fourierdlem33 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem33 47286

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 486	. . 3 ⊢ ((𝜑 ∧ 𝐷 = 𝐵) → 𝐿 ∈ (𝐹 lim_ℂ 𝐵))
3		fourierdlem33.y	. . . . 5 ⊢ 𝑌 = if(𝐷 = 𝐵, 𝐿, (𝐹‘𝐷))
4		iftrue 4487	. . . . 5 ⊢ (𝐷 = 𝐵 → if(𝐷 = 𝐵, 𝐿, (𝐹‘𝐷)) = 𝐿)
5	3, 4	eqtr2id 2808	. . . 4 ⊢ (𝐷 = 𝐵 → 𝐿 = 𝑌)
6	5	adantl 487	. . 3 ⊢ ((𝜑 ∧ 𝐷 = 𝐵) → 𝐿 = 𝑌)
7		oveq2 7432	. . . . 5 ⊢ (𝐷 = 𝐵 → ((𝐹 ↾ (𝐶(,)𝐷)) lim_ℂ 𝐷) = ((𝐹 ↾ (𝐶(,)𝐷)) lim_ℂ 𝐵))
8	7	adantl 487	. . . 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 25223	. . . . . . 7 ⊢ (𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ) → 𝐹:(𝐴(,)𝐵)⟶ℂ)
11	9, 10	syl 18	. . . . . 6 ⊢ (𝜑 → 𝐹:(𝐴(,)𝐵)⟶ℂ)
12	11	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝐷 = 𝐵) → 𝐹:(𝐴(,)𝐵)⟶ℂ)
13		fourierdlem33.ss	. . . . . 6 ⊢ (𝜑 → (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵))
14	13	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝐷 = 𝐵) → (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵))
15		ioosscn 13546	. . . . . 6 ⊢ (𝐴(,)𝐵) ⊆ ℂ
16	15	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝐷 = 𝐵) → (𝐴(,)𝐵) ⊆ ℂ)
17		eqid 2760	. . . . 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 11889	. . . . . . . . 9 ⊢ (𝜑 → 𝐷 ≤ 𝐷)
22		fourierdlem33.6	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐶 ∈ ℝ)
23	22	rexrd 11366	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ ℝ^*)
24		elioc2 13547	. . . . . . . . . 10 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐷 ∈ ℝ) → (𝐷 ∈ (𝐶(,]𝐷) ↔ (𝐷 ∈ ℝ ∧ 𝐶 < 𝐷 ∧ 𝐷 ≤ 𝐷)))
25	23, 19, 24	syl2anc 596	. . . . . . . . 9 ⊢ (𝜑 → (𝐷 ∈ (𝐶(,]𝐷) ↔ (𝐷 ∈ ℝ ∧ 𝐶 < 𝐷 ∧ 𝐷 ≤ 𝐷)))
26	19, 20, 21, 25	mpbir3and 1361	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ (𝐶(,]𝐷))
27	26	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐷 = 𝐵) → 𝐷 ∈ (𝐶(,]𝐷))
28		eqcom 2767	. . . . . . . 8 ⊢ (𝐷 = 𝐵 ↔ 𝐵 = 𝐷)
29	28	bilani 510	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐷 = 𝐵) → 𝐵 = 𝐷)
30	17	cnfldtop 25111	. . . . . . . . . . 11 ⊢ (TopOpen‘ℂ_fld) ∈ Top
31		fourierdlem33.1	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ∈ ℝ)
32	31	rexrd 11366	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
33		fourierdlem33.2	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐵 ∈ ℝ)
34	33	rexrd 11366	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
35		fourierdlem33.3	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 < 𝐵)
36		ioounsn 13615	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → ((𝐴(,)𝐵) ∪ {𝐵}) = (𝐴(,]𝐵))
37	32, 34, 35, 36	syl3anc 1398	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐴(,)𝐵) ∪ {𝐵}) = (𝐴(,]𝐵))
38		ovex 7457	. . . . . . . . . . . . 13 ⊢ (𝐴(,]𝐵) ∈ V
39	38	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴(,]𝐵) ∈ V)
40	37, 39	eqeltrd 2860	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐴(,)𝐵) ∪ {𝐵}) ∈ V)
41		resttop 23487	. . . . . . . . . . 11 ⊢ (((TopOpen‘ℂ_fld) ∈ Top ∧ ((𝐴(,)𝐵) ∪ {𝐵}) ∈ V) → ((TopOpen‘ℂ_fld) ↾_t ((𝐴(,)𝐵) ∪ {𝐵})) ∈ Top)
42	30, 40, 41	sylancr 599	. . . . . . . . . 10 ⊢ (𝜑 → ((TopOpen‘ℂ_fld) ↾_t ((𝐴(,)𝐵) ∪ {𝐵})) ∈ Top)
43	18, 42	eqeltrid 2864	. . . . . . . . 9 ⊢ (𝜑 → 𝐽 ∈ Top)
44	43	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐷 = 𝐵) → 𝐽 ∈ Top)
45		oveq2 7432	. . . . . . . . . . 11 ⊢ (𝐷 = 𝐵 → (𝐶(,]𝐷) = (𝐶(,]𝐵))
46	45	adantl 487	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐷 = 𝐵) → (𝐶(,]𝐷) = (𝐶(,]𝐵))
47	23	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,]𝐵)) → 𝐶 ∈ ℝ^*)
48		pnfxr 11370	. . . . . . . . . . . . . . . . 17 ⊢ +∞ ∈ ℝ^*
49	48	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,]𝐵)) → +∞ ∈ ℝ^*)
50		simpr 490	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,]𝐵)) → 𝑥 ∈ (𝐶(,]𝐵))
51	33	adantr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,]𝐵)) → 𝐵 ∈ ℝ)
52		elioc2 13547	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → (𝑥 ∈ (𝐶(,]𝐵) ↔ (𝑥 ∈ ℝ ∧ 𝐶 < 𝑥 ∧ 𝑥 ≤ 𝐵)))
53	47, 51, 52	syl2anc 596	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,]𝐵)) → (𝑥 ∈ (𝐶(,]𝐵) ↔ (𝑥 ∈ ℝ ∧ 𝐶 < 𝑥 ∧ 𝑥 ≤ 𝐵)))
54	50, 53	mpbid 235	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,]𝐵)) → (𝑥 ∈ ℝ ∧ 𝐶 < 𝑥 ∧ 𝑥 ≤ 𝐵))
55	54	simp1d 1160	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,]𝐵)) → 𝑥 ∈ ℝ)
56	54	simp2d 1161	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,]𝐵)) → 𝐶 < 𝑥)
57	55	ltpnfd 13257	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,]𝐵)) → 𝑥 < +∞)
58	47, 49, 55, 56, 57	eliood 46646	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,]𝐵)) → 𝑥 ∈ (𝐶(,)+∞))
59	31	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,]𝐵)) → 𝐴 ∈ ℝ)
60	22	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,]𝐵)) → 𝐶 ∈ ℝ)
61	31, 33, 22, 19, 20, 13	fourierdlem10 47263	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝐴 ≤ 𝐶 ∧ 𝐷 ≤ 𝐵))
62	61	simpld 500	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐴 ≤ 𝐶)
63	62	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,]𝐵)) → 𝐴 ≤ 𝐶)
64	59, 60, 55, 63, 56	lelttrd 11475	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,]𝐵)) → 𝐴 < 𝑥)
65	54	simp3d 1162	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,]𝐵)) → 𝑥 ≤ 𝐵)
66	32	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,]𝐵)) → 𝐴 ∈ ℝ^*)
67		elioc2 13547	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → (𝑥 ∈ (𝐴(,]𝐵) ↔ (𝑥 ∈ ℝ ∧ 𝐴 < 𝑥 ∧ 𝑥 ≤ 𝐵)))
68	66, 51, 67	syl2anc 596	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,]𝐵)) → (𝑥 ∈ (𝐴(,]𝐵) ↔ (𝑥 ∈ ℝ ∧ 𝐴 < 𝑥 ∧ 𝑥 ≤ 𝐵)))
69	55, 64, 65, 68	mpbir3and 1361	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,]𝐵)) → 𝑥 ∈ (𝐴(,]𝐵))
70	58, 69	elind 4096	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐶(,]𝐵)) → 𝑥 ∈ ((𝐶(,)+∞) ∩ (𝐴(,]𝐵)))
71		elinel1 4097	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ((𝐶(,)+∞) ∩ (𝐴(,]𝐵)) → 𝑥 ∈ (𝐶(,)+∞))
72		elioore 13513	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (𝐶(,)+∞) → 𝑥 ∈ ℝ)
73	71, 72	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ((𝐶(,)+∞) ∩ (𝐴(,]𝐵)) → 𝑥 ∈ ℝ)
74	73	adantl 487	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐶(,)+∞) ∩ (𝐴(,]𝐵))) → 𝑥 ∈ ℝ)
75	23	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐶(,)+∞) ∩ (𝐴(,]𝐵))) → 𝐶 ∈ ℝ^*)
76	48	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐶(,)+∞) ∩ (𝐴(,]𝐵))) → +∞ ∈ ℝ^*)
77	71	adantl 487	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐶(,)+∞) ∩ (𝐴(,]𝐵))) → 𝑥 ∈ (𝐶(,)+∞))
78		ioogtlb 46643	. . . . . . . . . . . . . . . 16 ⊢ ((𝐶 ∈ ℝ^* ∧ +∞ ∈ ℝ^* ∧ 𝑥 ∈ (𝐶(,)+∞)) → 𝐶 < 𝑥)
79	75, 76, 77, 78	syl3anc 1398	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐶(,)+∞) ∩ (𝐴(,]𝐵))) → 𝐶 < 𝑥)
80		elinel2 4098	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ((𝐶(,)+∞) ∩ (𝐴(,]𝐵)) → 𝑥 ∈ (𝐴(,]𝐵))
81	80	adantl 487	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐶(,)+∞) ∩ (𝐴(,]𝐵))) → 𝑥 ∈ (𝐴(,]𝐵))
82	32	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐶(,)+∞) ∩ (𝐴(,]𝐵))) → 𝐴 ∈ ℝ^*)
83	33	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐶(,)+∞) ∩ (𝐴(,]𝐵))) → 𝐵 ∈ ℝ)
84	82, 83, 67	syl2anc 596	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐶(,)+∞) ∩ (𝐴(,]𝐵))) → (𝑥 ∈ (𝐴(,]𝐵) ↔ (𝑥 ∈ ℝ ∧ 𝐴 < 𝑥 ∧ 𝑥 ≤ 𝐵)))
85	81, 84	mpbid 235	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐶(,)+∞) ∩ (𝐴(,]𝐵))) → (𝑥 ∈ ℝ ∧ 𝐴 < 𝑥 ∧ 𝑥 ≤ 𝐵))
86	85	simp3d 1162	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐶(,)+∞) ∩ (𝐴(,]𝐵))) → 𝑥 ≤ 𝐵)
87	75, 83, 52	syl2anc 596	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐶(,)+∞) ∩ (𝐴(,]𝐵))) → (𝑥 ∈ (𝐶(,]𝐵) ↔ (𝑥 ∈ ℝ ∧ 𝐶 < 𝑥 ∧ 𝑥 ≤ 𝐵)))
88	74, 79, 86, 87	mpbir3and 1361	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐶(,)+∞) ∩ (𝐴(,]𝐵))) → 𝑥 ∈ (𝐶(,]𝐵))
89	70, 88	impbida 813	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑥 ∈ (𝐶(,]𝐵) ↔ 𝑥 ∈ ((𝐶(,)+∞) ∩ (𝐴(,]𝐵))))
90	89	eqrdv 2758	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶(,]𝐵) = ((𝐶(,)+∞) ∩ (𝐴(,]𝐵)))
91		retop 25089	. . . . . . . . . . . . . 14 ⊢ (topGen‘ran (,)) ∈ Top
92	91	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → (topGen‘ran (,)) ∈ Top)
93		iooretop 25093	. . . . . . . . . . . . . 14 ⊢ (𝐶(,)+∞) ∈ (topGen‘ran (,))
94	93	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐶(,)+∞) ∈ (topGen‘ran (,)))
95		elrestr 17608	. . . . . . . . . . . . 13 ⊢ (((topGen‘ran (,)) ∈ Top ∧ (𝐴(,]𝐵) ∈ V ∧ (𝐶(,)+∞) ∈ (topGen‘ran (,))) → ((𝐶(,)+∞) ∩ (𝐴(,]𝐵)) ∈ ((topGen‘ran (,)) ↾_t (𝐴(,]𝐵)))
96	92, 39, 94, 95	syl3anc 1398	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐶(,)+∞) ∩ (𝐴(,]𝐵)) ∈ ((topGen‘ran (,)) ↾_t (𝐴(,]𝐵)))
97	90, 96	eqeltrd 2860	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐶(,]𝐵) ∈ ((topGen‘ran (,)) ↾_t (𝐴(,]𝐵)))
98	97	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐷 = 𝐵) → (𝐶(,]𝐵) ∈ ((topGen‘ran (,)) ↾_t (𝐴(,]𝐵)))
99	46, 98	eqeltrd 2860	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐷 = 𝐵) → (𝐶(,]𝐷) ∈ ((topGen‘ran (,)) ↾_t (𝐴(,]𝐵)))
100	18	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐽 = ((TopOpen‘ℂ_fld) ↾_t ((𝐴(,)𝐵) ∪ {𝐵})))
101	37	oveq2d 7440	. . . . . . . . . . 11 ⊢ (𝜑 → ((TopOpen‘ℂ_fld) ↾_t ((𝐴(,)𝐵) ∪ {𝐵})) = ((TopOpen‘ℂ_fld) ↾_t (𝐴(,]𝐵)))
102	30	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → (TopOpen‘ℂ_fld) ∈ Top)
103		iocssre 13565	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → (𝐴(,]𝐵) ⊆ ℝ)
104	32, 33, 103	syl2anc 596	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴(,]𝐵) ⊆ ℝ)
105		reex 11298	. . . . . . . . . . . . . 14 ⊢ ℝ ∈ V
106	105	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → ℝ ∈ V)
107		restabs 23492	. . . . . . . . . . . . 13 ⊢ (((TopOpen‘ℂ_fld) ∈ Top ∧ (𝐴(,]𝐵) ⊆ ℝ ∧ ℝ ∈ V) → (((TopOpen‘ℂ_fld) ↾_t ℝ) ↾_t (𝐴(,]𝐵)) = ((TopOpen‘ℂ_fld) ↾_t (𝐴(,]𝐵)))
108	102, 104, 106, 107	syl3anc 1398	. . . . . . . . . . . 12 ⊢ (𝜑 → (((TopOpen‘ℂ_fld) ↾_t ℝ) ↾_t (𝐴(,]𝐵)) = ((TopOpen‘ℂ_fld) ↾_t (𝐴(,]𝐵)))
109		tgioo4 25133	. . . . . . . . . . . . . 14 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
110	109	eqcomi 2769	. . . . . . . . . . . . 13 ⊢ ((TopOpen‘ℂ_fld) ↾_t ℝ) = (topGen‘ran (,))
111	110	oveq1i 7434	. . . . . . . . . . . 12 ⊢ (((TopOpen‘ℂ_fld) ↾_t ℝ) ↾_t (𝐴(,]𝐵)) = ((topGen‘ran (,)) ↾_t (𝐴(,]𝐵))
112	108, 111	eqtr3di 2810	. . . . . . . . . . 11 ⊢ (𝜑 → ((TopOpen‘ℂ_fld) ↾_t (𝐴(,]𝐵)) = ((topGen‘ran (,)) ↾_t (𝐴(,]𝐵)))
113	100, 101, 112	3eqtrrd 2800	. . . . . . . . . 10 ⊢ (𝜑 → ((topGen‘ran (,)) ↾_t (𝐴(,]𝐵)) = 𝐽)
114	113	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐷 = 𝐵) → ((topGen‘ran (,)) ↾_t (𝐴(,]𝐵)) = 𝐽)
115	99, 114	eleqtrd 2862	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐷 = 𝐵) → (𝐶(,]𝐷) ∈ 𝐽)
116		isopn3i 23409	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ (𝐶(,]𝐷) ∈ 𝐽) → ((int‘𝐽)‘(𝐶(,]𝐷)) = (𝐶(,]𝐷))
117	44, 115, 116	syl2anc 596	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐷 = 𝐵) → ((int‘𝐽)‘(𝐶(,]𝐷)) = (𝐶(,]𝐷))
118	27, 29, 117	3eltr4d 2875	. . . . . 6 ⊢ ((𝜑 ∧ 𝐷 = 𝐵) → 𝐵 ∈ ((int‘𝐽)‘(𝐶(,]𝐷)))
119		sneq 4590	. . . . . . . . . . 11 ⊢ (𝐷 = 𝐵 → {𝐷} = {𝐵})
120	119	eqcomd 2766	. . . . . . . . . 10 ⊢ (𝐷 = 𝐵 → {𝐵} = {𝐷})
121	120	uneq2d 4166	. . . . . . . . 9 ⊢ (𝐷 = 𝐵 → ((𝐶(,)𝐷) ∪ {𝐵}) = ((𝐶(,)𝐷) ∪ {𝐷}))
122	121	adantl 487	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐷 = 𝐵) → ((𝐶(,)𝐷) ∪ {𝐵}) = ((𝐶(,)𝐷) ∪ {𝐷}))
123	19	rexrd 11366	. . . . . . . . . 10 ⊢ (𝜑 → 𝐷 ∈ ℝ^*)
124		ioounsn 13615	. . . . . . . . . 10 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐷 ∈ ℝ^* ∧ 𝐶 < 𝐷) → ((𝐶(,)𝐷) ∪ {𝐷}) = (𝐶(,]𝐷))
125	23, 123, 20, 124	syl3anc 1398	. . . . . . . . 9 ⊢ (𝜑 → ((𝐶(,)𝐷) ∪ {𝐷}) = (𝐶(,]𝐷))
126	125	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐷 = 𝐵) → ((𝐶(,)𝐷) ∪ {𝐷}) = (𝐶(,]𝐷))
127	122, 126	eqtr2d 2796	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐷 = 𝐵) → (𝐶(,]𝐷) = ((𝐶(,)𝐷) ∪ {𝐵}))
128	127	fveq2d 6894	. . . . . 6 ⊢ ((𝜑 ∧ 𝐷 = 𝐵) → ((int‘𝐽)‘(𝐶(,]𝐷)) = ((int‘𝐽)‘((𝐶(,)𝐷) ∪ {𝐵})))
129	118, 128	eleqtrd 2862	. . . . 5 ⊢ ((𝜑 ∧ 𝐷 = 𝐵) → 𝐵 ∈ ((int‘𝐽)‘((𝐶(,)𝐷) ∪ {𝐵})))
130	12, 14, 16, 17, 18, 129	limcres 26215	. . . 4 ⊢ ((𝜑 ∧ 𝐷 = 𝐵) → ((𝐹 ↾ (𝐶(,)𝐷)) lim_ℂ 𝐵) = (𝐹 lim_ℂ 𝐵))
131	8, 130	eqtr2d 2796	. . 3 ⊢ ((𝜑 ∧ 𝐷 = 𝐵) → (𝐹 lim_ℂ 𝐵) = ((𝐹 ↾ (𝐶(,)𝐷)) lim_ℂ 𝐷))
132	2, 6, 131	3eltr3d 2874	. 2 ⊢ ((𝜑 ∧ 𝐷 = 𝐵) → 𝑌 ∈ ((𝐹 ↾ (𝐶(,)𝐷)) lim_ℂ 𝐷))
133		limcresi 26214	. . 3 ⊢ (𝐹 lim_ℂ 𝐷) ⊆ ((𝐹 ↾ (𝐶(,)𝐷)) lim_ℂ 𝐷)
134		iffalse 4490	. . . . . 6 ⊢ (¬ 𝐷 = 𝐵 → if(𝐷 = 𝐵, 𝐿, (𝐹‘𝐷)) = (𝐹‘𝐷))
135	3, 134	eqtrid 2807	. . . . 5 ⊢ (¬ 𝐷 = 𝐵 → 𝑌 = (𝐹‘𝐷))
136	135	adantl 487	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐷 = 𝐵) → 𝑌 = (𝐹‘𝐷))
137		ssid 3931	. . . . . . . . . . . . 13 ⊢ ℂ ⊆ ℂ
138	137	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → ℂ ⊆ ℂ)
139		eqid 2760	. . . . . . . . . . . . 13 ⊢ ((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) = ((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵))
140		unicntop 25113	. . . . . . . . . . . . . . . 16 ⊢ ℂ = ∪ (TopOpen‘ℂ_fld)
141	140	restid 17613	. . . . . . . . . . . . . . 15 ⊢ ((TopOpen‘ℂ_fld) ∈ Top → ((TopOpen‘ℂ_fld) ↾_t ℂ) = (TopOpen‘ℂ_fld))
142	30, 141	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ ((TopOpen‘ℂ_fld) ↾_t ℂ) = (TopOpen‘ℂ_fld)
143	142	eqcomi 2769	. . . . . . . . . . . . 13 ⊢ (TopOpen‘ℂ_fld) = ((TopOpen‘ℂ_fld) ↾_t ℂ)
144	17, 139, 143	cncfcn 25240	. . . . . . . . . . . 12 ⊢ (((𝐴(,)𝐵) ⊆ ℂ ∧ ℂ ⊆ ℂ) → ((𝐴(,)𝐵)–cn→ℂ) = (((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) Cn (TopOpen‘ℂ_fld)))
145	15, 138, 144	sylancr 599	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐴(,)𝐵)–cn→ℂ) = (((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) Cn (TopOpen‘ℂ_fld)))
146	9, 145	eleqtrd 2862	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ (((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) Cn (TopOpen‘ℂ_fld)))
147	17	cnfldtopon 25110	. . . . . . . . . . . 12 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
148	15	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℂ)
149		resttopon 23488	. . . . . . . . . . . 12 ⊢ (((TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ) ∧ (𝐴(,)𝐵) ⊆ ℂ) → ((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) ∈ (TopOn‘(𝐴(,)𝐵)))
150	147, 148, 149	sylancr 599	. . . . . . . . . . 11 ⊢ (𝜑 → ((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) ∈ (TopOn‘(𝐴(,)𝐵)))
151	147	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ))
152		cncnp 23607	. . . . . . . . . . 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))))
153	150, 151, 152	syl2anc 596	. . . . . . . . . 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))))
154	146, 153	mpbid 235	. . . . . . . . 9 ⊢ (𝜑 → (𝐹:(𝐴(,)𝐵)⟶ℂ ∧ ∀𝑥 ∈ (𝐴(,)𝐵)𝐹 ∈ ((((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) CnP (TopOpen‘ℂ_fld))‘𝑥)))
155	154	simprd 501	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ (𝐴(,)𝐵)𝐹 ∈ ((((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) CnP (TopOpen‘ℂ_fld))‘𝑥))
156	155	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐷 = 𝐵) → ∀𝑥 ∈ (𝐴(,)𝐵)𝐹 ∈ ((((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) CnP (TopOpen‘ℂ_fld))‘𝑥))
157	32	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐷 = 𝐵) → 𝐴 ∈ ℝ^*)
158	34	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐷 = 𝐵) → 𝐵 ∈ ℝ^*)
159	19	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐷 = 𝐵) → 𝐷 ∈ ℝ)
160	31, 22, 19, 62, 20	lelttrd 11475	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 < 𝐷)
161	160	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐷 = 𝐵) → 𝐴 < 𝐷)
162	33	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐷 = 𝐵) → 𝐵 ∈ ℝ)
163	61	simprd 501	. . . . . . . . . 10 ⊢ (𝜑 → 𝐷 ≤ 𝐵)
164	163	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐷 = 𝐵) → 𝐷 ≤ 𝐵)
165		neqne 2963	. . . . . . . . . . 11 ⊢ (¬ 𝐷 = 𝐵 → 𝐷 ≠ 𝐵)
166	165	necomd 3010	. . . . . . . . . 10 ⊢ (¬ 𝐷 = 𝐵 → 𝐵 ≠ 𝐷)
167	166	adantl 487	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐷 = 𝐵) → 𝐵 ≠ 𝐷)
168	159, 162, 164, 167	leneltd 11471	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐷 = 𝐵) → 𝐷 < 𝐵)
169	157, 158, 159, 161, 168	eliood 46646	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐷 = 𝐵) → 𝐷 ∈ (𝐴(,)𝐵))
170		fveq2 6890	. . . . . . . . 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))
171	170	eleq2d 2846	. . . . . . . 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)))
172	171	rspccva 3575	. . . . . . 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))
173	156, 169, 172	syl2anc 596	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐷 = 𝐵) → 𝐹 ∈ ((((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) CnP (TopOpen‘ℂ_fld))‘𝐷))
174	17, 139	cnplimc 26216	. . . . . . 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))))
175	15, 169, 174	sylancr 599	. . . . . 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))))
176	173, 175	mpbid 235	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐷 = 𝐵) → (𝐹:(𝐴(,)𝐵)⟶ℂ ∧ (𝐹‘𝐷) ∈ (𝐹 lim_ℂ 𝐷)))
177	176	simprd 501	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐷 = 𝐵) → (𝐹‘𝐷) ∈ (𝐹 lim_ℂ 𝐷))
178	136, 177	eqeltrd 2860	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐷 = 𝐵) → 𝑌 ∈ (𝐹 lim_ℂ 𝐷))
179	133, 178	sselid 3907	. 2 ⊢ ((𝜑 ∧ ¬ 𝐷 = 𝐵) → 𝑌 ∈ ((𝐹 ↾ (𝐶(,)𝐷)) lim_ℂ 𝐷))
180	132, 179	pm2.61dan 825	1 ⊢ (𝜑 → 𝑌 ∈ ((𝐹 ↾ (𝐶(,)𝐷)) lim_ℂ 𝐷))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1569 ∈ wcel 2144 ≠ wne 2955 ∀wral 3076 Vcvv 3450 ⊆ wss 3896 ∩ cin 4088 ∪ cun 4148 ifcif 4481 {csn 4584 class class class wbr 5102 ran crn 5652 ↾ cres 5653 ⟶wf 6539 ‘cfv 6543 (class class class)co 7424 ℂcc 11205 ℝcr 11206 +∞cpnf 11347 ℝ^*cxr 11349 < clt 11350 ≤ cle 11351 (,)cioo 13483 (,]cioc 13484 ↾_t crest 17600 TopOpenctopn 17601 topGenctg 17617 ℂ_fldccnfld 21687 Topctop 23220 TopOnctopon 23237 intcnt 23344 Cn ccn 23551 CnP ccnp 23552 –cn→ccncf 25206 lim_ℂ climc 26191

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5327 ax-pr 5391 ax-un 7755 ax-cnex 11263 ax-resscn 11264 ax-1cn 11265 ax-icn 11266 ax-addcl 11267 ax-addrcl 11268 ax-mulcl 11269 ax-mulrcl 11270 ax-mulcom 11271 ax-addass 11272 ax-mulass 11273 ax-distr 11274 ax-i2m1 11275 ax-1ne0 11276 ax-1rid 11277 ax-rnegex 11278 ax-rrecex 11279 ax-cnre 11280 ax-pre-lttri 11281 ax-pre-lttrn 11282 ax-pre-ltadd 11283 ax-pre-mulgt0 11284 ax-pre-sup 11285

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3848 df-ss 3897 df-pss 4013 df-dif 4046 df-in 4090 df-un 4150 df-nul 4278 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4835 df-uni 4867 df-int 4907 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6307 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6497 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7381 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7882 df-1st 8005 df-2nd 8006 df-frecs 8307 df-wrecs 8338 df-recs 8387 df-rdg 8426 df-1o 8484 df-er 8725 df-map 8857 df-pm 8858 df-en 8982 df-dom 8983 df-sdom 8984 df-fin 8985 df-fi 9410 df-sup 9441 df-inf 9442 df-pnf 11352 df-mnf 11353 df-xr 11354 df-ltxr 11355 df-le 11356 df-sub 11550 df-neg 11551 df-div 11981 df-nn 12343 df-2 12412 df-3 12413 df-4 12414 df-5 12415 df-6 12416 df-7 12417 df-8 12418 df-9 12419 df-n0 12614 df-z 12701 df-dec 12822 df-uz 12973 df-q 13083 df-rp 13128 df-xneg 13248 df-xadd 13249 df-xmul 13250 df-ioo 13487 df-ioc 13488 df-icc 13490 df-fz 13647 df-seq 14152 df-exp 14212 df-cj 15275 df-re 15276 df-im 15277 df-sqrt 15411 df-abs 15412 df-struct 17334 df-slot 17369 df-ndx 17381 df-base 17397 df-plusg 17450 df-mulr 17451 df-starv 17452 df-tset 17456 df-ple 17457 df-ds 17459 df-unif 17460 df-rest 17602 df-topn 17603 df-topgen 17623 df-psmet 21679 df-xmet 21680 df-met 21681 df-bl 21682 df-mopn 21683 df-cnfld 21688 df-top 23221 df-topon 23238 df-topsp 23260 df-bases 23273 df-ntr 23347 df-cn 23554 df-cnp 23555 df-xms 24648 df-ms 24649 df-cncf 25208 df-limc 26195

This theorem is used by: fourierdlem49 47301 fourierdlem76 47328 fourierdlem91 47343

W3C validator