fourierdlem77 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem77 46104

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 26518	. . . . . . . . . 10 ⊢ π ∈ ℝ
3	2	renegcli 11597	. . . . . . . . 9 ⊢ -π ∈ ℝ
4	3	a1i 11	. . . . . . . 8 ⊢ (⊤ → -π ∈ ℝ)
5	2	a1i 11	. . . . . . . 8 ⊢ (⊤ → π ∈ ℝ)
6		pirp 26521	. . . . . . . . . . 11 ⊢ π ∈ ℝ⁺
7		neglt 45199	. . . . . . . . . . 11 ⊢ (π ∈ ℝ⁺ → -π < π)
8	6, 7	ax-mp 5	. . . . . . . . . 10 ⊢ -π < π
9	3, 2, 8	ltleii 11413	. . . . . . . . 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 46089	. . . . . . . . 9 ⊢ 𝐾 ∈ ((-π[,]π)–cn→ℝ)
13	12	a1i 11	. . . . . . . 8 ⊢ (⊤ → 𝐾 ∈ ((-π[,]π)–cn→ℝ))
14	4, 5, 10, 13	evthiccabs 45414	. . . . . . 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 1544	. . . . . 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 46071	. . . . . . . . . . . . . 14 ⊢ 𝐾:(-π[,]π)⟶ℝ
20	19	ffvelcdmi 7117	. . . . . . . . . . . . 13 ⊢ (𝑐 ∈ (-π[,]π) → (𝐾‘𝑐) ∈ ℝ)
21	20	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ ℝ ∧ 𝑐 ∈ (-π[,]π)) → (𝐾‘𝑐) ∈ ℝ)
22	18, 21	remulcld 11320	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ ℝ ∧ 𝑐 ∈ (-π[,]π)) → (𝑎 · (𝐾‘𝑐)) ∈ ℝ)
23	22	recnd 11318	. . . . . . . . . 10 ⊢ ((𝑎 ∈ ℝ ∧ 𝑐 ∈ (-π[,]π)) → (𝑎 · (𝐾‘𝑐)) ∈ ℂ)
24	23	abscld 15485	. . . . . . . . 9 ⊢ ((𝑎 ∈ ℝ ∧ 𝑐 ∈ (-π[,]π)) → (abs‘(𝑎 · (𝐾‘𝑐))) ∈ ℝ)
25	23	absge0d 15493	. . . . . . . . 9 ⊢ ((𝑎 ∈ ℝ ∧ 𝑐 ∈ (-π[,]π)) → 0 ≤ (abs‘(𝑎 · (𝐾‘𝑐))))
26	24, 25	ge0p1rpd 13129	. . . . . . . 8 ⊢ ((𝑎 ∈ ℝ ∧ 𝑐 ∈ (-π[,]π)) → ((abs‘(𝑎 · (𝐾‘𝑐))) + 1) ∈ ℝ⁺)
27	26	3ad2antl2 1186	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ ∧ ∀𝑠 ∈ (-π[,]π)(abs‘(𝐻‘𝑠)) ≤ 𝑎) ∧ 𝑐 ∈ (-π[,]π)) → ((abs‘(𝑎 · (𝐾‘𝑐))) + 1) ∈ ℝ⁺)
28	27	3adant3 1132	. . . . . 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 1913	. . . . . . . . 9 ⊢ Ⅎ𝑠𝜑
30		nfv 1913	. . . . . . . . 9 ⊢ Ⅎ𝑠 𝑎 ∈ ℝ
31		nfra1 3290	. . . . . . . . 9 ⊢ Ⅎ𝑠∀𝑠 ∈ (-π[,]π)(abs‘(𝐻‘𝑠)) ≤ 𝑎
32	29, 30, 31	nf3an 1900	. . . . . . . 8 ⊢ Ⅎ𝑠(𝜑 ∧ 𝑎 ∈ ℝ ∧ ∀𝑠 ∈ (-π[,]π)(abs‘(𝐻‘𝑠)) ≤ 𝑎)
33		nfv 1913	. . . . . . . 8 ⊢ Ⅎ𝑠 𝑐 ∈ (-π[,]π)
34		nfra1 3290	. . . . . . . 8 ⊢ Ⅎ𝑠∀𝑠 ∈ (-π[,]π)(abs‘(𝐾‘𝑠)) ≤ (abs‘(𝐾‘𝑐))
35	32, 33, 34	nf3an 1900	. . . . . . 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 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)) → 𝜑)
37		simpl12 1249	. . . . . . . . . . 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 1250	. . . . . . . . . . 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 3254	. . . . . . . . . . 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 1192	. . . . . . . . . 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 3254	. . . . . . . . . 10 ⊢ ((∀𝑠 ∈ (-π[,]π)(abs‘(𝐾‘𝑠)) ≤ (abs‘(𝐾‘𝑐)) ∧ 𝑠 ∈ (-π[,]π)) → (abs‘(𝐾‘𝑠)) ≤ (abs‘(𝐾‘𝑐)))
45	44	3ad2antl3 1187	. . . . . . . . 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 784	. . . . . . . . . . 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 46037	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐻:(-π[,]π)⟶ℝ)
55	54	ffvelcdmda 7118	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑠 ∈ (-π[,]π)) → (𝐻‘𝑠) ∈ ℝ)
56	19	ffvelcdmi 7117	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 ∈ (-π[,]π) → (𝐾‘𝑠) ∈ ℝ)
57	56	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑠 ∈ (-π[,]π)) → (𝐾‘𝑠) ∈ ℝ)
58	55, 57	remulcld 11320	. . . . . . . . . . . . . . 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 7040	. . . . . . . . . . . . . . 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 2844	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 ∈ (-π[,]π)) → (𝑈‘𝑠) ∈ ℝ)
63	62	recnd 11318	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ (-π[,]π)) → (𝑈‘𝑠) ∈ ℂ)
64	63	abscld 15485	. . . . . . . . . . 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 785	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑎 ∈ ℝ) ∧ (abs‘(𝐻‘𝑠)) ≤ 𝑎) ∧ 𝑐 ∈ (-π[,]π)) ∧ (abs‘(𝐾‘𝑠)) ≤ (abs‘(𝐾‘𝑐))) ∧ 𝑠 ∈ (-π[,]π)) → 𝑎 ∈ ℝ)
67		simpllr 775	. . . . . . . . . . . 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 11463	. . . . . . . . . . 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 6924	. . . . . . . . . . . . 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 11318	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑠 ∈ (-π[,]π)) → (𝐻‘𝑠) ∈ ℂ)
74	73	abscld 15485	. . . . . . . . . . . . . . 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 11274	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 ∈ ℝ → 𝑎 ∈ ℂ)
77	76	abscld 15485	. . . . . . . . . . . . . . 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 11318	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 ∈ (-π[,]π) → (𝐾‘𝑠) ∈ ℂ)
80	79	abscld 15485	. . . . . . . . . . . . . . 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 11318	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 ∈ (-π[,]π) → (𝐾‘𝑐) ∈ ℂ)
83	82	abscld 15485	. . . . . . . . . . . . . . 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 15493	. . . . . . . . . . . . . . 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 15493	. . . . . . . . . . . . . . 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 751	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℝ) ∧ (abs‘(𝐻‘𝑠)) ≤ 𝑎) ∧ 𝑠 ∈ (-π[,]π)) → (abs‘(𝐻‘𝑠)) ∈ ℝ)
90		simpllr 775	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℝ) ∧ (abs‘(𝐻‘𝑠)) ≤ 𝑎) ∧ 𝑠 ∈ (-π[,]π)) → 𝑎 ∈ ℝ)
91	77	ad3antlr 730	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℝ) ∧ (abs‘(𝐻‘𝑠)) ≤ 𝑎) ∧ 𝑠 ∈ (-π[,]π)) → (abs‘𝑎) ∈ ℝ)
92		simplr 768	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℝ) ∧ (abs‘(𝐻‘𝑠)) ≤ 𝑎) ∧ 𝑠 ∈ (-π[,]π)) → (abs‘(𝐻‘𝑠)) ≤ 𝑎)
93	90	leabsd 15463	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℝ) ∧ (abs‘(𝐻‘𝑠)) ≤ 𝑎) ∧ 𝑠 ∈ (-π[,]π)) → 𝑎 ≤ (abs‘𝑎))
94	89, 90, 91, 92, 93	letrd 11447	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℝ) ∧ (abs‘(𝐻‘𝑠)) ≤ 𝑎) ∧ 𝑠 ∈ (-π[,]π)) → (abs‘(𝐻‘𝑠)) ≤ (abs‘𝑎))
95	94	ad4ant14 751	. . . . . . . . . . . . . 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 768	. . . . . . . . . . . . . 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 12238	. . . . . . . . . . . . 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 11318	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑠 ∈ (-π[,]π)) → (𝐾‘𝑠) ∈ ℂ)
99	73, 98	absmuld 15503	. . . . . . . . . . . . . 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 11318	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 ∈ ℝ ∧ 𝑐 ∈ (-π[,]π)) → (𝐾‘𝑐) ∈ ℂ)
103	101, 102	absmuld 15503	. . . . . . . . . . . . . 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 5198	. . . . . . . . . . . 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 5188	. . . . . . . . . . 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 12225	. . . . . . . . . . 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 11448	. . . . . . . . . 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 11438	. . . . . . . . 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 837	. . . . . . . 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 3263	. . . . . 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 5170	. . . . . . . 8 ⊢ (𝑏 = ((abs‘(𝑎 · (𝐾‘𝑐))) + 1) → ((abs‘(𝑈‘𝑠)) ≤ 𝑏 ↔ (abs‘(𝑈‘𝑠)) ≤ ((abs‘(𝑎 · (𝐾‘𝑐))) + 1)))
114	113	ralbidv 3184	. . . . . . 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 3635	. . . . . 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 3165	. . . 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 3165	. 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 1087 = wceq 1537 ⊤wtru 1538 ∈ wcel 2108 ∀wral 3067 ∃wrex 3076 ifcif 4548 class class class wbr 5166 ↦ cmpt 5249 ⟶wf 6569 ‘cfv 6573 (class class class)co 7448 ℂcc 11182 ℝcr 11183 0cc0 11184 1c1 11185 + caddc 11187 · cmul 11189 < clt 11324 ≤ cle 11325 − cmin 11520 -cneg 11521 / cdiv 11947 2c2 12348 ℝ⁺crp 13057 [,]cicc 13410 abscabs 15283 sincsin 16111 πcpi 16114 –cn→ccncf 24921

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-inf2 9710 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262 ax-addf 11263

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-om 7904 df-1st 8030 df-2nd 8031 df-supp 8202 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-er 8763 df-map 8886 df-pm 8887 df-ixp 8956 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fsupp 9432 df-fi 9480 df-sup 9511 df-inf 9512 df-oi 9579 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-z 12640 df-dec 12759 df-uz 12904 df-q 13014 df-rp 13058 df-xneg 13175 df-xadd 13176 df-xmul 13177 df-ioo 13411 df-ioc 13412 df-ico 13413 df-icc 13414 df-fz 13568 df-fzo 13712 df-fl 13843 df-mod 13921 df-seq 14053 df-exp 14113 df-fac 14323 df-bc 14352 df-hash 14380 df-shft 15116 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-limsup 15517 df-clim 15534 df-rlim 15535 df-sum 15735 df-ef 16115 df-sin 16117 df-cos 16118 df-pi 16120 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-mulr 17325 df-starv 17326 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 df-unif 17334 df-hom 17335 df-cco 17336 df-rest 17482 df-topn 17483 df-0g 17501 df-gsum 17502 df-topgen 17503 df-pt 17504 df-prds 17507 df-xrs 17562 df-qtop 17567 df-imas 17568 df-xps 17570 df-mre 17644 df-mrc 17645 df-acs 17647 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-submnd 18819 df-mulg 19108 df-cntz 19357 df-cmn 19824 df-psmet 21379 df-xmet 21380 df-met 21381 df-bl 21382 df-mopn 21383 df-fbas 21384 df-fg 21385 df-cnfld 21388 df-top 22921 df-topon 22938 df-topsp 22960 df-bases 22974 df-cld 23048 df-ntr 23049 df-cls 23050 df-nei 23127 df-lp 23165 df-perf 23166 df-cn 23256 df-cnp 23257 df-t1 23343 df-haus 23344 df-cmp 23416 df-tx 23591 df-hmeo 23784 df-fil 23875 df-fm 23967 df-flim 23968 df-flf 23969 df-xms 24351 df-ms 24352 df-tms 24353 df-cncf 24923 df-limc 25921 df-dv 25922

This theorem is referenced by: fourierdlem87 46114

W3C validator