dirkercncf - Mathbox for Glauco Siliprandi

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Theorem dirkercncf 45123

Description: For any natural number 𝑁, the Dirichlet Kernel (𝐷‘𝑁) is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Hypothesis**
Ref	Expression
dirkercncf.d	⊢ 𝐷 = (𝑛 ∈ ℕ ↦ (𝑦 ∈ ℝ ↦ if((𝑦 mod (2 · π)) = 0, (((2 · 𝑛) + 1) / (2 · π)), ((sin‘((𝑛 + (1 / 2)) · 𝑦)) / ((2 · π) · (sin‘(𝑦 / 2)))))))

**Assertion**
Ref	Expression
dirkercncf	⊢ (𝑁 ∈ ℕ → (𝐷‘𝑁) ∈ (ℝ–cn→ℝ))

Distinct variable groups: 𝑦,𝐷 𝑦,𝑁 𝑦,𝑛 Allowed substitution hints: 𝐷(𝑛) 𝑁(𝑛)

Proof of Theorem dirkercncf Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dirkercncf.d	. . . 4 ⊢ 𝐷 = (𝑛 ∈ ℕ ↦ (𝑦 ∈ ℝ ↦ if((𝑦 mod (2 · π)) = 0, (((2 · 𝑛) + 1) / (2 · π)), ((sin‘((𝑛 + (1 / 2)) · 𝑦)) / ((2 · π) · (sin‘(𝑦 / 2)))))))
2	1	dirkerf 45113	. . 3 ⊢ (𝑁 ∈ ℕ → (𝐷‘𝑁):ℝ⟶ℝ)
3		ax-resscn 11170	. . . . . . . . . . 11 ⊢ ℝ ⊆ ℂ
4	3	a1i 11	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → ℝ ⊆ ℂ)
5	2, 4	fssd 6736	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (𝐷‘𝑁):ℝ⟶ℂ)
6	5	ad2antrr 723	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝑦 ∈ ℝ) ∧ (𝑦 mod (2 · π)) = 0) → (𝐷‘𝑁):ℝ⟶ℂ)
7		oveq1 7419	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑤 → (𝑦 mod (2 · π)) = (𝑤 mod (2 · π)))
8	7	eqeq1d 2733	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑤 → ((𝑦 mod (2 · π)) = 0 ↔ (𝑤 mod (2 · π)) = 0))
9		oveq2 7420	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑤 → ((𝑛 + (1 / 2)) · 𝑦) = ((𝑛 + (1 / 2)) · 𝑤))
10	9	fveq2d 6896	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑤 → (sin‘((𝑛 + (1 / 2)) · 𝑦)) = (sin‘((𝑛 + (1 / 2)) · 𝑤)))
11		oveq1 7419	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑤 → (𝑦 / 2) = (𝑤 / 2))
12	11	fveq2d 6896	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑤 → (sin‘(𝑦 / 2)) = (sin‘(𝑤 / 2)))
13	12	oveq2d 7428	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑤 → ((2 · π) · (sin‘(𝑦 / 2))) = ((2 · π) · (sin‘(𝑤 / 2))))
14	10, 13	oveq12d 7430	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑤 → ((sin‘((𝑛 + (1 / 2)) · 𝑦)) / ((2 · π) · (sin‘(𝑦 / 2)))) = ((sin‘((𝑛 + (1 / 2)) · 𝑤)) / ((2 · π) · (sin‘(𝑤 / 2)))))
15	8, 14	ifbieq2d 4555	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑤 → if((𝑦 mod (2 · π)) = 0, (((2 · 𝑛) + 1) / (2 · π)), ((sin‘((𝑛 + (1 / 2)) · 𝑦)) / ((2 · π) · (sin‘(𝑦 / 2))))) = if((𝑤 mod (2 · π)) = 0, (((2 · 𝑛) + 1) / (2 · π)), ((sin‘((𝑛 + (1 / 2)) · 𝑤)) / ((2 · π) · (sin‘(𝑤 / 2))))))
16	15	cbvmptv 5262	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℝ ↦ if((𝑦 mod (2 · π)) = 0, (((2 · 𝑛) + 1) / (2 · π)), ((sin‘((𝑛 + (1 / 2)) · 𝑦)) / ((2 · π) · (sin‘(𝑦 / 2)))))) = (𝑤 ∈ ℝ ↦ if((𝑤 mod (2 · π)) = 0, (((2 · 𝑛) + 1) / (2 · π)), ((sin‘((𝑛 + (1 / 2)) · 𝑤)) / ((2 · π) · (sin‘(𝑤 / 2))))))
17	16	mpteq2i 5254	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ ↦ (𝑦 ∈ ℝ ↦ if((𝑦 mod (2 · π)) = 0, (((2 · 𝑛) + 1) / (2 · π)), ((sin‘((𝑛 + (1 / 2)) · 𝑦)) / ((2 · π) · (sin‘(𝑦 / 2))))))) = (𝑛 ∈ ℕ ↦ (𝑤 ∈ ℝ ↦ if((𝑤 mod (2 · π)) = 0, (((2 · 𝑛) + 1) / (2 · π)), ((sin‘((𝑛 + (1 / 2)) · 𝑤)) / ((2 · π) · (sin‘(𝑤 / 2)))))))
18	1, 17	eqtri 2759	. . . . . . . . 9 ⊢ 𝐷 = (𝑛 ∈ ℕ ↦ (𝑤 ∈ ℝ ↦ if((𝑤 mod (2 · π)) = 0, (((2 · 𝑛) + 1) / (2 · π)), ((sin‘((𝑛 + (1 / 2)) · 𝑤)) / ((2 · π) · (sin‘(𝑤 / 2)))))))
19		eqid 2731	. . . . . . . . 9 ⊢ (𝑦 − π) = (𝑦 − π)
20		eqid 2731	. . . . . . . . 9 ⊢ (𝑦 + π) = (𝑦 + π)
21		eqid 2731	. . . . . . . . 9 ⊢ (𝑤 ∈ ((𝑦 − π)(,)(𝑦 + π)) ↦ ((sin‘((𝑛 + (1 / 2)) · 𝑤)) / ((2 · π) · (sin‘(𝑤 / 2))))) = (𝑤 ∈ ((𝑦 − π)(,)(𝑦 + π)) ↦ ((sin‘((𝑛 + (1 / 2)) · 𝑤)) / ((2 · π) · (sin‘(𝑤 / 2)))))
22		eqid 2731	. . . . . . . . 9 ⊢ (𝑤 ∈ ((𝑦 − π)(,)(𝑦 + π)) ↦ ((2 · π) · (sin‘(𝑤 / 2)))) = (𝑤 ∈ ((𝑦 − π)(,)(𝑦 + π)) ↦ ((2 · π) · (sin‘(𝑤 / 2))))
23		simpll 764	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝑦 ∈ ℝ) ∧ (𝑦 mod (2 · π)) = 0) → 𝑁 ∈ ℕ)
24		simplr 766	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝑦 ∈ ℝ) ∧ (𝑦 mod (2 · π)) = 0) → 𝑦 ∈ ℝ)
25		simpr 484	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝑦 ∈ ℝ) ∧ (𝑦 mod (2 · π)) = 0) → (𝑦 mod (2 · π)) = 0)
26	18, 19, 20, 21, 22, 23, 24, 25	dirkercncflem3 45121	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝑦 ∈ ℝ) ∧ (𝑦 mod (2 · π)) = 0) → ((𝐷‘𝑁)‘𝑦) ∈ ((𝐷‘𝑁) lim_ℂ 𝑦))
27	3	jctl 523	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℝ → (ℝ ⊆ ℂ ∧ 𝑦 ∈ ℝ))
28	27	ad2antlr 724	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝑦 ∈ ℝ) ∧ (𝑦 mod (2 · π)) = 0) → (ℝ ⊆ ℂ ∧ 𝑦 ∈ ℝ))
29		eqid 2731	. . . . . . . . . 10 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
30	29	tgioo2 24540	. . . . . . . . . 10 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
31	29, 30	cnplimc 25637	. . . . . . . . 9 ⊢ ((ℝ ⊆ ℂ ∧ 𝑦 ∈ ℝ) → ((𝐷‘𝑁) ∈ (((topGen‘ran (,)) CnP (TopOpen‘ℂ_fld))‘𝑦) ↔ ((𝐷‘𝑁):ℝ⟶ℂ ∧ ((𝐷‘𝑁)‘𝑦) ∈ ((𝐷‘𝑁) lim_ℂ 𝑦))))
32	28, 31	syl 17	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝑦 ∈ ℝ) ∧ (𝑦 mod (2 · π)) = 0) → ((𝐷‘𝑁) ∈ (((topGen‘ran (,)) CnP (TopOpen‘ℂ_fld))‘𝑦) ↔ ((𝐷‘𝑁):ℝ⟶ℂ ∧ ((𝐷‘𝑁)‘𝑦) ∈ ((𝐷‘𝑁) lim_ℂ 𝑦))))
33	6, 26, 32	mpbir2and 710	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ 𝑦 ∈ ℝ) ∧ (𝑦 mod (2 · π)) = 0) → (𝐷‘𝑁) ∈ (((topGen‘ran (,)) CnP (TopOpen‘ℂ_fld))‘𝑦))
34	29	cnfldtop 24521	. . . . . . . . 9 ⊢ (TopOpen‘ℂ_fld) ∈ Top
35	34	a1i 11	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝑦 ∈ ℝ) ∧ (𝑦 mod (2 · π)) = 0) → (TopOpen‘ℂ_fld) ∈ Top)
36	2	ad2antrr 723	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝑦 ∈ ℝ) ∧ (𝑦 mod (2 · π)) = 0) → (𝐷‘𝑁):ℝ⟶ℝ)
37	3	a1i 11	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝑦 ∈ ℝ) ∧ (𝑦 mod (2 · π)) = 0) → ℝ ⊆ ℂ)
38		retopon 24501	. . . . . . . . . 10 ⊢ (topGen‘ran (,)) ∈ (TopOn‘ℝ)
39	38	toponunii 22639	. . . . . . . . 9 ⊢ ℝ = ∪ (topGen‘ran (,))
40	29	cnfldtopon 24520	. . . . . . . . . 10 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
41	40	toponunii 22639	. . . . . . . . 9 ⊢ ℂ = ∪ (TopOpen‘ℂ_fld)
42	39, 41	cnprest2 23015	. . . . . . . 8 ⊢ (((TopOpen‘ℂ_fld) ∈ Top ∧ (𝐷‘𝑁):ℝ⟶ℝ ∧ ℝ ⊆ ℂ) → ((𝐷‘𝑁) ∈ (((topGen‘ran (,)) CnP (TopOpen‘ℂ_fld))‘𝑦) ↔ (𝐷‘𝑁) ∈ (((topGen‘ran (,)) CnP ((TopOpen‘ℂ_fld) ↾_t ℝ))‘𝑦)))
43	35, 36, 37, 42	syl3anc 1370	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ 𝑦 ∈ ℝ) ∧ (𝑦 mod (2 · π)) = 0) → ((𝐷‘𝑁) ∈ (((topGen‘ran (,)) CnP (TopOpen‘ℂ_fld))‘𝑦) ↔ (𝐷‘𝑁) ∈ (((topGen‘ran (,)) CnP ((TopOpen‘ℂ_fld) ↾_t ℝ))‘𝑦)))
44	33, 43	mpbid 231	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ 𝑦 ∈ ℝ) ∧ (𝑦 mod (2 · π)) = 0) → (𝐷‘𝑁) ∈ (((topGen‘ran (,)) CnP ((TopOpen‘ℂ_fld) ↾_t ℝ))‘𝑦))
45	30	eqcomi 2740	. . . . . . . . 9 ⊢ ((TopOpen‘ℂ_fld) ↾_t ℝ) = (topGen‘ran (,))
46	45	a1i 11	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝑦 ∈ ℝ) ∧ (𝑦 mod (2 · π)) = 0) → ((TopOpen‘ℂ_fld) ↾_t ℝ) = (topGen‘ran (,)))
47	46	oveq2d 7428	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ 𝑦 ∈ ℝ) ∧ (𝑦 mod (2 · π)) = 0) → ((topGen‘ran (,)) CnP ((TopOpen‘ℂ_fld) ↾_t ℝ)) = ((topGen‘ran (,)) CnP (topGen‘ran (,))))
48	47	fveq1d 6894	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ 𝑦 ∈ ℝ) ∧ (𝑦 mod (2 · π)) = 0) → (((topGen‘ran (,)) CnP ((TopOpen‘ℂ_fld) ↾_t ℝ))‘𝑦) = (((topGen‘ran (,)) CnP (topGen‘ran (,)))‘𝑦))
49	44, 48	eleqtrd 2834	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ 𝑦 ∈ ℝ) ∧ (𝑦 mod (2 · π)) = 0) → (𝐷‘𝑁) ∈ (((topGen‘ran (,)) CnP (topGen‘ran (,)))‘𝑦))
50		simpll 764	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ 𝑦 ∈ ℝ) ∧ ¬ (𝑦 mod (2 · π)) = 0) → 𝑁 ∈ ℕ)
51		simplr 766	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ 𝑦 ∈ ℝ) ∧ ¬ (𝑦 mod (2 · π)) = 0) → 𝑦 ∈ ℝ)
52		neqne 2947	. . . . . . 7 ⊢ (¬ (𝑦 mod (2 · π)) = 0 → (𝑦 mod (2 · π)) ≠ 0)
53	52	adantl 481	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ 𝑦 ∈ ℝ) ∧ ¬ (𝑦 mod (2 · π)) = 0) → (𝑦 mod (2 · π)) ≠ 0)
54		eqid 2731	. . . . . 6 ⊢ (⌊‘(𝑦 / (2 · π))) = (⌊‘(𝑦 / (2 · π)))
55		eqid 2731	. . . . . 6 ⊢ ((⌊‘(𝑦 / (2 · π))) + 1) = ((⌊‘(𝑦 / (2 · π))) + 1)
56		eqid 2731	. . . . . 6 ⊢ ((⌊‘(𝑦 / (2 · π))) · (2 · π)) = ((⌊‘(𝑦 / (2 · π))) · (2 · π))
57		eqid 2731	. . . . . 6 ⊢ (((⌊‘(𝑦 / (2 · π))) + 1) · (2 · π)) = (((⌊‘(𝑦 / (2 · π))) + 1) · (2 · π))
58	18, 50, 51, 53, 54, 55, 56, 57	dirkercncflem4 45122	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ 𝑦 ∈ ℝ) ∧ ¬ (𝑦 mod (2 · π)) = 0) → (𝐷‘𝑁) ∈ (((topGen‘ran (,)) CnP (topGen‘ran (,)))‘𝑦))
59	49, 58	pm2.61dan 810	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝑦 ∈ ℝ) → (𝐷‘𝑁) ∈ (((topGen‘ran (,)) CnP (topGen‘ran (,)))‘𝑦))
60	59	ralrimiva 3145	. . 3 ⊢ (𝑁 ∈ ℕ → ∀𝑦 ∈ ℝ (𝐷‘𝑁) ∈ (((topGen‘ran (,)) CnP (topGen‘ran (,)))‘𝑦))
61		cncnp 23005	. . . 4 ⊢ (((topGen‘ran (,)) ∈ (TopOn‘ℝ) ∧ (topGen‘ran (,)) ∈ (TopOn‘ℝ)) → ((𝐷‘𝑁) ∈ ((topGen‘ran (,)) Cn (topGen‘ran (,))) ↔ ((𝐷‘𝑁):ℝ⟶ℝ ∧ ∀𝑦 ∈ ℝ (𝐷‘𝑁) ∈ (((topGen‘ran (,)) CnP (topGen‘ran (,)))‘𝑦))))
62	38, 38, 61	mp2an 689	. . 3 ⊢ ((𝐷‘𝑁) ∈ ((topGen‘ran (,)) Cn (topGen‘ran (,))) ↔ ((𝐷‘𝑁):ℝ⟶ℝ ∧ ∀𝑦 ∈ ℝ (𝐷‘𝑁) ∈ (((topGen‘ran (,)) CnP (topGen‘ran (,)))‘𝑦)))
63	2, 60, 62	sylanbrc 582	. 2 ⊢ (𝑁 ∈ ℕ → (𝐷‘𝑁) ∈ ((topGen‘ran (,)) Cn (topGen‘ran (,))))
64	29, 30, 30	cncfcn 24651	. . 3 ⊢ ((ℝ ⊆ ℂ ∧ ℝ ⊆ ℂ) → (ℝ–cn→ℝ) = ((topGen‘ran (,)) Cn (topGen‘ran (,))))
65	3, 3, 64	mp2an 689	. 2 ⊢ (ℝ–cn→ℝ) = ((topGen‘ran (,)) Cn (topGen‘ran (,)))
66	63, 65	eleqtrrdi 2843	1 ⊢ (𝑁 ∈ ℕ → (𝐷‘𝑁) ∈ (ℝ–cn→ℝ))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1540 ∈ wcel 2105 ≠ wne 2939 ∀wral 3060 ⊆ wss 3949 ifcif 4529 ↦ cmpt 5232 ran crn 5678 ⟶wf 6540 ‘cfv 6544 (class class class)co 7412 ℂcc 11111 ℝcr 11112 0cc0 11113 1c1 11114 + caddc 11116 · cmul 11118 − cmin 11449 / cdiv 11876 ℕcn 12217 2c2 12272 (,)cioo 13329 ⌊cfl 13760 mod cmo 13839 sincsin 16012 πcpi 16015 ↾_t crest 17371 TopOpenctopn 17372 topGenctg 17388 ℂ_fldccnfld 21145 Topctop 22616 TopOnctopon 22633 Cn ccn 22949 CnP ccnp 22950 –cn→ccncf 24617 lim_ℂ climc 25612

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7728 ax-inf2 9639 ax-cnex 11169 ax-resscn 11170 ax-1cn 11171 ax-icn 11172 ax-addcl 11173 ax-addrcl 11174 ax-mulcl 11175 ax-mulrcl 11176 ax-mulcom 11177 ax-addass 11178 ax-mulass 11179 ax-distr 11180 ax-i2m1 11181 ax-1ne0 11182 ax-1rid 11183 ax-rnegex 11184 ax-rrecex 11185 ax-cnre 11186 ax-pre-lttri 11187 ax-pre-lttrn 11188 ax-pre-ltadd 11189 ax-pre-mulgt0 11190 ax-pre-sup 11191 ax-addf 11192 ax-mulf 11193

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7673 df-om 7859 df-1st 7978 df-2nd 7979 df-supp 8150 df-frecs 8269 df-wrecs 8300 df-recs 8374 df-rdg 8413 df-1o 8469 df-2o 8470 df-er 8706 df-map 8825 df-pm 8826 df-ixp 8895 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-fsupp 9365 df-fi 9409 df-sup 9440 df-inf 9441 df-oi 9508 df-card 9937 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-div 11877 df-nn 12218 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-7 12285 df-8 12286 df-9 12287 df-n0 12478 df-z 12564 df-dec 12683 df-uz 12828 df-q 12938 df-rp 12980 df-xneg 13097 df-xadd 13098 df-xmul 13099 df-ioo 13333 df-ioc 13334 df-ico 13335 df-icc 13336 df-fz 13490 df-fzo 13633 df-fl 13762 df-mod 13840 df-seq 13972 df-exp 14033 df-fac 14239 df-bc 14268 df-hash 14296 df-shft 15019 df-cj 15051 df-re 15052 df-im 15053 df-sqrt 15187 df-abs 15188 df-limsup 15420 df-clim 15437 df-rlim 15438 df-sum 15638 df-ef 16016 df-sin 16018 df-cos 16019 df-pi 16021 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-mulr 17216 df-starv 17217 df-sca 17218 df-vsca 17219 df-ip 17220 df-tset 17221 df-ple 17222 df-ds 17224 df-unif 17225 df-hom 17226 df-cco 17227 df-rest 17373 df-topn 17374 df-0g 17392 df-gsum 17393 df-topgen 17394 df-pt 17395 df-prds 17398 df-xrs 17453 df-qtop 17458 df-imas 17459 df-xps 17461 df-mre 17535 df-mrc 17536 df-acs 17538 df-mgm 18566 df-sgrp 18645 df-mnd 18661 df-submnd 18707 df-mulg 18988 df-cntz 19223 df-cmn 19692 df-psmet 21137 df-xmet 21138 df-met 21139 df-bl 21140 df-mopn 21141 df-fbas 21142 df-fg 21143 df-cnfld 21146 df-top 22617 df-topon 22634 df-topsp 22656 df-bases 22670 df-cld 22744 df-ntr 22745 df-cls 22746 df-nei 22823 df-lp 22861 df-perf 22862 df-cn 22952 df-cnp 22953 df-t1 23039 df-haus 23040 df-cmp 23112 df-tx 23287 df-hmeo 23480 df-fil 23571 df-fm 23663 df-flim 23664 df-flf 23665 df-xms 24047 df-ms 24048 df-tms 24049 df-cncf 24619 df-limc 25616 df-dv 25617

This theorem is referenced by: fourierdlem83 45205 fourierdlem95 45217 fourierdlem101 45223 fourierdlem103 45225 fourierdlem104 45226 fourierdlem111 45233 fourierdlem112 45234

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