fourierdlem95 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem95 47484

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 490	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
2		fourierdlem95.ass	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ⊆ (( -π[,]π) ∖ {0}))
3	2	difss2d 4078	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ⊆ ( -π[,]π))
4	3	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴 ⊆ ( -π[,]π))
5	4	sselda 3913	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → 𝑠 ∈ ( -π[,]π))
6		fourierdlem95.f	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:ℝ⟶ℝ)
7	6	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐹:ℝ⟶ℝ)
8		fourierdlem95.xre	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ ℝ)
9	8	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑋 ∈ ℝ)
10		ioossre 13578	. . . . . . . . . . . . 13 ⊢ (𝑋(,)+∞) ⊆ ℝ
11	10	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑋(,)+∞) ⊆ ℝ)
12	6, 11	fssresd 6779	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹 ↾ (𝑋(,)+∞)):(𝑋(,)+∞)⟶ℝ)
13		ioosscn 13579	. . . . . . . . . . . 12 ⊢ (𝑋(,)+∞) ⊆ ℂ
14	13	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑋(,)+∞) ⊆ ℂ)
15		eqid 2763	. . . . . . . . . . . 12 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
16		pnfxr 11402	. . . . . . . . . . . . 13 ⊢ +∞ ∈ ℝ^*
17	16	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → +∞ ∈ ℝ^*)
18	8	ltpnfd 13290	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋 < +∞)
19	15, 17, 8, 18	lptioo1cn 46929	. . . . . . . . . . 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 46914	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ∈ ℝ)
22	21	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑌 ∈ ℝ)
23		ioossre 13578	. . . . . . . . . . . . 13 ⊢ (-∞(,)𝑋) ⊆ ℝ
24	23	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → (-∞(,)𝑋) ⊆ ℝ)
25	6, 24	fssresd 6779	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹 ↾ (-∞(,)𝑋)):(-∞(,)𝑋)⟶ℝ)
26		ioosscn 13579	. . . . . . . . . . . 12 ⊢ (-∞(,)𝑋) ⊆ ℂ
27	26	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → (-∞(,)𝑋) ⊆ ℂ)
28		mnfxr 11405	. . . . . . . . . . . . 13 ⊢ -∞ ∈ ℝ^*
29	28	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → -∞ ∈ ℝ^*)
30	8	mnfltd 13293	. . . . . . . . . . . 12 ⊢ (𝜑 → -∞ < 𝑋)
31	15, 29, 8, 30	lptioo2cn 46928	. . . . . . . . . . 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 46914	. . . . . . . . . 10 ⊢ (𝜑 → 𝑊 ∈ ℝ)
34	33	adantr 486	. . . . . . . . 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 12390	. . . . . . . . 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 47456	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐺:( -π[,]π)⟶ℝ)
42	41	ffvelcdmda 7116	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ ( -π[,]π)) → (𝐺‘𝑠) ∈ ℝ)
43	5, 42	syldan 603	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (𝐺‘𝑠) ∈ ℝ)
44		fourierdlem95.admvol	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ dom vol)
45	44	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴 ∈ dom vol)
46	41	feqmptd 6984	. . . . . . . 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 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑋 ∈ ran 𝑉)
50	20	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑌 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋))
51	32	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑊 ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋))
52		fourierdlem95.m	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℕ)
53	52	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑀 ∈ ℕ)
54		fourierdlem95.v	. . . . . . . . . 10 ⊢ (𝜑 → 𝑉 ∈ (𝑃‘𝑀))
55	54	adantr 486	. . . . . . . . 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 728	. . . . . . . . 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 728	. . . . . . . . 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 728	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) lim_ℂ (𝑉‘(𝑖 + 1))))
62		fveq2 6916	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑖 → (𝑉‘𝑗) = (𝑉‘𝑖))
63	62	oveq1d 7466	. . . . . . . . . 10 ⊢ (𝑗 = 𝑖 → ((𝑉‘𝑗) − 𝑋) = ((𝑉‘𝑖) − 𝑋))
64	63	cbvmptv 5215	. . . . . . . . 9 ⊢ (𝑗 ∈ (0...𝑀) ↦ ((𝑉‘𝑗) − 𝑋)) = (𝑖 ∈ (0...𝑀) ↦ ((𝑉‘𝑖) − 𝑋))
65		eqid 2763	. . . . . . . . 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 728	. . . . . . . . 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 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐵 ∈ ((𝐼 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋))
71		fourierdlem95.c	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ ((𝐼 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋))
72	71	adantr 486	. . . . . . . . 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 47477	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐺 ∈ 𝐿¹)
74	46, 73	eqeltrrd 2865	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑠 ∈ ( -π[,]π) ↦ (𝐺‘𝑠)) ∈ 𝐿¹)
75	4, 45, 42, 74	iblss 26180	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑠 ∈ 𝐴 ↦ (𝐺‘𝑠)) ∈ 𝐿¹)
76	43, 75	itgrecl 26173	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∫𝐴(𝐺‘𝑠) d𝑠 ∈ ℝ)
77		pire 26840	. . . . . 6 ⊢ π ∈ ℝ
78	77	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → π ∈ ℝ)
79		pipos 26844	. . . . . . 7 ⊢ 0 < π
80	77, 79	gt0ne0ii 11892	. . . . . 6 ⊢ π ≠ 0
81	80	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → π ≠ 0)
82	76, 78, 81	redivcld 12185	. . . 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 7036	. . . 4 ⊢ ((𝑛 ∈ ℕ ∧ (∫𝐴(𝐺‘𝑠) d𝑠 / π) ∈ ℝ) → (𝐸‘𝑛) = (∫𝐴(𝐺‘𝑠) d𝑠 / π))
85	1, 82, 84	syl2anc 596	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐸‘𝑛) = (∫𝐴(𝐺‘𝑠) d𝑠 / π))
86		fourierdlem95.o	. . . . . . 7 ⊢ (𝜑 → 𝑂 ∈ ℝ)
87	86	recnd 11376	. . . . . 6 ⊢ (𝜑 → 𝑂 ∈ ℂ)
88		2cnd 12461	. . . . . 6 ⊢ (𝜑 → 2 ∈ ℂ)
89		2ne0 12489	. . . . . . 7 ⊢ 2 ≠ 0
90	89	a1i 11	. . . . . 6 ⊢ (𝜑 → 2 ≠ 0)
91	87, 88, 90	divrecd 12136	. . . . 5 ⊢ (𝜑 → (𝑂 / 2) = (𝑂 · (1 / 2)))
92	91	adantr 486	. . . 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 2769	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1 / 2) = ∫𝐴((𝐷‘𝑛)‘𝑠) d𝑠)
95	94	oveq2d 7467	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑂 · (1 / 2)) = (𝑂 · ∫𝐴((𝐷‘𝑛)‘𝑠) d𝑠))
96	92, 95	eqtrd 2798	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑂 / 2) = (𝑂 · ∫𝐴((𝐷‘𝑛)‘𝑠) d𝑠))
97	85, 96	oveq12d 7469	. 2 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐸‘𝑛) + (𝑂 / 2)) = ((∫𝐴(𝐺‘𝑠) d𝑠 / π) + (𝑂 · ∫𝐴((𝐷‘𝑛)‘𝑠) d𝑠)))
98	2	sselda 3913	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 𝑠 ∈ (( -π[,]π) ∖ {0}))
99	98	adantlr 728	. . . . . . 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 2763	. . . . . . . 8 ⊢ (( -π[,]π) ∖ {0}) = (( -π[,]π) ∖ {0})
102	6, 8, 21, 33, 100, 35, 36, 37, 39, 40, 101	fourierdlem66 47455	. . . . . . 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 603	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (𝐺‘𝑠) = (π · (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) · ((𝐷‘𝑛)‘𝑠))))
104	103	itgeq2dv 26157	. . . . 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 7466	. . . 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 11376	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → π ∈ ℂ)
107	6	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 𝐹:ℝ⟶ℝ)
108	8	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 𝑋 ∈ ℝ)
109		difss 4075	. . . . . . . . . . . . . 14 ⊢ (( -π[,]π) ∖ {0}) ⊆ ( -π[,]π)
110	77	renegcli 11658	. . . . . . . . . . . . . . 15 ⊢ -π ∈ ℝ
111		iccssre 13600	. . . . . . . . . . . . . . 15 ⊢ (( -π ∈ ℝ ∧ π ∈ ℝ) → ( -π[,]π) ⊆ ℝ)
112	110, 77, 111	mp2an 705	. . . . . . . . . . . . . 14 ⊢ ( -π[,]π) ⊆ ℝ
113	109, 112	sstri 3922	. . . . . . . . . . . . 13 ⊢ (( -π[,]π) ∖ {0}) ⊆ ℝ
114	113, 98	sselid 3911	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 𝑠 ∈ ℝ)
115	108, 114	readdcld 11377	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (𝑋 + 𝑠) ∈ ℝ)
116	107, 115	ffvelcdmd 7117	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (𝐹‘(𝑋 + 𝑠)) ∈ ℝ)
117	21, 33	ifcld 4532	. . . . . . . . . . 11 ⊢ (𝜑 → if(0 < 𝑠, 𝑌, 𝑊) ∈ ℝ)
118	117	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → if(0 < 𝑠, 𝑌, 𝑊) ∈ ℝ)
119	116, 118	resubcld 11783	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → ((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) ∈ ℝ)
120	119	adantlr 728	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → ((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) ∈ ℝ)
121	1	adantr 486	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → 𝑛 ∈ ℕ)
122	114	adantlr 728	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → 𝑠 ∈ ℝ)
123	100	dirkerre 47378	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ ℝ) → ((𝐷‘𝑛)‘𝑠) ∈ ℝ)
124	121, 122, 123	syl2anc 596	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → ((𝐷‘𝑛)‘𝑠) ∈ ℝ)
125	120, 124	remulcld 11378	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) · ((𝐷‘𝑛)‘𝑠)) ∈ ℝ)
126	103	eqcomd 2769	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (π · (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) · ((𝐷‘𝑛)‘𝑠))) = (𝐺‘𝑠))
127	126	oveq1d 7466	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → ((π · (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) · ((𝐷‘𝑛)‘𝑠))) / π) = ((𝐺‘𝑠) / π))
128		picn 26842	. . . . . . . . . . . . 13 ⊢ π ∈ ℂ
129	128	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → π ∈ ℂ)
130	125	recnd 11376	. . . . . . . . . . . 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 12170	. . . . . . . . . . 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 11376	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (𝐺‘𝑠) ∈ ℂ)
134	133, 129, 131	divrec2d 12137	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → ((𝐺‘𝑠) / π) = ((1 / π) · (𝐺‘𝑠)))
135	127, 132, 134	3eqtr3rd 2807	. . . . . . . . . 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 12090	. . . . . . . . . . . 12 ⊢ (π / π) = 1
137	136	a1i 11	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (π / π) = 1)
138	137	oveq1d 7466	. . . . . . . . . 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 11366	. . . . . . . . . 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 2803	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) · ((𝐷‘𝑛)‘𝑠)) = ((1 / π) · (𝐺‘𝑠)))
141	140	mpteq2dva 5204	. . . . . . . 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 12126	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1 / π) ∈ ℂ)
143	142, 43, 75	iblmulc2 26206	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑠 ∈ 𝐴 ↦ ((1 / π) · (𝐺‘𝑠))) ∈ 𝐿¹)
144	141, 143	eqeltrd 2864	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑠 ∈ 𝐴 ↦ (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) · ((𝐷‘𝑛)‘𝑠))) ∈ 𝐿¹)
145	106, 125, 144	itgmulc2 26209	. . . . . 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 2769	. . . . 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 7466	. . . 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 26159	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∫𝐴(((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) · ((𝐷‘𝑛)‘𝑠)) d𝑠 ∈ ℂ)
149	148, 106, 81	divcan3d 12138	. . . 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 2802	. . 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 486	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑂 ∈ ℂ)
152	112	sseli 3909	. . . . . . 7 ⊢ (𝑠 ∈ ( -π[,]π) → 𝑠 ∈ ℝ)
153	152, 123	sylan2 605	. . . . . 6 ⊢ ((𝑛 ∈ ℕ ∧ 𝑠 ∈ ( -π[,]π)) → ((𝐷‘𝑛)‘𝑠) ∈ ℝ)
154	153	adantll 727	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ ( -π[,]π)) → ((𝐷‘𝑛)‘𝑠) ∈ ℝ)
155	110	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → -π ∈ ℝ)
156		ax-resscn 11296	. . . . . . . . . 10 ⊢ ℝ ⊆ ℂ
157	156	a1i 11	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → ℝ ⊆ ℂ)
158		ssid 3935	. . . . . . . . 9 ⊢ ℂ ⊆ ℂ
159		cncfss 25275	. . . . . . . . 9 ⊢ ((ℝ ⊆ ℂ ∧ ℂ ⊆ ℂ) → (( -π[,]π)–cn→ℝ) ⊆ (( -π[,]π)–cn→ℂ))
160	157, 158, 159	sylancl 598	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → (( -π[,]π)–cn→ℝ) ⊆ (( -π[,]π)–cn→ℂ))
161		eqid 2763	. . . . . . . . 9 ⊢ (𝑠 ∈ ℝ ↦ ((𝐷‘𝑛)‘𝑠)) = (𝑠 ∈ ℝ ↦ ((𝐷‘𝑛)‘𝑠))
162	100	dirkerf 47380	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (𝐷‘𝑛):ℝ⟶ℝ)
163	162	feqmptd 6984	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (𝐷‘𝑛) = (𝑠 ∈ ℝ ↦ ((𝐷‘𝑛)‘𝑠)))
164	100	dirkercncf 47390	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (𝐷‘𝑛) ∈ (ℝ–cn→ℝ))
165	163, 164	eqeltrrd 2865	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → (𝑠 ∈ ℝ ↦ ((𝐷‘𝑛)‘𝑠)) ∈ (ℝ–cn→ℝ))
166	112	a1i 11	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → ( -π[,]π) ⊆ ℝ)
167		ssid 3935	. . . . . . . . . 10 ⊢ ℝ ⊆ ℝ
168	167	a1i 11	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → ℝ ⊆ ℝ)
169	161, 165, 166, 168, 153	cncfmptssg 47154	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → (𝑠 ∈ ( -π[,]π) ↦ ((𝐷‘𝑛)‘𝑠)) ∈ (( -π[,]π)–cn→ℝ))
170	160, 169	sseldd 3914	. . . . . . 7 ⊢ (𝑛 ∈ ℕ → (𝑠 ∈ ( -π[,]π) ↦ ((𝐷‘𝑛)‘𝑠)) ∈ (( -π[,]π)–cn→ℂ))
171	170	adantl 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑠 ∈ ( -π[,]π) ↦ ((𝐷‘𝑛)‘𝑠)) ∈ (( -π[,]π)–cn→ℂ))
172		cniccibl 26216	. . . . . 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 1399	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑠 ∈ ( -π[,]π) ↦ ((𝐷‘𝑛)‘𝑠)) ∈ 𝐿¹)
174	4, 45, 154, 173	iblss 26180	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑠 ∈ 𝐴 ↦ ((𝐷‘𝑛)‘𝑠)) ∈ 𝐿¹)
175	151, 124, 174	itgmulc2 26209	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑂 · ∫𝐴((𝐷‘𝑛)‘𝑠) d𝑠) = ∫𝐴(𝑂 · ((𝐷‘𝑛)‘𝑠)) d𝑠)
176	150, 175	oveq12d 7469	. 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 739	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → 𝑂 ∈ ℝ)
178	177, 124	remulcld 11378	. . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (𝑂 · ((𝐷‘𝑛)‘𝑠)) ∈ ℝ)
179	151, 124, 174	iblmulc2 26206	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑠 ∈ 𝐴 ↦ (𝑂 · ((𝐷‘𝑛)‘𝑠))) ∈ 𝐿¹)
180	125, 144, 178, 179	itgadd 26200	. . 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 2769	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 𝑂 = if(0 < 𝑠, 𝑌, 𝑊))
183	182	adantlr 728	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → 𝑂 = if(0 < 𝑠, 𝑌, 𝑊))
184	183	oveq1d 7466	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (𝑂 · ((𝐷‘𝑛)‘𝑠)) = (if(0 < 𝑠, 𝑌, 𝑊) · ((𝐷‘𝑛)‘𝑠)))
185	184	oveq2d 7467	. . . . 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 11376	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (𝐹‘(𝑋 + 𝑠)) ∈ ℂ)
187	186	adantlr 728	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (𝐹‘(𝑋 + 𝑠)) ∈ ℂ)
188	118	recnd 11376	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → if(0 < 𝑠, 𝑌, 𝑊) ∈ ℂ)
189	188	adantlr 728	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → if(0 < 𝑠, 𝑌, 𝑊) ∈ ℂ)
190	124	recnd 11376	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → ((𝐷‘𝑛)‘𝑠) ∈ ℂ)
191	187, 189, 190	subdird 11812	. . . . . 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 7466	. . . . 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 11368	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → ((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠)) ∈ ℂ)
194	189, 190	mulcld 11368	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (if(0 < 𝑠, 𝑌, 𝑊) · ((𝐷‘𝑛)‘𝑠)) ∈ ℂ)
195	193, 194	npcand 11712	. . . . 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 2802	. . . 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 26157	. . 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 2800	. 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 2802	1 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐸‘𝑛) + (𝑂 / 2)) = ∫𝐴((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠)) d𝑠)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2147 ≠ wne 2959 ∀wral 3080 {crab 3416 ⊆ wss 3900 ∖ cdif 4048 ifcif 4485 {csn 4588 class class class wbr 5108 ↦ cmpt 5192 dom cdm 5658 ran crn 5659 ↾ cres 5660 ⟶wf 6562 ‘cfv 6566 (class class class)co 7451 ↑_m cmap 8885 ℂcc 11237 ℝcr 11238 0cc0 11239 1c1 11240 + caddc 11242 · cmul 11244 +∞cpnf 11379 -∞cmnf 11380 ℝ^*cxr 11381 < clt 11382 − cmin 11580 -cneg 11581 / cdiv 12013 ℕcn 12375 2c2 12437 (,)cioo 13516 [,]cicc 13519 ...cfz 13679 ..^cfzo 13828 mod cmo 14049 sincsin 16273 πcpi 16276 TopOpenctopn 17642 ℂ_fldccnfld 21729 –cn→ccncf 25252 volcvol 25839 𝐿¹cibl 25993 ∫citg 25994 lim_ℂ climc 26237 D cdv 26238

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5238 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783 ax-inf2 9679 ax-cc 10552 ax-cnex 11295 ax-resscn 11296 ax-1cn 11297 ax-icn 11298 ax-addcl 11299 ax-addrcl 11300 ax-mulcl 11301 ax-mulrcl 11302 ax-mulcom 11303 ax-addass 11304 ax-mulass 11305 ax-distr 11306 ax-i2m1 11307 ax-1ne0 11308 ax-1rid 11309 ax-rnegex 11310 ax-rrecex 11311 ax-cnre 11312 ax-pre-lttri 11313 ax-pre-lttrn 11314 ax-pre-ltadd 11315 ax-pre-mulgt0 11316 ax-pre-sup 11317 ax-addf 11318

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-symdif 4201 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4841 df-uni 4873 df-int 4913 df-iun 4958 df-iin 4959 df-disj 5076 df-br 5109 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-se 5612 df-we 5613 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-pred 6330 df-ord 6391 df-on 6392 df-lim 6393 df-suc 6394 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-isom 6575 df-riota 7408 df-ov 7454 df-oprab 7455 df-mpo 7456 df-of 7725 df-ofr 7726 df-om 7912 df-1st 8035 df-2nd 8036 df-supp 8215 df-frecs 8337 df-wrecs 8368 df-recs 8417 df-rdg 8456 df-1o 8514 df-2o 8515 df-oadd 8518 df-omul 8519 df-er 8755 df-map 8887 df-pm 8888 df-ixp 8964 df-en 9012 df-dom 9013 df-sdom 9014 df-fin 9015 df-fsupp 9392 df-fi 9440 df-sup 9471 df-inf 9472 df-oi 9541 df-dju 10019 df-card 10057 df-acn 10060 df-pnf 11384 df-mnf 11385 df-xr 11386 df-ltxr 11387 df-le 11388 df-sub 11582 df-neg 11583 df-div 12014 df-nn 12376 df-2 12445 df-3 12446 df-4 12447 df-5 12448 df-6 12449 df-7 12450 df-8 12451 df-9 12452 df-n0 12647 df-z 12734 df-dec 12855 df-uz 13006 df-q 13116 df-rp 13161 df-xneg 13281 df-xadd 13282 df-xmul 13283 df-ioo 13520 df-ioc 13521 df-ico 13522 df-icc 13523 df-fz 13680 df-fzo 13829 df-fl 13972 df-mod 14050 df-seq 14185 df-exp 14245 df-fac 14459 df-bc 14488 df-hash 14516 df-shft 15263 df-cj 15309 df-re 15310 df-im 15311 df-sqrt 15445 df-abs 15446 df-limsup 15682 df-clim 15699 df-rlim 15700 df-sum 15898 df-ef 16277 df-sin 16279 df-cos 16280 df-pi 16282 df-struct 17369 df-sets 17386 df-slot 17404 df-ndx 17416 df-base 17432 df-ress 17453 df-plusg 17486 df-mulr 17487 df-starv 17488 df-sca 17489 df-vsca 17490 df-ip 17491 df-tset 17492 df-ple 17493 df-ds 17495 df-unif 17496 df-hom 17497 df-cco 17498 df-euf 17499 df-rest 17643 df-topn 17644 df-0g 17662 df-gsum 17663 df-topgen 17664 df-pt 17665 df-prds 17668 df-xrs 17724 df-qtop 17729 df-imas 17730 df-xps 17732 df-mre 17806 df-mrc 17807 df-acs 17809 df-mgm 18866 df-sgrp 18958 df-mnd 18974 df-submnd 19029 df-mulg 19328 df-cntz 19582 df-cmn 20047 df-psmet 21721 df-xmet 21722 df-met 21723 df-bl 21724 df-mopn 21725 df-fbas 21726 df-fg 21727 df-cnfld 21730 df-top 23267 df-topon 23284 df-topsp 23306 df-bases 23319 df-cld 23392 df-ntr 23393 df-cls 23394 df-nei 23471 df-lp 23509 df-perf 23510 df-cn 23600 df-cnp 23601 df-t1 23687 df-haus 23688 df-cmp 23760 df-tx 23936 df-hmeo 24129 df-fil 24220 df-fm 24312 df-flim 24313 df-flf 24314 df-xms 24694 df-ms 24695 df-tms 24696 df-cncf 25254 df-ovol 25840 df-vol 25841 df-mbf 25995 df-itg1 25996 df-itg2 25997 df-ibl 25998 df-itg 25999 df-0p 26046 df-limc 26241 df-dv 26242

This theorem is used by: fourierdlem103 47492 fourierdlem104 47493

W3C validator