lptioo2 - Mathbox for Glauco Siliprandi

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Theorem lptioo2 38698

Description: The upper bound of an open interval is a limit point of the interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Hypotheses**
Ref	Expression
lptioo2.1	⊢ 𝐽 = (topGen‘ran (,))
lptioo2.2	⊢ (𝜑 → 𝐴 ∈ ℝ^*)
lptioo2.3	⊢ (𝜑 → 𝐵 ∈ ℝ)
lptioo2.4	⊢ (𝜑 → 𝐴 < 𝐵)

**Assertion**
Ref	Expression
lptioo2	⊢ (𝜑 → 𝐵 ∈ ((limPt‘𝐽)‘(𝐴(,)𝐵)))

Proof of Theorem lptioo2 Dummy variables 𝑎 𝑏 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		difssd 3700	. . . . . . . 8 ⊢ (𝜑 → ((𝐴(,)𝐵) ∖ {𝐵}) ⊆ (𝐴(,)𝐵))
2		simpr 476	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ (𝐴(,)𝐵))
3		ubioo 12078	. . . . . . . . . . . . . 14 ⊢ ¬ 𝐵 ∈ (𝐴(,)𝐵)
4		eleq1 2676	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ (𝐴(,)𝐵) ↔ 𝐵 ∈ (𝐴(,)𝐵)))
5	4	biimpcd 238	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (𝐴(,)𝐵) → (𝑥 = 𝐵 → 𝐵 ∈ (𝐴(,)𝐵)))
6	3, 5	mtoi 189	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝐴(,)𝐵) → ¬ 𝑥 = 𝐵)
7	6	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ¬ 𝑥 = 𝐵)
8		velsn 4141	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ {𝐵} ↔ 𝑥 = 𝐵)
9	7, 8	sylnibr 318	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ¬ 𝑥 ∈ {𝐵})
10	2, 9	eldifd 3551	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ ((𝐴(,)𝐵) ∖ {𝐵}))
11	10	ex 449	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) → 𝑥 ∈ ((𝐴(,)𝐵) ∖ {𝐵})))
12	11	ssrdv 3574	. . . . . . . 8 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ((𝐴(,)𝐵) ∖ {𝐵}))
13	1, 12	eqssd 3585	. . . . . . 7 ⊢ (𝜑 → ((𝐴(,)𝐵) ∖ {𝐵}) = (𝐴(,)𝐵))
14	13	ineq2d 3776	. . . . . 6 ⊢ (𝜑 → ((𝑎(,)𝑏) ∩ ((𝐴(,)𝐵) ∖ {𝐵})) = ((𝑎(,)𝑏) ∩ (𝐴(,)𝐵)))
15	14	ad2antrr 758	. . . . 5 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^*)) ∧ 𝐵 ∈ (𝑎(,)𝑏)) → ((𝑎(,)𝑏) ∩ ((𝐴(,)𝐵) ∖ {𝐵})) = ((𝑎(,)𝑏) ∩ (𝐴(,)𝐵)))
16		simplrl 796	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^)) ∧ 𝐵 ∈ (𝑎(,)𝑏)) → 𝑎 ∈ ℝ^)
17		simplrr 797	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^)) ∧ 𝐵 ∈ (𝑎(,)𝑏)) → 𝑏 ∈ ℝ^)
18		lptioo2.2	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
19	18	ad2antrr 758	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^)) ∧ 𝐵 ∈ (𝑎(,)𝑏)) → 𝐴 ∈ ℝ^)
20		elioo3g 12075	. . . . . . . . . . 11 ⊢ (𝐵 ∈ (𝑎(,)𝑏) ↔ ((𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝑎 < 𝐵 ∧ 𝐵 < 𝑏)))
21	20	biimpi 205	. . . . . . . . . 10 ⊢ (𝐵 ∈ (𝑎(,)𝑏) → ((𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝑎 < 𝐵 ∧ 𝐵 < 𝑏)))
22	21	simpld 474	. . . . . . . . 9 ⊢ (𝐵 ∈ (𝑎(,)𝑏) → (𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*))
23	22	simp3d 1068	. . . . . . . 8 ⊢ (𝐵 ∈ (𝑎(,)𝑏) → 𝐵 ∈ ℝ^*)
24	23	adantl 481	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^)) ∧ 𝐵 ∈ (𝑎(,)𝑏)) → 𝐵 ∈ ℝ^)
25		iooin 12080	. . . . . . 7 ⊢ (((𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^) ∧ (𝐴 ∈ ℝ^ ∧ 𝐵 ∈ ℝ^*)) → ((𝑎(,)𝑏) ∩ (𝐴(,)𝐵)) = (if(𝑎 ≤ 𝐴, 𝐴, 𝑎)(,)if(𝑏 ≤ 𝐵, 𝑏, 𝐵)))
26	16, 17, 19, 24, 25	syl22anc 1319	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^*)) ∧ 𝐵 ∈ (𝑎(,)𝑏)) → ((𝑎(,)𝑏) ∩ (𝐴(,)𝐵)) = (if(𝑎 ≤ 𝐴, 𝐴, 𝑎)(,)if(𝑏 ≤ 𝐵, 𝑏, 𝐵)))
27		iftrue 4042	. . . . . . . . . . 11 ⊢ (𝑎 ≤ 𝐴 → if(𝑎 ≤ 𝐴, 𝐴, 𝑎) = 𝐴)
28	27	adantl 481	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^*)) ∧ 𝐵 ∈ (𝑎(,)𝑏)) ∧ 𝑎 ≤ 𝐴) → if(𝑎 ≤ 𝐴, 𝐴, 𝑎) = 𝐴)
29		lptioo2.4	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 < 𝐵)
30	29	ad3antrrr 762	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^*)) ∧ 𝐵 ∈ (𝑎(,)𝑏)) ∧ 𝑎 ≤ 𝐴) → 𝐴 < 𝐵)
31	28, 30	eqbrtrd 4605	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^*)) ∧ 𝐵 ∈ (𝑎(,)𝑏)) ∧ 𝑎 ≤ 𝐴) → if(𝑎 ≤ 𝐴, 𝐴, 𝑎) < 𝐵)
32		iffalse 4045	. . . . . . . . . . 11 ⊢ (¬ 𝑎 ≤ 𝐴 → if(𝑎 ≤ 𝐴, 𝐴, 𝑎) = 𝑎)
33	32	adantl 481	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^*)) ∧ 𝐵 ∈ (𝑎(,)𝑏)) ∧ ¬ 𝑎 ≤ 𝐴) → if(𝑎 ≤ 𝐴, 𝐴, 𝑎) = 𝑎)
34	21	simprd 478	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ (𝑎(,)𝑏) → (𝑎 < 𝐵 ∧ 𝐵 < 𝑏))
35	34	simpld 474	. . . . . . . . . . 11 ⊢ (𝐵 ∈ (𝑎(,)𝑏) → 𝑎 < 𝐵)
36	35	ad2antlr 759	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^*)) ∧ 𝐵 ∈ (𝑎(,)𝑏)) ∧ ¬ 𝑎 ≤ 𝐴) → 𝑎 < 𝐵)
37	33, 36	eqbrtrd 4605	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^*)) ∧ 𝐵 ∈ (𝑎(,)𝑏)) ∧ ¬ 𝑎 ≤ 𝐴) → if(𝑎 ≤ 𝐴, 𝐴, 𝑎) < 𝐵)
38	31, 37	pm2.61dan 828	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^*)) ∧ 𝐵 ∈ (𝑎(,)𝑏)) → if(𝑎 ≤ 𝐴, 𝐴, 𝑎) < 𝐵)
39	34	simprd 478	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ (𝑎(,)𝑏) → 𝐵 < 𝑏)
40	22	simp2d 1067	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ (𝑎(,)𝑏) → 𝑏 ∈ ℝ^*)
41		xrltnle 9984	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^*) → (𝐵 < 𝑏 ↔ ¬ 𝑏 ≤ 𝐵))
42	23, 40, 41	syl2anc 691	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ (𝑎(,)𝑏) → (𝐵 < 𝑏 ↔ ¬ 𝑏 ≤ 𝐵))
43	39, 42	mpbid 221	. . . . . . . . . . 11 ⊢ (𝐵 ∈ (𝑎(,)𝑏) → ¬ 𝑏 ≤ 𝐵)
44		iffalse 4045	. . . . . . . . . . 11 ⊢ (¬ 𝑏 ≤ 𝐵 → if(𝑏 ≤ 𝐵, 𝑏, 𝐵) = 𝐵)
45	43, 44	syl 17	. . . . . . . . . 10 ⊢ (𝐵 ∈ (𝑎(,)𝑏) → if(𝑏 ≤ 𝐵, 𝑏, 𝐵) = 𝐵)
46	45	eqcomd 2616	. . . . . . . . 9 ⊢ (𝐵 ∈ (𝑎(,)𝑏) → 𝐵 = if(𝑏 ≤ 𝐵, 𝑏, 𝐵))
47	46	adantl 481	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^*)) ∧ 𝐵 ∈ (𝑎(,)𝑏)) → 𝐵 = if(𝑏 ≤ 𝐵, 𝑏, 𝐵))
48	38, 47	breqtrd 4609	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^*)) ∧ 𝐵 ∈ (𝑎(,)𝑏)) → if(𝑎 ≤ 𝐴, 𝐴, 𝑎) < if(𝑏 ≤ 𝐵, 𝑏, 𝐵))
49	19, 16	ifcld 4081	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^)) ∧ 𝐵 ∈ (𝑎(,)𝑏)) → if(𝑎 ≤ 𝐴, 𝐴, 𝑎) ∈ ℝ^)
50	47, 24	eqeltrrd 2689	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^)) ∧ 𝐵 ∈ (𝑎(,)𝑏)) → if(𝑏 ≤ 𝐵, 𝑏, 𝐵) ∈ ℝ^)
51		ioon0 12072	. . . . . . . 8 ⊢ ((if(𝑎 ≤ 𝐴, 𝐴, 𝑎) ∈ ℝ^* ∧ if(𝑏 ≤ 𝐵, 𝑏, 𝐵) ∈ ℝ^*) → ((if(𝑎 ≤ 𝐴, 𝐴, 𝑎)(,)if(𝑏 ≤ 𝐵, 𝑏, 𝐵)) ≠ ∅ ↔ if(𝑎 ≤ 𝐴, 𝐴, 𝑎) < if(𝑏 ≤ 𝐵, 𝑏, 𝐵)))
52	49, 50, 51	syl2anc 691	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^*)) ∧ 𝐵 ∈ (𝑎(,)𝑏)) → ((if(𝑎 ≤ 𝐴, 𝐴, 𝑎)(,)if(𝑏 ≤ 𝐵, 𝑏, 𝐵)) ≠ ∅ ↔ if(𝑎 ≤ 𝐴, 𝐴, 𝑎) < if(𝑏 ≤ 𝐵, 𝑏, 𝐵)))
53	48, 52	mpbird 246	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^*)) ∧ 𝐵 ∈ (𝑎(,)𝑏)) → (if(𝑎 ≤ 𝐴, 𝐴, 𝑎)(,)if(𝑏 ≤ 𝐵, 𝑏, 𝐵)) ≠ ∅)
54	26, 53	eqnetrd 2849	. . . . 5 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^*)) ∧ 𝐵 ∈ (𝑎(,)𝑏)) → ((𝑎(,)𝑏) ∩ (𝐴(,)𝐵)) ≠ ∅)
55	15, 54	eqnetrd 2849	. . . 4 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^*)) ∧ 𝐵 ∈ (𝑎(,)𝑏)) → ((𝑎(,)𝑏) ∩ ((𝐴(,)𝐵) ∖ {𝐵})) ≠ ∅)
56	55	ex 449	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^*)) → (𝐵 ∈ (𝑎(,)𝑏) → ((𝑎(,)𝑏) ∩ ((𝐴(,)𝐵) ∖ {𝐵})) ≠ ∅))
57	56	ralrimivva 2954	. 2 ⊢ (𝜑 → ∀𝑎 ∈ ℝ^* ∀𝑏 ∈ ℝ^* (𝐵 ∈ (𝑎(,)𝑏) → ((𝑎(,)𝑏) ∩ ((𝐴(,)𝐵) ∖ {𝐵})) ≠ ∅))
58		lptioo2.1	. . 3 ⊢ 𝐽 = (topGen‘ran (,))
59		ioossre 12106	. . . 4 ⊢ (𝐴(,)𝐵) ⊆ ℝ
60	59	a1i 11	. . 3 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℝ)
61		lptioo2.3	. . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ)
62	58, 60, 61	islptre 38686	. 2 ⊢ (𝜑 → (𝐵 ∈ ((limPt‘𝐽)‘(𝐴(,)𝐵)) ↔ ∀𝑎 ∈ ℝ^* ∀𝑏 ∈ ℝ^* (𝐵 ∈ (𝑎(,)𝑏) → ((𝑎(,)𝑏) ∩ ((𝐴(,)𝐵) ∖ {𝐵})) ≠ ∅)))
63	57, 62	mpbird 246	1 ⊢ (𝜑 → 𝐵 ∈ ((limPt‘𝐽)‘(𝐴(,)𝐵)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 ∖ cdif 3537 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 ifcif 4036 {csn 4125 class class class wbr 4583 ran crn 5039 ‘cfv 5804 (class class class)co 6549 ℝcr 9814 ℝ^*cxr 9952 < clt 9953 ≤ cle 9954 (,)cioo 12046 topGenctg 15921 limPtclp 20748

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-sup 8231 df-inf 8232 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-q 11665 df-ioo 12050 df-topgen 15927 df-top 20521 df-bases 20522 df-topon 20523 df-cld 20633 df-ntr 20634 df-cls 20635 df-nei 20712 df-lp 20750

This theorem is referenced by: lptioo2cn 38712 fouriersw 39124

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