fourierdlem66 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem66 40873

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 3867	. . . . . . 7 ⊢ 𝐴 ⊆ ((-π[,]π) ∖ {0})
3		difss 3947	. . . . . . 7 ⊢ ((-π[,]π) ∖ {0}) ⊆ (-π[,]π)
4	2, 3	sstri 3818	. . . . . 6 ⊢ 𝐴 ⊆ (-π[,]π)
5	4	a1i 11	. . . . 5 ⊢ (𝜑 → 𝐴 ⊆ (-π[,]π))
6	5	sselda 3809	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 𝑠 ∈ (-π[,]π))
7	6	adantlr 697	. . 3 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → 𝑠 ∈ (-π[,]π))
8		fourierdlem66.f	. . . . . . . 8 ⊢ (𝜑 → 𝐹:ℝ⟶ℝ)
9	8	adantr 468	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐹:ℝ⟶ℝ)
10		fourierdlem66.x	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ ℝ)
11	10	adantr 468	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑋 ∈ ℝ)
12		fourierdlem66.y	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ ℝ)
13	12	adantr 468	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑌 ∈ ℝ)
14		fourierdlem66.w	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ ℝ)
15	14	adantr 468	. . . . . . 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 40862	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑈:(-π[,]π)⟶ℝ)
20	19	adantr 468	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → 𝑈:(-π[,]π)⟶ℝ)
21	20, 7	ffvelrnd 6589	. . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (𝑈‘𝑠) ∈ ℝ)
22		nnre 11319	. . . . . . 7 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
23		fourierdlem66.s	. . . . . . . 8 ⊢ 𝑆 = (𝑠 ∈ (-π[,]π) ↦ (sin‘((𝑛 + (1 / 2)) · 𝑠)))
24	23	fourierdlem5 40813	. . . . . . 7 ⊢ (𝑛 ∈ ℝ → 𝑆:(-π[,]π)⟶ℝ)
25	22, 24	syl 17	. . . . . 6 ⊢ (𝑛 ∈ ℕ → 𝑆:(-π[,]π)⟶ℝ)
26	25	ad2antlr 709	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → 𝑆:(-π[,]π)⟶ℝ)
27	26, 7	ffvelrnd 6589	. . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (𝑆‘𝑠) ∈ ℝ)
28	21, 27	remulcld 10362	. . 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 6519	. . 3 ⊢ ((𝑠 ∈ (-π[,]π) ∧ ((𝑈‘𝑠) · (𝑆‘𝑠)) ∈ ℝ) → (𝐺‘𝑠) = ((𝑈‘𝑠) · (𝑆‘𝑠)))
31	7, 28, 30	syl2anc 575	. 2 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (𝐺‘𝑠) = ((𝑈‘𝑠) · (𝑆‘𝑠)))
32	8, 10, 12, 14, 16	fourierdlem9 40817	. . . . . . . . 9 ⊢ (𝜑 → 𝐻:(-π[,]π)⟶ℝ)
33	32	adantr 468	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 𝐻:(-π[,]π)⟶ℝ)
34	33, 6	ffvelrnd 6589	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (𝐻‘𝑠) ∈ ℝ)
35	17	fourierdlem43 40851	. . . . . . . . 9 ⊢ 𝐾:(-π[,]π)⟶ℝ
36	35	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 𝐾:(-π[,]π)⟶ℝ)
37	36, 6	ffvelrnd 6589	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (𝐾‘𝑠) ∈ ℝ)
38	34, 37	remulcld 10362	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → ((𝐻‘𝑠) · (𝐾‘𝑠)) ∈ ℝ)
39	18	fvmpt2 6519	. . . . . 6 ⊢ ((𝑠 ∈ (-π[,]π) ∧ ((𝐻‘𝑠) · (𝐾‘𝑠)) ∈ ℝ) → (𝑈‘𝑠) = ((𝐻‘𝑠) · (𝐾‘𝑠)))
40	6, 38, 39	syl2anc 575	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (𝑈‘𝑠) = ((𝐻‘𝑠) · (𝐾‘𝑠)))
41		0red 10335	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 0 ∈ ℝ)
42	8	adantr 468	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 𝐹:ℝ⟶ℝ)
43	10	adantr 468	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 𝑋 ∈ ℝ)
44		pire 24435	. . . . . . . . . . . . . . . . 17 ⊢ π ∈ ℝ
45	44	renegcli 10634	. . . . . . . . . . . . . . . 16 ⊢ -π ∈ ℝ
46		iccssre 12480	. . . . . . . . . . . . . . . 16 ⊢ ((-π ∈ ℝ ∧ π ∈ ℝ) → (-π[,]π) ⊆ ℝ)
47	45, 44, 46	mp2an 675	. . . . . . . . . . . . . . 15 ⊢ (-π[,]π) ⊆ ℝ
48	4	sseli 3805	. . . . . . . . . . . . . . 15 ⊢ (𝑠 ∈ 𝐴 → 𝑠 ∈ (-π[,]π))
49	47, 48	sseldi 3807	. . . . . . . . . . . . . 14 ⊢ (𝑠 ∈ 𝐴 → 𝑠 ∈ ℝ)
50	49	adantl 469	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 𝑠 ∈ ℝ)
51	43, 50	readdcld 10361	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (𝑋 + 𝑠) ∈ ℝ)
52	42, 51	ffvelrnd 6589	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (𝐹‘(𝑋 + 𝑠)) ∈ ℝ)
53	12, 14	ifcld 4335	. . . . . . . . . . . 12 ⊢ (𝜑 → if(0 < 𝑠, 𝑌, 𝑊) ∈ ℝ)
54	53	adantr 468	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → if(0 < 𝑠, 𝑌, 𝑊) ∈ ℝ)
55	52, 54	resubcld 10750	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → ((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) ∈ ℝ)
56		simpr 473	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 𝑠 ∈ 𝐴)
57	2, 56	sseldi 3807	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 𝑠 ∈ ((-π[,]π) ∖ {0}))
58	57	eldifbd 3793	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → ¬ 𝑠 ∈ {0})
59		velsn 4397	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ {0} ↔ 𝑠 = 0)
60	58, 59	sylnib 319	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → ¬ 𝑠 = 0)
61	60	neqned 2996	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 𝑠 ≠ 0)
62	55, 50, 61	redivcld 11145	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠) ∈ ℝ)
63	41, 62	ifcld 4335	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)) ∈ ℝ)
64	16	fvmpt2 6519	. . . . . . . 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 575	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (𝐻‘𝑠) = if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)))
66	60	iffalsed 4301	. . . . . . 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 2851	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (𝐻‘𝑠) = (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠))
68		1red 10333	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 1 ∈ ℝ)
69		2re 11381	. . . . . . . . . . . 12 ⊢ 2 ∈ ℝ
70	69	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 2 ∈ ℝ)
71	50	rehalfcld 11553	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (𝑠 / 2) ∈ ℝ)
72	71	resincld 15100	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (sin‘(𝑠 / 2)) ∈ ℝ)
73	70, 72	remulcld 10362	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (2 · (sin‘(𝑠 / 2))) ∈ ℝ)
74		2cnd 11384	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 2 ∈ ℂ)
75	72	recnd 10360	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (sin‘(𝑠 / 2)) ∈ ℂ)
76		2ne0 11403	. . . . . . . . . . . 12 ⊢ 2 ≠ 0
77	76	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 2 ≠ 0)
78		fourierdlem44 40852	. . . . . . . . . . . 12 ⊢ ((𝑠 ∈ (-π[,]π) ∧ 𝑠 ≠ 0) → (sin‘(𝑠 / 2)) ≠ 0)
79	6, 61, 78	syl2anc 575	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (sin‘(𝑠 / 2)) ≠ 0)
80	74, 75, 77, 79	mulne0d 10971	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (2 · (sin‘(𝑠 / 2))) ≠ 0)
81	50, 73, 80	redivcld 11145	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (𝑠 / (2 · (sin‘(𝑠 / 2)))) ∈ ℝ)
82	68, 81	ifcld 4335	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2))))) ∈ ℝ)
83	17	fvmpt2 6519	. . . . . . . 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 575	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (𝐾‘𝑠) = if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2))))))
85	60	iffalsed 4301	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2))))) = (𝑠 / (2 · (sin‘(𝑠 / 2)))))
86	84, 85	eqtrd 2851	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (𝐾‘𝑠) = (𝑠 / (2 · (sin‘(𝑠 / 2)))))
87	67, 86	oveq12d 6899	. . . . 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 10360	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → ((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) ∈ ℂ)
89	50	recnd 10360	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 𝑠 ∈ ℂ)
90	74, 75	mulcld 10352	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (2 · (sin‘(𝑠 / 2))) ∈ ℂ)
91	88, 89, 90, 61, 80	dmdcan2d 11123	. . . . 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 2855	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (𝑈‘𝑠) = (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / (2 · (sin‘(𝑠 / 2)))))
93	92	adantlr 697	. . 3 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (𝑈‘𝑠) = (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / (2 · (sin‘(𝑠 / 2)))))
94	22	ad2antlr 709	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → 𝑛 ∈ ℝ)
95		1red 10333	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → 1 ∈ ℝ)
96	95	rehalfcld 11553	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (1 / 2) ∈ ℝ)
97	94, 96	readdcld 10361	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (𝑛 + (1 / 2)) ∈ ℝ)
98	49	adantl 469	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → 𝑠 ∈ ℝ)
99	97, 98	remulcld 10362	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → ((𝑛 + (1 / 2)) · 𝑠) ∈ ℝ)
100	99	resincld 15100	. . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (sin‘((𝑛 + (1 / 2)) · 𝑠)) ∈ ℝ)
101	23	fvmpt2 6519	. . . 4 ⊢ ((𝑠 ∈ (-π[,]π) ∧ (sin‘((𝑛 + (1 / 2)) · 𝑠)) ∈ ℝ) → (𝑆‘𝑠) = (sin‘((𝑛 + (1 / 2)) · 𝑠)))
102	7, 100, 101	syl2anc 575	. . 3 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (𝑆‘𝑠) = (sin‘((𝑛 + (1 / 2)) · 𝑠)))
103	93, 102	oveq12d 6899	. 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 697	. . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → ((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) ∈ ℂ)
105	90	adantlr 697	. . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (2 · (sin‘(𝑠 / 2))) ∈ ℂ)
106	100	recnd 10360	. . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (sin‘((𝑛 + (1 / 2)) · 𝑠)) ∈ ℂ)
107	80	adantlr 697	. . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (2 · (sin‘(𝑠 / 2))) ≠ 0)
108	104, 105, 106, 107	div32d 11116	. . 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 468	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → 𝑛 ∈ ℝ)
110		halfre 11520	. . . . . . . . . . . . . 14 ⊢ (1 / 2) ∈ ℝ
111	110	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → (1 / 2) ∈ ℝ)
112	109, 111	readdcld 10361	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → (𝑛 + (1 / 2)) ∈ ℝ)
113	49	adantl 469	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → 𝑠 ∈ ℝ)
114	112, 113	remulcld 10362	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → ((𝑛 + (1 / 2)) · 𝑠) ∈ ℝ)
115	114	resincld 15100	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → (sin‘((𝑛 + (1 / 2)) · 𝑠)) ∈ ℝ)
116	115	recnd 10360	. . . . . . . . 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 11553	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → (𝑠 / 2) ∈ ℝ)
119	118	resincld 15100	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → (sin‘(𝑠 / 2)) ∈ ℝ)
120	117, 119	remulcld 10362	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → (2 · (sin‘(𝑠 / 2))) ∈ ℝ)
121	120	recnd 10360	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → (2 · (sin‘(𝑠 / 2))) ∈ ℂ)
122		picn 24436	. . . . . . . . . 10 ⊢ π ∈ ℂ
123	122	a1i 11	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → π ∈ ℂ)
124		2cnd 11384	. . . . . . . . . . 11 ⊢ (𝑠 ∈ 𝐴 → 2 ∈ ℂ)
125		rehalfcl 11532	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ ℝ → (𝑠 / 2) ∈ ℝ)
126		resincl 15097	. . . . . . . . . . . . 13 ⊢ ((𝑠 / 2) ∈ ℝ → (sin‘(𝑠 / 2)) ∈ ℝ)
127	49, 125, 126	3syl 18	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ 𝐴 → (sin‘(𝑠 / 2)) ∈ ℝ)
128	127	recnd 10360	. . . . . . . . . . 11 ⊢ (𝑠 ∈ 𝐴 → (sin‘(𝑠 / 2)) ∈ ℂ)
129	76	a1i 11	. . . . . . . . . . 11 ⊢ (𝑠 ∈ 𝐴 → 2 ≠ 0)
130		eldifsni 4523	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ ((-π[,]π) ∖ {0}) → 𝑠 ≠ 0)
131	130, 1	eleq2s 2914	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ 𝐴 → 𝑠 ≠ 0)
132	48, 131, 78	syl2anc 575	. . . . . . . . . . 11 ⊢ (𝑠 ∈ 𝐴 → (sin‘(𝑠 / 2)) ≠ 0)
133	124, 128, 129, 132	mulne0d 10971	. . . . . . . . . 10 ⊢ (𝑠 ∈ 𝐴 → (2 · (sin‘(𝑠 / 2))) ≠ 0)
134	133	adantl 469	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → (2 · (sin‘(𝑠 / 2))) ≠ 0)
135		0re 10334	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ
136		pipos 24437	. . . . . . . . . . 11 ⊢ 0 < π
137	135, 136	gtneii 10441	. . . . . . . . . 10 ⊢ π ≠ 0
138	137	a1i 11	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → π ≠ 0)
139	116, 121, 123, 134, 138	divdiv1d 11124	. . . . . . . 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 11384	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → 2 ∈ ℂ)
141	128	adantl 469	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → (sin‘(𝑠 / 2)) ∈ ℂ)
142	140, 141, 123	mulassd 10355	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → ((2 · (sin‘(𝑠 / 2))) · π) = (2 · ((sin‘(𝑠 / 2)) · π)))
143	142	oveq2d 6897	. . . . . . . 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 10353	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → ((sin‘(𝑠 / 2)) · π) = (π · (sin‘(𝑠 / 2))))
145	144	oveq2d 6897	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → (2 · ((sin‘(𝑠 / 2)) · π)) = (2 · (π · (sin‘(𝑠 / 2)))))
146	140, 123, 141	mulassd 10355	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → ((2 · π) · (sin‘(𝑠 / 2))) = (2 · (π · (sin‘(𝑠 / 2)))))
147	145, 146	eqtr4d 2854	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → (2 · ((sin‘(𝑠 / 2)) · π)) = ((2 · π) · (sin‘(𝑠 / 2))))
148	147	oveq2d 6897	. . . . . . . 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 2855	. . . . . . 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 6897	. . . . . 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 11145	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → ((sin‘((𝑛 + (1 / 2)) · 𝑠)) / (2 · (sin‘(𝑠 / 2)))) ∈ ℝ)
152	151	recnd 10360	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → ((sin‘((𝑛 + (1 / 2)) · 𝑠)) / (2 · (sin‘(𝑠 / 2)))) ∈ ℂ)
153	152, 123, 138	divcan2d 11095	. . . . . 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 40795	. . . . . . . . 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 582	. . . . . . . 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 40832	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ ((-π[,]π) ∖ {0}) → (𝑠 mod (2 · π)) ≠ 0)
158	157, 1	eleq2s 2914	. . . . . . . . . . 11 ⊢ (𝑠 ∈ 𝐴 → (𝑠 mod (2 · π)) ≠ 0)
159	158	neneqd 2994	. . . . . . . . . 10 ⊢ (𝑠 ∈ 𝐴 → ¬ (𝑠 mod (2 · π)) = 0)
160	159	adantl 469	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → ¬ (𝑠 mod (2 · π)) = 0)
161	160	iffalsed 4301	. . . . . . . 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 2852	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → ((sin‘((𝑛 + (1 / 2)) · 𝑠)) / ((2 · π) · (sin‘(𝑠 / 2)))) = ((𝐷‘𝑛)‘𝑠))
163	162	oveq2d 6897	. . . . . 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 2859	. . . . 5 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → ((sin‘((𝑛 + (1 / 2)) · 𝑠)) / (2 · (sin‘(𝑠 / 2)))) = (π · ((𝐷‘𝑛)‘𝑠)))
165	164	oveq2d 6897	. . . 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 696	. . 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 40796	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ ℝ) → ((𝐷‘𝑛)‘𝑠) ∈ ℝ)
169	49, 168	sylan2 582	. . . . . 6 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → ((𝐷‘𝑛)‘𝑠) ∈ ℝ)
170	169	recnd 10360	. . . . 5 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ 𝐴) → ((𝐷‘𝑛)‘𝑠) ∈ ℂ)
171	170	adantll 696	. . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → ((𝐷‘𝑛)‘𝑠) ∈ ℂ)
172	104, 167, 171	mul12d 10537	. . 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 2855	. 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 2855	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 384 = wceq 1637 ∈ wcel 2157 ≠ wne 2989 ∖ cdif 3777 ⊆ wss 3780 ifcif 4290 {csn 4381 class class class wbr 4855 ↦ cmpt 4934 ⟶wf 6104 ‘cfv 6108 (class class class)co 6881 ℂcc 10226 ℝcr 10227 0cc0 10228 1c1 10229 + caddc 10231 · cmul 10233 < clt 10366 − cmin 10558 -cneg 10559 / cdiv 10976 ℕcn 11312 2c2 11363 [,]cicc 12403 mod cmo 12899 sincsin 15021 πcpi 15024

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1877 ax-4 1894 ax-5 2001 ax-6 2069 ax-7 2105 ax-8 2159 ax-9 2166 ax-10 2186 ax-11 2202 ax-12 2215 ax-13 2422 ax-ext 2795 ax-rep 4975 ax-sep 4986 ax-nul 4994 ax-pow 5046 ax-pr 5107 ax-un 7186 ax-inf2 8792 ax-cnex 10284 ax-resscn 10285 ax-1cn 10286 ax-icn 10287 ax-addcl 10288 ax-addrcl 10289 ax-mulcl 10290 ax-mulrcl 10291 ax-mulcom 10292 ax-addass 10293 ax-mulass 10294 ax-distr 10295 ax-i2m1 10296 ax-1ne0 10297 ax-1rid 10298 ax-rnegex 10299 ax-rrecex 10300 ax-cnre 10301 ax-pre-lttri 10302 ax-pre-lttrn 10303 ax-pre-ltadd 10304 ax-pre-mulgt0 10305 ax-pre-sup 10306 ax-addf 10307 ax-mulf 10308

This theorem depends on definitions: df-bi 198 df-an 385 df-or 866 df-3or 1101 df-3an 1102 df-tru 1641 df-fal 1651 df-ex 1860 df-nf 1864 df-sb 2062 df-mo 2635 df-eu 2642 df-clab 2804 df-cleq 2810 df-clel 2813 df-nfc 2948 df-ne 2990 df-nel 3093 df-ral 3112 df-rex 3113 df-reu 3114 df-rmo 3115 df-rab 3116 df-v 3404 df-sbc 3645 df-csb 3740 df-dif 3783 df-un 3785 df-in 3787 df-ss 3794 df-pss 3796 df-nul 4128 df-if 4291 df-pw 4364 df-sn 4382 df-pr 4384 df-tp 4386 df-op 4388 df-uni 4642 df-int 4681 df-iun 4725 df-iin 4726 df-br 4856 df-opab 4918 df-mpt 4935 df-tr 4958 df-id 5230 df-eprel 5235 df-po 5243 df-so 5244 df-fr 5281 df-se 5282 df-we 5283 df-xp 5328 df-rel 5329 df-cnv 5330 df-co 5331 df-dm 5332 df-rn 5333 df-res 5334 df-ima 5335 df-pred 5904 df-ord 5950 df-on 5951 df-lim 5952 df-suc 5953 df-iota 6071 df-fun 6110 df-fn 6111 df-f 6112 df-f1 6113 df-fo 6114 df-f1o 6115 df-fv 6116 df-isom 6117 df-riota 6842 df-ov 6884 df-oprab 6885 df-mpt2 6886 df-of 7134 df-om 7303 df-1st 7405 df-2nd 7406 df-supp 7537 df-wrecs 7649 df-recs 7711 df-rdg 7749 df-1o 7803 df-2o 7804 df-oadd 7807 df-er 7986 df-map 8101 df-pm 8102 df-ixp 8153 df-en 8200 df-dom 8201 df-sdom 8202 df-fin 8203 df-fsupp 8522 df-fi 8563 df-sup 8594 df-inf 8595 df-oi 8661 df-card 9055 df-cda 9282 df-pnf 10368 df-mnf 10369 df-xr 10370 df-ltxr 10371 df-le 10372 df-sub 10560 df-neg 10561 df-div 10977 df-nn 11313 df-2 11371 df-3 11372 df-4 11373 df-5 11374 df-6 11375 df-7 11376 df-8 11377 df-9 11378 df-n0 11567 df-z 11651 df-dec 11767 df-uz 11912 df-q 12015 df-rp 12054 df-xneg 12169 df-xadd 12170 df-xmul 12171 df-ioo 12404 df-ioc 12405 df-ico 12406 df-icc 12407 df-fz 12557 df-fzo 12697 df-fl 12824 df-mod 12900 df-seq 13032 df-exp 13091 df-fac 13288 df-bc 13317 df-hash 13345 df-shft 14037 df-cj 14069 df-re 14070 df-im 14071 df-sqrt 14205 df-abs 14206 df-limsup 14432 df-clim 14449 df-rlim 14450 df-sum 14647 df-ef 15025 df-sin 15027 df-cos 15028 df-pi 15030 df-struct 16077 df-ndx 16078 df-slot 16079 df-base 16081 df-sets 16082 df-ress 16083 df-plusg 16173 df-mulr 16174 df-starv 16175 df-sca 16176 df-vsca 16177 df-ip 16178 df-tset 16179 df-ple 16180 df-ds 16182 df-unif 16183 df-hom 16184 df-cco 16185 df-rest 16295 df-topn 16296 df-0g 16314 df-gsum 16315 df-topgen 16316 df-pt 16317 df-prds 16320 df-xrs 16374 df-qtop 16379 df-imas 16380 df-xps 16382 df-mre 16458 df-mrc 16459 df-acs 16461 df-mgm 17454 df-sgrp 17496 df-mnd 17507 df-submnd 17548 df-mulg 17753 df-cntz 17958 df-cmn 18403 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 20920 df-topon 20937 df-topsp 20959 df-bases 20972 df-cld 21045 df-ntr 21046 df-cls 21047 df-nei 21124 df-lp 21162 df-perf 21163 df-cn 21253 df-cnp 21254 df-haus 21341 df-tx 21587 df-hmeo 21780 df-fil 21871 df-fm 21963 df-flim 21964 df-flf 21965 df-xms 22346 df-ms 22347 df-tms 22348 df-cncf 22902 df-limc 23854 df-dv 23855

This theorem is referenced by: fourierdlem95 40902

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