dirkercncf - Mathbox for Glauco Siliprandi

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Theorem dirkercncf 44809

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 44799	. . 3 ⊢ (𝑁 ∈ ℕ → (𝐷‘𝑁):ℝ⟶ℝ)
3		ax-resscn 11163	. . . . . . . . . . 11 ⊢ ℝ ⊆ ℂ
4	3	a1i 11	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → ℝ ⊆ ℂ)
5	2, 4	fssd 6732	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (𝐷‘𝑁):ℝ⟶ℂ)
6	5	ad2antrr 724	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝑦 ∈ ℝ) ∧ (𝑦 mod (2 · π)) = 0) → (𝐷‘𝑁):ℝ⟶ℂ)
7		oveq1 7412	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑤 → (𝑦 mod (2 · π)) = (𝑤 mod (2 · π)))
8	7	eqeq1d 2734	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑤 → ((𝑦 mod (2 · π)) = 0 ↔ (𝑤 mod (2 · π)) = 0))
9		oveq2 7413	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑤 → ((𝑛 + (1 / 2)) · 𝑦) = ((𝑛 + (1 / 2)) · 𝑤))
10	9	fveq2d 6892	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑤 → (sin‘((𝑛 + (1 / 2)) · 𝑦)) = (sin‘((𝑛 + (1 / 2)) · 𝑤)))
11		oveq1 7412	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑤 → (𝑦 / 2) = (𝑤 / 2))
12	11	fveq2d 6892	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑤 → (sin‘(𝑦 / 2)) = (sin‘(𝑤 / 2)))
13	12	oveq2d 7421	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑤 → ((2 · π) · (sin‘(𝑦 / 2))) = ((2 · π) · (sin‘(𝑤 / 2))))
14	10, 13	oveq12d 7423	. . . . . . . . . . . . 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 4553	. . . . . . . . . . . 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 5260	. . . . . . . . . . 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 5252	. . . . . . . . . 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 2760	. . . . . . . . 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 2732	. . . . . . . . 9 ⊢ (𝑦 − π) = (𝑦 − π)
20		eqid 2732	. . . . . . . . 9 ⊢ (𝑦 + π) = (𝑦 + π)
21		eqid 2732	. . . . . . . . 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 2732	. . . . . . . . 9 ⊢ (𝑤 ∈ ((𝑦 − π)(,)(𝑦 + π)) ↦ ((2 · π) · (sin‘(𝑤 / 2)))) = (𝑤 ∈ ((𝑦 − π)(,)(𝑦 + π)) ↦ ((2 · π) · (sin‘(𝑤 / 2))))
23		simpll 765	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝑦 ∈ ℝ) ∧ (𝑦 mod (2 · π)) = 0) → 𝑁 ∈ ℕ)
24		simplr 767	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝑦 ∈ ℝ) ∧ (𝑦 mod (2 · π)) = 0) → 𝑦 ∈ ℝ)
25		simpr 485	. . . . . . . . 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 44807	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝑦 ∈ ℝ) ∧ (𝑦 mod (2 · π)) = 0) → ((𝐷‘𝑁)‘𝑦) ∈ ((𝐷‘𝑁) lim_ℂ 𝑦))
27	3	jctl 524	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℝ → (ℝ ⊆ ℂ ∧ 𝑦 ∈ ℝ))
28	27	ad2antlr 725	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝑦 ∈ ℝ) ∧ (𝑦 mod (2 · π)) = 0) → (ℝ ⊆ ℂ ∧ 𝑦 ∈ ℝ))
29		eqid 2732	. . . . . . . . . 10 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
30	29	tgioo2 24310	. . . . . . . . . 10 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
31	29, 30	cnplimc 25395	. . . . . . . . 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 711	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ 𝑦 ∈ ℝ) ∧ (𝑦 mod (2 · π)) = 0) → (𝐷‘𝑁) ∈ (((topGen‘ran (,)) CnP (TopOpen‘ℂ_fld))‘𝑦))
34	29	cnfldtop 24291	. . . . . . . . 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 724	. . . . . . . 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 24271	. . . . . . . . . 10 ⊢ (topGen‘ran (,)) ∈ (TopOn‘ℝ)
39	38	toponunii 22409	. . . . . . . . 9 ⊢ ℝ = ∪ (topGen‘ran (,))
40	29	cnfldtopon 24290	. . . . . . . . . 10 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
41	40	toponunii 22409	. . . . . . . . 9 ⊢ ℂ = ∪ (TopOpen‘ℂ_fld)
42	39, 41	cnprest2 22785	. . . . . . . 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 1371	. . . . . . 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 2741	. . . . . . . . 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 7421	. . . . . . 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 6890	. . . . . 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 2835	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ 𝑦 ∈ ℝ) ∧ (𝑦 mod (2 · π)) = 0) → (𝐷‘𝑁) ∈ (((topGen‘ran (,)) CnP (topGen‘ran (,)))‘𝑦))
50		simpll 765	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ 𝑦 ∈ ℝ) ∧ ¬ (𝑦 mod (2 · π)) = 0) → 𝑁 ∈ ℕ)
51		simplr 767	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ 𝑦 ∈ ℝ) ∧ ¬ (𝑦 mod (2 · π)) = 0) → 𝑦 ∈ ℝ)
52		neqne 2948	. . . . . . 7 ⊢ (¬ (𝑦 mod (2 · π)) = 0 → (𝑦 mod (2 · π)) ≠ 0)
53	52	adantl 482	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ 𝑦 ∈ ℝ) ∧ ¬ (𝑦 mod (2 · π)) = 0) → (𝑦 mod (2 · π)) ≠ 0)
54		eqid 2732	. . . . . 6 ⊢ (⌊‘(𝑦 / (2 · π))) = (⌊‘(𝑦 / (2 · π)))
55		eqid 2732	. . . . . 6 ⊢ ((⌊‘(𝑦 / (2 · π))) + 1) = ((⌊‘(𝑦 / (2 · π))) + 1)
56		eqid 2732	. . . . . 6 ⊢ ((⌊‘(𝑦 / (2 · π))) · (2 · π)) = ((⌊‘(𝑦 / (2 · π))) · (2 · π))
57		eqid 2732	. . . . . 6 ⊢ (((⌊‘(𝑦 / (2 · π))) + 1) · (2 · π)) = (((⌊‘(𝑦 / (2 · π))) + 1) · (2 · π))
58	18, 50, 51, 53, 54, 55, 56, 57	dirkercncflem4 44808	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ 𝑦 ∈ ℝ) ∧ ¬ (𝑦 mod (2 · π)) = 0) → (𝐷‘𝑁) ∈ (((topGen‘ran (,)) CnP (topGen‘ran (,)))‘𝑦))
59	49, 58	pm2.61dan 811	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝑦 ∈ ℝ) → (𝐷‘𝑁) ∈ (((topGen‘ran (,)) CnP (topGen‘ran (,)))‘𝑦))
60	59	ralrimiva 3146	. . 3 ⊢ (𝑁 ∈ ℕ → ∀𝑦 ∈ ℝ (𝐷‘𝑁) ∈ (((topGen‘ran (,)) CnP (topGen‘ran (,)))‘𝑦))
61		cncnp 22775	. . . 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 690	. . 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 583	. 2 ⊢ (𝑁 ∈ ℕ → (𝐷‘𝑁) ∈ ((topGen‘ran (,)) Cn (topGen‘ran (,))))
64	29, 30, 30	cncfcn 24417	. . 3 ⊢ ((ℝ ⊆ ℂ ∧ ℝ ⊆ ℂ) → (ℝ–cn→ℝ) = ((topGen‘ran (,)) Cn (topGen‘ran (,))))
65	3, 3, 64	mp2an 690	. 2 ⊢ (ℝ–cn→ℝ) = ((topGen‘ran (,)) Cn (topGen‘ran (,)))
66	63, 65	eleqtrrdi 2844	1 ⊢ (𝑁 ∈ ℕ → (𝐷‘𝑁) ∈ (ℝ–cn→ℝ))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ∀wral 3061 ⊆ wss 3947 ifcif 4527 ↦ cmpt 5230 ran crn 5676 ⟶wf 6536 ‘cfv 6540 (class class class)co 7405 ℂcc 11104 ℝcr 11105 0cc0 11106 1c1 11107 + caddc 11109 · cmul 11111 − cmin 11440 / cdiv 11867 ℕcn 12208 2c2 12263 (,)cioo 13320 ⌊cfl 13751 mod cmo 13830 sincsin 16003 πcpi 16006 ↾_t crest 17362 TopOpenctopn 17363 topGenctg 17379 ℂ_fldccnfld 20936 Topctop 22386 TopOnctopon 22403 Cn ccn 22719 CnP ccnp 22720 –cn→ccncf 24383 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-inf2 9632 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 ax-addf 11185 ax-mulf 11186

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-se 5631 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-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8143 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-er 8699 df-map 8818 df-pm 8819 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-fi 9402 df-sup 9433 df-inf 9434 df-oi 9501 df-card 9930 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-ico 13326 df-icc 13327 df-fz 13481 df-fzo 13624 df-fl 13753 df-mod 13831 df-seq 13963 df-exp 14024 df-fac 14230 df-bc 14259 df-hash 14287 df-shft 15010 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-limsup 15411 df-clim 15428 df-rlim 15429 df-sum 15629 df-ef 16007 df-sin 16009 df-cos 16010 df-pi 16012 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-starv 17208 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-unif 17216 df-hom 17217 df-cco 17218 df-rest 17364 df-topn 17365 df-0g 17383 df-gsum 17384 df-topgen 17385 df-pt 17386 df-prds 17389 df-xrs 17444 df-qtop 17449 df-imas 17450 df-xps 17452 df-mre 17526 df-mrc 17527 df-acs 17529 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-submnd 18668 df-mulg 18945 df-cntz 19175 df-cmn 19644 df-psmet 20928 df-xmet 20929 df-met 20930 df-bl 20931 df-mopn 20932 df-fbas 20933 df-fg 20934 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-nei 22593 df-lp 22631 df-perf 22632 df-cn 22722 df-cnp 22723 df-t1 22809 df-haus 22810 df-cmp 22882 df-tx 23057 df-hmeo 23250 df-fil 23341 df-fm 23433 df-flim 23434 df-flf 23435 df-xms 23817 df-ms 23818 df-tms 23819 df-cncf 24385 df-limc 25374 df-dv 25375

This theorem is referenced by: fourierdlem83 44891 fourierdlem95 44903 fourierdlem101 44909 fourierdlem103 44911 fourierdlem104 44912 fourierdlem111 44919 fourierdlem112 44920

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