dirkercncf - Mathbox for Glauco Siliprandi

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Theorem dirkercncf 45121

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 45111	. . 3 ⊢ (𝑁 ∈ ℕ → (𝐷‘𝑁):ℝ⟶ℝ)
3		ax-resscn 11169	. . . . . . . . . . 11 ⊢ ℝ ⊆ ℂ
4	3	a1i 11	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → ℝ ⊆ ℂ)
5	2, 4	fssd 6734	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (𝐷‘𝑁):ℝ⟶ℂ)
6	5	ad2antrr 722	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝑦 ∈ ℝ) ∧ (𝑦 mod (2 · π)) = 0) → (𝐷‘𝑁):ℝ⟶ℂ)
7		oveq1 7418	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑤 → (𝑦 mod (2 · π)) = (𝑤 mod (2 · π)))
8	7	eqeq1d 2732	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑤 → ((𝑦 mod (2 · π)) = 0 ↔ (𝑤 mod (2 · π)) = 0))
9		oveq2 7419	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑤 → ((𝑛 + (1 / 2)) · 𝑦) = ((𝑛 + (1 / 2)) · 𝑤))
10	9	fveq2d 6894	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑤 → (sin‘((𝑛 + (1 / 2)) · 𝑦)) = (sin‘((𝑛 + (1 / 2)) · 𝑤)))
11		oveq1 7418	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑤 → (𝑦 / 2) = (𝑤 / 2))
12	11	fveq2d 6894	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑤 → (sin‘(𝑦 / 2)) = (sin‘(𝑤 / 2)))
13	12	oveq2d 7427	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑤 → ((2 · π) · (sin‘(𝑦 / 2))) = ((2 · π) · (sin‘(𝑤 / 2))))
14	10, 13	oveq12d 7429	. . . . . . . . . . . . 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 2758	. . . . . . . . 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 2730	. . . . . . . . 9 ⊢ (𝑦 − π) = (𝑦 − π)
20		eqid 2730	. . . . . . . . 9 ⊢ (𝑦 + π) = (𝑦 + π)
21		eqid 2730	. . . . . . . . 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 2730	. . . . . . . . 9 ⊢ (𝑤 ∈ ((𝑦 − π)(,)(𝑦 + π)) ↦ ((2 · π) · (sin‘(𝑤 / 2)))) = (𝑤 ∈ ((𝑦 − π)(,)(𝑦 + π)) ↦ ((2 · π) · (sin‘(𝑤 / 2))))
23		simpll 763	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝑦 ∈ ℝ) ∧ (𝑦 mod (2 · π)) = 0) → 𝑁 ∈ ℕ)
24		simplr 765	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝑦 ∈ ℝ) ∧ (𝑦 mod (2 · π)) = 0) → 𝑦 ∈ ℝ)
25		simpr 483	. . . . . . . . 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 45119	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝑦 ∈ ℝ) ∧ (𝑦 mod (2 · π)) = 0) → ((𝐷‘𝑁)‘𝑦) ∈ ((𝐷‘𝑁) lim_ℂ 𝑦))
27	3	jctl 522	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℝ → (ℝ ⊆ ℂ ∧ 𝑦 ∈ ℝ))
28	27	ad2antlr 723	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝑦 ∈ ℝ) ∧ (𝑦 mod (2 · π)) = 0) → (ℝ ⊆ ℂ ∧ 𝑦 ∈ ℝ))
29		eqid 2730	. . . . . . . . . 10 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
30	29	tgioo2 24539	. . . . . . . . . 10 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
31	29, 30	cnplimc 25636	. . . . . . . . 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 709	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ 𝑦 ∈ ℝ) ∧ (𝑦 mod (2 · π)) = 0) → (𝐷‘𝑁) ∈ (((topGen‘ran (,)) CnP (TopOpen‘ℂ_fld))‘𝑦))
34	29	cnfldtop 24520	. . . . . . . . 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 722	. . . . . . . 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 24500	. . . . . . . . . 10 ⊢ (topGen‘ran (,)) ∈ (TopOn‘ℝ)
39	38	toponunii 22638	. . . . . . . . 9 ⊢ ℝ = ∪ (topGen‘ran (,))
40	29	cnfldtopon 24519	. . . . . . . . . 10 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
41	40	toponunii 22638	. . . . . . . . 9 ⊢ ℂ = ∪ (TopOpen‘ℂ_fld)
42	39, 41	cnprest2 23014	. . . . . . . 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 1369	. . . . . . 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 2739	. . . . . . . . 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 7427	. . . . . . 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 6892	. . . . . 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 2833	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ 𝑦 ∈ ℝ) ∧ (𝑦 mod (2 · π)) = 0) → (𝐷‘𝑁) ∈ (((topGen‘ran (,)) CnP (topGen‘ran (,)))‘𝑦))
50		simpll 763	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ 𝑦 ∈ ℝ) ∧ ¬ (𝑦 mod (2 · π)) = 0) → 𝑁 ∈ ℕ)
51		simplr 765	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ 𝑦 ∈ ℝ) ∧ ¬ (𝑦 mod (2 · π)) = 0) → 𝑦 ∈ ℝ)
52		neqne 2946	. . . . . . 7 ⊢ (¬ (𝑦 mod (2 · π)) = 0 → (𝑦 mod (2 · π)) ≠ 0)
53	52	adantl 480	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ 𝑦 ∈ ℝ) ∧ ¬ (𝑦 mod (2 · π)) = 0) → (𝑦 mod (2 · π)) ≠ 0)
54		eqid 2730	. . . . . 6 ⊢ (⌊‘(𝑦 / (2 · π))) = (⌊‘(𝑦 / (2 · π)))
55		eqid 2730	. . . . . 6 ⊢ ((⌊‘(𝑦 / (2 · π))) + 1) = ((⌊‘(𝑦 / (2 · π))) + 1)
56		eqid 2730	. . . . . 6 ⊢ ((⌊‘(𝑦 / (2 · π))) · (2 · π)) = ((⌊‘(𝑦 / (2 · π))) · (2 · π))
57		eqid 2730	. . . . . 6 ⊢ (((⌊‘(𝑦 / (2 · π))) + 1) · (2 · π)) = (((⌊‘(𝑦 / (2 · π))) + 1) · (2 · π))
58	18, 50, 51, 53, 54, 55, 56, 57	dirkercncflem4 45120	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ 𝑦 ∈ ℝ) ∧ ¬ (𝑦 mod (2 · π)) = 0) → (𝐷‘𝑁) ∈ (((topGen‘ran (,)) CnP (topGen‘ran (,)))‘𝑦))
59	49, 58	pm2.61dan 809	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝑦 ∈ ℝ) → (𝐷‘𝑁) ∈ (((topGen‘ran (,)) CnP (topGen‘ran (,)))‘𝑦))
60	59	ralrimiva 3144	. . 3 ⊢ (𝑁 ∈ ℕ → ∀𝑦 ∈ ℝ (𝐷‘𝑁) ∈ (((topGen‘ran (,)) CnP (topGen‘ran (,)))‘𝑦))
61		cncnp 23004	. . . 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 688	. . 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 581	. 2 ⊢ (𝑁 ∈ ℕ → (𝐷‘𝑁) ∈ ((topGen‘ran (,)) Cn (topGen‘ran (,))))
64	29, 30, 30	cncfcn 24650	. . 3 ⊢ ((ℝ ⊆ ℂ ∧ ℝ ⊆ ℂ) → (ℝ–cn→ℝ) = ((topGen‘ran (,)) Cn (topGen‘ran (,))))
65	3, 3, 64	mp2an 688	. 2 ⊢ (ℝ–cn→ℝ) = ((topGen‘ran (,)) Cn (topGen‘ran (,)))
66	63, 65	eleqtrrdi 2842	1 ⊢ (𝑁 ∈ ℕ → (𝐷‘𝑁) ∈ (ℝ–cn→ℝ))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1539 ∈ wcel 2104 ≠ wne 2938 ∀wral 3059 ⊆ wss 3947 ifcif 4527 ↦ cmpt 5230 ran crn 5676 ⟶wf 6538 ‘cfv 6542 (class class class)co 7411 ℂcc 11110 ℝcr 11111 0cc0 11112 1c1 11113 + caddc 11115 · cmul 11117 − cmin 11448 / cdiv 11875 ℕcn 12216 2c2 12271 (,)cioo 13328 ⌊cfl 13759 mod cmo 13838 sincsin 16011 πcpi 16014 ↾_t crest 17370 TopOpenctopn 17371 topGenctg 17387 ℂ_fldccnfld 21144 Topctop 22615 TopOnctopon 22632 Cn ccn 22948 CnP ccnp 22949 –cn→ccncf 24616 lim_ℂ climc 25611

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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-inf2 9638 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 ax-addf 11191 ax-mulf 11192

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 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 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-isom 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7672 df-om 7858 df-1st 7977 df-2nd 7978 df-supp 8149 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-2o 8469 df-er 8705 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-fi 9408 df-sup 9439 df-inf 9440 df-oi 9507 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-q 12937 df-rp 12979 df-xneg 13096 df-xadd 13097 df-xmul 13098 df-ioo 13332 df-ioc 13333 df-ico 13334 df-icc 13335 df-fz 13489 df-fzo 13632 df-fl 13761 df-mod 13839 df-seq 13971 df-exp 14032 df-fac 14238 df-bc 14267 df-hash 14295 df-shft 15018 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-limsup 15419 df-clim 15436 df-rlim 15437 df-sum 15637 df-ef 16015 df-sin 16017 df-cos 16018 df-pi 16020 df-struct 17084 df-sets 17101 df-slot 17119 df-ndx 17131 df-base 17149 df-ress 17178 df-plusg 17214 df-mulr 17215 df-starv 17216 df-sca 17217 df-vsca 17218 df-ip 17219 df-tset 17220 df-ple 17221 df-ds 17223 df-unif 17224 df-hom 17225 df-cco 17226 df-rest 17372 df-topn 17373 df-0g 17391 df-gsum 17392 df-topgen 17393 df-pt 17394 df-prds 17397 df-xrs 17452 df-qtop 17457 df-imas 17458 df-xps 17460 df-mre 17534 df-mrc 17535 df-acs 17537 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-submnd 18706 df-mulg 18987 df-cntz 19222 df-cmn 19691 df-psmet 21136 df-xmet 21137 df-met 21138 df-bl 21139 df-mopn 21140 df-fbas 21141 df-fg 21142 df-cnfld 21145 df-top 22616 df-topon 22633 df-topsp 22655 df-bases 22669 df-cld 22743 df-ntr 22744 df-cls 22745 df-nei 22822 df-lp 22860 df-perf 22861 df-cn 22951 df-cnp 22952 df-t1 23038 df-haus 23039 df-cmp 23111 df-tx 23286 df-hmeo 23479 df-fil 23570 df-fm 23662 df-flim 23663 df-flf 23664 df-xms 24046 df-ms 24047 df-tms 24048 df-cncf 24618 df-limc 25615 df-dv 25616

This theorem is referenced by: fourierdlem83 45203 fourierdlem95 45215 fourierdlem101 45221 fourierdlem103 45223 fourierdlem104 45224 fourierdlem111 45231 fourierdlem112 45232

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