fourierdlem107 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem107 45660

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 45645 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 7424	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 − 𝑋) + 𝑇) = ((𝐴 − 𝑋) + (𝐵 − 𝐴))
3		fourierdlem107.a	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐴 ∈ ℝ)
4	3	recnd 11267	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐴 ∈ ℂ)
5		fourierdlem107.x	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑋 ∈ ℝ⁺)
6	5	rpred 13043	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑋 ∈ ℝ)
7	6	recnd 11267	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑋 ∈ ℂ)
8		fourierdlem107.b	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐵 ∈ ℝ)
9	8	recnd 11267	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐵 ∈ ℂ)
10	4, 7, 9, 4	subadd4b 44723	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝐴 − 𝑋) + (𝐵 − 𝐴)) = ((𝐴 − 𝐴) + (𝐵 − 𝑋)))
11	2, 10	eqtrid 2777	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝐴 − 𝑋) + 𝑇) = ((𝐴 − 𝐴) + (𝐵 − 𝑋)))
12	4	subidd 11584	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐴 − 𝐴) = 0)
13	12	oveq1d 7428	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝐴 − 𝐴) + (𝐵 − 𝑋)) = (0 + (𝐵 − 𝑋)))
14	8, 6	resubcld 11667	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐵 − 𝑋) ∈ ℝ)
15	14	recnd 11267	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐵 − 𝑋) ∈ ℂ)
16	15	addlidd 11440	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (0 + (𝐵 − 𝑋)) = (𝐵 − 𝑋))
17	11, 13, 16	3eqtrd 2769	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝐴 − 𝑋) + 𝑇) = (𝐵 − 𝑋))
18	1	oveq2i 7424	. . . . . . . . . . . . . . 15 ⊢ (𝐴 + 𝑇) = (𝐴 + (𝐵 − 𝐴))
19	4, 9	pncan3d 11599	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐴 + (𝐵 − 𝐴)) = 𝐵)
20	18, 19	eqtrid 2777	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐴 + 𝑇) = 𝐵)
21	17, 20	oveq12d 7431	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((𝐴 − 𝑋) + 𝑇)[,](𝐴 + 𝑇)) = ((𝐵 − 𝑋)[,]𝐵))
22	21	eqcomd 2731	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐵 − 𝑋)[,]𝐵) = (((𝐴 − 𝑋) + 𝑇)[,](𝐴 + 𝑇)))
23	22	itgeq1d 45404	. . . . . . . . . . 11 ⊢ (𝜑 → ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 = ∫(((𝐴 − 𝑋) + 𝑇)[,](𝐴 + 𝑇))(𝐹‘𝑥) d𝑥)
24	3, 6	resubcld 11667	. . . . . . . . . . . 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 6890	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 = 𝑗 → (𝑝‘𝑖) = (𝑝‘𝑗))
27		oveq1 7420	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 = 𝑗 → (𝑖 + 1) = (𝑗 + 1))
28	27	fveq2d 6894	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 = 𝑗 → (𝑝‘(𝑖 + 1)) = (𝑝‘(𝑗 + 1)))
29	26, 28	breq12d 5157	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 = 𝑗 → ((𝑝‘𝑖) < (𝑝‘(𝑖 + 1)) ↔ (𝑝‘𝑗) < (𝑝‘(𝑗 + 1))))
30	29	cbvralvw 3225	. . . . . . . . . . . . . . . . 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 628	. . . . . . . . . . . . . . 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 3427	. . . . . . . . . . . . . 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 5247	. . . . . . . . . . . . 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 2753	. . . . . . . . . . . 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 13075	. . . . . . . . . . . . . . 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 45607	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑁 ∈ ℕ ∧ 𝑆 ∈ (𝑂‘𝑁)) ∧ 𝑆 Isom < , < ((0...𝑁), 𝐻)))
44	43	simpld 493	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑁 ∈ ℕ ∧ 𝑆 ∈ (𝑂‘𝑁)))
45	44	simpld 493	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℕ)
46	8, 3	resubcld 11667	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ)
47	1, 46	eqeltrid 2829	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑇 ∈ ℝ)
48	44	simprd 494	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑆 ∈ (𝑂‘𝑁))
49	24	adantr 479	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 − 𝑋)[,]𝐴)) → (𝐴 − 𝑋) ∈ ℝ)
50	3	adantr 479	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 − 𝑋)[,]𝐴)) → 𝐴 ∈ ℝ)
51		simpr 483	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 − 𝑋)[,]𝐴)) → 𝑥 ∈ ((𝐴 − 𝑋)[,]𝐴))
52		eliccre 44949	. . . . . . . . . . . . . 14 ⊢ (((𝐴 − 𝑋) ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ 𝑥 ∈ ((𝐴 − 𝑋)[,]𝐴)) → 𝑥 ∈ ℝ)
53	49, 50, 51, 52	syl3anc 1368	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 − 𝑋)[,]𝐴)) → 𝑥 ∈ ℝ)
54		fourierdlem107.fper	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥))
55	53, 54	syldan 589	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 − 𝑋)[,]𝐴)) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥))
56		fveq2 6890	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑗 → (𝑆‘𝑖) = (𝑆‘𝑗))
57	56	oveq1d 7428	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑗 → ((𝑆‘𝑖) + 𝑇) = ((𝑆‘𝑗) + 𝑇))
58	57	cbvmptv 5257	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (0...𝑁) ↦ ((𝑆‘𝑖) + 𝑇)) = (𝑗 ∈ (0...𝑁) ↦ ((𝑆‘𝑗) + 𝑇))
59		eqid 2725	. . . . . . . . . . . 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 479	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝑀 ∈ ℕ)
62	38	adantr 479	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝑄 ∈ (𝑃‘𝑀))
63	60	adantr 479	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝐹:ℝ⟶ℂ)
64	54	adantlr 713	. . . . . . . . . . . . 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 713	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
67	24	adantr 479	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝐴 − 𝑋) ∈ ℝ)
68	67	rexrd 11289	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝐴 − 𝑋) ∈ ℝ^*)
69		pnfxr 11293	. . . . . . . . . . . . . . 15 ⊢ +∞ ∈ ℝ^*
70	69	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → +∞ ∈ ℝ^*)
71	3	adantr 479	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝐴 ∈ ℝ)
72	39	adantr 479	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝐴 − 𝑋) < 𝐴)
73	3	ltpnfd 13128	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐴 < +∞)
74	73	adantr 479	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝐴 < +∞)
75	68, 70, 71, 72, 74	eliood 44942	. . . . . . . . . . . . 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 483	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝑗 ∈ (0..^𝑁))
79		eqid 2725	. . . . . . . . . . . . 13 ⊢ ((𝑆‘(𝑗 + 1)) − (𝐸‘(𝑆‘(𝑗 + 1)))) = ((𝑆‘(𝑗 + 1)) − (𝐸‘(𝑆‘(𝑗 + 1))))
80		eqid 2725	. . . . . . . . . . . . 13 ⊢ (𝐹 ↾ ((𝑍‘(𝐸‘(𝑆‘𝑗)))(,)(𝐸‘(𝑆‘(𝑗 + 1))))) = (𝐹 ↾ ((𝑍‘(𝐸‘(𝑆‘𝑗)))(,)(𝐸‘(𝑆‘(𝑗 + 1)))))
81		eqid 2725	. . . . . . . . . . . . 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 45643	. . . . . . . . . . . 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 713	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)))
86		eqid 2725	. . . . . . . . . . . . 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 45642	. . . . . . . . . . . 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 713	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1))))
90		eqid 2725	. . . . . . . . . . . . 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 45644	. . . . . . . . . . . 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 45645	. . . . . . . . . . 11 ⊢ (𝜑 → ∫(((𝐴 − 𝑋) + 𝑇)[,](𝐴 + 𝑇))(𝐹‘𝑥) d𝑥 = ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥)
93	23, 92	eqtrd 2765	. . . . . . . . . 10 ⊢ (𝜑 → ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 = ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥)
94	60	adantr 479	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐵 − 𝑋)[,]𝐵)) → 𝐹:ℝ⟶ℂ)
95	14	adantr 479	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐵 − 𝑋)[,]𝐵)) → (𝐵 − 𝑋) ∈ ℝ)
96	8	adantr 479	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐵 − 𝑋)[,]𝐵)) → 𝐵 ∈ ℝ)
97		simpr 483	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐵 − 𝑋)[,]𝐵)) → 𝑥 ∈ ((𝐵 − 𝑋)[,]𝐵))
98		eliccre 44949	. . . . . . . . . . . . 13 ⊢ (((𝐵 − 𝑋) ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑥 ∈ ((𝐵 − 𝑋)[,]𝐵)) → 𝑥 ∈ ℝ)
99	95, 96, 97, 98	syl3anc 1368	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐵 − 𝑋)[,]𝐵)) → 𝑥 ∈ ℝ)
100	94, 99	ffvelcdmd 7088	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐵 − 𝑋)[,]𝐵)) → (𝐹‘𝑥) ∈ ℂ)
101	14	rexrd 11289	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐵 − 𝑋) ∈ ℝ^*)
102	69	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → +∞ ∈ ℝ^*)
103	8, 5	ltsubrpd 13075	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐵 − 𝑋) < 𝐵)
104	8	ltpnfd 13128	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 < +∞)
105	101, 102, 8, 103, 104	eliood 44942	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ∈ ((𝐵 − 𝑋)(,)+∞))
106	36, 1, 37, 38, 60, 54, 65, 84, 88, 14, 105	fourierdlem105 45658	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ ((𝐵 − 𝑋)[,]𝐵) ↦ (𝐹‘𝑥)) ∈ 𝐿¹)
107	100, 106	itgcl 25726	. . . . . . . . . 10 ⊢ (𝜑 → ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 ∈ ℂ)
108	93, 107	eqeltrrd 2826	. . . . . . . . 9 ⊢ (𝜑 → ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 ∈ ℂ)
109	108	subidd 11584	. . . . . . . 8 ⊢ (𝜑 → (∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥) = 0)
110	109	eqcomd 2731	. . . . . . 7 ⊢ (𝜑 → 0 = (∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥))
111	110	adantr 479	. . . . . 6 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → 0 = (∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥))
112	24	adantr 479	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (𝐴 − 𝑋) ∈ ℝ)
113	3	adantr 479	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → 𝐴 ∈ ℝ)
114	14	adantr 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (𝐵 − 𝑋) ∈ ℝ)
115	36, 37, 38	fourierdlem11 45565	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵))
116	115	simp3d 1141	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 < 𝐵)
117	3, 8, 116	ltled 11387	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
118	117	adantr 479	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → 𝐴 ≤ 𝐵)
119	3, 8, 6	lesub1d 11846	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ (𝐴 − 𝑋) ≤ (𝐵 − 𝑋)))
120	119	adantr 479	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (𝐴 ≤ 𝐵 ↔ (𝐴 − 𝑋) ≤ (𝐵 − 𝑋)))
121	118, 120	mpbid 231	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (𝐴 − 𝑋) ≤ (𝐵 − 𝑋))
122	8	adantr 479	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → 𝐵 ∈ ℝ)
123	6	adantr 479	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → 𝑋 ∈ ℝ)
124		simpr 483	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → 𝑇 < 𝑋)
125	1, 124	eqbrtrrid 5180	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (𝐵 − 𝐴) < 𝑋)
126	122, 113, 123, 125	ltsub23d 11844	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (𝐵 − 𝑋) < 𝐴)
127	114, 113, 126	ltled 11387	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (𝐵 − 𝑋) ≤ 𝐴)
128	112, 113, 114, 121, 127	eliccd 44948	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (𝐵 − 𝑋) ∈ ((𝐴 − 𝑋)[,]𝐴))
129	60	adantr 479	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 − 𝑋)[,]𝐴)) → 𝐹:ℝ⟶ℂ)
130	129, 53	ffvelcdmd 7088	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 − 𝑋)[,]𝐴)) → (𝐹‘𝑥) ∈ ℂ)
131	130	adantlr 713	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑇 < 𝑋) ∧ 𝑥 ∈ ((𝐴 − 𝑋)[,]𝐴)) → (𝐹‘𝑥) ∈ ℂ)
132	24	rexrd 11289	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 − 𝑋) ∈ ℝ^*)
133	3, 8, 6, 116	ltsub1dd 11851	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 − 𝑋) < (𝐵 − 𝑋))
134	14	ltpnfd 13128	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐵 − 𝑋) < +∞)
135	132, 102, 14, 133, 134	eliood 44942	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵 − 𝑋) ∈ ((𝐴 − 𝑋)(,)+∞))
136	36, 1, 37, 38, 60, 54, 65, 84, 88, 24, 135	fourierdlem105 45658	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋)) ↦ (𝐹‘𝑥)) ∈ 𝐿¹)
137	136	adantr 479	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (𝑥 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋)) ↦ (𝐹‘𝑥)) ∈ 𝐿¹)
138	37	adantr 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → 𝑀 ∈ ℕ)
139	38	adantr 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → 𝑄 ∈ (𝑃‘𝑀))
140	60	adantr 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → 𝐹:ℝ⟶ℂ)
141	54	adantlr 713	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑇 < 𝑋) ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥))
142	65	adantlr 713	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑇 < 𝑋) ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
143	84	adantlr 713	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑇 < 𝑋) ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)))
144	88	adantlr 713	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑇 < 𝑋) ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1))))
145	101	adantr 479	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (𝐵 − 𝑋) ∈ ℝ^*)
146	69	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → +∞ ∈ ℝ^*)
147	113	ltpnfd 13128	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → 𝐴 < +∞)
148	145, 146, 113, 126, 147	eliood 44942	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → 𝐴 ∈ ((𝐵 − 𝑋)(,)+∞))
149	36, 1, 138, 139, 140, 141, 142, 143, 144, 114, 148	fourierdlem105 45658	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (𝑥 ∈ ((𝐵 − 𝑋)[,]𝐴) ↦ (𝐹‘𝑥)) ∈ 𝐿¹)
150	112, 113, 128, 131, 137, 149	itgspliticc 25779	. . . . . . 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 7428	. . . . . 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 479	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → 𝐹:ℝ⟶ℂ)
153	24	adantr 479	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → (𝐴 − 𝑋) ∈ ℝ)
154	14	adantr 479	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → (𝐵 − 𝑋) ∈ ℝ)
155		simpr 483	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → 𝑥 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋)))
156		eliccre 44949	. . . . . . . . . . 11 ⊢ (((𝐴 − 𝑋) ∈ ℝ ∧ (𝐵 − 𝑋) ∈ ℝ ∧ 𝑥 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → 𝑥 ∈ ℝ)
157	153, 154, 155, 156	syl3anc 1368	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → 𝑥 ∈ ℝ)
158	152, 157	ffvelcdmd 7088	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → (𝐹‘𝑥) ∈ ℂ)
159	158, 136	itgcl 25726	. . . . . . . 8 ⊢ (𝜑 → ∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 ∈ ℂ)
160	159	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → ∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 ∈ ℂ)
161	60	adantr 479	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐵 − 𝑋)[,]𝐴)) → 𝐹:ℝ⟶ℂ)
162	14	adantr 479	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐵 − 𝑋)[,]𝐴)) → (𝐵 − 𝑋) ∈ ℝ)
163	3	adantr 479	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐵 − 𝑋)[,]𝐴)) → 𝐴 ∈ ℝ)
164		simpr 483	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐵 − 𝑋)[,]𝐴)) → 𝑥 ∈ ((𝐵 − 𝑋)[,]𝐴))
165		eliccre 44949	. . . . . . . . . . 11 ⊢ (((𝐵 − 𝑋) ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ 𝑥 ∈ ((𝐵 − 𝑋)[,]𝐴)) → 𝑥 ∈ ℝ)
166	162, 163, 164, 165	syl3anc 1368	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐵 − 𝑋)[,]𝐴)) → 𝑥 ∈ ℝ)
167	161, 166	ffvelcdmd 7088	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐵 − 𝑋)[,]𝐴)) → (𝐹‘𝑥) ∈ ℂ)
168	167	adantlr 713	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑇 < 𝑋) ∧ 𝑥 ∈ ((𝐵 − 𝑋)[,]𝐴)) → (𝐹‘𝑥) ∈ ℂ)
169	168, 149	itgcl 25726	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → ∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 ∈ ℂ)
170	108	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 ∈ ℂ)
171	160, 169, 170	addsubassd 11616	. . . . . 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 2769	. . . . 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 7429	. . . 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 11585	. . . 4 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 − 0) = ∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥)
175	159	subidd 11584	. . . . . . 7 ⊢ (𝜑 → (∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 − ∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥) = 0)
176	175	oveq1d 7428	. . . . . 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 479	. . . . 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 11596	. . . . . 6 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥) ∈ ℂ)
179	160, 160, 178	subsub4d 11627	. . . . 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 11472	. . . . . 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 11612	. . . . . 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 2779	. . . . 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 2773	. . . 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 2773	. . 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 11584	. . . . . . . 8 ⊢ (𝜑 → (∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 − ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥) = 0)
186	185	eqcomd 2731	. . . . . . 7 ⊢ (𝜑 → 0 = (∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 − ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥))
187	186	oveq2d 7429	. . . . . 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 479	. . . . 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 11439	. . . . 5 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 + 0) = ∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥)
190	114, 122, 113, 127, 118	eliccd 44948	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → 𝐴 ∈ ((𝐵 − 𝑋)[,]𝐵))
191	100	adantlr 713	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑇 < 𝑋) ∧ 𝑥 ∈ ((𝐵 − 𝑋)[,]𝐵)) → (𝐹‘𝑥) ∈ ℂ)
192	3, 8	iccssred 13438	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
193	60, 192	feqresmpt 6961	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹 ↾ (𝐴[,]𝐵)) = (𝑥 ∈ (𝐴[,]𝐵) ↦ (𝐹‘𝑥)))
194	60, 192	fssresd 6758	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹 ↾ (𝐴[,]𝐵)):(𝐴[,]𝐵)⟶ℂ)
195		ioossicc 13437	. . . . . . . . . . . . . . 15 ⊢ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))
196	3	rexrd 11289	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
197	196	adantr 479	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐴 ∈ ℝ^*)
198	8	rexrd 11289	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
199	198	adantr 479	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐵 ∈ ℝ^*)
200	36, 37, 38	fourierdlem15 45569	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶(𝐴[,]𝐵))
201	200	adantr 479	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶(𝐴[,]𝐵))
202		simpr 483	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0..^𝑀))
203	197, 199, 201, 202	fourierdlem8 45562	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))) ⊆ (𝐴[,]𝐵))
204	195, 203	sstrid 3985	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ (𝐴[,]𝐵))
205	204	resabs1d 6008	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ (𝐴[,]𝐵)) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))))
206	205, 65	eqeltrd 2825	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ (𝐴[,]𝐵)) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
207	205	eqcomd 2731	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝐹 ↾ (𝐴[,]𝐵)) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))))
208	207	oveq1d 7428	. . . . . . . . . . . . 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 2827	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ (((𝐹 ↾ (𝐴[,]𝐵)) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)))
210	207	oveq1d 7428	. . . . . . . . . . . . 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 2827	. . . . . . . . . . . 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 45622	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹 ↾ (𝐴[,]𝐵)) ∈ 𝐿¹)
213	193, 212	eqeltrrd 2826	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ (𝐹‘𝑥)) ∈ 𝐿¹)
214	213	adantr 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (𝑥 ∈ (𝐴[,]𝐵) ↦ (𝐹‘𝑥)) ∈ 𝐿¹)
215	114, 122, 190, 191, 149, 214	itgspliticc 25779	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 = (∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 + ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥))
216	215	oveq2d 7429	. . . . . . 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 7429	. . . . . 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 479	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 ∈ ℂ)
219	215, 218	eqeltrrd 2826	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 + ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥) ∈ ℂ)
220	169, 218, 219	addsub12d 11619	. . . . . 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 479	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐹:ℝ⟶ℂ)
222	3	adantr 479	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐴 ∈ ℝ)
223	8	adantr 479	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐵 ∈ ℝ)
224		simpr 483	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ∈ (𝐴[,]𝐵))
225		eliccre 44949	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ∈ ℝ)
226	222, 223, 224, 225	syl3anc 1368	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ∈ ℝ)
227	221, 226	ffvelcdmd 7088	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℂ)
228	227, 213	itgcl 25726	. . . . . . . . . . 11 ⊢ (𝜑 → ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥 ∈ ℂ)
229	228	adantr 479	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥 ∈ ℂ)
230	169, 169, 229	subsub4d 11627	. . . . . . . . 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 2731	. . . . . . . 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 7429	. . . . . . 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 11584	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥) = 0)
234	233	oveq1d 7428	. . . . . . . . 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 11472	. . . . . . . . 9 ⊢ -∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥 = (0 − ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)
236	234, 235	eqtr4di 2783	. . . . . . . 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 7429	. . . . . . 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 11602	. . . . . . 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 2769	. . . . . 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 2769	. . . . 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 2773	. . . 4 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → ∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 = (∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 − ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥))
242	241	oveq2d 7429	. . 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 11624	. . . . 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 7429	. . . . . . 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 2765	. . . . . 6 ⊢ (𝜑 → (∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥) = 0)
246	245	oveq1d 7428	. . . . 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 11440	. . . . 5 ⊢ (𝜑 → (0 + ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥) = ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)
248	243, 246, 247	3eqtrd 2769	. . . 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 479	. . 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 2769	. 2 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → ∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 = ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)
251	24	adantr 479	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → (𝐴 − 𝑋) ∈ ℝ)
252	14	adantr 479	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → (𝐵 − 𝑋) ∈ ℝ)
253	3	adantr 479	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → 𝐴 ∈ ℝ)
254	24, 3, 39	ltled 11387	. . . . . . 7 ⊢ (𝜑 → (𝐴 − 𝑋) ≤ 𝐴)
255	254	adantr 479	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → (𝐴 − 𝑋) ≤ 𝐴)
256	6	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → 𝑋 ∈ ℝ)
257	8	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → 𝐵 ∈ ℝ)
258		id 22	. . . . . . . . 9 ⊢ (𝑋 ≤ 𝑇 → 𝑋 ≤ 𝑇)
259	258, 1	breqtrdi 5185	. . . . . . . 8 ⊢ (𝑋 ≤ 𝑇 → 𝑋 ≤ (𝐵 − 𝐴))
260	259	adantl 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → 𝑋 ≤ (𝐵 − 𝐴))
261	256, 257, 253, 260	lesubd 11843	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → 𝐴 ≤ (𝐵 − 𝑋))
262	251, 252, 253, 255, 261	eliccd 44948	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → 𝐴 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋)))
263	158	adantlr 713	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ≤ 𝑇) ∧ 𝑥 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → (𝐹‘𝑥) ∈ ℂ)
264	132, 102, 3, 39, 73	eliood 44942	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ((𝐴 − 𝑋)(,)+∞))
265	36, 1, 37, 38, 60, 54, 65, 84, 88, 24, 264	fourierdlem105 45658	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ((𝐴 − 𝑋)[,]𝐴) ↦ (𝐹‘𝑥)) ∈ 𝐿¹)
266	265	adantr 479	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → (𝑥 ∈ ((𝐴 − 𝑋)[,]𝐴) ↦ (𝐹‘𝑥)) ∈ 𝐿¹)
267	3	leidd 11805	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ≤ 𝐴)
268	5	rpge0d 13047	. . . . . . . . 9 ⊢ (𝜑 → 0 ≤ 𝑋)
269	8, 6	subge02d 11831	. . . . . . . . 9 ⊢ (𝜑 → (0 ≤ 𝑋 ↔ (𝐵 − 𝑋) ≤ 𝐵))
270	268, 269	mpbid 231	. . . . . . . 8 ⊢ (𝜑 → (𝐵 − 𝑋) ≤ 𝐵)
271		iccss 13419	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐴 ≤ 𝐴 ∧ (𝐵 − 𝑋) ≤ 𝐵)) → (𝐴[,](𝐵 − 𝑋)) ⊆ (𝐴[,]𝐵))
272	3, 8, 267, 270, 271	syl22anc 837	. . . . . . 7 ⊢ (𝜑 → (𝐴[,](𝐵 − 𝑋)) ⊆ (𝐴[,]𝐵))
273		iccmbl 25508	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ (𝐵 − 𝑋) ∈ ℝ) → (𝐴[,](𝐵 − 𝑋)) ∈ dom vol)
274	3, 14, 273	syl2anc 582	. . . . . . 7 ⊢ (𝜑 → (𝐴[,](𝐵 − 𝑋)) ∈ dom vol)
275	272, 274, 227, 213	iblss 25747	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (𝐴[,](𝐵 − 𝑋)) ↦ (𝐹‘𝑥)) ∈ 𝐿¹)
276	275	adantr 479	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → (𝑥 ∈ (𝐴[,](𝐵 − 𝑋)) ↦ (𝐹‘𝑥)) ∈ 𝐿¹)
277	251, 252, 262, 263, 266, 276	itgspliticc 25779	. . . 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 479	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → 0 ≤ 𝑋)
279	269	adantr 479	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → (0 ≤ 𝑋 ↔ (𝐵 − 𝑋) ≤ 𝐵))
280	278, 279	mpbid 231	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → (𝐵 − 𝑋) ≤ 𝐵)
281	253, 257, 252, 261, 280	eliccd 44948	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → (𝐵 − 𝑋) ∈ (𝐴[,]𝐵))
282	227	adantlr 713	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑋 ≤ 𝑇) ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℂ)
283	8	leidd 11805	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ≤ 𝐵)
284	283	adantr 479	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → 𝐵 ≤ 𝐵)
285		iccss 13419	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐴 ≤ (𝐵 − 𝑋) ∧ 𝐵 ≤ 𝐵)) → ((𝐵 − 𝑋)[,]𝐵) ⊆ (𝐴[,]𝐵))
286	253, 257, 261, 284, 285	syl22anc 837	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → ((𝐵 − 𝑋)[,]𝐵) ⊆ (𝐴[,]𝐵))
287		iccmbl 25508	. . . . . . . . . . 11 ⊢ (((𝐵 − 𝑋) ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐵 − 𝑋)[,]𝐵) ∈ dom vol)
288	14, 8, 287	syl2anc 582	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐵 − 𝑋)[,]𝐵) ∈ dom vol)
289	288	adantr 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → ((𝐵 − 𝑋)[,]𝐵) ∈ dom vol)
290	213	adantr 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → (𝑥 ∈ (𝐴[,]𝐵) ↦ (𝐹‘𝑥)) ∈ 𝐿¹)
291	286, 289, 282, 290	iblss 25747	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → (𝑥 ∈ ((𝐵 − 𝑋)[,]𝐵) ↦ (𝐹‘𝑥)) ∈ 𝐿¹)
292	253, 257, 281, 282, 276, 291	itgspliticc 25779	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥 = (∫(𝐴[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 + ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥))
293	292	oveq1d 7428	. . . . . 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 479	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,](𝐵 − 𝑋))) → 𝐹:ℝ⟶ℂ)
295	3	adantr 479	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,](𝐵 − 𝑋))) → 𝐴 ∈ ℝ)
296	14	adantr 479	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,](𝐵 − 𝑋))) → (𝐵 − 𝑋) ∈ ℝ)
297		simpr 483	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,](𝐵 − 𝑋))) → 𝑥 ∈ (𝐴[,](𝐵 − 𝑋)))
298		eliccre 44949	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ (𝐵 − 𝑋) ∈ ℝ ∧ 𝑥 ∈ (𝐴[,](𝐵 − 𝑋))) → 𝑥 ∈ ℝ)
299	295, 296, 297, 298	syl3anc 1368	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,](𝐵 − 𝑋))) → 𝑥 ∈ ℝ)
300	294, 299	ffvelcdmd 7088	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,](𝐵 − 𝑋))) → (𝐹‘𝑥) ∈ ℂ)
301	300, 275	itgcl 25726	. . . . . . . 8 ⊢ (𝜑 → ∫(𝐴[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 ∈ ℂ)
302	301, 107, 107	addsubassd 11616	. . . . . . 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 479	. . . . . 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 7429	. . . . . . . 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 11439	. . . . . . . 8 ⊢ (𝜑 → (∫(𝐴[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 + 0) = ∫(𝐴[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥)
306	304, 305	eqtrd 2765	. . . . . . 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 479	. . . . . 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 2770	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → ∫(𝐴[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 = (∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥 − ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥))
309	308	oveq2d 7429	. . . 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 479	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 = ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥)
311	107	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 ∈ ℂ)
312	310, 311	eqeltrrd 2826	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 ∈ ℂ)
313	282, 290	itgcl 25726	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥 ∈ ℂ)
314	312, 313, 311	addsub12d 11619	. . . . 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 11616	. . . . 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 2768	. . . 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 2769	. . 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 7429	. . 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 11597	. . 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 2769	. 2 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → ∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 = ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)
321	250, 320, 47, 6	ltlecasei 11347	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 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∀wral 3051 ∃wrex 3060 {crab 3419 ∪ cun 3939 ⊆ wss 3941 ifcif 4525 {cpr 4627 class class class wbr 5144 ↦ cmpt 5227 dom cdm 5673 ran crn 5674 ↾ cres 5675 ℩cio 6493 ⟶wf 6539 ‘cfv 6543 Isom wiso 6544 (class class class)co 7413 ↑_m cmap 8838 supcsup 9458 ℂcc 11131 ℝcr 11132 0cc0 11133 1c1 11134 + caddc 11136 · cmul 11138 +∞cpnf 11270 ℝ^*cxr 11272 < clt 11273 ≤ cle 11274 − cmin 11469 -cneg 11470 / cdiv 11896 ℕcn 12237 ℤcz 12583 ℝ⁺crp 13001 (,)cioo 13351 (,]cioc 13352 [,]cicc 13354 ...cfz 13511 ..^cfzo 13654 ⌊cfl 13782 ♯chash 14316 –cn→ccncf 24809 volcvol 25405 𝐿¹cibl 25559 ∫citg 25560 lim_ℂ climc 25804

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-inf2 9659 ax-cc 10453 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 ax-pre-sup 11211 ax-addf 11212

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3961 df-symdif 4238 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4905 df-int 4946 df-iun 4994 df-iin 4995 df-disj 5110 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-se 5629 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-of 7679 df-ofr 7680 df-om 7866 df-1st 7987 df-2nd 7988 df-supp 8159 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-oadd 8484 df-omul 8485 df-er 8718 df-map 8840 df-pm 8841 df-ixp 8910 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-fsupp 9381 df-fi 9429 df-sup 9460 df-inf 9461 df-oi 9528 df-dju 9919 df-card 9957 df-acn 9960 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-div 11897 df-nn 12238 df-2 12300 df-3 12301 df-4 12302 df-5 12303 df-6 12304 df-7 12305 df-8 12306 df-9 12307 df-n0 12498 df-xnn0 12570 df-z 12584 df-dec 12703 df-uz 12848 df-q 12958 df-rp 13002 df-xneg 13119 df-xadd 13120 df-xmul 13121 df-ioo 13355 df-ioc 13356 df-ico 13357 df-icc 13358 df-fz 13512 df-fzo 13655 df-fl 13784 df-mod 13862 df-seq 13994 df-exp 14054 df-hash 14317 df-cj 15073 df-re 15074 df-im 15075 df-sqrt 15209 df-abs 15210 df-limsup 15442 df-clim 15459 df-rlim 15460 df-sum 15660 df-struct 17110 df-sets 17127 df-slot 17145 df-ndx 17157 df-base 17175 df-ress 17204 df-plusg 17240 df-mulr 17241 df-starv 17242 df-sca 17243 df-vsca 17244 df-ip 17245 df-tset 17246 df-ple 17247 df-ds 17249 df-unif 17250 df-hom 17251 df-cco 17252 df-rest 17398 df-topn 17399 df-0g 17417 df-gsum 17418 df-topgen 17419 df-pt 17420 df-prds 17423 df-xrs 17478 df-qtop 17483 df-imas 17484 df-xps 17486 df-mre 17560 df-mrc 17561 df-acs 17563 df-mgm 18594 df-sgrp 18673 df-mnd 18689 df-submnd 18735 df-mulg 19023 df-cntz 19267 df-cmn 19736 df-psmet 21270 df-xmet 21271 df-met 21272 df-bl 21273 df-mopn 21274 df-fbas 21275 df-fg 21276 df-cnfld 21279 df-top 22809 df-topon 22826 df-topsp 22848 df-bases 22862 df-cld 22936 df-ntr 22937 df-cls 22938 df-nei 23015 df-lp 23053 df-perf 23054 df-cn 23144 df-cnp 23145 df-haus 23232 df-cmp 23304 df-tx 23479 df-hmeo 23672 df-fil 23763 df-fm 23855 df-flim 23856 df-flf 23857 df-xms 24239 df-ms 24240 df-tms 24241 df-cncf 24811 df-ovol 25406 df-vol 25407 df-mbf 25561 df-itg1 25562 df-itg2 25563 df-ibl 25564 df-itg 25565 df-0p 25612 df-ditg 25789 df-limc 25808 df-dv 25809

This theorem is referenced by: fourierdlem108 45661

W3C validator