fourierdlem77 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem77 46626

Description: If 𝐻 is bounded, then 𝑈 is bounded. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Hypotheses**
Ref	Expression
fourierdlem77.f	⊢ (𝜑 → 𝐹:ℝ⟶ℝ)
fourierdlem77.x	⊢ (𝜑 → 𝑋 ∈ ℝ)
fourierdlem77.y	⊢ (𝜑 → 𝑌 ∈ ℝ)
fourierdlem77.w	⊢ (𝜑 → 𝑊 ∈ ℝ)
fourierdlem77.h	⊢ 𝐻 = (𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)))
fourierdlem77.k	⊢ 𝐾 = (𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2))))))
fourierdlem77.u	⊢ 𝑈 = (𝑠 ∈ (-π[,]π) ↦ ((𝐻‘𝑠) · (𝐾‘𝑠)))
fourierdlem77.bd	⊢ (𝜑 → ∃𝑎 ∈ ℝ ∀𝑠 ∈ (-π[,]π)(abs‘(𝐻‘𝑠)) ≤ 𝑎)

**Assertion**
Ref	Expression
fourierdlem77	⊢ (𝜑 → ∃𝑏 ∈ ℝ⁺ ∀𝑠 ∈ (-π[,]π)(abs‘(𝑈‘𝑠)) ≤ 𝑏)

Distinct variable groups: 𝐾,𝑏,𝑠 𝑈,𝑎,𝑏 𝜑,𝑎,𝑠 Allowed substitution hints: 𝜑(𝑏) 𝑈(𝑠) 𝐹(𝑠,𝑎,𝑏) 𝐻(𝑠,𝑎,𝑏) 𝐾(𝑎) 𝑊(𝑠,𝑎,𝑏) 𝑋(𝑠,𝑎,𝑏) 𝑌(𝑠,𝑎,𝑏)

Proof of Theorem fourierdlem77 Dummy variables 𝑐 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fourierdlem77.bd	. 2 ⊢ (𝜑 → ∃𝑎 ∈ ℝ ∀𝑠 ∈ (-π[,]π)(abs‘(𝐻‘𝑠)) ≤ 𝑎)
2		pire 26439	. . . . . . . . . 10 ⊢ π ∈ ℝ
3	2	renegcli 11446	. . . . . . . . 9 ⊢ -π ∈ ℝ
4	3	a1i 11	. . . . . . . 8 ⊢ (⊤ → -π ∈ ℝ)
5	2	a1i 11	. . . . . . . 8 ⊢ (⊤ → π ∈ ℝ)
6		pirp 26443	. . . . . . . . . . 11 ⊢ π ∈ ℝ⁺
7		neglt 12953	. . . . . . . . . . 11 ⊢ (π ∈ ℝ⁺ → -π < π)
8	6, 7	ax-mp 5	. . . . . . . . . 10 ⊢ -π < π
9	3, 2, 8	ltleii 11260	. . . . . . . . 9 ⊢ -π ≤ π
10	9	a1i 11	. . . . . . . 8 ⊢ (⊤ → -π ≤ π)
11		fourierdlem77.k	. . . . . . . . . 10 ⊢ 𝐾 = (𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2))))))
12	11	fourierdlem62 46611	. . . . . . . . 9 ⊢ 𝐾 ∈ ((-π[,]π)–cn→ℝ)
13	12	a1i 11	. . . . . . . 8 ⊢ (⊤ → 𝐾 ∈ ((-π[,]π)–cn→ℝ))
14	4, 5, 10, 13	evthiccabs 45941	. . . . . . 7 ⊢ (⊤ → (∃𝑐 ∈ (-π[,]π)∀𝑠 ∈ (-π[,]π)(abs‘(𝐾‘𝑠)) ≤ (abs‘(𝐾‘𝑐)) ∧ ∃𝑥 ∈ (-π[,]π)∀𝑦 ∈ (-π[,]π)(abs‘(𝐾‘𝑥)) ≤ (abs‘(𝐾‘𝑦))))
15	14	mptru 1554	. . . . . 6 ⊢ (∃𝑐 ∈ (-π[,]π)∀𝑠 ∈ (-π[,]π)(abs‘(𝐾‘𝑠)) ≤ (abs‘(𝐾‘𝑐)) ∧ ∃𝑥 ∈ (-π[,]π)∀𝑦 ∈ (-π[,]π)(abs‘(𝐾‘𝑥)) ≤ (abs‘(𝐾‘𝑦)))
16	15	simpli 484	. . . . 5 ⊢ ∃𝑐 ∈ (-π[,]π)∀𝑠 ∈ (-π[,]π)(abs‘(𝐾‘𝑠)) ≤ (abs‘(𝐾‘𝑐))
17	16	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ ∧ ∀𝑠 ∈ (-π[,]π)(abs‘(𝐻‘𝑠)) ≤ 𝑎) → ∃𝑐 ∈ (-π[,]π)∀𝑠 ∈ (-π[,]π)(abs‘(𝐾‘𝑠)) ≤ (abs‘(𝐾‘𝑐)))
18		simpl 483	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ ℝ ∧ 𝑐 ∈ (-π[,]π)) → 𝑎 ∈ ℝ)
19	11	fourierdlem43 46593	. . . . . . . . . . . . . 14 ⊢ 𝐾:(-π[,]π)⟶ℝ
20	19	ffvelcdmi 7024	. . . . . . . . . . . . 13 ⊢ (𝑐 ∈ (-π[,]π) → (𝐾‘𝑐) ∈ ℝ)
21	20	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ ℝ ∧ 𝑐 ∈ (-π[,]π)) → (𝐾‘𝑐) ∈ ℝ)
22	18, 21	remulcld 11166	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ ℝ ∧ 𝑐 ∈ (-π[,]π)) → (𝑎 · (𝐾‘𝑐)) ∈ ℝ)
23	22	recnd 11164	. . . . . . . . . 10 ⊢ ((𝑎 ∈ ℝ ∧ 𝑐 ∈ (-π[,]π)) → (𝑎 · (𝐾‘𝑐)) ∈ ℂ)
24	23	abscld 15392	. . . . . . . . 9 ⊢ ((𝑎 ∈ ℝ ∧ 𝑐 ∈ (-π[,]π)) → (abs‘(𝑎 · (𝐾‘𝑐))) ∈ ℝ)
25	23	absge0d 15400	. . . . . . . . 9 ⊢ ((𝑎 ∈ ℝ ∧ 𝑐 ∈ (-π[,]π)) → 0 ≤ (abs‘(𝑎 · (𝐾‘𝑐))))
26	24, 25	ge0p1rpd 13007	. . . . . . . 8 ⊢ ((𝑎 ∈ ℝ ∧ 𝑐 ∈ (-π[,]π)) → ((abs‘(𝑎 · (𝐾‘𝑐))) + 1) ∈ ℝ⁺)
27	26	3ad2antl2 1193	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ ∧ ∀𝑠 ∈ (-π[,]π)(abs‘(𝐻‘𝑠)) ≤ 𝑎) ∧ 𝑐 ∈ (-π[,]π)) → ((abs‘(𝑎 · (𝐾‘𝑐))) + 1) ∈ ℝ⁺)
28	27	3adant3 1138	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ ∧ ∀𝑠 ∈ (-π[,]π)(abs‘(𝐻‘𝑠)) ≤ 𝑎) ∧ 𝑐 ∈ (-π[,]π) ∧ ∀𝑠 ∈ (-π[,]π)(abs‘(𝐾‘𝑠)) ≤ (abs‘(𝐾‘𝑐))) → ((abs‘(𝑎 · (𝐾‘𝑐))) + 1) ∈ ℝ⁺)
29		nfv 1921	. . . . . . . . 9 ⊢ Ⅎ𝑠𝜑
30		nfv 1921	. . . . . . . . 9 ⊢ Ⅎ𝑠 𝑎 ∈ ℝ
31		nfra1 3263	. . . . . . . . 9 ⊢ Ⅎ𝑠∀𝑠 ∈ (-π[,]π)(abs‘(𝐻‘𝑠)) ≤ 𝑎
32	29, 30, 31	nf3an 1908	. . . . . . . 8 ⊢ Ⅎ𝑠(𝜑 ∧ 𝑎 ∈ ℝ ∧ ∀𝑠 ∈ (-π[,]π)(abs‘(𝐻‘𝑠)) ≤ 𝑎)
33		nfv 1921	. . . . . . . 8 ⊢ Ⅎ𝑠 𝑐 ∈ (-π[,]π)
34		nfra1 3263	. . . . . . . 8 ⊢ Ⅎ𝑠∀𝑠 ∈ (-π[,]π)(abs‘(𝐾‘𝑠)) ≤ (abs‘(𝐾‘𝑐))
35	32, 33, 34	nf3an 1908	. . . . . . 7 ⊢ Ⅎ𝑠((𝜑 ∧ 𝑎 ∈ ℝ ∧ ∀𝑠 ∈ (-π[,]π)(abs‘(𝐻‘𝑠)) ≤ 𝑎) ∧ 𝑐 ∈ (-π[,]π) ∧ ∀𝑠 ∈ (-π[,]π)(abs‘(𝐾‘𝑠)) ≤ (abs‘(𝐾‘𝑐)))
36		simpl11 1255	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℝ ∧ ∀𝑠 ∈ (-π[,]π)(abs‘(𝐻‘𝑠)) ≤ 𝑎) ∧ 𝑐 ∈ (-π[,]π) ∧ ∀𝑠 ∈ (-π[,]π)(abs‘(𝐾‘𝑠)) ≤ (abs‘(𝐾‘𝑐))) ∧ 𝑠 ∈ (-π[,]π)) → 𝜑)
37		simpl12 1256	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℝ ∧ ∀𝑠 ∈ (-π[,]π)(abs‘(𝐻‘𝑠)) ≤ 𝑎) ∧ 𝑐 ∈ (-π[,]π) ∧ ∀𝑠 ∈ (-π[,]π)(abs‘(𝐾‘𝑠)) ≤ (abs‘(𝐾‘𝑐))) ∧ 𝑠 ∈ (-π[,]π)) → 𝑎 ∈ ℝ)
38	36, 37	jca 516	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℝ ∧ ∀𝑠 ∈ (-π[,]π)(abs‘(𝐻‘𝑠)) ≤ 𝑎) ∧ 𝑐 ∈ (-π[,]π) ∧ ∀𝑠 ∈ (-π[,]π)(abs‘(𝐾‘𝑠)) ≤ (abs‘(𝐾‘𝑐))) ∧ 𝑠 ∈ (-π[,]π)) → (𝜑 ∧ 𝑎 ∈ ℝ))
39		simpl13 1257	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℝ ∧ ∀𝑠 ∈ (-π[,]π)(abs‘(𝐻‘𝑠)) ≤ 𝑎) ∧ 𝑐 ∈ (-π[,]π) ∧ ∀𝑠 ∈ (-π[,]π)(abs‘(𝐾‘𝑠)) ≤ (abs‘(𝐾‘𝑐))) ∧ 𝑠 ∈ (-π[,]π)) → ∀𝑠 ∈ (-π[,]π)(abs‘(𝐻‘𝑠)) ≤ 𝑎)
40		rspa 3228	. . . . . . . . . . 11 ⊢ ((∀𝑠 ∈ (-π[,]π)(abs‘(𝐻‘𝑠)) ≤ 𝑎 ∧ 𝑠 ∈ (-π[,]π)) → (abs‘(𝐻‘𝑠)) ≤ 𝑎)
41	39, 40	sylancom 594	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℝ ∧ ∀𝑠 ∈ (-π[,]π)(abs‘(𝐻‘𝑠)) ≤ 𝑎) ∧ 𝑐 ∈ (-π[,]π) ∧ ∀𝑠 ∈ (-π[,]π)(abs‘(𝐾‘𝑠)) ≤ (abs‘(𝐾‘𝑐))) ∧ 𝑠 ∈ (-π[,]π)) → (abs‘(𝐻‘𝑠)) ≤ 𝑎)
42		simpl2 1199	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℝ ∧ ∀𝑠 ∈ (-π[,]π)(abs‘(𝐻‘𝑠)) ≤ 𝑎) ∧ 𝑐 ∈ (-π[,]π) ∧ ∀𝑠 ∈ (-π[,]π)(abs‘(𝐾‘𝑠)) ≤ (abs‘(𝐾‘𝑐))) ∧ 𝑠 ∈ (-π[,]π)) → 𝑐 ∈ (-π[,]π))
43	38, 41, 42	jca31 519	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℝ ∧ ∀𝑠 ∈ (-π[,]π)(abs‘(𝐻‘𝑠)) ≤ 𝑎) ∧ 𝑐 ∈ (-π[,]π) ∧ ∀𝑠 ∈ (-π[,]π)(abs‘(𝐾‘𝑠)) ≤ (abs‘(𝐾‘𝑐))) ∧ 𝑠 ∈ (-π[,]π)) → (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ (abs‘(𝐻‘𝑠)) ≤ 𝑎) ∧ 𝑐 ∈ (-π[,]π)))
44		rspa 3228	. . . . . . . . . 10 ⊢ ((∀𝑠 ∈ (-π[,]π)(abs‘(𝐾‘𝑠)) ≤ (abs‘(𝐾‘𝑐)) ∧ 𝑠 ∈ (-π[,]π)) → (abs‘(𝐾‘𝑠)) ≤ (abs‘(𝐾‘𝑐)))
45	44	3ad2antl3 1194	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℝ ∧ ∀𝑠 ∈ (-π[,]π)(abs‘(𝐻‘𝑠)) ≤ 𝑎) ∧ 𝑐 ∈ (-π[,]π) ∧ ∀𝑠 ∈ (-π[,]π)(abs‘(𝐾‘𝑠)) ≤ (abs‘(𝐾‘𝑐))) ∧ 𝑠 ∈ (-π[,]π)) → (abs‘(𝐾‘𝑠)) ≤ (abs‘(𝐾‘𝑐)))
46		simpr 485	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℝ ∧ ∀𝑠 ∈ (-π[,]π)(abs‘(𝐻‘𝑠)) ≤ 𝑎) ∧ 𝑐 ∈ (-π[,]π) ∧ ∀𝑠 ∈ (-π[,]π)(abs‘(𝐾‘𝑠)) ≤ (abs‘(𝐾‘𝑐))) ∧ 𝑠 ∈ (-π[,]π)) → 𝑠 ∈ (-π[,]π))
47		simp-5l 790	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ℝ) ∧ (abs‘(𝐻‘𝑠)) ≤ 𝑎) ∧ 𝑐 ∈ (-π[,]π)) ∧ (abs‘(𝐾‘𝑠)) ≤ (abs‘(𝐾‘𝑐))) ∧ 𝑠 ∈ (-π[,]π)) → 𝜑)
48		simpr 485	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑠 ∈ (-π[,]π)) → 𝑠 ∈ (-π[,]π))
49		fourierdlem77.f	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐹:ℝ⟶ℝ)
50		fourierdlem77.x	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑋 ∈ ℝ)
51		fourierdlem77.y	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑌 ∈ ℝ)
52		fourierdlem77.w	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑊 ∈ ℝ)
53		fourierdlem77.h	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐻 = (𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)))
54	49, 50, 51, 52, 53	fourierdlem9 46559	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐻:(-π[,]π)⟶ℝ)
55	54	ffvelcdmda 7025	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑠 ∈ (-π[,]π)) → (𝐻‘𝑠) ∈ ℝ)
56	19	ffvelcdmi 7024	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 ∈ (-π[,]π) → (𝐾‘𝑠) ∈ ℝ)
57	56	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑠 ∈ (-π[,]π)) → (𝐾‘𝑠) ∈ ℝ)
58	55, 57	remulcld 11166	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑠 ∈ (-π[,]π)) → ((𝐻‘𝑠) · (𝐾‘𝑠)) ∈ ℝ)
59		fourierdlem77.u	. . . . . . . . . . . . . . . 16 ⊢ 𝑈 = (𝑠 ∈ (-π[,]π) ↦ ((𝐻‘𝑠) · (𝐾‘𝑠)))
60	59	fvmpt2 6947	. . . . . . . . . . . . . . 15 ⊢ ((𝑠 ∈ (-π[,]π) ∧ ((𝐻‘𝑠) · (𝐾‘𝑠)) ∈ ℝ) → (𝑈‘𝑠) = ((𝐻‘𝑠) · (𝐾‘𝑠)))
61	48, 58, 60	syl2anc 590	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑠 ∈ (-π[,]π)) → (𝑈‘𝑠) = ((𝐻‘𝑠) · (𝐾‘𝑠)))
62	61, 58	eqeltrd 2839	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 ∈ (-π[,]π)) → (𝑈‘𝑠) ∈ ℝ)
63	62	recnd 11164	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ (-π[,]π)) → (𝑈‘𝑠) ∈ ℂ)
64	63	abscld 15392	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ (-π[,]π)) → (abs‘(𝑈‘𝑠)) ∈ ℝ)
65	47, 64	sylancom 594	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ℝ) ∧ (abs‘(𝐻‘𝑠)) ≤ 𝑎) ∧ 𝑐 ∈ (-π[,]π)) ∧ (abs‘(𝐾‘𝑠)) ≤ (abs‘(𝐾‘𝑐))) ∧ 𝑠 ∈ (-π[,]π)) → (abs‘(𝑈‘𝑠)) ∈ ℝ)
66		simp-5r 791	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ℝ) ∧ (abs‘(𝐻‘𝑠)) ≤ 𝑎) ∧ 𝑐 ∈ (-π[,]π)) ∧ (abs‘(𝐾‘𝑠)) ≤ (abs‘(𝐾‘𝑐))) ∧ 𝑠 ∈ (-π[,]π)) → 𝑎 ∈ ℝ)
67		simpllr 781	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ℝ) ∧ (abs‘(𝐻‘𝑠)) ≤ 𝑎) ∧ 𝑐 ∈ (-π[,]π)) ∧ (abs‘(𝐾‘𝑠)) ≤ (abs‘(𝐾‘𝑐))) ∧ 𝑠 ∈ (-π[,]π)) → 𝑐 ∈ (-π[,]π))
68	66, 67, 24	syl2anc 590	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ℝ) ∧ (abs‘(𝐻‘𝑠)) ≤ 𝑎) ∧ 𝑐 ∈ (-π[,]π)) ∧ (abs‘(𝐾‘𝑠)) ≤ (abs‘(𝐾‘𝑐))) ∧ 𝑠 ∈ (-π[,]π)) → (abs‘(𝑎 · (𝐾‘𝑐))) ∈ ℝ)
69		peano2re 11310	. . . . . . . . . . 11 ⊢ ((abs‘(𝑎 · (𝐾‘𝑐))) ∈ ℝ → ((abs‘(𝑎 · (𝐾‘𝑐))) + 1) ∈ ℝ)
70	68, 69	syl 17	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ℝ) ∧ (abs‘(𝐻‘𝑠)) ≤ 𝑎) ∧ 𝑐 ∈ (-π[,]π)) ∧ (abs‘(𝐾‘𝑠)) ≤ (abs‘(𝐾‘𝑐))) ∧ 𝑠 ∈ (-π[,]π)) → ((abs‘(𝑎 · (𝐾‘𝑐))) + 1) ∈ ℝ)
71	61	fveq2d 6831	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 ∈ (-π[,]π)) → (abs‘(𝑈‘𝑠)) = (abs‘((𝐻‘𝑠) · (𝐾‘𝑠))))
72	47, 71	sylancom 594	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ℝ) ∧ (abs‘(𝐻‘𝑠)) ≤ 𝑎) ∧ 𝑐 ∈ (-π[,]π)) ∧ (abs‘(𝐾‘𝑠)) ≤ (abs‘(𝐾‘𝑐))) ∧ 𝑠 ∈ (-π[,]π)) → (abs‘(𝑈‘𝑠)) = (abs‘((𝐻‘𝑠) · (𝐾‘𝑠))))
73	55	recnd 11164	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑠 ∈ (-π[,]π)) → (𝐻‘𝑠) ∈ ℂ)
74	73	abscld 15392	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑠 ∈ (-π[,]π)) → (abs‘(𝐻‘𝑠)) ∈ ℝ)
75	47, 74	sylancom 594	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ℝ) ∧ (abs‘(𝐻‘𝑠)) ≤ 𝑎) ∧ 𝑐 ∈ (-π[,]π)) ∧ (abs‘(𝐾‘𝑠)) ≤ (abs‘(𝐾‘𝑐))) ∧ 𝑠 ∈ (-π[,]π)) → (abs‘(𝐻‘𝑠)) ∈ ℝ)
76		recn 11119	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 ∈ ℝ → 𝑎 ∈ ℂ)
77	76	abscld 15392	. . . . . . . . . . . . . . 15 ⊢ (𝑎 ∈ ℝ → (abs‘𝑎) ∈ ℝ)
78	66, 77	syl 17	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ℝ) ∧ (abs‘(𝐻‘𝑠)) ≤ 𝑎) ∧ 𝑐 ∈ (-π[,]π)) ∧ (abs‘(𝐾‘𝑠)) ≤ (abs‘(𝐾‘𝑐))) ∧ 𝑠 ∈ (-π[,]π)) → (abs‘𝑎) ∈ ℝ)
79	56	recnd 11164	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 ∈ (-π[,]π) → (𝐾‘𝑠) ∈ ℂ)
80	79	abscld 15392	. . . . . . . . . . . . . . 15 ⊢ (𝑠 ∈ (-π[,]π) → (abs‘(𝐾‘𝑠)) ∈ ℝ)
81	80	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ℝ) ∧ (abs‘(𝐻‘𝑠)) ≤ 𝑎) ∧ 𝑐 ∈ (-π[,]π)) ∧ (abs‘(𝐾‘𝑠)) ≤ (abs‘(𝐾‘𝑐))) ∧ 𝑠 ∈ (-π[,]π)) → (abs‘(𝐾‘𝑠)) ∈ ℝ)
82	20	recnd 11164	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 ∈ (-π[,]π) → (𝐾‘𝑐) ∈ ℂ)
83	82	abscld 15392	. . . . . . . . . . . . . . 15 ⊢ (𝑐 ∈ (-π[,]π) → (abs‘(𝐾‘𝑐)) ∈ ℝ)
84	67, 83	syl 17	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ℝ) ∧ (abs‘(𝐻‘𝑠)) ≤ 𝑎) ∧ 𝑐 ∈ (-π[,]π)) ∧ (abs‘(𝐾‘𝑠)) ≤ (abs‘(𝐾‘𝑐))) ∧ 𝑠 ∈ (-π[,]π)) → (abs‘(𝐾‘𝑐)) ∈ ℝ)
85	73	absge0d 15400	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑠 ∈ (-π[,]π)) → 0 ≤ (abs‘(𝐻‘𝑠)))
86	47, 85	sylancom 594	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ℝ) ∧ (abs‘(𝐻‘𝑠)) ≤ 𝑎) ∧ 𝑐 ∈ (-π[,]π)) ∧ (abs‘(𝐾‘𝑠)) ≤ (abs‘(𝐾‘𝑐))) ∧ 𝑠 ∈ (-π[,]π)) → 0 ≤ (abs‘(𝐻‘𝑠)))
87	82	absge0d 15400	. . . . . . . . . . . . . . 15 ⊢ (𝑐 ∈ (-π[,]π) → 0 ≤ (abs‘(𝐾‘𝑐)))
88	67, 87	syl 17	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ℝ) ∧ (abs‘(𝐻‘𝑠)) ≤ 𝑎) ∧ 𝑐 ∈ (-π[,]π)) ∧ (abs‘(𝐾‘𝑠)) ≤ (abs‘(𝐾‘𝑐))) ∧ 𝑠 ∈ (-π[,]π)) → 0 ≤ (abs‘(𝐾‘𝑐)))
89	74	ad4ant14 758	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℝ) ∧ (abs‘(𝐻‘𝑠)) ≤ 𝑎) ∧ 𝑠 ∈ (-π[,]π)) → (abs‘(𝐻‘𝑠)) ∈ ℝ)
90		simpllr 781	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℝ) ∧ (abs‘(𝐻‘𝑠)) ≤ 𝑎) ∧ 𝑠 ∈ (-π[,]π)) → 𝑎 ∈ ℝ)
91	77	ad3antlr 737	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℝ) ∧ (abs‘(𝐻‘𝑠)) ≤ 𝑎) ∧ 𝑠 ∈ (-π[,]π)) → (abs‘𝑎) ∈ ℝ)
92		simplr 774	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℝ) ∧ (abs‘(𝐻‘𝑠)) ≤ 𝑎) ∧ 𝑠 ∈ (-π[,]π)) → (abs‘(𝐻‘𝑠)) ≤ 𝑎)
93	90	leabsd 15368	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℝ) ∧ (abs‘(𝐻‘𝑠)) ≤ 𝑎) ∧ 𝑠 ∈ (-π[,]π)) → 𝑎 ≤ (abs‘𝑎))
94	89, 90, 91, 92, 93	letrd 11294	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℝ) ∧ (abs‘(𝐻‘𝑠)) ≤ 𝑎) ∧ 𝑠 ∈ (-π[,]π)) → (abs‘(𝐻‘𝑠)) ≤ (abs‘𝑎))
95	94	ad4ant14 758	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ℝ) ∧ (abs‘(𝐻‘𝑠)) ≤ 𝑎) ∧ 𝑐 ∈ (-π[,]π)) ∧ (abs‘(𝐾‘𝑠)) ≤ (abs‘(𝐾‘𝑐))) ∧ 𝑠 ∈ (-π[,]π)) → (abs‘(𝐻‘𝑠)) ≤ (abs‘𝑎))
96		simplr 774	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ℝ) ∧ (abs‘(𝐻‘𝑠)) ≤ 𝑎) ∧ 𝑐 ∈ (-π[,]π)) ∧ (abs‘(𝐾‘𝑠)) ≤ (abs‘(𝐾‘𝑐))) ∧ 𝑠 ∈ (-π[,]π)) → (abs‘(𝐾‘𝑠)) ≤ (abs‘(𝐾‘𝑐)))
97	75, 78, 81, 84, 86, 88, 95, 96	lemul12bd 12090	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ℝ) ∧ (abs‘(𝐻‘𝑠)) ≤ 𝑎) ∧ 𝑐 ∈ (-π[,]π)) ∧ (abs‘(𝐾‘𝑠)) ≤ (abs‘(𝐾‘𝑐))) ∧ 𝑠 ∈ (-π[,]π)) → ((abs‘(𝐻‘𝑠)) · (abs‘(𝐾‘𝑠))) ≤ ((abs‘𝑎) · (abs‘(𝐾‘𝑐))))
98	57	recnd 11164	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑠 ∈ (-π[,]π)) → (𝐾‘𝑠) ∈ ℂ)
99	73, 98	absmuld 15410	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑠 ∈ (-π[,]π)) → (abs‘((𝐻‘𝑠) · (𝐾‘𝑠))) = ((abs‘(𝐻‘𝑠)) · (abs‘(𝐾‘𝑠))))
100	47, 99	sylancom 594	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ℝ) ∧ (abs‘(𝐻‘𝑠)) ≤ 𝑎) ∧ 𝑐 ∈ (-π[,]π)) ∧ (abs‘(𝐾‘𝑠)) ≤ (abs‘(𝐾‘𝑐))) ∧ 𝑠 ∈ (-π[,]π)) → (abs‘((𝐻‘𝑠) · (𝐾‘𝑠))) = ((abs‘(𝐻‘𝑠)) · (abs‘(𝐾‘𝑠))))
101	76	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 ∈ ℝ ∧ 𝑐 ∈ (-π[,]π)) → 𝑎 ∈ ℂ)
102	21	recnd 11164	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 ∈ ℝ ∧ 𝑐 ∈ (-π[,]π)) → (𝐾‘𝑐) ∈ ℂ)
103	101, 102	absmuld 15410	. . . . . . . . . . . . . 14 ⊢ ((𝑎 ∈ ℝ ∧ 𝑐 ∈ (-π[,]π)) → (abs‘(𝑎 · (𝐾‘𝑐))) = ((abs‘𝑎) · (abs‘(𝐾‘𝑐))))
104	66, 67, 103	syl2anc 590	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ℝ) ∧ (abs‘(𝐻‘𝑠)) ≤ 𝑎) ∧ 𝑐 ∈ (-π[,]π)) ∧ (abs‘(𝐾‘𝑠)) ≤ (abs‘(𝐾‘𝑐))) ∧ 𝑠 ∈ (-π[,]π)) → (abs‘(𝑎 · (𝐾‘𝑐))) = ((abs‘𝑎) · (abs‘(𝐾‘𝑐))))
105	97, 100, 104	3brtr4d 5104	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ℝ) ∧ (abs‘(𝐻‘𝑠)) ≤ 𝑎) ∧ 𝑐 ∈ (-π[,]π)) ∧ (abs‘(𝐾‘𝑠)) ≤ (abs‘(𝐾‘𝑐))) ∧ 𝑠 ∈ (-π[,]π)) → (abs‘((𝐻‘𝑠) · (𝐾‘𝑠))) ≤ (abs‘(𝑎 · (𝐾‘𝑐))))
106	72, 105	eqbrtrd 5094	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ℝ) ∧ (abs‘(𝐻‘𝑠)) ≤ 𝑎) ∧ 𝑐 ∈ (-π[,]π)) ∧ (abs‘(𝐾‘𝑠)) ≤ (abs‘(𝐾‘𝑐))) ∧ 𝑠 ∈ (-π[,]π)) → (abs‘(𝑈‘𝑠)) ≤ (abs‘(𝑎 · (𝐾‘𝑐))))
107	68	ltp1d 12077	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ℝ) ∧ (abs‘(𝐻‘𝑠)) ≤ 𝑎) ∧ 𝑐 ∈ (-π[,]π)) ∧ (abs‘(𝐾‘𝑠)) ≤ (abs‘(𝐾‘𝑐))) ∧ 𝑠 ∈ (-π[,]π)) → (abs‘(𝑎 · (𝐾‘𝑐))) < ((abs‘(𝑎 · (𝐾‘𝑐))) + 1))
108	65, 68, 70, 106, 107	lelttrd 11295	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ℝ) ∧ (abs‘(𝐻‘𝑠)) ≤ 𝑎) ∧ 𝑐 ∈ (-π[,]π)) ∧ (abs‘(𝐾‘𝑠)) ≤ (abs‘(𝐾‘𝑐))) ∧ 𝑠 ∈ (-π[,]π)) → (abs‘(𝑈‘𝑠)) < ((abs‘(𝑎 · (𝐾‘𝑐))) + 1))
109	65, 70, 108	ltled 11285	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ℝ) ∧ (abs‘(𝐻‘𝑠)) ≤ 𝑎) ∧ 𝑐 ∈ (-π[,]π)) ∧ (abs‘(𝐾‘𝑠)) ≤ (abs‘(𝐾‘𝑐))) ∧ 𝑠 ∈ (-π[,]π)) → (abs‘(𝑈‘𝑠)) ≤ ((abs‘(𝑎 · (𝐾‘𝑐))) + 1))
110	43, 45, 46, 109	syl21anc 843	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℝ ∧ ∀𝑠 ∈ (-π[,]π)(abs‘(𝐻‘𝑠)) ≤ 𝑎) ∧ 𝑐 ∈ (-π[,]π) ∧ ∀𝑠 ∈ (-π[,]π)(abs‘(𝐾‘𝑠)) ≤ (abs‘(𝐾‘𝑐))) ∧ 𝑠 ∈ (-π[,]π)) → (abs‘(𝑈‘𝑠)) ≤ ((abs‘(𝑎 · (𝐾‘𝑐))) + 1))
111	110	ex 413	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ ∧ ∀𝑠 ∈ (-π[,]π)(abs‘(𝐻‘𝑠)) ≤ 𝑎) ∧ 𝑐 ∈ (-π[,]π) ∧ ∀𝑠 ∈ (-π[,]π)(abs‘(𝐾‘𝑠)) ≤ (abs‘(𝐾‘𝑐))) → (𝑠 ∈ (-π[,]π) → (abs‘(𝑈‘𝑠)) ≤ ((abs‘(𝑎 · (𝐾‘𝑐))) + 1)))
112	35, 111	ralrimi 3237	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ ∧ ∀𝑠 ∈ (-π[,]π)(abs‘(𝐻‘𝑠)) ≤ 𝑎) ∧ 𝑐 ∈ (-π[,]π) ∧ ∀𝑠 ∈ (-π[,]π)(abs‘(𝐾‘𝑠)) ≤ (abs‘(𝐾‘𝑐))) → ∀𝑠 ∈ (-π[,]π)(abs‘(𝑈‘𝑠)) ≤ ((abs‘(𝑎 · (𝐾‘𝑐))) + 1))
113		breq2 5076	. . . . . . . 8 ⊢ (𝑏 = ((abs‘(𝑎 · (𝐾‘𝑐))) + 1) → ((abs‘(𝑈‘𝑠)) ≤ 𝑏 ↔ (abs‘(𝑈‘𝑠)) ≤ ((abs‘(𝑎 · (𝐾‘𝑐))) + 1)))
114	113	ralbidv 3162	. . . . . . 7 ⊢ (𝑏 = ((abs‘(𝑎 · (𝐾‘𝑐))) + 1) → (∀𝑠 ∈ (-π[,]π)(abs‘(𝑈‘𝑠)) ≤ 𝑏 ↔ ∀𝑠 ∈ (-π[,]π)(abs‘(𝑈‘𝑠)) ≤ ((abs‘(𝑎 · (𝐾‘𝑐))) + 1)))
115	114	rspcev 3560	. . . . . 6 ⊢ ((((abs‘(𝑎 · (𝐾‘𝑐))) + 1) ∈ ℝ⁺ ∧ ∀𝑠 ∈ (-π[,]π)(abs‘(𝑈‘𝑠)) ≤ ((abs‘(𝑎 · (𝐾‘𝑐))) + 1)) → ∃𝑏 ∈ ℝ⁺ ∀𝑠 ∈ (-π[,]π)(abs‘(𝑈‘𝑠)) ≤ 𝑏)
116	28, 112, 115	syl2anc 590	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ ∧ ∀𝑠 ∈ (-π[,]π)(abs‘(𝐻‘𝑠)) ≤ 𝑎) ∧ 𝑐 ∈ (-π[,]π) ∧ ∀𝑠 ∈ (-π[,]π)(abs‘(𝐾‘𝑠)) ≤ (abs‘(𝐾‘𝑐))) → ∃𝑏 ∈ ℝ⁺ ∀𝑠 ∈ (-π[,]π)(abs‘(𝑈‘𝑠)) ≤ 𝑏)
117	116	rexlimdv3a 3144	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ ∧ ∀𝑠 ∈ (-π[,]π)(abs‘(𝐻‘𝑠)) ≤ 𝑎) → (∃𝑐 ∈ (-π[,]π)∀𝑠 ∈ (-π[,]π)(abs‘(𝐾‘𝑠)) ≤ (abs‘(𝐾‘𝑐)) → ∃𝑏 ∈ ℝ⁺ ∀𝑠 ∈ (-π[,]π)(abs‘(𝑈‘𝑠)) ≤ 𝑏))
118	17, 117	mpd 15	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ ∧ ∀𝑠 ∈ (-π[,]π)(abs‘(𝐻‘𝑠)) ≤ 𝑎) → ∃𝑏 ∈ ℝ⁺ ∀𝑠 ∈ (-π[,]π)(abs‘(𝑈‘𝑠)) ≤ 𝑏)
119	118	rexlimdv3a 3144	. 2 ⊢ (𝜑 → (∃𝑎 ∈ ℝ ∀𝑠 ∈ (-π[,]π)(abs‘(𝐻‘𝑠)) ≤ 𝑎 → ∃𝑏 ∈ ℝ⁺ ∀𝑠 ∈ (-π[,]π)(abs‘(𝑈‘𝑠)) ≤ 𝑏))
120	1, 119	mpd 15	1 ⊢ (𝜑 → ∃𝑏 ∈ ℝ⁺ ∀𝑠 ∈ (-π[,]π)(abs‘(𝑈‘𝑠)) ≤ 𝑏)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1092 = wceq 1547 ⊤wtru 1548 ∈ wcel 2119 ∀wral 3053 ∃wrex 3063 ifcif 4454 class class class wbr 5072 ↦ cmpt 5153 ⟶wf 6481 ‘cfv 6485 (class class class)co 7356 ℂcc 11027 ℝcr 11028 0cc0 11029 1c1 11030 + caddc 11032 · cmul 11034 < clt 11170 ≤ cle 11171 − cmin 11368 -cneg 11369 / cdiv 11798 2c2 12227 ℝ⁺crp 12933 [,]cicc 13292 abscabs 15187 sincsin 16019 πcpi 16022 –cn→ccncf 24861

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5199 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 ax-inf2 9553 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-pre-sup 11107 ax-addf 11108

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-tp 4560 df-op 4562 df-uni 4839 df-int 4878 df-iun 4923 df-iin 4924 df-br 5073 df-opab 5135 df-mpt 5154 df-tr 5180 df-id 5513 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5571 df-se 5572 df-we 5573 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-pred 6252 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-isom 6494 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-of 7620 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8101 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-er 8633 df-map 8765 df-pm 8766 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9265 df-fi 9314 df-sup 9345 df-inf 9346 df-oi 9415 df-card 9854 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-7 12240 df-8 12241 df-9 12242 df-n0 12429 df-z 12516 df-dec 12636 df-uz 12780 df-q 12890 df-rp 12934 df-xneg 13054 df-xadd 13055 df-xmul 13056 df-ioo 13293 df-ioc 13294 df-ico 13295 df-icc 13296 df-fz 13453 df-fzo 13600 df-fl 13742 df-mod 13820 df-seq 13955 df-exp 14015 df-fac 14227 df-bc 14256 df-hash 14284 df-shft 15020 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-limsup 15424 df-clim 15441 df-rlim 15442 df-sum 15640 df-ef 16023 df-sin 16025 df-cos 16026 df-pi 16028 df-struct 17108 df-sets 17125 df-slot 17143 df-ndx 17155 df-base 17171 df-ress 17192 df-plusg 17224 df-mulr 17225 df-starv 17226 df-sca 17227 df-vsca 17228 df-ip 17229 df-tset 17230 df-ple 17231 df-ds 17233 df-unif 17234 df-hom 17235 df-cco 17236 df-rest 17376 df-topn 17377 df-0g 17395 df-gsum 17396 df-topgen 17397 df-pt 17398 df-prds 17401 df-xrs 17457 df-qtop 17462 df-imas 17463 df-xps 17465 df-mre 17539 df-mrc 17540 df-acs 17542 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-submnd 18743 df-mulg 19035 df-cntz 19283 df-cmn 19748 df-psmet 21339 df-xmet 21340 df-met 21341 df-bl 21342 df-mopn 21343 df-fbas 21344 df-fg 21345 df-cnfld 21348 df-top 22877 df-topon 22894 df-topsp 22916 df-bases 22929 df-cld 23002 df-ntr 23003 df-cls 23004 df-nei 23081 df-lp 23119 df-perf 23120 df-cn 23210 df-cnp 23211 df-t1 23297 df-haus 23298 df-cmp 23370 df-tx 23545 df-hmeo 23738 df-fil 23829 df-fm 23921 df-flim 23922 df-flf 23923 df-xms 24303 df-ms 24304 df-tms 24305 df-cncf 24863 df-limc 25851 df-dv 25852

This theorem is referenced by: fourierdlem87 46636

W3C validator