limcicciooub - Mathbox for Glauco Siliprandi

	Mathbox for Glauco Siliprandi		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > limcicciooub		Structured version Visualization version GIF version

Theorem limcicciooub 44938

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 13428	. . 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 13429	. . 3 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
7		ax-resscn 11181	. . 3 ⊢ ℝ ⊆ ℂ
8	6, 7	sstrdi 3990	. 2 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℂ)
9		eqid 2727	. 2 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
10		eqid 2727	. 2 ⊢ ((TopOpen‘ℂ_fld) ↾_t ((𝐴[,]𝐵) ∪ {𝐵})) = ((TopOpen‘ℂ_fld) ↾_t ((𝐴[,]𝐵) ∪ {𝐵}))
11		retop 24652	. . . . . . . . 9 ⊢ (topGen‘ran (,)) ∈ Top
12	11	a1i 11	. . . . . . . 8 ⊢ (𝜑 → (topGen‘ran (,)) ∈ Top)
13	4	rexrd 11280	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
14		iocssre 13422	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → (𝐴(,]𝐵) ⊆ ℝ)
15	13, 5, 14	syl2anc 583	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴(,]𝐵) ⊆ ℝ)
16		difssd 4128	. . . . . . . . . 10 ⊢ (𝜑 → (ℝ ∖ (𝐴[,]𝐵)) ⊆ ℝ)
17	15, 16	unssd 4182	. . . . . . . . 9 ⊢ (𝜑 → ((𝐴(,]𝐵) ∪ (ℝ ∖ (𝐴[,]𝐵))) ⊆ ℝ)
18		uniretop 24653	. . . . . . . . 9 ⊢ ℝ = ∪ (topGen‘ran (,))
19	17, 18	sseqtrdi 4028	. . . . . . . 8 ⊢ (𝜑 → ((𝐴(,]𝐵) ∪ (ℝ ∖ (𝐴[,]𝐵))) ⊆ ∪ (topGen‘ran (,)))
20		elioore 13372	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (𝐴(,)+∞) → 𝑥 ∈ ℝ)
21	20	ad2antlr 726	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)+∞)) ∧ 𝑥 ≤ 𝐵) → 𝑥 ∈ ℝ)
22		simplr 768	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)+∞)) ∧ 𝑥 ≤ 𝐵) → 𝑥 ∈ (𝐴(,)+∞))
23	13	ad2antrr 725	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)+∞)) ∧ 𝑥 ≤ 𝐵) → 𝐴 ∈ ℝ^*)
24		pnfxr 11284	. . . . . . . . . . . . . . . . 17 ⊢ +∞ ∈ ℝ^*
25		elioo2 13383	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → (𝑥 ∈ (𝐴(,)+∞) ↔ (𝑥 ∈ ℝ ∧ 𝐴 < 𝑥 ∧ 𝑥 < +∞)))
26	23, 24, 25	sylancl 585	. . . . . . . . . . . . . . . 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 1141	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)+∞)) ∧ 𝑥 ≤ 𝐵) → 𝐴 < 𝑥)
29		simpr 484	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)+∞)) ∧ 𝑥 ≤ 𝐵) → 𝑥 ≤ 𝐵)
30	5	ad2antrr 725	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)+∞)) ∧ 𝑥 ≤ 𝐵) → 𝐵 ∈ ℝ)
31		elioc2 13405	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → (𝑥 ∈ (𝐴(,]𝐵) ↔ (𝑥 ∈ ℝ ∧ 𝐴 < 𝑥 ∧ 𝑥 ≤ 𝐵)))
32	23, 30, 31	syl2anc 583	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)+∞)) ∧ 𝑥 ≤ 𝐵) → (𝑥 ∈ (𝐴(,]𝐵) ↔ (𝑥 ∈ ℝ ∧ 𝐴 < 𝑥 ∧ 𝑥 ≤ 𝐵)))
33	21, 28, 29, 32	mpbir3and 1340	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)+∞)) ∧ 𝑥 ≤ 𝐵) → 𝑥 ∈ (𝐴(,]𝐵))
34	33	orcd 872	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)+∞)) ∧ 𝑥 ≤ 𝐵) → (𝑥 ∈ (𝐴(,]𝐵) ∨ 𝑥 ∈ (ℝ ∖ (𝐴[,]𝐵))))
35	20	ad2antlr 726	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)+∞)) ∧ ¬ 𝑥 ≤ 𝐵) → 𝑥 ∈ ℝ)
36		3mix3 1330	. . . . . . . . . . . . . . . . 17 ⊢ (¬ 𝑥 ≤ 𝐵 → (¬ 𝑥 ∈ ℝ ∨ ¬ 𝐴 ≤ 𝑥 ∨ ¬ 𝑥 ≤ 𝐵))
37		3ianor 1105	. . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)+∞)) ∧ ¬ 𝑥 ≤ 𝐵) → ¬ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ∧ 𝑥 ≤ 𝐵))
40	4	ad2antrr 725	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)+∞)) ∧ ¬ 𝑥 ≤ 𝐵) → 𝐴 ∈ ℝ)
41	5	ad2antrr 725	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)+∞)) ∧ ¬ 𝑥 ≤ 𝐵) → 𝐵 ∈ ℝ)
42		elicc2 13407	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝑥 ∈ (𝐴[,]𝐵) ↔ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ∧ 𝑥 ≤ 𝐵)))
43	40, 41, 42	syl2anc 583	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)+∞)) ∧ ¬ 𝑥 ≤ 𝐵) → (𝑥 ∈ (𝐴[,]𝐵) ↔ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ∧ 𝑥 ≤ 𝐵)))
44	39, 43	mtbird 325	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)+∞)) ∧ ¬ 𝑥 ≤ 𝐵) → ¬ 𝑥 ∈ (𝐴[,]𝐵))
45	35, 44	eldifd 3955	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)+∞)) ∧ ¬ 𝑥 ≤ 𝐵) → 𝑥 ∈ (ℝ ∖ (𝐴[,]𝐵)))
46	45	olcd 873	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)+∞)) ∧ ¬ 𝑥 ≤ 𝐵) → (𝑥 ∈ (𝐴(,]𝐵) ∨ 𝑥 ∈ (ℝ ∖ (𝐴[,]𝐵))))
47	34, 46	pm2.61dan 812	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)+∞)) → (𝑥 ∈ (𝐴(,]𝐵) ∨ 𝑥 ∈ (ℝ ∖ (𝐴[,]𝐵))))
48		elun 4144	. . . . . . . . . . 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 3141	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑥 ∈ (𝐴(,)+∞)𝑥 ∈ ((𝐴(,]𝐵) ∪ (ℝ ∖ (𝐴[,]𝐵))))
51		dfss3 3966	. . . . . . . . 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 2727	. . . . . . . . 9 ⊢ ∪ (topGen‘ran (,)) = ∪ (topGen‘ran (,))
54	53	ntrss 22933	. . . . . . . 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 1369	. . . . . . 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 13119	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 < +∞)
59	13, 56, 5, 57, 58	eliood 44796	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ (𝐴(,)+∞))
60		iooretop 24656	. . . . . . . . 9 ⊢ (𝐴(,)+∞) ∈ (topGen‘ran (,))
61		isopn3i 22960	. . . . . . . . 9 ⊢ (((topGen‘ran (,)) ∈ Top ∧ (𝐴(,)+∞) ∈ (topGen‘ran (,))) → ((int‘(topGen‘ran (,)))‘(𝐴(,)+∞)) = (𝐴(,)+∞))
62	12, 60, 61	sylancl 585	. . . . . . . 8 ⊢ (𝜑 → ((int‘(topGen‘ran (,)))‘(𝐴(,)+∞)) = (𝐴(,)+∞))
63	59, 62	eleqtrrd 2831	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ((int‘(topGen‘ran (,)))‘(𝐴(,)+∞)))
64	55, 63	sseldd 3979	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ((int‘(topGen‘ran (,)))‘((𝐴(,]𝐵) ∪ (ℝ ∖ (𝐴[,]𝐵)))))
65	5	rexrd 11280	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
66	4, 5, 57	ltled 11378	. . . . . . 7 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
67		ubicc2 13460	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ (𝐴[,]𝐵))
68	13, 65, 66, 67	syl3anc 1369	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ (𝐴[,]𝐵))
69	64, 68	elind 4190	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ (((int‘(topGen‘ran (,)))‘((𝐴(,]𝐵) ∪ (ℝ ∖ (𝐴[,]𝐵)))) ∩ (𝐴[,]𝐵)))
70		iocssicc 13432	. . . . . . 7 ⊢ (𝐴(,]𝐵) ⊆ (𝐴[,]𝐵)
71	70	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝐴(,]𝐵) ⊆ (𝐴[,]𝐵))
72		eqid 2727	. . . . . . 7 ⊢ ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)) = ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵))
73	18, 72	restntr 23060	. . . . . 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 1369	. . . . 5 ⊢ (𝜑 → ((int‘((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)))‘(𝐴(,]𝐵)) = (((int‘(topGen‘ran (,)))‘((𝐴(,]𝐵) ∪ (ℝ ∖ (𝐴[,]𝐵)))) ∩ (𝐴[,]𝐵)))
75	69, 74	eleqtrrd 2831	. . . 4 ⊢ (𝜑 → 𝐵 ∈ ((int‘((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)))‘(𝐴(,]𝐵)))
76		eqid 2727	. . . . . . . . 9 ⊢ (topGen‘ran (,)) = (topGen‘ran (,))
77	9, 76	rerest 24694	. . . . . . . 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 2733	. . . . . 6 ⊢ (𝜑 → ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)) = ((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵)))
80	79	fveq2d 6895	. . . . 5 ⊢ (𝜑 → (int‘((topGen‘ran (,)) ↾_t (𝐴[,]𝐵))) = (int‘((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵))))
81	80	fveq1d 6893	. . . 4 ⊢ (𝜑 → ((int‘((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)))‘(𝐴(,]𝐵)) = ((int‘((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵)))‘(𝐴(,]𝐵)))
82	75, 81	eleqtrd 2830	. . 3 ⊢ (𝜑 → 𝐵 ∈ ((int‘((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵)))‘(𝐴(,]𝐵)))
83	68	snssd 4808	. . . . . . . 8 ⊢ (𝜑 → {𝐵} ⊆ (𝐴[,]𝐵))
84		ssequn2 4179	. . . . . . . 8 ⊢ ({𝐵} ⊆ (𝐴[,]𝐵) ↔ ((𝐴[,]𝐵) ∪ {𝐵}) = (𝐴[,]𝐵))
85	83, 84	sylib 217	. . . . . . 7 ⊢ (𝜑 → ((𝐴[,]𝐵) ∪ {𝐵}) = (𝐴[,]𝐵))
86	85	eqcomd 2733	. . . . . 6 ⊢ (𝜑 → (𝐴[,]𝐵) = ((𝐴[,]𝐵) ∪ {𝐵}))
87	86	oveq2d 7430	. . . . 5 ⊢ (𝜑 → ((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵)) = ((TopOpen‘ℂ_fld) ↾_t ((𝐴[,]𝐵) ∪ {𝐵})))
88	87	fveq2d 6895	. . . 4 ⊢ (𝜑 → (int‘((TopOpen‘ℂ_fld) ↾_t (𝐴[,]𝐵))) = (int‘((TopOpen‘ℂ_fld) ↾_t ((𝐴[,]𝐵) ∪ {𝐵}))))
89		ioounsn 13472	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → ((𝐴(,)𝐵) ∪ {𝐵}) = (𝐴(,]𝐵))
90	13, 65, 57, 89	syl3anc 1369	. . . . 5 ⊢ (𝜑 → ((𝐴(,)𝐵) ∪ {𝐵}) = (𝐴(,]𝐵))
91	90	eqcomd 2733	. . . 4 ⊢ (𝜑 → (𝐴(,]𝐵) = ((𝐴(,)𝐵) ∪ {𝐵}))
92	88, 91	fveq12d 6898	. . 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 2830	. 2 ⊢ (𝜑 → 𝐵 ∈ ((int‘((TopOpen‘ℂ_fld) ↾_t ((𝐴[,]𝐵) ∪ {𝐵})))‘((𝐴(,)𝐵) ∪ {𝐵})))
94	1, 3, 8, 9, 10, 93	limcres 25789	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 395 ∨ wo 846 ∨ w3o 1084 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ∀wral 3056 ∖ cdif 3941 ∪ cun 3942 ∩ cin 3943 ⊆ wss 3944 {csn 4624 ∪ cuni 4903 class class class wbr 5142 ran crn 5673 ↾ cres 5674 ⟶wf 6538 ‘cfv 6542 (class class class)co 7414 ℂcc 11122 ℝcr 11123 +∞cpnf 11261 ℝ^*cxr 11263 < clt 11264 ≤ cle 11265 (,)cioo 13342 (,]cioc 13343 [,]cicc 13345 ↾_t crest 17387 TopOpenctopn 17388 topGenctg 17404 ℂ_fldccnfld 21259 Topctop 22769 intcnt 22895 lim_ℂ climc 25765

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 ax-pre-sup 11202

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7863 df-1st 7985 df-2nd 7986 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-er 8716 df-map 8836 df-pm 8837 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-fi 9420 df-sup 9451 df-inf 9452 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 df-div 11888 df-nn 12229 df-2 12291 df-3 12292 df-4 12293 df-5 12294 df-6 12295 df-7 12296 df-8 12297 df-9 12298 df-n0 12489 df-z 12575 df-dec 12694 df-uz 12839 df-q 12949 df-rp 12993 df-xneg 13110 df-xadd 13111 df-xmul 13112 df-ioo 13346 df-ioc 13347 df-icc 13349 df-fz 13503 df-seq 13985 df-exp 14045 df-cj 15064 df-re 15065 df-im 15066 df-sqrt 15200 df-abs 15201 df-struct 17101 df-slot 17136 df-ndx 17148 df-base 17166 df-plusg 17231 df-mulr 17232 df-starv 17233 df-tset 17237 df-ple 17238 df-ds 17240 df-unif 17241 df-rest 17389 df-topn 17390 df-topgen 17410 df-psmet 21251 df-xmet 21252 df-met 21253 df-bl 21254 df-mopn 21255 df-cnfld 21260 df-top 22770 df-topon 22787 df-topsp 22809 df-bases 22823 df-cld 22897 df-ntr 22898 df-cls 22899 df-cnp 23106 df-xms 24200 df-ms 24201 df-limc 25769

This theorem is referenced by: cncfiooicclem1 45194 fourierdlem82 45489 fourierdlem93 45500 fourierdlem111 45518

W3C validator