fourierdlem77 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem77 43614

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 25520	. . . . . . . . . 10 ⊢ π ∈ ℝ
3	2	renegcli 11212	. . . . . . . . 9 ⊢ -π ∈ ℝ
4	3	a1i 11	. . . . . . . 8 ⊢ (⊤ → -π ∈ ℝ)
5	2	a1i 11	. . . . . . . 8 ⊢ (⊤ → π ∈ ℝ)
6		pirp 25523	. . . . . . . . . . 11 ⊢ π ∈ ℝ⁺
7		neglt 42712	. . . . . . . . . . 11 ⊢ (π ∈ ℝ⁺ → -π < π)
8	6, 7	ax-mp 5	. . . . . . . . . 10 ⊢ -π < π
9	3, 2, 8	ltleii 11028	. . . . . . . . 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 43599	. . . . . . . . 9 ⊢ 𝐾 ∈ ((-π[,]π)–cn→ℝ)
13	12	a1i 11	. . . . . . . 8 ⊢ (⊤ → 𝐾 ∈ ((-π[,]π)–cn→ℝ))
14	4, 5, 10, 13	evthiccabs 42924	. . . . . . 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 1546	. . . . . 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 483	. . . . 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 482	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ ℝ ∧ 𝑐 ∈ (-π[,]π)) → 𝑎 ∈ ℝ)
19	11	fourierdlem43 43581	. . . . . . . . . . . . . 14 ⊢ 𝐾:(-π[,]π)⟶ℝ
20	19	ffvelrni 6942	. . . . . . . . . . . . 13 ⊢ (𝑐 ∈ (-π[,]π) → (𝐾‘𝑐) ∈ ℝ)
21	20	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ ℝ ∧ 𝑐 ∈ (-π[,]π)) → (𝐾‘𝑐) ∈ ℝ)
22	18, 21	remulcld 10936	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ ℝ ∧ 𝑐 ∈ (-π[,]π)) → (𝑎 · (𝐾‘𝑐)) ∈ ℝ)
23	22	recnd 10934	. . . . . . . . . 10 ⊢ ((𝑎 ∈ ℝ ∧ 𝑐 ∈ (-π[,]π)) → (𝑎 · (𝐾‘𝑐)) ∈ ℂ)
24	23	abscld 15076	. . . . . . . . 9 ⊢ ((𝑎 ∈ ℝ ∧ 𝑐 ∈ (-π[,]π)) → (abs‘(𝑎 · (𝐾‘𝑐))) ∈ ℝ)
25	23	absge0d 15084	. . . . . . . . 9 ⊢ ((𝑎 ∈ ℝ ∧ 𝑐 ∈ (-π[,]π)) → 0 ≤ (abs‘(𝑎 · (𝐾‘𝑐))))
26	24, 25	ge0p1rpd 12731	. . . . . . . 8 ⊢ ((𝑎 ∈ ℝ ∧ 𝑐 ∈ (-π[,]π)) → ((abs‘(𝑎 · (𝐾‘𝑐))) + 1) ∈ ℝ⁺)
27	26	3ad2antl2 1184	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ ∧ ∀𝑠 ∈ (-π[,]π)(abs‘(𝐻‘𝑠)) ≤ 𝑎) ∧ 𝑐 ∈ (-π[,]π)) → ((abs‘(𝑎 · (𝐾‘𝑐))) + 1) ∈ ℝ⁺)
28	27	3adant3 1130	. . . . . 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 1918	. . . . . . . . 9 ⊢ Ⅎ𝑠𝜑
30		nfv 1918	. . . . . . . . 9 ⊢ Ⅎ𝑠 𝑎 ∈ ℝ
31		nfra1 3142	. . . . . . . . 9 ⊢ Ⅎ𝑠∀𝑠 ∈ (-π[,]π)(abs‘(𝐻‘𝑠)) ≤ 𝑎
32	29, 30, 31	nf3an 1905	. . . . . . . 8 ⊢ Ⅎ𝑠(𝜑 ∧ 𝑎 ∈ ℝ ∧ ∀𝑠 ∈ (-π[,]π)(abs‘(𝐻‘𝑠)) ≤ 𝑎)
33		nfv 1918	. . . . . . . 8 ⊢ Ⅎ𝑠 𝑐 ∈ (-π[,]π)
34		nfra1 3142	. . . . . . . 8 ⊢ Ⅎ𝑠∀𝑠 ∈ (-π[,]π)(abs‘(𝐾‘𝑠)) ≤ (abs‘(𝐾‘𝑐))
35	32, 33, 34	nf3an 1905	. . . . . . 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 1246	. . . . . . . . . . 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 1247	. . . . . . . . . . 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 511	. . . . . . . . . 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 1248	. . . . . . . . . . 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 3130	. . . . . . . . . . 11 ⊢ ((∀𝑠 ∈ (-π[,]π)(abs‘(𝐻‘𝑠)) ≤ 𝑎 ∧ 𝑠 ∈ (-π[,]π)) → (abs‘(𝐻‘𝑠)) ≤ 𝑎)
41	39, 40	sylancom 587	. . . . . . . . . 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 1190	. . . . . . . . . 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 514	. . . . . . . . 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 3130	. . . . . . . . . 10 ⊢ ((∀𝑠 ∈ (-π[,]π)(abs‘(𝐾‘𝑠)) ≤ (abs‘(𝐾‘𝑐)) ∧ 𝑠 ∈ (-π[,]π)) → (abs‘(𝐾‘𝑠)) ≤ (abs‘(𝐾‘𝑐)))
45	44	3ad2antl3 1185	. . . . . . . . 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 484	. . . . . . . . 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 781	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ℝ) ∧ (abs‘(𝐻‘𝑠)) ≤ 𝑎) ∧ 𝑐 ∈ (-π[,]π)) ∧ (abs‘(𝐾‘𝑠)) ≤ (abs‘(𝐾‘𝑐))) ∧ 𝑠 ∈ (-π[,]π)) → 𝜑)
48		simpr 484	. . . . . . . . . . . . . . 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 43547	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐻:(-π[,]π)⟶ℝ)
55	54	ffvelrnda 6943	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑠 ∈ (-π[,]π)) → (𝐻‘𝑠) ∈ ℝ)
56	19	ffvelrni 6942	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 ∈ (-π[,]π) → (𝐾‘𝑠) ∈ ℝ)
57	56	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑠 ∈ (-π[,]π)) → (𝐾‘𝑠) ∈ ℝ)
58	55, 57	remulcld 10936	. . . . . . . . . . . . . . 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 6868	. . . . . . . . . . . . . . 15 ⊢ ((𝑠 ∈ (-π[,]π) ∧ ((𝐻‘𝑠) · (𝐾‘𝑠)) ∈ ℝ) → (𝑈‘𝑠) = ((𝐻‘𝑠) · (𝐾‘𝑠)))
61	48, 58, 60	syl2anc 583	. . . . . . . . . . . . . 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 10934	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ (-π[,]π)) → (𝑈‘𝑠) ∈ ℂ)
64	63	abscld 15076	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ (-π[,]π)) → (abs‘(𝑈‘𝑠)) ∈ ℝ)
65	47, 64	sylancom 587	. . . . . . . . . 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 782	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ℝ) ∧ (abs‘(𝐻‘𝑠)) ≤ 𝑎) ∧ 𝑐 ∈ (-π[,]π)) ∧ (abs‘(𝐾‘𝑠)) ≤ (abs‘(𝐾‘𝑐))) ∧ 𝑠 ∈ (-π[,]π)) → 𝑎 ∈ ℝ)
67		simpllr 772	. . . . . . . . . . . 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 583	. . . . . . . . . . 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 11078	. . . . . . . . . . 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 6760	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 ∈ (-π[,]π)) → (abs‘(𝑈‘𝑠)) = (abs‘((𝐻‘𝑠) · (𝐾‘𝑠))))
72	47, 71	sylancom 587	. . . . . . . . . . . 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 10934	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑠 ∈ (-π[,]π)) → (𝐻‘𝑠) ∈ ℂ)
74	73	abscld 15076	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑠 ∈ (-π[,]π)) → (abs‘(𝐻‘𝑠)) ∈ ℝ)
75	47, 74	sylancom 587	. . . . . . . . . . . . . 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 10892	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 ∈ ℝ → 𝑎 ∈ ℂ)
77	76	abscld 15076	. . . . . . . . . . . . . . 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 10934	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 ∈ (-π[,]π) → (𝐾‘𝑠) ∈ ℂ)
80	79	abscld 15076	. . . . . . . . . . . . . . 15 ⊢ (𝑠 ∈ (-π[,]π) → (abs‘(𝐾‘𝑠)) ∈ ℝ)
81	80	adantl 481	. . . . . . . . . . . . . 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 10934	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 ∈ (-π[,]π) → (𝐾‘𝑐) ∈ ℂ)
83	82	abscld 15076	. . . . . . . . . . . . . . 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 15084	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑠 ∈ (-π[,]π)) → 0 ≤ (abs‘(𝐻‘𝑠)))
86	47, 85	sylancom 587	. . . . . . . . . . . . . 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 15084	. . . . . . . . . . . . . . 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 748	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℝ) ∧ (abs‘(𝐻‘𝑠)) ≤ 𝑎) ∧ 𝑠 ∈ (-π[,]π)) → (abs‘(𝐻‘𝑠)) ∈ ℝ)
90		simpllr 772	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℝ) ∧ (abs‘(𝐻‘𝑠)) ≤ 𝑎) ∧ 𝑠 ∈ (-π[,]π)) → 𝑎 ∈ ℝ)
91	77	ad3antlr 727	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℝ) ∧ (abs‘(𝐻‘𝑠)) ≤ 𝑎) ∧ 𝑠 ∈ (-π[,]π)) → (abs‘𝑎) ∈ ℝ)
92		simplr 765	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℝ) ∧ (abs‘(𝐻‘𝑠)) ≤ 𝑎) ∧ 𝑠 ∈ (-π[,]π)) → (abs‘(𝐻‘𝑠)) ≤ 𝑎)
93	90	leabsd 15054	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℝ) ∧ (abs‘(𝐻‘𝑠)) ≤ 𝑎) ∧ 𝑠 ∈ (-π[,]π)) → 𝑎 ≤ (abs‘𝑎))
94	89, 90, 91, 92, 93	letrd 11062	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℝ) ∧ (abs‘(𝐻‘𝑠)) ≤ 𝑎) ∧ 𝑠 ∈ (-π[,]π)) → (abs‘(𝐻‘𝑠)) ≤ (abs‘𝑎))
95	94	ad4ant14 748	. . . . . . . . . . . . . 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 765	. . . . . . . . . . . . . 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 11848	. . . . . . . . . . . . 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 10934	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑠 ∈ (-π[,]π)) → (𝐾‘𝑠) ∈ ℂ)
99	73, 98	absmuld 15094	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑠 ∈ (-π[,]π)) → (abs‘((𝐻‘𝑠) · (𝐾‘𝑠))) = ((abs‘(𝐻‘𝑠)) · (abs‘(𝐾‘𝑠))))
100	47, 99	sylancom 587	. . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 ∈ ℝ ∧ 𝑐 ∈ (-π[,]π)) → 𝑎 ∈ ℂ)
102	21	recnd 10934	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 ∈ ℝ ∧ 𝑐 ∈ (-π[,]π)) → (𝐾‘𝑐) ∈ ℂ)
103	101, 102	absmuld 15094	. . . . . . . . . . . . . 14 ⊢ ((𝑎 ∈ ℝ ∧ 𝑐 ∈ (-π[,]π)) → (abs‘(𝑎 · (𝐾‘𝑐))) = ((abs‘𝑎) · (abs‘(𝐾‘𝑐))))
104	66, 67, 103	syl2anc 583	. . . . . . . . . . . . 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 5102	. . . . . . . . . . . 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 5092	. . . . . . . . . . 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 11835	. . . . . . . . . . 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 11063	. . . . . . . . . 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 11053	. . . . . . . . 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 834	. . . . . . . 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 412	. . . . . . 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 3139	. . . . . 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 5074	. . . . . . . 8 ⊢ (𝑏 = ((abs‘(𝑎 · (𝐾‘𝑐))) + 1) → ((abs‘(𝑈‘𝑠)) ≤ 𝑏 ↔ (abs‘(𝑈‘𝑠)) ≤ ((abs‘(𝑎 · (𝐾‘𝑐))) + 1)))
114	113	ralbidv 3120	. . . . . . 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 3552	. . . . . 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 583	. . . . 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 3214	. . . 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 3214	. 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 395 ∧ w3a 1085 = wceq 1539 ⊤wtru 1540 ∈ wcel 2108 ∀wral 3063 ∃wrex 3064 ifcif 4456 class class class wbr 5070 ↦ cmpt 5153 ⟶wf 6414 ‘cfv 6418 (class class class)co 7255 ℂcc 10800 ℝcr 10801 0cc0 10802 1c1 10803 + caddc 10805 · cmul 10807 < clt 10940 ≤ cle 10941 − cmin 11135 -cneg 11136 / cdiv 11562 2c2 11958 ℝ⁺crp 12659 [,]cicc 13011 abscabs 14873 sincsin 15701 πcpi 15704 –cn→ccncf 23945

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-inf2 9329 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880 ax-addf 10881 ax-mulf 10882

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-om 7688 df-1st 7804 df-2nd 7805 df-supp 7949 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-2o 8268 df-er 8456 df-map 8575 df-pm 8576 df-ixp 8644 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fsupp 9059 df-fi 9100 df-sup 9131 df-inf 9132 df-oi 9199 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-z 12250 df-dec 12367 df-uz 12512 df-q 12618 df-rp 12660 df-xneg 12777 df-xadd 12778 df-xmul 12779 df-ioo 13012 df-ioc 13013 df-ico 13014 df-icc 13015 df-fz 13169 df-fzo 13312 df-fl 13440 df-mod 13518 df-seq 13650 df-exp 13711 df-fac 13916 df-bc 13945 df-hash 13973 df-shft 14706 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-limsup 15108 df-clim 15125 df-rlim 15126 df-sum 15326 df-ef 15705 df-sin 15707 df-cos 15708 df-pi 15710 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-starv 16903 df-sca 16904 df-vsca 16905 df-ip 16906 df-tset 16907 df-ple 16908 df-ds 16910 df-unif 16911 df-hom 16912 df-cco 16913 df-rest 17050 df-topn 17051 df-0g 17069 df-gsum 17070 df-topgen 17071 df-pt 17072 df-prds 17075 df-xrs 17130 df-qtop 17135 df-imas 17136 df-xps 17138 df-mre 17212 df-mrc 17213 df-acs 17215 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-submnd 18346 df-mulg 18616 df-cntz 18838 df-cmn 19303 df-psmet 20502 df-xmet 20503 df-met 20504 df-bl 20505 df-mopn 20506 df-fbas 20507 df-fg 20508 df-cnfld 20511 df-top 21951 df-topon 21968 df-topsp 21990 df-bases 22004 df-cld 22078 df-ntr 22079 df-cls 22080 df-nei 22157 df-lp 22195 df-perf 22196 df-cn 22286 df-cnp 22287 df-t1 22373 df-haus 22374 df-cmp 22446 df-tx 22621 df-hmeo 22814 df-fil 22905 df-fm 22997 df-flim 22998 df-flf 22999 df-xms 23381 df-ms 23382 df-tms 23383 df-cncf 23947 df-limc 24935 df-dv 24936

This theorem is referenced by: fourierdlem87 43624

W3C validator