fourierdlem77 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem77 46295

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 26403	. . . . . . . . . 10 ⊢ π ∈ ℝ
3	2	renegcli 11432	. . . . . . . . 9 ⊢ -π ∈ ℝ
4	3	a1i 11	. . . . . . . 8 ⊢ (⊤ → -π ∈ ℝ)
5	2	a1i 11	. . . . . . . 8 ⊢ (⊤ → π ∈ ℝ)
6		pirp 26407	. . . . . . . . . . 11 ⊢ π ∈ ℝ⁺
7		neglt 12920	. . . . . . . . . . 11 ⊢ (π ∈ ℝ⁺ → -π < π)
8	6, 7	ax-mp 5	. . . . . . . . . 10 ⊢ -π < π
9	3, 2, 8	ltleii 11246	. . . . . . . . 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 46280	. . . . . . . . 9 ⊢ 𝐾 ∈ ((-π[,]π)–cn→ℝ)
13	12	a1i 11	. . . . . . . 8 ⊢ (⊤ → 𝐾 ∈ ((-π[,]π)–cn→ℝ))
14	4, 5, 10, 13	evthiccabs 45610	. . . . . . 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 1548	. . . . . 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 46262	. . . . . . . . . . . . . 14 ⊢ 𝐾:(-π[,]π)⟶ℝ
20	19	ffvelcdmi 7025	. . . . . . . . . . . . 13 ⊢ (𝑐 ∈ (-π[,]π) → (𝐾‘𝑐) ∈ ℝ)
21	20	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ ℝ ∧ 𝑐 ∈ (-π[,]π)) → (𝐾‘𝑐) ∈ ℝ)
22	18, 21	remulcld 11152	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ ℝ ∧ 𝑐 ∈ (-π[,]π)) → (𝑎 · (𝐾‘𝑐)) ∈ ℝ)
23	22	recnd 11150	. . . . . . . . . 10 ⊢ ((𝑎 ∈ ℝ ∧ 𝑐 ∈ (-π[,]π)) → (𝑎 · (𝐾‘𝑐)) ∈ ℂ)
24	23	abscld 15356	. . . . . . . . 9 ⊢ ((𝑎 ∈ ℝ ∧ 𝑐 ∈ (-π[,]π)) → (abs‘(𝑎 · (𝐾‘𝑐))) ∈ ℝ)
25	23	absge0d 15364	. . . . . . . . 9 ⊢ ((𝑎 ∈ ℝ ∧ 𝑐 ∈ (-π[,]π)) → 0 ≤ (abs‘(𝑎 · (𝐾‘𝑐))))
26	24, 25	ge0p1rpd 12974	. . . . . . . 8 ⊢ ((𝑎 ∈ ℝ ∧ 𝑐 ∈ (-π[,]π)) → ((abs‘(𝑎 · (𝐾‘𝑐))) + 1) ∈ ℝ⁺)
27	26	3ad2antl2 1187	. . . . . . 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 1915	. . . . . . . . 9 ⊢ Ⅎ𝑠𝜑
30		nfv 1915	. . . . . . . . 9 ⊢ Ⅎ𝑠 𝑎 ∈ ℝ
31		nfra1 3258	. . . . . . . . 9 ⊢ Ⅎ𝑠∀𝑠 ∈ (-π[,]π)(abs‘(𝐻‘𝑠)) ≤ 𝑎
32	29, 30, 31	nf3an 1902	. . . . . . . 8 ⊢ Ⅎ𝑠(𝜑 ∧ 𝑎 ∈ ℝ ∧ ∀𝑠 ∈ (-π[,]π)(abs‘(𝐻‘𝑠)) ≤ 𝑎)
33		nfv 1915	. . . . . . . 8 ⊢ Ⅎ𝑠 𝑐 ∈ (-π[,]π)
34		nfra1 3258	. . . . . . . 8 ⊢ Ⅎ𝑠∀𝑠 ∈ (-π[,]π)(abs‘(𝐾‘𝑠)) ≤ (abs‘(𝐾‘𝑐))
35	32, 33, 34	nf3an 1902	. . . . . . 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 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)) → 𝜑)
37		simpl12 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)) → 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 1251	. . . . . . . . . . 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 3223	. . . . . . . . . . 11 ⊢ ((∀𝑠 ∈ (-π[,]π)(abs‘(𝐻‘𝑠)) ≤ 𝑎 ∧ 𝑠 ∈ (-π[,]π)) → (abs‘(𝐻‘𝑠)) ≤ 𝑎)
41	39, 40	sylancom 588	. . . . . . . . . 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 1193	. . . . . . . . . 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 3223	. . . . . . . . . 10 ⊢ ((∀𝑠 ∈ (-π[,]π)(abs‘(𝐾‘𝑠)) ≤ (abs‘(𝐾‘𝑐)) ∧ 𝑠 ∈ (-π[,]π)) → (abs‘(𝐾‘𝑠)) ≤ (abs‘(𝐾‘𝑐)))
45	44	3ad2antl3 1188	. . . . . . . . 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 46228	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐻:(-π[,]π)⟶ℝ)
55	54	ffvelcdmda 7026	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑠 ∈ (-π[,]π)) → (𝐻‘𝑠) ∈ ℝ)
56	19	ffvelcdmi 7025	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 ∈ (-π[,]π) → (𝐾‘𝑠) ∈ ℝ)
57	56	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑠 ∈ (-π[,]π)) → (𝐾‘𝑠) ∈ ℝ)
58	55, 57	remulcld 11152	. . . . . . . . . . . . . . 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 6949	. . . . . . . . . . . . . . 15 ⊢ ((𝑠 ∈ (-π[,]π) ∧ ((𝐻‘𝑠) · (𝐾‘𝑠)) ∈ ℝ) → (𝑈‘𝑠) = ((𝐻‘𝑠) · (𝐾‘𝑠)))
61	48, 58, 60	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑠 ∈ (-π[,]π)) → (𝑈‘𝑠) = ((𝐻‘𝑠) · (𝐾‘𝑠)))
62	61, 58	eqeltrd 2833	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 ∈ (-π[,]π)) → (𝑈‘𝑠) ∈ ℝ)
63	62	recnd 11150	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ (-π[,]π)) → (𝑈‘𝑠) ∈ ℂ)
64	63	abscld 15356	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ (-π[,]π)) → (abs‘(𝑈‘𝑠)) ∈ ℝ)
65	47, 64	sylancom 588	. . . . . . . . . 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 584	. . . . . . . . . . 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 11296	. . . . . . . . . . 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 6835	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 ∈ (-π[,]π)) → (abs‘(𝑈‘𝑠)) = (abs‘((𝐻‘𝑠) · (𝐾‘𝑠))))
72	47, 71	sylancom 588	. . . . . . . . . . . 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 11150	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑠 ∈ (-π[,]π)) → (𝐻‘𝑠) ∈ ℂ)
74	73	abscld 15356	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑠 ∈ (-π[,]π)) → (abs‘(𝐻‘𝑠)) ∈ ℝ)
75	47, 74	sylancom 588	. . . . . . . . . . . . . 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 11106	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 ∈ ℝ → 𝑎 ∈ ℂ)
77	76	abscld 15356	. . . . . . . . . . . . . . 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 11150	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 ∈ (-π[,]π) → (𝐾‘𝑠) ∈ ℂ)
80	79	abscld 15356	. . . . . . . . . . . . . . 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 11150	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 ∈ (-π[,]π) → (𝐾‘𝑐) ∈ ℂ)
83	82	abscld 15356	. . . . . . . . . . . . . . 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 15364	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑠 ∈ (-π[,]π)) → 0 ≤ (abs‘(𝐻‘𝑠)))
86	47, 85	sylancom 588	. . . . . . . . . . . . . 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 15364	. . . . . . . . . . . . . . 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 752	. . . . . . . . . . . . . . . 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 731	. . . . . . . . . . . . . . . 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 15332	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℝ) ∧ (abs‘(𝐻‘𝑠)) ≤ 𝑎) ∧ 𝑠 ∈ (-π[,]π)) → 𝑎 ≤ (abs‘𝑎))
94	89, 90, 91, 92, 93	letrd 11280	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℝ) ∧ (abs‘(𝐻‘𝑠)) ≤ 𝑎) ∧ 𝑠 ∈ (-π[,]π)) → (abs‘(𝐻‘𝑠)) ≤ (abs‘𝑎))
95	94	ad4ant14 752	. . . . . . . . . . . . . 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 12075	. . . . . . . . . . . . 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 11150	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑠 ∈ (-π[,]π)) → (𝐾‘𝑠) ∈ ℂ)
99	73, 98	absmuld 15374	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑠 ∈ (-π[,]π)) → (abs‘((𝐻‘𝑠) · (𝐾‘𝑠))) = ((abs‘(𝐻‘𝑠)) · (abs‘(𝐾‘𝑠))))
100	47, 99	sylancom 588	. . . . . . . . . . . . 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 11150	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 ∈ ℝ ∧ 𝑐 ∈ (-π[,]π)) → (𝐾‘𝑐) ∈ ℂ)
103	101, 102	absmuld 15374	. . . . . . . . . . . . . 14 ⊢ ((𝑎 ∈ ℝ ∧ 𝑐 ∈ (-π[,]π)) → (abs‘(𝑎 · (𝐾‘𝑐))) = ((abs‘𝑎) · (abs‘(𝐾‘𝑐))))
104	66, 67, 103	syl2anc 584	. . . . . . . . . . . . 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 5127	. . . . . . . . . . . 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 5117	. . . . . . . . . . 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 12062	. . . . . . . . . . 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 11281	. . . . . . . . . 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 11271	. . . . . . . . 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 3232	. . . . . 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 5099	. . . . . . . 8 ⊢ (𝑏 = ((abs‘(𝑎 · (𝐾‘𝑐))) + 1) → ((abs‘(𝑈‘𝑠)) ≤ 𝑏 ↔ (abs‘(𝑈‘𝑠)) ≤ ((abs‘(𝑎 · (𝐾‘𝑐))) + 1)))
114	113	ralbidv 3157	. . . . . . 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 3574	. . . . . 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 584	. . . . 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 3139	. . . 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 3139	. 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 1086 = wceq 1541 ⊤wtru 1542 ∈ wcel 2113 ∀wral 3049 ∃wrex 3058 ifcif 4476 class class class wbr 5095 ↦ cmpt 5176 ⟶wf 6485 ‘cfv 6489 (class class class)co 7355 ℂcc 11014 ℝcr 11015 0cc0 11016 1c1 11017 + caddc 11019 · cmul 11021 < clt 11156 ≤ cle 11157 − cmin 11354 -cneg 11355 / cdiv 11784 2c2 12190 ℝ⁺crp 12900 [,]cicc 13258 abscabs 15151 sincsin 15980 πcpi 15983 –cn→ccncf 24806

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7677 ax-inf2 9541 ax-cnex 11072 ax-resscn 11073 ax-1cn 11074 ax-icn 11075 ax-addcl 11076 ax-addrcl 11077 ax-mulcl 11078 ax-mulrcl 11079 ax-mulcom 11080 ax-addass 11081 ax-mulass 11082 ax-distr 11083 ax-i2m1 11084 ax-1ne0 11085 ax-1rid 11086 ax-rnegex 11087 ax-rrecex 11088 ax-cnre 11089 ax-pre-lttri 11090 ax-pre-lttrn 11091 ax-pre-ltadd 11092 ax-pre-mulgt0 11093 ax-pre-sup 11094 ax-addf 11095

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2883 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3059 df-rmo 3348 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-uni 4861 df-int 4900 df-iun 4945 df-iin 4946 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-se 5575 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-isom 6498 df-riota 7312 df-ov 7358 df-oprab 7359 df-mpo 7360 df-of 7619 df-om 7806 df-1st 7930 df-2nd 7931 df-supp 8100 df-frecs 8220 df-wrecs 8251 df-recs 8300 df-rdg 8338 df-1o 8394 df-2o 8395 df-er 8631 df-map 8761 df-pm 8762 df-ixp 8831 df-en 8879 df-dom 8880 df-sdom 8881 df-fin 8882 df-fsupp 9256 df-fi 9305 df-sup 9336 df-inf 9337 df-oi 9406 df-card 9842 df-pnf 11158 df-mnf 11159 df-xr 11160 df-ltxr 11161 df-le 11162 df-sub 11356 df-neg 11357 df-div 11785 df-nn 12136 df-2 12198 df-3 12199 df-4 12200 df-5 12201 df-6 12202 df-7 12203 df-8 12204 df-9 12205 df-n0 12392 df-z 12479 df-dec 12599 df-uz 12743 df-q 12857 df-rp 12901 df-xneg 13021 df-xadd 13022 df-xmul 13023 df-ioo 13259 df-ioc 13260 df-ico 13261 df-icc 13262 df-fz 13418 df-fzo 13565 df-fl 13706 df-mod 13784 df-seq 13919 df-exp 13979 df-fac 14191 df-bc 14220 df-hash 14248 df-shft 14984 df-cj 15016 df-re 15017 df-im 15018 df-sqrt 15152 df-abs 15153 df-limsup 15388 df-clim 15405 df-rlim 15406 df-sum 15604 df-ef 15984 df-sin 15986 df-cos 15987 df-pi 15989 df-struct 17068 df-sets 17085 df-slot 17103 df-ndx 17115 df-base 17131 df-ress 17152 df-plusg 17184 df-mulr 17185 df-starv 17186 df-sca 17187 df-vsca 17188 df-ip 17189 df-tset 17190 df-ple 17191 df-ds 17193 df-unif 17194 df-hom 17195 df-cco 17196 df-rest 17336 df-topn 17337 df-0g 17355 df-gsum 17356 df-topgen 17357 df-pt 17358 df-prds 17361 df-xrs 17416 df-qtop 17421 df-imas 17422 df-xps 17424 df-mre 17498 df-mrc 17499 df-acs 17501 df-mgm 18558 df-sgrp 18637 df-mnd 18653 df-submnd 18702 df-mulg 18991 df-cntz 19239 df-cmn 19704 df-psmet 21293 df-xmet 21294 df-met 21295 df-bl 21296 df-mopn 21297 df-fbas 21298 df-fg 21299 df-cnfld 21302 df-top 22819 df-topon 22836 df-topsp 22858 df-bases 22871 df-cld 22944 df-ntr 22945 df-cls 22946 df-nei 23023 df-lp 23061 df-perf 23062 df-cn 23152 df-cnp 23153 df-t1 23239 df-haus 23240 df-cmp 23312 df-tx 23487 df-hmeo 23680 df-fil 23771 df-fm 23863 df-flim 23864 df-flf 23865 df-xms 24245 df-ms 24246 df-tms 24247 df-cncf 24808 df-limc 25804 df-dv 25805

This theorem is referenced by: fourierdlem87 46305

W3C validator