fourierdlem95 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem95 44903

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 485	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
2		fourierdlem95.ass	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ⊆ ((-π[,]π) ∖ {0}))
3	2	difss2d 4133	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ⊆ (-π[,]π))
4	3	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴 ⊆ (-π[,]π))
5	4	sselda 3981	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → 𝑠 ∈ (-π[,]π))
6		fourierdlem95.f	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:ℝ⟶ℝ)
7	6	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐹:ℝ⟶ℝ)
8		fourierdlem95.xre	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ ℝ)
9	8	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑋 ∈ ℝ)
10		ioossre 13381	. . . . . . . . . . . . 13 ⊢ (𝑋(,)+∞) ⊆ ℝ
11	10	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑋(,)+∞) ⊆ ℝ)
12	6, 11	fssresd 6755	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹 ↾ (𝑋(,)+∞)):(𝑋(,)+∞)⟶ℝ)
13		ioosscn 13382	. . . . . . . . . . . 12 ⊢ (𝑋(,)+∞) ⊆ ℂ
14	13	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑋(,)+∞) ⊆ ℂ)
15		eqid 2732	. . . . . . . . . . . 12 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
16		pnfxr 11264	. . . . . . . . . . . . 13 ⊢ +∞ ∈ ℝ^*
17	16	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → +∞ ∈ ℝ^*)
18	8	ltpnfd 13097	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋 < +∞)
19	15, 17, 8, 18	lptioo1cn 44348	. . . . . . . . . . 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 44331	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ∈ ℝ)
22	21	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑌 ∈ ℝ)
23		ioossre 13381	. . . . . . . . . . . . 13 ⊢ (-∞(,)𝑋) ⊆ ℝ
24	23	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → (-∞(,)𝑋) ⊆ ℝ)
25	6, 24	fssresd 6755	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹 ↾ (-∞(,)𝑋)):(-∞(,)𝑋)⟶ℝ)
26		ioosscn 13382	. . . . . . . . . . . 12 ⊢ (-∞(,)𝑋) ⊆ ℂ
27	26	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → (-∞(,)𝑋) ⊆ ℂ)
28		mnfxr 11267	. . . . . . . . . . . . 13 ⊢ -∞ ∈ ℝ^*
29	28	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → -∞ ∈ ℝ^*)
30	8	mnfltd 13100	. . . . . . . . . . . 12 ⊢ (𝜑 → -∞ < 𝑋)
31	15, 29, 8, 30	lptioo2cn 44347	. . . . . . . . . . 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 44331	. . . . . . . . . 10 ⊢ (𝜑 → 𝑊 ∈ ℝ)
34	33	adantr 481	. . . . . . . . 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 12223	. . . . . . . . 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 44875	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐺:(-π[,]π)⟶ℝ)
42	41	ffvelcdmda 7083	. . . . . . 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 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴 ∈ dom vol)
46	41	feqmptd 6957	. . . . . . . 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 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑋 ∈ ran 𝑉)
50	20	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑌 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋))
51	32	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑊 ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋))
52		fourierdlem95.m	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℕ)
53	52	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑀 ∈ ℕ)
54		fourierdlem95.v	. . . . . . . . . 10 ⊢ (𝜑 → 𝑉 ∈ (𝑃‘𝑀))
55	54	adantr 481	. . . . . . . . 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 713	. . . . . . . . 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 713	. . . . . . . . 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 713	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) lim_ℂ (𝑉‘(𝑖 + 1))))
62		fveq2 6888	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑖 → (𝑉‘𝑗) = (𝑉‘𝑖))
63	62	oveq1d 7420	. . . . . . . . . 10 ⊢ (𝑗 = 𝑖 → ((𝑉‘𝑗) − 𝑋) = ((𝑉‘𝑖) − 𝑋))
64	63	cbvmptv 5260	. . . . . . . . 9 ⊢ (𝑗 ∈ (0...𝑀) ↦ ((𝑉‘𝑗) − 𝑋)) = (𝑖 ∈ (0...𝑀) ↦ ((𝑉‘𝑖) − 𝑋))
65		eqid 2732	. . . . . . . . 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 713	. . . . . . . . 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 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐵 ∈ ((𝐼 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋))
71		fourierdlem95.c	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ ((𝐼 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋))
72	71	adantr 481	. . . . . . . . 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 44896	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐺 ∈ 𝐿¹)
74	46, 73	eqeltrrd 2834	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑠 ∈ (-π[,]π) ↦ (𝐺‘𝑠)) ∈ 𝐿¹)
75	4, 45, 42, 74	iblss 25313	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑠 ∈ 𝐴 ↦ (𝐺‘𝑠)) ∈ 𝐿¹)
76	43, 75	itgrecl 25306	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∫𝐴(𝐺‘𝑠) d𝑠 ∈ ℝ)
77		pire 25959	. . . . . 6 ⊢ π ∈ ℝ
78	77	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → π ∈ ℝ)
79		pipos 25961	. . . . . . 7 ⊢ 0 < π
80	77, 79	gt0ne0ii 11746	. . . . . 6 ⊢ π ≠ 0
81	80	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → π ≠ 0)
82	76, 78, 81	redivcld 12038	. . . 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 7006	. . . 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 11238	. . . . . 6 ⊢ (𝜑 → 𝑂 ∈ ℂ)
88		2cnd 12286	. . . . . 6 ⊢ (𝜑 → 2 ∈ ℂ)
89		2ne0 12312	. . . . . . 7 ⊢ 2 ≠ 0
90	89	a1i 11	. . . . . 6 ⊢ (𝜑 → 2 ≠ 0)
91	87, 88, 90	divrecd 11989	. . . . 5 ⊢ (𝜑 → (𝑂 / 2) = (𝑂 · (1 / 2)))
92	91	adantr 481	. . . 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 2738	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1 / 2) = ∫𝐴((𝐷‘𝑛)‘𝑠) d𝑠)
95	94	oveq2d 7421	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑂 · (1 / 2)) = (𝑂 · ∫𝐴((𝐷‘𝑛)‘𝑠) d𝑠))
96	92, 95	eqtrd 2772	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑂 / 2) = (𝑂 · ∫𝐴((𝐷‘𝑛)‘𝑠) d𝑠))
97	85, 96	oveq12d 7423	. 2 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐸‘𝑛) + (𝑂 / 2)) = ((∫𝐴(𝐺‘𝑠) d𝑠 / π) + (𝑂 · ∫𝐴((𝐷‘𝑛)‘𝑠) d𝑠)))
98	2	sselda 3981	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 𝑠 ∈ ((-π[,]π) ∖ {0}))
99	98	adantlr 713	. . . . . . 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 2732	. . . . . . . 8 ⊢ ((-π[,]π) ∖ {0}) = ((-π[,]π) ∖ {0})
102	6, 8, 21, 33, 100, 35, 36, 37, 39, 40, 101	fourierdlem66 44874	. . . . . . 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 25290	. . . . 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 7420	. . . 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 11238	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → π ∈ ℂ)
107	6	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 𝐹:ℝ⟶ℝ)
108	8	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 𝑋 ∈ ℝ)
109		difss 4130	. . . . . . . . . . . . . 14 ⊢ ((-π[,]π) ∖ {0}) ⊆ (-π[,]π)
110	77	renegcli 11517	. . . . . . . . . . . . . . 15 ⊢ -π ∈ ℝ
111		iccssre 13402	. . . . . . . . . . . . . . 15 ⊢ ((-π ∈ ℝ ∧ π ∈ ℝ) → (-π[,]π) ⊆ ℝ)
112	110, 77, 111	mp2an 690	. . . . . . . . . . . . . 14 ⊢ (-π[,]π) ⊆ ℝ
113	109, 112	sstri 3990	. . . . . . . . . . . . 13 ⊢ ((-π[,]π) ∖ {0}) ⊆ ℝ
114	113, 98	sselid 3979	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 𝑠 ∈ ℝ)
115	108, 114	readdcld 11239	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (𝑋 + 𝑠) ∈ ℝ)
116	107, 115	ffvelcdmd 7084	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (𝐹‘(𝑋 + 𝑠)) ∈ ℝ)
117	21, 33	ifcld 4573	. . . . . . . . . . 11 ⊢ (𝜑 → if(0 < 𝑠, 𝑌, 𝑊) ∈ ℝ)
118	117	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → if(0 < 𝑠, 𝑌, 𝑊) ∈ ℝ)
119	116, 118	resubcld 11638	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → ((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) ∈ ℝ)
120	119	adantlr 713	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → ((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) ∈ ℝ)
121	1	adantr 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → 𝑛 ∈ ℕ)
122	114	adantlr 713	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → 𝑠 ∈ ℝ)
123	100	dirkerre 44797	. . . . . . . . 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 11240	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) · ((𝐷‘𝑛)‘𝑠)) ∈ ℝ)
126	103	eqcomd 2738	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (π · (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) · ((𝐷‘𝑛)‘𝑠))) = (𝐺‘𝑠))
127	126	oveq1d 7420	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → ((π · (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) · ((𝐷‘𝑛)‘𝑠))) / π) = ((𝐺‘𝑠) / π))
128		picn 25960	. . . . . . . . . . . . 13 ⊢ π ∈ ℂ
129	128	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → π ∈ ℂ)
130	125	recnd 11238	. . . . . . . . . . . 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 12023	. . . . . . . . . . 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 11238	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (𝐺‘𝑠) ∈ ℂ)
134	133, 129, 131	divrec2d 11990	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → ((𝐺‘𝑠) / π) = ((1 / π) · (𝐺‘𝑠)))
135	127, 132, 134	3eqtr3rd 2781	. . . . . . . . . 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 11943	. . . . . . . . . . . 12 ⊢ (π / π) = 1
137	136	a1i 11	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (π / π) = 1)
138	137	oveq1d 7420	. . . . . . . . . 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 11228	. . . . . . . . . 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 2777	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) · ((𝐷‘𝑛)‘𝑠)) = ((1 / π) · (𝐺‘𝑠)))
141	140	mpteq2dva 5247	. . . . . . . 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 11979	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1 / π) ∈ ℂ)
143	142, 43, 75	iblmulc2 25339	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑠 ∈ 𝐴 ↦ ((1 / π) · (𝐺‘𝑠))) ∈ 𝐿¹)
144	141, 143	eqeltrd 2833	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑠 ∈ 𝐴 ↦ (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) · ((𝐷‘𝑛)‘𝑠))) ∈ 𝐿¹)
145	106, 125, 144	itgmulc2 25342	. . . . . 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 2738	. . . . 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 7420	. . . 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 25292	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∫𝐴(((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) · ((𝐷‘𝑛)‘𝑠)) d𝑠 ∈ ℂ)
149	148, 106, 81	divcan3d 11991	. . . 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 2776	. . 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 481	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑂 ∈ ℂ)
152	112	sseli 3977	. . . . . . 7 ⊢ (𝑠 ∈ (-π[,]π) → 𝑠 ∈ ℝ)
153	152, 123	sylan2 593	. . . . . 6 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ (-π[,]π)) → ((𝐷‘𝑛)‘𝑠) ∈ ℝ)
154	153	adantll 712	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ (-π[,]π)) → ((𝐷‘𝑛)‘𝑠) ∈ ℝ)
155	110	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → -π ∈ ℝ)
156		ax-resscn 11163	. . . . . . . . . 10 ⊢ ℝ ⊆ ℂ
157	156	a1i 11	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → ℝ ⊆ ℂ)
158		ssid 4003	. . . . . . . . 9 ⊢ ℂ ⊆ ℂ
159		cncfss 24406	. . . . . . . . 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 2732	. . . . . . . . 9 ⊢ (𝑠 ∈ ℝ ↦ ((𝐷‘𝑛)‘𝑠)) = (𝑠 ∈ ℝ ↦ ((𝐷‘𝑛)‘𝑠))
162	100	dirkerf 44799	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (𝐷‘𝑛):ℝ⟶ℝ)
163	162	feqmptd 6957	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (𝐷‘𝑛) = (𝑠 ∈ ℝ ↦ ((𝐷‘𝑛)‘𝑠)))
164	100	dirkercncf 44809	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (𝐷‘𝑛) ∈ (ℝ–cn→ℝ))
165	163, 164	eqeltrrd 2834	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → (𝑠 ∈ ℝ ↦ ((𝐷‘𝑛)‘𝑠)) ∈ (ℝ–cn→ℝ))
166	112	a1i 11	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → (-π[,]π) ⊆ ℝ)
167		ssid 4003	. . . . . . . . . 10 ⊢ ℝ ⊆ ℝ
168	167	a1i 11	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → ℝ ⊆ ℝ)
169	161, 165, 166, 168, 153	cncfmptssg 44573	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → (𝑠 ∈ (-π[,]π) ↦ ((𝐷‘𝑛)‘𝑠)) ∈ ((-π[,]π)–cn→ℝ))
170	160, 169	sseldd 3982	. . . . . . 7 ⊢ (𝑛 ∈ ℕ → (𝑠 ∈ (-π[,]π) ↦ ((𝐷‘𝑛)‘𝑠)) ∈ ((-π[,]π)–cn→ℂ))
171	170	adantl 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑠 ∈ (-π[,]π) ↦ ((𝐷‘𝑛)‘𝑠)) ∈ ((-π[,]π)–cn→ℂ))
172		cniccibl 25349	. . . . . 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 1371	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑠 ∈ (-π[,]π) ↦ ((𝐷‘𝑛)‘𝑠)) ∈ 𝐿¹)
174	4, 45, 154, 173	iblss 25313	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑠 ∈ 𝐴 ↦ ((𝐷‘𝑛)‘𝑠)) ∈ 𝐿¹)
175	151, 124, 174	itgmulc2 25342	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑂 · ∫𝐴((𝐷‘𝑛)‘𝑠) d𝑠) = ∫𝐴(𝑂 · ((𝐷‘𝑛)‘𝑠)) d𝑠)
176	150, 175	oveq12d 7423	. 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 724	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → 𝑂 ∈ ℝ)
178	177, 124	remulcld 11240	. . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (𝑂 · ((𝐷‘𝑛)‘𝑠)) ∈ ℝ)
179	151, 124, 174	iblmulc2 25339	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑠 ∈ 𝐴 ↦ (𝑂 · ((𝐷‘𝑛)‘𝑠))) ∈ 𝐿¹)
180	125, 144, 178, 179	itgadd 25333	. . 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 2738	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 𝑂 = if(0 < 𝑠, 𝑌, 𝑊))
183	182	adantlr 713	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → 𝑂 = if(0 < 𝑠, 𝑌, 𝑊))
184	183	oveq1d 7420	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (𝑂 · ((𝐷‘𝑛)‘𝑠)) = (if(0 < 𝑠, 𝑌, 𝑊) · ((𝐷‘𝑛)‘𝑠)))
185	184	oveq2d 7421	. . . . 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 11238	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (𝐹‘(𝑋 + 𝑠)) ∈ ℂ)
187	186	adantlr 713	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (𝐹‘(𝑋 + 𝑠)) ∈ ℂ)
188	118	recnd 11238	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → if(0 < 𝑠, 𝑌, 𝑊) ∈ ℂ)
189	188	adantlr 713	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → if(0 < 𝑠, 𝑌, 𝑊) ∈ ℂ)
190	124	recnd 11238	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → ((𝐷‘𝑛)‘𝑠) ∈ ℂ)
191	187, 189, 190	subdird 11667	. . . . . 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 7420	. . . . 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 11230	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → ((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠)) ∈ ℂ)
194	189, 190	mulcld 11230	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (if(0 < 𝑠, 𝑌, 𝑊) · ((𝐷‘𝑛)‘𝑠)) ∈ ℂ)
195	193, 194	npcand 11571	. . . . 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 2776	. . . 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 25290	. . 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 2774	. 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 2776	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 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ∀wral 3061 {crab 3432 ∖ cdif 3944 ⊆ wss 3947 ifcif 4527 {csn 4627 class class class wbr 5147 ↦ cmpt 5230 dom cdm 5675 ran crn 5676 ↾ cres 5677 ⟶wf 6536 ‘cfv 6540 (class class class)co 7405 ↑_m cmap 8816 ℂcc 11104 ℝcr 11105 0cc0 11106 1c1 11107 + caddc 11109 · cmul 11111 +∞cpnf 11241 -∞cmnf 11242 ℝ^*cxr 11243 < clt 11244 − cmin 11440 -cneg 11441 / cdiv 11867 ℕcn 12208 2c2 12263 (,)cioo 13320 [,]cicc 13323 ...cfz 13480 ..^cfzo 13623 mod cmo 13830 sincsin 16003 πcpi 16006 TopOpenctopn 17363 ℂ_fldccnfld 20936 –cn→ccncf 24383 volcvol 24971 𝐿¹cibl 25125 ∫citg 25126 lim_ℂ climc 25370 D cdv 25371

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-inf2 9632 ax-cc 10426 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185 ax-mulf 11186

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-symdif 4241 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-disj 5113 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7666 df-ofr 7667 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8143 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-oadd 8466 df-omul 8467 df-er 8699 df-map 8818 df-pm 8819 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-fi 9402 df-sup 9433 df-inf 9434 df-oi 9501 df-dju 9892 df-card 9930 df-acn 9933 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-ioo 13324 df-ioc 13325 df-ico 13326 df-icc 13327 df-fz 13481 df-fzo 13624 df-fl 13753 df-mod 13831 df-seq 13963 df-exp 14024 df-fac 14230 df-bc 14259 df-hash 14287 df-shft 15010 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-limsup 15411 df-clim 15428 df-rlim 15429 df-sum 15629 df-ef 16007 df-sin 16009 df-cos 16010 df-pi 16012 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-starv 17208 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-unif 17216 df-hom 17217 df-cco 17218 df-rest 17364 df-topn 17365 df-0g 17383 df-gsum 17384 df-topgen 17385 df-pt 17386 df-prds 17389 df-xrs 17444 df-qtop 17449 df-imas 17450 df-xps 17452 df-mre 17526 df-mrc 17527 df-acs 17529 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-submnd 18668 df-mulg 18945 df-cntz 19175 df-cmn 19644 df-psmet 20928 df-xmet 20929 df-met 20930 df-bl 20931 df-mopn 20932 df-fbas 20933 df-fg 20934 df-cnfld 20937 df-top 22387 df-topon 22404 df-topsp 22426 df-bases 22440 df-cld 22514 df-ntr 22515 df-cls 22516 df-nei 22593 df-lp 22631 df-perf 22632 df-cn 22722 df-cnp 22723 df-t1 22809 df-haus 22810 df-cmp 22882 df-tx 23057 df-hmeo 23250 df-fil 23341 df-fm 23433 df-flim 23434 df-flf 23435 df-xms 23817 df-ms 23818 df-tms 23819 df-cncf 24385 df-ovol 24972 df-vol 24973 df-mbf 25127 df-itg1 25128 df-itg2 25129 df-ibl 25130 df-itg 25131 df-0p 25178 df-limc 25374 df-dv 25375

This theorem is referenced by: fourierdlem103 44911 fourierdlem104 44912

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