limcresioolb - Mathbox for Glauco Siliprandi

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Theorem limcresioolb 46089

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

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

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

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

Step	Hyp	Ref	Expression
1		limcresioolb.d	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ ℝ^*)
2		limcresioolb.cled	. . . . . 6 ⊢ (𝜑 → 𝐶 ≤ 𝐷)
3		iooss2 13325	. . . . . 6 ⊢ ((𝐷 ∈ ℝ^* ∧ 𝐶 ≤ 𝐷) → (𝐵(,)𝐶) ⊆ (𝐵(,)𝐷))
4	1, 2, 3	syl2anc 585	. . . . 5 ⊢ (𝜑 → (𝐵(,)𝐶) ⊆ (𝐵(,)𝐷))
5	4	resabs1d 5967	. . . 4 ⊢ (𝜑 → ((𝐹 ↾ (𝐵(,)𝐷)) ↾ (𝐵(,)𝐶)) = (𝐹 ↾ (𝐵(,)𝐶)))
6	5	eqcomd 2743	. . 3 ⊢ (𝜑 → (𝐹 ↾ (𝐵(,)𝐶)) = ((𝐹 ↾ (𝐵(,)𝐷)) ↾ (𝐵(,)𝐶)))
7	6	oveq1d 7375	. 2 ⊢ (𝜑 → ((𝐹 ↾ (𝐵(,)𝐶)) lim_ℂ 𝐵) = (((𝐹 ↾ (𝐵(,)𝐷)) ↾ (𝐵(,)𝐶)) lim_ℂ 𝐵))
8		limcresioolb.f	. . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶ℂ)
9		fresin 6703	. . . 4 ⊢ (𝐹:𝐴⟶ℂ → (𝐹 ↾ (𝐵(,)𝐷)):(𝐴 ∩ (𝐵(,)𝐷))⟶ℂ)
10	8, 9	syl 17	. . 3 ⊢ (𝜑 → (𝐹 ↾ (𝐵(,)𝐷)):(𝐴 ∩ (𝐵(,)𝐷))⟶ℂ)
11		limcresioolb.bcss	. . . 4 ⊢ (𝜑 → (𝐵(,)𝐶) ⊆ 𝐴)
12	11, 4	ssind 4182	. . 3 ⊢ (𝜑 → (𝐵(,)𝐶) ⊆ (𝐴 ∩ (𝐵(,)𝐷)))
13		inss2 4179	. . . . 5 ⊢ (𝐴 ∩ (𝐵(,)𝐷)) ⊆ (𝐵(,)𝐷)
14		ioosscn 13352	. . . . 5 ⊢ (𝐵(,)𝐷) ⊆ ℂ
15	13, 14	sstri 3932	. . . 4 ⊢ (𝐴 ∩ (𝐵(,)𝐷)) ⊆ ℂ
16	15	a1i 11	. . 3 ⊢ (𝜑 → (𝐴 ∩ (𝐵(,)𝐷)) ⊆ ℂ)
17		eqid 2737	. . 3 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
18		eqid 2737	. . 3 ⊢ ((TopOpen‘ℂ_fld) ↾_t ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵})) = ((TopOpen‘ℂ_fld) ↾_t ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵}))
19		limcresioolb.b	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℝ)
20	19	rexrd 11186	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
21		limcresioolb.c	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℝ^*)
22		limcresioolb.bltc	. . . . 5 ⊢ (𝜑 → 𝐵 < 𝐶)
23		lbico1 13344	. . . . 5 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝐵 < 𝐶) → 𝐵 ∈ (𝐵[,)𝐶))
24	20, 21, 22, 23	syl3anc 1374	. . . 4 ⊢ (𝜑 → 𝐵 ∈ (𝐵[,)𝐶))
25		snunioo1 45960	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝐵 < 𝐶) → ((𝐵(,)𝐶) ∪ {𝐵}) = (𝐵[,)𝐶))
26	20, 21, 22, 25	syl3anc 1374	. . . . . 6 ⊢ (𝜑 → ((𝐵(,)𝐶) ∪ {𝐵}) = (𝐵[,)𝐶))
27	26	fveq2d 6838	. . . . 5 ⊢ (𝜑 → ((int‘((TopOpen‘ℂ_fld) ↾_t ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵})))‘((𝐵(,)𝐶) ∪ {𝐵})) = ((int‘((TopOpen‘ℂ_fld) ↾_t ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵})))‘(𝐵[,)𝐶)))
28	17	cnfldtop 24758	. . . . . . . 8 ⊢ (TopOpen‘ℂ_fld) ∈ Top
29		ovex 7393	. . . . . . . . . 10 ⊢ (𝐵(,)𝐷) ∈ V
30	29	inex2 5255	. . . . . . . . 9 ⊢ (𝐴 ∩ (𝐵(,)𝐷)) ∈ V
31		snex 5376	. . . . . . . . 9 ⊢ {𝐵} ∈ V
32	30, 31	unex 7691	. . . . . . . 8 ⊢ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵}) ∈ V
33		resttop 23135	. . . . . . . 8 ⊢ (((TopOpen‘ℂ_fld) ∈ Top ∧ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵}) ∈ V) → ((TopOpen‘ℂ_fld) ↾_t ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵})) ∈ Top)
34	28, 32, 33	mp2an 693	. . . . . . 7 ⊢ ((TopOpen‘ℂ_fld) ↾_t ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵})) ∈ Top
35	34	a1i 11	. . . . . 6 ⊢ (𝜑 → ((TopOpen‘ℂ_fld) ↾_t ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵})) ∈ Top)
36		mnfxr 11193	. . . . . . . . . . . . 13 ⊢ -∞ ∈ ℝ^*
37	36	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵[,)𝐶)) → -∞ ∈ ℝ^*)
38	21	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵[,)𝐶)) → 𝐶 ∈ ℝ^*)
39		icossre 13372	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ^*) → (𝐵[,)𝐶) ⊆ ℝ)
40	19, 21, 39	syl2anc 585	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐵[,)𝐶) ⊆ ℝ)
41	40	sselda 3922	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵[,)𝐶)) → 𝑥 ∈ ℝ)
42	41	mnfltd 13066	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵[,)𝐶)) → -∞ < 𝑥)
43	20	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵[,)𝐶)) → 𝐵 ∈ ℝ^*)
44		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵[,)𝐶)) → 𝑥 ∈ (𝐵[,)𝐶))
45		icoltub 45956	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝑥 ∈ (𝐵[,)𝐶)) → 𝑥 < 𝐶)
46	43, 38, 44, 45	syl3anc 1374	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵[,)𝐶)) → 𝑥 < 𝐶)
47	37, 38, 41, 42, 46	eliood 45946	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵[,)𝐶)) → 𝑥 ∈ (-∞(,)𝐶))
48		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → 𝑥 = 𝐵)
49		snidg 4605	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ {𝐵})
50		elun2 4124	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∈ {𝐵} → 𝐵 ∈ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵}))
51	19, 49, 50	3syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐵 ∈ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵}))
52	51	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → 𝐵 ∈ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵}))
53	48, 52	eqeltrd 2837	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → 𝑥 ∈ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵}))
54	53	adantlr 716	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵[,)𝐶)) ∧ 𝑥 = 𝐵) → 𝑥 ∈ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵}))
55		simpll 767	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵[,)𝐶)) ∧ ¬ 𝑥 = 𝐵) → 𝜑)
56	43	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵[,)𝐶)) ∧ ¬ 𝑥 = 𝐵) → 𝐵 ∈ ℝ^*)
57	38	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵[,)𝐶)) ∧ ¬ 𝑥 = 𝐵) → 𝐶 ∈ ℝ^*)
58	41	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵[,)𝐶)) ∧ ¬ 𝑥 = 𝐵) → 𝑥 ∈ ℝ)
59	19	ad2antrr 727	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵[,)𝐶)) ∧ ¬ 𝑥 = 𝐵) → 𝐵 ∈ ℝ)
60		icogelb 13340	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝑥 ∈ (𝐵[,)𝐶)) → 𝐵 ≤ 𝑥)
61	43, 38, 44, 60	syl3anc 1374	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵[,)𝐶)) → 𝐵 ≤ 𝑥)
62	61	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵[,)𝐶)) ∧ ¬ 𝑥 = 𝐵) → 𝐵 ≤ 𝑥)
63		neqne 2941	. . . . . . . . . . . . . . . 16 ⊢ (¬ 𝑥 = 𝐵 → 𝑥 ≠ 𝐵)
64	63	adantl 481	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵[,)𝐶)) ∧ ¬ 𝑥 = 𝐵) → 𝑥 ≠ 𝐵)
65	59, 58, 62, 64	leneltd 11291	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵[,)𝐶)) ∧ ¬ 𝑥 = 𝐵) → 𝐵 < 𝑥)
66	46	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵[,)𝐶)) ∧ ¬ 𝑥 = 𝐵) → 𝑥 < 𝐶)
67	56, 57, 58, 65, 66	eliood 45946	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵[,)𝐶)) ∧ ¬ 𝑥 = 𝐵) → 𝑥 ∈ (𝐵(,)𝐶))
68	12	sselda 3922	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,)𝐶)) → 𝑥 ∈ (𝐴 ∩ (𝐵(,)𝐷)))
69		elun1 4123	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (𝐴 ∩ (𝐵(,)𝐷)) → 𝑥 ∈ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵}))
70	68, 69	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,)𝐶)) → 𝑥 ∈ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵}))
71	55, 67, 70	syl2anc 585	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵[,)𝐶)) ∧ ¬ 𝑥 = 𝐵) → 𝑥 ∈ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵}))
72	54, 71	pm2.61dan 813	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵[,)𝐶)) → 𝑥 ∈ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵}))
73	47, 72	elind 4141	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵[,)𝐶)) → 𝑥 ∈ ((-∞(,)𝐶) ∩ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵})))
74	24	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → 𝐵 ∈ (𝐵[,)𝐶))
75	48, 74	eqeltrd 2837	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → 𝑥 ∈ (𝐵[,)𝐶))
76	75	adantlr 716	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ((-∞(,)𝐶) ∩ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵}))) ∧ 𝑥 = 𝐵) → 𝑥 ∈ (𝐵[,)𝐶))
77		ioossico 13382	. . . . . . . . . . . 12 ⊢ (𝐵(,)𝐶) ⊆ (𝐵[,)𝐶)
78	20	ad2antrr 727	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ((-∞(,)𝐶) ∩ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵}))) ∧ ¬ 𝑥 = 𝐵) → 𝐵 ∈ ℝ^*)
79	21	ad2antrr 727	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ((-∞(,)𝐶) ∩ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵}))) ∧ ¬ 𝑥 = 𝐵) → 𝐶 ∈ ℝ^*)
80		elinel1 4142	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ((-∞(,)𝐶) ∩ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵})) → 𝑥 ∈ (-∞(,)𝐶))
81	80	elioored 45997	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ((-∞(,)𝐶) ∩ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵})) → 𝑥 ∈ ℝ)
82	81	ad2antlr 728	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ((-∞(,)𝐶) ∩ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵}))) ∧ ¬ 𝑥 = 𝐵) → 𝑥 ∈ ℝ)
83	1	ad2antrr 727	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ((-∞(,)𝐶) ∩ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵}))) ∧ ¬ 𝑥 = 𝐵) → 𝐷 ∈ ℝ^*)
84		elinel2 4143	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ((-∞(,)𝐶) ∩ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵})) → 𝑥 ∈ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵}))
85		id 22	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ 𝑥 = 𝐵 → ¬ 𝑥 = 𝐵)
86		velsn 4584	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ {𝐵} ↔ 𝑥 = 𝐵)
87	85, 86	sylnibr 329	. . . . . . . . . . . . . . . . 17 ⊢ (¬ 𝑥 = 𝐵 → ¬ 𝑥 ∈ {𝐵})
88		elunnel2 4096	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵}) ∧ ¬ 𝑥 ∈ {𝐵}) → 𝑥 ∈ (𝐴 ∩ (𝐵(,)𝐷)))
89	84, 87, 88	syl2an 597	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ((-∞(,)𝐶) ∩ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵})) ∧ ¬ 𝑥 = 𝐵) → 𝑥 ∈ (𝐴 ∩ (𝐵(,)𝐷)))
90	13, 89	sselid 3920	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ((-∞(,)𝐶) ∩ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵})) ∧ ¬ 𝑥 = 𝐵) → 𝑥 ∈ (𝐵(,)𝐷))
91	90	adantll 715	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ((-∞(,)𝐶) ∩ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵}))) ∧ ¬ 𝑥 = 𝐵) → 𝑥 ∈ (𝐵(,)𝐷))
92		ioogtlb 45943	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐷 ∈ ℝ^* ∧ 𝑥 ∈ (𝐵(,)𝐷)) → 𝐵 < 𝑥)
93	78, 83, 91, 92	syl3anc 1374	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ((-∞(,)𝐶) ∩ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵}))) ∧ ¬ 𝑥 = 𝐵) → 𝐵 < 𝑥)
94	36	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ((-∞(,)𝐶) ∩ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵}))) → -∞ ∈ ℝ^*)
95	21	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ((-∞(,)𝐶) ∩ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵}))) → 𝐶 ∈ ℝ^*)
96	80	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ((-∞(,)𝐶) ∩ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵}))) → 𝑥 ∈ (-∞(,)𝐶))
97		iooltub 45958	. . . . . . . . . . . . . . 15 ⊢ ((-∞ ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝑥 ∈ (-∞(,)𝐶)) → 𝑥 < 𝐶)
98	94, 95, 96, 97	syl3anc 1374	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ((-∞(,)𝐶) ∩ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵}))) → 𝑥 < 𝐶)
99	98	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ((-∞(,)𝐶) ∩ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵}))) ∧ ¬ 𝑥 = 𝐵) → 𝑥 < 𝐶)
100	78, 79, 82, 93, 99	eliood 45946	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ((-∞(,)𝐶) ∩ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵}))) ∧ ¬ 𝑥 = 𝐵) → 𝑥 ∈ (𝐵(,)𝐶))
101	77, 100	sselid 3920	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ((-∞(,)𝐶) ∩ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵}))) ∧ ¬ 𝑥 = 𝐵) → 𝑥 ∈ (𝐵[,)𝐶))
102	76, 101	pm2.61dan 813	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ((-∞(,)𝐶) ∩ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵}))) → 𝑥 ∈ (𝐵[,)𝐶))
103	73, 102	impbida 801	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (𝐵[,)𝐶) ↔ 𝑥 ∈ ((-∞(,)𝐶) ∩ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵}))))
104	103	eqrdv 2735	. . . . . . . 8 ⊢ (𝜑 → (𝐵[,)𝐶) = ((-∞(,)𝐶) ∩ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵})))
105		retop 24736	. . . . . . . . . 10 ⊢ (topGen‘ran (,)) ∈ Top
106	105	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → (topGen‘ran (,)) ∈ Top)
107	32	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵}) ∈ V)
108		iooretop 24740	. . . . . . . . . 10 ⊢ (-∞(,)𝐶) ∈ (topGen‘ran (,))
109	108	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → (-∞(,)𝐶) ∈ (topGen‘ran (,)))
110		elrestr 17382	. . . . . . . . 9 ⊢ (((topGen‘ran (,)) ∈ Top ∧ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵}) ∈ V ∧ (-∞(,)𝐶) ∈ (topGen‘ran (,))) → ((-∞(,)𝐶) ∩ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵})) ∈ ((topGen‘ran (,)) ↾_t ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵})))
111	106, 107, 109, 110	syl3anc 1374	. . . . . . . 8 ⊢ (𝜑 → ((-∞(,)𝐶) ∩ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵})) ∈ ((topGen‘ran (,)) ↾_t ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵})))
112	104, 111	eqeltrd 2837	. . . . . . 7 ⊢ (𝜑 → (𝐵[,)𝐶) ∈ ((topGen‘ran (,)) ↾_t ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵})))
113		tgioo4 24780	. . . . . . . . 9 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
114	113	oveq1i 7370	. . . . . . . 8 ⊢ ((topGen‘ran (,)) ↾_t ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵})) = (((TopOpen‘ℂ_fld) ↾_t ℝ) ↾_t ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵}))
115	28	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → (TopOpen‘ℂ_fld) ∈ Top)
116		ioossre 13351	. . . . . . . . . . . 12 ⊢ (𝐵(,)𝐷) ⊆ ℝ
117	13, 116	sstri 3932	. . . . . . . . . . 11 ⊢ (𝐴 ∩ (𝐵(,)𝐷)) ⊆ ℝ
118	117	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 ∩ (𝐵(,)𝐷)) ⊆ ℝ)
119	19	snssd 4753	. . . . . . . . . 10 ⊢ (𝜑 → {𝐵} ⊆ ℝ)
120	118, 119	unssd 4133	. . . . . . . . 9 ⊢ (𝜑 → ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵}) ⊆ ℝ)
121		reex 11120	. . . . . . . . . 10 ⊢ ℝ ∈ V
122	121	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → ℝ ∈ V)
123		restabs 23140	. . . . . . . . 9 ⊢ (((TopOpen‘ℂ_fld) ∈ Top ∧ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵}) ⊆ ℝ ∧ ℝ ∈ V) → (((TopOpen‘ℂ_fld) ↾_t ℝ) ↾_t ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵})) = ((TopOpen‘ℂ_fld) ↾_t ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵})))
124	115, 120, 122, 123	syl3anc 1374	. . . . . . . 8 ⊢ (𝜑 → (((TopOpen‘ℂ_fld) ↾_t ℝ) ↾_t ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵})) = ((TopOpen‘ℂ_fld) ↾_t ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵})))
125	114, 124	eqtrid 2784	. . . . . . 7 ⊢ (𝜑 → ((topGen‘ran (,)) ↾_t ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵})) = ((TopOpen‘ℂ_fld) ↾_t ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵})))
126	112, 125	eleqtrd 2839	. . . . . 6 ⊢ (𝜑 → (𝐵[,)𝐶) ∈ ((TopOpen‘ℂ_fld) ↾_t ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵})))
127		isopn3i 23057	. . . . . 6 ⊢ ((((TopOpen‘ℂ_fld) ↾_t ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵})) ∈ Top ∧ (𝐵[,)𝐶) ∈ ((TopOpen‘ℂ_fld) ↾_t ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵}))) → ((int‘((TopOpen‘ℂ_fld) ↾_t ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵})))‘(𝐵[,)𝐶)) = (𝐵[,)𝐶))
128	35, 126, 127	syl2anc 585	. . . . 5 ⊢ (𝜑 → ((int‘((TopOpen‘ℂ_fld) ↾_t ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵})))‘(𝐵[,)𝐶)) = (𝐵[,)𝐶))
129	27, 128	eqtr2d 2773	. . . 4 ⊢ (𝜑 → (𝐵[,)𝐶) = ((int‘((TopOpen‘ℂ_fld) ↾_t ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵})))‘((𝐵(,)𝐶) ∪ {𝐵})))
130	24, 129	eleqtrd 2839	. . 3 ⊢ (𝜑 → 𝐵 ∈ ((int‘((TopOpen‘ℂ_fld) ↾_t ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵})))‘((𝐵(,)𝐶) ∪ {𝐵})))
131	10, 12, 16, 17, 18, 130	limcres 25863	. 2 ⊢ (𝜑 → (((𝐹 ↾ (𝐵(,)𝐷)) ↾ (𝐵(,)𝐶)) lim_ℂ 𝐵) = ((𝐹 ↾ (𝐵(,)𝐷)) lim_ℂ 𝐵))
132	7, 131	eqtrd 2772	1 ⊢ (𝜑 → ((𝐹 ↾ (𝐵(,)𝐶)) lim_ℂ 𝐵) = ((𝐹 ↾ (𝐵(,)𝐷)) lim_ℂ 𝐵))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 Vcvv 3430 ∪ cun 3888 ∩ cin 3889 ⊆ wss 3890 {csn 4568 class class class wbr 5086 ran crn 5625 ↾ cres 5626 ⟶wf 6488 ‘cfv 6492 (class class class)co 7360 ℂcc 11027 ℝcr 11028 -∞cmnf 11168 ℝ^*cxr 11169 < clt 11170 ≤ cle 11171 (,)cioo 13289 [,)cico 13291 ↾_t crest 17374 TopOpenctopn 17375 topGenctg 17391 ℂ_fldccnfld 21344 Topctop 22868 intcnt 22992 lim_ℂ climc 25839

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 5212 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-pre-sup 11107

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 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-1st 7935 df-2nd 7936 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8342 df-1o 8398 df-er 8636 df-map 8768 df-pm 8769 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-fi 9317 df-sup 9348 df-inf 9349 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-7 12240 df-8 12241 df-9 12242 df-n0 12429 df-z 12516 df-dec 12636 df-uz 12780 df-q 12890 df-rp 12934 df-xneg 13054 df-xadd 13055 df-xmul 13056 df-ioo 13293 df-ico 13295 df-icc 13296 df-fz 13453 df-seq 13955 df-exp 14015 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-struct 17108 df-slot 17143 df-ndx 17155 df-base 17171 df-plusg 17224 df-mulr 17225 df-starv 17226 df-tset 17230 df-ple 17231 df-ds 17233 df-unif 17234 df-rest 17376 df-topn 17377 df-topgen 17397 df-psmet 21336 df-xmet 21337 df-met 21338 df-bl 21339 df-mopn 21340 df-cnfld 21345 df-top 22869 df-topon 22886 df-topsp 22908 df-bases 22921 df-ntr 22995 df-cnp 23203 df-xms 24295 df-ms 24296 df-limc 25843

This theorem is referenced by: fouriersw 46677

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