limcresiooub - Mathbox for Glauco Siliprandi

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Theorem limcresiooub 46356

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 13402	. . . . . 6 ⊢ ((𝐷 ∈ ℝ^* ∧ 𝐷 ≤ 𝐵) → (𝐵(,)𝐶) ⊆ (𝐷(,)𝐶))
4	1, 2, 3	syl2anc 595	. . . . 5 ⊢ (𝜑 → (𝐵(,)𝐶) ⊆ (𝐷(,)𝐶))
5	4	resabs1d 6007	. . . 4 ⊢ (𝜑 → ((𝐹 ↾ (𝐷(,)𝐶)) ↾ (𝐵(,)𝐶)) = (𝐹 ↾ (𝐵(,)𝐶)))
6	5	eqcomd 2769	. . 3 ⊢ (𝜑 → (𝐹 ↾ (𝐵(,)𝐶)) = ((𝐹 ↾ (𝐷(,)𝐶)) ↾ (𝐵(,)𝐶)))
7	6	oveq1d 7425	. 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 6747	. . . 4 ⊢ (𝐹:𝐴⟶ℂ → (𝐹 ↾ (𝐷(,)𝐶)):(𝐴 ∩ (𝐷(,)𝐶))⟶ℂ)
10	8, 9	syl 18	. . 3 ⊢ (𝜑 → (𝐹 ↾ (𝐷(,)𝐶)):(𝐴 ∩ (𝐷(,)𝐶))⟶ℂ)
11		limcresiooub.bcss	. . . 4 ⊢ (𝜑 → (𝐵(,)𝐶) ⊆ 𝐴)
12	11, 4	ssind 4193	. . 3 ⊢ (𝜑 → (𝐵(,)𝐶) ⊆ (𝐴 ∩ (𝐷(,)𝐶)))
13		inss2 4190	. . . . 5 ⊢ (𝐴 ∩ (𝐷(,)𝐶)) ⊆ (𝐷(,)𝐶)
14		ioosscn 13430	. . . . 5 ⊢ (𝐷(,)𝐶) ⊆ ℂ
15	13, 14	sstri 3946	. . . 4 ⊢ (𝐴 ∩ (𝐷(,)𝐶)) ⊆ ℂ
16	15	a1i 11	. . 3 ⊢ (𝜑 → (𝐴 ∩ (𝐷(,)𝐶)) ⊆ ℂ)
17		eqid 2763	. . 3 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
18		eqid 2763	. . 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 11254	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℝ^*)
22		limcresiooub.bltc	. . . . 5 ⊢ (𝜑 → 𝐵 < 𝐶)
23		ubioc1 13421	. . . . 5 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝐵 < 𝐶) → 𝐶 ∈ (𝐵(,]𝐶))
24	19, 21, 22, 23	syl3anc 1398	. . . 4 ⊢ (𝜑 → 𝐶 ∈ (𝐵(,]𝐶))
25		ioounsn 13499	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝐵 < 𝐶) → ((𝐵(,)𝐶) ∪ {𝐶}) = (𝐵(,]𝐶))
26	19, 21, 22, 25	syl3anc 1398	. . . . . 6 ⊢ (𝜑 → ((𝐵(,)𝐶) ∪ {𝐶}) = (𝐵(,]𝐶))
27	26	fveq2d 6885	. . . . 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 24940	. . . . . . . 8 ⊢ (TopOpen‘ℂ_fld) ∈ Top
29		ovex 7443	. . . . . . . . . 10 ⊢ (𝐷(,)𝐶) ∈ V
30	29	inex2 5287	. . . . . . . . 9 ⊢ (𝐴 ∩ (𝐷(,)𝐶)) ∈ V
31		snex 5410	. . . . . . . . 9 ⊢ {𝐶} ∈ V
32	30, 31	unex 7742	. . . . . . . 8 ⊢ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}) ∈ V
33		resttop 23317	. . . . . . . 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 11258	. . . . . . . . . . . . . 14 ⊢ +∞ ∈ ℝ^*
37	36	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → +∞ ∈ ℝ^*)
38	19	xrleidd 13172	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 ≤ 𝐵)
39	20	ltpnfd 13141	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐶 < +∞)
40		iocssioo 13461	. . . . . . . . . . . . 13 ⊢ (((𝐵 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) ∧ (𝐵 ≤ 𝐵 ∧ 𝐶 < +∞)) → (𝐵(,]𝐶) ⊆ (𝐵(,)+∞))
41	19, 37, 38, 39, 40	syl22anc 851	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐵(,]𝐶) ⊆ (𝐵(,)+∞))
42		simpr 489	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → 𝑥 = 𝐶)
43		snidg 4626	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐶 ∈ ℝ → 𝐶 ∈ {𝐶})
44		elun2 4136	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐶 ∈ {𝐶} → 𝐶 ∈ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))
45	20, 43, 44	3syl 19	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐶 ∈ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))
46	45	adantr 485	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → 𝐶 ∈ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))
47	42, 46	eqeltrd 2863	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → 𝑥 ∈ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))
48	47	adantlr 727	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) ∧ 𝑥 = 𝐶) → 𝑥 ∈ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))
49		simpll 778	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) ∧ ¬ 𝑥 = 𝐶) → 𝜑)
50	19	adantr 485	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) → 𝐵 ∈ ℝ^*)
51	50	adantr 485	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) ∧ ¬ 𝑥 = 𝐶) → 𝐵 ∈ ℝ^*)
52	21	adantr 485	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) → 𝐶 ∈ ℝ^*)
53	52	adantr 485	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) ∧ ¬ 𝑥 = 𝐶) → 𝐶 ∈ ℝ^*)
54		iocssre 13449	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ) → (𝐵(,]𝐶) ⊆ ℝ)
55	19, 20, 54	syl2anc 595	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝐵(,]𝐶) ⊆ ℝ)
56	55	sselda 3937	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) → 𝑥 ∈ ℝ)
57	56	adantr 485	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) ∧ ¬ 𝑥 = 𝐶) → 𝑥 ∈ ℝ)
58		simpr 489	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) → 𝑥 ∈ (𝐵(,]𝐶))
59		iocgtlb 46218	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝑥 ∈ (𝐵(,]𝐶)) → 𝐵 < 𝑥)
60	50, 52, 58, 59	syl3anc 1398	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) → 𝐵 < 𝑥)
61	60	adantr 485	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) ∧ ¬ 𝑥 = 𝐶) → 𝐵 < 𝑥)
62	20	ad2antrr 738	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) ∧ ¬ 𝑥 = 𝐶) → 𝐶 ∈ ℝ)
63		iocleub 46219	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝑥 ∈ (𝐵(,]𝐶)) → 𝑥 ≤ 𝐶)
64	50, 52, 58, 63	syl3anc 1398	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) → 𝑥 ≤ 𝐶)
65	64	adantr 485	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) ∧ ¬ 𝑥 = 𝐶) → 𝑥 ≤ 𝐶)
66		neqne 2966	. . . . . . . . . . . . . . . . . . . 20 ⊢ (¬ 𝑥 = 𝐶 → 𝑥 ≠ 𝐶)
67	66	adantl 486	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) ∧ ¬ 𝑥 = 𝐶) → 𝑥 ≠ 𝐶)
68	67	necomd 3013	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) ∧ ¬ 𝑥 = 𝐶) → 𝐶 ≠ 𝑥)
69	57, 62, 65, 68	leneltd 11359	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) ∧ ¬ 𝑥 = 𝐶) → 𝑥 < 𝐶)
70	51, 53, 57, 61, 69	eliood 46214	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) ∧ ¬ 𝑥 = 𝐶) → 𝑥 ∈ (𝐵(,)𝐶))
71	12	sselda 3937	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,)𝐶)) → 𝑥 ∈ (𝐴 ∩ (𝐷(,)𝐶)))
72		elun1 4135	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (𝐴 ∩ (𝐷(,)𝐶)) → 𝑥 ∈ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))
73	71, 72	syl 18	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,)𝐶)) → 𝑥 ∈ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))
74	49, 70, 73	syl2anc 595	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) ∧ ¬ 𝑥 = 𝐶) → 𝑥 ∈ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))
75	48, 74	pm2.61dan 824	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) → 𝑥 ∈ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))
76	75	ralrimiva 3157	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑥 ∈ (𝐵(,]𝐶)𝑥 ∈ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))
77		dfss3 3926	. . . . . . . . . . . . 13 ⊢ ((𝐵(,]𝐶) ⊆ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}) ↔ ∀𝑥 ∈ (𝐵(,]𝐶)𝑥 ∈ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))
78	76, 77	sylibr 237	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐵(,]𝐶) ⊆ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))
79	41, 78	ssind 4193	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐵(,]𝐶) ⊆ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})))
80	79	sseld 3936	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ (𝐵(,]𝐶) → 𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))))
81	24	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → 𝐶 ∈ (𝐵(,]𝐶))
82	42, 81	eqeltrd 2863	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → 𝑥 ∈ (𝐵(,]𝐶))
83	82	adantlr 727	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))) ∧ 𝑥 = 𝐶) → 𝑥 ∈ (𝐵(,]𝐶))
84		ioossioc 46208	. . . . . . . . . . . . 13 ⊢ (𝐵(,)𝐶) ⊆ (𝐵(,]𝐶)
85	19	ad2antrr 738	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))) ∧ ¬ 𝑥 = 𝐶) → 𝐵 ∈ ℝ^*)
86	21	ad2antrr 738	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))) ∧ ¬ 𝑥 = 𝐶) → 𝐶 ∈ ℝ^*)
87		elinel1 4154	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})) → 𝑥 ∈ (𝐵(,)+∞))
88	87	elioored 46265	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})) → 𝑥 ∈ ℝ)
89	88	ad2antlr 739	. . . . . . . . . . . . . 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 739	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))) ∧ ¬ 𝑥 = 𝐶) → 𝑥 ∈ (𝐵(,)+∞))
92		ioogtlb 46211	. . . . . . . . . . . . . . 15 ⊢ ((𝐵 ∈ ℝ^* ∧ +∞ ∈ ℝ^* ∧ 𝑥 ∈ (𝐵(,)+∞)) → 𝐵 < 𝑥)
93	85, 90, 91, 92	syl3anc 1398	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))) ∧ ¬ 𝑥 = 𝐶) → 𝐵 < 𝑥)
94	1	ad2antrr 738	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))) ∧ ¬ 𝑥 = 𝐶) → 𝐷 ∈ ℝ^*)
95		elinel2 4155	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})) → 𝑥 ∈ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))
96		id 23	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ 𝑥 = 𝐶 → ¬ 𝑥 = 𝐶)
97		velsn 4605	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ {𝐶} ↔ 𝑥 = 𝐶)
98	96, 97	sylnibr 332	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ 𝑥 = 𝐶 → ¬ 𝑥 ∈ {𝐶})
99		elunnel2 4109	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}) ∧ ¬ 𝑥 ∈ {𝐶}) → 𝑥 ∈ (𝐴 ∩ (𝐷(,)𝐶)))
100	95, 98, 99	syl2an 607	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})) ∧ ¬ 𝑥 = 𝐶) → 𝑥 ∈ (𝐴 ∩ (𝐷(,)𝐶)))
101	13, 100	sselid 3935	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})) ∧ ¬ 𝑥 = 𝐶) → 𝑥 ∈ (𝐷(,)𝐶))
102	101	adantll 726	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))) ∧ ¬ 𝑥 = 𝐶) → 𝑥 ∈ (𝐷(,)𝐶))
103		iooltub 46226	. . . . . . . . . . . . . . 15 ⊢ ((𝐷 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝑥 ∈ (𝐷(,)𝐶)) → 𝑥 < 𝐶)
104	94, 86, 102, 103	syl3anc 1398	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))) ∧ ¬ 𝑥 = 𝐶) → 𝑥 < 𝐶)
105	85, 86, 89, 93, 104	eliood 46214	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))) ∧ ¬ 𝑥 = 𝐶) → 𝑥 ∈ (𝐵(,)𝐶))
106	84, 105	sselid 3935	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))) ∧ ¬ 𝑥 = 𝐶) → 𝑥 ∈ (𝐵(,]𝐶))
107	83, 106	pm2.61dan 824	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))) → 𝑥 ∈ (𝐵(,]𝐶))
108	107	ex 417	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})) → 𝑥 ∈ (𝐵(,]𝐶)))
109	80, 108	impbid 215	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (𝐵(,]𝐶) ↔ 𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))))
110	109	eqrdv 2761	. . . . . . . 8 ⊢ (𝜑 → (𝐵(,]𝐶) = ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})))
111		retop 24918	. . . . . . . . . 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 24922	. . . . . . . . . 10 ⊢ (𝐵(,)+∞) ∈ (topGen‘ran (,))
115	114	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → (𝐵(,)+∞) ∈ (topGen‘ran (,)))
116		elrestr 17476	. . . . . . . . 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 1398	. . . . . . . 8 ⊢ (𝜑 → ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})) ∈ ((topGen‘ran (,)) ↾_t ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})))
118	110, 117	eqeltrd 2863	. . . . . . 7 ⊢ (𝜑 → (𝐵(,]𝐶) ∈ ((topGen‘ran (,)) ↾_t ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})))
119		tgioo4 24962	. . . . . . . . 9 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
120	119	oveq1i 7420	. . . . . . . 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 13429	. . . . . . . . . . . 12 ⊢ (𝐷(,)𝐶) ⊆ ℝ
123	13, 122	sstri 3946	. . . . . . . . . . 11 ⊢ (𝐴 ∩ (𝐷(,)𝐶)) ⊆ ℝ
124	123	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 ∩ (𝐷(,)𝐶)) ⊆ ℝ)
125	20	snssd 4752	. . . . . . . . . 10 ⊢ (𝜑 → {𝐶} ⊆ ℝ)
126	124, 125	unssd 4145	. . . . . . . . 9 ⊢ (𝜑 → ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}) ⊆ ℝ)
127		reex 11186	. . . . . . . . . 10 ⊢ ℝ ∈ V
128	127	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → ℝ ∈ V)
129		restabs 23322	. . . . . . . . 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 1398	. . . . . . . 8 ⊢ (𝜑 → (((TopOpen‘ℂ_fld) ↾_t ℝ) ↾_t ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})) = ((TopOpen‘ℂ_fld) ↾_t ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})))
131	120, 130	eqtrid 2810	. . . . . . 7 ⊢ (𝜑 → ((topGen‘ran (,)) ↾_t ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})) = ((TopOpen‘ℂ_fld) ↾_t ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})))
132	118, 131	eleqtrd 2865	. . . . . 6 ⊢ (𝜑 → (𝐵(,]𝐶) ∈ ((TopOpen‘ℂ_fld) ↾_t ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})))
133		isopn3i 23239	. . . . . 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 595	. . . . 5 ⊢ (𝜑 → ((int‘((TopOpen‘ℂ_fld) ↾_t ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})))‘(𝐵(,]𝐶)) = (𝐵(,]𝐶))
135	27, 134	eqtr2d 2799	. . . 4 ⊢ (𝜑 → (𝐵(,]𝐶) = ((int‘((TopOpen‘ℂ_fld) ↾_t ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})))‘((𝐵(,)𝐶) ∪ {𝐶})))
136	24, 135	eleqtrd 2865	. . 3 ⊢ (𝜑 → 𝐶 ∈ ((int‘((TopOpen‘ℂ_fld) ↾_t ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})))‘((𝐵(,)𝐶) ∪ {𝐶})))
137	10, 12, 16, 17, 18, 136	limcres 26045	. 2 ⊢ (𝜑 → (((𝐹 ↾ (𝐷(,)𝐶)) ↾ (𝐵(,)𝐶)) lim_ℂ 𝐶) = ((𝐹 ↾ (𝐷(,)𝐶)) lim_ℂ 𝐶))
138	7, 137	eqtrd 2798	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 400 = wceq 1570 ∈ wcel 2143 ≠ wne 2958 ∀wral 3079 Vcvv 3455 ∪ cun 3903 ∩ cin 3904 ⊆ wss 3905 {csn 4589 class class class wbr 5109 ran crn 5662 ↾ cres 5663 ⟶wf 6532 ‘cfv 6536 (class class class)co 7410 ℂcc 11093 ℝcr 11094 +∞cpnf 11235 ℝ^*cxr 11237 < clt 11238 ≤ cle 11239 (,)cioo 13367 (,]cioc 13368 ↾_t crest 17468 TopOpenctopn 17469 topGenctg 17485 ℂ_fldccnfld 21522 Topctop 23050 intcnt 23174 lim_ℂ climc 26021

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5238 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-cnex 11151 ax-resscn 11152 ax-1cn 11153 ax-icn 11154 ax-addcl 11155 ax-addrcl 11156 ax-mulcl 11157 ax-mulrcl 11158 ax-mulcom 11159 ax-addass 11160 ax-mulass 11161 ax-distr 11162 ax-i2m1 11163 ax-1ne0 11164 ax-1rid 11165 ax-rnegex 11166 ax-rrecex 11167 ax-cnre 11168 ax-pre-lttri 11169 ax-pre-lttrn 11170 ax-pre-ltadd 11171 ax-pre-mulgt0 11172 ax-pre-sup 11173

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4873 df-int 4913 df-iun 4958 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7859 df-1st 7982 df-2nd 7983 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-er 8690 df-map 8822 df-pm 8823 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fi 9367 df-sup 9398 df-inf 9399 df-pnf 11240 df-mnf 11241 df-xr 11242 df-ltxr 11243 df-le 11244 df-sub 11438 df-neg 11439 df-div 11867 df-nn 12229 df-2 12298 df-3 12299 df-4 12300 df-5 12301 df-6 12302 df-7 12303 df-8 12304 df-9 12305 df-n0 12500 df-z 12587 df-dec 12707 df-uz 12858 df-q 12968 df-rp 13012 df-xneg 13132 df-xadd 13133 df-xmul 13134 df-ioo 13371 df-ioc 13372 df-icc 13374 df-fz 13531 df-seq 14034 df-exp 14094 df-cj 15146 df-re 15147 df-im 15148 df-sqrt 15282 df-abs 15283 df-struct 17202 df-slot 17237 df-ndx 17249 df-base 17265 df-plusg 17318 df-mulr 17319 df-starv 17320 df-tset 17324 df-ple 17325 df-ds 17327 df-unif 17328 df-rest 17470 df-topn 17471 df-topgen 17491 df-psmet 21514 df-xmet 21515 df-met 21516 df-bl 21517 df-mopn 21518 df-cnfld 21523 df-top 23051 df-topon 23068 df-topsp 23090 df-bases 23103 df-ntr 23177 df-cnp 23385 df-xms 24477 df-ms 24478 df-limc 26025

This theorem is referenced by: fouriersw 46945

W3C validator