fourierdlem66 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem66 46530

Description: Value of the 𝐺 function when the argument is not zero. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Hypotheses**
Ref	Expression
fourierdlem66.f	⊢ (𝜑 → 𝐹:ℝ⟶ℝ)
fourierdlem66.x	⊢ (𝜑 → 𝑋 ∈ ℝ)
fourierdlem66.y	⊢ (𝜑 → 𝑌 ∈ ℝ)
fourierdlem66.w	⊢ (𝜑 → 𝑊 ∈ ℝ)
fourierdlem66.d	⊢ 𝐷 = (𝑛 ∈ ℕ ↦ (𝑠 ∈ ℝ ↦ if((𝑠 mod (2 · π)) = 0, (((2 · 𝑛) + 1) / (2 · π)), ((sin‘((𝑛 + (1 / 2)) · 𝑠)) / ((2 · π) · (sin‘(𝑠 / 2)))))))
fourierdlem66.h	⊢ 𝐻 = (𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)))
fourierdlem66.k	⊢ 𝐾 = (𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2))))))
fourierdlem66.u	⊢ 𝑈 = (𝑠 ∈ (-π[,]π) ↦ ((𝐻‘𝑠) · (𝐾‘𝑠)))
fourierdlem66.s	⊢ 𝑆 = (𝑠 ∈ (-π[,]π) ↦ (sin‘((𝑛 + (1 / 2)) · 𝑠)))
fourierdlem66.g	⊢ 𝐺 = (𝑠 ∈ (-π[,]π) ↦ ((𝑈‘𝑠) · (𝑆‘𝑠)))
fourierdlem66.a	⊢ 𝐴 = ((-π[,]π) ∖ {0})

**Assertion**
Ref	Expression
fourierdlem66	⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (𝐺‘𝑠) = (π · (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) · ((𝐷‘𝑛)‘𝑠))))

Distinct variable groups: 𝑛,𝑠 𝜑,𝑠 Allowed substitution hints: 𝜑(𝑛) 𝐴(𝑛,𝑠) 𝐷(𝑛,𝑠) 𝑆(𝑛,𝑠) 𝑈(𝑛,𝑠) 𝐹(𝑛,𝑠) 𝐺(𝑛,𝑠) 𝐻(𝑛,𝑠) 𝐾(𝑛,𝑠) 𝑊(𝑛,𝑠) 𝑋(𝑛,𝑠) 𝑌(𝑛,𝑠)

**Proof of Theorem fourierdlem66**
Step	Hyp	Ref	Expression
1		fourierdlem66.a	. . . . . . . 8 ⊢ 𝐴 = ((-π[,]π) ∖ {0})
2	1	eqimssi 3996	. . . . . . 7 ⊢ 𝐴 ⊆ ((-π[,]π) ∖ {0})
3		difss 4090	. . . . . . 7 ⊢ ((-π[,]π) ∖ {0}) ⊆ (-π[,]π)
4	2, 3	sstri 3945	. . . . . 6 ⊢ 𝐴 ⊆ (-π[,]π)
5	4	a1i 11	. . . . 5 ⊢ (𝜑 → 𝐴 ⊆ (-π[,]π))
6	5	sselda 3935	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 𝑠 ∈ (-π[,]π))
7	6	adantlr 716	. . 3 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → 𝑠 ∈ (-π[,]π))
8		fourierdlem66.f	. . . . . . . 8 ⊢ (𝜑 → 𝐹:ℝ⟶ℝ)
9	8	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐹:ℝ⟶ℝ)
10		fourierdlem66.x	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ ℝ)
11	10	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑋 ∈ ℝ)
12		fourierdlem66.y	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ ℝ)
13	12	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑌 ∈ ℝ)
14		fourierdlem66.w	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ ℝ)
15	14	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑊 ∈ ℝ)
16		fourierdlem66.h	. . . . . . 7 ⊢ 𝐻 = (𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)))
17		fourierdlem66.k	. . . . . . 7 ⊢ 𝐾 = (𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2))))))
18		fourierdlem66.u	. . . . . . 7 ⊢ 𝑈 = (𝑠 ∈ (-π[,]π) ↦ ((𝐻‘𝑠) · (𝐾‘𝑠)))
19	9, 11, 13, 15, 16, 17, 18	fourierdlem55 46519	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑈:(-π[,]π)⟶ℝ)
20	19	adantr 480	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → 𝑈:(-π[,]π)⟶ℝ)
21	20, 7	ffvelcdmd 7039	. . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (𝑈‘𝑠) ∈ ℝ)
22		nnre 12164	. . . . . . 7 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
23		fourierdlem66.s	. . . . . . . 8 ⊢ 𝑆 = (𝑠 ∈ (-π[,]π) ↦ (sin‘((𝑛 + (1 / 2)) · 𝑠)))
24	23	fourierdlem5 46470	. . . . . . 7 ⊢ (𝑛 ∈ ℝ → 𝑆:(-π[,]π)⟶ℝ)
25	22, 24	syl 17	. . . . . 6 ⊢ (𝑛 ∈ ℕ → 𝑆:(-π[,]π)⟶ℝ)
26	25	ad2antlr 728	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → 𝑆:(-π[,]π)⟶ℝ)
27	26, 7	ffvelcdmd 7039	. . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (𝑆‘𝑠) ∈ ℝ)
28	21, 27	remulcld 11174	. . 3 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → ((𝑈‘𝑠) · (𝑆‘𝑠)) ∈ ℝ)
29		fourierdlem66.g	. . . 4 ⊢ 𝐺 = (𝑠 ∈ (-π[,]π) ↦ ((𝑈‘𝑠) · (𝑆‘𝑠)))
30	29	fvmpt2 6961	. . 3 ⊢ ((𝑠 ∈ (-π[,]π) ∧ ((𝑈‘𝑠) · (𝑆‘𝑠)) ∈ ℝ) → (𝐺‘𝑠) = ((𝑈‘𝑠) · (𝑆‘𝑠)))
31	7, 28, 30	syl2anc 585	. 2 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (𝐺‘𝑠) = ((𝑈‘𝑠) · (𝑆‘𝑠)))
32	8, 10, 12, 14, 16	fourierdlem9 46474	. . . . . . . . 9 ⊢ (𝜑 → 𝐻:(-π[,]π)⟶ℝ)
33	32	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 𝐻:(-π[,]π)⟶ℝ)
34	33, 6	ffvelcdmd 7039	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (𝐻‘𝑠) ∈ ℝ)
35	17	fourierdlem43 46508	. . . . . . . . 9 ⊢ 𝐾:(-π[,]π)⟶ℝ
36	35	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 𝐾:(-π[,]π)⟶ℝ)
37	36, 6	ffvelcdmd 7039	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (𝐾‘𝑠) ∈ ℝ)
38	34, 37	remulcld 11174	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → ((𝐻‘𝑠) · (𝐾‘𝑠)) ∈ ℝ)
39	18	fvmpt2 6961	. . . . . 6 ⊢ ((𝑠 ∈ (-π[,]π) ∧ ((𝐻‘𝑠) · (𝐾‘𝑠)) ∈ ℝ) → (𝑈‘𝑠) = ((𝐻‘𝑠) · (𝐾‘𝑠)))
40	6, 38, 39	syl2anc 585	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (𝑈‘𝑠) = ((𝐻‘𝑠) · (𝐾‘𝑠)))
41		0red 11147	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 0 ∈ ℝ)
42	8	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 𝐹:ℝ⟶ℝ)
43	10	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 𝑋 ∈ ℝ)
44		pire 26434	. . . . . . . . . . . . . . . . 17 ⊢ π ∈ ℝ
45	44	renegcli 11454	. . . . . . . . . . . . . . . 16 ⊢ -π ∈ ℝ
46		iccssre 13357	. . . . . . . . . . . . . . . 16 ⊢ ((-π ∈ ℝ ∧ π ∈ ℝ) → (-π[,]π) ⊆ ℝ)
47	45, 44, 46	mp2an 693	. . . . . . . . . . . . . . 15 ⊢ (-π[,]π) ⊆ ℝ
48	4	sseli 3931	. . . . . . . . . . . . . . 15 ⊢ (𝑠 ∈ 𝐴 → 𝑠 ∈ (-π[,]π))
49	47, 48	sselid 3933	. . . . . . . . . . . . . 14 ⊢ (𝑠 ∈ 𝐴 → 𝑠 ∈ ℝ)
50	49	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 𝑠 ∈ ℝ)
51	43, 50	readdcld 11173	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (𝑋 + 𝑠) ∈ ℝ)
52	42, 51	ffvelcdmd 7039	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (𝐹‘(𝑋 + 𝑠)) ∈ ℝ)
53	12, 14	ifcld 4528	. . . . . . . . . . . 12 ⊢ (𝜑 → if(0 < 𝑠, 𝑌, 𝑊) ∈ ℝ)
54	53	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → if(0 < 𝑠, 𝑌, 𝑊) ∈ ℝ)
55	52, 54	resubcld 11577	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → ((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) ∈ ℝ)
56		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 𝑠 ∈ 𝐴)
57	2, 56	sselid 3933	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 𝑠 ∈ ((-π[,]π) ∖ {0}))
58	57	eldifbd 3916	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → ¬ 𝑠 ∈ {0})
59		velsn 4598	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ {0} ↔ 𝑠 = 0)
60	58, 59	sylnib 328	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → ¬ 𝑠 = 0)
61	60	neqned 2940	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 𝑠 ≠ 0)
62	55, 50, 61	redivcld 11981	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠) ∈ ℝ)
63	41, 62	ifcld 4528	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)) ∈ ℝ)
64	16	fvmpt2 6961	. . . . . . . 8 ⊢ ((𝑠 ∈ (-π[,]π) ∧ if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)) ∈ ℝ) → (𝐻‘𝑠) = if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)))
65	6, 63, 64	syl2anc 585	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (𝐻‘𝑠) = if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)))
66	60	iffalsed 4492	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)) = (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠))
67	65, 66	eqtrd 2772	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (𝐻‘𝑠) = (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠))
68		1red 11145	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 1 ∈ ℝ)
69		2re 12231	. . . . . . . . . . . 12 ⊢ 2 ∈ ℝ
70	69	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 2 ∈ ℝ)
71	50	rehalfcld 12400	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (𝑠 / 2) ∈ ℝ)
72	71	resincld 16080	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (sin‘(𝑠 / 2)) ∈ ℝ)
73	70, 72	remulcld 11174	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (2 · (sin‘(𝑠 / 2))) ∈ ℝ)
74		2cnd 12235	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 2 ∈ ℂ)
75	72	recnd 11172	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (sin‘(𝑠 / 2)) ∈ ℂ)
76		2ne0 12261	. . . . . . . . . . . 12 ⊢ 2 ≠ 0
77	76	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 2 ≠ 0)
78		fourierdlem44 46509	. . . . . . . . . . . 12 ⊢ ((𝑠 ∈ (-π[,]π) ∧ 𝑠 ≠ 0) → (sin‘(𝑠 / 2)) ≠ 0)
79	6, 61, 78	syl2anc 585	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (sin‘(𝑠 / 2)) ≠ 0)
80	74, 75, 77, 79	mulne0d 11801	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (2 · (sin‘(𝑠 / 2))) ≠ 0)
81	50, 73, 80	redivcld 11981	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (𝑠 / (2 · (sin‘(𝑠 / 2)))) ∈ ℝ)
82	68, 81	ifcld 4528	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2))))) ∈ ℝ)
83	17	fvmpt2 6961	. . . . . . . 8 ⊢ ((𝑠 ∈ (-π[,]π) ∧ if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2))))) ∈ ℝ) → (𝐾‘𝑠) = if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2))))))
84	6, 82, 83	syl2anc 585	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (𝐾‘𝑠) = if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2))))))
85	60	iffalsed 4492	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2))))) = (𝑠 / (2 · (sin‘(𝑠 / 2)))))
86	84, 85	eqtrd 2772	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (𝐾‘𝑠) = (𝑠 / (2 · (sin‘(𝑠 / 2)))))
87	67, 86	oveq12d 7386	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → ((𝐻‘𝑠) · (𝐾‘𝑠)) = ((((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠) · (𝑠 / (2 · (sin‘(𝑠 / 2))))))
88	55	recnd 11172	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → ((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) ∈ ℂ)
89	50	recnd 11172	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 𝑠 ∈ ℂ)
90	74, 75	mulcld 11164	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (2 · (sin‘(𝑠 / 2))) ∈ ℂ)
91	88, 89, 90, 61, 80	dmdcan2d 11959	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → ((((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠) · (𝑠 / (2 · (sin‘(𝑠 / 2))))) = (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / (2 · (sin‘(𝑠 / 2)))))
92	40, 87, 91	3eqtrd 2776	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (𝑈‘𝑠) = (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / (2 · (sin‘(𝑠 / 2)))))
93	92	adantlr 716	. . 3 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (𝑈‘𝑠) = (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / (2 · (sin‘(𝑠 / 2)))))
94	22	ad2antlr 728	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → 𝑛 ∈ ℝ)
95		1red 11145	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → 1 ∈ ℝ)
96	95	rehalfcld 12400	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (1 / 2) ∈ ℝ)
97	94, 96	readdcld 11173	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (𝑛 + (1 / 2)) ∈ ℝ)
98	49	adantl 481	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → 𝑠 ∈ ℝ)
99	97, 98	remulcld 11174	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → ((𝑛 + (1 / 2)) · 𝑠) ∈ ℝ)
100	99	resincld 16080	. . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (sin‘((𝑛 + (1 / 2)) · 𝑠)) ∈ ℝ)
101	23	fvmpt2 6961	. . . 4 ⊢ ((𝑠 ∈ (-π[,]π) ∧ (sin‘((𝑛 + (1 / 2)) · 𝑠)) ∈ ℝ) → (𝑆‘𝑠) = (sin‘((𝑛 + (1 / 2)) · 𝑠)))
102	7, 100, 101	syl2anc 585	. . 3 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (𝑆‘𝑠) = (sin‘((𝑛 + (1 / 2)) · 𝑠)))
103	93, 102	oveq12d 7386	. 2 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → ((𝑈‘𝑠) · (𝑆‘𝑠)) = ((((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / (2 · (sin‘(𝑠 / 2)))) · (sin‘((𝑛 + (1 / 2)) · 𝑠))))
104	88	adantlr 716	. . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → ((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) ∈ ℂ)
105	90	adantlr 716	. . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (2 · (sin‘(𝑠 / 2))) ∈ ℂ)
106	100	recnd 11172	. . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (sin‘((𝑛 + (1 / 2)) · 𝑠)) ∈ ℂ)
107	80	adantlr 716	. . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (2 · (sin‘(𝑠 / 2))) ≠ 0)
108	104, 105, 106, 107	div32d 11952	. . 3 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → ((((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / (2 · (sin‘(𝑠 / 2)))) · (sin‘((𝑛 + (1 / 2)) · 𝑠))) = (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) · ((sin‘((𝑛 + (1 / 2)) · 𝑠)) / (2 · (sin‘(𝑠 / 2))))))
109	22	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → 𝑛 ∈ ℝ)
110		halfre 12366	. . . . . . . . . . . . . 14 ⊢ (1 / 2) ∈ ℝ
111	110	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → (1 / 2) ∈ ℝ)
112	109, 111	readdcld 11173	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → (𝑛 + (1 / 2)) ∈ ℝ)
113	49	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → 𝑠 ∈ ℝ)
114	112, 113	remulcld 11174	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → ((𝑛 + (1 / 2)) · 𝑠) ∈ ℝ)
115	114	resincld 16080	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → (sin‘((𝑛 + (1 / 2)) · 𝑠)) ∈ ℝ)
116	115	recnd 11172	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → (sin‘((𝑛 + (1 / 2)) · 𝑠)) ∈ ℂ)
117	69	a1i 11	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → 2 ∈ ℝ)
118	113	rehalfcld 12400	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → (𝑠 / 2) ∈ ℝ)
119	118	resincld 16080	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → (sin‘(𝑠 / 2)) ∈ ℝ)
120	117, 119	remulcld 11174	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → (2 · (sin‘(𝑠 / 2))) ∈ ℝ)
121	120	recnd 11172	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → (2 · (sin‘(𝑠 / 2))) ∈ ℂ)
122		picn 26435	. . . . . . . . . 10 ⊢ π ∈ ℂ
123	122	a1i 11	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → π ∈ ℂ)
124		2cnd 12235	. . . . . . . . . . 11 ⊢ (𝑠 ∈ 𝐴 → 2 ∈ ℂ)
125		rehalfcl 12380	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ ℝ → (𝑠 / 2) ∈ ℝ)
126		resincl 16077	. . . . . . . . . . . . 13 ⊢ ((𝑠 / 2) ∈ ℝ → (sin‘(𝑠 / 2)) ∈ ℝ)
127	49, 125, 126	3syl 18	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ 𝐴 → (sin‘(𝑠 / 2)) ∈ ℝ)
128	127	recnd 11172	. . . . . . . . . . 11 ⊢ (𝑠 ∈ 𝐴 → (sin‘(𝑠 / 2)) ∈ ℂ)
129	76	a1i 11	. . . . . . . . . . 11 ⊢ (𝑠 ∈ 𝐴 → 2 ≠ 0)
130		eldifsni 4748	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ ((-π[,]π) ∖ {0}) → 𝑠 ≠ 0)
131	130, 1	eleq2s 2855	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ 𝐴 → 𝑠 ≠ 0)
132	48, 131, 78	syl2anc 585	. . . . . . . . . . 11 ⊢ (𝑠 ∈ 𝐴 → (sin‘(𝑠 / 2)) ≠ 0)
133	124, 128, 129, 132	mulne0d 11801	. . . . . . . . . 10 ⊢ (𝑠 ∈ 𝐴 → (2 · (sin‘(𝑠 / 2))) ≠ 0)
134	133	adantl 481	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → (2 · (sin‘(𝑠 / 2))) ≠ 0)
135		0re 11146	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ
136		pipos 26436	. . . . . . . . . . 11 ⊢ 0 < π
137	135, 136	gtneii 11257	. . . . . . . . . 10 ⊢ π ≠ 0
138	137	a1i 11	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → π ≠ 0)
139	116, 121, 123, 134, 138	divdiv1d 11960	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → (((sin‘((𝑛 + (1 / 2)) · 𝑠)) / (2 · (sin‘(𝑠 / 2)))) / π) = ((sin‘((𝑛 + (1 / 2)) · 𝑠)) / ((2 · (sin‘(𝑠 / 2))) · π)))
140		2cnd 12235	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → 2 ∈ ℂ)
141	128	adantl 481	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → (sin‘(𝑠 / 2)) ∈ ℂ)
142	140, 141, 123	mulassd 11167	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → ((2 · (sin‘(𝑠 / 2))) · π) = (2 · ((sin‘(𝑠 / 2)) · π)))
143	142	oveq2d 7384	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → ((sin‘((𝑛 + (1 / 2)) · 𝑠)) / ((2 · (sin‘(𝑠 / 2))) · π)) = ((sin‘((𝑛 + (1 / 2)) · 𝑠)) / (2 · ((sin‘(𝑠 / 2)) · π))))
144	141, 123	mulcomd 11165	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → ((sin‘(𝑠 / 2)) · π) = (π · (sin‘(𝑠 / 2))))
145	144	oveq2d 7384	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → (2 · ((sin‘(𝑠 / 2)) · π)) = (2 · (π · (sin‘(𝑠 / 2)))))
146	140, 123, 141	mulassd 11167	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → ((2 · π) · (sin‘(𝑠 / 2))) = (2 · (π · (sin‘(𝑠 / 2)))))
147	145, 146	eqtr4d 2775	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → (2 · ((sin‘(𝑠 / 2)) · π)) = ((2 · π) · (sin‘(𝑠 / 2))))
148	147	oveq2d 7384	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → ((sin‘((𝑛 + (1 / 2)) · 𝑠)) / (2 · ((sin‘(𝑠 / 2)) · π))) = ((sin‘((𝑛 + (1 / 2)) · 𝑠)) / ((2 · π) · (sin‘(𝑠 / 2)))))
149	139, 143, 148	3eqtrd 2776	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → (((sin‘((𝑛 + (1 / 2)) · 𝑠)) / (2 · (sin‘(𝑠 / 2)))) / π) = ((sin‘((𝑛 + (1 / 2)) · 𝑠)) / ((2 · π) · (sin‘(𝑠 / 2)))))
150	149	oveq2d 7384	. . . . . 6 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → (π · (((sin‘((𝑛 + (1 / 2)) · 𝑠)) / (2 · (sin‘(𝑠 / 2)))) / π)) = (π · ((sin‘((𝑛 + (1 / 2)) · 𝑠)) / ((2 · π) · (sin‘(𝑠 / 2))))))
151	115, 120, 134	redivcld 11981	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → ((sin‘((𝑛 + (1 / 2)) · 𝑠)) / (2 · (sin‘(𝑠 / 2)))) ∈ ℝ)
152	151	recnd 11172	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → ((sin‘((𝑛 + (1 / 2)) · 𝑠)) / (2 · (sin‘(𝑠 / 2)))) ∈ ℂ)
153	152, 123, 138	divcan2d 11931	. . . . . 6 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → (π · (((sin‘((𝑛 + (1 / 2)) · 𝑠)) / (2 · (sin‘(𝑠 / 2)))) / π)) = ((sin‘((𝑛 + (1 / 2)) · 𝑠)) / (2 · (sin‘(𝑠 / 2)))))
154		fourierdlem66.d	. . . . . . . . . 10 ⊢ 𝐷 = (𝑛 ∈ ℕ ↦ (𝑠 ∈ ℝ ↦ if((𝑠 mod (2 · π)) = 0, (((2 · 𝑛) + 1) / (2 · π)), ((sin‘((𝑛 + (1 / 2)) · 𝑠)) / ((2 · π) · (sin‘(𝑠 / 2)))))))
155	154	dirkerval2 46452	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ ℝ) → ((𝐷‘𝑛)‘𝑠) = if((𝑠 mod (2 · π)) = 0, (((2 · 𝑛) + 1) / (2 · π)), ((sin‘((𝑛 + (1 / 2)) · 𝑠)) / ((2 · π) · (sin‘(𝑠 / 2))))))
156	49, 155	sylan2 594	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → ((𝐷‘𝑛)‘𝑠) = if((𝑠 mod (2 · π)) = 0, (((2 · 𝑛) + 1) / (2 · π)), ((sin‘((𝑛 + (1 / 2)) · 𝑠)) / ((2 · π) · (sin‘(𝑠 / 2))))))
157		fourierdlem24 46489	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ ((-π[,]π) ∖ {0}) → (𝑠 mod (2 · π)) ≠ 0)
158	157, 1	eleq2s 2855	. . . . . . . . . . 11 ⊢ (𝑠 ∈ 𝐴 → (𝑠 mod (2 · π)) ≠ 0)
159	158	neneqd 2938	. . . . . . . . . 10 ⊢ (𝑠 ∈ 𝐴 → ¬ (𝑠 mod (2 · π)) = 0)
160	159	adantl 481	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → ¬ (𝑠 mod (2 · π)) = 0)
161	160	iffalsed 4492	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → if((𝑠 mod (2 · π)) = 0, (((2 · 𝑛) + 1) / (2 · π)), ((sin‘((𝑛 + (1 / 2)) · 𝑠)) / ((2 · π) · (sin‘(𝑠 / 2))))) = ((sin‘((𝑛 + (1 / 2)) · 𝑠)) / ((2 · π) · (sin‘(𝑠 / 2)))))
162	156, 161	eqtr2d 2773	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → ((sin‘((𝑛 + (1 / 2)) · 𝑠)) / ((2 · π) · (sin‘(𝑠 / 2)))) = ((𝐷‘𝑛)‘𝑠))
163	162	oveq2d 7384	. . . . . 6 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → (π · ((sin‘((𝑛 + (1 / 2)) · 𝑠)) / ((2 · π) · (sin‘(𝑠 / 2))))) = (π · ((𝐷‘𝑛)‘𝑠)))
164	150, 153, 163	3eqtr3d 2780	. . . . 5 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → ((sin‘((𝑛 + (1 / 2)) · 𝑠)) / (2 · (sin‘(𝑠 / 2)))) = (π · ((𝐷‘𝑛)‘𝑠)))
165	164	oveq2d 7384	. . . 4 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) · ((sin‘((𝑛 + (1 / 2)) · 𝑠)) / (2 · (sin‘(𝑠 / 2))))) = (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) · (π · ((𝐷‘𝑛)‘𝑠))))
166	165	adantll 715	. . 3 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) · ((sin‘((𝑛 + (1 / 2)) · 𝑠)) / (2 · (sin‘(𝑠 / 2))))) = (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) · (π · ((𝐷‘𝑛)‘𝑠))))
167	122	a1i 11	. . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → π ∈ ℂ)
168	154	dirkerre 46453	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ ℝ) → ((𝐷‘𝑛)‘𝑠) ∈ ℝ)
169	49, 168	sylan2 594	. . . . . 6 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → ((𝐷‘𝑛)‘𝑠) ∈ ℝ)
170	169	recnd 11172	. . . . 5 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → ((𝐷‘𝑛)‘𝑠) ∈ ℂ)
171	170	adantll 715	. . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → ((𝐷‘𝑛)‘𝑠) ∈ ℂ)
172	104, 167, 171	mul12d 11354	. . 3 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) · (π · ((𝐷‘𝑛)‘𝑠))) = (π · (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) · ((𝐷‘𝑛)‘𝑠))))
173	108, 166, 172	3eqtrd 2776	. 2 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → ((((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / (2 · (sin‘(𝑠 / 2)))) · (sin‘((𝑛 + (1 / 2)) · 𝑠))) = (π · (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) · ((𝐷‘𝑛)‘𝑠))))
174	31, 103, 173	3eqtrd 2776	1 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (𝐺‘𝑠) = (π · (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) · ((𝐷‘𝑛)‘𝑠))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 ∖ cdif 3900 ⊆ wss 3903 ifcif 4481 {csn 4582 class class class wbr 5100 ↦ cmpt 5181 ⟶wf 6496 ‘cfv 6500 (class class class)co 7368 ℂcc 11036 ℝcr 11037 0cc0 11038 1c1 11039 + caddc 11041 · cmul 11043 < clt 11178 − cmin 11376 -cneg 11377 / cdiv 11806 ℕcn 12157 2c2 12212 [,]cicc 13276 mod cmo 13801 sincsin 15998 πcpi 16001

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-inf2 9562 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 ax-addf 11117

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-iin 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-se 5586 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-of 7632 df-om 7819 df-1st 7943 df-2nd 7944 df-supp 8113 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-2o 8408 df-er 8645 df-map 8777 df-pm 8778 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9277 df-fi 9326 df-sup 9357 df-inf 9358 df-oi 9427 df-card 9863 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-div 11807 df-nn 12158 df-2 12220 df-3 12221 df-4 12222 df-5 12223 df-6 12224 df-7 12225 df-8 12226 df-9 12227 df-n0 12414 df-z 12501 df-dec 12620 df-uz 12764 df-q 12874 df-rp 12918 df-xneg 13038 df-xadd 13039 df-xmul 13040 df-ioo 13277 df-ioc 13278 df-ico 13279 df-icc 13280 df-fz 13436 df-fzo 13583 df-fl 13724 df-mod 13802 df-seq 13937 df-exp 13997 df-fac 14209 df-bc 14238 df-hash 14266 df-shft 15002 df-cj 15034 df-re 15035 df-im 15036 df-sqrt 15170 df-abs 15171 df-limsup 15406 df-clim 15423 df-rlim 15424 df-sum 15622 df-ef 16002 df-sin 16004 df-cos 16005 df-pi 16007 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17149 df-ress 17170 df-plusg 17202 df-mulr 17203 df-starv 17204 df-sca 17205 df-vsca 17206 df-ip 17207 df-tset 17208 df-ple 17209 df-ds 17211 df-unif 17212 df-hom 17213 df-cco 17214 df-rest 17354 df-topn 17355 df-0g 17373 df-gsum 17374 df-topgen 17375 df-pt 17376 df-prds 17379 df-xrs 17435 df-qtop 17440 df-imas 17441 df-xps 17443 df-mre 17517 df-mrc 17518 df-acs 17520 df-mgm 18577 df-sgrp 18656 df-mnd 18672 df-submnd 18721 df-mulg 19010 df-cntz 19258 df-cmn 19723 df-psmet 21313 df-xmet 21314 df-met 21315 df-bl 21316 df-mopn 21317 df-fbas 21318 df-fg 21319 df-cnfld 21322 df-top 22850 df-topon 22867 df-topsp 22889 df-bases 22902 df-cld 22975 df-ntr 22976 df-cls 22977 df-nei 23054 df-lp 23092 df-perf 23093 df-cn 23183 df-cnp 23184 df-haus 23271 df-tx 23518 df-hmeo 23711 df-fil 23802 df-fm 23894 df-flim 23895 df-flf 23896 df-xms 24276 df-ms 24277 df-tms 24278 df-cncf 24839 df-limc 25835 df-dv 25836

This theorem is referenced by: fourierdlem95 46559

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