limcicciooub - Mathbox for Glauco Siliprandi

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Theorem limcicciooub 44339

Description: The limit of a function at the upper bound of a closed interval only depends on the values in the inner open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Hypotheses**
Ref	Expression
limcicciooub.1	⊢ (𝜑 → 𝐴 ∈ ℝ)
limcicciooub.2	⊢ (𝜑 → 𝐵 ∈ ℝ)
limcicciooub.3	⊢ (𝜑 → 𝐴 < 𝐵)
limcicciooub.4	⊢ (𝜑 → 𝐹:(𝐴[,]𝐵)⟶ℂ)

**Assertion**
Ref	Expression
limcicciooub	⊢ (𝜑 → ((𝐹 ↾ (𝐴(,)𝐵)) lim_ℂ 𝐵) = (𝐹 lim_ℂ 𝐵))

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

Step	Hyp	Ref	Expression
1		limcicciooub.4	. 2 ⊢ (𝜑 → 𝐹:(𝐴[,]𝐵)⟶ℂ)
2		ioossicc 13406	. . 3 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
3	2	a1i 11	. 2 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵))
4		limcicciooub.1	. . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ)
5		limcicciooub.2	. . . 4 ⊢ (𝜑 → 𝐵 ∈ ℝ)
6	4, 5	iccssred 13407	. . 3 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
7		ax-resscn 11163	. . 3 ⊢ ℝ ⊆ ℂ
8	6, 7	sstrdi 3993	. 2 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℂ)
9		eqid 2732	. 2 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
10		eqid 2732	. 2 ⊢ ((TopOpen‘ℂ_fld) ↾_t ((𝐴[,]𝐵) ∪ {𝐵})) = ((TopOpen‘ℂ_fld) ↾_t ((𝐴[,]𝐵) ∪ {𝐵}))
11		retop 24269	. . . . . . . . 9 ⊢ (topGen‘ran (,)) ∈ Top
12	11	a1i 11	. . . . . . . 8 ⊢ (𝜑 → (topGen‘ran (,)) ∈ Top)
13	4	rexrd 11260	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
14		iocssre 13400	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → (𝐴(,]𝐵) ⊆ ℝ)
15	13, 5, 14	syl2anc 584	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴(,]𝐵) ⊆ ℝ)
16		difssd 4131	. . . . . . . . . 10 ⊢ (𝜑 → (ℝ ∖ (𝐴[,]𝐵)) ⊆ ℝ)
17	15, 16	unssd 4185	. . . . . . . . 9 ⊢ (𝜑 → ((𝐴(,]𝐵) ∪ (ℝ ∖ (𝐴[,]𝐵))) ⊆ ℝ)
18		uniretop 24270	. . . . . . . . 9 ⊢ ℝ = ∪ (topGen‘ran (,))
19	17, 18	sseqtrdi 4031	. . . . . . . 8 ⊢ (𝜑 → ((𝐴(,]𝐵) ∪ (ℝ ∖ (𝐴[,]𝐵))) ⊆ ∪ (topGen‘ran (,)))
20		elioore 13350	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (𝐴(,)+∞) → 𝑥 ∈ ℝ)
21	20	ad2antlr 725	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)+∞)) ∧ 𝑥 ≤ 𝐵) → 𝑥 ∈ ℝ)
22		simplr 767	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)+∞)) ∧ 𝑥 ≤ 𝐵) → 𝑥 ∈ (𝐴(,)+∞))
23	13	ad2antrr 724	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)+∞)) ∧ 𝑥 ≤ 𝐵) → 𝐴 ∈ ℝ^*)
24		pnfxr 11264	. . . . . . . . . . . . . . . . 17 ⊢ +∞ ∈ ℝ^*
25		elioo2 13361	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → (𝑥 ∈ (𝐴(,)+∞) ↔ (𝑥 ∈ ℝ ∧ 𝐴 < 𝑥 ∧ 𝑥 < +∞)))
26	23, 24, 25	sylancl 586	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)+∞)) ∧ 𝑥 ≤ 𝐵) → (𝑥 ∈ (𝐴(,)+∞) ↔ (𝑥 ∈ ℝ ∧ 𝐴 < 𝑥 ∧ 𝑥 < +∞)))
27	22, 26	mpbid 231	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)+∞)) ∧ 𝑥 ≤ 𝐵) → (𝑥 ∈ ℝ ∧ 𝐴 < 𝑥 ∧ 𝑥 < +∞))
28	27	simp2d 1143	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)+∞)) ∧ 𝑥 ≤ 𝐵) → 𝐴 < 𝑥)
29		simpr 485	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)+∞)) ∧ 𝑥 ≤ 𝐵) → 𝑥 ≤ 𝐵)
30	5	ad2antrr 724	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)+∞)) ∧ 𝑥 ≤ 𝐵) → 𝐵 ∈ ℝ)
31		elioc2 13383	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → (𝑥 ∈ (𝐴(,]𝐵) ↔ (𝑥 ∈ ℝ ∧ 𝐴 < 𝑥 ∧ 𝑥 ≤ 𝐵)))
32	23, 30, 31	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)+∞)) ∧ 𝑥 ≤ 𝐵) → (𝑥 ∈ (𝐴(,]𝐵) ↔ (𝑥 ∈ ℝ ∧ 𝐴 < 𝑥 ∧ 𝑥 ≤ 𝐵)))
33	21, 28, 29, 32	mpbir3and 1342	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)+∞)) ∧ 𝑥 ≤ 𝐵) → 𝑥 ∈ (𝐴(,]𝐵))
34	33	orcd 871	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)+∞)) ∧ 𝑥 ≤ 𝐵) → (𝑥 ∈ (𝐴(,]𝐵) ∨ 𝑥 ∈ (ℝ ∖ (𝐴[,]𝐵))))
35	20	ad2antlr 725	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)+∞)) ∧ ¬ 𝑥 ≤ 𝐵) → 𝑥 ∈ ℝ)
36		3mix3 1332	. . . . . . . . . . . . . . . . 17 ⊢ (¬ 𝑥 ≤ 𝐵 → (¬ 𝑥 ∈ ℝ ∨ ¬ 𝐴 ≤ 𝑥 ∨ ¬ 𝑥 ≤ 𝐵))
37		3ianor 1107	. . . . . . . . . . . . . . . . 17 ⊢ (¬ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ∧ 𝑥 ≤ 𝐵) ↔ (¬ 𝑥 ∈ ℝ ∨ ¬ 𝐴 ≤ 𝑥 ∨ ¬ 𝑥 ≤ 𝐵))
38	36, 37	sylibr 233	. . . . . . . . . . . . . . . 16 ⊢ (¬ 𝑥 ≤ 𝐵 → ¬ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ∧ 𝑥 ≤ 𝐵))
39	38	adantl 482	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)+∞)) ∧ ¬ 𝑥 ≤ 𝐵) → ¬ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ∧ 𝑥 ≤ 𝐵))
40	4	ad2antrr 724	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)+∞)) ∧ ¬ 𝑥 ≤ 𝐵) → 𝐴 ∈ ℝ)
41	5	ad2antrr 724	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)+∞)) ∧ ¬ 𝑥 ≤ 𝐵) → 𝐵 ∈ ℝ)
42		elicc2 13385	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝑥 ∈ (𝐴[,]𝐵) ↔ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ∧ 𝑥 ≤ 𝐵)))
43	40, 41, 42	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)+∞)) ∧ ¬ 𝑥 ≤ 𝐵) → (𝑥 ∈ (𝐴[,]𝐵) ↔ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ∧ 𝑥 ≤ 𝐵)))
44	39, 43	mtbird 324	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)+∞)) ∧ ¬ 𝑥 ≤ 𝐵) → ¬ 𝑥 ∈ (𝐴[,]𝐵))
45	35, 44	eldifd 3958	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)+∞)) ∧ ¬ 𝑥 ≤ 𝐵) → 𝑥 ∈ (ℝ ∖ (𝐴[,]𝐵)))
46	45	olcd 872	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)+∞)) ∧ ¬ 𝑥 ≤ 𝐵) → (𝑥 ∈ (𝐴(,]𝐵) ∨ 𝑥 ∈ (ℝ ∖ (𝐴[,]𝐵))))
47	34, 46	pm2.61dan 811	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)+∞)) → (𝑥 ∈ (𝐴(,]𝐵) ∨ 𝑥 ∈ (ℝ ∖ (𝐴[,]𝐵))))
48		elun 4147	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ((𝐴(,]𝐵) ∪ (ℝ ∖ (𝐴[,]𝐵))) ↔ (𝑥 ∈ (𝐴(,]𝐵) ∨ 𝑥 ∈ (ℝ ∖ (𝐴[,]𝐵))))
49	47, 48	sylibr 233	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)+∞)) → 𝑥 ∈ ((𝐴(,]𝐵) ∪ (ℝ ∖ (𝐴[,]𝐵))))
50	49	ralrimiva 3146	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑥 ∈ (𝐴(,)+∞)𝑥 ∈ ((𝐴(,]𝐵) ∪ (ℝ ∖ (𝐴[,]𝐵))))
51		dfss3 3969	. . . . . . . . 9 ⊢ ((𝐴(,)+∞) ⊆ ((𝐴(,]𝐵) ∪ (ℝ ∖ (𝐴[,]𝐵))) ↔ ∀𝑥 ∈ (𝐴(,)+∞)𝑥 ∈ ((𝐴(,]𝐵) ∪ (ℝ ∖ (𝐴[,]𝐵))))
52	50, 51	sylibr 233	. . . . . . . 8 ⊢ (𝜑 → (𝐴(,)+∞) ⊆ ((𝐴(,]𝐵) ∪ (ℝ ∖ (𝐴[,]𝐵))))
53		eqid 2732	. . . . . . . . 9 ⊢ ∪ (topGen‘ran (,)) = ∪ (topGen‘ran (,))
54	53	ntrss 22550	. . . . . . . 8 ⊢ (((topGen‘ran (,)) ∈ Top ∧ ((𝐴(,]𝐵) ∪ (ℝ ∖ (𝐴[,]𝐵))) ⊆ ∪ (topGen‘ran (,)) ∧ (𝐴(,)+∞) ⊆ ((𝐴(,]𝐵) ∪ (ℝ ∖ (𝐴[,]𝐵)))) → ((int‘(topGen‘ran (,)))‘(𝐴(,)+∞)) ⊆ ((int‘(topGen‘ran (,)))‘((𝐴(,]𝐵) ∪ (ℝ ∖ (𝐴[,]𝐵)))))
55	12, 19, 52, 54	syl3anc 1371	. . . . . . 7 ⊢ (𝜑 → ((int‘(topGen‘ran (,)))‘(𝐴(,)+∞)) ⊆ ((int‘(topGen‘ran (,)))‘((𝐴(,]𝐵) ∪ (ℝ ∖ (𝐴[,]𝐵)))))
56	24	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → +∞ ∈ ℝ^*)
57		limcicciooub.3	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 < 𝐵)
58	5	ltpnfd 13097	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 < +∞)
59	13, 56, 5, 57, 58	eliood 44197	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ (𝐴(,)+∞))
60		iooretop 24273	. . . . . . . . 9 ⊢ (𝐴(,)+∞) ∈ (topGen‘ran (,))
61		isopn3i 22577	. . . . . . . . 9 ⊢ (((topGen‘ran (,)) ∈ Top ∧ (𝐴(,)+∞) ∈ (topGen‘ran (,))) → ((int‘(topGen‘ran (,)))‘(𝐴(,)+∞)) = (𝐴(,)+∞))
62	12, 60, 61	sylancl 586	. . . . . . . 8 ⊢ (𝜑 → ((int‘(topGen‘ran (,)))‘(𝐴(,)+∞)) = (𝐴(,)+∞))
63	59, 62	eleqtrrd 2836	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ((int‘(topGen‘ran (,)))‘(𝐴(,)+∞)))
64	55, 63	sseldd 3982	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ((int‘(topGen‘ran (,)))‘((𝐴(,]𝐵) ∪ (ℝ ∖ (𝐴[,]𝐵)))))
65	5	rexrd 11260	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
66	4, 5, 57	ltled 11358	. . . . . . 7 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
67		ubicc2 13438	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ (𝐴[,]𝐵))
68	13, 65, 66, 67	syl3anc 1371	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ (𝐴[,]𝐵))
69	64, 68	elind 4193	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ (((int‘(topGen‘ran (,)))‘((𝐴(,]𝐵) ∪ (ℝ ∖ (𝐴[,]𝐵)))) ∩ (𝐴[,]𝐵)))
70		iocssicc 13410	. . . . . . 7 ⊢ (𝐴(,]𝐵) ⊆ (𝐴[,]𝐵)
71	70	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝐴(,]𝐵) ⊆ (𝐴[,]𝐵))
72		eqid 2732	. . . . . . 7 ⊢ ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)) = ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵))
73	18, 72	restntr 22677	. . . . . 6 ⊢ (((topGen‘ran (,)) ∈ Top ∧ (𝐴[,]𝐵) ⊆ ℝ ∧ (𝐴(,]𝐵) ⊆ (𝐴[,]𝐵)) → ((int‘((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)))‘(𝐴(,]𝐵)) = (((int‘(topGen‘ran (,)))‘((𝐴(,]𝐵) ∪ (ℝ ∖ (𝐴[,]𝐵)))) ∩ (𝐴[,]𝐵)))
74	12, 6, 71, 73	syl3anc 1371	. . . . 5 ⊢ (𝜑 → ((int‘((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)))‘(𝐴(,]𝐵)) = (((int‘(topGen‘ran (,)))‘((𝐴(,]𝐵) ∪ (ℝ ∖ (𝐴[,]𝐵)))) ∩ (𝐴[,]𝐵)))
75	69, 74	eleqtrrd 2836	. . . 4 ⊢ (𝜑 → 𝐵 ∈ ((int‘((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)))‘(𝐴(,]𝐵)))
76		eqid 2732	. . . . . . . . 9 ⊢ (topGen‘ran (,)) = (topGen‘ran (,))
77	9, 76	rerest 24311	. . . . . . . 8 ⊢ ((𝐴[,]𝐵) ⊆ ℝ → ((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵)) = ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)))
78	6, 77	syl 17	. . . . . . 7 ⊢ (𝜑 → ((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵)) = ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)))
79	78	eqcomd 2738	. . . . . 6 ⊢ (𝜑 → ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)) = ((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵)))
80	79	fveq2d 6892	. . . . 5 ⊢ (𝜑 → (int‘((topGen‘ran (,)) ↾_t (𝐴[,]𝐵))) = (int‘((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵))))
81	80	fveq1d 6890	. . . 4 ⊢ (𝜑 → ((int‘((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)))‘(𝐴(,]𝐵)) = ((int‘((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵)))‘(𝐴(,]𝐵)))
82	75, 81	eleqtrd 2835	. . 3 ⊢ (𝜑 → 𝐵 ∈ ((int‘((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵)))‘(𝐴(,]𝐵)))
83	68	snssd 4811	. . . . . . . 8 ⊢ (𝜑 → {𝐵} ⊆ (𝐴[,]𝐵))
84		ssequn2 4182	. . . . . . . 8 ⊢ ({𝐵} ⊆ (𝐴[,]𝐵) ↔ ((𝐴[,]𝐵) ∪ {𝐵}) = (𝐴[,]𝐵))
85	83, 84	sylib 217	. . . . . . 7 ⊢ (𝜑 → ((𝐴[,]𝐵) ∪ {𝐵}) = (𝐴[,]𝐵))
86	85	eqcomd 2738	. . . . . 6 ⊢ (𝜑 → (𝐴[,]𝐵) = ((𝐴[,]𝐵) ∪ {𝐵}))
87	86	oveq2d 7421	. . . . 5 ⊢ (𝜑 → ((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵)) = ((TopOpen‘ℂ_fld) ↾_t ((𝐴[,]𝐵) ∪ {𝐵})))
88	87	fveq2d 6892	. . . 4 ⊢ (𝜑 → (int‘((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵))) = (int‘((TopOpen‘ℂ_fld) ↾_t ((𝐴[,]𝐵) ∪ {𝐵}))))
89		ioounsn 13450	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → ((𝐴(,)𝐵) ∪ {𝐵}) = (𝐴(,]𝐵))
90	13, 65, 57, 89	syl3anc 1371	. . . . 5 ⊢ (𝜑 → ((𝐴(,)𝐵) ∪ {𝐵}) = (𝐴(,]𝐵))
91	90	eqcomd 2738	. . . 4 ⊢ (𝜑 → (𝐴(,]𝐵) = ((𝐴(,)𝐵) ∪ {𝐵}))
92	88, 91	fveq12d 6895	. . 3 ⊢ (𝜑 → ((int‘((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵)))‘(𝐴(,]𝐵)) = ((int‘((TopOpen‘ℂ_fld) ↾_t ((𝐴[,]𝐵) ∪ {𝐵})))‘((𝐴(,)𝐵) ∪ {𝐵})))
93	82, 92	eleqtrd 2835	. 2 ⊢ (𝜑 → 𝐵 ∈ ((int‘((TopOpen‘ℂ_fld) ↾_t ((𝐴[,]𝐵) ∪ {𝐵})))‘((𝐴(,)𝐵) ∪ {𝐵})))
94	1, 3, 8, 9, 10, 93	limcres 25394	1 ⊢ (𝜑 → ((𝐹 ↾ (𝐴(,)𝐵)) lim_ℂ 𝐵) = (𝐹 lim_ℂ 𝐵))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∨ wo 845 ∨ w3o 1086 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∀wral 3061 ∖ cdif 3944 ∪ cun 3945 ∩ cin 3946 ⊆ wss 3947 {csn 4627 ∪ cuni 4907 class class class wbr 5147 ran crn 5676 ↾ cres 5677 ⟶wf 6536 ‘cfv 6540 (class class class)co 7405 ℂcc 11104 ℝcr 11105 +∞cpnf 11241 ℝ^*cxr 11243 < clt 11244 ≤ cle 11245 (,)cioo 13320 (,]cioc 13321 [,]cicc 13323 ↾_t crest 17362 TopOpenctopn 17363 topGenctg 17379 ℂ_fldccnfld 20936 Topctop 22386 intcnt 22512 lim_ℂ climc 25370

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8699 df-map 8818 df-pm 8819 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fi 9402 df-sup 9433 df-inf 9434 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-ioo 13324 df-ioc 13325 df-icc 13327 df-fz 13481 df-seq 13963 df-exp 14024 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-struct 17076 df-slot 17111 df-ndx 17123 df-base 17141 df-plusg 17206 df-mulr 17207 df-starv 17208 df-tset 17212 df-ple 17213 df-ds 17215 df-unif 17216 df-rest 17364 df-topn 17365 df-topgen 17385 df-psmet 20928 df-xmet 20929 df-met 20930 df-bl 20931 df-mopn 20932 df-cnfld 20937 df-top 22387 df-topon 22404 df-topsp 22426 df-bases 22440 df-cld 22514 df-ntr 22515 df-cls 22516 df-cnp 22723 df-xms 23817 df-ms 23818 df-limc 25374

This theorem is referenced by: cncfiooicclem1 44595 fourierdlem82 44890 fourierdlem93 44901 fourierdlem111 44919

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