fourierdlem95 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem95 46247

Description: Algebraic manipulation of integrals, used by other lemmas. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Hypotheses**
Ref	Expression
fourierdlem95.f	⊢ (𝜑 → 𝐹:ℝ⟶ℝ)
fourierdlem95.xre	⊢ (𝜑 → 𝑋 ∈ ℝ)
fourierdlem95.p	⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = (-π + 𝑋) ∧ (𝑝‘𝑚) = (π + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
fourierdlem95.m	⊢ (𝜑 → 𝑀 ∈ ℕ)
fourierdlem95.v	⊢ (𝜑 → 𝑉 ∈ (𝑃‘𝑀))
fourierdlem95.x	⊢ (𝜑 → 𝑋 ∈ ran 𝑉)
fourierdlem95.fcn	⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) ∈ (((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))–cn→ℂ))
fourierdlem95.r	⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) lim_ℂ (𝑉‘𝑖)))
fourierdlem95.l	⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) lim_ℂ (𝑉‘(𝑖 + 1))))
fourierdlem95.h	⊢ 𝐻 = (𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)))
fourierdlem95.k	⊢ 𝐾 = (𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2))))))
fourierdlem95.u	⊢ 𝑈 = (𝑠 ∈ (-π[,]π) ↦ ((𝐻‘𝑠) · (𝐾‘𝑠)))
fourierdlem95.s	⊢ 𝑆 = (𝑠 ∈ (-π[,]π) ↦ (sin‘((𝑛 + (1 / 2)) · 𝑠)))
fourierdlem95.g	⊢ 𝐺 = (𝑠 ∈ (-π[,]π) ↦ ((𝑈‘𝑠) · (𝑆‘𝑠)))
fourierdlem95.i	⊢ 𝐼 = (ℝ D 𝐹)
fourierdlem95.ifn	⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐼 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))):((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))⟶ℝ)
fourierdlem95.b	⊢ (𝜑 → 𝐵 ∈ ((𝐼 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋))
fourierdlem95.c	⊢ (𝜑 → 𝐶 ∈ ((𝐼 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋))
fourierdlem95.y	⊢ (𝜑 → 𝑌 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋))
fourierdlem95.w	⊢ (𝜑 → 𝑊 ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋))
fourierdlem95.admvol	⊢ (𝜑 → 𝐴 ∈ dom vol)
fourierdlem95.ass	⊢ (𝜑 → 𝐴 ⊆ ((-π[,]π) ∖ {0}))
fourierlemenplusacver2eqitgdirker.e	⊢ 𝐸 = (𝑛 ∈ ℕ ↦ (∫𝐴(𝐺‘𝑠) d𝑠 / π))
fourierdlem95.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)))))))
fourierdlem95.o	⊢ (𝜑 → 𝑂 ∈ ℝ)
fourierdlem95.ifeqo	⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → if(0 < 𝑠, 𝑌, 𝑊) = 𝑂)
fourierdlem95.itgdirker	⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∫𝐴((𝐷‘𝑛)‘𝑠) d𝑠 = (1 / 2))

**Assertion**
Ref	Expression
fourierdlem95	⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐸‘𝑛) + (𝑂 / 2)) = ∫𝐴((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠)) d𝑠)

Distinct variable groups: 𝐴,𝑠 𝐵,𝑠 𝐶,𝑠 𝐷,𝑠 𝐹,𝑠 𝑖,𝐺,𝑠 𝐻,𝑠 𝐾,𝑠 𝐿,𝑠 𝑖,𝑀,𝑝,𝑚 𝑀,𝑠 𝑂,𝑠 𝑅,𝑠 𝑆,𝑠 𝑖,𝑉,𝑝 𝑉,𝑠 𝑊,𝑠 𝑖,𝑋,𝑝,𝑚 𝑋,𝑠 𝑌,𝑠 𝑖,𝑛,𝑠 𝜑,𝑖,𝑠 Allowed substitution hints: 𝜑(𝑚,𝑛,𝑝) 𝐴(𝑖,𝑚,𝑛,𝑝) 𝐵(𝑖,𝑚,𝑛,𝑝) 𝐶(𝑖,𝑚,𝑛,𝑝) 𝐷(𝑖,𝑚,𝑛,𝑝) 𝑃(𝑖,𝑚,𝑛,𝑠,𝑝) 𝑅(𝑖,𝑚,𝑛,𝑝) 𝑆(𝑖,𝑚,𝑛,𝑝) 𝑈(𝑖,𝑚,𝑛,𝑠,𝑝) 𝐸(𝑖,𝑚,𝑛,𝑠,𝑝) 𝐹(𝑖,𝑚,𝑛,𝑝) 𝐺(𝑚,𝑛,𝑝) 𝐻(𝑖,𝑚,𝑛,𝑝) 𝐼(𝑖,𝑚,𝑛,𝑠,𝑝) 𝐾(𝑖,𝑚,𝑛,𝑝) 𝐿(𝑖,𝑚,𝑛,𝑝) 𝑀(𝑛) 𝑂(𝑖,𝑚,𝑛,𝑝) 𝑉(𝑚,𝑛) 𝑊(𝑖,𝑚,𝑛,𝑝) 𝑋(𝑛) 𝑌(𝑖,𝑚,𝑛,𝑝)

Proof of Theorem fourierdlem95 Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
2		fourierdlem95.ass	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ⊆ ((-π[,]π) ∖ {0}))
3	2	difss2d 4086	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ⊆ (-π[,]π))
4	3	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴 ⊆ (-π[,]π))
5	4	sselda 3929	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → 𝑠 ∈ (-π[,]π))
6		fourierdlem95.f	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:ℝ⟶ℝ)
7	6	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐹:ℝ⟶ℝ)
8		fourierdlem95.xre	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ ℝ)
9	8	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑋 ∈ ℝ)
10		ioossre 13307	. . . . . . . . . . . . 13 ⊢ (𝑋(,)+∞) ⊆ ℝ
11	10	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑋(,)+∞) ⊆ ℝ)
12	6, 11	fssresd 6690	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹 ↾ (𝑋(,)+∞)):(𝑋(,)+∞)⟶ℝ)
13		ioosscn 13308	. . . . . . . . . . . 12 ⊢ (𝑋(,)+∞) ⊆ ℂ
14	13	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑋(,)+∞) ⊆ ℂ)
15		eqid 2731	. . . . . . . . . . . 12 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
16		pnfxr 11166	. . . . . . . . . . . . 13 ⊢ +∞ ∈ ℝ^*
17	16	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → +∞ ∈ ℝ^*)
18	8	ltpnfd 13020	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋 < +∞)
19	15, 17, 8, 18	lptioo1cn 45692	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ∈ ((limPt‘(TopOpen‘ℂ_fld))‘(𝑋(,)+∞)))
20		fourierdlem95.y	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑌 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋))
21	12, 14, 19, 20	limcrecl 45677	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ∈ ℝ)
22	21	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑌 ∈ ℝ)
23		ioossre 13307	. . . . . . . . . . . . 13 ⊢ (-∞(,)𝑋) ⊆ ℝ
24	23	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → (-∞(,)𝑋) ⊆ ℝ)
25	6, 24	fssresd 6690	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹 ↾ (-∞(,)𝑋)):(-∞(,)𝑋)⟶ℝ)
26		ioosscn 13308	. . . . . . . . . . . 12 ⊢ (-∞(,)𝑋) ⊆ ℂ
27	26	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → (-∞(,)𝑋) ⊆ ℂ)
28		mnfxr 11169	. . . . . . . . . . . . 13 ⊢ -∞ ∈ ℝ^*
29	28	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → -∞ ∈ ℝ^*)
30	8	mnfltd 13023	. . . . . . . . . . . 12 ⊢ (𝜑 → -∞ < 𝑋)
31	15, 29, 8, 30	lptioo2cn 45691	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ∈ ((limPt‘(TopOpen‘ℂ_fld))‘(-∞(,)𝑋)))
32		fourierdlem95.w	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑊 ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋))
33	25, 27, 31, 32	limcrecl 45677	. . . . . . . . . 10 ⊢ (𝜑 → 𝑊 ∈ ℝ)
34	33	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑊 ∈ ℝ)
35		fourierdlem95.h	. . . . . . . . 9 ⊢ 𝐻 = (𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)))
36		fourierdlem95.k	. . . . . . . . 9 ⊢ 𝐾 = (𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2))))))
37		fourierdlem95.u	. . . . . . . . 9 ⊢ 𝑈 = (𝑠 ∈ (-π[,]π) ↦ ((𝐻‘𝑠) · (𝐾‘𝑠)))
38	1	nnred 12140	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℝ)
39		fourierdlem95.s	. . . . . . . . 9 ⊢ 𝑆 = (𝑠 ∈ (-π[,]π) ↦ (sin‘((𝑛 + (1 / 2)) · 𝑠)))
40		fourierdlem95.g	. . . . . . . . 9 ⊢ 𝐺 = (𝑠 ∈ (-π[,]π) ↦ ((𝑈‘𝑠) · (𝑆‘𝑠)))
41	7, 9, 22, 34, 35, 36, 37, 38, 39, 40	fourierdlem67 46219	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐺:(-π[,]π)⟶ℝ)
42	41	ffvelcdmda 7017	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ (-π[,]π)) → (𝐺‘𝑠) ∈ ℝ)
43	5, 42	syldan 591	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (𝐺‘𝑠) ∈ ℝ)
44		fourierdlem95.admvol	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ dom vol)
45	44	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴 ∈ dom vol)
46	41	feqmptd 6890	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐺 = (𝑠 ∈ (-π[,]π) ↦ (𝐺‘𝑠)))
47		fourierdlem95.p	. . . . . . . . 9 ⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = (-π + 𝑋) ∧ (𝑝‘𝑚) = (π + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
48		fourierdlem95.x	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ ran 𝑉)
49	48	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑋 ∈ ran 𝑉)
50	20	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑌 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋))
51	32	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑊 ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋))
52		fourierdlem95.m	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℕ)
53	52	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑀 ∈ ℕ)
54		fourierdlem95.v	. . . . . . . . . 10 ⊢ (𝜑 → 𝑉 ∈ (𝑃‘𝑀))
55	54	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑉 ∈ (𝑃‘𝑀))
56		fourierdlem95.fcn	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) ∈ (((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))–cn→ℂ))
57	56	adantlr 715	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) ∈ (((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))–cn→ℂ))
58		fourierdlem95.r	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) lim_ℂ (𝑉‘𝑖)))
59	58	adantlr 715	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) lim_ℂ (𝑉‘𝑖)))
60		fourierdlem95.l	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) lim_ℂ (𝑉‘(𝑖 + 1))))
61	60	adantlr 715	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) lim_ℂ (𝑉‘(𝑖 + 1))))
62		fveq2 6822	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑖 → (𝑉‘𝑗) = (𝑉‘𝑖))
63	62	oveq1d 7361	. . . . . . . . . 10 ⊢ (𝑗 = 𝑖 → ((𝑉‘𝑗) − 𝑋) = ((𝑉‘𝑖) − 𝑋))
64	63	cbvmptv 5193	. . . . . . . . 9 ⊢ (𝑗 ∈ (0...𝑀) ↦ ((𝑉‘𝑗) − 𝑋)) = (𝑖 ∈ (0...𝑀) ↦ ((𝑉‘𝑖) − 𝑋))
65		eqid 2731	. . . . . . . . 9 ⊢ (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = -π ∧ (𝑝‘𝑚) = π) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = -π ∧ (𝑝‘𝑚) = π) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
66		fourierdlem95.i	. . . . . . . . 9 ⊢ 𝐼 = (ℝ D 𝐹)
67		fourierdlem95.ifn	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐼 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))):((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))⟶ℝ)
68	67	adantlr 715	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → (𝐼 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))):((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))⟶ℝ)
69		fourierdlem95.b	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ ((𝐼 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋))
70	69	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐵 ∈ ((𝐼 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋))
71		fourierdlem95.c	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ ((𝐼 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋))
72	71	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐶 ∈ ((𝐼 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋))
73	47, 7, 49, 50, 51, 35, 36, 37, 38, 39, 40, 53, 55, 57, 59, 61, 64, 65, 66, 68, 70, 72	fourierdlem88 46240	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐺 ∈ 𝐿¹)
74	46, 73	eqeltrrd 2832	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑠 ∈ (-π[,]π) ↦ (𝐺‘𝑠)) ∈ 𝐿¹)
75	4, 45, 42, 74	iblss 25733	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑠 ∈ 𝐴 ↦ (𝐺‘𝑠)) ∈ 𝐿¹)
76	43, 75	itgrecl 25726	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∫𝐴(𝐺‘𝑠) d𝑠 ∈ ℝ)
77		pire 26393	. . . . . 6 ⊢ π ∈ ℝ
78	77	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → π ∈ ℝ)
79		pipos 26395	. . . . . . 7 ⊢ 0 < π
80	77, 79	gt0ne0ii 11653	. . . . . 6 ⊢ π ≠ 0
81	80	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → π ≠ 0)
82	76, 78, 81	redivcld 11949	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (∫𝐴(𝐺‘𝑠) d𝑠 / π) ∈ ℝ)
83		fourierlemenplusacver2eqitgdirker.e	. . . . 5 ⊢ 𝐸 = (𝑛 ∈ ℕ ↦ (∫𝐴(𝐺‘𝑠) d𝑠 / π))
84	83	fvmpt2 6940	. . . 4 ⊢ ((𝑛 ∈ ℕ ∧ (∫𝐴(𝐺‘𝑠) d𝑠 / π) ∈ ℝ) → (𝐸‘𝑛) = (∫𝐴(𝐺‘𝑠) d𝑠 / π))
85	1, 82, 84	syl2anc 584	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐸‘𝑛) = (∫𝐴(𝐺‘𝑠) d𝑠 / π))
86		fourierdlem95.o	. . . . . . 7 ⊢ (𝜑 → 𝑂 ∈ ℝ)
87	86	recnd 11140	. . . . . 6 ⊢ (𝜑 → 𝑂 ∈ ℂ)
88		2cnd 12203	. . . . . 6 ⊢ (𝜑 → 2 ∈ ℂ)
89		2ne0 12229	. . . . . . 7 ⊢ 2 ≠ 0
90	89	a1i 11	. . . . . 6 ⊢ (𝜑 → 2 ≠ 0)
91	87, 88, 90	divrecd 11900	. . . . 5 ⊢ (𝜑 → (𝑂 / 2) = (𝑂 · (1 / 2)))
92	91	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑂 / 2) = (𝑂 · (1 / 2)))
93		fourierdlem95.itgdirker	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∫𝐴((𝐷‘𝑛)‘𝑠) d𝑠 = (1 / 2))
94	93	eqcomd 2737	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1 / 2) = ∫𝐴((𝐷‘𝑛)‘𝑠) d𝑠)
95	94	oveq2d 7362	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑂 · (1 / 2)) = (𝑂 · ∫𝐴((𝐷‘𝑛)‘𝑠) d𝑠))
96	92, 95	eqtrd 2766	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑂 / 2) = (𝑂 · ∫𝐴((𝐷‘𝑛)‘𝑠) d𝑠))
97	85, 96	oveq12d 7364	. 2 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐸‘𝑛) + (𝑂 / 2)) = ((∫𝐴(𝐺‘𝑠) d𝑠 / π) + (𝑂 · ∫𝐴((𝐷‘𝑛)‘𝑠) d𝑠)))
98	2	sselda 3929	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 𝑠 ∈ ((-π[,]π) ∖ {0}))
99	98	adantlr 715	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → 𝑠 ∈ ((-π[,]π) ∖ {0}))
100		fourierdlem95.d	. . . . . . . 8 ⊢ 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)))))))
101		eqid 2731	. . . . . . . 8 ⊢ ((-π[,]π) ∖ {0}) = ((-π[,]π) ∖ {0})
102	6, 8, 21, 33, 100, 35, 36, 37, 39, 40, 101	fourierdlem66 46218	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ ((-π[,]π) ∖ {0})) → (𝐺‘𝑠) = (π · (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) · ((𝐷‘𝑛)‘𝑠))))
103	99, 102	syldan 591	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (𝐺‘𝑠) = (π · (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) · ((𝐷‘𝑛)‘𝑠))))
104	103	itgeq2dv 25710	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∫𝐴(𝐺‘𝑠) d𝑠 = ∫𝐴(π · (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) · ((𝐷‘𝑛)‘𝑠))) d𝑠)
105	104	oveq1d 7361	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (∫𝐴(𝐺‘𝑠) d𝑠 / π) = (∫𝐴(π · (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) · ((𝐷‘𝑛)‘𝑠))) d𝑠 / π))
106	78	recnd 11140	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → π ∈ ℂ)
107	6	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 𝐹:ℝ⟶ℝ)
108	8	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 𝑋 ∈ ℝ)
109		difss 4083	. . . . . . . . . . . . . 14 ⊢ ((-π[,]π) ∖ {0}) ⊆ (-π[,]π)
110	77	renegcli 11422	. . . . . . . . . . . . . . 15 ⊢ -π ∈ ℝ
111		iccssre 13329	. . . . . . . . . . . . . . 15 ⊢ ((-π ∈ ℝ ∧ π ∈ ℝ) → (-π[,]π) ⊆ ℝ)
112	110, 77, 111	mp2an 692	. . . . . . . . . . . . . 14 ⊢ (-π[,]π) ⊆ ℝ
113	109, 112	sstri 3939	. . . . . . . . . . . . 13 ⊢ ((-π[,]π) ∖ {0}) ⊆ ℝ
114	113, 98	sselid 3927	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 𝑠 ∈ ℝ)
115	108, 114	readdcld 11141	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (𝑋 + 𝑠) ∈ ℝ)
116	107, 115	ffvelcdmd 7018	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (𝐹‘(𝑋 + 𝑠)) ∈ ℝ)
117	21, 33	ifcld 4519	. . . . . . . . . . 11 ⊢ (𝜑 → if(0 < 𝑠, 𝑌, 𝑊) ∈ ℝ)
118	117	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → if(0 < 𝑠, 𝑌, 𝑊) ∈ ℝ)
119	116, 118	resubcld 11545	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → ((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) ∈ ℝ)
120	119	adantlr 715	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → ((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) ∈ ℝ)
121	1	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → 𝑛 ∈ ℕ)
122	114	adantlr 715	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → 𝑠 ∈ ℝ)
123	100	dirkerre 46141	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ ℝ) → ((𝐷‘𝑛)‘𝑠) ∈ ℝ)
124	121, 122, 123	syl2anc 584	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → ((𝐷‘𝑛)‘𝑠) ∈ ℝ)
125	120, 124	remulcld 11142	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) · ((𝐷‘𝑛)‘𝑠)) ∈ ℝ)
126	103	eqcomd 2737	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (π · (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) · ((𝐷‘𝑛)‘𝑠))) = (𝐺‘𝑠))
127	126	oveq1d 7361	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → ((π · (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) · ((𝐷‘𝑛)‘𝑠))) / π) = ((𝐺‘𝑠) / π))
128		picn 26394	. . . . . . . . . . . . 13 ⊢ π ∈ ℂ
129	128	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → π ∈ ℂ)
130	125	recnd 11140	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) · ((𝐷‘𝑛)‘𝑠)) ∈ ℂ)
131	80	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → π ≠ 0)
132	129, 130, 129, 131	div23d 11934	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → ((π · (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) · ((𝐷‘𝑛)‘𝑠))) / π) = ((π / π) · (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) · ((𝐷‘𝑛)‘𝑠))))
133	43	recnd 11140	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (𝐺‘𝑠) ∈ ℂ)
134	133, 129, 131	divrec2d 11901	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → ((𝐺‘𝑠) / π) = ((1 / π) · (𝐺‘𝑠)))
135	127, 132, 134	3eqtr3rd 2775	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → ((1 / π) · (𝐺‘𝑠)) = ((π / π) · (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) · ((𝐷‘𝑛)‘𝑠))))
136	128, 80	dividi 11854	. . . . . . . . . . . 12 ⊢ (π / π) = 1
137	136	a1i 11	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (π / π) = 1)
138	137	oveq1d 7361	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → ((π / π) · (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) · ((𝐷‘𝑛)‘𝑠))) = (1 · (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) · ((𝐷‘𝑛)‘𝑠))))
139	130	mullidd 11130	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (1 · (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) · ((𝐷‘𝑛)‘𝑠))) = (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) · ((𝐷‘𝑛)‘𝑠)))
140	135, 138, 139	3eqtrrd 2771	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) · ((𝐷‘𝑛)‘𝑠)) = ((1 / π) · (𝐺‘𝑠)))
141	140	mpteq2dva 5182	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑠 ∈ 𝐴 ↦ (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) · ((𝐷‘𝑛)‘𝑠))) = (𝑠 ∈ 𝐴 ↦ ((1 / π) · (𝐺‘𝑠))))
142	106, 81	reccld 11890	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1 / π) ∈ ℂ)
143	142, 43, 75	iblmulc2 25759	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑠 ∈ 𝐴 ↦ ((1 / π) · (𝐺‘𝑠))) ∈ 𝐿¹)
144	141, 143	eqeltrd 2831	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑠 ∈ 𝐴 ↦ (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) · ((𝐷‘𝑛)‘𝑠))) ∈ 𝐿¹)
145	106, 125, 144	itgmulc2 25762	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (π · ∫𝐴(((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) · ((𝐷‘𝑛)‘𝑠)) d𝑠) = ∫𝐴(π · (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) · ((𝐷‘𝑛)‘𝑠))) d𝑠)
146	145	eqcomd 2737	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∫𝐴(π · (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) · ((𝐷‘𝑛)‘𝑠))) d𝑠 = (π · ∫𝐴(((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) · ((𝐷‘𝑛)‘𝑠)) d𝑠))
147	146	oveq1d 7361	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (∫𝐴(π · (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) · ((𝐷‘𝑛)‘𝑠))) d𝑠 / π) = ((π · ∫𝐴(((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) · ((𝐷‘𝑛)‘𝑠)) d𝑠) / π))
148	125, 144	itgcl 25712	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∫𝐴(((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) · ((𝐷‘𝑛)‘𝑠)) d𝑠 ∈ ℂ)
149	148, 106, 81	divcan3d 11902	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((π · ∫𝐴(((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) · ((𝐷‘𝑛)‘𝑠)) d𝑠) / π) = ∫𝐴(((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) · ((𝐷‘𝑛)‘𝑠)) d𝑠)
150	105, 147, 149	3eqtrd 2770	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (∫𝐴(𝐺‘𝑠) d𝑠 / π) = ∫𝐴(((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) · ((𝐷‘𝑛)‘𝑠)) d𝑠)
151	87	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑂 ∈ ℂ)
152	112	sseli 3925	. . . . . . 7 ⊢ (𝑠 ∈ (-π[,]π) → 𝑠 ∈ ℝ)
153	152, 123	sylan2 593	. . . . . 6 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ (-π[,]π)) → ((𝐷‘𝑛)‘𝑠) ∈ ℝ)
154	153	adantll 714	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ (-π[,]π)) → ((𝐷‘𝑛)‘𝑠) ∈ ℝ)
155	110	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → -π ∈ ℝ)
156		ax-resscn 11063	. . . . . . . . . 10 ⊢ ℝ ⊆ ℂ
157	156	a1i 11	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → ℝ ⊆ ℂ)
158		ssid 3952	. . . . . . . . 9 ⊢ ℂ ⊆ ℂ
159		cncfss 24819	. . . . . . . . 9 ⊢ ((ℝ ⊆ ℂ ∧ ℂ ⊆ ℂ) → ((-π[,]π)–cn→ℝ) ⊆ ((-π[,]π)–cn→ℂ))
160	157, 158, 159	sylancl 586	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → ((-π[,]π)–cn→ℝ) ⊆ ((-π[,]π)–cn→ℂ))
161		eqid 2731	. . . . . . . . 9 ⊢ (𝑠 ∈ ℝ ↦ ((𝐷‘𝑛)‘𝑠)) = (𝑠 ∈ ℝ ↦ ((𝐷‘𝑛)‘𝑠))
162	100	dirkerf 46143	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (𝐷‘𝑛):ℝ⟶ℝ)
163	162	feqmptd 6890	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (𝐷‘𝑛) = (𝑠 ∈ ℝ ↦ ((𝐷‘𝑛)‘𝑠)))
164	100	dirkercncf 46153	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (𝐷‘𝑛) ∈ (ℝ–cn→ℝ))
165	163, 164	eqeltrrd 2832	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → (𝑠 ∈ ℝ ↦ ((𝐷‘𝑛)‘𝑠)) ∈ (ℝ–cn→ℝ))
166	112	a1i 11	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → (-π[,]π) ⊆ ℝ)
167		ssid 3952	. . . . . . . . . 10 ⊢ ℝ ⊆ ℝ
168	167	a1i 11	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → ℝ ⊆ ℝ)
169	161, 165, 166, 168, 153	cncfmptssg 45917	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → (𝑠 ∈ (-π[,]π) ↦ ((𝐷‘𝑛)‘𝑠)) ∈ ((-π[,]π)–cn→ℝ))
170	160, 169	sseldd 3930	. . . . . . 7 ⊢ (𝑛 ∈ ℕ → (𝑠 ∈ (-π[,]π) ↦ ((𝐷‘𝑛)‘𝑠)) ∈ ((-π[,]π)–cn→ℂ))
171	170	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑠 ∈ (-π[,]π) ↦ ((𝐷‘𝑛)‘𝑠)) ∈ ((-π[,]π)–cn→ℂ))
172		cniccibl 25769	. . . . . 6 ⊢ ((-π ∈ ℝ ∧ π ∈ ℝ ∧ (𝑠 ∈ (-π[,]π) ↦ ((𝐷‘𝑛)‘𝑠)) ∈ ((-π[,]π)–cn→ℂ)) → (𝑠 ∈ (-π[,]π) ↦ ((𝐷‘𝑛)‘𝑠)) ∈ 𝐿¹)
173	155, 78, 171, 172	syl3anc 1373	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑠 ∈ (-π[,]π) ↦ ((𝐷‘𝑛)‘𝑠)) ∈ 𝐿¹)
174	4, 45, 154, 173	iblss 25733	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑠 ∈ 𝐴 ↦ ((𝐷‘𝑛)‘𝑠)) ∈ 𝐿¹)
175	151, 124, 174	itgmulc2 25762	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑂 · ∫𝐴((𝐷‘𝑛)‘𝑠) d𝑠) = ∫𝐴(𝑂 · ((𝐷‘𝑛)‘𝑠)) d𝑠)
176	150, 175	oveq12d 7364	. 2 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((∫𝐴(𝐺‘𝑠) d𝑠 / π) + (𝑂 · ∫𝐴((𝐷‘𝑛)‘𝑠) d𝑠)) = (∫𝐴(((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) · ((𝐷‘𝑛)‘𝑠)) d𝑠 + ∫𝐴(𝑂 · ((𝐷‘𝑛)‘𝑠)) d𝑠))
177	86	ad2antrr 726	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → 𝑂 ∈ ℝ)
178	177, 124	remulcld 11142	. . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (𝑂 · ((𝐷‘𝑛)‘𝑠)) ∈ ℝ)
179	151, 124, 174	iblmulc2 25759	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑠 ∈ 𝐴 ↦ (𝑂 · ((𝐷‘𝑛)‘𝑠))) ∈ 𝐿¹)
180	125, 144, 178, 179	itgadd 25753	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∫𝐴((((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) · ((𝐷‘𝑛)‘𝑠)) + (𝑂 · ((𝐷‘𝑛)‘𝑠))) d𝑠 = (∫𝐴(((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) · ((𝐷‘𝑛)‘𝑠)) d𝑠 + ∫𝐴(𝑂 · ((𝐷‘𝑛)‘𝑠)) d𝑠))
181		fourierdlem95.ifeqo	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → if(0 < 𝑠, 𝑌, 𝑊) = 𝑂)
182	181	eqcomd 2737	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 𝑂 = if(0 < 𝑠, 𝑌, 𝑊))
183	182	adantlr 715	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → 𝑂 = if(0 < 𝑠, 𝑌, 𝑊))
184	183	oveq1d 7361	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (𝑂 · ((𝐷‘𝑛)‘𝑠)) = (if(0 < 𝑠, 𝑌, 𝑊) · ((𝐷‘𝑛)‘𝑠)))
185	184	oveq2d 7362	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → ((((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) · ((𝐷‘𝑛)‘𝑠)) + (𝑂 · ((𝐷‘𝑛)‘𝑠))) = ((((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) · ((𝐷‘𝑛)‘𝑠)) + (if(0 < 𝑠, 𝑌, 𝑊) · ((𝐷‘𝑛)‘𝑠))))
186	116	recnd 11140	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (𝐹‘(𝑋 + 𝑠)) ∈ ℂ)
187	186	adantlr 715	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (𝐹‘(𝑋 + 𝑠)) ∈ ℂ)
188	118	recnd 11140	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → if(0 < 𝑠, 𝑌, 𝑊) ∈ ℂ)
189	188	adantlr 715	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → if(0 < 𝑠, 𝑌, 𝑊) ∈ ℂ)
190	124	recnd 11140	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → ((𝐷‘𝑛)‘𝑠) ∈ ℂ)
191	187, 189, 190	subdird 11574	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) · ((𝐷‘𝑛)‘𝑠)) = (((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠)) − (if(0 < 𝑠, 𝑌, 𝑊) · ((𝐷‘𝑛)‘𝑠))))
192	191	oveq1d 7361	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → ((((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) · ((𝐷‘𝑛)‘𝑠)) + (if(0 < 𝑠, 𝑌, 𝑊) · ((𝐷‘𝑛)‘𝑠))) = ((((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠)) − (if(0 < 𝑠, 𝑌, 𝑊) · ((𝐷‘𝑛)‘𝑠))) + (if(0 < 𝑠, 𝑌, 𝑊) · ((𝐷‘𝑛)‘𝑠))))
193	187, 190	mulcld 11132	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → ((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠)) ∈ ℂ)
194	189, 190	mulcld 11132	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (if(0 < 𝑠, 𝑌, 𝑊) · ((𝐷‘𝑛)‘𝑠)) ∈ ℂ)
195	193, 194	npcand 11476	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → ((((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠)) − (if(0 < 𝑠, 𝑌, 𝑊) · ((𝐷‘𝑛)‘𝑠))) + (if(0 < 𝑠, 𝑌, 𝑊) · ((𝐷‘𝑛)‘𝑠))) = ((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠)))
196	185, 192, 195	3eqtrd 2770	. . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → ((((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) · ((𝐷‘𝑛)‘𝑠)) + (𝑂 · ((𝐷‘𝑛)‘𝑠))) = ((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠)))
197	196	itgeq2dv 25710	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∫𝐴((((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) · ((𝐷‘𝑛)‘𝑠)) + (𝑂 · ((𝐷‘𝑛)‘𝑠))) d𝑠 = ∫𝐴((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠)) d𝑠)
198	180, 197	eqtr3d 2768	. 2 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (∫𝐴(((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) · ((𝐷‘𝑛)‘𝑠)) d𝑠 + ∫𝐴(𝑂 · ((𝐷‘𝑛)‘𝑠)) d𝑠) = ∫𝐴((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠)) d𝑠)
199	97, 176, 198	3eqtrd 2770	1 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐸‘𝑛) + (𝑂 / 2)) = ∫𝐴((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠)) d𝑠)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2111 ≠ wne 2928 ∀wral 3047 {crab 3395 ∖ cdif 3894 ⊆ wss 3897 ifcif 4472 {csn 4573 class class class wbr 5089 ↦ cmpt 5170 dom cdm 5614 ran crn 5615 ↾ cres 5616 ⟶wf 6477 ‘cfv 6481 (class class class)co 7346 ↑_m cmap 8750 ℂcc 11004 ℝcr 11005 0cc0 11006 1c1 11007 + caddc 11009 · cmul 11011 +∞cpnf 11143 -∞cmnf 11144 ℝ^*cxr 11145 < clt 11146 − cmin 11344 -cneg 11345 / cdiv 11774 ℕcn 12125 2c2 12180 (,)cioo 13245 [,]cicc 13248 ...cfz 13407 ..^cfzo 13554 mod cmo 13773 sincsin 15970 πcpi 15973 TopOpenctopn 17325 ℂ_fldccnfld 21291 –cn→ccncf 24796 volcvol 25391 𝐿¹cibl 25545 ∫citg 25546 lim_ℂ climc 25790 D cdv 25791

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 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-inf2 9531 ax-cc 10326 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083 ax-pre-sup 11084 ax-addf 11085

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 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-symdif 4200 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-tp 4578 df-op 4580 df-uni 4857 df-int 4896 df-iun 4941 df-iin 4942 df-disj 5057 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-se 5568 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-isom 6490 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-of 7610 df-ofr 7611 df-om 7797 df-1st 7921 df-2nd 7922 df-supp 8091 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-2o 8386 df-oadd 8389 df-omul 8390 df-er 8622 df-map 8752 df-pm 8753 df-ixp 8822 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-fsupp 9246 df-fi 9295 df-sup 9326 df-inf 9327 df-oi 9396 df-dju 9794 df-card 9832 df-acn 9835 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-div 11775 df-nn 12126 df-2 12188 df-3 12189 df-4 12190 df-5 12191 df-6 12192 df-7 12193 df-8 12194 df-9 12195 df-n0 12382 df-z 12469 df-dec 12589 df-uz 12733 df-q 12847 df-rp 12891 df-xneg 13011 df-xadd 13012 df-xmul 13013 df-ioo 13249 df-ioc 13250 df-ico 13251 df-icc 13252 df-fz 13408 df-fzo 13555 df-fl 13696 df-mod 13774 df-seq 13909 df-exp 13969 df-fac 14181 df-bc 14210 df-hash 14238 df-shft 14974 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-limsup 15378 df-clim 15395 df-rlim 15396 df-sum 15594 df-ef 15974 df-sin 15976 df-cos 15977 df-pi 15979 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-starv 17176 df-sca 17177 df-vsca 17178 df-ip 17179 df-tset 17180 df-ple 17181 df-ds 17183 df-unif 17184 df-hom 17185 df-cco 17186 df-rest 17326 df-topn 17327 df-0g 17345 df-gsum 17346 df-topgen 17347 df-pt 17348 df-prds 17351 df-xrs 17406 df-qtop 17411 df-imas 17412 df-xps 17414 df-mre 17488 df-mrc 17489 df-acs 17491 df-mgm 18548 df-sgrp 18627 df-mnd 18643 df-submnd 18692 df-mulg 18981 df-cntz 19229 df-cmn 19694 df-psmet 21283 df-xmet 21284 df-met 21285 df-bl 21286 df-mopn 21287 df-fbas 21288 df-fg 21289 df-cnfld 21292 df-top 22809 df-topon 22826 df-topsp 22848 df-bases 22861 df-cld 22934 df-ntr 22935 df-cls 22936 df-nei 23013 df-lp 23051 df-perf 23052 df-cn 23142 df-cnp 23143 df-t1 23229 df-haus 23230 df-cmp 23302 df-tx 23477 df-hmeo 23670 df-fil 23761 df-fm 23853 df-flim 23854 df-flf 23855 df-xms 24235 df-ms 24236 df-tms 24237 df-cncf 24798 df-ovol 25392 df-vol 25393 df-mbf 25547 df-itg1 25548 df-itg2 25549 df-ibl 25550 df-itg 25551 df-0p 25598 df-limc 25794 df-dv 25795

This theorem is referenced by: fourierdlem103 46255 fourierdlem104 46256

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