑚))) = ((𝑥 / (2↑𝑚))[,)((𝑥 + 1) / (2↑𝑚))))
14		oveq2 7421	. . . . . . 7 ⊢ (𝑚 = 𝑛 → (2↑𝑚) = (2↑𝑛))
15	14	oveq2d 7429	. . . . . 6 ⊢ (𝑚 = 𝑛 → (𝑥 / (2↑𝑚)) = (𝑥 / (2↑𝑛)))
16	14	oveq2d 7429	. . . . . 6 ⊢ (𝑚 = 𝑛 → ((𝑥 + 1) / (2↑𝑚)) = ((𝑥 + 1) / (2↑𝑛)))
17	15, 16	oveq12d 7431	. . . . 5 ⊢ (𝑚 = 𝑛 → ((𝑥 / (2↑𝑚))[,)((𝑥 + 1) / (2↑𝑚))) = ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))))
18	13, 17	cbvmpov 7508	. . . 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 2786	. . 3 ⊢ 𝐼 = (𝑢 ∈ ℤ, 𝑚 ∈ ℤ ↦ ((𝑢 / (2↑𝑚))[,)((𝑢 + 1) / (2↑𝑚))))
20		ovex 7446	. . 3 ⊢ ((𝑋 / (2↑𝑁))[,)((𝑋 + 1) / (2↑𝑁))) ∈ V
21	4, 8, 19, 20	ovmpo 7573	. 2 ⊢ ((𝑋 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑋𝐼𝑁) = ((𝑋 / (2↑𝑁))[,)((𝑋 + 1) / (2↑𝑁))))
22	21	ancoms 464	1 ⊢ ((𝑁 ∈ ℤ ∧ 𝑋 ∈ ℤ) → (𝑋𝐼𝑁) = ((𝑋 / (2↑𝑁))[,)((𝑋 + 1) / (2↑𝑁))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ran crn 5656  ‘cfv 6533  (class class class)co 7413   ∈ cmpo 7415  1c1 11125   + caddc 11127   / cdiv 11895  2c2 12319  ℤcz 12615  (,)cioo 13398  [,)cico 13400  ↑cexp 14125  topGenctg 17522

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3740  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-iota 6489  df-fun 6535  df-fv 6541  df-ov 7416  df-oprab 7417  df-mpo 7418

This theorem is used by:  dya2iocress  34785  dya2iocbrsiga  34786  dya2icoseg  34788