fourierdlem32 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem32 46567

Description: Limit of a continuous function on an open subinterval. Lower bound version. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Hypotheses**
Ref	Expression
fourierdlem32.a	⊢ (𝜑 → 𝐴 ∈ ℝ)
fourierdlem32.b	⊢ (𝜑 → 𝐵 ∈ ℝ)
fourierdlem32.altb	⊢ (𝜑 → 𝐴 < 𝐵)
fourierdlem32.f	⊢ (𝜑 → 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ))
fourierdlem32.l	⊢ (𝜑 → 𝑅 ∈ (𝐹 lim_ℂ 𝐴))
fourierdlem32.c	⊢ (𝜑 → 𝐶 ∈ ℝ)
fourierdlem32.d	⊢ (𝜑 → 𝐷 ∈ ℝ)
fourierdlem32.cltd	⊢ (𝜑 → 𝐶 < 𝐷)
fourierdlem32.ss	⊢ (𝜑 → (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵))
fourierdlem32.y	⊢ 𝑌 = if(𝐶 = 𝐴, 𝑅, (𝐹‘𝐶))
fourierdlem32.j	⊢ 𝐽 = ((TopOpen‘ℂ_fld) ↾_t (𝐴[,)𝐵))

**Assertion**
Ref	Expression
fourierdlem32	⊢ (𝜑 → 𝑌 ∈ ((𝐹 ↾ (𝐶(,)𝐷)) lim_ℂ 𝐶))

Proof of Theorem fourierdlem32 Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fourierdlem32.l	. . . 4 ⊢ (𝜑 → 𝑅 ∈ (𝐹 lim_ℂ 𝐴))
2	1	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝐶 = 𝐴) → 𝑅 ∈ (𝐹 lim_ℂ 𝐴))
3		fourierdlem32.y	. . . . 5 ⊢ 𝑌 = if(𝐶 = 𝐴, 𝑅, (𝐹‘𝐶))
4		iftrue 4472	. . . . 5 ⊢ (𝐶 = 𝐴 → if(𝐶 = 𝐴, 𝑅, (𝐹‘𝐶)) = 𝑅)
5	3, 4	eqtr2id 2784	. . . 4 ⊢ (𝐶 = 𝐴 → 𝑅 = 𝑌)
6	5	adantl 481	. . 3 ⊢ ((𝜑 ∧ 𝐶 = 𝐴) → 𝑅 = 𝑌)
7		oveq2 7375	. . . . 5 ⊢ (𝐶 = 𝐴 → ((𝐹 ↾ (𝐶(,)𝐷)) lim_ℂ 𝐶) = ((𝐹 ↾ (𝐶(,)𝐷)) lim_ℂ 𝐴))
8	7	adantl 481	. . . 4 ⊢ ((𝜑 ∧ 𝐶 = 𝐴) → ((𝐹 ↾ (𝐶(,)𝐷)) lim_ℂ 𝐶) = ((𝐹 ↾ (𝐶(,)𝐷)) lim_ℂ 𝐴))
9		fourierdlem32.f	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ))
10		cncff 24860	. . . . . . 7 ⊢ (𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ) → 𝐹:(𝐴(,)𝐵)⟶ℂ)
11	9, 10	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐹:(𝐴(,)𝐵)⟶ℂ)
12	11	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 = 𝐴) → 𝐹:(𝐴(,)𝐵)⟶ℂ)
13		fourierdlem32.ss	. . . . . 6 ⊢ (𝜑 → (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵))
14	13	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 = 𝐴) → (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵))
15		ioosscn 13361	. . . . . 6 ⊢ (𝐴(,)𝐵) ⊆ ℂ
16	15	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 = 𝐴) → (𝐴(,)𝐵) ⊆ ℂ)
17		eqid 2736	. . . . 5 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
18		eqid 2736	. . . . 5 ⊢ ((TopOpen‘ℂ_fld) ↾_t ((𝐴(,)𝐵) ∪ {𝐴})) = ((TopOpen‘ℂ_fld) ↾_t ((𝐴(,)𝐵) ∪ {𝐴}))
19		fourierdlem32.c	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ ℝ)
20	19	leidd 11716	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ≤ 𝐶)
21		fourierdlem32.cltd	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 < 𝐷)
22		fourierdlem32.d	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷 ∈ ℝ)
23	22	rexrd 11195	. . . . . . . . . 10 ⊢ (𝜑 → 𝐷 ∈ ℝ^*)
24		elico2 13363	. . . . . . . . . 10 ⊢ ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ^*) → (𝐶 ∈ (𝐶[,)𝐷) ↔ (𝐶 ∈ ℝ ∧ 𝐶 ≤ 𝐶 ∧ 𝐶 < 𝐷)))
25	19, 23, 24	syl2anc 585	. . . . . . . . 9 ⊢ (𝜑 → (𝐶 ∈ (𝐶[,)𝐷) ↔ (𝐶 ∈ ℝ ∧ 𝐶 ≤ 𝐶 ∧ 𝐶 < 𝐷)))
26	19, 20, 21, 25	mpbir3and 1344	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ (𝐶[,)𝐷))
27	26	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 = 𝐴) → 𝐶 ∈ (𝐶[,)𝐷))
28		fourierdlem32.j	. . . . . . . . 9 ⊢ 𝐽 = ((TopOpen‘ℂ_fld) ↾_t (𝐴[,)𝐵))
29	17	cnfldtop 24748	. . . . . . . . . 10 ⊢ (TopOpen‘ℂ_fld) ∈ Top
30		ovex 7400	. . . . . . . . . . 11 ⊢ (𝐴[,)𝐵) ∈ V
31	30	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐶 = 𝐴) → (𝐴[,)𝐵) ∈ V)
32		resttop 23125	. . . . . . . . . 10 ⊢ (((TopOpen‘ℂ_fld) ∈ Top ∧ (𝐴[,)𝐵) ∈ V) → ((TopOpen‘ℂ_fld) ↾_t (𝐴[,)𝐵)) ∈ Top)
33	29, 31, 32	sylancr 588	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 = 𝐴) → ((TopOpen‘ℂ_fld) ↾_t (𝐴[,)𝐵)) ∈ Top)
34	28, 33	eqeltrid 2840	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 = 𝐴) → 𝐽 ∈ Top)
35		mnfxr 11202	. . . . . . . . . . . . . . . 16 ⊢ -∞ ∈ ℝ^*
36	35	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,)𝐷)) → -∞ ∈ ℝ^*)
37	23	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,)𝐷)) → 𝐷 ∈ ℝ^*)
38		simpr 484	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,)𝐷)) → 𝑥 ∈ (𝐴[,)𝐷))
39		fourierdlem32.a	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐴 ∈ ℝ)
40	39	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,)𝐷)) → 𝐴 ∈ ℝ)
41		elico2 13363	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ ℝ ∧ 𝐷 ∈ ℝ^*) → (𝑥 ∈ (𝐴[,)𝐷) ↔ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐷)))
42	40, 37, 41	syl2anc 585	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,)𝐷)) → (𝑥 ∈ (𝐴[,)𝐷) ↔ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐷)))
43	38, 42	mpbid 232	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,)𝐷)) → (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐷))
44	43	simp1d 1143	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,)𝐷)) → 𝑥 ∈ ℝ)
45	44	mnfltd 13075	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,)𝐷)) → -∞ < 𝑥)
46	43	simp3d 1145	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,)𝐷)) → 𝑥 < 𝐷)
47	36, 37, 44, 45, 46	eliood 45928	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,)𝐷)) → 𝑥 ∈ (-∞(,)𝐷))
48	43	simp2d 1144	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,)𝐷)) → 𝐴 ≤ 𝑥)
49	22	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,)𝐷)) → 𝐷 ∈ ℝ)
50		fourierdlem32.b	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐵 ∈ ℝ)
51	50	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,)𝐷)) → 𝐵 ∈ ℝ)
52	39, 50, 19, 22, 21, 13	fourierdlem10 46545	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐴 ≤ 𝐶 ∧ 𝐷 ≤ 𝐵))
53	52	simprd 495	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐷 ≤ 𝐵)
54	53	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,)𝐷)) → 𝐷 ≤ 𝐵)
55	44, 49, 51, 46, 54	ltletrd 11306	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,)𝐷)) → 𝑥 < 𝐵)
56	50	rexrd 11195	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
57	56	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,)𝐷)) → 𝐵 ∈ ℝ^*)
58		elico2 13363	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) → (𝑥 ∈ (𝐴[,)𝐵) ↔ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵)))
59	40, 57, 58	syl2anc 585	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,)𝐷)) → (𝑥 ∈ (𝐴[,)𝐵) ↔ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵)))
60	44, 48, 55, 59	mpbir3and 1344	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,)𝐷)) → 𝑥 ∈ (𝐴[,)𝐵))
61	47, 60	elind 4140	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,)𝐷)) → 𝑥 ∈ ((-∞(,)𝐷) ∩ (𝐴[,)𝐵)))
62		elinel1 4141	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ((-∞(,)𝐷) ∩ (𝐴[,)𝐵)) → 𝑥 ∈ (-∞(,)𝐷))
63		elioore 13328	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (-∞(,)𝐷) → 𝑥 ∈ ℝ)
64	62, 63	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ((-∞(,)𝐷) ∩ (𝐴[,)𝐵)) → 𝑥 ∈ ℝ)
65	64	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ((-∞(,)𝐷) ∩ (𝐴[,)𝐵))) → 𝑥 ∈ ℝ)
66		elinel2 4142	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ((-∞(,)𝐷) ∩ (𝐴[,)𝐵)) → 𝑥 ∈ (𝐴[,)𝐵))
67	66	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ((-∞(,)𝐷) ∩ (𝐴[,)𝐵))) → 𝑥 ∈ (𝐴[,)𝐵))
68	39	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ ((-∞(,)𝐷) ∩ (𝐴[,)𝐵))) → 𝐴 ∈ ℝ)
69	56	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ ((-∞(,)𝐷) ∩ (𝐴[,)𝐵))) → 𝐵 ∈ ℝ^*)
70	68, 69, 58	syl2anc 585	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ((-∞(,)𝐷) ∩ (𝐴[,)𝐵))) → (𝑥 ∈ (𝐴[,)𝐵) ↔ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵)))
71	67, 70	mpbid 232	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ((-∞(,)𝐷) ∩ (𝐴[,)𝐵))) → (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵))
72	71	simp2d 1144	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ((-∞(,)𝐷) ∩ (𝐴[,)𝐵))) → 𝐴 ≤ 𝑥)
73	62	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ((-∞(,)𝐷) ∩ (𝐴[,)𝐵))) → 𝑥 ∈ (-∞(,)𝐷))
74	23	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ ((-∞(,)𝐷) ∩ (𝐴[,)𝐵))) → 𝐷 ∈ ℝ^*)
75		elioo2 13339	. . . . . . . . . . . . . . . . 17 ⊢ ((-∞ ∈ ℝ^* ∧ 𝐷 ∈ ℝ^*) → (𝑥 ∈ (-∞(,)𝐷) ↔ (𝑥 ∈ ℝ ∧ -∞ < 𝑥 ∧ 𝑥 < 𝐷)))
76	35, 74, 75	sylancr 588	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ((-∞(,)𝐷) ∩ (𝐴[,)𝐵))) → (𝑥 ∈ (-∞(,)𝐷) ↔ (𝑥 ∈ ℝ ∧ -∞ < 𝑥 ∧ 𝑥 < 𝐷)))
77	73, 76	mpbid 232	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ((-∞(,)𝐷) ∩ (𝐴[,)𝐵))) → (𝑥 ∈ ℝ ∧ -∞ < 𝑥 ∧ 𝑥 < 𝐷))
78	77	simp3d 1145	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ((-∞(,)𝐷) ∩ (𝐴[,)𝐵))) → 𝑥 < 𝐷)
79	68, 74, 41	syl2anc 585	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ((-∞(,)𝐷) ∩ (𝐴[,)𝐵))) → (𝑥 ∈ (𝐴[,)𝐷) ↔ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐷)))
80	65, 72, 78, 79	mpbir3and 1344	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ((-∞(,)𝐷) ∩ (𝐴[,)𝐵))) → 𝑥 ∈ (𝐴[,)𝐷))
81	61, 80	impbida 801	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑥 ∈ (𝐴[,)𝐷) ↔ 𝑥 ∈ ((-∞(,)𝐷) ∩ (𝐴[,)𝐵))))
82	81	eqrdv 2734	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴[,)𝐷) = ((-∞(,)𝐷) ∩ (𝐴[,)𝐵)))
83		retop 24726	. . . . . . . . . . . . 13 ⊢ (topGen‘ran (,)) ∈ Top
84	83	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → (topGen‘ran (,)) ∈ Top)
85	30	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴[,)𝐵) ∈ V)
86		iooretop 24730	. . . . . . . . . . . . 13 ⊢ (-∞(,)𝐷) ∈ (topGen‘ran (,))
87	86	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → (-∞(,)𝐷) ∈ (topGen‘ran (,)))
88		elrestr 17391	. . . . . . . . . . . 12 ⊢ (((topGen‘ran (,)) ∈ Top ∧ (𝐴[,)𝐵) ∈ V ∧ (-∞(,)𝐷) ∈ (topGen‘ran (,))) → ((-∞(,)𝐷) ∩ (𝐴[,)𝐵)) ∈ ((topGen‘ran (,)) ↾_t (𝐴[,)𝐵)))
89	84, 85, 87, 88	syl3anc 1374	. . . . . . . . . . 11 ⊢ (𝜑 → ((-∞(,)𝐷) ∩ (𝐴[,)𝐵)) ∈ ((topGen‘ran (,)) ↾_t (𝐴[,)𝐵)))
90	82, 89	eqeltrd 2836	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴[,)𝐷) ∈ ((topGen‘ran (,)) ↾_t (𝐴[,)𝐵)))
91	90	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 = 𝐴) → (𝐴[,)𝐷) ∈ ((topGen‘ran (,)) ↾_t (𝐴[,)𝐵)))
92		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐶 = 𝐴) → 𝐶 = 𝐴)
93	92	oveq1d 7382	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 = 𝐴) → (𝐶[,)𝐷) = (𝐴[,)𝐷))
94	28	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐽 = ((TopOpen‘ℂ_fld) ↾_t (𝐴[,)𝐵)))
95	29	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → (TopOpen‘ℂ_fld) ∈ Top)
96		icossre 13381	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) → (𝐴[,)𝐵) ⊆ ℝ)
97	39, 56, 96	syl2anc 585	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴[,)𝐵) ⊆ ℝ)
98		reex 11129	. . . . . . . . . . . . 13 ⊢ ℝ ∈ V
99	98	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → ℝ ∈ V)
100		restabs 23130	. . . . . . . . . . . 12 ⊢ (((TopOpen‘ℂ_fld) ∈ Top ∧ (𝐴[,)𝐵) ⊆ ℝ ∧ ℝ ∈ V) → (((TopOpen‘ℂ_fld) ↾_t ℝ) ↾_t (𝐴[,)𝐵)) = ((TopOpen‘ℂ_fld) ↾_t (𝐴[,)𝐵)))
101	95, 97, 99, 100	syl3anc 1374	. . . . . . . . . . 11 ⊢ (𝜑 → (((TopOpen‘ℂ_fld) ↾_t ℝ) ↾_t (𝐴[,)𝐵)) = ((TopOpen‘ℂ_fld) ↾_t (𝐴[,)𝐵)))
102		tgioo4 24770	. . . . . . . . . . . . . 14 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
103	102	eqcomi 2745	. . . . . . . . . . . . 13 ⊢ ((TopOpen‘ℂ_fld) ↾_t ℝ) = (topGen‘ran (,))
104	103	oveq1i 7377	. . . . . . . . . . . 12 ⊢ (((TopOpen‘ℂ_fld) ↾_t ℝ) ↾_t (𝐴[,)𝐵)) = ((topGen‘ran (,)) ↾_t (𝐴[,)𝐵))
105	104	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → (((TopOpen‘ℂ_fld) ↾_t ℝ) ↾_t (𝐴[,)𝐵)) = ((topGen‘ran (,)) ↾_t (𝐴[,)𝐵)))
106	94, 101, 105	3eqtr2d 2777	. . . . . . . . . 10 ⊢ (𝜑 → 𝐽 = ((topGen‘ran (,)) ↾_t (𝐴[,)𝐵)))
107	106	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 = 𝐴) → 𝐽 = ((topGen‘ran (,)) ↾_t (𝐴[,)𝐵)))
108	91, 93, 107	3eltr4d 2851	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 = 𝐴) → (𝐶[,)𝐷) ∈ 𝐽)
109		isopn3i 23047	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ (𝐶[,)𝐷) ∈ 𝐽) → ((int‘𝐽)‘(𝐶[,)𝐷)) = (𝐶[,)𝐷))
110	34, 108, 109	syl2anc 585	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 = 𝐴) → ((int‘𝐽)‘(𝐶[,)𝐷)) = (𝐶[,)𝐷))
111	27, 110	eleqtrrd 2839	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶 = 𝐴) → 𝐶 ∈ ((int‘𝐽)‘(𝐶[,)𝐷)))
112		id 22	. . . . . . . 8 ⊢ (𝐶 = 𝐴 → 𝐶 = 𝐴)
113	112	eqcomd 2742	. . . . . . 7 ⊢ (𝐶 = 𝐴 → 𝐴 = 𝐶)
114	113	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶 = 𝐴) → 𝐴 = 𝐶)
115		uncom 4098	. . . . . . . . . . . 12 ⊢ ((𝐴(,)𝐵) ∪ {𝐴}) = ({𝐴} ∪ (𝐴(,)𝐵))
116	39	rexrd 11195	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
117		fourierdlem32.altb	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 < 𝐵)
118		snunioo 13431	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → ({𝐴} ∪ (𝐴(,)𝐵)) = (𝐴[,)𝐵))
119	116, 56, 117, 118	syl3anc 1374	. . . . . . . . . . . 12 ⊢ (𝜑 → ({𝐴} ∪ (𝐴(,)𝐵)) = (𝐴[,)𝐵))
120	115, 119	eqtrid 2783	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐴(,)𝐵) ∪ {𝐴}) = (𝐴[,)𝐵))
121	120	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐶 = 𝐴) → ((𝐴(,)𝐵) ∪ {𝐴}) = (𝐴[,)𝐵))
122	121	oveq2d 7383	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 = 𝐴) → ((TopOpen‘ℂ_fld) ↾_t ((𝐴(,)𝐵) ∪ {𝐴})) = ((TopOpen‘ℂ_fld) ↾_t (𝐴[,)𝐵)))
123	122, 28	eqtr4di 2789	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 = 𝐴) → ((TopOpen‘ℂ_fld) ↾_t ((𝐴(,)𝐵) ∪ {𝐴})) = 𝐽)
124	123	fveq2d 6844	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 = 𝐴) → (int‘((TopOpen‘ℂ_fld) ↾_t ((𝐴(,)𝐵) ∪ {𝐴}))) = (int‘𝐽))
125		uncom 4098	. . . . . . . . 9 ⊢ ((𝐶(,)𝐷) ∪ {𝐴}) = ({𝐴} ∪ (𝐶(,)𝐷))
126		sneq 4577	. . . . . . . . . . 11 ⊢ (𝐶 = 𝐴 → {𝐶} = {𝐴})
127	126	eqcomd 2742	. . . . . . . . . 10 ⊢ (𝐶 = 𝐴 → {𝐴} = {𝐶})
128	127	uneq1d 4107	. . . . . . . . 9 ⊢ (𝐶 = 𝐴 → ({𝐴} ∪ (𝐶(,)𝐷)) = ({𝐶} ∪ (𝐶(,)𝐷)))
129	125, 128	eqtrid 2783	. . . . . . . 8 ⊢ (𝐶 = 𝐴 → ((𝐶(,)𝐷) ∪ {𝐴}) = ({𝐶} ∪ (𝐶(,)𝐷)))
130	19	rexrd 11195	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ ℝ^*)
131		snunioo 13431	. . . . . . . . 9 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐷 ∈ ℝ^* ∧ 𝐶 < 𝐷) → ({𝐶} ∪ (𝐶(,)𝐷)) = (𝐶[,)𝐷))
132	130, 23, 21, 131	syl3anc 1374	. . . . . . . 8 ⊢ (𝜑 → ({𝐶} ∪ (𝐶(,)𝐷)) = (𝐶[,)𝐷))
133	129, 132	sylan9eqr 2793	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 = 𝐴) → ((𝐶(,)𝐷) ∪ {𝐴}) = (𝐶[,)𝐷))
134	124, 133	fveq12d 6847	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶 = 𝐴) → ((int‘((TopOpen‘ℂ_fld) ↾_t ((𝐴(,)𝐵) ∪ {𝐴})))‘((𝐶(,)𝐷) ∪ {𝐴})) = ((int‘𝐽)‘(𝐶[,)𝐷)))
135	111, 114, 134	3eltr4d 2851	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 = 𝐴) → 𝐴 ∈ ((int‘((TopOpen‘ℂ_fld) ↾_t ((𝐴(,)𝐵) ∪ {𝐴})))‘((𝐶(,)𝐷) ∪ {𝐴})))
136	12, 14, 16, 17, 18, 135	limcres 25853	. . . 4 ⊢ ((𝜑 ∧ 𝐶 = 𝐴) → ((𝐹 ↾ (𝐶(,)𝐷)) lim_ℂ 𝐴) = (𝐹 lim_ℂ 𝐴))
137	8, 136	eqtr2d 2772	. . 3 ⊢ ((𝜑 ∧ 𝐶 = 𝐴) → (𝐹 lim_ℂ 𝐴) = ((𝐹 ↾ (𝐶(,)𝐷)) lim_ℂ 𝐶))
138	2, 6, 137	3eltr3d 2850	. 2 ⊢ ((𝜑 ∧ 𝐶 = 𝐴) → 𝑌 ∈ ((𝐹 ↾ (𝐶(,)𝐷)) lim_ℂ 𝐶))
139		limcresi 25852	. . 3 ⊢ (𝐹 lim_ℂ 𝐶) ⊆ ((𝐹 ↾ (𝐶(,)𝐷)) lim_ℂ 𝐶)
140		iffalse 4475	. . . . . 6 ⊢ (¬ 𝐶 = 𝐴 → if(𝐶 = 𝐴, 𝑅, (𝐹‘𝐶)) = (𝐹‘𝐶))
141	3, 140	eqtrid 2783	. . . . 5 ⊢ (¬ 𝐶 = 𝐴 → 𝑌 = (𝐹‘𝐶))
142	141	adantl 481	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐶 = 𝐴) → 𝑌 = (𝐹‘𝐶))
143		ssid 3944	. . . . . . . . . . . . 13 ⊢ ℂ ⊆ ℂ
144	143	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → ℂ ⊆ ℂ)
145		eqid 2736	. . . . . . . . . . . . 13 ⊢ ((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) = ((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵))
146		unicntop 24750	. . . . . . . . . . . . . . . 16 ⊢ ℂ = ∪ (TopOpen‘ℂ_fld)
147	146	restid 17396	. . . . . . . . . . . . . . 15 ⊢ ((TopOpen‘ℂ_fld) ∈ Top → ((TopOpen‘ℂ_fld) ↾_t ℂ) = (TopOpen‘ℂ_fld))
148	29, 147	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ ((TopOpen‘ℂ_fld) ↾_t ℂ) = (TopOpen‘ℂ_fld)
149	148	eqcomi 2745	. . . . . . . . . . . . 13 ⊢ (TopOpen‘ℂ_fld) = ((TopOpen‘ℂ_fld) ↾_t ℂ)
150	17, 145, 149	cncfcn 24877	. . . . . . . . . . . 12 ⊢ (((𝐴(,)𝐵) ⊆ ℂ ∧ ℂ ⊆ ℂ) → ((𝐴(,)𝐵)–cn→ℂ) = (((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) Cn (TopOpen‘ℂ_fld)))
151	15, 144, 150	sylancr 588	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐴(,)𝐵)–cn→ℂ) = (((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) Cn (TopOpen‘ℂ_fld)))
152	9, 151	eleqtrd 2838	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ (((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) Cn (TopOpen‘ℂ_fld)))
153	17	cnfldtopon 24747	. . . . . . . . . . . 12 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
154		resttopon 23126	. . . . . . . . . . . 12 ⊢ (((TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ) ∧ (𝐴(,)𝐵) ⊆ ℂ) → ((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) ∈ (TopOn‘(𝐴(,)𝐵)))
155	153, 15, 154	mp2an 693	. . . . . . . . . . 11 ⊢ ((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) ∈ (TopOn‘(𝐴(,)𝐵))
156		cncnp 23245	. . . . . . . . . . 11 ⊢ ((((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) ∈ (TopOn‘(𝐴(,)𝐵)) ∧ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)) → (𝐹 ∈ (((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) Cn (TopOpen‘ℂ_fld)) ↔ (𝐹:(𝐴(,)𝐵)⟶ℂ ∧ ∀𝑥 ∈ (𝐴(,)𝐵)𝐹 ∈ ((((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) CnP (TopOpen‘ℂ_fld))‘𝑥))))
157	155, 153, 156	mp2an 693	. . . . . . . . . 10 ⊢ (𝐹 ∈ (((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) Cn (TopOpen‘ℂ_fld)) ↔ (𝐹:(𝐴(,)𝐵)⟶ℂ ∧ ∀𝑥 ∈ (𝐴(,)𝐵)𝐹 ∈ ((((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) CnP (TopOpen‘ℂ_fld))‘𝑥)))
158	152, 157	sylib 218	. . . . . . . . 9 ⊢ (𝜑 → (𝐹:(𝐴(,)𝐵)⟶ℂ ∧ ∀𝑥 ∈ (𝐴(,)𝐵)𝐹 ∈ ((((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) CnP (TopOpen‘ℂ_fld))‘𝑥)))
159	158	simprd 495	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ (𝐴(,)𝐵)𝐹 ∈ ((((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) CnP (TopOpen‘ℂ_fld))‘𝑥))
160	159	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐶 = 𝐴) → ∀𝑥 ∈ (𝐴(,)𝐵)𝐹 ∈ ((((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) CnP (TopOpen‘ℂ_fld))‘𝑥))
161	116	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐶 = 𝐴) → 𝐴 ∈ ℝ^*)
162	56	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐶 = 𝐴) → 𝐵 ∈ ℝ^*)
163	19	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐶 = 𝐴) → 𝐶 ∈ ℝ)
164	39	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐶 = 𝐴) → 𝐴 ∈ ℝ)
165	52	simpld 494	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ≤ 𝐶)
166	165	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐶 = 𝐴) → 𝐴 ≤ 𝐶)
167	112	eqcoms 2744	. . . . . . . . . . . 12 ⊢ (𝐴 = 𝐶 → 𝐶 = 𝐴)
168	167	necon3bi 2958	. . . . . . . . . . 11 ⊢ (¬ 𝐶 = 𝐴 → 𝐴 ≠ 𝐶)
169	168	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐶 = 𝐴) → 𝐴 ≠ 𝐶)
170	169	necomd 2987	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐶 = 𝐴) → 𝐶 ≠ 𝐴)
171	164, 163, 166, 170	leneltd 11300	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐶 = 𝐴) → 𝐴 < 𝐶)
172	19, 22, 50, 21, 53	ltletrd 11306	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 < 𝐵)
173	172	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐶 = 𝐴) → 𝐶 < 𝐵)
174	161, 162, 163, 171, 173	eliood 45928	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐶 = 𝐴) → 𝐶 ∈ (𝐴(,)𝐵))
175		fveq2 6840	. . . . . . . . 9 ⊢ (𝑥 = 𝐶 → ((((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) CnP (TopOpen‘ℂ_fld))‘𝑥) = ((((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) CnP (TopOpen‘ℂ_fld))‘𝐶))
176	175	eleq2d 2822	. . . . . . . 8 ⊢ (𝑥 = 𝐶 → (𝐹 ∈ ((((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) CnP (TopOpen‘ℂ_fld))‘𝑥) ↔ 𝐹 ∈ ((((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) CnP (TopOpen‘ℂ_fld))‘𝐶)))
177	176	rspccva 3563	. . . . . . 7 ⊢ ((∀𝑥 ∈ (𝐴(,)𝐵)𝐹 ∈ ((((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) CnP (TopOpen‘ℂ_fld))‘𝑥) ∧ 𝐶 ∈ (𝐴(,)𝐵)) → 𝐹 ∈ ((((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) CnP (TopOpen‘ℂ_fld))‘𝐶))
178	160, 174, 177	syl2anc 585	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐶 = 𝐴) → 𝐹 ∈ ((((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) CnP (TopOpen‘ℂ_fld))‘𝐶))
179	17, 145	cnplimc 25854	. . . . . . 7 ⊢ (((𝐴(,)𝐵) ⊆ ℂ ∧ 𝐶 ∈ (𝐴(,)𝐵)) → (𝐹 ∈ ((((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) CnP (TopOpen‘ℂ_fld))‘𝐶) ↔ (𝐹:(𝐴(,)𝐵)⟶ℂ ∧ (𝐹‘𝐶) ∈ (𝐹 lim_ℂ 𝐶))))
180	15, 174, 179	sylancr 588	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐶 = 𝐴) → (𝐹 ∈ ((((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) CnP (TopOpen‘ℂ_fld))‘𝐶) ↔ (𝐹:(𝐴(,)𝐵)⟶ℂ ∧ (𝐹‘𝐶) ∈ (𝐹 lim_ℂ 𝐶))))
181	178, 180	mpbid 232	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐶 = 𝐴) → (𝐹:(𝐴(,)𝐵)⟶ℂ ∧ (𝐹‘𝐶) ∈ (𝐹 lim_ℂ 𝐶)))
182	181	simprd 495	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐶 = 𝐴) → (𝐹‘𝐶) ∈ (𝐹 lim_ℂ 𝐶))
183	142, 182	eqeltrd 2836	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐶 = 𝐴) → 𝑌 ∈ (𝐹 lim_ℂ 𝐶))
184	139, 183	sselid 3919	. 2 ⊢ ((𝜑 ∧ ¬ 𝐶 = 𝐴) → 𝑌 ∈ ((𝐹 ↾ (𝐶(,)𝐷)) lim_ℂ 𝐶))
185	138, 184	pm2.61dan 813	1 ⊢ (𝜑 → 𝑌 ∈ ((𝐹 ↾ (𝐶(,)𝐷)) lim_ℂ 𝐶))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ≠ wne 2932 ∀wral 3051 Vcvv 3429 ∪ cun 3887 ∩ cin 3888 ⊆ wss 3889 ifcif 4466 {csn 4567 class class class wbr 5085 ran crn 5632 ↾ cres 5633 ⟶wf 6494 ‘cfv 6498 (class class class)co 7367 ℂcc 11036 ℝcr 11037 -∞cmnf 11177 ℝ^*cxr 11178 < clt 11179 ≤ cle 11180 (,)cioo 13298 [,)cico 13300 ↾_t crest 17383 TopOpenctopn 17384 topGenctg 17400 ℂ_fldccnfld 21352 Topctop 22858 TopOnctopon 22875 intcnt 22982 Cn ccn 23189 CnP ccnp 23190 –cn→ccncf 24843 lim_ℂ climc 25829

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-om 7818 df-1st 7942 df-2nd 7943 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-er 8643 df-map 8775 df-pm 8776 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-fi 9324 df-sup 9355 df-inf 9356 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-7 12249 df-8 12250 df-9 12251 df-n0 12438 df-z 12525 df-dec 12645 df-uz 12789 df-q 12899 df-rp 12943 df-xneg 13063 df-xadd 13064 df-xmul 13065 df-ioo 13302 df-ico 13304 df-icc 13305 df-fz 13462 df-seq 13964 df-exp 14024 df-cj 15061 df-re 15062 df-im 15063 df-sqrt 15197 df-abs 15198 df-struct 17117 df-slot 17152 df-ndx 17164 df-base 17180 df-plusg 17233 df-mulr 17234 df-starv 17235 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-rest 17385 df-topn 17386 df-topgen 17406 df-psmet 21344 df-xmet 21345 df-met 21346 df-bl 21347 df-mopn 21348 df-cnfld 21353 df-top 22859 df-topon 22876 df-topsp 22898 df-bases 22911 df-ntr 22985 df-cn 23192 df-cnp 23193 df-xms 24285 df-ms 24286 df-cncf 24845 df-limc 25833

This theorem is referenced by: fourierdlem48 46582 fourierdlem76 46610 fourierdlem89 46623

W3C validator