fourierdlem107 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem107 46169

Description: The integral of a piecewise continuous periodic function 𝐹 is unchanged if the domain is shifted by any positive value 𝑋. This lemma generalizes fourierdlem92 46154 where the integral was shifted by the exact period. This lemma uses local definitions, so that the proof is more readable. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Hypotheses**
Ref	Expression
fourierdlem107.a	⊢ (𝜑 → 𝐴 ∈ ℝ)
fourierdlem107.b	⊢ (𝜑 → 𝐵 ∈ ℝ)
fourierdlem107.t	⊢ 𝑇 = (𝐵 − 𝐴)
fourierdlem107.x	⊢ (𝜑 → 𝑋 ∈ ℝ⁺)
fourierdlem107.p	⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
fourierdlem107.m	⊢ (𝜑 → 𝑀 ∈ ℕ)
fourierdlem107.q	⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀))
fourierdlem107.f	⊢ (𝜑 → 𝐹:ℝ⟶ℂ)
fourierdlem107.fper	⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥))
fourierdlem107.fcn	⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
fourierdlem107.r	⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)))
fourierdlem107.l	⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1))))
fourierdlem107.o	⊢ 𝑂 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = (𝐴 − 𝑋) ∧ (𝑝‘𝑚) = 𝐴) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
fourierdlem107.h	⊢ 𝐻 = ({(𝐴 − 𝑋), 𝐴} ∪ {𝑦 ∈ ((𝐴 − 𝑋)[,]𝐴) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})
fourierdlem107.n	⊢ 𝑁 = ((♯‘𝐻) − 1)
fourierdlem107.s	⊢ 𝑆 = (℩𝑓𝑓 Isom < , < ((0...𝑁), 𝐻))
fourierdlem107.e	⊢ 𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)))
fourierdlem107.z	⊢ 𝑍 = (𝑦 ∈ (𝐴(,]𝐵) ↦ if(𝑦 = 𝐵, 𝐴, 𝑦))
fourierdlem107.i	⊢ 𝐼 = (𝑥 ∈ ℝ ↦ sup({𝑖 ∈ (0..^𝑀) ∣ (𝑄‘𝑖) ≤ (𝑍‘(𝐸‘𝑥))}, ℝ, < ))

**Assertion**
Ref	Expression
fourierdlem107	⊢ (𝜑 → ∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 = ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)

Distinct variable groups: 𝐴,𝑓,𝑘,𝑦 𝐴,𝑖,𝑥,𝑘,𝑦 𝐴,𝑚,𝑝,𝑖 𝐵,𝑓,𝑘,𝑦 𝐵,𝑖,𝑥 𝐵,𝑚,𝑝 𝑓,𝐸,𝑘,𝑦 𝑖,𝐸,𝑥 𝑖,𝐹,𝑥,𝑦 𝑓,𝐻,𝑦 𝑥,𝐻 𝑓,𝐼,𝑘,𝑦 𝑖,𝐼,𝑥 𝑥,𝐿,𝑦 𝑖,𝑀,𝑥,𝑦 𝑚,𝑀,𝑝 𝑓,𝑁,𝑘,𝑦 𝑖,𝑁,𝑥 𝑚,𝑁,𝑝 𝑄,𝑓,𝑘,𝑦 𝑄,𝑖,𝑥 𝑄,𝑚,𝑝 𝑥,𝑅,𝑦 𝑆,𝑓,𝑘,𝑦 𝑆,𝑖,𝑥 𝑆,𝑝 𝑇,𝑓,𝑘,𝑦 𝑇,𝑖,𝑥 𝑇,𝑚,𝑝 𝑓,𝑋,𝑦 𝑖,𝑋,𝑚,𝑝 𝑥,𝑋 𝑖,𝑍,𝑥,𝑦 𝜑,𝑓,𝑘,𝑦 𝜑,𝑖,𝑥 Allowed substitution hints: 𝜑(𝑚,𝑝) 𝑃(𝑥,𝑦,𝑓,𝑖,𝑘,𝑚,𝑝) 𝑅(𝑓,𝑖,𝑘,𝑚,𝑝) 𝑆(𝑚) 𝐸(𝑚,𝑝) 𝐹(𝑓,𝑘,𝑚,𝑝) 𝐻(𝑖,𝑘,𝑚,𝑝) 𝐼(𝑚,𝑝) 𝐿(𝑓,𝑖,𝑘,𝑚,𝑝) 𝑀(𝑓,𝑘) 𝑂(𝑥,𝑦,𝑓,𝑖,𝑘,𝑚,𝑝) 𝑋(𝑘) 𝑍(𝑓,𝑘,𝑚,𝑝)

Proof of Theorem fourierdlem107 Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fourierdlem107.t	. . . . . . . . . . . . . . . . 17 ⊢ 𝑇 = (𝐵 − 𝐴)
2	1	oveq2i 7442	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 − 𝑋) + 𝑇) = ((𝐴 − 𝑋) + (𝐵 − 𝐴))
3		fourierdlem107.a	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐴 ∈ ℝ)
4	3	recnd 11287	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐴 ∈ ℂ)
5		fourierdlem107.x	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑋 ∈ ℝ⁺)
6	5	rpred 13075	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑋 ∈ ℝ)
7	6	recnd 11287	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑋 ∈ ℂ)
8		fourierdlem107.b	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐵 ∈ ℝ)
9	8	recnd 11287	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐵 ∈ ℂ)
10	4, 7, 9, 4	subadd4b 45233	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝐴 − 𝑋) + (𝐵 − 𝐴)) = ((𝐴 − 𝐴) + (𝐵 − 𝑋)))
11	2, 10	eqtrid 2787	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝐴 − 𝑋) + 𝑇) = ((𝐴 − 𝐴) + (𝐵 − 𝑋)))
12	4	subidd 11606	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐴 − 𝐴) = 0)
13	12	oveq1d 7446	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝐴 − 𝐴) + (𝐵 − 𝑋)) = (0 + (𝐵 − 𝑋)))
14	8, 6	resubcld 11689	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐵 − 𝑋) ∈ ℝ)
15	14	recnd 11287	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐵 − 𝑋) ∈ ℂ)
16	15	addlidd 11460	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (0 + (𝐵 − 𝑋)) = (𝐵 − 𝑋))
17	11, 13, 16	3eqtrd 2779	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝐴 − 𝑋) + 𝑇) = (𝐵 − 𝑋))
18	1	oveq2i 7442	. . . . . . . . . . . . . . 15 ⊢ (𝐴 + 𝑇) = (𝐴 + (𝐵 − 𝐴))
19	4, 9	pncan3d 11621	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐴 + (𝐵 − 𝐴)) = 𝐵)
20	18, 19	eqtrid 2787	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐴 + 𝑇) = 𝐵)
21	17, 20	oveq12d 7449	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((𝐴 − 𝑋) + 𝑇)[,](𝐴 + 𝑇)) = ((𝐵 − 𝑋)[,]𝐵))
22	21	eqcomd 2741	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐵 − 𝑋)[,]𝐵) = (((𝐴 − 𝑋) + 𝑇)[,](𝐴 + 𝑇)))
23	22	itgeq1d 45913	. . . . . . . . . . 11 ⊢ (𝜑 → ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 = ∫(((𝐴 − 𝑋) + 𝑇)[,](𝐴 + 𝑇))(𝐹‘𝑥) d𝑥)
24	3, 6	resubcld 11689	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 − 𝑋) ∈ ℝ)
25		fourierdlem107.o	. . . . . . . . . . . . 13 ⊢ 𝑂 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = (𝐴 − 𝑋) ∧ (𝑝‘𝑚) = 𝐴) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
26		fveq2 6907	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 = 𝑗 → (𝑝‘𝑖) = (𝑝‘𝑗))
27		oveq1 7438	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 = 𝑗 → (𝑖 + 1) = (𝑗 + 1))
28	27	fveq2d 6911	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 = 𝑗 → (𝑝‘(𝑖 + 1)) = (𝑝‘(𝑗 + 1)))
29	26, 28	breq12d 5161	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 = 𝑗 → ((𝑝‘𝑖) < (𝑝‘(𝑖 + 1)) ↔ (𝑝‘𝑗) < (𝑝‘(𝑗 + 1))))
30	29	cbvralvw 3235	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)) ↔ ∀𝑗 ∈ (0..^𝑚)(𝑝‘𝑗) < (𝑝‘(𝑗 + 1)))
31	30	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ ℕ → (∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)) ↔ ∀𝑗 ∈ (0..^𝑚)(𝑝‘𝑗) < (𝑝‘(𝑗 + 1))))
32	31	anbi2d 630	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ ℕ → ((((𝑝‘0) = (𝐴 − 𝑋) ∧ (𝑝‘𝑚) = 𝐴) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1))) ↔ (((𝑝‘0) = (𝐴 − 𝑋) ∧ (𝑝‘𝑚) = 𝐴) ∧ ∀𝑗 ∈ (0..^𝑚)(𝑝‘𝑗) < (𝑝‘(𝑗 + 1)))))
33	32	rabbidv 3441	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ ℕ → {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = (𝐴 − 𝑋) ∧ (𝑝‘𝑚) = 𝐴) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))} = {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = (𝐴 − 𝑋) ∧ (𝑝‘𝑚) = 𝐴) ∧ ∀𝑗 ∈ (0..^𝑚)(𝑝‘𝑗) < (𝑝‘(𝑗 + 1)))})
34	33	mpteq2ia 5251	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = (𝐴 − 𝑋) ∧ (𝑝‘𝑚) = 𝐴) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = (𝐴 − 𝑋) ∧ (𝑝‘𝑚) = 𝐴) ∧ ∀𝑗 ∈ (0..^𝑚)(𝑝‘𝑗) < (𝑝‘(𝑗 + 1)))})
35	25, 34	eqtri 2763	. . . . . . . . . . . 12 ⊢ 𝑂 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = (𝐴 − 𝑋) ∧ (𝑝‘𝑚) = 𝐴) ∧ ∀𝑗 ∈ (0..^𝑚)(𝑝‘𝑗) < (𝑝‘(𝑗 + 1)))})
36		fourierdlem107.p	. . . . . . . . . . . . . . 15 ⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
37		fourierdlem107.m	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑀 ∈ ℕ)
38		fourierdlem107.q	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀))
39	3, 5	ltsubrpd 13107	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐴 − 𝑋) < 𝐴)
40		fourierdlem107.h	. . . . . . . . . . . . . . 15 ⊢ 𝐻 = ({(𝐴 − 𝑋), 𝐴} ∪ {𝑦 ∈ ((𝐴 − 𝑋)[,]𝐴) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})
41		fourierdlem107.n	. . . . . . . . . . . . . . 15 ⊢ 𝑁 = ((♯‘𝐻) − 1)
42		fourierdlem107.s	. . . . . . . . . . . . . . 15 ⊢ 𝑆 = (℩𝑓𝑓 Isom < , < ((0...𝑁), 𝐻))
43	1, 36, 37, 38, 24, 3, 39, 25, 40, 41, 42	fourierdlem54 46116	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑁 ∈ ℕ ∧ 𝑆 ∈ (𝑂‘𝑁)) ∧ 𝑆 Isom < , < ((0...𝑁), 𝐻)))
44	43	simpld 494	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑁 ∈ ℕ ∧ 𝑆 ∈ (𝑂‘𝑁)))
45	44	simpld 494	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℕ)
46	8, 3	resubcld 11689	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ)
47	1, 46	eqeltrid 2843	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑇 ∈ ℝ)
48	44	simprd 495	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑆 ∈ (𝑂‘𝑁))
49	24	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 − 𝑋)[,]𝐴)) → (𝐴 − 𝑋) ∈ ℝ)
50	3	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 − 𝑋)[,]𝐴)) → 𝐴 ∈ ℝ)
51		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 − 𝑋)[,]𝐴)) → 𝑥 ∈ ((𝐴 − 𝑋)[,]𝐴))
52		eliccre 45458	. . . . . . . . . . . . . 14 ⊢ (((𝐴 − 𝑋) ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ 𝑥 ∈ ((𝐴 − 𝑋)[,]𝐴)) → 𝑥 ∈ ℝ)
53	49, 50, 51, 52	syl3anc 1370	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 − 𝑋)[,]𝐴)) → 𝑥 ∈ ℝ)
54		fourierdlem107.fper	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥))
55	53, 54	syldan 591	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 − 𝑋)[,]𝐴)) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥))
56		fveq2 6907	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑗 → (𝑆‘𝑖) = (𝑆‘𝑗))
57	56	oveq1d 7446	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑗 → ((𝑆‘𝑖) + 𝑇) = ((𝑆‘𝑗) + 𝑇))
58	57	cbvmptv 5261	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (0...𝑁) ↦ ((𝑆‘𝑖) + 𝑇)) = (𝑗 ∈ (0...𝑁) ↦ ((𝑆‘𝑗) + 𝑇))
59		eqid 2735	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = ((𝐴 − 𝑋) + 𝑇) ∧ (𝑝‘𝑚) = (𝐴 + 𝑇)) ∧ ∀𝑗 ∈ (0..^𝑚)(𝑝‘𝑗) < (𝑝‘(𝑗 + 1)))}) = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = ((𝐴 − 𝑋) + 𝑇) ∧ (𝑝‘𝑚) = (𝐴 + 𝑇)) ∧ ∀𝑗 ∈ (0..^𝑚)(𝑝‘𝑗) < (𝑝‘(𝑗 + 1)))})
60		fourierdlem107.f	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹:ℝ⟶ℂ)
61	37	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝑀 ∈ ℕ)
62	38	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝑄 ∈ (𝑃‘𝑀))
63	60	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝐹:ℝ⟶ℂ)
64	54	adantlr 715	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥))
65		fourierdlem107.fcn	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
66	65	adantlr 715	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
67	24	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝐴 − 𝑋) ∈ ℝ)
68	67	rexrd 11309	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝐴 − 𝑋) ∈ ℝ^*)
69		pnfxr 11313	. . . . . . . . . . . . . . 15 ⊢ +∞ ∈ ℝ^*
70	69	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → +∞ ∈ ℝ^*)
71	3	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝐴 ∈ ℝ)
72	39	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝐴 − 𝑋) < 𝐴)
73	3	ltpnfd 13161	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐴 < +∞)
74	73	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝐴 < +∞)
75	68, 70, 71, 72, 74	eliood 45451	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝐴 ∈ ((𝐴 − 𝑋)(,)+∞))
76		fourierdlem107.e	. . . . . . . . . . . . 13 ⊢ 𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)))
77		fourierdlem107.z	. . . . . . . . . . . . 13 ⊢ 𝑍 = (𝑦 ∈ (𝐴(,]𝐵) ↦ if(𝑦 = 𝐵, 𝐴, 𝑦))
78		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝑗 ∈ (0..^𝑁))
79		eqid 2735	. . . . . . . . . . . . 13 ⊢ ((𝑆‘(𝑗 + 1)) − (𝐸‘(𝑆‘(𝑗 + 1)))) = ((𝑆‘(𝑗 + 1)) − (𝐸‘(𝑆‘(𝑗 + 1))))
80		eqid 2735	. . . . . . . . . . . . 13 ⊢ (𝐹 ↾ ((𝑍‘(𝐸‘(𝑆‘𝑗)))(,)(𝐸‘(𝑆‘(𝑗 + 1))))) = (𝐹 ↾ ((𝑍‘(𝐸‘(𝑆‘𝑗)))(,)(𝐸‘(𝑆‘(𝑗 + 1)))))
81		eqid 2735	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (((𝑍‘(𝐸‘(𝑆‘𝑗))) + ((𝑆‘(𝑗 + 1)) − (𝐸‘(𝑆‘(𝑗 + 1)))))(,)((𝐸‘(𝑆‘(𝑗 + 1))) + ((𝑆‘(𝑗 + 1)) − (𝐸‘(𝑆‘(𝑗 + 1)))))) ↦ ((𝐹 ↾ ((𝑍‘(𝐸‘(𝑆‘𝑗)))(,)(𝐸‘(𝑆‘(𝑗 + 1)))))‘(𝑦 − ((𝑆‘(𝑗 + 1)) − (𝐸‘(𝑆‘(𝑗 + 1))))))) = (𝑦 ∈ (((𝑍‘(𝐸‘(𝑆‘𝑗))) + ((𝑆‘(𝑗 + 1)) − (𝐸‘(𝑆‘(𝑗 + 1)))))(,)((𝐸‘(𝑆‘(𝑗 + 1))) + ((𝑆‘(𝑗 + 1)) − (𝐸‘(𝑆‘(𝑗 + 1)))))) ↦ ((𝐹 ↾ ((𝑍‘(𝐸‘(𝑆‘𝑗)))(,)(𝐸‘(𝑆‘(𝑗 + 1)))))‘(𝑦 − ((𝑆‘(𝑗 + 1)) − (𝐸‘(𝑆‘(𝑗 + 1)))))))
82		fourierdlem107.i	. . . . . . . . . . . . 13 ⊢ 𝐼 = (𝑥 ∈ ℝ ↦ sup({𝑖 ∈ (0..^𝑀) ∣ (𝑄‘𝑖) ≤ (𝑍‘(𝐸‘𝑥))}, ℝ, < ))
83	36, 1, 61, 62, 63, 64, 66, 67, 75, 25, 40, 41, 42, 76, 77, 78, 79, 80, 81, 82	fourierdlem90 46152	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝐹 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) ∈ (((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))–cn→ℂ))
84		fourierdlem107.r	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)))
85	84	adantlr 715	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)))
86		eqid 2735	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ (0..^𝑀) ↦ 𝑅) = (𝑖 ∈ (0..^𝑀) ↦ 𝑅)
87	36, 1, 61, 62, 63, 64, 66, 85, 67, 75, 25, 40, 41, 42, 76, 77, 78, 79, 82, 86	fourierdlem89 46151	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → if((𝑍‘(𝐸‘(𝑆‘𝑗))) = (𝑄‘(𝐼‘(𝑆‘𝑗))), ((𝑖 ∈ (0..^𝑀) ↦ 𝑅)‘(𝐼‘(𝑆‘𝑗))), (𝐹‘(𝑍‘(𝐸‘(𝑆‘𝑗))))) ∈ ((𝐹 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) lim_ℂ (𝑆‘𝑗)))
88		fourierdlem107.l	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1))))
89	88	adantlr 715	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1))))
90		eqid 2735	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ (0..^𝑀) ↦ 𝐿) = (𝑖 ∈ (0..^𝑀) ↦ 𝐿)
91	36, 1, 61, 62, 63, 64, 66, 89, 67, 75, 25, 40, 41, 42, 76, 77, 78, 79, 82, 90	fourierdlem91 46153	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → if((𝐸‘(𝑆‘(𝑗 + 1))) = (𝑄‘((𝐼‘(𝑆‘𝑗)) + 1)), ((𝑖 ∈ (0..^𝑀) ↦ 𝐿)‘(𝐼‘(𝑆‘𝑗))), (𝐹‘(𝐸‘(𝑆‘(𝑗 + 1))))) ∈ ((𝐹 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) lim_ℂ (𝑆‘(𝑗 + 1))))
92	24, 3, 35, 45, 47, 48, 55, 58, 59, 60, 83, 87, 91	fourierdlem92 46154	. . . . . . . . . . 11 ⊢ (𝜑 → ∫(((𝐴 − 𝑋) + 𝑇)[,](𝐴 + 𝑇))(𝐹‘𝑥) d𝑥 = ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥)
93	23, 92	eqtrd 2775	. . . . . . . . . 10 ⊢ (𝜑 → ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 = ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥)
94	60	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐵 − 𝑋)[,]𝐵)) → 𝐹:ℝ⟶ℂ)
95	14	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐵 − 𝑋)[,]𝐵)) → (𝐵 − 𝑋) ∈ ℝ)
96	8	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐵 − 𝑋)[,]𝐵)) → 𝐵 ∈ ℝ)
97		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐵 − 𝑋)[,]𝐵)) → 𝑥 ∈ ((𝐵 − 𝑋)[,]𝐵))
98		eliccre 45458	. . . . . . . . . . . . 13 ⊢ (((𝐵 − 𝑋) ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑥 ∈ ((𝐵 − 𝑋)[,]𝐵)) → 𝑥 ∈ ℝ)
99	95, 96, 97, 98	syl3anc 1370	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐵 − 𝑋)[,]𝐵)) → 𝑥 ∈ ℝ)
100	94, 99	ffvelcdmd 7105	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐵 − 𝑋)[,]𝐵)) → (𝐹‘𝑥) ∈ ℂ)
101	14	rexrd 11309	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐵 − 𝑋) ∈ ℝ^*)
102	69	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → +∞ ∈ ℝ^*)
103	8, 5	ltsubrpd 13107	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐵 − 𝑋) < 𝐵)
104	8	ltpnfd 13161	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 < +∞)
105	101, 102, 8, 103, 104	eliood 45451	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ∈ ((𝐵 − 𝑋)(,)+∞))
106	36, 1, 37, 38, 60, 54, 65, 84, 88, 14, 105	fourierdlem105 46167	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ ((𝐵 − 𝑋)[,]𝐵) ↦ (𝐹‘𝑥)) ∈ 𝐿¹)
107	100, 106	itgcl 25834	. . . . . . . . . 10 ⊢ (𝜑 → ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 ∈ ℂ)
108	93, 107	eqeltrrd 2840	. . . . . . . . 9 ⊢ (𝜑 → ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 ∈ ℂ)
109	108	subidd 11606	. . . . . . . 8 ⊢ (𝜑 → (∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥) = 0)
110	109	eqcomd 2741	. . . . . . 7 ⊢ (𝜑 → 0 = (∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥))
111	110	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → 0 = (∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥))
112	24	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (𝐴 − 𝑋) ∈ ℝ)
113	3	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → 𝐴 ∈ ℝ)
114	14	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (𝐵 − 𝑋) ∈ ℝ)
115	36, 37, 38	fourierdlem11 46074	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵))
116	115	simp3d 1143	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 < 𝐵)
117	3, 8, 116	ltled 11407	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
118	117	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → 𝐴 ≤ 𝐵)
119	3, 8, 6	lesub1d 11868	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ (𝐴 − 𝑋) ≤ (𝐵 − 𝑋)))
120	119	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (𝐴 ≤ 𝐵 ↔ (𝐴 − 𝑋) ≤ (𝐵 − 𝑋)))
121	118, 120	mpbid 232	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (𝐴 − 𝑋) ≤ (𝐵 − 𝑋))
122	8	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → 𝐵 ∈ ℝ)
123	6	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → 𝑋 ∈ ℝ)
124		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → 𝑇 < 𝑋)
125	1, 124	eqbrtrrid 5184	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (𝐵 − 𝐴) < 𝑋)
126	122, 113, 123, 125	ltsub23d 11866	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (𝐵 − 𝑋) < 𝐴)
127	114, 113, 126	ltled 11407	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (𝐵 − 𝑋) ≤ 𝐴)
128	112, 113, 114, 121, 127	eliccd 45457	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (𝐵 − 𝑋) ∈ ((𝐴 − 𝑋)[,]𝐴))
129	60	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 − 𝑋)[,]𝐴)) → 𝐹:ℝ⟶ℂ)
130	129, 53	ffvelcdmd 7105	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 − 𝑋)[,]𝐴)) → (𝐹‘𝑥) ∈ ℂ)
131	130	adantlr 715	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑇 < 𝑋) ∧ 𝑥 ∈ ((𝐴 − 𝑋)[,]𝐴)) → (𝐹‘𝑥) ∈ ℂ)
132	24	rexrd 11309	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 − 𝑋) ∈ ℝ^*)
133	3, 8, 6, 116	ltsub1dd 11873	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 − 𝑋) < (𝐵 − 𝑋))
134	14	ltpnfd 13161	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐵 − 𝑋) < +∞)
135	132, 102, 14, 133, 134	eliood 45451	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵 − 𝑋) ∈ ((𝐴 − 𝑋)(,)+∞))
136	36, 1, 37, 38, 60, 54, 65, 84, 88, 24, 135	fourierdlem105 46167	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋)) ↦ (𝐹‘𝑥)) ∈ 𝐿¹)
137	136	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (𝑥 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋)) ↦ (𝐹‘𝑥)) ∈ 𝐿¹)
138	37	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → 𝑀 ∈ ℕ)
139	38	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → 𝑄 ∈ (𝑃‘𝑀))
140	60	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → 𝐹:ℝ⟶ℂ)
141	54	adantlr 715	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑇 < 𝑋) ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥))
142	65	adantlr 715	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑇 < 𝑋) ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
143	84	adantlr 715	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑇 < 𝑋) ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)))
144	88	adantlr 715	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑇 < 𝑋) ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1))))
145	101	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (𝐵 − 𝑋) ∈ ℝ^*)
146	69	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → +∞ ∈ ℝ^*)
147	113	ltpnfd 13161	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → 𝐴 < +∞)
148	145, 146, 113, 126, 147	eliood 45451	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → 𝐴 ∈ ((𝐵 − 𝑋)(,)+∞))
149	36, 1, 138, 139, 140, 141, 142, 143, 144, 114, 148	fourierdlem105 46167	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (𝑥 ∈ ((𝐵 − 𝑋)[,]𝐴) ↦ (𝐹‘𝑥)) ∈ 𝐿¹)
150	112, 113, 128, 131, 137, 149	itgspliticc 25887	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 = (∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 + ∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥))
151	150	oveq1d 7446	. . . . . 6 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥) = ((∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 + ∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥) − ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥))
152	60	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → 𝐹:ℝ⟶ℂ)
153	24	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → (𝐴 − 𝑋) ∈ ℝ)
154	14	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → (𝐵 − 𝑋) ∈ ℝ)
155		simpr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → 𝑥 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋)))
156		eliccre 45458	. . . . . . . . . . 11 ⊢ (((𝐴 − 𝑋) ∈ ℝ ∧ (𝐵 − 𝑋) ∈ ℝ ∧ 𝑥 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → 𝑥 ∈ ℝ)
157	153, 154, 155, 156	syl3anc 1370	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → 𝑥 ∈ ℝ)
158	152, 157	ffvelcdmd 7105	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → (𝐹‘𝑥) ∈ ℂ)
159	158, 136	itgcl 25834	. . . . . . . 8 ⊢ (𝜑 → ∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 ∈ ℂ)
160	159	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → ∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 ∈ ℂ)
161	60	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐵 − 𝑋)[,]𝐴)) → 𝐹:ℝ⟶ℂ)
162	14	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐵 − 𝑋)[,]𝐴)) → (𝐵 − 𝑋) ∈ ℝ)
163	3	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐵 − 𝑋)[,]𝐴)) → 𝐴 ∈ ℝ)
164		simpr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐵 − 𝑋)[,]𝐴)) → 𝑥 ∈ ((𝐵 − 𝑋)[,]𝐴))
165		eliccre 45458	. . . . . . . . . . 11 ⊢ (((𝐵 − 𝑋) ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ 𝑥 ∈ ((𝐵 − 𝑋)[,]𝐴)) → 𝑥 ∈ ℝ)
166	162, 163, 164, 165	syl3anc 1370	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐵 − 𝑋)[,]𝐴)) → 𝑥 ∈ ℝ)
167	161, 166	ffvelcdmd 7105	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐵 − 𝑋)[,]𝐴)) → (𝐹‘𝑥) ∈ ℂ)
168	167	adantlr 715	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑇 < 𝑋) ∧ 𝑥 ∈ ((𝐵 − 𝑋)[,]𝐴)) → (𝐹‘𝑥) ∈ ℂ)
169	168, 149	itgcl 25834	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → ∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 ∈ ℂ)
170	108	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 ∈ ℂ)
171	160, 169, 170	addsubassd 11638	. . . . . 6 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → ((∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 + ∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥) − ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥) = (∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 + (∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥)))
172	111, 151, 171	3eqtrd 2779	. . . . 5 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → 0 = (∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 + (∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥)))
173	172	oveq2d 7447	. . . 4 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 − 0) = (∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 − (∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 + (∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥))))
174	160	subid1d 11607	. . . 4 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 − 0) = ∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥)
175	159	subidd 11606	. . . . . . 7 ⊢ (𝜑 → (∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 − ∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥) = 0)
176	175	oveq1d 7446	. . . . . 6 ⊢ (𝜑 → ((∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 − ∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥) − (∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥)) = (0 − (∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥)))
177	176	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → ((∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 − ∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥) − (∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥)) = (0 − (∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥)))
178	169, 170	subcld 11618	. . . . . 6 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥) ∈ ℂ)
179	160, 160, 178	subsub4d 11649	. . . . 5 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → ((∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 − ∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥) − (∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥)) = (∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 − (∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 + (∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥))))
180		df-neg 11493	. . . . . 6 ⊢ -(∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥) = (0 − (∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥))
181	169, 170	negsubdi2d 11634	. . . . . 6 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → -(∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥) = (∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥))
182	180, 181	eqtr3id 2789	. . . . 5 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (0 − (∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥)) = (∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥))
183	177, 179, 182	3eqtr3d 2783	. . . 4 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 − (∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 + (∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥))) = (∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥))
184	173, 174, 183	3eqtr3d 2783	. . 3 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → ∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 = (∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥))
185	107	subidd 11606	. . . . . . . 8 ⊢ (𝜑 → (∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 − ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥) = 0)
186	185	eqcomd 2741	. . . . . . 7 ⊢ (𝜑 → 0 = (∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 − ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥))
187	186	oveq2d 7447	. . . . . 6 ⊢ (𝜑 → (∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 + 0) = (∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 + (∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 − ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥)))
188	187	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 + 0) = (∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 + (∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 − ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥)))
189	169	addridd 11459	. . . . 5 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 + 0) = ∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥)
190	114, 122, 113, 127, 118	eliccd 45457	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → 𝐴 ∈ ((𝐵 − 𝑋)[,]𝐵))
191	100	adantlr 715	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑇 < 𝑋) ∧ 𝑥 ∈ ((𝐵 − 𝑋)[,]𝐵)) → (𝐹‘𝑥) ∈ ℂ)
192	3, 8	iccssred 13471	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
193	60, 192	feqresmpt 6978	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹 ↾ (𝐴[,]𝐵)) = (𝑥 ∈ (𝐴[,]𝐵) ↦ (𝐹‘𝑥)))
194	60, 192	fssresd 6776	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹 ↾ (𝐴[,]𝐵)):(𝐴[,]𝐵)⟶ℂ)
195		ioossicc 13470	. . . . . . . . . . . . . . 15 ⊢ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))
196	3	rexrd 11309	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
197	196	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐴 ∈ ℝ^*)
198	8	rexrd 11309	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
199	198	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐵 ∈ ℝ^*)
200	36, 37, 38	fourierdlem15 46078	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶(𝐴[,]𝐵))
201	200	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶(𝐴[,]𝐵))
202		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0..^𝑀))
203	197, 199, 201, 202	fourierdlem8 46071	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))) ⊆ (𝐴[,]𝐵))
204	195, 203	sstrid 4007	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ (𝐴[,]𝐵))
205	204	resabs1d 6028	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ (𝐴[,]𝐵)) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))))
206	205, 65	eqeltrd 2839	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ (𝐴[,]𝐵)) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
207	205	eqcomd 2741	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝐹 ↾ (𝐴[,]𝐵)) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))))
208	207	oveq1d 7446	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)) = (((𝐹 ↾ (𝐴[,]𝐵)) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)))
209	84, 208	eleqtrd 2841	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ (((𝐹 ↾ (𝐴[,]𝐵)) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)))
210	207	oveq1d 7446	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1))) = (((𝐹 ↾ (𝐴[,]𝐵)) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1))))
211	88, 210	eleqtrd 2841	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ (((𝐹 ↾ (𝐴[,]𝐵)) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1))))
212	36, 37, 38, 194, 206, 209, 211	fourierdlem69 46131	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹 ↾ (𝐴[,]𝐵)) ∈ 𝐿¹)
213	193, 212	eqeltrrd 2840	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ (𝐹‘𝑥)) ∈ 𝐿¹)
214	213	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (𝑥 ∈ (𝐴[,]𝐵) ↦ (𝐹‘𝑥)) ∈ 𝐿¹)
215	114, 122, 190, 191, 149, 214	itgspliticc 25887	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 = (∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 + ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥))
216	215	oveq2d 7447	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 − ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥) = (∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 − (∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 + ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)))
217	216	oveq2d 7447	. . . . . 6 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 + (∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 − ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥)) = (∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 + (∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 − (∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 + ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥))))
218	107	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 ∈ ℂ)
219	215, 218	eqeltrrd 2840	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 + ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥) ∈ ℂ)
220	169, 218, 219	addsub12d 11641	. . . . . 6 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 + (∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 − (∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 + ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥))) = (∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 + (∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − (∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 + ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥))))
221	60	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐹:ℝ⟶ℂ)
222	3	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐴 ∈ ℝ)
223	8	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐵 ∈ ℝ)
224		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ∈ (𝐴[,]𝐵))
225		eliccre 45458	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ∈ ℝ)
226	222, 223, 224, 225	syl3anc 1370	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ∈ ℝ)
227	221, 226	ffvelcdmd 7105	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℂ)
228	227, 213	itgcl 25834	. . . . . . . . . . 11 ⊢ (𝜑 → ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥 ∈ ℂ)
229	228	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥 ∈ ℂ)
230	169, 169, 229	subsub4d 11649	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → ((∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥) − ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥) = (∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − (∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 + ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)))
231	230	eqcomd 2741	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − (∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 + ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)) = ((∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥) − ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥))
232	231	oveq2d 7447	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 + (∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − (∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 + ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥))) = (∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 + ((∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥) − ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)))
233	169	subidd 11606	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥) = 0)
234	233	oveq1d 7446	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → ((∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥) − ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥) = (0 − ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥))
235		df-neg 11493	. . . . . . . . 9 ⊢ -∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥 = (0 − ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)
236	234, 235	eqtr4di 2793	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → ((∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥) − ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥) = -∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)
237	236	oveq2d 7447	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 + ((∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥) − ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)) = (∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 + -∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥))
238	218, 229	negsubd 11624	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 + -∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥) = (∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 − ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥))
239	232, 237, 238	3eqtrd 2779	. . . . . 6 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 + (∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − (∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 + ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥))) = (∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 − ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥))
240	217, 220, 239	3eqtrd 2779	. . . . 5 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 + (∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 − ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥)) = (∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 − ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥))
241	188, 189, 240	3eqtr3d 2783	. . . 4 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → ∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 = (∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 − ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥))
242	241	oveq2d 7447	. . 3 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥) = (∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − (∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 − ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)))
243	108, 107, 228	subsubd 11646	. . . . 5 ⊢ (𝜑 → (∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − (∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 − ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)) = ((∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥) + ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥))
244	93	oveq2d 7447	. . . . . . 7 ⊢ (𝜑 → (∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥) = (∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥))
245	244, 109	eqtrd 2775	. . . . . 6 ⊢ (𝜑 → (∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥) = 0)
246	245	oveq1d 7446	. . . . 5 ⊢ (𝜑 → ((∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥) + ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥) = (0 + ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥))
247	228	addlidd 11460	. . . . 5 ⊢ (𝜑 → (0 + ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥) = ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)
248	243, 246, 247	3eqtrd 2779	. . . 4 ⊢ (𝜑 → (∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − (∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 − ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)) = ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)
249	248	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − (∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 − ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)) = ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)
250	184, 242, 249	3eqtrd 2779	. 2 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → ∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 = ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)
251	24	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → (𝐴 − 𝑋) ∈ ℝ)
252	14	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → (𝐵 − 𝑋) ∈ ℝ)
253	3	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → 𝐴 ∈ ℝ)
254	24, 3, 39	ltled 11407	. . . . . . 7 ⊢ (𝜑 → (𝐴 − 𝑋) ≤ 𝐴)
255	254	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → (𝐴 − 𝑋) ≤ 𝐴)
256	6	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → 𝑋 ∈ ℝ)
257	8	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → 𝐵 ∈ ℝ)
258		id 22	. . . . . . . . 9 ⊢ (𝑋 ≤ 𝑇 → 𝑋 ≤ 𝑇)
259	258, 1	breqtrdi 5189	. . . . . . . 8 ⊢ (𝑋 ≤ 𝑇 → 𝑋 ≤ (𝐵 − 𝐴))
260	259	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → 𝑋 ≤ (𝐵 − 𝐴))
261	256, 257, 253, 260	lesubd 11865	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → 𝐴 ≤ (𝐵 − 𝑋))
262	251, 252, 253, 255, 261	eliccd 45457	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → 𝐴 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋)))
263	158	adantlr 715	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ≤ 𝑇) ∧ 𝑥 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → (𝐹‘𝑥) ∈ ℂ)
264	132, 102, 3, 39, 73	eliood 45451	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ((𝐴 − 𝑋)(,)+∞))
265	36, 1, 37, 38, 60, 54, 65, 84, 88, 24, 264	fourierdlem105 46167	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ((𝐴 − 𝑋)[,]𝐴) ↦ (𝐹‘𝑥)) ∈ 𝐿¹)
266	265	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → (𝑥 ∈ ((𝐴 − 𝑋)[,]𝐴) ↦ (𝐹‘𝑥)) ∈ 𝐿¹)
267	3	leidd 11827	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ≤ 𝐴)
268	5	rpge0d 13079	. . . . . . . . 9 ⊢ (𝜑 → 0 ≤ 𝑋)
269	8, 6	subge02d 11853	. . . . . . . . 9 ⊢ (𝜑 → (0 ≤ 𝑋 ↔ (𝐵 − 𝑋) ≤ 𝐵))
270	268, 269	mpbid 232	. . . . . . . 8 ⊢ (𝜑 → (𝐵 − 𝑋) ≤ 𝐵)
271		iccss 13452	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐴 ≤ 𝐴 ∧ (𝐵 − 𝑋) ≤ 𝐵)) → (𝐴[,](𝐵 − 𝑋)) ⊆ (𝐴[,]𝐵))
272	3, 8, 267, 270, 271	syl22anc 839	. . . . . . 7 ⊢ (𝜑 → (𝐴[,](𝐵 − 𝑋)) ⊆ (𝐴[,]𝐵))
273		iccmbl 25615	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ (𝐵 − 𝑋) ∈ ℝ) → (𝐴[,](𝐵 − 𝑋)) ∈ dom vol)
274	3, 14, 273	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → (𝐴[,](𝐵 − 𝑋)) ∈ dom vol)
275	272, 274, 227, 213	iblss 25855	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (𝐴[,](𝐵 − 𝑋)) ↦ (𝐹‘𝑥)) ∈ 𝐿¹)
276	275	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → (𝑥 ∈ (𝐴[,](𝐵 − 𝑋)) ↦ (𝐹‘𝑥)) ∈ 𝐿¹)
277	251, 252, 262, 263, 266, 276	itgspliticc 25887	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → ∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 = (∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 + ∫(𝐴[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥))
278	268	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → 0 ≤ 𝑋)
279	269	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → (0 ≤ 𝑋 ↔ (𝐵 − 𝑋) ≤ 𝐵))
280	278, 279	mpbid 232	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → (𝐵 − 𝑋) ≤ 𝐵)
281	253, 257, 252, 261, 280	eliccd 45457	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → (𝐵 − 𝑋) ∈ (𝐴[,]𝐵))
282	227	adantlr 715	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑋 ≤ 𝑇) ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℂ)
283	8	leidd 11827	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ≤ 𝐵)
284	283	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → 𝐵 ≤ 𝐵)
285		iccss 13452	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐴 ≤ (𝐵 − 𝑋) ∧ 𝐵 ≤ 𝐵)) → ((𝐵 − 𝑋)[,]𝐵) ⊆ (𝐴[,]𝐵))
286	253, 257, 261, 284, 285	syl22anc 839	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → ((𝐵 − 𝑋)[,]𝐵) ⊆ (𝐴[,]𝐵))
287		iccmbl 25615	. . . . . . . . . . 11 ⊢ (((𝐵 − 𝑋) ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐵 − 𝑋)[,]𝐵) ∈ dom vol)
288	14, 8, 287	syl2anc 584	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐵 − 𝑋)[,]𝐵) ∈ dom vol)
289	288	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → ((𝐵 − 𝑋)[,]𝐵) ∈ dom vol)
290	213	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → (𝑥 ∈ (𝐴[,]𝐵) ↦ (𝐹‘𝑥)) ∈ 𝐿¹)
291	286, 289, 282, 290	iblss 25855	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → (𝑥 ∈ ((𝐵 − 𝑋)[,]𝐵) ↦ (𝐹‘𝑥)) ∈ 𝐿¹)
292	253, 257, 281, 282, 276, 291	itgspliticc 25887	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥 = (∫(𝐴[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 + ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥))
293	292	oveq1d 7446	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → (∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥 − ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥) = ((∫(𝐴[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 + ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥) − ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥))
294	60	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,](𝐵 − 𝑋))) → 𝐹:ℝ⟶ℂ)
295	3	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,](𝐵 − 𝑋))) → 𝐴 ∈ ℝ)
296	14	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,](𝐵 − 𝑋))) → (𝐵 − 𝑋) ∈ ℝ)
297		simpr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,](𝐵 − 𝑋))) → 𝑥 ∈ (𝐴[,](𝐵 − 𝑋)))
298		eliccre 45458	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ (𝐵 − 𝑋) ∈ ℝ ∧ 𝑥 ∈ (𝐴[,](𝐵 − 𝑋))) → 𝑥 ∈ ℝ)
299	295, 296, 297, 298	syl3anc 1370	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,](𝐵 − 𝑋))) → 𝑥 ∈ ℝ)
300	294, 299	ffvelcdmd 7105	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,](𝐵 − 𝑋))) → (𝐹‘𝑥) ∈ ℂ)
301	300, 275	itgcl 25834	. . . . . . . 8 ⊢ (𝜑 → ∫(𝐴[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 ∈ ℂ)
302	301, 107, 107	addsubassd 11638	. . . . . . 7 ⊢ (𝜑 → ((∫(𝐴[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 + ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥) − ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥) = (∫(𝐴[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 + (∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 − ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥)))
303	302	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → ((∫(𝐴[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 + ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥) − ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥) = (∫(𝐴[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 + (∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 − ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥)))
304	185	oveq2d 7447	. . . . . . . 8 ⊢ (𝜑 → (∫(𝐴[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 + (∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 − ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥)) = (∫(𝐴[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 + 0))
305	301	addridd 11459	. . . . . . . 8 ⊢ (𝜑 → (∫(𝐴[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 + 0) = ∫(𝐴[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥)
306	304, 305	eqtrd 2775	. . . . . . 7 ⊢ (𝜑 → (∫(𝐴[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 + (∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 − ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥)) = ∫(𝐴[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥)
307	306	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → (∫(𝐴[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 + (∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 − ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥)) = ∫(𝐴[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥)
308	293, 303, 307	3eqtrrd 2780	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → ∫(𝐴[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 = (∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥 − ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥))
309	308	oveq2d 7447	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → (∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 + ∫(𝐴[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥) = (∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 + (∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥 − ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥)))
310	93	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 = ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥)
311	107	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 ∈ ℂ)
312	310, 311	eqeltrrd 2840	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 ∈ ℂ)
313	282, 290	itgcl 25834	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥 ∈ ℂ)
314	312, 313, 311	addsub12d 11641	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → (∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 + (∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥 − ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥)) = (∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥 + (∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥)))
315	313, 312, 311	addsubassd 11638	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → ((∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥 + ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥) − ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥) = (∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥 + (∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥)))
316	314, 315	eqtr4d 2778	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → (∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 + (∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥 − ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥)) = ((∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥 + ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥) − ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥))
317	277, 309, 316	3eqtrd 2779	. . 3 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → ∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 = ((∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥 + ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥) − ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥))
318	310	oveq2d 7447	. . 3 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → ((∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥 + ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥) − ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥) = ((∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥 + ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥) − ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥))
319	313, 312	pncand 11619	. . 3 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → ((∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥 + ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥) − ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥) = ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)
320	317, 318, 319	3eqtrd 2779	. 2 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → ∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 = ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)
321	250, 320, 47, 6	ltlecasei 11367	1 ⊢ (𝜑 → ∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 = ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∀wral 3059 ∃wrex 3068 {crab 3433 ∪ cun 3961 ⊆ wss 3963 ifcif 4531 {cpr 4633 class class class wbr 5148 ↦ cmpt 5231 dom cdm 5689 ran crn 5690 ↾ cres 5691 ℩cio 6514 ⟶wf 6559 ‘cfv 6563 Isom wiso 6564 (class class class)co 7431 ↑_m cmap 8865 supcsup 9478 ℂcc 11151 ℝcr 11152 0cc0 11153 1c1 11154 + caddc 11156 · cmul 11158 +∞cpnf 11290 ℝ^*cxr 11292 < clt 11293 ≤ cle 11294 − cmin 11490 -cneg 11491 / cdiv 11918 ℕcn 12264 ℤcz 12611 ℝ⁺crp 13032 (,)cioo 13384 (,]cioc 13385 [,]cicc 13387 ...cfz 13544 ..^cfzo 13691 ⌊cfl 13827 ♯chash 14366 –cn→ccncf 24916 volcvol 25512 𝐿¹cibl 25666 ∫citg 25667 lim_ℂ climc 25912

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cc 10473 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-addf 11232

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-symdif 4259 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-disj 5116 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-ofr 7698 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-oadd 8509 df-omul 8510 df-er 8744 df-map 8867 df-pm 8868 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-fi 9449 df-sup 9480 df-inf 9481 df-oi 9548 df-dju 9939 df-card 9977 df-acn 9980 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-xnn0 12598 df-z 12612 df-dec 12732 df-uz 12877 df-q 12989 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-ioo 13388 df-ioc 13389 df-ico 13390 df-icc 13391 df-fz 13545 df-fzo 13692 df-fl 13829 df-mod 13907 df-seq 14040 df-exp 14100 df-hash 14367 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-limsup 15504 df-clim 15521 df-rlim 15522 df-sum 15720 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-rest 17469 df-topn 17470 df-0g 17488 df-gsum 17489 df-topgen 17490 df-pt 17491 df-prds 17494 df-xrs 17549 df-qtop 17554 df-imas 17555 df-xps 17557 df-mre 17631 df-mrc 17632 df-acs 17634 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-mulg 19099 df-cntz 19348 df-cmn 19815 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-fbas 21379 df-fg 21380 df-cnfld 21383 df-top 22916 df-topon 22933 df-topsp 22955 df-bases 22969 df-cld 23043 df-ntr 23044 df-cls 23045 df-nei 23122 df-lp 23160 df-perf 23161 df-cn 23251 df-cnp 23252 df-haus 23339 df-cmp 23411 df-tx 23586 df-hmeo 23779 df-fil 23870 df-fm 23962 df-flim 23963 df-flf 23964 df-xms 24346 df-ms 24347 df-tms 24348 df-cncf 24918 df-ovol 25513 df-vol 25514 df-mbf 25668 df-itg1 25669 df-itg2 25670 df-ibl 25671 df-itg 25672 df-0p 25719 df-ditg 25897 df-limc 25916 df-dv 25917

This theorem is referenced by: fourierdlem108 46170

W3C validator