limcresiooub - Mathbox for Glauco Siliprandi

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Theorem limcresiooub 46091

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 13327	. . . . . 6 ⊢ ((𝐷 ∈ ℝ^* ∧ 𝐷 ≤ 𝐵) → (𝐵(,)𝐶) ⊆ (𝐷(,)𝐶))
4	1, 2, 3	syl2anc 585	. . . . 5 ⊢ (𝜑 → (𝐵(,)𝐶) ⊆ (𝐷(,)𝐶))
5	4	resabs1d 5968	. . . 4 ⊢ (𝜑 → ((𝐹 ↾ (𝐷(,)𝐶)) ↾ (𝐵(,)𝐶)) = (𝐹 ↾ (𝐵(,)𝐶)))
6	5	eqcomd 2743	. . 3 ⊢ (𝜑 → (𝐹 ↾ (𝐵(,)𝐶)) = ((𝐹 ↾ (𝐷(,)𝐶)) ↾ (𝐵(,)𝐶)))
7	6	oveq1d 7376	. 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 6704	. . . 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 4182	. . 3 ⊢ (𝜑 → (𝐵(,)𝐶) ⊆ (𝐴 ∩ (𝐷(,)𝐶)))
13		inss2 4179	. . . . 5 ⊢ (𝐴 ∩ (𝐷(,)𝐶)) ⊆ (𝐷(,)𝐶)
14		ioosscn 13355	. . . . 5 ⊢ (𝐷(,)𝐶) ⊆ ℂ
15	13, 14	sstri 3932	. . . 4 ⊢ (𝐴 ∩ (𝐷(,)𝐶)) ⊆ ℂ
16	15	a1i 11	. . 3 ⊢ (𝜑 → (𝐴 ∩ (𝐷(,)𝐶)) ⊆ ℂ)
17		eqid 2737	. . 3 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
18		eqid 2737	. . 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 11189	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℝ^*)
22		limcresiooub.bltc	. . . . 5 ⊢ (𝜑 → 𝐵 < 𝐶)
23		ubioc1 13346	. . . . 5 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝐵 < 𝐶) → 𝐶 ∈ (𝐵(,]𝐶))
24	19, 21, 22, 23	syl3anc 1374	. . . 4 ⊢ (𝜑 → 𝐶 ∈ (𝐵(,]𝐶))
25		ioounsn 13424	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝐵 < 𝐶) → ((𝐵(,)𝐶) ∪ {𝐶}) = (𝐵(,]𝐶))
26	19, 21, 22, 25	syl3anc 1374	. . . . . 6 ⊢ (𝜑 → ((𝐵(,)𝐶) ∪ {𝐶}) = (𝐵(,]𝐶))
27	26	fveq2d 6839	. . . . 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 24761	. . . . . . . 8 ⊢ (TopOpen‘ℂ_fld) ∈ Top
29		ovex 7394	. . . . . . . . . 10 ⊢ (𝐷(,)𝐶) ∈ V
30	29	inex2 5256	. . . . . . . . 9 ⊢ (𝐴 ∩ (𝐷(,)𝐶)) ∈ V
31		snex 5377	. . . . . . . . 9 ⊢ {𝐶} ∈ V
32	30, 31	unex 7692	. . . . . . . 8 ⊢ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}) ∈ V
33		resttop 23138	. . . . . . . 8 ⊢ (((TopOpen‘ℂ_fld) ∈ Top ∧ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}) ∈ V) → ((TopOpen‘ℂ_fld) ↾_t ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})) ∈ Top)
34	28, 32, 33	mp2an 693	. . . . . . 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 11193	. . . . . . . . . . . . . 14 ⊢ +∞ ∈ ℝ^*
37	36	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → +∞ ∈ ℝ^*)
38	19	xrleidd 13097	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 ≤ 𝐵)
39	20	ltpnfd 13066	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐶 < +∞)
40		iocssioo 13386	. . . . . . . . . . . . 13 ⊢ (((𝐵 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) ∧ (𝐵 ≤ 𝐵 ∧ 𝐶 < +∞)) → (𝐵(,]𝐶) ⊆ (𝐵(,)+∞))
41	19, 37, 38, 39, 40	syl22anc 839	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐵(,]𝐶) ⊆ (𝐵(,)+∞))
42		simpr 484	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → 𝑥 = 𝐶)
43		snidg 4605	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐶 ∈ ℝ → 𝐶 ∈ {𝐶})
44		elun2 4124	. . . . . . . . . . . . . . . . . . 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 2837	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → 𝑥 ∈ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))
48	47	adantlr 716	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) ∧ 𝑥 = 𝐶) → 𝑥 ∈ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))
49		simpll 767	. . . . . . . . . . . . . . . 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 13374	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ) → (𝐵(,]𝐶) ⊆ ℝ)
55	19, 20, 54	syl2anc 585	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝐵(,]𝐶) ⊆ ℝ)
56	55	sselda 3922	. . . . . . . . . . . . . . . . . 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 45953	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝑥 ∈ (𝐵(,]𝐶)) → 𝐵 < 𝑥)
60	50, 52, 58, 59	syl3anc 1374	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) → 𝐵 < 𝑥)
61	60	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) ∧ ¬ 𝑥 = 𝐶) → 𝐵 < 𝑥)
62	20	ad2antrr 727	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) ∧ ¬ 𝑥 = 𝐶) → 𝐶 ∈ ℝ)
63		iocleub 45954	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝑥 ∈ (𝐵(,]𝐶)) → 𝑥 ≤ 𝐶)
64	50, 52, 58, 63	syl3anc 1374	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) → 𝑥 ≤ 𝐶)
65	64	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) ∧ ¬ 𝑥 = 𝐶) → 𝑥 ≤ 𝐶)
66		neqne 2941	. . . . . . . . . . . . . . . . . . . 20 ⊢ (¬ 𝑥 = 𝐶 → 𝑥 ≠ 𝐶)
67	66	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) ∧ ¬ 𝑥 = 𝐶) → 𝑥 ≠ 𝐶)
68	67	necomd 2988	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) ∧ ¬ 𝑥 = 𝐶) → 𝐶 ≠ 𝑥)
69	57, 62, 65, 68	leneltd 11294	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) ∧ ¬ 𝑥 = 𝐶) → 𝑥 < 𝐶)
70	51, 53, 57, 61, 69	eliood 45949	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) ∧ ¬ 𝑥 = 𝐶) → 𝑥 ∈ (𝐵(,)𝐶))
71	12	sselda 3922	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,)𝐶)) → 𝑥 ∈ (𝐴 ∩ (𝐷(,)𝐶)))
72		elun1 4123	. . . . . . . . . . . . . . . . 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 585	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) ∧ ¬ 𝑥 = 𝐶) → 𝑥 ∈ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))
75	48, 74	pm2.61dan 813	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) → 𝑥 ∈ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))
76	75	ralrimiva 3130	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑥 ∈ (𝐵(,]𝐶)𝑥 ∈ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))
77		dfss3 3911	. . . . . . . . . . . . 13 ⊢ ((𝐵(,]𝐶) ⊆ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}) ↔ ∀𝑥 ∈ (𝐵(,]𝐶)𝑥 ∈ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))
78	76, 77	sylibr 234	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐵(,]𝐶) ⊆ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))
79	41, 78	ssind 4182	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐵(,]𝐶) ⊆ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})))
80	79	sseld 3921	. . . . . . . . . 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 2837	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → 𝑥 ∈ (𝐵(,]𝐶))
83	82	adantlr 716	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))) ∧ 𝑥 = 𝐶) → 𝑥 ∈ (𝐵(,]𝐶))
84		ioossioc 45943	. . . . . . . . . . . . 13 ⊢ (𝐵(,)𝐶) ⊆ (𝐵(,]𝐶)
85	19	ad2antrr 727	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))) ∧ ¬ 𝑥 = 𝐶) → 𝐵 ∈ ℝ^*)
86	21	ad2antrr 727	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))) ∧ ¬ 𝑥 = 𝐶) → 𝐶 ∈ ℝ^*)
87		elinel1 4142	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})) → 𝑥 ∈ (𝐵(,)+∞))
88	87	elioored 46000	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})) → 𝑥 ∈ ℝ)
89	88	ad2antlr 728	. . . . . . . . . . . . . 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 728	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))) ∧ ¬ 𝑥 = 𝐶) → 𝑥 ∈ (𝐵(,)+∞))
92		ioogtlb 45946	. . . . . . . . . . . . . . 15 ⊢ ((𝐵 ∈ ℝ^* ∧ +∞ ∈ ℝ^* ∧ 𝑥 ∈ (𝐵(,)+∞)) → 𝐵 < 𝑥)
93	85, 90, 91, 92	syl3anc 1374	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))) ∧ ¬ 𝑥 = 𝐶) → 𝐵 < 𝑥)
94	1	ad2antrr 727	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))) ∧ ¬ 𝑥 = 𝐶) → 𝐷 ∈ ℝ^*)
95		elinel2 4143	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})) → 𝑥 ∈ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))
96		id 22	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ 𝑥 = 𝐶 → ¬ 𝑥 = 𝐶)
97		velsn 4584	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ {𝐶} ↔ 𝑥 = 𝐶)
98	96, 97	sylnibr 329	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ 𝑥 = 𝐶 → ¬ 𝑥 ∈ {𝐶})
99		elunnel2 4096	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}) ∧ ¬ 𝑥 ∈ {𝐶}) → 𝑥 ∈ (𝐴 ∩ (𝐷(,)𝐶)))
100	95, 98, 99	syl2an 597	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})) ∧ ¬ 𝑥 = 𝐶) → 𝑥 ∈ (𝐴 ∩ (𝐷(,)𝐶)))
101	13, 100	sselid 3920	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})) ∧ ¬ 𝑥 = 𝐶) → 𝑥 ∈ (𝐷(,)𝐶))
102	101	adantll 715	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))) ∧ ¬ 𝑥 = 𝐶) → 𝑥 ∈ (𝐷(,)𝐶))
103		iooltub 45961	. . . . . . . . . . . . . . 15 ⊢ ((𝐷 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝑥 ∈ (𝐷(,)𝐶)) → 𝑥 < 𝐶)
104	94, 86, 102, 103	syl3anc 1374	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))) ∧ ¬ 𝑥 = 𝐶) → 𝑥 < 𝐶)
105	85, 86, 89, 93, 104	eliood 45949	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))) ∧ ¬ 𝑥 = 𝐶) → 𝑥 ∈ (𝐵(,)𝐶))
106	84, 105	sselid 3920	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))) ∧ ¬ 𝑥 = 𝐶) → 𝑥 ∈ (𝐵(,]𝐶))
107	83, 106	pm2.61dan 813	. . . . . . . . . . 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 212	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (𝐵(,]𝐶) ↔ 𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))))
110	109	eqrdv 2735	. . . . . . . 8 ⊢ (𝜑 → (𝐵(,]𝐶) = ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})))
111		retop 24739	. . . . . . . . . 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 24743	. . . . . . . . . 10 ⊢ (𝐵(,)+∞) ∈ (topGen‘ran (,))
115	114	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → (𝐵(,)+∞) ∈ (topGen‘ran (,)))
116		elrestr 17385	. . . . . . . . 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 1374	. . . . . . . 8 ⊢ (𝜑 → ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})) ∈ ((topGen‘ran (,)) ↾_t ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})))
118	110, 117	eqeltrd 2837	. . . . . . 7 ⊢ (𝜑 → (𝐵(,]𝐶) ∈ ((topGen‘ran (,)) ↾_t ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})))
119		tgioo4 24783	. . . . . . . . 9 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
120	119	oveq1i 7371	. . . . . . . 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 13354	. . . . . . . . . . . 12 ⊢ (𝐷(,)𝐶) ⊆ ℝ
123	13, 122	sstri 3932	. . . . . . . . . . 11 ⊢ (𝐴 ∩ (𝐷(,)𝐶)) ⊆ ℝ
124	123	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 ∩ (𝐷(,)𝐶)) ⊆ ℝ)
125	20	snssd 4753	. . . . . . . . . 10 ⊢ (𝜑 → {𝐶} ⊆ ℝ)
126	124, 125	unssd 4133	. . . . . . . . 9 ⊢ (𝜑 → ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}) ⊆ ℝ)
127		reex 11123	. . . . . . . . . 10 ⊢ ℝ ∈ V
128	127	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → ℝ ∈ V)
129		restabs 23143	. . . . . . . . 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 1374	. . . . . . . 8 ⊢ (𝜑 → (((TopOpen‘ℂ_fld) ↾_t ℝ) ↾_t ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})) = ((TopOpen‘ℂ_fld) ↾_t ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})))
131	120, 130	eqtrid 2784	. . . . . . 7 ⊢ (𝜑 → ((topGen‘ran (,)) ↾_t ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})) = ((TopOpen‘ℂ_fld) ↾_t ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})))
132	118, 131	eleqtrd 2839	. . . . . 6 ⊢ (𝜑 → (𝐵(,]𝐶) ∈ ((TopOpen‘ℂ_fld) ↾_t ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})))
133		isopn3i 23060	. . . . . 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 585	. . . . 5 ⊢ (𝜑 → ((int‘((TopOpen‘ℂ_fld) ↾_t ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})))‘(𝐵(,]𝐶)) = (𝐵(,]𝐶))
135	27, 134	eqtr2d 2773	. . . 4 ⊢ (𝜑 → (𝐵(,]𝐶) = ((int‘((TopOpen‘ℂ_fld) ↾_t ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})))‘((𝐵(,)𝐶) ∪ {𝐶})))
136	24, 135	eleqtrd 2839	. . 3 ⊢ (𝜑 → 𝐶 ∈ ((int‘((TopOpen‘ℂ_fld) ↾_t ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})))‘((𝐵(,)𝐶) ∪ {𝐶})))
137	10, 12, 16, 17, 18, 136	limcres 25866	. 2 ⊢ (𝜑 → (((𝐹 ↾ (𝐷(,)𝐶)) ↾ (𝐵(,)𝐶)) lim_ℂ 𝐶) = ((𝐹 ↾ (𝐷(,)𝐶)) lim_ℂ 𝐶))
138	7, 137	eqtrd 2772	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 1542 ∈ wcel 2114 ≠ wne 2933 ∀wral 3052 Vcvv 3430 ∪ cun 3888 ∩ cin 3889 ⊆ wss 3890 {csn 4568 class class class wbr 5086 ran crn 5626 ↾ cres 5627 ⟶wf 6489 ‘cfv 6493 (class class class)co 7361 ℂcc 11030 ℝcr 11031 +∞cpnf 11170 ℝ^*cxr 11172 < clt 11173 ≤ cle 11174 (,)cioo 13292 (,]cioc 13293 ↾_t crest 17377 TopOpenctopn 17378 topGenctg 17394 ℂ_fldccnfld 21347 Topctop 22871 intcnt 22995 lim_ℂ climc 25842

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 2709 ax-rep 5213 ax-sep 5232 ax-nul 5242 ax-pow 5303 ax-pr 5371 ax-un 7683 ax-cnex 11088 ax-resscn 11089 ax-1cn 11090 ax-icn 11091 ax-addcl 11092 ax-addrcl 11093 ax-mulcl 11094 ax-mulrcl 11095 ax-mulcom 11096 ax-addass 11097 ax-mulass 11098 ax-distr 11099 ax-i2m1 11100 ax-1ne0 11101 ax-1rid 11102 ax-rnegex 11103 ax-rrecex 11104 ax-cnre 11105 ax-pre-lttri 11106 ax-pre-lttrn 11107 ax-pre-ltadd 11108 ax-pre-mulgt0 11109 ax-pre-sup 11110

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 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7318 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7812 df-1st 7936 df-2nd 7937 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-er 8637 df-map 8769 df-pm 8770 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-fi 9318 df-sup 9349 df-inf 9350 df-pnf 11175 df-mnf 11176 df-xr 11177 df-ltxr 11178 df-le 11179 df-sub 11373 df-neg 11374 df-div 11802 df-nn 12169 df-2 12238 df-3 12239 df-4 12240 df-5 12241 df-6 12242 df-7 12243 df-8 12244 df-9 12245 df-n0 12432 df-z 12519 df-dec 12639 df-uz 12783 df-q 12893 df-rp 12937 df-xneg 13057 df-xadd 13058 df-xmul 13059 df-ioo 13296 df-ioc 13297 df-icc 13299 df-fz 13456 df-seq 13958 df-exp 14018 df-cj 15055 df-re 15056 df-im 15057 df-sqrt 15191 df-abs 15192 df-struct 17111 df-slot 17146 df-ndx 17158 df-base 17174 df-plusg 17227 df-mulr 17228 df-starv 17229 df-tset 17233 df-ple 17234 df-ds 17236 df-unif 17237 df-rest 17379 df-topn 17380 df-topgen 17400 df-psmet 21339 df-xmet 21340 df-met 21341 df-bl 21342 df-mopn 21343 df-cnfld 21348 df-top 22872 df-topon 22889 df-topsp 22911 df-bases 22924 df-ntr 22998 df-cnp 23206 df-xms 24298 df-ms 24299 df-limc 25846

This theorem is referenced by: fouriersw 46680

W3C validator