limcresiooub - Mathbox for Glauco Siliprandi

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Theorem limcresiooub 44953

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 13383	. . . . . 6 ⊢ ((𝐷 ∈ ℝ^* ∧ 𝐷 ≤ 𝐵) → (𝐵(,)𝐶) ⊆ (𝐷(,)𝐶))
4	1, 2, 3	syl2anc 583	. . . . 5 ⊢ (𝜑 → (𝐵(,)𝐶) ⊆ (𝐷(,)𝐶))
5	4	resabs1d 6010	. . . 4 ⊢ (𝜑 → ((𝐹 ↾ (𝐷(,)𝐶)) ↾ (𝐵(,)𝐶)) = (𝐹 ↾ (𝐵(,)𝐶)))
6	5	eqcomd 2733	. . 3 ⊢ (𝜑 → (𝐹 ↾ (𝐵(,)𝐶)) = ((𝐹 ↾ (𝐷(,)𝐶)) ↾ (𝐵(,)𝐶)))
7	6	oveq1d 7429	. 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 6760	. . . 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 4228	. . 3 ⊢ (𝜑 → (𝐵(,)𝐶) ⊆ (𝐴 ∩ (𝐷(,)𝐶)))
13		inss2 4225	. . . . 5 ⊢ (𝐴 ∩ (𝐷(,)𝐶)) ⊆ (𝐷(,)𝐶)
14		ioosscn 13410	. . . . 5 ⊢ (𝐷(,)𝐶) ⊆ ℂ
15	13, 14	sstri 3987	. . . 4 ⊢ (𝐴 ∩ (𝐷(,)𝐶)) ⊆ ℂ
16	15	a1i 11	. . 3 ⊢ (𝜑 → (𝐴 ∩ (𝐷(,)𝐶)) ⊆ ℂ)
17		eqid 2727	. . 3 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
18		eqid 2727	. . 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 11286	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℝ^*)
22		limcresiooub.bltc	. . . . 5 ⊢ (𝜑 → 𝐵 < 𝐶)
23		ubioc1 13401	. . . . 5 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝐵 < 𝐶) → 𝐶 ∈ (𝐵(,]𝐶))
24	19, 21, 22, 23	syl3anc 1369	. . . 4 ⊢ (𝜑 → 𝐶 ∈ (𝐵(,]𝐶))
25		ioounsn 13478	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝐵 < 𝐶) → ((𝐵(,)𝐶) ∪ {𝐶}) = (𝐵(,]𝐶))
26	19, 21, 22, 25	syl3anc 1369	. . . . . 6 ⊢ (𝜑 → ((𝐵(,)𝐶) ∪ {𝐶}) = (𝐵(,]𝐶))
27	26	fveq2d 6895	. . . . 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 24687	. . . . . . . 8 ⊢ (TopOpen‘ℂ_fld) ∈ Top
29		ovex 7447	. . . . . . . . . 10 ⊢ (𝐷(,)𝐶) ∈ V
30	29	inex2 5312	. . . . . . . . 9 ⊢ (𝐴 ∩ (𝐷(,)𝐶)) ∈ V
31		snex 5427	. . . . . . . . 9 ⊢ {𝐶} ∈ V
32	30, 31	unex 7742	. . . . . . . 8 ⊢ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}) ∈ V
33		resttop 23051	. . . . . . . 8 ⊢ (((TopOpen‘ℂ_fld) ∈ Top ∧ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}) ∈ V) → ((TopOpen‘ℂ_fld) ↾_t ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})) ∈ Top)
34	28, 32, 33	mp2an 691	. . . . . . 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 11290	. . . . . . . . . . . . . 14 ⊢ +∞ ∈ ℝ^*
37	36	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → +∞ ∈ ℝ^*)
38	19	xrleidd 13155	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 ≤ 𝐵)
39	20	ltpnfd 13125	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐶 < +∞)
40		iocssioo 13440	. . . . . . . . . . . . 13 ⊢ (((𝐵 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) ∧ (𝐵 ≤ 𝐵 ∧ 𝐶 < +∞)) → (𝐵(,]𝐶) ⊆ (𝐵(,)+∞))
41	19, 37, 38, 39, 40	syl22anc 838	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐵(,]𝐶) ⊆ (𝐵(,)+∞))
42		simpr 484	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → 𝑥 = 𝐶)
43		snidg 4658	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐶 ∈ ℝ → 𝐶 ∈ {𝐶})
44		elun2 4173	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐶 ∈ {𝐶} → 𝐶 ∈ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))
45	20, 43, 44	3syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐶 ∈ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))
46	45	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → 𝐶 ∈ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))
47	42, 46	eqeltrd 2828	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → 𝑥 ∈ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))
48	47	adantlr 714	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) ∧ 𝑥 = 𝐶) → 𝑥 ∈ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))
49		simpll 766	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) ∧ ¬ 𝑥 = 𝐶) → 𝜑)
50	19	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) → 𝐵 ∈ ℝ^*)
51	50	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) ∧ ¬ 𝑥 = 𝐶) → 𝐵 ∈ ℝ^*)
52	21	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) → 𝐶 ∈ ℝ^*)
53	52	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) ∧ ¬ 𝑥 = 𝐶) → 𝐶 ∈ ℝ^*)
54		iocssre 13428	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ) → (𝐵(,]𝐶) ⊆ ℝ)
55	19, 20, 54	syl2anc 583	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝐵(,]𝐶) ⊆ ℝ)
56	55	sselda 3978	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) → 𝑥 ∈ ℝ)
57	56	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) ∧ ¬ 𝑥 = 𝐶) → 𝑥 ∈ ℝ)
58		simpr 484	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) → 𝑥 ∈ (𝐵(,]𝐶))
59		iocgtlb 44810	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝑥 ∈ (𝐵(,]𝐶)) → 𝐵 < 𝑥)
60	50, 52, 58, 59	syl3anc 1369	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) → 𝐵 < 𝑥)
61	60	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) ∧ ¬ 𝑥 = 𝐶) → 𝐵 < 𝑥)
62	20	ad2antrr 725	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) ∧ ¬ 𝑥 = 𝐶) → 𝐶 ∈ ℝ)
63		iocleub 44811	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝑥 ∈ (𝐵(,]𝐶)) → 𝑥 ≤ 𝐶)
64	50, 52, 58, 63	syl3anc 1369	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) → 𝑥 ≤ 𝐶)
65	64	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) ∧ ¬ 𝑥 = 𝐶) → 𝑥 ≤ 𝐶)
66		neqne 2943	. . . . . . . . . . . . . . . . . . . 20 ⊢ (¬ 𝑥 = 𝐶 → 𝑥 ≠ 𝐶)
67	66	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) ∧ ¬ 𝑥 = 𝐶) → 𝑥 ≠ 𝐶)
68	67	necomd 2991	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) ∧ ¬ 𝑥 = 𝐶) → 𝐶 ≠ 𝑥)
69	57, 62, 65, 68	leneltd 11390	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) ∧ ¬ 𝑥 = 𝐶) → 𝑥 < 𝐶)
70	51, 53, 57, 61, 69	eliood 44806	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) ∧ ¬ 𝑥 = 𝐶) → 𝑥 ∈ (𝐵(,)𝐶))
71	12	sselda 3978	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,)𝐶)) → 𝑥 ∈ (𝐴 ∩ (𝐷(,)𝐶)))
72		elun1 4172	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (𝐴 ∩ (𝐷(,)𝐶)) → 𝑥 ∈ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))
73	71, 72	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,)𝐶)) → 𝑥 ∈ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))
74	49, 70, 73	syl2anc 583	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) ∧ ¬ 𝑥 = 𝐶) → 𝑥 ∈ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))
75	48, 74	pm2.61dan 812	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) → 𝑥 ∈ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))
76	75	ralrimiva 3141	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑥 ∈ (𝐵(,]𝐶)𝑥 ∈ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))
77		dfss3 3966	. . . . . . . . . . . . 13 ⊢ ((𝐵(,]𝐶) ⊆ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}) ↔ ∀𝑥 ∈ (𝐵(,]𝐶)𝑥 ∈ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))
78	76, 77	sylibr 233	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐵(,]𝐶) ⊆ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))
79	41, 78	ssind 4228	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐵(,]𝐶) ⊆ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})))
80	79	sseld 3977	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ (𝐵(,]𝐶) → 𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))))
81	24	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → 𝐶 ∈ (𝐵(,]𝐶))
82	42, 81	eqeltrd 2828	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → 𝑥 ∈ (𝐵(,]𝐶))
83	82	adantlr 714	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))) ∧ 𝑥 = 𝐶) → 𝑥 ∈ (𝐵(,]𝐶))
84		ioossioc 44800	. . . . . . . . . . . . 13 ⊢ (𝐵(,)𝐶) ⊆ (𝐵(,]𝐶)
85	19	ad2antrr 725	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))) ∧ ¬ 𝑥 = 𝐶) → 𝐵 ∈ ℝ^*)
86	21	ad2antrr 725	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))) ∧ ¬ 𝑥 = 𝐶) → 𝐶 ∈ ℝ^*)
87		elinel1 4191	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})) → 𝑥 ∈ (𝐵(,)+∞))
88	87	elioored 44857	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})) → 𝑥 ∈ ℝ)
89	88	ad2antlr 726	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))) ∧ ¬ 𝑥 = 𝐶) → 𝑥 ∈ ℝ)
90	36	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))) ∧ ¬ 𝑥 = 𝐶) → +∞ ∈ ℝ^*)
91	87	ad2antlr 726	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))) ∧ ¬ 𝑥 = 𝐶) → 𝑥 ∈ (𝐵(,)+∞))
92		ioogtlb 44803	. . . . . . . . . . . . . . 15 ⊢ ((𝐵 ∈ ℝ^* ∧ +∞ ∈ ℝ^* ∧ 𝑥 ∈ (𝐵(,)+∞)) → 𝐵 < 𝑥)
93	85, 90, 91, 92	syl3anc 1369	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))) ∧ ¬ 𝑥 = 𝐶) → 𝐵 < 𝑥)
94	1	ad2antrr 725	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))) ∧ ¬ 𝑥 = 𝐶) → 𝐷 ∈ ℝ^*)
95		elinel2 4192	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})) → 𝑥 ∈ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))
96		id 22	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ 𝑥 = 𝐶 → ¬ 𝑥 = 𝐶)
97		velsn 4640	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ {𝐶} ↔ 𝑥 = 𝐶)
98	96, 97	sylnibr 329	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ 𝑥 = 𝐶 → ¬ 𝑥 ∈ {𝐶})
99		elunnel2 4146	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}) ∧ ¬ 𝑥 ∈ {𝐶}) → 𝑥 ∈ (𝐴 ∩ (𝐷(,)𝐶)))
100	95, 98, 99	syl2an 595	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})) ∧ ¬ 𝑥 = 𝐶) → 𝑥 ∈ (𝐴 ∩ (𝐷(,)𝐶)))
101	13, 100	sselid 3976	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})) ∧ ¬ 𝑥 = 𝐶) → 𝑥 ∈ (𝐷(,)𝐶))
102	101	adantll 713	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))) ∧ ¬ 𝑥 = 𝐶) → 𝑥 ∈ (𝐷(,)𝐶))
103		iooltub 44818	. . . . . . . . . . . . . . 15 ⊢ ((𝐷 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝑥 ∈ (𝐷(,)𝐶)) → 𝑥 < 𝐶)
104	94, 86, 102, 103	syl3anc 1369	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))) ∧ ¬ 𝑥 = 𝐶) → 𝑥 < 𝐶)
105	85, 86, 89, 93, 104	eliood 44806	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))) ∧ ¬ 𝑥 = 𝐶) → 𝑥 ∈ (𝐵(,)𝐶))
106	84, 105	sselid 3976	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))) ∧ ¬ 𝑥 = 𝐶) → 𝑥 ∈ (𝐵(,]𝐶))
107	83, 106	pm2.61dan 812	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))) → 𝑥 ∈ (𝐵(,]𝐶))
108	107	ex 412	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})) → 𝑥 ∈ (𝐵(,]𝐶)))
109	80, 108	impbid 211	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (𝐵(,]𝐶) ↔ 𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))))
110	109	eqrdv 2725	. . . . . . . 8 ⊢ (𝜑 → (𝐵(,]𝐶) = ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})))
111		retop 24665	. . . . . . . . . 10 ⊢ (topGen‘ran (,)) ∈ Top
112	111	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → (topGen‘ran (,)) ∈ Top)
113	32	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}) ∈ V)
114		iooretop 24669	. . . . . . . . . 10 ⊢ (𝐵(,)+∞) ∈ (topGen‘ran (,))
115	114	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → (𝐵(,)+∞) ∈ (topGen‘ran (,)))
116		elrestr 17401	. . . . . . . . 9 ⊢ (((topGen‘ran (,)) ∈ Top ∧ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}) ∈ V ∧ (𝐵(,)+∞) ∈ (topGen‘ran (,))) → ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})) ∈ ((topGen‘ran (,)) ↾_t ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})))
117	112, 113, 115, 116	syl3anc 1369	. . . . . . . 8 ⊢ (𝜑 → ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})) ∈ ((topGen‘ran (,)) ↾_t ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})))
118	110, 117	eqeltrd 2828	. . . . . . 7 ⊢ (𝜑 → (𝐵(,]𝐶) ∈ ((topGen‘ran (,)) ↾_t ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})))
119	17	tgioo2 24706	. . . . . . . . 9 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
120	119	oveq1i 7424	. . . . . . . 8 ⊢ ((topGen‘ran (,)) ↾_t ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})) = (((TopOpen‘ℂ_fld) ↾_t ℝ) ↾_t ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))
121	28	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → (TopOpen‘ℂ_fld) ∈ Top)
122		ioossre 13409	. . . . . . . . . . . 12 ⊢ (𝐷(,)𝐶) ⊆ ℝ
123	13, 122	sstri 3987	. . . . . . . . . . 11 ⊢ (𝐴 ∩ (𝐷(,)𝐶)) ⊆ ℝ
124	123	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 ∩ (𝐷(,)𝐶)) ⊆ ℝ)
125	20	snssd 4808	. . . . . . . . . 10 ⊢ (𝜑 → {𝐶} ⊆ ℝ)
126	124, 125	unssd 4182	. . . . . . . . 9 ⊢ (𝜑 → ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}) ⊆ ℝ)
127		reex 11221	. . . . . . . . . 10 ⊢ ℝ ∈ V
128	127	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → ℝ ∈ V)
129		restabs 23056	. . . . . . . . 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})))
130	121, 126, 128, 129	syl3anc 1369	. . . . . . . 8 ⊢ (𝜑 → (((TopOpen‘ℂ_fld) ↾_t ℝ) ↾_t ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})) = ((TopOpen‘ℂ_fld) ↾_t ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})))
131	120, 130	eqtrid 2779	. . . . . . 7 ⊢ (𝜑 → ((topGen‘ran (,)) ↾_t ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})) = ((TopOpen‘ℂ_fld) ↾_t ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})))
132	118, 131	eleqtrd 2830	. . . . . 6 ⊢ (𝜑 → (𝐵(,]𝐶) ∈ ((TopOpen‘ℂ_fld) ↾_t ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})))
133		isopn3i 22973	. . . . . 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))
134	35, 132, 133	syl2anc 583	. . . . 5 ⊢ (𝜑 → ((int‘((TopOpen‘ℂ_fld) ↾_t ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})))‘(𝐵(,]𝐶)) = (𝐵(,]𝐶))
135	27, 134	eqtr2d 2768	. . . 4 ⊢ (𝜑 → (𝐵(,]𝐶) = ((int‘((TopOpen‘ℂ_fld) ↾_t ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})))‘((𝐵(,)𝐶) ∪ {𝐶})))
136	24, 135	eleqtrd 2830	. . 3 ⊢ (𝜑 → 𝐶 ∈ ((int‘((TopOpen‘ℂ_fld) ↾_t ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})))‘((𝐵(,)𝐶) ∪ {𝐶})))
137	10, 12, 16, 17, 18, 136	limcres 25802	. 2 ⊢ (𝜑 → (((𝐹 ↾ (𝐷(,)𝐶)) ↾ (𝐵(,)𝐶)) lim_ℂ 𝐶) = ((𝐹 ↾ (𝐷(,)𝐶)) lim_ℂ 𝐶))
138	7, 137	eqtrd 2767	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 395 = wceq 1534 ∈ wcel 2099 ≠ wne 2935 ∀wral 3056 Vcvv 3469 ∪ cun 3942 ∩ cin 3943 ⊆ wss 3944 {csn 4624 class class class wbr 5142 ran crn 5673 ↾ cres 5674 ⟶wf 6538 ‘cfv 6542 (class class class)co 7414 ℂcc 11128 ℝcr 11129 +∞cpnf 11267 ℝ^*cxr 11269 < clt 11270 ≤ cle 11271 (,)cioo 13348 (,]cioc 13349 ↾_t crest 17393 TopOpenctopn 17394 topGenctg 17410 ℂ_fldccnfld 21266 Topctop 22782 intcnt 22908 lim_ℂ climc 25778

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 ax-pre-sup 11208

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-map 8838 df-pm 8839 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-fi 9426 df-sup 9457 df-inf 9458 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-div 11894 df-nn 12235 df-2 12297 df-3 12298 df-4 12299 df-5 12300 df-6 12301 df-7 12302 df-8 12303 df-9 12304 df-n0 12495 df-z 12581 df-dec 12700 df-uz 12845 df-q 12955 df-rp 12999 df-xneg 13116 df-xadd 13117 df-xmul 13118 df-ioo 13352 df-ioc 13353 df-icc 13355 df-fz 13509 df-seq 13991 df-exp 14051 df-cj 15070 df-re 15071 df-im 15072 df-sqrt 15206 df-abs 15207 df-struct 17107 df-slot 17142 df-ndx 17154 df-base 17172 df-plusg 17237 df-mulr 17238 df-starv 17239 df-tset 17243 df-ple 17244 df-ds 17246 df-unif 17247 df-rest 17395 df-topn 17396 df-topgen 17416 df-psmet 21258 df-xmet 21259 df-met 21260 df-bl 21261 df-mopn 21262 df-cnfld 21267 df-top 22783 df-topon 22800 df-topsp 22822 df-bases 22836 df-ntr 22911 df-cnp 23119 df-xms 24213 df-ms 24214 df-limc 25782

This theorem is referenced by: fouriersw 45542

W3C validator