limcresiooub - Mathbox for Glauco Siliprandi

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Theorem limcresiooub 38709

Description: The left limit doesn't change if the function is restricted to a smaller open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Hypotheses**
Ref	Expression
limcresiooub.f	⊢ (𝜑 → 𝐹:𝐴⟶ℂ)
limcresiooub.b	⊢ (𝜑 → 𝐵 ∈ ℝ^*)
limcresiooub.c	⊢ (𝜑 → 𝐶 ∈ ℝ)
limcresiooub.bltc	⊢ (𝜑 → 𝐵 < 𝐶)
limcresiooub.bcss	⊢ (𝜑 → (𝐵(,)𝐶) ⊆ 𝐴)
limcresiooub.d	⊢ (𝜑 → 𝐷 ∈ ℝ^*)
limcresiooub.cled	⊢ (𝜑 → 𝐷 ≤ 𝐵)

**Assertion**
Ref	Expression
limcresiooub	⊢ (𝜑 → ((𝐹 ↾ (𝐵(,)𝐶)) lim_ℂ 𝐶) = ((𝐹 ↾ (𝐷(,)𝐶)) lim_ℂ 𝐶))

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

Step	Hyp	Ref	Expression
1		limcresiooub.d	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ ℝ^*)
2		limcresiooub.cled	. . . . . 6 ⊢ (𝜑 → 𝐷 ≤ 𝐵)
3		iooss1 12081	. . . . . 6 ⊢ ((𝐷 ∈ ℝ^* ∧ 𝐷 ≤ 𝐵) → (𝐵(,)𝐶) ⊆ (𝐷(,)𝐶))
4	1, 2, 3	syl2anc 691	. . . . 5 ⊢ (𝜑 → (𝐵(,)𝐶) ⊆ (𝐷(,)𝐶))
5	4	resabs1d 5348	. . . 4 ⊢ (𝜑 → ((𝐹 ↾ (𝐷(,)𝐶)) ↾ (𝐵(,)𝐶)) = (𝐹 ↾ (𝐵(,)𝐶)))
6	5	eqcomd 2616	. . 3 ⊢ (𝜑 → (𝐹 ↾ (𝐵(,)𝐶)) = ((𝐹 ↾ (𝐷(,)𝐶)) ↾ (𝐵(,)𝐶)))
7	6	oveq1d 6564	. 2 ⊢ (𝜑 → ((𝐹 ↾ (𝐵(,)𝐶)) lim_ℂ 𝐶) = (((𝐹 ↾ (𝐷(,)𝐶)) ↾ (𝐵(,)𝐶)) lim_ℂ 𝐶))
8		limcresiooub.f	. . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶ℂ)
9		fresin 5986	. . . 4 ⊢ (𝐹:𝐴⟶ℂ → (𝐹 ↾ (𝐷(,)𝐶)):(𝐴 ∩ (𝐷(,)𝐶))⟶ℂ)
10	8, 9	syl 17	. . 3 ⊢ (𝜑 → (𝐹 ↾ (𝐷(,)𝐶)):(𝐴 ∩ (𝐷(,)𝐶))⟶ℂ)
11		limcresiooub.bcss	. . . 4 ⊢ (𝜑 → (𝐵(,)𝐶) ⊆ 𝐴)
12	11, 4	ssind 3799	. . 3 ⊢ (𝜑 → (𝐵(,)𝐶) ⊆ (𝐴 ∩ (𝐷(,)𝐶)))
13		inss2 3796	. . . . 5 ⊢ (𝐴 ∩ (𝐷(,)𝐶)) ⊆ (𝐷(,)𝐶)
14		ioosscn 38563	. . . . 5 ⊢ (𝐷(,)𝐶) ⊆ ℂ
15	13, 14	sstri 3577	. . . 4 ⊢ (𝐴 ∩ (𝐷(,)𝐶)) ⊆ ℂ
16	15	a1i 11	. . 3 ⊢ (𝜑 → (𝐴 ∩ (𝐷(,)𝐶)) ⊆ ℂ)
17		eqid 2610	. . 3 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
18		eqid 2610	. . 3 ⊢ ((TopOpen‘ℂ_fld) ↾_t ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})) = ((TopOpen‘ℂ_fld) ↾_t ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))
19		limcresiooub.b	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
20		limcresiooub.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ ℝ)
21	20	rexrd 9968	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℝ^*)
22		limcresiooub.bltc	. . . . 5 ⊢ (𝜑 → 𝐵 < 𝐶)
23		ubioc1 12098	. . . . 5 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝐵 < 𝐶) → 𝐶 ∈ (𝐵(,]𝐶))
24	19, 21, 22, 23	syl3anc 1318	. . . 4 ⊢ (𝜑 → 𝐶 ∈ (𝐵(,]𝐶))
25		snunioo2 38578	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝐵 < 𝐶) → ((𝐵(,)𝐶) ∪ {𝐶}) = (𝐵(,]𝐶))
26	19, 21, 22, 25	syl3anc 1318	. . . . . 6 ⊢ (𝜑 → ((𝐵(,)𝐶) ∪ {𝐶}) = (𝐵(,]𝐶))
27	26	fveq2d 6107	. . . . 5 ⊢ (𝜑 → ((int‘((TopOpen‘ℂ_fld) ↾_t ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})))‘((𝐵(,)𝐶) ∪ {𝐶})) = ((int‘((TopOpen‘ℂ_fld) ↾_t ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})))‘(𝐵(,]𝐶)))
28	17	cnfldtop 22397	. . . . . . . 8 ⊢ (TopOpen‘ℂ_fld) ∈ Top
29		ovex 6577	. . . . . . . . . 10 ⊢ (𝐷(,)𝐶) ∈ V
30	29	inex2 4728	. . . . . . . . 9 ⊢ (𝐴 ∩ (𝐷(,)𝐶)) ∈ V
31		snex 4835	. . . . . . . . 9 ⊢ {𝐶} ∈ V
32	30, 31	unex 6854	. . . . . . . 8 ⊢ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}) ∈ V
33		resttop 20774	. . . . . . . 8 ⊢ (((TopOpen‘ℂ_fld) ∈ Top ∧ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}) ∈ V) → ((TopOpen‘ℂ_fld) ↾_t ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})) ∈ Top)
34	28, 32, 33	mp2an 704	. . . . . . 7 ⊢ ((TopOpen‘ℂ_fld) ↾_t ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})) ∈ Top
35	34	a1i 11	. . . . . 6 ⊢ (𝜑 → ((TopOpen‘ℂ_fld) ↾_t ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})) ∈ Top)
36		pnfxr 9971	. . . . . . . . . . . . . 14 ⊢ +∞ ∈ ℝ^*
37	36	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → +∞ ∈ ℝ^*)
38		xrleid 11859	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ ℝ^* → 𝐵 ≤ 𝐵)
39	19, 38	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 ≤ 𝐵)
40	20	ltpnfd 11831	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐶 < +∞)
41		iocssioo 12134	. . . . . . . . . . . . 13 ⊢ (((𝐵 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) ∧ (𝐵 ≤ 𝐵 ∧ 𝐶 < +∞)) → (𝐵(,]𝐶) ⊆ (𝐵(,)+∞))
42	19, 37, 39, 40, 41	syl22anc 1319	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐵(,]𝐶) ⊆ (𝐵(,)+∞))
43		simpr 476	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → 𝑥 = 𝐶)
44		snidg 4153	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐶 ∈ ℝ → 𝐶 ∈ {𝐶})
45		elun2 3743	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐶 ∈ {𝐶} → 𝐶 ∈ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))
46	20, 44, 45	3syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐶 ∈ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))
47	46	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → 𝐶 ∈ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))
48	43, 47	eqeltrd 2688	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → 𝑥 ∈ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))
49	48	adantlr 747	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) ∧ 𝑥 = 𝐶) → 𝑥 ∈ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))
50		simpll 786	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) ∧ ¬ 𝑥 = 𝐶) → 𝜑)
51	19	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) → 𝐵 ∈ ℝ^*)
52	51	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) ∧ ¬ 𝑥 = 𝐶) → 𝐵 ∈ ℝ^*)
53	21	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) → 𝐶 ∈ ℝ^*)
54	53	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) ∧ ¬ 𝑥 = 𝐶) → 𝐶 ∈ ℝ^*)
55		iocssre 12124	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ) → (𝐵(,]𝐶) ⊆ ℝ)
56	19, 20, 55	syl2anc 691	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝐵(,]𝐶) ⊆ ℝ)
57	56	sselda 3568	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) → 𝑥 ∈ ℝ)
58	57	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) ∧ ¬ 𝑥 = 𝐶) → 𝑥 ∈ ℝ)
59		simpr 476	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) → 𝑥 ∈ (𝐵(,]𝐶))
60		iocgtlb 38571	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝑥 ∈ (𝐵(,]𝐶)) → 𝐵 < 𝑥)
61	51, 53, 59, 60	syl3anc 1318	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) → 𝐵 < 𝑥)
62	61	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) ∧ ¬ 𝑥 = 𝐶) → 𝐵 < 𝑥)
63	20	ad2antrr 758	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) ∧ ¬ 𝑥 = 𝐶) → 𝐶 ∈ ℝ)
64		iocleub 38572	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝑥 ∈ (𝐵(,]𝐶)) → 𝑥 ≤ 𝐶)
65	51, 53, 59, 64	syl3anc 1318	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) → 𝑥 ≤ 𝐶)
66	65	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) ∧ ¬ 𝑥 = 𝐶) → 𝑥 ≤ 𝐶)
67		neqne 2790	. . . . . . . . . . . . . . . . . . . 20 ⊢ (¬ 𝑥 = 𝐶 → 𝑥 ≠ 𝐶)
68	67	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) ∧ ¬ 𝑥 = 𝐶) → 𝑥 ≠ 𝐶)
69	68	necomd 2837	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) ∧ ¬ 𝑥 = 𝐶) → 𝐶 ≠ 𝑥)
70	58, 63, 66, 69	leneltd 10070	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) ∧ ¬ 𝑥 = 𝐶) → 𝑥 < 𝐶)
71	52, 54, 58, 62, 70	eliood 38567	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) ∧ ¬ 𝑥 = 𝐶) → 𝑥 ∈ (𝐵(,)𝐶))
72	12	sselda 3568	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,)𝐶)) → 𝑥 ∈ (𝐴 ∩ (𝐷(,)𝐶)))
73		elun1 3742	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (𝐴 ∩ (𝐷(,)𝐶)) → 𝑥 ∈ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))
74	72, 73	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,)𝐶)) → 𝑥 ∈ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))
75	50, 71, 74	syl2anc 691	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) ∧ ¬ 𝑥 = 𝐶) → 𝑥 ∈ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))
76	49, 75	pm2.61dan 828	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) → 𝑥 ∈ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))
77	76	ralrimiva 2949	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑥 ∈ (𝐵(,]𝐶)𝑥 ∈ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))
78		dfss3 3558	. . . . . . . . . . . . 13 ⊢ ((𝐵(,]𝐶) ⊆ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}) ↔ ∀𝑥 ∈ (𝐵(,]𝐶)𝑥 ∈ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))
79	77, 78	sylibr 223	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐵(,]𝐶) ⊆ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))
80	42, 79	ssind 3799	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐵(,]𝐶) ⊆ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})))
81	80	sseld 3567	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ (𝐵(,]𝐶) → 𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))))
82	24	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → 𝐶 ∈ (𝐵(,]𝐶))
83	43, 82	eqeltrd 2688	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → 𝑥 ∈ (𝐵(,]𝐶))
84	83	adantlr 747	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))) ∧ 𝑥 = 𝐶) → 𝑥 ∈ (𝐵(,]𝐶))
85		ioossioc 38560	. . . . . . . . . . . . 13 ⊢ (𝐵(,)𝐶) ⊆ (𝐵(,]𝐶)
86	19	ad2antrr 758	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))) ∧ ¬ 𝑥 = 𝐶) → 𝐵 ∈ ℝ^*)
87	21	ad2antrr 758	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))) ∧ ¬ 𝑥 = 𝐶) → 𝐶 ∈ ℝ^*)
88		elinel1 3761	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})) → 𝑥 ∈ (𝐵(,)+∞))
89		elioore 12076	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (𝐵(,)+∞) → 𝑥 ∈ ℝ)
90	88, 89	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})) → 𝑥 ∈ ℝ)
91	90	ad2antlr 759	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))) ∧ ¬ 𝑥 = 𝐶) → 𝑥 ∈ ℝ)
92	36	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))) ∧ ¬ 𝑥 = 𝐶) → +∞ ∈ ℝ^*)
93	88	ad2antlr 759	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))) ∧ ¬ 𝑥 = 𝐶) → 𝑥 ∈ (𝐵(,)+∞))
94		ioogtlb 38564	. . . . . . . . . . . . . . 15 ⊢ ((𝐵 ∈ ℝ^* ∧ +∞ ∈ ℝ^* ∧ 𝑥 ∈ (𝐵(,)+∞)) → 𝐵 < 𝑥)
95	86, 92, 93, 94	syl3anc 1318	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))) ∧ ¬ 𝑥 = 𝐶) → 𝐵 < 𝑥)
96	1	ad2antrr 758	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))) ∧ ¬ 𝑥 = 𝐶) → 𝐷 ∈ ℝ^*)
97		elinel2 3762	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})) → 𝑥 ∈ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))
98		id 22	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ 𝑥 = 𝐶 → ¬ 𝑥 = 𝐶)
99		velsn 4141	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ {𝐶} ↔ 𝑥 = 𝐶)
100	98, 99	sylnibr 318	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ 𝑥 = 𝐶 → ¬ 𝑥 ∈ {𝐶})
101		elunnel2 38221	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}) ∧ ¬ 𝑥 ∈ {𝐶}) → 𝑥 ∈ (𝐴 ∩ (𝐷(,)𝐶)))
102	97, 100, 101	syl2an 493	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})) ∧ ¬ 𝑥 = 𝐶) → 𝑥 ∈ (𝐴 ∩ (𝐷(,)𝐶)))
103	13, 102	sseldi 3566	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})) ∧ ¬ 𝑥 = 𝐶) → 𝑥 ∈ (𝐷(,)𝐶))
104	103	adantll 746	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))) ∧ ¬ 𝑥 = 𝐶) → 𝑥 ∈ (𝐷(,)𝐶))
105		iooltub 38582	. . . . . . . . . . . . . . 15 ⊢ ((𝐷 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝑥 ∈ (𝐷(,)𝐶)) → 𝑥 < 𝐶)
106	96, 87, 104, 105	syl3anc 1318	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))) ∧ ¬ 𝑥 = 𝐶) → 𝑥 < 𝐶)
107	86, 87, 91, 95, 106	eliood 38567	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))) ∧ ¬ 𝑥 = 𝐶) → 𝑥 ∈ (𝐵(,)𝐶))
108	85, 107	sseldi 3566	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))) ∧ ¬ 𝑥 = 𝐶) → 𝑥 ∈ (𝐵(,]𝐶))
109	84, 108	pm2.61dan 828	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))) → 𝑥 ∈ (𝐵(,]𝐶))
110	109	ex 449	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})) → 𝑥 ∈ (𝐵(,]𝐶)))
111	81, 110	impbid 201	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (𝐵(,]𝐶) ↔ 𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))))
112	111	eqrdv 2608	. . . . . . . 8 ⊢ (𝜑 → (𝐵(,]𝐶) = ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})))
113		retop 22375	. . . . . . . . . 10 ⊢ (topGen‘ran (,)) ∈ Top
114	113	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → (topGen‘ran (,)) ∈ Top)
115	32	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}) ∈ V)
116		iooretop 22379	. . . . . . . . . 10 ⊢ (𝐵(,)+∞) ∈ (topGen‘ran (,))
117	116	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → (𝐵(,)+∞) ∈ (topGen‘ran (,)))
118		elrestr 15912	. . . . . . . . 9 ⊢ (((topGen‘ran (,)) ∈ Top ∧ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}) ∈ V ∧ (𝐵(,)+∞) ∈ (topGen‘ran (,))) → ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})) ∈ ((topGen‘ran (,)) ↾_t ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})))
119	114, 115, 117, 118	syl3anc 1318	. . . . . . . 8 ⊢ (𝜑 → ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})) ∈ ((topGen‘ran (,)) ↾_t ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})))
120	112, 119	eqeltrd 2688	. . . . . . 7 ⊢ (𝜑 → (𝐵(,]𝐶) ∈ ((topGen‘ran (,)) ↾_t ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})))
121	17	tgioo2 22414	. . . . . . . . 9 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
122	121	oveq1i 6559	. . . . . . . 8 ⊢ ((topGen‘ran (,)) ↾_t ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})) = (((TopOpen‘ℂ_fld) ↾_t ℝ) ↾_t ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))
123	28	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → (TopOpen‘ℂ_fld) ∈ Top)
124		ioossre 12106	. . . . . . . . . . . 12 ⊢ (𝐷(,)𝐶) ⊆ ℝ
125	13, 124	sstri 3577	. . . . . . . . . . 11 ⊢ (𝐴 ∩ (𝐷(,)𝐶)) ⊆ ℝ
126	125	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 ∩ (𝐷(,)𝐶)) ⊆ ℝ)
127	20	snssd 4281	. . . . . . . . . 10 ⊢ (𝜑 → {𝐶} ⊆ ℝ)
128	126, 127	unssd 3751	. . . . . . . . 9 ⊢ (𝜑 → ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}) ⊆ ℝ)
129		reex 9906	. . . . . . . . . 10 ⊢ ℝ ∈ V
130	129	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → ℝ ∈ V)
131		restabs 20779	. . . . . . . . 9 ⊢ (((TopOpen‘ℂ_fld) ∈ Top ∧ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}) ⊆ ℝ ∧ ℝ ∈ V) → (((TopOpen‘ℂ_fld) ↾_t ℝ) ↾_t ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})) = ((TopOpen‘ℂ_fld) ↾_t ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})))
132	123, 128, 130, 131	syl3anc 1318	. . . . . . . 8 ⊢ (𝜑 → (((TopOpen‘ℂ_fld) ↾_t ℝ) ↾_t ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})) = ((TopOpen‘ℂ_fld) ↾_t ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})))
133	122, 132	syl5eq 2656	. . . . . . 7 ⊢ (𝜑 → ((topGen‘ran (,)) ↾_t ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})) = ((TopOpen‘ℂ_fld) ↾_t ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})))
134	120, 133	eleqtrd 2690	. . . . . 6 ⊢ (𝜑 → (𝐵(,]𝐶) ∈ ((TopOpen‘ℂ_fld) ↾_t ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})))
135		isopn3i 20696	. . . . . 6 ⊢ ((((TopOpen‘ℂ_fld) ↾_t ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})) ∈ Top ∧ (𝐵(,]𝐶) ∈ ((TopOpen‘ℂ_fld) ↾_t ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))) → ((int‘((TopOpen‘ℂ_fld) ↾_t ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})))‘(𝐵(,]𝐶)) = (𝐵(,]𝐶))
136	35, 134, 135	syl2anc 691	. . . . 5 ⊢ (𝜑 → ((int‘((TopOpen‘ℂ_fld) ↾_t ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})))‘(𝐵(,]𝐶)) = (𝐵(,]𝐶))
137	27, 136	eqtr2d 2645	. . . 4 ⊢ (𝜑 → (𝐵(,]𝐶) = ((int‘((TopOpen‘ℂ_fld) ↾_t ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})))‘((𝐵(,)𝐶) ∪ {𝐶})))
138	24, 137	eleqtrd 2690	. . 3 ⊢ (𝜑 → 𝐶 ∈ ((int‘((TopOpen‘ℂ_fld) ↾_t ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})))‘((𝐵(,)𝐶) ∪ {𝐶})))
139	10, 12, 16, 17, 18, 138	limcres 23456	. 2 ⊢ (𝜑 → (((𝐹 ↾ (𝐷(,)𝐶)) ↾ (𝐵(,)𝐶)) lim_ℂ 𝐶) = ((𝐹 ↾ (𝐷(,)𝐶)) lim_ℂ 𝐶))
140	7, 139	eqtrd 2644	1 ⊢ (𝜑 → ((𝐹 ↾ (𝐵(,)𝐶)) lim_ℂ 𝐶) = ((𝐹 ↾ (𝐷(,)𝐶)) lim_ℂ 𝐶))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 Vcvv 3173 ∪ cun 3538 ∩ cin 3539 ⊆ wss 3540 {csn 4125 class class class wbr 4583 ran crn 5039 ↾ cres 5040 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 ℝcr 9814 +∞cpnf 9950 ℝ^*cxr 9952 < clt 9953 ≤ cle 9954 (,)cioo 12046 (,]cioc 12047 ↾_t crest 15904 TopOpenctopn 15905 topGenctg 15921 ℂ_fldccnfld 19567 Topctop 20517 intcnt 20631 lim_ℂ climc 23432

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fi 8200 df-sup 8231 df-inf 8232 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-ioo 12050 df-ioc 12051 df-icc 12053 df-fz 12198 df-seq 12664 df-exp 12723 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-plusg 15781 df-mulr 15782 df-starv 15783 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-rest 15906 df-topn 15907 df-topgen 15927 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-cnfld 19568 df-top 20521 df-bases 20522 df-topon 20523 df-topsp 20524 df-ntr 20634 df-cnp 20842 df-xms 21935 df-ms 21936 df-limc 23436

This theorem is referenced by: fouriersw 39124

W3C validator