limcresiooub - Mathbox for Glauco Siliprandi

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Theorem limcresiooub 46925

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 13551	. . . . . 6 ⊢ ((𝐷 ∈ ℝ^* ∧ 𝐷 ≤ 𝐵) → (𝐵(,)𝐶) ⊆ (𝐷(,)𝐶))
4	1, 2, 3	syl2anc 596	. . . . 5 ⊢ (𝜑 → (𝐵(,)𝐶) ⊆ (𝐷(,)𝐶))
5	4	resabs1d 6016	. . . 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 7466	. 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 6781	. . . 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 4140	. . 3 ⊢ (𝜑 → (𝐵(,)𝐶) ⊆ (𝐴 ∩ (𝐷(,)𝐶)))
13		inss2 4137	. . . . 5 ⊢ (𝐴 ∩ (𝐷(,)𝐶)) ⊆ (𝐷(,)𝐶)
14		ioosscn 13579	. . . . 5 ⊢ (𝐷(,)𝐶) ⊆ ℂ
15	13, 14	sstri 3922	. . . 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 11398	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℝ^*)
22		limcresiooub.bltc	. . . . 5 ⊢ (𝜑 → 𝐵 < 𝐶)
23		ubioc1 13570	. . . . 5 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝐵 < 𝐶) → 𝐶 ∈ (𝐵(,]𝐶))
24	19, 21, 22, 23	syl3anc 1399	. . . 4 ⊢ (𝜑 → 𝐶 ∈ (𝐵(,]𝐶))
25		ioounsn 13648	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝐵 < 𝐶) → ((𝐵(,)𝐶) ∪ {𝐶}) = (𝐵(,]𝐶))
26	19, 21, 22, 25	syl3anc 1399	. . . . . 6 ⊢ (𝜑 → ((𝐵(,)𝐶) ∪ {𝐶}) = (𝐵(,]𝐶))
27	26	fveq2d 6920	. . . . 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 25157	. . . . . . . 8 ⊢ (TopOpen‘ℂ_fld) ∈ Top
29		ovex 7484	. . . . . . . . . 10 ⊢ (𝐷(,)𝐶) ∈ V
30	29	inex2 5285	. . . . . . . . 9 ⊢ (𝐴 ∩ (𝐷(,)𝐶)) ∈ V
31		snex 5404	. . . . . . . . 9 ⊢ {𝐶} ∈ V
32	30, 31	unex 7793	. . . . . . . 8 ⊢ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}) ∈ V
33		resttop 23533	. . . . . . . 8 ⊢ (((TopOpen‘ℂ_fld) ∈ Top ∧ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}) ∈ V) → ((TopOpen‘ℂ_fld) ↾_t ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})) ∈ Top)
34	28, 32, 33	mp2an 705	. . . . . . 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 11402	. . . . . . . . . . . . . 14 ⊢ +∞ ∈ ℝ^*
37	36	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → +∞ ∈ ℝ^*)
38	19	xrleidd 13321	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 ≤ 𝐵)
39	20	ltpnfd 13290	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐶 < +∞)
40		iocssioo 13610	. . . . . . . . . . . . 13 ⊢ (((𝐵 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) ∧ (𝐵 ≤ 𝐵 ∧ 𝐶 < +∞)) → (𝐵(,]𝐶) ⊆ (𝐵(,)+∞))
41	19, 37, 38, 39, 40	syl22anc 852	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐵(,]𝐶) ⊆ (𝐵(,)+∞))
42		simpr 490	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → 𝑥 = 𝐶)
43		snidg 4622	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐶 ∈ ℝ → 𝐶 ∈ {𝐶})
44		elun2 4184	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐶 ∈ {𝐶} → 𝐶 ∈ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))
45	20, 43, 44	3syl 19	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐶 ∈ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))
46	45	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → 𝐶 ∈ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))
47	42, 46	eqeltrd 2864	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → 𝑥 ∈ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))
48	47	adantlr 728	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) ∧ 𝑥 = 𝐶) → 𝑥 ∈ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))
49		simpll 779	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) ∧ ¬ 𝑥 = 𝐶) → 𝜑)
50	19	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) → 𝐵 ∈ ℝ^*)
51	50	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) ∧ ¬ 𝑥 = 𝐶) → 𝐵 ∈ ℝ^*)
52	21	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) → 𝐶 ∈ ℝ^*)
53	52	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) ∧ ¬ 𝑥 = 𝐶) → 𝐶 ∈ ℝ^*)
54		iocssre 13598	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ) → (𝐵(,]𝐶) ⊆ ℝ)
55	19, 20, 54	syl2anc 596	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝐵(,]𝐶) ⊆ ℝ)
56	55	sselda 3913	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) → 𝑥 ∈ ℝ)
57	56	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) ∧ ¬ 𝑥 = 𝐶) → 𝑥 ∈ ℝ)
58		simpr 490	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) → 𝑥 ∈ (𝐵(,]𝐶))
59		iocgtlb 46787	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝑥 ∈ (𝐵(,]𝐶)) → 𝐵 < 𝑥)
60	50, 52, 58, 59	syl3anc 1399	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) → 𝐵 < 𝑥)
61	60	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) ∧ ¬ 𝑥 = 𝐶) → 𝐵 < 𝑥)
62	20	ad2antrr 739	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) ∧ ¬ 𝑥 = 𝐶) → 𝐶 ∈ ℝ)
63		iocleub 46788	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝑥 ∈ (𝐵(,]𝐶)) → 𝑥 ≤ 𝐶)
64	50, 52, 58, 63	syl3anc 1399	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) → 𝑥 ≤ 𝐶)
65	64	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) ∧ ¬ 𝑥 = 𝐶) → 𝑥 ≤ 𝐶)
66		neqne 2967	. . . . . . . . . . . . . . . . . . . 20 ⊢ (¬ 𝑥 = 𝐶 → 𝑥 ≠ 𝐶)
67	66	adantl 487	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) ∧ ¬ 𝑥 = 𝐶) → 𝑥 ≠ 𝐶)
68	67	necomd 3014	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) ∧ ¬ 𝑥 = 𝐶) → 𝐶 ≠ 𝑥)
69	57, 62, 65, 68	leneltd 11503	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) ∧ ¬ 𝑥 = 𝐶) → 𝑥 < 𝐶)
70	51, 53, 57, 61, 69	eliood 46783	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) ∧ ¬ 𝑥 = 𝐶) → 𝑥 ∈ (𝐵(,)𝐶))
71	12	sselda 3913	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,)𝐶)) → 𝑥 ∈ (𝐴 ∩ (𝐷(,)𝐶)))
72		elun1 4183	. . . . . . . . . . . . . . . . 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 596	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) ∧ ¬ 𝑥 = 𝐶) → 𝑥 ∈ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))
75	48, 74	pm2.61dan 825	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,]𝐶)) → 𝑥 ∈ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))
76	75	ralrimiva 3158	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑥 ∈ (𝐵(,]𝐶)𝑥 ∈ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))
77		dfss3 3902	. . . . . . . . . . . . 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 4140	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐵(,]𝐶) ⊆ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})))
80	79	sseld 3912	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ (𝐵(,]𝐶) → 𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))))
81	24	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → 𝐶 ∈ (𝐵(,]𝐶))
82	42, 81	eqeltrd 2864	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → 𝑥 ∈ (𝐵(,]𝐶))
83	82	adantlr 728	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))) ∧ 𝑥 = 𝐶) → 𝑥 ∈ (𝐵(,]𝐶))
84		ioossioc 46777	. . . . . . . . . . . . 13 ⊢ (𝐵(,)𝐶) ⊆ (𝐵(,]𝐶)
85	19	ad2antrr 739	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))) ∧ ¬ 𝑥 = 𝐶) → 𝐵 ∈ ℝ^*)
86	21	ad2antrr 739	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))) ∧ ¬ 𝑥 = 𝐶) → 𝐶 ∈ ℝ^*)
87		elinel1 4101	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})) → 𝑥 ∈ (𝐵(,)+∞))
88	87	elioored 46834	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})) → 𝑥 ∈ ℝ)
89	88	ad2antlr 740	. . . . . . . . . . . . . 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 740	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))) ∧ ¬ 𝑥 = 𝐶) → 𝑥 ∈ (𝐵(,)+∞))
92		ioogtlb 46780	. . . . . . . . . . . . . . 15 ⊢ ((𝐵 ∈ ℝ^* ∧ +∞ ∈ ℝ^* ∧ 𝑥 ∈ (𝐵(,)+∞)) → 𝐵 < 𝑥)
93	85, 90, 91, 92	syl3anc 1399	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))) ∧ ¬ 𝑥 = 𝐶) → 𝐵 < 𝑥)
94	1	ad2antrr 739	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))) ∧ ¬ 𝑥 = 𝐶) → 𝐷 ∈ ℝ^*)
95		elinel2 4102	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})) → 𝑥 ∈ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))
96		id 23	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ 𝑥 = 𝐶 → ¬ 𝑥 = 𝐶)
97		velsn 4600	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ {𝐶} ↔ 𝑥 = 𝐶)
98	96, 97	sylnibr 332	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ 𝑥 = 𝐶 → ¬ 𝑥 ∈ {𝐶})
99		elunnel2 4157	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}) ∧ ¬ 𝑥 ∈ {𝐶}) → 𝑥 ∈ (𝐴 ∩ (𝐷(,)𝐶)))
100	95, 98, 99	syl2an 608	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})) ∧ ¬ 𝑥 = 𝐶) → 𝑥 ∈ (𝐴 ∩ (𝐷(,)𝐶)))
101	13, 100	sselid 3911	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})) ∧ ¬ 𝑥 = 𝐶) → 𝑥 ∈ (𝐷(,)𝐶))
102	101	adantll 727	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))) ∧ ¬ 𝑥 = 𝐶) → 𝑥 ∈ (𝐷(,)𝐶))
103		iooltub 46795	. . . . . . . . . . . . . . 15 ⊢ ((𝐷 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝑥 ∈ (𝐷(,)𝐶)) → 𝑥 < 𝐶)
104	94, 86, 102, 103	syl3anc 1399	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))) ∧ ¬ 𝑥 = 𝐶) → 𝑥 < 𝐶)
105	85, 86, 89, 93, 104	eliood 46783	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))) ∧ ¬ 𝑥 = 𝐶) → 𝑥 ∈ (𝐵(,)𝐶))
106	84, 105	sselid 3911	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))) ∧ ¬ 𝑥 = 𝐶) → 𝑥 ∈ (𝐵(,]𝐶))
107	83, 106	pm2.61dan 825	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}))) → 𝑥 ∈ (𝐵(,]𝐶))
108	107	ex 418	. . . . . . . . . 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 25135	. . . . . . . . . 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 25139	. . . . . . . . . 10 ⊢ (𝐵(,)+∞) ∈ (topGen‘ran (,))
115	114	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → (𝐵(,)+∞) ∈ (topGen‘ran (,)))
116		elrestr 17649	. . . . . . . . 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 1399	. . . . . . . 8 ⊢ (𝜑 → ((𝐵(,)+∞) ∩ ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})) ∈ ((topGen‘ran (,)) ↾_t ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})))
118	110, 117	eqeltrd 2864	. . . . . . 7 ⊢ (𝜑 → (𝐵(,]𝐶) ∈ ((topGen‘ran (,)) ↾_t ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})))
119		tgioo4 25179	. . . . . . . . 9 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
120	119	oveq1i 7461	. . . . . . . 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 13578	. . . . . . . . . . . 12 ⊢ (𝐷(,)𝐶) ⊆ ℝ
123	13, 122	sstri 3922	. . . . . . . . . . 11 ⊢ (𝐴 ∩ (𝐷(,)𝐶)) ⊆ ℝ
124	123	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 ∩ (𝐷(,)𝐶)) ⊆ ℝ)
125	20	snssd 4749	. . . . . . . . . 10 ⊢ (𝜑 → {𝐶} ⊆ ℝ)
126	124, 125	unssd 4193	. . . . . . . . 9 ⊢ (𝜑 → ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶}) ⊆ ℝ)
127		reex 11330	. . . . . . . . . 10 ⊢ ℝ ∈ V
128	127	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → ℝ ∈ V)
129		restabs 23538	. . . . . . . . 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 1399	. . . . . . . 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 2866	. . . . . 6 ⊢ (𝜑 → (𝐵(,]𝐶) ∈ ((TopOpen‘ℂ_fld) ↾_t ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})))
133		isopn3i 23455	. . . . . 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 596	. . . . 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 2866	. . 3 ⊢ (𝜑 → 𝐶 ∈ ((int‘((TopOpen‘ℂ_fld) ↾_t ((𝐴 ∩ (𝐷(,)𝐶)) ∪ {𝐶})))‘((𝐵(,)𝐶) ∪ {𝐶})))
137	10, 12, 16, 17, 18, 136	limcres 26261	. 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

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2147 ≠ wne 2959 ∀wral 3080 Vcvv 3454 ⊆ wss 3900 ∩ cin 4092 ∪ cun 4152 {csn 4588 class class class wbr 5108 ran crn 5659 ↾ cres 5660 ⟶wf 6562 ‘cfv 6566 (class class class)co 7451 ℂcc 11237 ℝcr 11238 +∞cpnf 11379 ℝ^*cxr 11381 < clt 11382 ≤ cle 11383 (,)cioo 13516 (,]cioc 13517 ↾_t crest 17641 TopOpenctopn 17642 topGenctg 17658 ℂ_fldccnfld 21729 Topctop 23266 intcnt 23390 lim_ℂ climc 26237

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5238 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783 ax-cnex 11295 ax-resscn 11296 ax-1cn 11297 ax-icn 11298 ax-addcl 11299 ax-addrcl 11300 ax-mulcl 11301 ax-mulrcl 11302 ax-mulcom 11303 ax-addass 11304 ax-mulass 11305 ax-distr 11306 ax-i2m1 11307 ax-1ne0 11308 ax-1rid 11309 ax-rnegex 11310 ax-rrecex 11311 ax-cnre 11312 ax-pre-lttri 11313 ax-pre-lttrn 11314 ax-pre-ltadd 11315 ax-pre-mulgt0 11316 ax-pre-sup 11317

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4841 df-uni 4873 df-int 4913 df-iun 4958 df-br 5109 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-we 5613 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-pred 6330 df-ord 6391 df-on 6392 df-lim 6393 df-suc 6394 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-riota 7408 df-ov 7454 df-oprab 7455 df-mpo 7456 df-om 7912 df-1st 8035 df-2nd 8036 df-frecs 8337 df-wrecs 8368 df-recs 8417 df-rdg 8456 df-1o 8514 df-er 8755 df-map 8887 df-pm 8888 df-en 9012 df-dom 9013 df-sdom 9014 df-fin 9015 df-fi 9440 df-sup 9471 df-inf 9472 df-pnf 11384 df-mnf 11385 df-xr 11386 df-ltxr 11387 df-le 11388 df-sub 11582 df-neg 11583 df-div 12014 df-nn 12376 df-2 12445 df-3 12446 df-4 12447 df-5 12448 df-6 12449 df-7 12450 df-8 12451 df-9 12452 df-n0 12647 df-z 12734 df-dec 12855 df-uz 13006 df-q 13116 df-rp 13161 df-xneg 13281 df-xadd 13282 df-xmul 13283 df-ioo 13520 df-ioc 13521 df-icc 13523 df-fz 13680 df-seq 14185 df-exp 14245 df-cj 15309 df-re 15310 df-im 15311 df-sqrt 15445 df-abs 15446 df-struct 17369 df-slot 17404 df-ndx 17416 df-base 17432 df-plusg 17486 df-mulr 17487 df-starv 17488 df-tset 17492 df-ple 17493 df-ds 17495 df-unif 17496 df-euf 17499 df-rest 17643 df-topn 17644 df-topgen 17664 df-psmet 21721 df-xmet 21722 df-met 21723 df-bl 21724 df-mopn 21725 df-cnfld 21730 df-top 23267 df-topon 23284 df-topsp 23306 df-bases 23319 df-ntr 23393 df-cnp 23601 df-xms 24694 df-ms 24695 df-limc 26241

This theorem is used by: fouriersw 47514

W3C validator