cncfiooicclem1 - Mathbox for Glauco Siliprandi

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Theorem cncfiooicclem1 42196

Description: A continuous function 𝐹 on an open interval (𝐴(,)𝐵) can be extended to a continuous function 𝐺 on the corresponding closed interval, if it has a finite right limit 𝑅 in 𝐴 and a finite left limit 𝐿 in 𝐵. 𝐹 can be complex-valued. This lemma assumes 𝐴 < 𝐵, the invoking theorem drops this assumption. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Hypotheses**
Ref	Expression
cncfiooicclem1.x	⊢ Ⅎ𝑥𝜑
cncfiooicclem1.g	⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))))
cncfiooicclem1.a	⊢ (𝜑 → 𝐴 ∈ ℝ)
cncfiooicclem1.b	⊢ (𝜑 → 𝐵 ∈ ℝ)
cncfiooicclem1.altb	⊢ (𝜑 → 𝐴 < 𝐵)
cncfiooicclem1.f	⊢ (𝜑 → 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ))
cncfiooicclem1.l	⊢ (𝜑 → 𝐿 ∈ (𝐹 lim_ℂ 𝐵))
cncfiooicclem1.r	⊢ (𝜑 → 𝑅 ∈ (𝐹 lim_ℂ 𝐴))

**Assertion**
Ref	Expression
cncfiooicclem1	⊢ (𝜑 → 𝐺 ∈ ((𝐴[,]𝐵)–cn→ℂ))

Distinct variable groups: 𝑥,𝐴 𝑥,𝐵 𝑥,𝐹 𝑥,𝐿 𝑥,𝑅 Allowed substitution hints: 𝜑(𝑥) 𝐺(𝑥)

Proof of Theorem cncfiooicclem1 Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cncfiooicclem1.x	. . . 4 ⊢ Ⅎ𝑥𝜑
2		limccl 24473	. . . . . . 7 ⊢ (𝐹 lim_ℂ 𝐴) ⊆ ℂ
3		cncfiooicclem1.r	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ (𝐹 lim_ℂ 𝐴))
4	2, 3	sseldi 3965	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ ℂ)
5	4	ad2antrr 724	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ 𝑥 = 𝐴) → 𝑅 ∈ ℂ)
6		limccl 24473	. . . . . . . 8 ⊢ (𝐹 lim_ℂ 𝐵) ⊆ ℂ
7		cncfiooicclem1.l	. . . . . . . 8 ⊢ (𝜑 → 𝐿 ∈ (𝐹 lim_ℂ 𝐵))
8	6, 7	sseldi 3965	. . . . . . 7 ⊢ (𝜑 → 𝐿 ∈ ℂ)
9	8	ad3antrrr 728	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ 𝑥 = 𝐵) → 𝐿 ∈ ℂ)
10		simplll 773	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ ¬ 𝑥 = 𝐵) → 𝜑)
11		orel1 885	. . . . . . . . . . 11 ⊢ (¬ 𝑥 = 𝐴 → ((𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → 𝑥 = 𝐵))
12	11	con3dimp 411	. . . . . . . . . 10 ⊢ ((¬ 𝑥 = 𝐴 ∧ ¬ 𝑥 = 𝐵) → ¬ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵))
13		vex 3497	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
14	13	elpr 4590	. . . . . . . . . 10 ⊢ (𝑥 ∈ {𝐴, 𝐵} ↔ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵))
15	12, 14	sylnibr 331	. . . . . . . . 9 ⊢ ((¬ 𝑥 = 𝐴 ∧ ¬ 𝑥 = 𝐵) → ¬ 𝑥 ∈ {𝐴, 𝐵})
16	15	adantll 712	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ ¬ 𝑥 = 𝐵) → ¬ 𝑥 ∈ {𝐴, 𝐵})
17		simpllr 774	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ ¬ 𝑥 = 𝐵) → 𝑥 ∈ (𝐴[,]𝐵))
18		cncfiooicclem1.a	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 ∈ ℝ)
19	18	rexrd 10691	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
20	10, 19	syl 17	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ ¬ 𝑥 = 𝐵) → 𝐴 ∈ ℝ^*)
21		cncfiooicclem1.b	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 ∈ ℝ)
22	21	rexrd 10691	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
23	10, 22	syl 17	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ ¬ 𝑥 = 𝐵) → 𝐵 ∈ ℝ^*)
24		cncfiooicclem1.altb	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 < 𝐵)
25	18, 21, 24	ltled 10788	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
26	10, 25	syl 17	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ ¬ 𝑥 = 𝐵) → 𝐴 ≤ 𝐵)
27		prunioo 12868	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → ((𝐴(,)𝐵) ∪ {𝐴, 𝐵}) = (𝐴[,]𝐵))
28	20, 23, 26, 27	syl3anc 1367	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ ¬ 𝑥 = 𝐵) → ((𝐴(,)𝐵) ∪ {𝐴, 𝐵}) = (𝐴[,]𝐵))
29	17, 28	eleqtrrd 2916	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ ¬ 𝑥 = 𝐵) → 𝑥 ∈ ((𝐴(,)𝐵) ∪ {𝐴, 𝐵}))
30		elun 4125	. . . . . . . . 9 ⊢ (𝑥 ∈ ((𝐴(,)𝐵) ∪ {𝐴, 𝐵}) ↔ (𝑥 ∈ (𝐴(,)𝐵) ∨ 𝑥 ∈ {𝐴, 𝐵}))
31	29, 30	sylib 220	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ ¬ 𝑥 = 𝐵) → (𝑥 ∈ (𝐴(,)𝐵) ∨ 𝑥 ∈ {𝐴, 𝐵}))
32		orel2 887	. . . . . . . 8 ⊢ (¬ 𝑥 ∈ {𝐴, 𝐵} → ((𝑥 ∈ (𝐴(,)𝐵) ∨ 𝑥 ∈ {𝐴, 𝐵}) → 𝑥 ∈ (𝐴(,)𝐵)))
33	16, 31, 32	sylc 65	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ ¬ 𝑥 = 𝐵) → 𝑥 ∈ (𝐴(,)𝐵))
34		cncfiooicclem1.f	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ))
35		cncff 23501	. . . . . . . . 9 ⊢ (𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ) → 𝐹:(𝐴(,)𝐵)⟶ℂ)
36	34, 35	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐹:(𝐴(,)𝐵)⟶ℂ)
37	36	ffvelrnda 6851	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑥) ∈ ℂ)
38	10, 33, 37	syl2anc 586	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ ¬ 𝑥 = 𝐵) → (𝐹‘𝑥) ∈ ℂ)
39	9, 38	ifclda 4501	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) → if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥)) ∈ ℂ)
40	5, 39	ifclda 4501	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))) ∈ ℂ)
41		cncfiooicclem1.g	. . . 4 ⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))))
42	1, 40, 41	fmptdf 6881	. . 3 ⊢ (𝜑 → 𝐺:(𝐴[,]𝐵)⟶ℂ)
43		elun 4125	. . . . . . 7 ⊢ (𝑦 ∈ ((𝐴(,)𝐵) ∪ {𝐴, 𝐵}) ↔ (𝑦 ∈ (𝐴(,)𝐵) ∨ 𝑦 ∈ {𝐴, 𝐵}))
44	19, 22, 25, 27	syl3anc 1367	. . . . . . . 8 ⊢ (𝜑 → ((𝐴(,)𝐵) ∪ {𝐴, 𝐵}) = (𝐴[,]𝐵))
45	44	eleq2d 2898	. . . . . . 7 ⊢ (𝜑 → (𝑦 ∈ ((𝐴(,)𝐵) ∪ {𝐴, 𝐵}) ↔ 𝑦 ∈ (𝐴[,]𝐵)))
46	43, 45	syl5bbr 287	. . . . . 6 ⊢ (𝜑 → ((𝑦 ∈ (𝐴(,)𝐵) ∨ 𝑦 ∈ {𝐴, 𝐵}) ↔ 𝑦 ∈ (𝐴[,]𝐵)))
47	46	biimpar 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝑦 ∈ (𝐴(,)𝐵) ∨ 𝑦 ∈ {𝐴, 𝐵}))
48		ioossicc 12823	. . . . . . . . . . . . 13 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
49		fssres 6544	. . . . . . . . . . . . 13 ⊢ ((𝐺:(𝐴[,]𝐵)⟶ℂ ∧ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)) → (𝐺 ↾ (𝐴(,)𝐵)):(𝐴(,)𝐵)⟶ℂ)
50	42, 48, 49	sylancl 588	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐺 ↾ (𝐴(,)𝐵)):(𝐴(,)𝐵)⟶ℂ)
51	50	feqmptd 6733	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐺 ↾ (𝐴(,)𝐵)) = (𝑦 ∈ (𝐴(,)𝐵) ↦ ((𝐺 ↾ (𝐴(,)𝐵))‘𝑦)))
52		nfmpt1 5164	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥(𝑥 ∈ (𝐴[,]𝐵) ↦ if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))))
53	41, 52	nfcxfr 2975	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥𝐺
54		nfcv 2977	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥(𝐴(,)𝐵)
55	53, 54	nfres 5855	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥(𝐺 ↾ (𝐴(,)𝐵))
56		nfcv 2977	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥𝑦
57	55, 56	nffv 6680	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥((𝐺 ↾ (𝐴(,)𝐵))‘𝑦)
58		nfcv 2977	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑦(𝐺 ↾ (𝐴(,)𝐵))
59		nfcv 2977	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑦𝑥
60	58, 59	nffv 6680	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑦((𝐺 ↾ (𝐴(,)𝐵))‘𝑥)
61		fveq2 6670	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑥 → ((𝐺 ↾ (𝐴(,)𝐵))‘𝑦) = ((𝐺 ↾ (𝐴(,)𝐵))‘𝑥))
62	57, 60, 61	cbvmpt 5167	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (𝐴(,)𝐵) ↦ ((𝐺 ↾ (𝐴(,)𝐵))‘𝑦)) = (𝑥 ∈ (𝐴(,)𝐵) ↦ ((𝐺 ↾ (𝐴(,)𝐵))‘𝑥))
63	62	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑦 ∈ (𝐴(,)𝐵) ↦ ((𝐺 ↾ (𝐴(,)𝐵))‘𝑦)) = (𝑥 ∈ (𝐴(,)𝐵) ↦ ((𝐺 ↾ (𝐴(,)𝐵))‘𝑥)))
64		fvres 6689	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (𝐴(,)𝐵) → ((𝐺 ↾ (𝐴(,)𝐵))‘𝑥) = (𝐺‘𝑥))
65	64	adantl 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((𝐺 ↾ (𝐴(,)𝐵))‘𝑥) = (𝐺‘𝑥))
66		simpr 487	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ (𝐴(,)𝐵))
67	48, 66	sseldi 3965	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ (𝐴[,]𝐵))
68	4	adantr 483	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑅 ∈ ℂ)
69	8	ad2antrr 724	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝐵) → 𝐿 ∈ ℂ)
70	37	adantr 483	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ ¬ 𝑥 = 𝐵) → (𝐹‘𝑥) ∈ ℂ)
71	69, 70	ifclda 4501	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥)) ∈ ℂ)
72	68, 71	ifcld 4512	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))) ∈ ℂ)
73	41	fvmpt2 6779	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (𝐴[,]𝐵) ∧ if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))) ∈ ℂ) → (𝐺‘𝑥) = if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))))
74	67, 72, 73	syl2anc 586	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝐺‘𝑥) = if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))))
75		elioo4g 12798	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ (𝐴(,)𝐵) ↔ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝑥 ∈ ℝ) ∧ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)))
76	75	biimpi 218	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ (𝐴(,)𝐵) → ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝑥 ∈ ℝ) ∧ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)))
77	76	simpld 497	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ (𝐴(,)𝐵) → (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝑥 ∈ ℝ))
78	77	simp1d 1138	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ (𝐴(,)𝐵) → 𝐴 ∈ ℝ^*)
79		elioore 12769	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ (𝐴(,)𝐵) → 𝑥 ∈ ℝ)
80	79	rexrd 10691	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ (𝐴(,)𝐵) → 𝑥 ∈ ℝ^*)
81		eliooord 12797	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ (𝐴(,)𝐵) → (𝐴 < 𝑥 ∧ 𝑥 < 𝐵))
82	81	simpld 497	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ (𝐴(,)𝐵) → 𝐴 < 𝑥)
83		xrltne 12557	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝑥 ∈ ℝ^* ∧ 𝐴 < 𝑥) → 𝑥 ≠ 𝐴)
84	78, 80, 82, 83	syl3anc 1367	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (𝐴(,)𝐵) → 𝑥 ≠ 𝐴)
85	84	adantl 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ≠ 𝐴)
86	85	neneqd 3021	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ¬ 𝑥 = 𝐴)
87	86	iffalsed 4478	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))) = if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥)))
88	81	simprd 498	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ (𝐴(,)𝐵) → 𝑥 < 𝐵)
89	79, 88	ltned 10776	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (𝐴(,)𝐵) → 𝑥 ≠ 𝐵)
90	89	neneqd 3021	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (𝐴(,)𝐵) → ¬ 𝑥 = 𝐵)
91	90	iffalsed 4478	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (𝐴(,)𝐵) → if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥)) = (𝐹‘𝑥))
92	91	adantl 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥)) = (𝐹‘𝑥))
93	87, 92	eqtrd 2856	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))) = (𝐹‘𝑥))
94	65, 74, 93	3eqtrd 2860	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((𝐺 ↾ (𝐴(,)𝐵))‘𝑥) = (𝐹‘𝑥))
95	1, 94	mpteq2da 5160	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) ↦ ((𝐺 ↾ (𝐴(,)𝐵))‘𝑥)) = (𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐹‘𝑥)))
96	51, 63, 95	3eqtrd 2860	. . . . . . . . . 10 ⊢ (𝜑 → (𝐺 ↾ (𝐴(,)𝐵)) = (𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐹‘𝑥)))
97	36	feqmptd 6733	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐹‘𝑥)))
98		ioosscn 41789	. . . . . . . . . . . . 13 ⊢ (𝐴(,)𝐵) ⊆ ℂ
99	98	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℂ)
100		ssid 3989	. . . . . . . . . . . 12 ⊢ ℂ ⊆ ℂ
101		eqid 2821	. . . . . . . . . . . . 13 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
102		eqid 2821	. . . . . . . . . . . . 13 ⊢ ((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) = ((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵))
103	101	cnfldtop 23392	. . . . . . . . . . . . . . 15 ⊢ (TopOpen‘ℂ_fld) ∈ Top
104		unicntop 23394	. . . . . . . . . . . . . . . 16 ⊢ ℂ = ∪ (TopOpen‘ℂ_fld)
105	104	restid 16707	. . . . . . . . . . . . . . 15 ⊢ ((TopOpen‘ℂ_fld) ∈ Top → ((TopOpen‘ℂ_fld) ↾_t ℂ) = (TopOpen‘ℂ_fld))
106	103, 105	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ ((TopOpen‘ℂ_fld) ↾_t ℂ) = (TopOpen‘ℂ_fld)
107	106	eqcomi 2830	. . . . . . . . . . . . 13 ⊢ (TopOpen‘ℂ_fld) = ((TopOpen‘ℂ_fld) ↾_t ℂ)
108	101, 102, 107	cncfcn 23517	. . . . . . . . . . . 12 ⊢ (((𝐴(,)𝐵) ⊆ ℂ ∧ ℂ ⊆ ℂ) → ((𝐴(,)𝐵)–cn→ℂ) = (((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) Cn (TopOpen‘ℂ_fld)))
109	99, 100, 108	sylancl 588	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐴(,)𝐵)–cn→ℂ) = (((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) Cn (TopOpen‘ℂ_fld)))
110	34, 97, 109	3eltr3d 2927	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐹‘𝑥)) ∈ (((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) Cn (TopOpen‘ℂ_fld)))
111	96, 110	eqeltrd 2913	. . . . . . . . 9 ⊢ (𝜑 → (𝐺 ↾ (𝐴(,)𝐵)) ∈ (((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) Cn (TopOpen‘ℂ_fld)))
112	104	restuni 21770	. . . . . . . . . . 11 ⊢ (((TopOpen‘ℂ_fld) ∈ Top ∧ (𝐴(,)𝐵) ⊆ ℂ) → (𝐴(,)𝐵) = ∪ ((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)))
113	103, 98, 112	mp2an 690	. . . . . . . . . 10 ⊢ (𝐴(,)𝐵) = ∪ ((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵))
114	113	cncnpi 21886	. . . . . . . . 9 ⊢ (((𝐺 ↾ (𝐴(,)𝐵)) ∈ (((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) Cn (TopOpen‘ℂ_fld)) ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (𝐺 ↾ (𝐴(,)𝐵)) ∈ ((((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) CnP (TopOpen‘ℂ_fld))‘𝑦))
115	111, 114	sylan 582	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (𝐺 ↾ (𝐴(,)𝐵)) ∈ ((((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) CnP (TopOpen‘ℂ_fld))‘𝑦))
116	103	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (TopOpen‘ℂ_fld) ∈ Top)
117	48	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵))
118		ovex 7189	. . . . . . . . . . . . 13 ⊢ (𝐴[,]𝐵) ∈ V
119	118	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (𝐴[,]𝐵) ∈ V)
120		restabs 21773	. . . . . . . . . . . 12 ⊢ (((TopOpen‘ℂ_fld) ∈ Top ∧ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵) ∧ (𝐴[,]𝐵) ∈ V) → (((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵)) ↾_t (𝐴(,)𝐵)) = ((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)))
121	116, 117, 119, 120	syl3anc 1367	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵)) ↾_t (𝐴(,)𝐵)) = ((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)))
122	121	eqcomd 2827	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → ((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) = (((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵)) ↾_t (𝐴(,)𝐵)))
123	122	oveq1d 7171	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) CnP (TopOpen‘ℂ_fld)) = ((((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵)) ↾_t (𝐴(,)𝐵)) CnP (TopOpen‘ℂ_fld)))
124	123	fveq1d 6672	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → ((((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) CnP (TopOpen‘ℂ_fld))‘𝑦) = (((((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵)) ↾_t (𝐴(,)𝐵)) CnP (TopOpen‘ℂ_fld))‘𝑦))
125	115, 124	eleqtrd 2915	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (𝐺 ↾ (𝐴(,)𝐵)) ∈ (((((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵)) ↾_t (𝐴(,)𝐵)) CnP (TopOpen‘ℂ_fld))‘𝑦))
126		resttop 21768	. . . . . . . . . 10 ⊢ (((TopOpen‘ℂ_fld) ∈ Top ∧ (𝐴[,]𝐵) ∈ V) → ((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵)) ∈ Top)
127	103, 118, 126	mp2an 690	. . . . . . . . 9 ⊢ ((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵)) ∈ Top
128	127	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → ((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵)) ∈ Top)
129	48	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵))
130	18, 21	iccssred 41800	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
131		ax-resscn 10594	. . . . . . . . . . . 12 ⊢ ℝ ⊆ ℂ
132	130, 131	sstrdi 3979	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℂ)
133	104	restuni 21770	. . . . . . . . . . 11 ⊢ (((TopOpen‘ℂ_fld) ∈ Top ∧ (𝐴[,]𝐵) ⊆ ℂ) → (𝐴[,]𝐵) = ∪ ((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵)))
134	103, 132, 133	sylancr 589	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴[,]𝐵) = ∪ ((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵)))
135	129, 134	sseqtrd 4007	. . . . . . . . 9 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ∪ ((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵)))
136	135	adantr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (𝐴(,)𝐵) ⊆ ∪ ((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵)))
137		retop 23370	. . . . . . . . . . . . . 14 ⊢ (topGen‘ran (,)) ∈ Top
138	137	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (topGen‘ran (,)) ∈ Top)
139		ioossre 12799	. . . . . . . . . . . . . . 15 ⊢ (𝐴(,)𝐵) ⊆ ℝ
140		difss 4108	. . . . . . . . . . . . . . 15 ⊢ (ℝ ∖ (𝐴[,]𝐵)) ⊆ ℝ
141	139, 140	unssi 4161	. . . . . . . . . . . . . 14 ⊢ ((𝐴(,)𝐵) ∪ (ℝ ∖ (𝐴[,]𝐵))) ⊆ ℝ
142	141	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → ((𝐴(,)𝐵) ∪ (ℝ ∖ (𝐴[,]𝐵))) ⊆ ℝ)
143		ssun1 4148	. . . . . . . . . . . . . 14 ⊢ (𝐴(,)𝐵) ⊆ ((𝐴(,)𝐵) ∪ (ℝ ∖ (𝐴[,]𝐵)))
144	143	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (𝐴(,)𝐵) ⊆ ((𝐴(,)𝐵) ∪ (ℝ ∖ (𝐴[,]𝐵))))
145		uniretop 23371	. . . . . . . . . . . . . 14 ⊢ ℝ = ∪ (topGen‘ran (,))
146	145	ntrss 21663	. . . . . . . . . . . . 13 ⊢ (((topGen‘ran (,)) ∈ Top ∧ ((𝐴(,)𝐵) ∪ (ℝ ∖ (𝐴[,]𝐵))) ⊆ ℝ ∧ (𝐴(,)𝐵) ⊆ ((𝐴(,)𝐵) ∪ (ℝ ∖ (𝐴[,]𝐵)))) → ((int‘(topGen‘ran (,)))‘(𝐴(,)𝐵)) ⊆ ((int‘(topGen‘ran (,)))‘((𝐴(,)𝐵) ∪ (ℝ ∖ (𝐴[,]𝐵)))))
147	138, 142, 144, 146	syl3anc 1367	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → ((int‘(topGen‘ran (,)))‘(𝐴(,)𝐵)) ⊆ ((int‘(topGen‘ran (,)))‘((𝐴(,)𝐵) ∪ (ℝ ∖ (𝐴[,]𝐵)))))
148		simpr 487	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝑦 ∈ (𝐴(,)𝐵))
149		ioontr 41807	. . . . . . . . . . . . 13 ⊢ ((int‘(topGen‘ran (,)))‘(𝐴(,)𝐵)) = (𝐴(,)𝐵)
150	148, 149	eleqtrrdi 2924	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝑦 ∈ ((int‘(topGen‘ran (,)))‘(𝐴(,)𝐵)))
151	147, 150	sseldd 3968	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝑦 ∈ ((int‘(topGen‘ran (,)))‘((𝐴(,)𝐵) ∪ (ℝ ∖ (𝐴[,]𝐵)))))
152	48, 148	sseldi 3965	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝑦 ∈ (𝐴[,]𝐵))
153	151, 152	elind 4171	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝑦 ∈ (((int‘(topGen‘ran (,)))‘((𝐴(,)𝐵) ∪ (ℝ ∖ (𝐴[,]𝐵)))) ∩ (𝐴[,]𝐵)))
154	130	adantr 483	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (𝐴[,]𝐵) ⊆ ℝ)
155		eqid 2821	. . . . . . . . . . . 12 ⊢ ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)) = ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵))
156	145, 155	restntr 21790	. . . . . . . . . . 11 ⊢ (((topGen‘ran (,)) ∈ Top ∧ (𝐴[,]𝐵) ⊆ ℝ ∧ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)) → ((int‘((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)))‘(𝐴(,)𝐵)) = (((int‘(topGen‘ran (,)))‘((𝐴(,)𝐵) ∪ (ℝ ∖ (𝐴[,]𝐵)))) ∩ (𝐴[,]𝐵)))
157	138, 154, 117, 156	syl3anc 1367	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → ((int‘((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)))‘(𝐴(,)𝐵)) = (((int‘(topGen‘ran (,)))‘((𝐴(,)𝐵) ∪ (ℝ ∖ (𝐴[,]𝐵)))) ∩ (𝐴[,]𝐵)))
158	153, 157	eleqtrrd 2916	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝑦 ∈ ((int‘((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)))‘(𝐴(,)𝐵)))
159	101	tgioo2 23411	. . . . . . . . . . . . . . 15 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
160	159	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ))
161	160	oveq1d 7171	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)) = (((TopOpen‘ℂ_fld) ↾_t ℝ) ↾_t (𝐴[,]𝐵)))
162	103	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (TopOpen‘ℂ_fld) ∈ Top)
163		reex 10628	. . . . . . . . . . . . . . 15 ⊢ ℝ ∈ V
164	163	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ℝ ∈ V)
165		restabs 21773	. . . . . . . . . . . . . 14 ⊢ (((TopOpen‘ℂ_fld) ∈ Top ∧ (𝐴[,]𝐵) ⊆ ℝ ∧ ℝ ∈ V) → (((TopOpen‘ℂ_fld) ↾_t ℝ) ↾_t (𝐴[,]𝐵)) = ((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵)))
166	162, 130, 164, 165	syl3anc 1367	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((TopOpen‘ℂ_fld) ↾_t ℝ) ↾_t (𝐴[,]𝐵)) = ((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵)))
167	161, 166	eqtrd 2856	. . . . . . . . . . . 12 ⊢ (𝜑 → ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)) = ((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵)))
168	167	fveq2d 6674	. . . . . . . . . . 11 ⊢ (𝜑 → (int‘((topGen‘ran (,)) ↾_t (𝐴[,]𝐵))) = (int‘((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵))))
169	168	fveq1d 6672	. . . . . . . . . 10 ⊢ (𝜑 → ((int‘((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)))‘(𝐴(,)𝐵)) = ((int‘((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵)))‘(𝐴(,)𝐵)))
170	169	adantr 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → ((int‘((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)))‘(𝐴(,)𝐵)) = ((int‘((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵)))‘(𝐴(,)𝐵)))
171	158, 170	eleqtrd 2915	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝑦 ∈ ((int‘((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵)))‘(𝐴(,)𝐵)))
172	134	feq2d 6500	. . . . . . . . . 10 ⊢ (𝜑 → (𝐺:(𝐴[,]𝐵)⟶ℂ ↔ 𝐺:∪ ((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵))⟶ℂ))
173	42, 172	mpbid 234	. . . . . . . . 9 ⊢ (𝜑 → 𝐺:∪ ((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵))⟶ℂ)
174	173	adantr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝐺:∪ ((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵))⟶ℂ)
175		eqid 2821	. . . . . . . . 9 ⊢ ∪ ((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵)) = ∪ ((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵))
176	175, 104	cnprest 21897	. . . . . . . 8 ⊢ (((((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵)) ∈ Top ∧ (𝐴(,)𝐵) ⊆ ∪ ((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵))) ∧ (𝑦 ∈ ((int‘((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵)))‘(𝐴(,)𝐵)) ∧ 𝐺:∪ ((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵))⟶ℂ)) → (𝐺 ∈ ((((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵)) CnP (TopOpen‘ℂ_fld))‘𝑦) ↔ (𝐺 ↾ (𝐴(,)𝐵)) ∈ (((((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵)) ↾_t (𝐴(,)𝐵)) CnP (TopOpen‘ℂ_fld))‘𝑦)))
177	128, 136, 171, 174, 176	syl22anc 836	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (𝐺 ∈ ((((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵)) CnP (TopOpen‘ℂ_fld))‘𝑦) ↔ (𝐺 ↾ (𝐴(,)𝐵)) ∈ (((((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵)) ↾_t (𝐴(,)𝐵)) CnP (TopOpen‘ℂ_fld))‘𝑦)))
178	125, 177	mpbird 259	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝐺 ∈ ((((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵)) CnP (TopOpen‘ℂ_fld))‘𝑦))
179		elpri 4589	. . . . . . 7 ⊢ (𝑦 ∈ {𝐴, 𝐵} → (𝑦 = 𝐴 ∨ 𝑦 = 𝐵))
180		iftrue 4473	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝐴 → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))) = 𝑅)
181		lbicc2 12853	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ (𝐴[,]𝐵))
182	19, 22, 25, 181	syl3anc 1367	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 ∈ (𝐴[,]𝐵))
183	41, 180, 182, 3	fvmptd3 6791	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐺‘𝐴) = 𝑅)
184	97	eqcomd 2827	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐹‘𝑥)) = 𝐹)
185	96, 184	eqtr2d 2857	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐹 = (𝐺 ↾ (𝐴(,)𝐵)))
186	185	oveq1d 7171	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐹 lim_ℂ 𝐴) = ((𝐺 ↾ (𝐴(,)𝐵)) lim_ℂ 𝐴))
187	3, 186	eleqtrd 2915	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑅 ∈ ((𝐺 ↾ (𝐴(,)𝐵)) lim_ℂ 𝐴))
188	18, 21, 24, 42	limciccioolb 41922	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐺 ↾ (𝐴(,)𝐵)) lim_ℂ 𝐴) = (𝐺 lim_ℂ 𝐴))
189	187, 188	eleqtrd 2915	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑅 ∈ (𝐺 lim_ℂ 𝐴))
190	183, 189	eqeltrd 2913	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐺‘𝐴) ∈ (𝐺 lim_ℂ 𝐴))
191		eqid 2821	. . . . . . . . . . . . 13 ⊢ ((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵)) = ((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵))
192	101, 191	cnplimc 24485	. . . . . . . . . . . 12 ⊢ (((𝐴[,]𝐵) ⊆ ℂ ∧ 𝐴 ∈ (𝐴[,]𝐵)) → (𝐺 ∈ ((((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵)) CnP (TopOpen‘ℂ_fld))‘𝐴) ↔ (𝐺:(𝐴[,]𝐵)⟶ℂ ∧ (𝐺‘𝐴) ∈ (𝐺 lim_ℂ 𝐴))))
193	132, 182, 192	syl2anc 586	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐺 ∈ ((((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵)) CnP (TopOpen‘ℂ_fld))‘𝐴) ↔ (𝐺:(𝐴[,]𝐵)⟶ℂ ∧ (𝐺‘𝐴) ∈ (𝐺 lim_ℂ 𝐴))))
194	42, 190, 193	mpbir2and 711	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 ∈ ((((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵)) CnP (TopOpen‘ℂ_fld))‘𝐴))
195	194	adantr 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 = 𝐴) → 𝐺 ∈ ((((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵)) CnP (TopOpen‘ℂ_fld))‘𝐴))
196		fveq2 6670	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐴 → ((((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵)) CnP (TopOpen‘ℂ_fld))‘𝑦) = ((((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵)) CnP (TopOpen‘ℂ_fld))‘𝐴))
197	196	eqcomd 2827	. . . . . . . . . 10 ⊢ (𝑦 = 𝐴 → ((((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵)) CnP (TopOpen‘ℂ_fld))‘𝐴) = ((((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵)) CnP (TopOpen‘ℂ_fld))‘𝑦))
198	197	adantl 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 = 𝐴) → ((((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵)) CnP (TopOpen‘ℂ_fld))‘𝐴) = ((((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵)) CnP (TopOpen‘ℂ_fld))‘𝑦))
199	195, 198	eleqtrd 2915	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 = 𝐴) → 𝐺 ∈ ((((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵)) CnP (TopOpen‘ℂ_fld))‘𝑦))
200	180	adantl 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 = 𝐵 ∧ 𝑥 = 𝐴) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))) = 𝑅)
201		eqtr2 2842	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 = 𝐵 ∧ 𝑥 = 𝐴) → 𝐵 = 𝐴)
202		iftrue 4473	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐵 = 𝐴 → if(𝐵 = 𝐴, 𝑅, if(𝐵 = 𝐵, 𝐿, (𝐹‘𝐵))) = 𝑅)
203	202	eqcomd 2827	. . . . . . . . . . . . . . . . 17 ⊢ (𝐵 = 𝐴 → 𝑅 = if(𝐵 = 𝐴, 𝑅, if(𝐵 = 𝐵, 𝐿, (𝐹‘𝐵))))
204	201, 203	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 = 𝐵 ∧ 𝑥 = 𝐴) → 𝑅 = if(𝐵 = 𝐴, 𝑅, if(𝐵 = 𝐵, 𝐿, (𝐹‘𝐵))))
205	200, 204	eqtrd 2856	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 = 𝐵 ∧ 𝑥 = 𝐴) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))) = if(𝐵 = 𝐴, 𝑅, if(𝐵 = 𝐵, 𝐿, (𝐹‘𝐵))))
206		iffalse 4476	. . . . . . . . . . . . . . . . 17 ⊢ (¬ 𝑥 = 𝐴 → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))) = if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥)))
207	206	adantl 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 = 𝐵 ∧ ¬ 𝑥 = 𝐴) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))) = if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥)))
208		iftrue 4473	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝐵 → if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥)) = 𝐿)
209	208	adantr 483	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 = 𝐵 ∧ ¬ 𝑥 = 𝐴) → if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥)) = 𝐿)
210		df-ne 3017	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ≠ 𝐴 ↔ ¬ 𝑥 = 𝐴)
211		pm13.18 3097	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 = 𝐵 ∧ 𝑥 ≠ 𝐴) → 𝐵 ≠ 𝐴)
212	210, 211	sylan2br 596	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 = 𝐵 ∧ ¬ 𝑥 = 𝐴) → 𝐵 ≠ 𝐴)
213	212	neneqd 3021	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 = 𝐵 ∧ ¬ 𝑥 = 𝐴) → ¬ 𝐵 = 𝐴)
214	213	iffalsed 4478	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 = 𝐵 ∧ ¬ 𝑥 = 𝐴) → if(𝐵 = 𝐴, 𝑅, if(𝐵 = 𝐵, 𝐿, (𝐹‘𝐵))) = if(𝐵 = 𝐵, 𝐿, (𝐹‘𝐵)))
215		eqid 2821	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐵 = 𝐵
216	215	iftruei 4474	. . . . . . . . . . . . . . . . 17 ⊢ if(𝐵 = 𝐵, 𝐿, (𝐹‘𝐵)) = 𝐿
217	214, 216	syl6req 2873	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 = 𝐵 ∧ ¬ 𝑥 = 𝐴) → 𝐿 = if(𝐵 = 𝐴, 𝑅, if(𝐵 = 𝐵, 𝐿, (𝐹‘𝐵))))
218	207, 209, 217	3eqtrd 2860	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 = 𝐵 ∧ ¬ 𝑥 = 𝐴) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))) = if(𝐵 = 𝐴, 𝑅, if(𝐵 = 𝐵, 𝐿, (𝐹‘𝐵))))
219	205, 218	pm2.61dan 811	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝐵 → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))) = if(𝐵 = 𝐴, 𝑅, if(𝐵 = 𝐵, 𝐿, (𝐹‘𝐵))))
220	21	leidd 11206	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐵 ≤ 𝐵)
221	18, 21, 21, 25, 220	eliccd 41799	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐵 ∈ (𝐴[,]𝐵))
222	216, 8	eqeltrid 2917	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → if(𝐵 = 𝐵, 𝐿, (𝐹‘𝐵)) ∈ ℂ)
223	4, 222	ifcld 4512	. . . . . . . . . . . . . 14 ⊢ (𝜑 → if(𝐵 = 𝐴, 𝑅, if(𝐵 = 𝐵, 𝐿, (𝐹‘𝐵))) ∈ ℂ)
224	41, 219, 221, 223	fvmptd3 6791	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐺‘𝐵) = if(𝐵 = 𝐴, 𝑅, if(𝐵 = 𝐵, 𝐿, (𝐹‘𝐵))))
225	18, 24	gtned 10775	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐵 ≠ 𝐴)
226	225	neneqd 3021	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ¬ 𝐵 = 𝐴)
227	226	iffalsed 4478	. . . . . . . . . . . . 13 ⊢ (𝜑 → if(𝐵 = 𝐴, 𝑅, if(𝐵 = 𝐵, 𝐿, (𝐹‘𝐵))) = if(𝐵 = 𝐵, 𝐿, (𝐹‘𝐵)))
228	216	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → if(𝐵 = 𝐵, 𝐿, (𝐹‘𝐵)) = 𝐿)
229	224, 227, 228	3eqtrd 2860	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐺‘𝐵) = 𝐿)
230	185	oveq1d 7171	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐹 lim_ℂ 𝐵) = ((𝐺 ↾ (𝐴(,)𝐵)) lim_ℂ 𝐵))
231	7, 230	eleqtrd 2915	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐿 ∈ ((𝐺 ↾ (𝐴(,)𝐵)) lim_ℂ 𝐵))
232	18, 21, 24, 42	limcicciooub 41938	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐺 ↾ (𝐴(,)𝐵)) lim_ℂ 𝐵) = (𝐺 lim_ℂ 𝐵))
233	231, 232	eleqtrd 2915	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐿 ∈ (𝐺 lim_ℂ 𝐵))
234	229, 233	eqeltrd 2913	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐺‘𝐵) ∈ (𝐺 lim_ℂ 𝐵))
235	101, 191	cnplimc 24485	. . . . . . . . . . . 12 ⊢ (((𝐴[,]𝐵) ⊆ ℂ ∧ 𝐵 ∈ (𝐴[,]𝐵)) → (𝐺 ∈ ((((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵)) CnP (TopOpen‘ℂ_fld))‘𝐵) ↔ (𝐺:(𝐴[,]𝐵)⟶ℂ ∧ (𝐺‘𝐵) ∈ (𝐺 lim_ℂ 𝐵))))
236	132, 221, 235	syl2anc 586	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐺 ∈ ((((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵)) CnP (TopOpen‘ℂ_fld))‘𝐵) ↔ (𝐺:(𝐴[,]𝐵)⟶ℂ ∧ (𝐺‘𝐵) ∈ (𝐺 lim_ℂ 𝐵))))
237	42, 234, 236	mpbir2and 711	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 ∈ ((((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵)) CnP (TopOpen‘ℂ_fld))‘𝐵))
238	237	adantr 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 = 𝐵) → 𝐺 ∈ ((((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵)) CnP (TopOpen‘ℂ_fld))‘𝐵))
239		fveq2 6670	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐵 → ((((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵)) CnP (TopOpen‘ℂ_fld))‘𝑦) = ((((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵)) CnP (TopOpen‘ℂ_fld))‘𝐵))
240	239	eqcomd 2827	. . . . . . . . . 10 ⊢ (𝑦 = 𝐵 → ((((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵)) CnP (TopOpen‘ℂ_fld))‘𝐵) = ((((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵)) CnP (TopOpen‘ℂ_fld))‘𝑦))
241	240	adantl 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 = 𝐵) → ((((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵)) CnP (TopOpen‘ℂ_fld))‘𝐵) = ((((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵)) CnP (TopOpen‘ℂ_fld))‘𝑦))
242	238, 241	eleqtrd 2915	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 = 𝐵) → 𝐺 ∈ ((((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵)) CnP (TopOpen‘ℂ_fld))‘𝑦))
243	199, 242	jaodan 954	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 = 𝐴 ∨ 𝑦 = 𝐵)) → 𝐺 ∈ ((((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵)) CnP (TopOpen‘ℂ_fld))‘𝑦))
244	179, 243	sylan2 594	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ {𝐴, 𝐵}) → 𝐺 ∈ ((((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵)) CnP (TopOpen‘ℂ_fld))‘𝑦))
245	178, 244	jaodan 954	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝐴(,)𝐵) ∨ 𝑦 ∈ {𝐴, 𝐵})) → 𝐺 ∈ ((((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵)) CnP (TopOpen‘ℂ_fld))‘𝑦))
246	47, 245	syldan 593	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → 𝐺 ∈ ((((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵)) CnP (TopOpen‘ℂ_fld))‘𝑦))
247	246	ralrimiva 3182	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ (𝐴[,]𝐵)𝐺 ∈ ((((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵)) CnP (TopOpen‘ℂ_fld))‘𝑦))
248	101	cnfldtopon 23391	. . . . 5 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
249		resttopon 21769	. . . . 5 ⊢ (((TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ) ∧ (𝐴[,]𝐵) ⊆ ℂ) → ((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵)) ∈ (TopOn‘(𝐴[,]𝐵)))
250	248, 132, 249	sylancr 589	. . . 4 ⊢ (𝜑 → ((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵)) ∈ (TopOn‘(𝐴[,]𝐵)))
251		cncnp 21888	. . . 4 ⊢ ((((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵)) ∈ (TopOn‘(𝐴[,]𝐵)) ∧ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)) → (𝐺 ∈ (((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵)) Cn (TopOpen‘ℂ_fld)) ↔ (𝐺:(𝐴[,]𝐵)⟶ℂ ∧ ∀𝑦 ∈ (𝐴[,]𝐵)𝐺 ∈ ((((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵)) CnP (TopOpen‘ℂ_fld))‘𝑦))))
252	250, 248, 251	sylancl 588	. . 3 ⊢ (𝜑 → (𝐺 ∈ (((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵)) Cn (TopOpen‘ℂ_fld)) ↔ (𝐺:(𝐴[,]𝐵)⟶ℂ ∧ ∀𝑦 ∈ (𝐴[,]𝐵)𝐺 ∈ ((((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵)) CnP (TopOpen‘ℂ_fld))‘𝑦))))
253	42, 247, 252	mpbir2and 711	. 2 ⊢ (𝜑 → 𝐺 ∈ (((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵)) Cn (TopOpen‘ℂ_fld)))
254	101, 191, 107	cncfcn 23517	. . 3 ⊢ (((𝐴[,]𝐵) ⊆ ℂ ∧ ℂ ⊆ ℂ) → ((𝐴[,]𝐵)–cn→ℂ) = (((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵)) Cn (TopOpen‘ℂ_fld)))
255	132, 100, 254	sylancl 588	. 2 ⊢ (𝜑 → ((𝐴[,]𝐵)–cn→ℂ) = (((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵)) Cn (TopOpen‘ℂ_fld)))
256	253, 255	eleqtrrd 2916	1 ⊢ (𝜑 → 𝐺 ∈ ((𝐴[,]𝐵)–cn→ℂ))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 ∧ w3a 1083 = wceq 1537 Ⅎwnf 1784 ∈ wcel 2114 ≠ wne 3016 ∀wral 3138 Vcvv 3494 ∖ cdif 3933 ∪ cun 3934 ∩ cin 3935 ⊆ wss 3936 ifcif 4467 {cpr 4569 ∪ cuni 4838 class class class wbr 5066 ↦ cmpt 5146 ran crn 5556 ↾ cres 5557 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 ℂcc 10535 ℝcr 10536 ℝ^*cxr 10674 < clt 10675 ≤ cle 10676 (,)cioo 12739 [,]cicc 12742 ↾_t crest 16694 TopOpenctopn 16695 topGenctg 16711 ℂ_fldccnfld 20545 Topctop 21501 TopOnctopon 21518 intcnt 21625 Cn ccn 21832 CnP ccnp 21833 –cn→ccncf 23484 lim_ℂ climc 24460

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-map 8408 df-pm 8409 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fi 8875 df-sup 8906 df-inf 8907 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-q 12350 df-rp 12391 df-xneg 12508 df-xadd 12509 df-xmul 12510 df-ioo 12743 df-ioc 12744 df-ico 12745 df-icc 12746 df-fz 12894 df-seq 13371 df-exp 13431 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-plusg 16578 df-mulr 16579 df-starv 16580 df-tset 16584 df-ple 16585 df-ds 16587 df-unif 16588 df-rest 16696 df-topn 16697 df-topgen 16717 df-psmet 20537 df-xmet 20538 df-met 20539 df-bl 20540 df-mopn 20541 df-cnfld 20546 df-top 21502 df-topon 21519 df-topsp 21541 df-bases 21554 df-cld 21627 df-ntr 21628 df-cls 21629 df-cn 21835 df-cnp 21836 df-xms 22930 df-ms 22931 df-cncf 23486 df-limc 24464

This theorem is referenced by: cncfiooicc 42197

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