fourierdlem33 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem33 46096

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

**Hypotheses**
Ref	Expression
fourierdlem33.1	⊢ (𝜑 → 𝐴 ∈ ℝ)
fourierdlem33.2	⊢ (𝜑 → 𝐵 ∈ ℝ)
fourierdlem33.3	⊢ (𝜑 → 𝐴 < 𝐵)
fourierdlem33.4	⊢ (𝜑 → 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ))
fourierdlem33.5	⊢ (𝜑 → 𝐿 ∈ (𝐹 lim_ℂ 𝐵))
fourierdlem33.6	⊢ (𝜑 → 𝐶 ∈ ℝ)
fourierdlem33.7	⊢ (𝜑 → 𝐷 ∈ ℝ)
fourierdlem33.8	⊢ (𝜑 → 𝐶 < 𝐷)
fourierdlem33.ss	⊢ (𝜑 → (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵))
fourierdlem33.y	⊢ 𝑌 = if(𝐷 = 𝐵, 𝐿, (𝐹‘𝐷))
fourierdlem33.10	⊢ 𝐽 = ((TopOpen‘ℂ_fld) ↾_t ((𝐴(,)𝐵) ∪ {𝐵}))

**Assertion**
Ref	Expression
fourierdlem33	⊢ (𝜑 → 𝑌 ∈ ((𝐹 ↾ (𝐶(,)𝐷)) lim_ℂ 𝐷))

Proof of Theorem fourierdlem33 Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fourierdlem33.5	. . . 4 ⊢ (𝜑 → 𝐿 ∈ (𝐹 lim_ℂ 𝐵))
2	1	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝐷 = 𝐵) → 𝐿 ∈ (𝐹 lim_ℂ 𝐵))
3		fourierdlem33.y	. . . . 5 ⊢ 𝑌 = if(𝐷 = 𝐵, 𝐿, (𝐹‘𝐷))
4		iftrue 4537	. . . . 5 ⊢ (𝐷 = 𝐵 → if(𝐷 = 𝐵, 𝐿, (𝐹‘𝐷)) = 𝐿)
5	3, 4	eqtr2id 2788	. . . 4 ⊢ (𝐷 = 𝐵 → 𝐿 = 𝑌)
6	5	adantl 481	. . 3 ⊢ ((𝜑 ∧ 𝐷 = 𝐵) → 𝐿 = 𝑌)
7		oveq2 7439	. . . . 5 ⊢ (𝐷 = 𝐵 → ((𝐹 ↾ (𝐶(,)𝐷)) lim_ℂ 𝐷) = ((𝐹 ↾ (𝐶(,)𝐷)) lim_ℂ 𝐵))
8	7	adantl 481	. . . 4 ⊢ ((𝜑 ∧ 𝐷 = 𝐵) → ((𝐹 ↾ (𝐶(,)𝐷)) lim_ℂ 𝐷) = ((𝐹 ↾ (𝐶(,)𝐷)) lim_ℂ 𝐵))
9		fourierdlem33.4	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ))
10		cncff 24933	. . . . . . 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 ∧ 𝐷 = 𝐵) → 𝐹:(𝐴(,)𝐵)⟶ℂ)
13		fourierdlem33.ss	. . . . . 6 ⊢ (𝜑 → (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵))
14	13	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝐷 = 𝐵) → (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵))
15		ioosscn 13446	. . . . . 6 ⊢ (𝐴(,)𝐵) ⊆ ℂ
16	15	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝐷 = 𝐵) → (𝐴(,)𝐵) ⊆ ℂ)
17		eqid 2735	. . . . 5 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
18		fourierdlem33.10	. . . . 5 ⊢ 𝐽 = ((TopOpen‘ℂ_fld) ↾_t ((𝐴(,)𝐵) ∪ {𝐵}))
19		fourierdlem33.7	. . . . . . . . 9 ⊢ (𝜑 → 𝐷 ∈ ℝ)
20		fourierdlem33.8	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 < 𝐷)
21	19	leidd 11827	. . . . . . . . 9 ⊢ (𝜑 → 𝐷 ≤ 𝐷)
22		fourierdlem33.6	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐶 ∈ ℝ)
23	22	rexrd 11309	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ ℝ^*)
24		elioc2 13447	. . . . . . . . . 10 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐷 ∈ ℝ) → (𝐷 ∈ (𝐶(,]𝐷) ↔ (𝐷 ∈ ℝ ∧ 𝐶 < 𝐷 ∧ 𝐷 ≤ 𝐷)))
25	23, 19, 24	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → (𝐷 ∈ (𝐶(,]𝐷) ↔ (𝐷 ∈ ℝ ∧ 𝐶 < 𝐷 ∧ 𝐷 ≤ 𝐷)))
26	19, 20, 21, 25	mpbir3and 1341	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ (𝐶(,]𝐷))
27	26	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐷 = 𝐵) → 𝐷 ∈ (𝐶(,]𝐷))
28		eqcom 2742	. . . . . . . . 9 ⊢ (𝐷 = 𝐵 ↔ 𝐵 = 𝐷)
29	28	biimpi 216	. . . . . . . 8 ⊢ (𝐷 = 𝐵 → 𝐵 = 𝐷)
30	29	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐷 = 𝐵) → 𝐵 = 𝐷)
31	17	cnfldtop 24820	. . . . . . . . . . 11 ⊢ (TopOpen‘ℂ_fld) ∈ Top
32		fourierdlem33.1	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ∈ ℝ)
33	32	rexrd 11309	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
34		fourierdlem33.2	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐵 ∈ ℝ)
35	34	rexrd 11309	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
36		fourierdlem33.3	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 < 𝐵)
37		ioounsn 13514	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → ((𝐴(,)𝐵) ∪ {𝐵}) = (𝐴(,]𝐵))
38	33, 35, 36, 37	syl3anc 1370	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐴(,)𝐵) ∪ {𝐵}) = (𝐴(,]𝐵))
39		ovex 7464	. . . . . . . . . . . . 13 ⊢ (𝐴(,]𝐵) ∈ V
40	39	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴(,]𝐵) ∈ V)
41	38, 40	eqeltrd 2839	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐴(,)𝐵) ∪ {𝐵}) ∈ V)
42		resttop 23184	. . . . . . . . . . 11 ⊢ (((TopOpen‘ℂ_fld) ∈ Top ∧ ((𝐴(,)𝐵) ∪ {𝐵}) ∈ V) → ((TopOpen‘ℂ_fld) ↾_t ((𝐴(,)𝐵) ∪ {𝐵})) ∈ Top)
43	31, 41, 42	sylancr 587	. . . . . . . . . 10 ⊢ (𝜑 → ((TopOpen‘ℂ_fld) ↾_t ((𝐴(,)𝐵) ∪ {𝐵})) ∈ Top)
44	18, 43	eqeltrid 2843	. . . . . . . . 9 ⊢ (𝜑 → 𝐽 ∈ Top)
45	44	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐷 = 𝐵) → 𝐽 ∈ Top)
46		oveq2 7439	. . . . . . . . . . 11 ⊢ (𝐷 = 𝐵 → (𝐶(,]𝐷) = (𝐶(,]𝐵))
47	46	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐷 = 𝐵) → (𝐶(,]𝐷) = (𝐶(,]𝐵))
48	23	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,]𝐵)) → 𝐶 ∈ ℝ^*)
49		pnfxr 11313	. . . . . . . . . . . . . . . . 17 ⊢ +∞ ∈ ℝ^*
50	49	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,]𝐵)) → +∞ ∈ ℝ^*)
51		simpr 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,]𝐵)) → 𝑥 ∈ (𝐶(,]𝐵))
52	34	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,]𝐵)) → 𝐵 ∈ ℝ)
53		elioc2 13447	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → (𝑥 ∈ (𝐶(,]𝐵) ↔ (𝑥 ∈ ℝ ∧ 𝐶 < 𝑥 ∧ 𝑥 ≤ 𝐵)))
54	48, 52, 53	syl2anc 584	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,]𝐵)) → (𝑥 ∈ (𝐶(,]𝐵) ↔ (𝑥 ∈ ℝ ∧ 𝐶 < 𝑥 ∧ 𝑥 ≤ 𝐵)))
55	51, 54	mpbid 232	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,]𝐵)) → (𝑥 ∈ ℝ ∧ 𝐶 < 𝑥 ∧ 𝑥 ≤ 𝐵))
56	55	simp1d 1141	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,]𝐵)) → 𝑥 ∈ ℝ)
57	55	simp2d 1142	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,]𝐵)) → 𝐶 < 𝑥)
58	56	ltpnfd 13161	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,]𝐵)) → 𝑥 < +∞)
59	48, 50, 56, 57, 58	eliood 45451	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,]𝐵)) → 𝑥 ∈ (𝐶(,)+∞))
60	32	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,]𝐵)) → 𝐴 ∈ ℝ)
61	22	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,]𝐵)) → 𝐶 ∈ ℝ)
62	32, 34, 22, 19, 20, 13	fourierdlem10 46073	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝐴 ≤ 𝐶 ∧ 𝐷 ≤ 𝐵))
63	62	simpld 494	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐴 ≤ 𝐶)
64	63	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,]𝐵)) → 𝐴 ≤ 𝐶)
65	60, 61, 56, 64, 57	lelttrd 11417	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,]𝐵)) → 𝐴 < 𝑥)
66	55	simp3d 1143	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,]𝐵)) → 𝑥 ≤ 𝐵)
67	33	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,]𝐵)) → 𝐴 ∈ ℝ^*)
68		elioc2 13447	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → (𝑥 ∈ (𝐴(,]𝐵) ↔ (𝑥 ∈ ℝ ∧ 𝐴 < 𝑥 ∧ 𝑥 ≤ 𝐵)))
69	67, 52, 68	syl2anc 584	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,]𝐵)) → (𝑥 ∈ (𝐴(,]𝐵) ↔ (𝑥 ∈ ℝ ∧ 𝐴 < 𝑥 ∧ 𝑥 ≤ 𝐵)))
70	56, 65, 66, 69	mpbir3and 1341	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,]𝐵)) → 𝑥 ∈ (𝐴(,]𝐵))
71	59, 70	elind 4210	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,]𝐵)) → 𝑥 ∈ ((𝐶(,)+∞) ∩ (𝐴(,]𝐵)))
72		elinel1 4211	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ((𝐶(,)+∞) ∩ (𝐴(,]𝐵)) → 𝑥 ∈ (𝐶(,)+∞))
73		elioore 13414	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (𝐶(,)+∞) → 𝑥 ∈ ℝ)
74	72, 73	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ((𝐶(,)+∞) ∩ (𝐴(,]𝐵)) → 𝑥 ∈ ℝ)
75	74	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐶(,)+∞) ∩ (𝐴(,]𝐵))) → 𝑥 ∈ ℝ)
76	23	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐶(,)+∞) ∩ (𝐴(,]𝐵))) → 𝐶 ∈ ℝ^*)
77	49	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐶(,)+∞) ∩ (𝐴(,]𝐵))) → +∞ ∈ ℝ^*)
78	72	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐶(,)+∞) ∩ (𝐴(,]𝐵))) → 𝑥 ∈ (𝐶(,)+∞))
79		ioogtlb 45448	. . . . . . . . . . . . . . . 16 ⊢ ((𝐶 ∈ ℝ^* ∧ +∞ ∈ ℝ^* ∧ 𝑥 ∈ (𝐶(,)+∞)) → 𝐶 < 𝑥)
80	76, 77, 78, 79	syl3anc 1370	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐶(,)+∞) ∩ (𝐴(,]𝐵))) → 𝐶 < 𝑥)
81		elinel2 4212	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ((𝐶(,)+∞) ∩ (𝐴(,]𝐵)) → 𝑥 ∈ (𝐴(,]𝐵))
82	81	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐶(,)+∞) ∩ (𝐴(,]𝐵))) → 𝑥 ∈ (𝐴(,]𝐵))
83	33	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐶(,)+∞) ∩ (𝐴(,]𝐵))) → 𝐴 ∈ ℝ^*)
84	34	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐶(,)+∞) ∩ (𝐴(,]𝐵))) → 𝐵 ∈ ℝ)
85	83, 84, 68	syl2anc 584	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐶(,)+∞) ∩ (𝐴(,]𝐵))) → (𝑥 ∈ (𝐴(,]𝐵) ↔ (𝑥 ∈ ℝ ∧ 𝐴 < 𝑥 ∧ 𝑥 ≤ 𝐵)))
86	82, 85	mpbid 232	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐶(,)+∞) ∩ (𝐴(,]𝐵))) → (𝑥 ∈ ℝ ∧ 𝐴 < 𝑥 ∧ 𝑥 ≤ 𝐵))
87	86	simp3d 1143	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐶(,)+∞) ∩ (𝐴(,]𝐵))) → 𝑥 ≤ 𝐵)
88	76, 84, 53	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐶(,)+∞) ∩ (𝐴(,]𝐵))) → (𝑥 ∈ (𝐶(,]𝐵) ↔ (𝑥 ∈ ℝ ∧ 𝐶 < 𝑥 ∧ 𝑥 ≤ 𝐵)))
89	75, 80, 87, 88	mpbir3and 1341	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐶(,)+∞) ∩ (𝐴(,]𝐵))) → 𝑥 ∈ (𝐶(,]𝐵))
90	71, 89	impbida 801	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑥 ∈ (𝐶(,]𝐵) ↔ 𝑥 ∈ ((𝐶(,)+∞) ∩ (𝐴(,]𝐵))))
91	90	eqrdv 2733	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶(,]𝐵) = ((𝐶(,)+∞) ∩ (𝐴(,]𝐵)))
92		retop 24798	. . . . . . . . . . . . . 14 ⊢ (topGen‘ran (,)) ∈ Top
93	92	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → (topGen‘ran (,)) ∈ Top)
94		iooretop 24802	. . . . . . . . . . . . . 14 ⊢ (𝐶(,)+∞) ∈ (topGen‘ran (,))
95	94	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐶(,)+∞) ∈ (topGen‘ran (,)))
96		elrestr 17475	. . . . . . . . . . . . 13 ⊢ (((topGen‘ran (,)) ∈ Top ∧ (𝐴(,]𝐵) ∈ V ∧ (𝐶(,)+∞) ∈ (topGen‘ran (,))) → ((𝐶(,)+∞) ∩ (𝐴(,]𝐵)) ∈ ((topGen‘ran (,)) ↾_t (𝐴(,]𝐵)))
97	93, 40, 95, 96	syl3anc 1370	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐶(,)+∞) ∩ (𝐴(,]𝐵)) ∈ ((topGen‘ran (,)) ↾_t (𝐴(,]𝐵)))
98	91, 97	eqeltrd 2839	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐶(,]𝐵) ∈ ((topGen‘ran (,)) ↾_t (𝐴(,]𝐵)))
99	98	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐷 = 𝐵) → (𝐶(,]𝐵) ∈ ((topGen‘ran (,)) ↾_t (𝐴(,]𝐵)))
100	47, 99	eqeltrd 2839	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐷 = 𝐵) → (𝐶(,]𝐷) ∈ ((topGen‘ran (,)) ↾_t (𝐴(,]𝐵)))
101	18	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐽 = ((TopOpen‘ℂ_fld) ↾_t ((𝐴(,)𝐵) ∪ {𝐵})))
102	38	oveq2d 7447	. . . . . . . . . . 11 ⊢ (𝜑 → ((TopOpen‘ℂ_fld) ↾_t ((𝐴(,)𝐵) ∪ {𝐵})) = ((TopOpen‘ℂ_fld) ↾_t (𝐴(,]𝐵)))
103	31	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → (TopOpen‘ℂ_fld) ∈ Top)
104		iocssre 13464	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → (𝐴(,]𝐵) ⊆ ℝ)
105	33, 34, 104	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴(,]𝐵) ⊆ ℝ)
106		reex 11244	. . . . . . . . . . . . . 14 ⊢ ℝ ∈ V
107	106	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → ℝ ∈ V)
108		restabs 23189	. . . . . . . . . . . . 13 ⊢ (((TopOpen‘ℂ_fld) ∈ Top ∧ (𝐴(,]𝐵) ⊆ ℝ ∧ ℝ ∈ V) → (((TopOpen‘ℂ_fld) ↾_t ℝ) ↾_t (𝐴(,]𝐵)) = ((TopOpen‘ℂ_fld) ↾_t (𝐴(,]𝐵)))
109	103, 105, 107, 108	syl3anc 1370	. . . . . . . . . . . 12 ⊢ (𝜑 → (((TopOpen‘ℂ_fld) ↾_t ℝ) ↾_t (𝐴(,]𝐵)) = ((TopOpen‘ℂ_fld) ↾_t (𝐴(,]𝐵)))
110	17	tgioo2 24839	. . . . . . . . . . . . . 14 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
111	110	eqcomi 2744	. . . . . . . . . . . . 13 ⊢ ((TopOpen‘ℂ_fld) ↾_t ℝ) = (topGen‘ran (,))
112	111	oveq1i 7441	. . . . . . . . . . . 12 ⊢ (((TopOpen‘ℂ_fld) ↾_t ℝ) ↾_t (𝐴(,]𝐵)) = ((topGen‘ran (,)) ↾_t (𝐴(,]𝐵))
113	109, 112	eqtr3di 2790	. . . . . . . . . . 11 ⊢ (𝜑 → ((TopOpen‘ℂ_fld) ↾_t (𝐴(,]𝐵)) = ((topGen‘ran (,)) ↾_t (𝐴(,]𝐵)))
114	101, 102, 113	3eqtrrd 2780	. . . . . . . . . 10 ⊢ (𝜑 → ((topGen‘ran (,)) ↾_t (𝐴(,]𝐵)) = 𝐽)
115	114	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐷 = 𝐵) → ((topGen‘ran (,)) ↾_t (𝐴(,]𝐵)) = 𝐽)
116	100, 115	eleqtrd 2841	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐷 = 𝐵) → (𝐶(,]𝐷) ∈ 𝐽)
117		isopn3i 23106	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ (𝐶(,]𝐷) ∈ 𝐽) → ((int‘𝐽)‘(𝐶(,]𝐷)) = (𝐶(,]𝐷))
118	45, 116, 117	syl2anc 584	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐷 = 𝐵) → ((int‘𝐽)‘(𝐶(,]𝐷)) = (𝐶(,]𝐷))
119	27, 30, 118	3eltr4d 2854	. . . . . 6 ⊢ ((𝜑 ∧ 𝐷 = 𝐵) → 𝐵 ∈ ((int‘𝐽)‘(𝐶(,]𝐷)))
120		sneq 4641	. . . . . . . . . . 11 ⊢ (𝐷 = 𝐵 → {𝐷} = {𝐵})
121	120	eqcomd 2741	. . . . . . . . . 10 ⊢ (𝐷 = 𝐵 → {𝐵} = {𝐷})
122	121	uneq2d 4178	. . . . . . . . 9 ⊢ (𝐷 = 𝐵 → ((𝐶(,)𝐷) ∪ {𝐵}) = ((𝐶(,)𝐷) ∪ {𝐷}))
123	122	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐷 = 𝐵) → ((𝐶(,)𝐷) ∪ {𝐵}) = ((𝐶(,)𝐷) ∪ {𝐷}))
124	19	rexrd 11309	. . . . . . . . . 10 ⊢ (𝜑 → 𝐷 ∈ ℝ^*)
125		ioounsn 13514	. . . . . . . . . 10 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐷 ∈ ℝ^* ∧ 𝐶 < 𝐷) → ((𝐶(,)𝐷) ∪ {𝐷}) = (𝐶(,]𝐷))
126	23, 124, 20, 125	syl3anc 1370	. . . . . . . . 9 ⊢ (𝜑 → ((𝐶(,)𝐷) ∪ {𝐷}) = (𝐶(,]𝐷))
127	126	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐷 = 𝐵) → ((𝐶(,)𝐷) ∪ {𝐷}) = (𝐶(,]𝐷))
128	123, 127	eqtr2d 2776	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐷 = 𝐵) → (𝐶(,]𝐷) = ((𝐶(,)𝐷) ∪ {𝐵}))
129	128	fveq2d 6911	. . . . . 6 ⊢ ((𝜑 ∧ 𝐷 = 𝐵) → ((int‘𝐽)‘(𝐶(,]𝐷)) = ((int‘𝐽)‘((𝐶(,)𝐷) ∪ {𝐵})))
130	119, 129	eleqtrd 2841	. . . . 5 ⊢ ((𝜑 ∧ 𝐷 = 𝐵) → 𝐵 ∈ ((int‘𝐽)‘((𝐶(,)𝐷) ∪ {𝐵})))
131	12, 14, 16, 17, 18, 130	limcres 25936	. . . 4 ⊢ ((𝜑 ∧ 𝐷 = 𝐵) → ((𝐹 ↾ (𝐶(,)𝐷)) lim_ℂ 𝐵) = (𝐹 lim_ℂ 𝐵))
132	8, 131	eqtr2d 2776	. . 3 ⊢ ((𝜑 ∧ 𝐷 = 𝐵) → (𝐹 lim_ℂ 𝐵) = ((𝐹 ↾ (𝐶(,)𝐷)) lim_ℂ 𝐷))
133	2, 6, 132	3eltr3d 2853	. 2 ⊢ ((𝜑 ∧ 𝐷 = 𝐵) → 𝑌 ∈ ((𝐹 ↾ (𝐶(,)𝐷)) lim_ℂ 𝐷))
134		limcresi 25935	. . 3 ⊢ (𝐹 lim_ℂ 𝐷) ⊆ ((𝐹 ↾ (𝐶(,)𝐷)) lim_ℂ 𝐷)
135		iffalse 4540	. . . . . 6 ⊢ (¬ 𝐷 = 𝐵 → if(𝐷 = 𝐵, 𝐿, (𝐹‘𝐷)) = (𝐹‘𝐷))
136	3, 135	eqtrid 2787	. . . . 5 ⊢ (¬ 𝐷 = 𝐵 → 𝑌 = (𝐹‘𝐷))
137	136	adantl 481	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐷 = 𝐵) → 𝑌 = (𝐹‘𝐷))
138		ssid 4018	. . . . . . . . . . . . 13 ⊢ ℂ ⊆ ℂ
139	138	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → ℂ ⊆ ℂ)
140		eqid 2735	. . . . . . . . . . . . 13 ⊢ ((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) = ((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵))
141		unicntop 24822	. . . . . . . . . . . . . . . 16 ⊢ ℂ = ∪ (TopOpen‘ℂ_fld)
142	141	restid 17480	. . . . . . . . . . . . . . 15 ⊢ ((TopOpen‘ℂ_fld) ∈ Top → ((TopOpen‘ℂ_fld) ↾_t ℂ) = (TopOpen‘ℂ_fld))
143	31, 142	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ ((TopOpen‘ℂ_fld) ↾_t ℂ) = (TopOpen‘ℂ_fld)
144	143	eqcomi 2744	. . . . . . . . . . . . 13 ⊢ (TopOpen‘ℂ_fld) = ((TopOpen‘ℂ_fld) ↾_t ℂ)
145	17, 140, 144	cncfcn 24950	. . . . . . . . . . . 12 ⊢ (((𝐴(,)𝐵) ⊆ ℂ ∧ ℂ ⊆ ℂ) → ((𝐴(,)𝐵)–cn→ℂ) = (((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) Cn (TopOpen‘ℂ_fld)))
146	15, 139, 145	sylancr 587	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐴(,)𝐵)–cn→ℂ) = (((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) Cn (TopOpen‘ℂ_fld)))
147	9, 146	eleqtrd 2841	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ (((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) Cn (TopOpen‘ℂ_fld)))
148	17	cnfldtopon 24819	. . . . . . . . . . . 12 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
149	15	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℂ)
150		resttopon 23185	. . . . . . . . . . . 12 ⊢ (((TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ) ∧ (𝐴(,)𝐵) ⊆ ℂ) → ((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) ∈ (TopOn‘(𝐴(,)𝐵)))
151	148, 149, 150	sylancr 587	. . . . . . . . . . 11 ⊢ (𝜑 → ((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) ∈ (TopOn‘(𝐴(,)𝐵)))
152	148	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ))
153		cncnp 23304	. . . . . . . . . . 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))))
154	151, 152, 153	syl2anc 584	. . . . . . . . . 10 ⊢ (𝜑 → (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))))
155	147, 154	mpbid 232	. . . . . . . . 9 ⊢ (𝜑 → (𝐹:(𝐴(,)𝐵)⟶ℂ ∧ ∀𝑥 ∈ (𝐴(,)𝐵)𝐹 ∈ ((((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) CnP (TopOpen‘ℂ_fld))‘𝑥)))
156	155	simprd 495	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ (𝐴(,)𝐵)𝐹 ∈ ((((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) CnP (TopOpen‘ℂ_fld))‘𝑥))
157	156	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐷 = 𝐵) → ∀𝑥 ∈ (𝐴(,)𝐵)𝐹 ∈ ((((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) CnP (TopOpen‘ℂ_fld))‘𝑥))
158	33	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐷 = 𝐵) → 𝐴 ∈ ℝ^*)
159	35	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐷 = 𝐵) → 𝐵 ∈ ℝ^*)
160	19	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐷 = 𝐵) → 𝐷 ∈ ℝ)
161	32, 22, 19, 63, 20	lelttrd 11417	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 < 𝐷)
162	161	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐷 = 𝐵) → 𝐴 < 𝐷)
163	34	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐷 = 𝐵) → 𝐵 ∈ ℝ)
164	62	simprd 495	. . . . . . . . . 10 ⊢ (𝜑 → 𝐷 ≤ 𝐵)
165	164	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐷 = 𝐵) → 𝐷 ≤ 𝐵)
166		neqne 2946	. . . . . . . . . . 11 ⊢ (¬ 𝐷 = 𝐵 → 𝐷 ≠ 𝐵)
167	166	necomd 2994	. . . . . . . . . 10 ⊢ (¬ 𝐷 = 𝐵 → 𝐵 ≠ 𝐷)
168	167	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐷 = 𝐵) → 𝐵 ≠ 𝐷)
169	160, 163, 165, 168	leneltd 11413	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐷 = 𝐵) → 𝐷 < 𝐵)
170	158, 159, 160, 162, 169	eliood 45451	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐷 = 𝐵) → 𝐷 ∈ (𝐴(,)𝐵))
171		fveq2 6907	. . . . . . . . 9 ⊢ (𝑥 = 𝐷 → ((((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) CnP (TopOpen‘ℂ_fld))‘𝑥) = ((((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) CnP (TopOpen‘ℂ_fld))‘𝐷))
172	171	eleq2d 2825	. . . . . . . 8 ⊢ (𝑥 = 𝐷 → (𝐹 ∈ ((((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) CnP (TopOpen‘ℂ_fld))‘𝑥) ↔ 𝐹 ∈ ((((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) CnP (TopOpen‘ℂ_fld))‘𝐷)))
173	172	rspccva 3621	. . . . . . 7 ⊢ ((∀𝑥 ∈ (𝐴(,)𝐵)𝐹 ∈ ((((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) CnP (TopOpen‘ℂ_fld))‘𝑥) ∧ 𝐷 ∈ (𝐴(,)𝐵)) → 𝐹 ∈ ((((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) CnP (TopOpen‘ℂ_fld))‘𝐷))
174	157, 170, 173	syl2anc 584	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐷 = 𝐵) → 𝐹 ∈ ((((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) CnP (TopOpen‘ℂ_fld))‘𝐷))
175	17, 140	cnplimc 25937	. . . . . . 7 ⊢ (((𝐴(,)𝐵) ⊆ ℂ ∧ 𝐷 ∈ (𝐴(,)𝐵)) → (𝐹 ∈ ((((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) CnP (TopOpen‘ℂ_fld))‘𝐷) ↔ (𝐹:(𝐴(,)𝐵)⟶ℂ ∧ (𝐹‘𝐷) ∈ (𝐹 lim_ℂ 𝐷))))
176	15, 170, 175	sylancr 587	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐷 = 𝐵) → (𝐹 ∈ ((((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) CnP (TopOpen‘ℂ_fld))‘𝐷) ↔ (𝐹:(𝐴(,)𝐵)⟶ℂ ∧ (𝐹‘𝐷) ∈ (𝐹 lim_ℂ 𝐷))))
177	174, 176	mpbid 232	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐷 = 𝐵) → (𝐹:(𝐴(,)𝐵)⟶ℂ ∧ (𝐹‘𝐷) ∈ (𝐹 lim_ℂ 𝐷)))
178	177	simprd 495	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐷 = 𝐵) → (𝐹‘𝐷) ∈ (𝐹 lim_ℂ 𝐷))
179	137, 178	eqeltrd 2839	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐷 = 𝐵) → 𝑌 ∈ (𝐹 lim_ℂ 𝐷))
180	134, 179	sselid 3993	. 2 ⊢ ((𝜑 ∧ ¬ 𝐷 = 𝐵) → 𝑌 ∈ ((𝐹 ↾ (𝐶(,)𝐷)) lim_ℂ 𝐷))
181	133, 180	pm2.61dan 813	1 ⊢ (𝜑 → 𝑌 ∈ ((𝐹 ↾ (𝐶(,)𝐷)) lim_ℂ 𝐷))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∀wral 3059 Vcvv 3478 ∪ cun 3961 ∩ cin 3962 ⊆ wss 3963 ifcif 4531 {csn 4631 class class class wbr 5148 ran crn 5690 ↾ cres 5691 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ℂcc 11151 ℝcr 11152 +∞cpnf 11290 ℝ^*cxr 11292 < clt 11293 ≤ cle 11294 (,)cioo 13384 (,]cioc 13385 ↾_t crest 17467 TopOpenctopn 17468 topGenctg 17484 ℂ_fldccnfld 21382 Topctop 22915 TopOnctopon 22932 intcnt 23041 Cn ccn 23248 CnP ccnp 23249 –cn→ccncf 24916 lim_ℂ climc 25912

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-map 8867 df-pm 8868 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fi 9449 df-sup 9480 df-inf 9481 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-q 12989 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-ioo 13388 df-ioc 13389 df-icc 13391 df-fz 13545 df-seq 14040 df-exp 14100 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-struct 17181 df-slot 17216 df-ndx 17228 df-base 17246 df-plusg 17311 df-mulr 17312 df-starv 17313 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-rest 17469 df-topn 17470 df-topgen 17490 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-cnfld 21383 df-top 22916 df-topon 22933 df-topsp 22955 df-bases 22969 df-ntr 23044 df-cn 23251 df-cnp 23252 df-xms 24346 df-ms 24347 df-cncf 24918 df-limc 25916

This theorem is referenced by: fourierdlem49 46111 fourierdlem76 46138 fourierdlem91 46153

W3C validator