fourierdlem66 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem66 43603

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 3975	. . . . . . 7 ⊢ 𝐴 ⊆ ((-π[,]π) ∖ {0})
3		difss 4062	. . . . . . 7 ⊢ ((-π[,]π) ∖ {0}) ⊆ (-π[,]π)
4	2, 3	sstri 3926	. . . . . 6 ⊢ 𝐴 ⊆ (-π[,]π)
5	4	a1i 11	. . . . 5 ⊢ (𝜑 → 𝐴 ⊆ (-π[,]π))
6	5	sselda 3917	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 𝑠 ∈ (-π[,]π))
7	6	adantlr 711	. . 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 43592	. . . . . 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	ffvelrnd 6944	. . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (𝑈‘𝑠) ∈ ℝ)
22		nnre 11910	. . . . . . 7 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
23		fourierdlem66.s	. . . . . . . 8 ⊢ 𝑆 = (𝑠 ∈ (-π[,]π) ↦ (sin‘((𝑛 + (1 / 2)) · 𝑠)))
24	23	fourierdlem5 43543	. . . . . . 7 ⊢ (𝑛 ∈ ℝ → 𝑆:(-π[,]π)⟶ℝ)
25	22, 24	syl 17	. . . . . 6 ⊢ (𝑛 ∈ ℕ → 𝑆:(-π[,]π)⟶ℝ)
26	25	ad2antlr 723	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → 𝑆:(-π[,]π)⟶ℝ)
27	26, 7	ffvelrnd 6944	. . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (𝑆‘𝑠) ∈ ℝ)
28	21, 27	remulcld 10936	. . 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 6868	. . 3 ⊢ ((𝑠 ∈ (-π[,]π) ∧ ((𝑈‘𝑠) · (𝑆‘𝑠)) ∈ ℝ) → (𝐺‘𝑠) = ((𝑈‘𝑠) · (𝑆‘𝑠)))
31	7, 28, 30	syl2anc 583	. 2 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (𝐺‘𝑠) = ((𝑈‘𝑠) · (𝑆‘𝑠)))
32	8, 10, 12, 14, 16	fourierdlem9 43547	. . . . . . . . 9 ⊢ (𝜑 → 𝐻:(-π[,]π)⟶ℝ)
33	32	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 𝐻:(-π[,]π)⟶ℝ)
34	33, 6	ffvelrnd 6944	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (𝐻‘𝑠) ∈ ℝ)
35	17	fourierdlem43 43581	. . . . . . . . 9 ⊢ 𝐾:(-π[,]π)⟶ℝ
36	35	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 𝐾:(-π[,]π)⟶ℝ)
37	36, 6	ffvelrnd 6944	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (𝐾‘𝑠) ∈ ℝ)
38	34, 37	remulcld 10936	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → ((𝐻‘𝑠) · (𝐾‘𝑠)) ∈ ℝ)
39	18	fvmpt2 6868	. . . . . 6 ⊢ ((𝑠 ∈ (-π[,]π) ∧ ((𝐻‘𝑠) · (𝐾‘𝑠)) ∈ ℝ) → (𝑈‘𝑠) = ((𝐻‘𝑠) · (𝐾‘𝑠)))
40	6, 38, 39	syl2anc 583	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (𝑈‘𝑠) = ((𝐻‘𝑠) · (𝐾‘𝑠)))
41		0red 10909	. . . . . . . . 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 25520	. . . . . . . . . . . . . . . . 17 ⊢ π ∈ ℝ
45	44	renegcli 11212	. . . . . . . . . . . . . . . 16 ⊢ -π ∈ ℝ
46		iccssre 13090	. . . . . . . . . . . . . . . 16 ⊢ ((-π ∈ ℝ ∧ π ∈ ℝ) → (-π[,]π) ⊆ ℝ)
47	45, 44, 46	mp2an 688	. . . . . . . . . . . . . . 15 ⊢ (-π[,]π) ⊆ ℝ
48	4	sseli 3913	. . . . . . . . . . . . . . 15 ⊢ (𝑠 ∈ 𝐴 → 𝑠 ∈ (-π[,]π))
49	47, 48	sselid 3915	. . . . . . . . . . . . . 14 ⊢ (𝑠 ∈ 𝐴 → 𝑠 ∈ ℝ)
50	49	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 𝑠 ∈ ℝ)
51	43, 50	readdcld 10935	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (𝑋 + 𝑠) ∈ ℝ)
52	42, 51	ffvelrnd 6944	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (𝐹‘(𝑋 + 𝑠)) ∈ ℝ)
53	12, 14	ifcld 4502	. . . . . . . . . . . 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 11333	. . . . . . . . . 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 3915	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 𝑠 ∈ ((-π[,]π) ∖ {0}))
58	57	eldifbd 3896	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → ¬ 𝑠 ∈ {0})
59		velsn 4574	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ {0} ↔ 𝑠 = 0)
60	58, 59	sylnib 327	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → ¬ 𝑠 = 0)
61	60	neqned 2949	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 𝑠 ≠ 0)
62	55, 50, 61	redivcld 11733	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠) ∈ ℝ)
63	41, 62	ifcld 4502	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)) ∈ ℝ)
64	16	fvmpt2 6868	. . . . . . . 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 583	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (𝐻‘𝑠) = if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)))
66	60	iffalsed 4467	. . . . . . 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 2778	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (𝐻‘𝑠) = (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠))
68		1red 10907	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 1 ∈ ℝ)
69		2re 11977	. . . . . . . . . . . 12 ⊢ 2 ∈ ℝ
70	69	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 2 ∈ ℝ)
71	50	rehalfcld 12150	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (𝑠 / 2) ∈ ℝ)
72	71	resincld 15780	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (sin‘(𝑠 / 2)) ∈ ℝ)
73	70, 72	remulcld 10936	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (2 · (sin‘(𝑠 / 2))) ∈ ℝ)
74		2cnd 11981	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 2 ∈ ℂ)
75	72	recnd 10934	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (sin‘(𝑠 / 2)) ∈ ℂ)
76		2ne0 12007	. . . . . . . . . . . 12 ⊢ 2 ≠ 0
77	76	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 2 ≠ 0)
78		fourierdlem44 43582	. . . . . . . . . . . 12 ⊢ ((𝑠 ∈ (-π[,]π) ∧ 𝑠 ≠ 0) → (sin‘(𝑠 / 2)) ≠ 0)
79	6, 61, 78	syl2anc 583	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (sin‘(𝑠 / 2)) ≠ 0)
80	74, 75, 77, 79	mulne0d 11557	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (2 · (sin‘(𝑠 / 2))) ≠ 0)
81	50, 73, 80	redivcld 11733	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (𝑠 / (2 · (sin‘(𝑠 / 2)))) ∈ ℝ)
82	68, 81	ifcld 4502	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2))))) ∈ ℝ)
83	17	fvmpt2 6868	. . . . . . . 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 583	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (𝐾‘𝑠) = if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2))))))
85	60	iffalsed 4467	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2))))) = (𝑠 / (2 · (sin‘(𝑠 / 2)))))
86	84, 85	eqtrd 2778	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (𝐾‘𝑠) = (𝑠 / (2 · (sin‘(𝑠 / 2)))))
87	67, 86	oveq12d 7273	. . . . 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 10934	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → ((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) ∈ ℂ)
89	50	recnd 10934	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 𝑠 ∈ ℂ)
90	74, 75	mulcld 10926	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (2 · (sin‘(𝑠 / 2))) ∈ ℂ)
91	88, 89, 90, 61, 80	dmdcan2d 11711	. . . . 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 2782	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (𝑈‘𝑠) = (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / (2 · (sin‘(𝑠 / 2)))))
93	92	adantlr 711	. . 3 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (𝑈‘𝑠) = (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / (2 · (sin‘(𝑠 / 2)))))
94	22	ad2antlr 723	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → 𝑛 ∈ ℝ)
95		1red 10907	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → 1 ∈ ℝ)
96	95	rehalfcld 12150	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (1 / 2) ∈ ℝ)
97	94, 96	readdcld 10935	. . . . . 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 10936	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → ((𝑛 + (1 / 2)) · 𝑠) ∈ ℝ)
100	99	resincld 15780	. . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (sin‘((𝑛 + (1 / 2)) · 𝑠)) ∈ ℝ)
101	23	fvmpt2 6868	. . . 4 ⊢ ((𝑠 ∈ (-π[,]π) ∧ (sin‘((𝑛 + (1 / 2)) · 𝑠)) ∈ ℝ) → (𝑆‘𝑠) = (sin‘((𝑛 + (1 / 2)) · 𝑠)))
102	7, 100, 101	syl2anc 583	. . 3 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (𝑆‘𝑠) = (sin‘((𝑛 + (1 / 2)) · 𝑠)))
103	93, 102	oveq12d 7273	. 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 711	. . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → ((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) ∈ ℂ)
105	90	adantlr 711	. . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (2 · (sin‘(𝑠 / 2))) ∈ ℂ)
106	100	recnd 10934	. . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (sin‘((𝑛 + (1 / 2)) · 𝑠)) ∈ ℂ)
107	80	adantlr 711	. . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (2 · (sin‘(𝑠 / 2))) ≠ 0)
108	104, 105, 106, 107	div32d 11704	. . 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 12117	. . . . . . . . . . . . . 14 ⊢ (1 / 2) ∈ ℝ
111	110	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → (1 / 2) ∈ ℝ)
112	109, 111	readdcld 10935	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → (𝑛 + (1 / 2)) ∈ ℝ)
113	49	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → 𝑠 ∈ ℝ)
114	112, 113	remulcld 10936	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → ((𝑛 + (1 / 2)) · 𝑠) ∈ ℝ)
115	114	resincld 15780	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → (sin‘((𝑛 + (1 / 2)) · 𝑠)) ∈ ℝ)
116	115	recnd 10934	. . . . . . . . 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 12150	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → (𝑠 / 2) ∈ ℝ)
119	118	resincld 15780	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → (sin‘(𝑠 / 2)) ∈ ℝ)
120	117, 119	remulcld 10936	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → (2 · (sin‘(𝑠 / 2))) ∈ ℝ)
121	120	recnd 10934	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → (2 · (sin‘(𝑠 / 2))) ∈ ℂ)
122		picn 25521	. . . . . . . . . 10 ⊢ π ∈ ℂ
123	122	a1i 11	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → π ∈ ℂ)
124		2cnd 11981	. . . . . . . . . . 11 ⊢ (𝑠 ∈ 𝐴 → 2 ∈ ℂ)
125		rehalfcl 12129	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ ℝ → (𝑠 / 2) ∈ ℝ)
126		resincl 15777	. . . . . . . . . . . . 13 ⊢ ((𝑠 / 2) ∈ ℝ → (sin‘(𝑠 / 2)) ∈ ℝ)
127	49, 125, 126	3syl 18	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ 𝐴 → (sin‘(𝑠 / 2)) ∈ ℝ)
128	127	recnd 10934	. . . . . . . . . . 11 ⊢ (𝑠 ∈ 𝐴 → (sin‘(𝑠 / 2)) ∈ ℂ)
129	76	a1i 11	. . . . . . . . . . 11 ⊢ (𝑠 ∈ 𝐴 → 2 ≠ 0)
130		eldifsni 4720	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ ((-π[,]π) ∖ {0}) → 𝑠 ≠ 0)
131	130, 1	eleq2s 2857	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ 𝐴 → 𝑠 ≠ 0)
132	48, 131, 78	syl2anc 583	. . . . . . . . . . 11 ⊢ (𝑠 ∈ 𝐴 → (sin‘(𝑠 / 2)) ≠ 0)
133	124, 128, 129, 132	mulne0d 11557	. . . . . . . . . 10 ⊢ (𝑠 ∈ 𝐴 → (2 · (sin‘(𝑠 / 2))) ≠ 0)
134	133	adantl 481	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → (2 · (sin‘(𝑠 / 2))) ≠ 0)
135		0re 10908	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ
136		pipos 25522	. . . . . . . . . . 11 ⊢ 0 < π
137	135, 136	gtneii 11017	. . . . . . . . . 10 ⊢ π ≠ 0
138	137	a1i 11	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → π ≠ 0)
139	116, 121, 123, 134, 138	divdiv1d 11712	. . . . . . . 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 11981	. . . . . . . . . 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 10929	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → ((2 · (sin‘(𝑠 / 2))) · π) = (2 · ((sin‘(𝑠 / 2)) · π)))
143	142	oveq2d 7271	. . . . . . . 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 10927	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → ((sin‘(𝑠 / 2)) · π) = (π · (sin‘(𝑠 / 2))))
145	144	oveq2d 7271	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → (2 · ((sin‘(𝑠 / 2)) · π)) = (2 · (π · (sin‘(𝑠 / 2)))))
146	140, 123, 141	mulassd 10929	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → ((2 · π) · (sin‘(𝑠 / 2))) = (2 · (π · (sin‘(𝑠 / 2)))))
147	145, 146	eqtr4d 2781	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → (2 · ((sin‘(𝑠 / 2)) · π)) = ((2 · π) · (sin‘(𝑠 / 2))))
148	147	oveq2d 7271	. . . . . . . 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 2782	. . . . . . 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 7271	. . . . . 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 11733	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → ((sin‘((𝑛 + (1 / 2)) · 𝑠)) / (2 · (sin‘(𝑠 / 2)))) ∈ ℝ)
152	151	recnd 10934	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → ((sin‘((𝑛 + (1 / 2)) · 𝑠)) / (2 · (sin‘(𝑠 / 2)))) ∈ ℂ)
153	152, 123, 138	divcan2d 11683	. . . . . 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 43525	. . . . . . . . 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 592	. . . . . . . 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 43562	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ ((-π[,]π) ∖ {0}) → (𝑠 mod (2 · π)) ≠ 0)
158	157, 1	eleq2s 2857	. . . . . . . . . . 11 ⊢ (𝑠 ∈ 𝐴 → (𝑠 mod (2 · π)) ≠ 0)
159	158	neneqd 2947	. . . . . . . . . 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 4467	. . . . . . . 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 2779	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → ((sin‘((𝑛 + (1 / 2)) · 𝑠)) / ((2 · π) · (sin‘(𝑠 / 2)))) = ((𝐷‘𝑛)‘𝑠))
163	162	oveq2d 7271	. . . . . 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 2786	. . . . 5 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → ((sin‘((𝑛 + (1 / 2)) · 𝑠)) / (2 · (sin‘(𝑠 / 2)))) = (π · ((𝐷‘𝑛)‘𝑠)))
165	164	oveq2d 7271	. . . 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 710	. . 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 43526	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ ℝ) → ((𝐷‘𝑛)‘𝑠) ∈ ℝ)
169	49, 168	sylan2 592	. . . . . 6 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → ((𝐷‘𝑛)‘𝑠) ∈ ℝ)
170	169	recnd 10934	. . . . 5 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → ((𝐷‘𝑛)‘𝑠) ∈ ℂ)
171	170	adantll 710	. . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → ((𝐷‘𝑛)‘𝑠) ∈ ℂ)
172	104, 167, 171	mul12d 11114	. . 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 2782	. 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 2782	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 1539 ∈ wcel 2108 ≠ wne 2942 ∖ cdif 3880 ⊆ wss 3883 ifcif 4456 {csn 4558 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 − cmin 11135 -cneg 11136 / cdiv 11562 ℕcn 11903 2c2 11958 [,]cicc 13011 mod cmo 13517 sincsin 15701 πcpi 15704

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-haus 22374 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: fourierdlem95 43632

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