limcresioolb - Mathbox for Glauco Siliprandi

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Theorem limcresioolb 45624

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 13284	. . . . . 6 ⊢ ((𝐷 ∈ ℝ^* ∧ 𝐶 ≤ 𝐷) → (𝐵(,)𝐶) ⊆ (𝐵(,)𝐷))
4	1, 2, 3	syl2anc 584	. . . . 5 ⊢ (𝜑 → (𝐵(,)𝐶) ⊆ (𝐵(,)𝐷))
5	4	resabs1d 5959	. . . 4 ⊢ (𝜑 → ((𝐹 ↾ (𝐵(,)𝐷)) ↾ (𝐵(,)𝐶)) = (𝐹 ↾ (𝐵(,)𝐶)))
6	5	eqcomd 2735	. . 3 ⊢ (𝜑 → (𝐹 ↾ (𝐵(,)𝐶)) = ((𝐹 ↾ (𝐵(,)𝐷)) ↾ (𝐵(,)𝐶)))
7	6	oveq1d 7364	. 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 6693	. . . 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 4192	. . 3 ⊢ (𝜑 → (𝐵(,)𝐶) ⊆ (𝐴 ∩ (𝐵(,)𝐷)))
13		inss2 4189	. . . . 5 ⊢ (𝐴 ∩ (𝐵(,)𝐷)) ⊆ (𝐵(,)𝐷)
14		ioosscn 13311	. . . . 5 ⊢ (𝐵(,)𝐷) ⊆ ℂ
15	13, 14	sstri 3945	. . . 4 ⊢ (𝐴 ∩ (𝐵(,)𝐷)) ⊆ ℂ
16	15	a1i 11	. . 3 ⊢ (𝜑 → (𝐴 ∩ (𝐵(,)𝐷)) ⊆ ℂ)
17		eqid 2729	. . 3 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
18		eqid 2729	. . 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 11165	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
21		limcresioolb.c	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℝ^*)
22		limcresioolb.bltc	. . . . 5 ⊢ (𝜑 → 𝐵 < 𝐶)
23		lbico1 13303	. . . . 5 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝐵 < 𝐶) → 𝐵 ∈ (𝐵[,)𝐶))
24	20, 21, 22, 23	syl3anc 1373	. . . 4 ⊢ (𝜑 → 𝐵 ∈ (𝐵[,)𝐶))
25		snunioo1 45493	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝐵 < 𝐶) → ((𝐵(,)𝐶) ∪ {𝐵}) = (𝐵[,)𝐶))
26	20, 21, 22, 25	syl3anc 1373	. . . . . 6 ⊢ (𝜑 → ((𝐵(,)𝐶) ∪ {𝐵}) = (𝐵[,)𝐶))
27	26	fveq2d 6826	. . . . 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 24669	. . . . . . . 8 ⊢ (TopOpen‘ℂ_fld) ∈ Top
29		ovex 7382	. . . . . . . . . 10 ⊢ (𝐵(,)𝐷) ∈ V
30	29	inex2 5257	. . . . . . . . 9 ⊢ (𝐴 ∩ (𝐵(,)𝐷)) ∈ V
31		snex 5375	. . . . . . . . 9 ⊢ {𝐵} ∈ V
32	30, 31	unex 7680	. . . . . . . 8 ⊢ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵}) ∈ V
33		resttop 23045	. . . . . . . 8 ⊢ (((TopOpen‘ℂ_fld) ∈ Top ∧ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵}) ∈ V) → ((TopOpen‘ℂ_fld) ↾_t ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵})) ∈ Top)
34	28, 32, 33	mp2an 692	. . . . . . 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 11172	. . . . . . . . . . . . 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 13331	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ^*) → (𝐵[,)𝐶) ⊆ ℝ)
40	19, 21, 39	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐵[,)𝐶) ⊆ ℝ)
41	40	sselda 3935	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵[,)𝐶)) → 𝑥 ∈ ℝ)
42	41	mnfltd 13026	. . . . . . . . . . . 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 45489	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝑥 ∈ (𝐵[,)𝐶)) → 𝑥 < 𝐶)
46	43, 38, 44, 45	syl3anc 1373	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵[,)𝐶)) → 𝑥 < 𝐶)
47	37, 38, 41, 42, 46	eliood 45479	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵[,)𝐶)) → 𝑥 ∈ (-∞(,)𝐶))
48		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → 𝑥 = 𝐵)
49		snidg 4612	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ {𝐵})
50		elun2 4134	. . . . . . . . . . . . . . . 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 2828	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → 𝑥 ∈ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵}))
54	53	adantlr 715	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵[,)𝐶)) ∧ 𝑥 = 𝐵) → 𝑥 ∈ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵}))
55		simpll 766	. . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵[,)𝐶)) ∧ ¬ 𝑥 = 𝐵) → 𝐵 ∈ ℝ)
60		icogelb 13299	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝑥 ∈ (𝐵[,)𝐶)) → 𝐵 ≤ 𝑥)
61	43, 38, 44, 60	syl3anc 1373	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵[,)𝐶)) → 𝐵 ≤ 𝑥)
62	61	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵[,)𝐶)) ∧ ¬ 𝑥 = 𝐵) → 𝐵 ≤ 𝑥)
63		neqne 2933	. . . . . . . . . . . . . . . 16 ⊢ (¬ 𝑥 = 𝐵 → 𝑥 ≠ 𝐵)
64	63	adantl 481	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵[,)𝐶)) ∧ ¬ 𝑥 = 𝐵) → 𝑥 ≠ 𝐵)
65	59, 58, 62, 64	leneltd 11270	. . . . . . . . . . . . . 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 45479	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵[,)𝐶)) ∧ ¬ 𝑥 = 𝐵) → 𝑥 ∈ (𝐵(,)𝐶))
68	12	sselda 3935	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,)𝐶)) → 𝑥 ∈ (𝐴 ∩ (𝐵(,)𝐷)))
69		elun1 4133	. . . . . . . . . . . . . 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 584	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵[,)𝐶)) ∧ ¬ 𝑥 = 𝐵) → 𝑥 ∈ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵}))
72	54, 71	pm2.61dan 812	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵[,)𝐶)) → 𝑥 ∈ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵}))
73	47, 72	elind 4151	. . . . . . . . . 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 2828	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → 𝑥 ∈ (𝐵[,)𝐶))
76	75	adantlr 715	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ((-∞(,)𝐶) ∩ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵}))) ∧ 𝑥 = 𝐵) → 𝑥 ∈ (𝐵[,)𝐶))
77		ioossico 13341	. . . . . . . . . . . 12 ⊢ (𝐵(,)𝐶) ⊆ (𝐵[,)𝐶)
78	20	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ((-∞(,)𝐶) ∩ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵}))) ∧ ¬ 𝑥 = 𝐵) → 𝐵 ∈ ℝ^*)
79	21	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ((-∞(,)𝐶) ∩ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵}))) ∧ ¬ 𝑥 = 𝐵) → 𝐶 ∈ ℝ^*)
80		elinel1 4152	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ((-∞(,)𝐶) ∩ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵})) → 𝑥 ∈ (-∞(,)𝐶))
81	80	elioored 45530	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ((-∞(,)𝐶) ∩ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵})) → 𝑥 ∈ ℝ)
82	81	ad2antlr 727	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ((-∞(,)𝐶) ∩ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵}))) ∧ ¬ 𝑥 = 𝐵) → 𝑥 ∈ ℝ)
83	1	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ((-∞(,)𝐶) ∩ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵}))) ∧ ¬ 𝑥 = 𝐵) → 𝐷 ∈ ℝ^*)
84		elinel2 4153	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ((-∞(,)𝐶) ∩ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵})) → 𝑥 ∈ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵}))
85		id 22	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ 𝑥 = 𝐵 → ¬ 𝑥 = 𝐵)
86		velsn 4593	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ {𝐵} ↔ 𝑥 = 𝐵)
87	85, 86	sylnibr 329	. . . . . . . . . . . . . . . . 17 ⊢ (¬ 𝑥 = 𝐵 → ¬ 𝑥 ∈ {𝐵})
88		elunnel2 4106	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵}) ∧ ¬ 𝑥 ∈ {𝐵}) → 𝑥 ∈ (𝐴 ∩ (𝐵(,)𝐷)))
89	84, 87, 88	syl2an 596	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ((-∞(,)𝐶) ∩ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵})) ∧ ¬ 𝑥 = 𝐵) → 𝑥 ∈ (𝐴 ∩ (𝐵(,)𝐷)))
90	13, 89	sselid 3933	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ((-∞(,)𝐶) ∩ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵})) ∧ ¬ 𝑥 = 𝐵) → 𝑥 ∈ (𝐵(,)𝐷))
91	90	adantll 714	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ((-∞(,)𝐶) ∩ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵}))) ∧ ¬ 𝑥 = 𝐵) → 𝑥 ∈ (𝐵(,)𝐷))
92		ioogtlb 45476	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐷 ∈ ℝ^* ∧ 𝑥 ∈ (𝐵(,)𝐷)) → 𝐵 < 𝑥)
93	78, 83, 91, 92	syl3anc 1373	. . . . . . . . . . . . 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 45491	. . . . . . . . . . . . . . 15 ⊢ ((-∞ ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝑥 ∈ (-∞(,)𝐶)) → 𝑥 < 𝐶)
98	94, 95, 96, 97	syl3anc 1373	. . . . . . . . . . . . . 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 45479	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ((-∞(,)𝐶) ∩ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵}))) ∧ ¬ 𝑥 = 𝐵) → 𝑥 ∈ (𝐵(,)𝐶))
101	77, 100	sselid 3933	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ((-∞(,)𝐶) ∩ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵}))) ∧ ¬ 𝑥 = 𝐵) → 𝑥 ∈ (𝐵[,)𝐶))
102	76, 101	pm2.61dan 812	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ((-∞(,)𝐶) ∩ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵}))) → 𝑥 ∈ (𝐵[,)𝐶))
103	73, 102	impbida 800	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (𝐵[,)𝐶) ↔ 𝑥 ∈ ((-∞(,)𝐶) ∩ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵}))))
104	103	eqrdv 2727	. . . . . . . 8 ⊢ (𝜑 → (𝐵[,)𝐶) = ((-∞(,)𝐶) ∩ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵})))
105		retop 24647	. . . . . . . . . 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 24651	. . . . . . . . . 10 ⊢ (-∞(,)𝐶) ∈ (topGen‘ran (,))
109	108	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → (-∞(,)𝐶) ∈ (topGen‘ran (,)))
110		elrestr 17332	. . . . . . . . 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 1373	. . . . . . . 8 ⊢ (𝜑 → ((-∞(,)𝐶) ∩ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵})) ∈ ((topGen‘ran (,)) ↾_t ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵})))
112	104, 111	eqeltrd 2828	. . . . . . 7 ⊢ (𝜑 → (𝐵[,)𝐶) ∈ ((topGen‘ran (,)) ↾_t ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵})))
113		tgioo4 24691	. . . . . . . . 9 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
114	113	oveq1i 7359	. . . . . . . 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 13310	. . . . . . . . . . . 12 ⊢ (𝐵(,)𝐷) ⊆ ℝ
117	13, 116	sstri 3945	. . . . . . . . . . 11 ⊢ (𝐴 ∩ (𝐵(,)𝐷)) ⊆ ℝ
118	117	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 ∩ (𝐵(,)𝐷)) ⊆ ℝ)
119	19	snssd 4760	. . . . . . . . . 10 ⊢ (𝜑 → {𝐵} ⊆ ℝ)
120	118, 119	unssd 4143	. . . . . . . . 9 ⊢ (𝜑 → ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵}) ⊆ ℝ)
121		reex 11100	. . . . . . . . . 10 ⊢ ℝ ∈ V
122	121	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → ℝ ∈ V)
123		restabs 23050	. . . . . . . . 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 1373	. . . . . . . 8 ⊢ (𝜑 → (((TopOpen‘ℂ_fld) ↾_t ℝ) ↾_t ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵})) = ((TopOpen‘ℂ_fld) ↾_t ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵})))
125	114, 124	eqtrid 2776	. . . . . . 7 ⊢ (𝜑 → ((topGen‘ran (,)) ↾_t ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵})) = ((TopOpen‘ℂ_fld) ↾_t ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵})))
126	112, 125	eleqtrd 2830	. . . . . 6 ⊢ (𝜑 → (𝐵[,)𝐶) ∈ ((TopOpen‘ℂ_fld) ↾_t ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵})))
127		isopn3i 22967	. . . . . 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 584	. . . . 5 ⊢ (𝜑 → ((int‘((TopOpen‘ℂ_fld) ↾_t ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵})))‘(𝐵[,)𝐶)) = (𝐵[,)𝐶))
129	27, 128	eqtr2d 2765	. . . 4 ⊢ (𝜑 → (𝐵[,)𝐶) = ((int‘((TopOpen‘ℂ_fld) ↾_t ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵})))‘((𝐵(,)𝐶) ∪ {𝐵})))
130	24, 129	eleqtrd 2830	. . 3 ⊢ (𝜑 → 𝐵 ∈ ((int‘((TopOpen‘ℂ_fld) ↾_t ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵})))‘((𝐵(,)𝐶) ∪ {𝐵})))
131	10, 12, 16, 17, 18, 130	limcres 25785	. 2 ⊢ (𝜑 → (((𝐹 ↾ (𝐵(,)𝐷)) ↾ (𝐵(,)𝐶)) lim_ℂ 𝐵) = ((𝐹 ↾ (𝐵(,)𝐷)) lim_ℂ 𝐵))
132	7, 131	eqtrd 2764	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 1540 ∈ wcel 2109 ≠ wne 2925 Vcvv 3436 ∪ cun 3901 ∩ cin 3902 ⊆ wss 3903 {csn 4577 class class class wbr 5092 ran crn 5620 ↾ cres 5621 ⟶wf 6478 ‘cfv 6482 (class class class)co 7349 ℂcc 11007 ℝcr 11008 -∞cmnf 11147 ℝ^*cxr 11148 < clt 11149 ≤ cle 11150 (,)cioo 13248 [,)cico 13250 ↾_t crest 17324 TopOpenctopn 17325 topGenctg 17341 ℂ_fldccnfld 21261 Topctop 22778 intcnt 22902 lim_ℂ climc 25761

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5218 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-pre-sup 11087

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3343 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-uni 4859 df-int 4897 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6249 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-om 7800 df-1st 7924 df-2nd 7925 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-1o 8388 df-er 8625 df-map 8755 df-pm 8756 df-en 8873 df-dom 8874 df-sdom 8875 df-fin 8876 df-fi 9301 df-sup 9332 df-inf 9333 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-sub 11349 df-neg 11350 df-div 11778 df-nn 12129 df-2 12191 df-3 12192 df-4 12193 df-5 12194 df-6 12195 df-7 12196 df-8 12197 df-9 12198 df-n0 12385 df-z 12472 df-dec 12592 df-uz 12736 df-q 12850 df-rp 12894 df-xneg 13014 df-xadd 13015 df-xmul 13016 df-ioo 13252 df-ico 13254 df-icc 13255 df-fz 13411 df-seq 13909 df-exp 13969 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-struct 17058 df-slot 17093 df-ndx 17105 df-base 17121 df-plusg 17174 df-mulr 17175 df-starv 17176 df-tset 17180 df-ple 17181 df-ds 17183 df-unif 17184 df-rest 17326 df-topn 17327 df-topgen 17347 df-psmet 21253 df-xmet 21254 df-met 21255 df-bl 21256 df-mopn 21257 df-cnfld 21262 df-top 22779 df-topon 22796 df-topsp 22818 df-bases 22831 df-ntr 22905 df-cnp 23113 df-xms 24206 df-ms 24207 df-limc 25765

This theorem is referenced by: fouriersw 46212

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