fourierdlem66 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem66 40906

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 3808	. . . . . . 7 ⊢ 𝐴 ⊆ ((-π[,]π) ∖ {0})
3		difss 3888	. . . . . . 7 ⊢ ((-π[,]π) ∖ {0}) ⊆ (-π[,]π)
4	2, 3	sstri 3761	. . . . . 6 ⊢ 𝐴 ⊆ (-π[,]π)
5	4	a1i 11	. . . . 5 ⊢ (𝜑 → 𝐴 ⊆ (-π[,]π))
6	5	sselda 3752	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 𝑠 ∈ (-π[,]π))
7	6	adantlr 694	. . 3 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → 𝑠 ∈ (-π[,]π))
8		fourierdlem66.f	. . . . . . . 8 ⊢ (𝜑 → 𝐹:ℝ⟶ℝ)
9	8	adantr 466	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐹:ℝ⟶ℝ)
10		fourierdlem66.x	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ ℝ)
11	10	adantr 466	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑋 ∈ ℝ)
12		fourierdlem66.y	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ ℝ)
13	12	adantr 466	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑌 ∈ ℝ)
14		fourierdlem66.w	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ ℝ)
15	14	adantr 466	. . . . . . 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 40895	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑈:(-π[,]π)⟶ℝ)
20	19	adantr 466	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → 𝑈:(-π[,]π)⟶ℝ)
21	20, 7	ffvelrnd 6503	. . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (𝑈‘𝑠) ∈ ℝ)
22		nnre 11229	. . . . . . 7 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
23		fourierdlem66.s	. . . . . . . 8 ⊢ 𝑆 = (𝑠 ∈ (-π[,]π) ↦ (sin‘((𝑛 + (1 / 2)) · 𝑠)))
24	23	fourierdlem5 40846	. . . . . . 7 ⊢ (𝑛 ∈ ℝ → 𝑆:(-π[,]π)⟶ℝ)
25	22, 24	syl 17	. . . . . 6 ⊢ (𝑛 ∈ ℕ → 𝑆:(-π[,]π)⟶ℝ)
26	25	ad2antlr 706	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → 𝑆:(-π[,]π)⟶ℝ)
27	26, 7	ffvelrnd 6503	. . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (𝑆‘𝑠) ∈ ℝ)
28	21, 27	remulcld 10272	. . 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 6433	. . 3 ⊢ ((𝑠 ∈ (-π[,]π) ∧ ((𝑈‘𝑠) · (𝑆‘𝑠)) ∈ ℝ) → (𝐺‘𝑠) = ((𝑈‘𝑠) · (𝑆‘𝑠)))
31	7, 28, 30	syl2anc 573	. 2 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (𝐺‘𝑠) = ((𝑈‘𝑠) · (𝑆‘𝑠)))
32	8, 10, 12, 14, 16	fourierdlem9 40850	. . . . . . . . 9 ⊢ (𝜑 → 𝐻:(-π[,]π)⟶ℝ)
33	32	adantr 466	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 𝐻:(-π[,]π)⟶ℝ)
34	33, 6	ffvelrnd 6503	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (𝐻‘𝑠) ∈ ℝ)
35	17	fourierdlem43 40884	. . . . . . . . 9 ⊢ 𝐾:(-π[,]π)⟶ℝ
36	35	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 𝐾:(-π[,]π)⟶ℝ)
37	36, 6	ffvelrnd 6503	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (𝐾‘𝑠) ∈ ℝ)
38	34, 37	remulcld 10272	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → ((𝐻‘𝑠) · (𝐾‘𝑠)) ∈ ℝ)
39	18	fvmpt2 6433	. . . . . 6 ⊢ ((𝑠 ∈ (-π[,]π) ∧ ((𝐻‘𝑠) · (𝐾‘𝑠)) ∈ ℝ) → (𝑈‘𝑠) = ((𝐻‘𝑠) · (𝐾‘𝑠)))
40	6, 38, 39	syl2anc 573	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (𝑈‘𝑠) = ((𝐻‘𝑠) · (𝐾‘𝑠)))
41		0red 10243	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 0 ∈ ℝ)
42	8	adantr 466	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 𝐹:ℝ⟶ℝ)
43	10	adantr 466	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 𝑋 ∈ ℝ)
44		pire 24431	. . . . . . . . . . . . . . . . 17 ⊢ π ∈ ℝ
45	44	renegcli 10544	. . . . . . . . . . . . . . . 16 ⊢ -π ∈ ℝ
46		iccssre 12460	. . . . . . . . . . . . . . . 16 ⊢ ((-π ∈ ℝ ∧ π ∈ ℝ) → (-π[,]π) ⊆ ℝ)
47	45, 44, 46	mp2an 672	. . . . . . . . . . . . . . 15 ⊢ (-π[,]π) ⊆ ℝ
48	4	sseli 3748	. . . . . . . . . . . . . . 15 ⊢ (𝑠 ∈ 𝐴 → 𝑠 ∈ (-π[,]π))
49	47, 48	sseldi 3750	. . . . . . . . . . . . . 14 ⊢ (𝑠 ∈ 𝐴 → 𝑠 ∈ ℝ)
50	49	adantl 467	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 𝑠 ∈ ℝ)
51	43, 50	readdcld 10271	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (𝑋 + 𝑠) ∈ ℝ)
52	42, 51	ffvelrnd 6503	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (𝐹‘(𝑋 + 𝑠)) ∈ ℝ)
53	12, 14	ifcld 4270	. . . . . . . . . . . 12 ⊢ (𝜑 → if(0 < 𝑠, 𝑌, 𝑊) ∈ ℝ)
54	53	adantr 466	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → if(0 < 𝑠, 𝑌, 𝑊) ∈ ℝ)
55	52, 54	resubcld 10660	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → ((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) ∈ ℝ)
56		simpr 471	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 𝑠 ∈ 𝐴)
57	2, 56	sseldi 3750	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 𝑠 ∈ ((-π[,]π) ∖ {0}))
58	57	eldifbd 3736	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → ¬ 𝑠 ∈ {0})
59		velsn 4332	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ {0} ↔ 𝑠 = 0)
60	58, 59	sylnib 317	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → ¬ 𝑠 = 0)
61	60	neqned 2950	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 𝑠 ≠ 0)
62	55, 50, 61	redivcld 11055	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠) ∈ ℝ)
63	41, 62	ifcld 4270	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)) ∈ ℝ)
64	16	fvmpt2 6433	. . . . . . . 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 573	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (𝐻‘𝑠) = if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)))
66	60	iffalsed 4236	. . . . . . 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 2805	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (𝐻‘𝑠) = (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠))
68		1red 10257	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 1 ∈ ℝ)
69		2re 11292	. . . . . . . . . . . 12 ⊢ 2 ∈ ℝ
70	69	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 2 ∈ ℝ)
71	50	rehalfcld 11481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (𝑠 / 2) ∈ ℝ)
72	71	resincld 15079	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (sin‘(𝑠 / 2)) ∈ ℝ)
73	70, 72	remulcld 10272	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (2 · (sin‘(𝑠 / 2))) ∈ ℝ)
74		2cnd 11295	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 2 ∈ ℂ)
75	72	recnd 10270	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (sin‘(𝑠 / 2)) ∈ ℂ)
76		2ne0 11315	. . . . . . . . . . . 12 ⊢ 2 ≠ 0
77	76	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 2 ≠ 0)
78		fourierdlem44 40885	. . . . . . . . . . . 12 ⊢ ((𝑠 ∈ (-π[,]π) ∧ 𝑠 ≠ 0) → (sin‘(𝑠 / 2)) ≠ 0)
79	6, 61, 78	syl2anc 573	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (sin‘(𝑠 / 2)) ≠ 0)
80	74, 75, 77, 79	mulne0d 10881	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (2 · (sin‘(𝑠 / 2))) ≠ 0)
81	50, 73, 80	redivcld 11055	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (𝑠 / (2 · (sin‘(𝑠 / 2)))) ∈ ℝ)
82	68, 81	ifcld 4270	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2))))) ∈ ℝ)
83	17	fvmpt2 6433	. . . . . . . 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 573	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (𝐾‘𝑠) = if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2))))))
85	60	iffalsed 4236	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2))))) = (𝑠 / (2 · (sin‘(𝑠 / 2)))))
86	84, 85	eqtrd 2805	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (𝐾‘𝑠) = (𝑠 / (2 · (sin‘(𝑠 / 2)))))
87	67, 86	oveq12d 6811	. . . . 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 10270	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → ((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) ∈ ℂ)
89	50	recnd 10270	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 𝑠 ∈ ℂ)
90	74, 75	mulcld 10262	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (2 · (sin‘(𝑠 / 2))) ∈ ℂ)
91	88, 89, 90, 61, 80	dmdcan2d 11033	. . . . 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 2809	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (𝑈‘𝑠) = (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / (2 · (sin‘(𝑠 / 2)))))
93	92	adantlr 694	. . 3 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (𝑈‘𝑠) = (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / (2 · (sin‘(𝑠 / 2)))))
94	22	ad2antlr 706	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → 𝑛 ∈ ℝ)
95		1red 10257	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → 1 ∈ ℝ)
96	95	rehalfcld 11481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (1 / 2) ∈ ℝ)
97	94, 96	readdcld 10271	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (𝑛 + (1 / 2)) ∈ ℝ)
98	49	adantl 467	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → 𝑠 ∈ ℝ)
99	97, 98	remulcld 10272	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → ((𝑛 + (1 / 2)) · 𝑠) ∈ ℝ)
100	99	resincld 15079	. . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (sin‘((𝑛 + (1 / 2)) · 𝑠)) ∈ ℝ)
101	23	fvmpt2 6433	. . . 4 ⊢ ((𝑠 ∈ (-π[,]π) ∧ (sin‘((𝑛 + (1 / 2)) · 𝑠)) ∈ ℝ) → (𝑆‘𝑠) = (sin‘((𝑛 + (1 / 2)) · 𝑠)))
102	7, 100, 101	syl2anc 573	. . 3 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (𝑆‘𝑠) = (sin‘((𝑛 + (1 / 2)) · 𝑠)))
103	93, 102	oveq12d 6811	. 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 694	. . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → ((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) ∈ ℂ)
105	90	adantlr 694	. . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (2 · (sin‘(𝑠 / 2))) ∈ ℂ)
106	100	recnd 10270	. . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (sin‘((𝑛 + (1 / 2)) · 𝑠)) ∈ ℂ)
107	80	adantlr 694	. . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (2 · (sin‘(𝑠 / 2))) ≠ 0)
108	104, 105, 106, 107	div32d 11026	. . 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 466	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → 𝑛 ∈ ℝ)
110		halfre 11448	. . . . . . . . . . . . . 14 ⊢ (1 / 2) ∈ ℝ
111	110	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → (1 / 2) ∈ ℝ)
112	109, 111	readdcld 10271	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → (𝑛 + (1 / 2)) ∈ ℝ)
113	49	adantl 467	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → 𝑠 ∈ ℝ)
114	112, 113	remulcld 10272	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → ((𝑛 + (1 / 2)) · 𝑠) ∈ ℝ)
115	114	resincld 15079	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → (sin‘((𝑛 + (1 / 2)) · 𝑠)) ∈ ℝ)
116	115	recnd 10270	. . . . . . . . 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 11481	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → (𝑠 / 2) ∈ ℝ)
119	118	resincld 15079	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → (sin‘(𝑠 / 2)) ∈ ℝ)
120	117, 119	remulcld 10272	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → (2 · (sin‘(𝑠 / 2))) ∈ ℝ)
121	120	recnd 10270	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → (2 · (sin‘(𝑠 / 2))) ∈ ℂ)
122		picn 24432	. . . . . . . . . 10 ⊢ π ∈ ℂ
123	122	a1i 11	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → π ∈ ℂ)
124		2cnd 11295	. . . . . . . . . . 11 ⊢ (𝑠 ∈ 𝐴 → 2 ∈ ℂ)
125		rehalfcl 11460	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ ℝ → (𝑠 / 2) ∈ ℝ)
126		resincl 15076	. . . . . . . . . . . . 13 ⊢ ((𝑠 / 2) ∈ ℝ → (sin‘(𝑠 / 2)) ∈ ℝ)
127	49, 125, 126	3syl 18	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ 𝐴 → (sin‘(𝑠 / 2)) ∈ ℝ)
128	127	recnd 10270	. . . . . . . . . . 11 ⊢ (𝑠 ∈ 𝐴 → (sin‘(𝑠 / 2)) ∈ ℂ)
129	76	a1i 11	. . . . . . . . . . 11 ⊢ (𝑠 ∈ 𝐴 → 2 ≠ 0)
130		eldifsni 4457	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ ((-π[,]π) ∖ {0}) → 𝑠 ≠ 0)
131	130, 1	eleq2s 2868	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ 𝐴 → 𝑠 ≠ 0)
132	48, 131, 78	syl2anc 573	. . . . . . . . . . 11 ⊢ (𝑠 ∈ 𝐴 → (sin‘(𝑠 / 2)) ≠ 0)
133	124, 128, 129, 132	mulne0d 10881	. . . . . . . . . 10 ⊢ (𝑠 ∈ 𝐴 → (2 · (sin‘(𝑠 / 2))) ≠ 0)
134	133	adantl 467	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → (2 · (sin‘(𝑠 / 2))) ≠ 0)
135		0re 10242	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ
136		pipos 24433	. . . . . . . . . . 11 ⊢ 0 < π
137	135, 136	gtneii 10351	. . . . . . . . . 10 ⊢ π ≠ 0
138	137	a1i 11	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → π ≠ 0)
139	116, 121, 123, 134, 138	divdiv1d 11034	. . . . . . . 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 11295	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → 2 ∈ ℂ)
141	128	adantl 467	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → (sin‘(𝑠 / 2)) ∈ ℂ)
142	140, 141, 123	mulassd 10265	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → ((2 · (sin‘(𝑠 / 2))) · π) = (2 · ((sin‘(𝑠 / 2)) · π)))
143	142	oveq2d 6809	. . . . . . . 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 10263	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → ((sin‘(𝑠 / 2)) · π) = (π · (sin‘(𝑠 / 2))))
145	144	oveq2d 6809	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → (2 · ((sin‘(𝑠 / 2)) · π)) = (2 · (π · (sin‘(𝑠 / 2)))))
146	140, 123, 141	mulassd 10265	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → ((2 · π) · (sin‘(𝑠 / 2))) = (2 · (π · (sin‘(𝑠 / 2)))))
147	145, 146	eqtr4d 2808	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → (2 · ((sin‘(𝑠 / 2)) · π)) = ((2 · π) · (sin‘(𝑠 / 2))))
148	147	oveq2d 6809	. . . . . . . 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 2809	. . . . . . 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 6809	. . . . . 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 11055	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → ((sin‘((𝑛 + (1 / 2)) · 𝑠)) / (2 · (sin‘(𝑠 / 2)))) ∈ ℝ)
152	151	recnd 10270	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → ((sin‘((𝑛 + (1 / 2)) · 𝑠)) / (2 · (sin‘(𝑠 / 2)))) ∈ ℂ)
153	152, 123, 138	divcan2d 11005	. . . . . 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 40828	. . . . . . . . 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 580	. . . . . . . 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 40865	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ ((-π[,]π) ∖ {0}) → (𝑠 mod (2 · π)) ≠ 0)
158	157, 1	eleq2s 2868	. . . . . . . . . . 11 ⊢ (𝑠 ∈ 𝐴 → (𝑠 mod (2 · π)) ≠ 0)
159	158	neneqd 2948	. . . . . . . . . 10 ⊢ (𝑠 ∈ 𝐴 → ¬ (𝑠 mod (2 · π)) = 0)
160	159	adantl 467	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → ¬ (𝑠 mod (2 · π)) = 0)
161	160	iffalsed 4236	. . . . . . . 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 2806	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → ((sin‘((𝑛 + (1 / 2)) · 𝑠)) / ((2 · π) · (sin‘(𝑠 / 2)))) = ((𝐷‘𝑛)‘𝑠))
163	162	oveq2d 6809	. . . . . 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 2813	. . . . 5 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → ((sin‘((𝑛 + (1 / 2)) · 𝑠)) / (2 · (sin‘(𝑠 / 2)))) = (π · ((𝐷‘𝑛)‘𝑠)))
165	164	oveq2d 6809	. . . 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 693	. . 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 40829	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ ℝ) → ((𝐷‘𝑛)‘𝑠) ∈ ℝ)
169	49, 168	sylan2 580	. . . . . 6 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → ((𝐷‘𝑛)‘𝑠) ∈ ℝ)
170	169	recnd 10270	. . . . 5 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → ((𝐷‘𝑛)‘𝑠) ∈ ℂ)
171	170	adantll 693	. . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → ((𝐷‘𝑛)‘𝑠) ∈ ℂ)
172	104, 167, 171	mul12d 10447	. . 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 2809	. 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 2809	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 382 = wceq 1631 ∈ wcel 2145 ≠ wne 2943 ∖ cdif 3720 ⊆ wss 3723 ifcif 4225 {csn 4316 class class class wbr 4786 ↦ cmpt 4863 ⟶wf 6027 ‘cfv 6031 (class class class)co 6793 ℂcc 10136 ℝcr 10137 0cc0 10138 1c1 10139 + caddc 10141 · cmul 10143 < clt 10276 − cmin 10468 -cneg 10469 / cdiv 10886 ℕcn 11222 2c2 11272 [,]cicc 12383 mod cmo 12876 sincsin 15000 πcpi 15003

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 ax-inf2 8702 ax-cnex 10194 ax-resscn 10195 ax-1cn 10196 ax-icn 10197 ax-addcl 10198 ax-addrcl 10199 ax-mulcl 10200 ax-mulrcl 10201 ax-mulcom 10202 ax-addass 10203 ax-mulass 10204 ax-distr 10205 ax-i2m1 10206 ax-1ne0 10207 ax-1rid 10208 ax-rnegex 10209 ax-rrecex 10210 ax-cnre 10211 ax-pre-lttri 10212 ax-pre-lttrn 10213 ax-pre-ltadd 10214 ax-pre-mulgt0 10215 ax-pre-sup 10216 ax-addf 10217 ax-mulf 10218

This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-fal 1637 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-int 4612 df-iun 4656 df-iin 4657 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-se 5209 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-isom 6040 df-riota 6754 df-ov 6796 df-oprab 6797 df-mpt2 6798 df-of 7044 df-om 7213 df-1st 7315 df-2nd 7316 df-supp 7447 df-wrecs 7559 df-recs 7621 df-rdg 7659 df-1o 7713 df-2o 7714 df-oadd 7717 df-er 7896 df-map 8011 df-pm 8012 df-ixp 8063 df-en 8110 df-dom 8111 df-sdom 8112 df-fin 8113 df-fsupp 8432 df-fi 8473 df-sup 8504 df-inf 8505 df-oi 8571 df-card 8965 df-cda 9192 df-pnf 10278 df-mnf 10279 df-xr 10280 df-ltxr 10281 df-le 10282 df-sub 10470 df-neg 10471 df-div 10887 df-nn 11223 df-2 11281 df-3 11282 df-4 11283 df-5 11284 df-6 11285 df-7 11286 df-8 11287 df-9 11288 df-n0 11495 df-z 11580 df-dec 11696 df-uz 11889 df-q 11992 df-rp 12036 df-xneg 12151 df-xadd 12152 df-xmul 12153 df-ioo 12384 df-ioc 12385 df-ico 12386 df-icc 12387 df-fz 12534 df-fzo 12674 df-fl 12801 df-mod 12877 df-seq 13009 df-exp 13068 df-fac 13265 df-bc 13294 df-hash 13322 df-shft 14015 df-cj 14047 df-re 14048 df-im 14049 df-sqrt 14183 df-abs 14184 df-limsup 14410 df-clim 14427 df-rlim 14428 df-sum 14625 df-ef 15004 df-sin 15006 df-cos 15007 df-pi 15009 df-struct 16066 df-ndx 16067 df-slot 16068 df-base 16070 df-sets 16071 df-ress 16072 df-plusg 16162 df-mulr 16163 df-starv 16164 df-sca 16165 df-vsca 16166 df-ip 16167 df-tset 16168 df-ple 16169 df-ds 16172 df-unif 16173 df-hom 16174 df-cco 16175 df-rest 16291 df-topn 16292 df-0g 16310 df-gsum 16311 df-topgen 16312 df-pt 16313 df-prds 16316 df-xrs 16370 df-qtop 16375 df-imas 16376 df-xps 16378 df-mre 16454 df-mrc 16455 df-acs 16457 df-mgm 17450 df-sgrp 17492 df-mnd 17503 df-submnd 17544 df-mulg 17749 df-cntz 17957 df-cmn 18402 df-psmet 19953 df-xmet 19954 df-met 19955 df-bl 19956 df-mopn 19957 df-fbas 19958 df-fg 19959 df-cnfld 19962 df-top 20919 df-topon 20936 df-topsp 20958 df-bases 20971 df-cld 21044 df-ntr 21045 df-cls 21046 df-nei 21123 df-lp 21161 df-perf 21162 df-cn 21252 df-cnp 21253 df-haus 21340 df-tx 21586 df-hmeo 21779 df-fil 21870 df-fm 21962 df-flim 21963 df-flf 21964 df-xms 22345 df-ms 22346 df-tms 22347 df-cncf 22901 df-limc 23850 df-dv 23851

This theorem is referenced by: fourierdlem95 40935

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