lptioo2 - Mathbox for Glauco Siliprandi

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Theorem lptioo2 41918

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 4112	. . . . . . . 8 ⊢ (𝜑 → ((𝐴(,)𝐵) ∖ {𝐵}) ⊆ (𝐴(,)𝐵))
2		simpr 487	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ (𝐴(,)𝐵))
3		ubioo 12773	. . . . . . . . . . . 12 ⊢ ¬ 𝐵 ∈ (𝐴(,)𝐵)
4		eleq1 2903	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ (𝐴(,)𝐵) ↔ 𝐵 ∈ (𝐴(,)𝐵)))
5	4	biimpcd 251	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝐴(,)𝐵) → (𝑥 = 𝐵 → 𝐵 ∈ (𝐴(,)𝐵)))
6	3, 5	mtoi 201	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐴(,)𝐵) → ¬ 𝑥 = 𝐵)
7	6	adantl 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ¬ 𝑥 = 𝐵)
8		velsn 4586	. . . . . . . . . 10 ⊢ (𝑥 ∈ {𝐵} ↔ 𝑥 = 𝐵)
9	7, 8	sylnibr 331	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ¬ 𝑥 ∈ {𝐵})
10	2, 9	eldifd 3950	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ ((𝐴(,)𝐵) ∖ {𝐵}))
11	1, 10	eqelssd 3991	. . . . . . 7 ⊢ (𝜑 → ((𝐴(,)𝐵) ∖ {𝐵}) = (𝐴(,)𝐵))
12	11	ineq2d 4192	. . . . . 6 ⊢ (𝜑 → ((𝑎(,)𝑏) ∩ ((𝐴(,)𝐵) ∖ {𝐵})) = ((𝑎(,)𝑏) ∩ (𝐴(,)𝐵)))
13	12	ad2antrr 724	. . . . 5 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^*)) ∧ 𝐵 ∈ (𝑎(,)𝑏)) → ((𝑎(,)𝑏) ∩ ((𝐴(,)𝐵) ∖ {𝐵})) = ((𝑎(,)𝑏) ∩ (𝐴(,)𝐵)))
14		simplrl 775	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^)) ∧ 𝐵 ∈ (𝑎(,)𝑏)) → 𝑎 ∈ ℝ^)
15		simplrr 776	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^)) ∧ 𝐵 ∈ (𝑎(,)𝑏)) → 𝑏 ∈ ℝ^)
16		lptioo2.2	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
17	16	ad2antrr 724	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^)) ∧ 𝐵 ∈ (𝑎(,)𝑏)) → 𝐴 ∈ ℝ^)
18		elioo3g 12770	. . . . . . . . . . 11 ⊢ (𝐵 ∈ (𝑎(,)𝑏) ↔ ((𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝑎 < 𝐵 ∧ 𝐵 < 𝑏)))
19	18	biimpi 218	. . . . . . . . . 10 ⊢ (𝐵 ∈ (𝑎(,)𝑏) → ((𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝑎 < 𝐵 ∧ 𝐵 < 𝑏)))
20	19	simpld 497	. . . . . . . . 9 ⊢ (𝐵 ∈ (𝑎(,)𝑏) → (𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*))
21	20	simp3d 1140	. . . . . . . 8 ⊢ (𝐵 ∈ (𝑎(,)𝑏) → 𝐵 ∈ ℝ^*)
22	21	adantl 484	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^)) ∧ 𝐵 ∈ (𝑎(,)𝑏)) → 𝐵 ∈ ℝ^)
23		iooin 12775	. . . . . . 7 ⊢ (((𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^) ∧ (𝐴 ∈ ℝ^ ∧ 𝐵 ∈ ℝ^*)) → ((𝑎(,)𝑏) ∩ (𝐴(,)𝐵)) = (if(𝑎 ≤ 𝐴, 𝐴, 𝑎)(,)if(𝑏 ≤ 𝐵, 𝑏, 𝐵)))
24	14, 15, 17, 22, 23	syl22anc 836	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^*)) ∧ 𝐵 ∈ (𝑎(,)𝑏)) → ((𝑎(,)𝑏) ∩ (𝐴(,)𝐵)) = (if(𝑎 ≤ 𝐴, 𝐴, 𝑎)(,)if(𝑏 ≤ 𝐵, 𝑏, 𝐵)))
25		iftrue 4476	. . . . . . . . . . 11 ⊢ (𝑎 ≤ 𝐴 → if(𝑎 ≤ 𝐴, 𝐴, 𝑎) = 𝐴)
26	25	adantl 484	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^*)) ∧ 𝐵 ∈ (𝑎(,)𝑏)) ∧ 𝑎 ≤ 𝐴) → if(𝑎 ≤ 𝐴, 𝐴, 𝑎) = 𝐴)
27		lptioo2.4	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 < 𝐵)
28	27	ad3antrrr 728	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^*)) ∧ 𝐵 ∈ (𝑎(,)𝑏)) ∧ 𝑎 ≤ 𝐴) → 𝐴 < 𝐵)
29	26, 28	eqbrtrd 5091	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^*)) ∧ 𝐵 ∈ (𝑎(,)𝑏)) ∧ 𝑎 ≤ 𝐴) → if(𝑎 ≤ 𝐴, 𝐴, 𝑎) < 𝐵)
30		iffalse 4479	. . . . . . . . . . 11 ⊢ (¬ 𝑎 ≤ 𝐴 → if(𝑎 ≤ 𝐴, 𝐴, 𝑎) = 𝑎)
31	30	adantl 484	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^*)) ∧ 𝐵 ∈ (𝑎(,)𝑏)) ∧ ¬ 𝑎 ≤ 𝐴) → if(𝑎 ≤ 𝐴, 𝐴, 𝑎) = 𝑎)
32	19	simprd 498	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ (𝑎(,)𝑏) → (𝑎 < 𝐵 ∧ 𝐵 < 𝑏))
33	32	simpld 497	. . . . . . . . . . 11 ⊢ (𝐵 ∈ (𝑎(,)𝑏) → 𝑎 < 𝐵)
34	33	ad2antlr 725	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^*)) ∧ 𝐵 ∈ (𝑎(,)𝑏)) ∧ ¬ 𝑎 ≤ 𝐴) → 𝑎 < 𝐵)
35	31, 34	eqbrtrd 5091	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^*)) ∧ 𝐵 ∈ (𝑎(,)𝑏)) ∧ ¬ 𝑎 ≤ 𝐴) → if(𝑎 ≤ 𝐴, 𝐴, 𝑎) < 𝐵)
36	29, 35	pm2.61dan 811	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^*)) ∧ 𝐵 ∈ (𝑎(,)𝑏)) → if(𝑎 ≤ 𝐴, 𝐴, 𝑎) < 𝐵)
37	32	simprd 498	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ (𝑎(,)𝑏) → 𝐵 < 𝑏)
38	20	simp2d 1139	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ (𝑎(,)𝑏) → 𝑏 ∈ ℝ^*)
39		xrltnle 10711	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^*) → (𝐵 < 𝑏 ↔ ¬ 𝑏 ≤ 𝐵))
40	21, 38, 39	syl2anc 586	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ (𝑎(,)𝑏) → (𝐵 < 𝑏 ↔ ¬ 𝑏 ≤ 𝐵))
41	37, 40	mpbid 234	. . . . . . . . . . 11 ⊢ (𝐵 ∈ (𝑎(,)𝑏) → ¬ 𝑏 ≤ 𝐵)
42		iffalse 4479	. . . . . . . . . . 11 ⊢ (¬ 𝑏 ≤ 𝐵 → if(𝑏 ≤ 𝐵, 𝑏, 𝐵) = 𝐵)
43	41, 42	syl 17	. . . . . . . . . 10 ⊢ (𝐵 ∈ (𝑎(,)𝑏) → if(𝑏 ≤ 𝐵, 𝑏, 𝐵) = 𝐵)
44	43	eqcomd 2830	. . . . . . . . 9 ⊢ (𝐵 ∈ (𝑎(,)𝑏) → 𝐵 = if(𝑏 ≤ 𝐵, 𝑏, 𝐵))
45	44	adantl 484	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^*)) ∧ 𝐵 ∈ (𝑎(,)𝑏)) → 𝐵 = if(𝑏 ≤ 𝐵, 𝑏, 𝐵))
46	36, 45	breqtrd 5095	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^*)) ∧ 𝐵 ∈ (𝑎(,)𝑏)) → if(𝑎 ≤ 𝐴, 𝐴, 𝑎) < if(𝑏 ≤ 𝐵, 𝑏, 𝐵))
47	17, 14	ifcld 4515	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^)) ∧ 𝐵 ∈ (𝑎(,)𝑏)) → if(𝑎 ≤ 𝐴, 𝐴, 𝑎) ∈ ℝ^)
48	45, 22	eqeltrrd 2917	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^)) ∧ 𝐵 ∈ (𝑎(,)𝑏)) → if(𝑏 ≤ 𝐵, 𝑏, 𝐵) ∈ ℝ^)
49		ioon0 12767	. . . . . . . 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)))
50	47, 48, 49	syl2anc 586	. . . . . . 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)))
51	46, 50	mpbird 259	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^*)) ∧ 𝐵 ∈ (𝑎(,)𝑏)) → (if(𝑎 ≤ 𝐴, 𝐴, 𝑎)(,)if(𝑏 ≤ 𝐵, 𝑏, 𝐵)) ≠ ∅)
52	24, 51	eqnetrd 3086	. . . . 5 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^*)) ∧ 𝐵 ∈ (𝑎(,)𝑏)) → ((𝑎(,)𝑏) ∩ (𝐴(,)𝐵)) ≠ ∅)
53	13, 52	eqnetrd 3086	. . . 4 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^*)) ∧ 𝐵 ∈ (𝑎(,)𝑏)) → ((𝑎(,)𝑏) ∩ ((𝐴(,)𝐵) ∖ {𝐵})) ≠ ∅)
54	53	ex 415	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ^* ∧ 𝑏 ∈ ℝ^*)) → (𝐵 ∈ (𝑎(,)𝑏) → ((𝑎(,)𝑏) ∩ ((𝐴(,)𝐵) ∖ {𝐵})) ≠ ∅))
55	54	ralrimivva 3194	. 2 ⊢ (𝜑 → ∀𝑎 ∈ ℝ^* ∀𝑏 ∈ ℝ^* (𝐵 ∈ (𝑎(,)𝑏) → ((𝑎(,)𝑏) ∩ ((𝐴(,)𝐵) ∖ {𝐵})) ≠ ∅))
56		lptioo2.1	. . 3 ⊢ 𝐽 = (topGen‘ran (,))
57		ioossre 12801	. . . 4 ⊢ (𝐴(,)𝐵) ⊆ ℝ
58	57	a1i 11	. . 3 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℝ)
59		lptioo2.3	. . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ)
60	56, 58, 59	islptre 41906	. 2 ⊢ (𝜑 → (𝐵 ∈ ((limPt‘𝐽)‘(𝐴(,)𝐵)) ↔ ∀𝑎 ∈ ℝ^* ∀𝑏 ∈ ℝ^* (𝐵 ∈ (𝑎(,)𝑏) → ((𝑎(,)𝑏) ∩ ((𝐴(,)𝐵) ∖ {𝐵})) ≠ ∅)))
61	55, 60	mpbird 259	1 ⊢ (𝜑 → 𝐵 ∈ ((limPt‘𝐽)‘(𝐴(,)𝐵)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1536 ∈ wcel 2113 ≠ wne 3019 ∀wral 3141 ∖ cdif 3936 ∩ cin 3938 ⊆ wss 3939 ∅c0 4294 ifcif 4470 {csn 4570 class class class wbr 5069 ran crn 5559 ‘cfv 6358 (class class class)co 7159 ℝcr 10539 ℝ^*cxr 10677 < clt 10678 ≤ cle 10679 (,)cioo 12741 topGenctg 16714 limPtclp 21745

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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 ax-pre-sup 10618

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-iin 4925 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-sup 8909 df-inf 8910 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-div 11301 df-nn 11642 df-n0 11901 df-z 11985 df-uz 12247 df-q 12352 df-ioo 12745 df-topgen 16720 df-top 21505 df-topon 21522 df-bases 21557 df-cld 21630 df-ntr 21631 df-cls 21632 df-nei 21709 df-lp 21747

This theorem is referenced by: lptioo2cn 41932 fouriersw 42523

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