dya2icoseg2 - Mathbox for Thierry Arnoux

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Theorem dya2icoseg2 34265

Description: For any point and any open interval of ℝ containing that point, there is a closed-below open-above dyadic rational interval which contains that point and is included in the original interval. (Contributed by Thierry Arnoux, 12-Oct-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
dya2icoseg2	⊢ ((𝑋 ∈ ℝ ∧ 𝐸 ∈ ran (,) ∧ 𝑋 ∈ 𝐸) → ∃𝑏 ∈ ran 𝐼(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ 𝐸))

Distinct variable groups: 𝑥,𝑛 𝑥,𝐼 𝑛,𝑏,𝑥 𝐸,𝑏,𝑥 𝐼,𝑏 𝑋,𝑏,𝑥 Allowed substitution hints: 𝐸(𝑛) 𝐼(𝑛) 𝐽(𝑥,𝑛,𝑏) 𝑋(𝑛)

Proof of Theorem dya2icoseg2 Dummy variable 𝑑 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sxbrsiga.0	. . . . . 6 ⊢ 𝐽 = (topGen‘ran (,))
2		dya2ioc.1	. . . . . 6 ⊢ 𝐼 = (𝑥 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))))
3		eqid 2729	. . . . . 6 ⊢ (⌊‘(1 − (2 log_b 𝑑))) = (⌊‘(1 − (2 log_b 𝑑)))
4	1, 2, 3	dya2icoseg 34264	. . . . 5 ⊢ ((𝑋 ∈ ℝ ∧ 𝑑 ∈ ℝ⁺) → ∃𝑏 ∈ ran 𝐼(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑))))
5	4	ralrimiva 3121	. . . 4 ⊢ (𝑋 ∈ ℝ → ∀𝑑 ∈ ℝ⁺ ∃𝑏 ∈ ran 𝐼(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑))))
6	5	3ad2ant1 1133	. . 3 ⊢ ((𝑋 ∈ ℝ ∧ 𝐸 ∈ ran (,) ∧ 𝑋 ∈ 𝐸) → ∀𝑑 ∈ ℝ⁺ ∃𝑏 ∈ ran 𝐼(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑))))
7		simp3 1138	. . . . 5 ⊢ ((𝑋 ∈ ℝ ∧ 𝐸 ∈ ran (,) ∧ 𝑋 ∈ 𝐸) → 𝑋 ∈ 𝐸)
8		iooex 13290	. . . . . . . . . 10 ⊢ (,) ∈ V
9	8	rnex 7850	. . . . . . . . 9 ⊢ ran (,) ∈ V
10		bastg 22870	. . . . . . . . 9 ⊢ (ran (,) ∈ V → ran (,) ⊆ (topGen‘ran (,)))
11	9, 10	ax-mp 5	. . . . . . . 8 ⊢ ran (,) ⊆ (topGen‘ran (,))
12		simp2 1137	. . . . . . . 8 ⊢ ((𝑋 ∈ ℝ ∧ 𝐸 ∈ ran (,) ∧ 𝑋 ∈ 𝐸) → 𝐸 ∈ ran (,))
13	11, 12	sselid 3935	. . . . . . 7 ⊢ ((𝑋 ∈ ℝ ∧ 𝐸 ∈ ran (,) ∧ 𝑋 ∈ 𝐸) → 𝐸 ∈ (topGen‘ran (,)))
14	13, 1	eleqtrrdi 2839	. . . . . 6 ⊢ ((𝑋 ∈ ℝ ∧ 𝐸 ∈ ran (,) ∧ 𝑋 ∈ 𝐸) → 𝐸 ∈ 𝐽)
15		eqid 2729	. . . . . . . . 9 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) = ((abs ∘ − ) ↾ (ℝ × ℝ))
16	15	rexmet 24696	. . . . . . . 8 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) ∈ (∞Met‘ℝ)
17		recms 25297	. . . . . . . . . . 11 ⊢ ℝ_fld ∈ CMetSp
18		cmsms 25265	. . . . . . . . . . 11 ⊢ (ℝ_fld ∈ CMetSp → ℝ_fld ∈ MetSp)
19		msxms 24359	. . . . . . . . . . 11 ⊢ (ℝ_fld ∈ MetSp → ℝ_fld ∈ ∞MetSp)
20	17, 18, 19	mp2b 10	. . . . . . . . . 10 ⊢ ℝ_fld ∈ ∞MetSp
21		retopn 25296	. . . . . . . . . . . 12 ⊢ (topGen‘ran (,)) = (TopOpen‘ℝ_fld)
22	1, 21	eqtri 2752	. . . . . . . . . . 11 ⊢ 𝐽 = (TopOpen‘ℝ_fld)
23		rebase 21532	. . . . . . . . . . 11 ⊢ ℝ = (Base‘ℝ_fld)
24		reds 21542	. . . . . . . . . . . 12 ⊢ (abs ∘ − ) = (dist‘ℝ_fld)
25	24	reseq1i 5930	. . . . . . . . . . 11 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) = ((dist‘ℝ_fld) ↾ (ℝ × ℝ))
26	22, 23, 25	xmstopn 24356	. . . . . . . . . 10 ⊢ (ℝ_fld ∈ ∞MetSp → 𝐽 = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ))))
27	20, 26	ax-mp 5	. . . . . . . . 9 ⊢ 𝐽 = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ)))
28	27	elmopn2 24350	. . . . . . . 8 ⊢ (((abs ∘ − ) ↾ (ℝ × ℝ)) ∈ (∞Met‘ℝ) → (𝐸 ∈ 𝐽 ↔ (𝐸 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐸 ∃𝑑 ∈ ℝ⁺ (𝑥(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑑) ⊆ 𝐸)))
29	16, 28	ax-mp 5	. . . . . . 7 ⊢ (𝐸 ∈ 𝐽 ↔ (𝐸 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐸 ∃𝑑 ∈ ℝ⁺ (𝑥(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑑) ⊆ 𝐸))
30	29	simprbi 496	. . . . . 6 ⊢ (𝐸 ∈ 𝐽 → ∀𝑥 ∈ 𝐸 ∃𝑑 ∈ ℝ⁺ (𝑥(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑑) ⊆ 𝐸)
31	14, 30	syl 17	. . . . 5 ⊢ ((𝑋 ∈ ℝ ∧ 𝐸 ∈ ran (,) ∧ 𝑋 ∈ 𝐸) → ∀𝑥 ∈ 𝐸 ∃𝑑 ∈ ℝ⁺ (𝑥(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑑) ⊆ 𝐸)
32		oveq1 7360	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑥(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑑) = (𝑋(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑑))
33	32	sseq1d 3969	. . . . . . 7 ⊢ (𝑥 = 𝑋 → ((𝑥(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑑) ⊆ 𝐸 ↔ (𝑋(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑑) ⊆ 𝐸))
34	33	rexbidv 3153	. . . . . 6 ⊢ (𝑥 = 𝑋 → (∃𝑑 ∈ ℝ⁺ (𝑥(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑑) ⊆ 𝐸 ↔ ∃𝑑 ∈ ℝ⁺ (𝑋(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑑) ⊆ 𝐸))
35	34	rspcva 3577	. . . . 5 ⊢ ((𝑋 ∈ 𝐸 ∧ ∀𝑥 ∈ 𝐸 ∃𝑑 ∈ ℝ⁺ (𝑥(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑑) ⊆ 𝐸) → ∃𝑑 ∈ ℝ⁺ (𝑋(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑑) ⊆ 𝐸)
36	7, 31, 35	syl2anc 584	. . . 4 ⊢ ((𝑋 ∈ ℝ ∧ 𝐸 ∈ ran (,) ∧ 𝑋 ∈ 𝐸) → ∃𝑑 ∈ ℝ⁺ (𝑋(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑑) ⊆ 𝐸)
37		rpre 12921	. . . . . . 7 ⊢ (𝑑 ∈ ℝ⁺ → 𝑑 ∈ ℝ)
38	15	bl2ioo 24697	. . . . . . . 8 ⊢ ((𝑋 ∈ ℝ ∧ 𝑑 ∈ ℝ) → (𝑋(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑑) = ((𝑋 − 𝑑)(,)(𝑋 + 𝑑)))
39	38	sseq1d 3969	. . . . . . 7 ⊢ ((𝑋 ∈ ℝ ∧ 𝑑 ∈ ℝ) → ((𝑋(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑑) ⊆ 𝐸 ↔ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑)) ⊆ 𝐸))
40	37, 39	sylan2 593	. . . . . 6 ⊢ ((𝑋 ∈ ℝ ∧ 𝑑 ∈ ℝ⁺) → ((𝑋(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑑) ⊆ 𝐸 ↔ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑)) ⊆ 𝐸))
41	40	rexbidva 3151	. . . . 5 ⊢ (𝑋 ∈ ℝ → (∃𝑑 ∈ ℝ⁺ (𝑋(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑑) ⊆ 𝐸 ↔ ∃𝑑 ∈ ℝ⁺ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑)) ⊆ 𝐸))
42	41	3ad2ant1 1133	. . . 4 ⊢ ((𝑋 ∈ ℝ ∧ 𝐸 ∈ ran (,) ∧ 𝑋 ∈ 𝐸) → (∃𝑑 ∈ ℝ⁺ (𝑋(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑑) ⊆ 𝐸 ↔ ∃𝑑 ∈ ℝ⁺ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑)) ⊆ 𝐸))
43	36, 42	mpbid 232	. . 3 ⊢ ((𝑋 ∈ ℝ ∧ 𝐸 ∈ ran (,) ∧ 𝑋 ∈ 𝐸) → ∃𝑑 ∈ ℝ⁺ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑)) ⊆ 𝐸)
44		r19.29 3092	. . 3 ⊢ ((∀𝑑 ∈ ℝ⁺ ∃𝑏 ∈ ran 𝐼(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑))) ∧ ∃𝑑 ∈ ℝ⁺ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑)) ⊆ 𝐸) → ∃𝑑 ∈ ℝ⁺ (∃𝑏 ∈ ran 𝐼(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑))) ∧ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑)) ⊆ 𝐸))
45	6, 43, 44	syl2anc 584	. 2 ⊢ ((𝑋 ∈ ℝ ∧ 𝐸 ∈ ran (,) ∧ 𝑋 ∈ 𝐸) → ∃𝑑 ∈ ℝ⁺ (∃𝑏 ∈ ran 𝐼(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑))) ∧ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑)) ⊆ 𝐸))
46		r19.41v 3159	. . . 4 ⊢ (∃𝑏 ∈ ran 𝐼((𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑))) ∧ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑)) ⊆ 𝐸) ↔ (∃𝑏 ∈ ran 𝐼(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑))) ∧ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑)) ⊆ 𝐸))
47		sstr 3946	. . . . . . 7 ⊢ ((𝑏 ⊆ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑)) ∧ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑)) ⊆ 𝐸) → 𝑏 ⊆ 𝐸)
48	47	anim2i 617	. . . . . 6 ⊢ ((𝑋 ∈ 𝑏 ∧ (𝑏 ⊆ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑)) ∧ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑)) ⊆ 𝐸)) → (𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ 𝐸))
49	48	anassrs 467	. . . . 5 ⊢ (((𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑))) ∧ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑)) ⊆ 𝐸) → (𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ 𝐸))
50	49	reximi 3067	. . . 4 ⊢ (∃𝑏 ∈ ran 𝐼((𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑))) ∧ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑)) ⊆ 𝐸) → ∃𝑏 ∈ ran 𝐼(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ 𝐸))
51	46, 50	sylbir 235	. . 3 ⊢ ((∃𝑏 ∈ ran 𝐼(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑))) ∧ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑)) ⊆ 𝐸) → ∃𝑏 ∈ ran 𝐼(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ 𝐸))
52	51	rexlimivw 3126	. 2 ⊢ (∃𝑑 ∈ ℝ⁺ (∃𝑏 ∈ ran 𝐼(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑))) ∧ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑)) ⊆ 𝐸) → ∃𝑏 ∈ ran 𝐼(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ 𝐸))
53	45, 52	syl 17	1 ⊢ ((𝑋 ∈ ℝ ∧ 𝐸 ∈ ran (,) ∧ 𝑋 ∈ 𝐸) → ∃𝑏 ∈ ran 𝐼(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ 𝐸))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ∀wral 3044 ∃wrex 3053 Vcvv 3438 ⊆ wss 3905 × cxp 5621 ran crn 5624 ↾ cres 5625 ∘ ccom 5627 ‘cfv 6486 (class class class)co 7353 ∈ cmpo 7355 ℝcr 11027 1c1 11029 + caddc 11031 − cmin 11366 / cdiv 11796 2c2 12202 ℤcz 12490 ℝ⁺crp 12912 (,)cioo 13267 [,)cico 13269 ⌊cfl 13713 ↑cexp 13987 abscabs 15160 distcds 17189 TopOpenctopn 17344 topGenctg 17360 ∞Metcxmet 21265 ballcbl 21267 MetOpencmopn 21270 ℝ_fldcrefld 21530 ∞MetSpcxms 24222 MetSpcms 24223 CMetSpccms 25249 log_b clogb 26691

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675 ax-inf2 9556 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105 ax-pre-sup 11106 ax-addf 11107

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3345 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-uni 4862 df-int 4900 df-iun 4946 df-iin 4947 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-se 5577 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-isom 6495 df-riota 7310 df-ov 7356 df-oprab 7357 df-mpo 7358 df-of 7617 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8101 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-er 8632 df-map 8762 df-pm 8763 df-ixp 8832 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-fsupp 9271 df-fi 9320 df-sup 9351 df-inf 9352 df-oi 9421 df-card 9854 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-div 11797 df-nn 12148 df-2 12210 df-3 12211 df-4 12212 df-5 12213 df-6 12214 df-7 12215 df-8 12216 df-9 12217 df-n0 12404 df-z 12491 df-dec 12611 df-uz 12755 df-q 12869 df-rp 12913 df-xneg 13033 df-xadd 13034 df-xmul 13035 df-ioo 13271 df-ioc 13272 df-ico 13273 df-icc 13274 df-fz 13430 df-fzo 13577 df-fl 13715 df-mod 13793 df-seq 13928 df-exp 13988 df-fac 14200 df-bc 14229 df-hash 14257 df-shft 14993 df-cj 15025 df-re 15026 df-im 15027 df-sqrt 15161 df-abs 15162 df-limsup 15397 df-clim 15414 df-rlim 15415 df-sum 15613 df-ef 15993 df-sin 15995 df-cos 15996 df-pi 15998 df-struct 17077 df-sets 17094 df-slot 17112 df-ndx 17124 df-base 17140 df-ress 17161 df-plusg 17193 df-mulr 17194 df-starv 17195 df-sca 17196 df-vsca 17197 df-ip 17198 df-tset 17199 df-ple 17200 df-ds 17202 df-unif 17203 df-hom 17204 df-cco 17205 df-rest 17345 df-topn 17346 df-0g 17364 df-gsum 17365 df-topgen 17366 df-pt 17367 df-prds 17370 df-xrs 17425 df-qtop 17430 df-imas 17431 df-xps 17433 df-mre 17507 df-mrc 17508 df-acs 17510 df-mgm 18533 df-sgrp 18612 df-mnd 18628 df-submnd 18677 df-mulg 18966 df-cntz 19215 df-cmn 19680 df-psmet 21272 df-xmet 21273 df-met 21274 df-bl 21275 df-mopn 21276 df-fbas 21277 df-fg 21278 df-cnfld 21281 df-refld 21531 df-top 22798 df-topon 22815 df-topsp 22837 df-bases 22850 df-cld 22923 df-ntr 22924 df-cls 22925 df-nei 23002 df-lp 23040 df-perf 23041 df-cn 23131 df-cnp 23132 df-haus 23219 df-cmp 23291 df-tx 23466 df-hmeo 23659 df-fil 23750 df-fm 23842 df-flim 23843 df-flf 23844 df-fcls 23845 df-xms 24225 df-ms 24226 df-tms 24227 df-cncf 24788 df-cfil 25172 df-cmet 25174 df-cms 25252 df-limc 25784 df-dv 25785 df-log 26482 df-cxp 26483 df-logb 26692

This theorem is referenced by: dya2iocnrect 34268

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