fourierdlem107 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem107 46457

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 46442 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 7369	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 − 𝑋) + 𝑇) = ((𝐴 − 𝑋) + (𝐵 − 𝐴))
3		fourierdlem107.a	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐴 ∈ ℝ)
4	3	recnd 11160	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐴 ∈ ℂ)
5		fourierdlem107.x	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑋 ∈ ℝ⁺)
6	5	rpred 12949	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑋 ∈ ℝ)
7	6	recnd 11160	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑋 ∈ ℂ)
8		fourierdlem107.b	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐵 ∈ ℝ)
9	8	recnd 11160	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐵 ∈ ℂ)
10	4, 7, 9, 4	subadd4b 45531	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝐴 − 𝑋) + (𝐵 − 𝐴)) = ((𝐴 − 𝐴) + (𝐵 − 𝑋)))
11	2, 10	eqtrid 2783	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝐴 − 𝑋) + 𝑇) = ((𝐴 − 𝐴) + (𝐵 − 𝑋)))
12	4	subidd 11480	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐴 − 𝐴) = 0)
13	12	oveq1d 7373	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝐴 − 𝐴) + (𝐵 − 𝑋)) = (0 + (𝐵 − 𝑋)))
14	8, 6	resubcld 11565	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐵 − 𝑋) ∈ ℝ)
15	14	recnd 11160	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐵 − 𝑋) ∈ ℂ)
16	15	addlidd 11334	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (0 + (𝐵 − 𝑋)) = (𝐵 − 𝑋))
17	11, 13, 16	3eqtrd 2775	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝐴 − 𝑋) + 𝑇) = (𝐵 − 𝑋))
18	1	oveq2i 7369	. . . . . . . . . . . . . . 15 ⊢ (𝐴 + 𝑇) = (𝐴 + (𝐵 − 𝐴))
19	4, 9	pncan3d 11495	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐴 + (𝐵 − 𝐴)) = 𝐵)
20	18, 19	eqtrid 2783	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐴 + 𝑇) = 𝐵)
21	17, 20	oveq12d 7376	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((𝐴 − 𝑋) + 𝑇)[,](𝐴 + 𝑇)) = ((𝐵 − 𝑋)[,]𝐵))
22	21	eqcomd 2742	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐵 − 𝑋)[,]𝐵) = (((𝐴 − 𝑋) + 𝑇)[,](𝐴 + 𝑇)))
23	22	itgeq1d 46201	. . . . . . . . . . 11 ⊢ (𝜑 → ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 = ∫(((𝐴 − 𝑋) + 𝑇)[,](𝐴 + 𝑇))(𝐹‘𝑥) d𝑥)
24	3, 6	resubcld 11565	. . . . . . . . . . . 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 6834	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 = 𝑗 → (𝑝‘𝑖) = (𝑝‘𝑗))
27		oveq1 7365	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 = 𝑗 → (𝑖 + 1) = (𝑗 + 1))
28	27	fveq2d 6838	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 = 𝑗 → (𝑝‘(𝑖 + 1)) = (𝑝‘(𝑗 + 1)))
29	26, 28	breq12d 5111	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 = 𝑗 → ((𝑝‘𝑖) < (𝑝‘(𝑖 + 1)) ↔ (𝑝‘𝑗) < (𝑝‘(𝑗 + 1))))
30	29	cbvralvw 3214	. . . . . . . . . . . . . . . . 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 3406	. . . . . . . . . . . . . 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 5193	. . . . . . . . . . . . 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 2759	. . . . . . . . . . . 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 12981	. . . . . . . . . . . . . . 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 46404	. . . . . . . . . . . . . 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 11565	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ)
47	1, 46	eqeltrid 2840	. . . . . . . . . . . 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 45751	. . . . . . . . . . . . . 14 ⊢ (((𝐴 − 𝑋) ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ 𝑥 ∈ ((𝐴 − 𝑋)[,]𝐴)) → 𝑥 ∈ ℝ)
53	49, 50, 51, 52	syl3anc 1373	. . . . . . . . . . . . 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 6834	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑗 → (𝑆‘𝑖) = (𝑆‘𝑗))
57	56	oveq1d 7373	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑗 → ((𝑆‘𝑖) + 𝑇) = ((𝑆‘𝑗) + 𝑇))
58	57	cbvmptv 5202	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (0...𝑁) ↦ ((𝑆‘𝑖) + 𝑇)) = (𝑗 ∈ (0...𝑁) ↦ ((𝑆‘𝑗) + 𝑇))
59		eqid 2736	. . . . . . . . . . . 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 11182	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝐴 − 𝑋) ∈ ℝ^*)
69		pnfxr 11186	. . . . . . . . . . . . . . 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 13035	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐴 < +∞)
74	73	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝐴 < +∞)
75	68, 70, 71, 72, 74	eliood 45744	. . . . . . . . . . . . 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 2736	. . . . . . . . . . . . 13 ⊢ ((𝑆‘(𝑗 + 1)) − (𝐸‘(𝑆‘(𝑗 + 1)))) = ((𝑆‘(𝑗 + 1)) − (𝐸‘(𝑆‘(𝑗 + 1))))
80		eqid 2736	. . . . . . . . . . . . 13 ⊢ (𝐹 ↾ ((𝑍‘(𝐸‘(𝑆‘𝑗)))(,)(𝐸‘(𝑆‘(𝑗 + 1))))) = (𝐹 ↾ ((𝑍‘(𝐸‘(𝑆‘𝑗)))(,)(𝐸‘(𝑆‘(𝑗 + 1)))))
81		eqid 2736	. . . . . . . . . . . . 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 46440	. . . . . . . . . . . 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 2736	. . . . . . . . . . . . 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 46439	. . . . . . . . . . . 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 2736	. . . . . . . . . . . . 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 46441	. . . . . . . . . . . 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 46442	. . . . . . . . . . 11 ⊢ (𝜑 → ∫(((𝐴 − 𝑋) + 𝑇)[,](𝐴 + 𝑇))(𝐹‘𝑥) d𝑥 = ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥)
93	23, 92	eqtrd 2771	. . . . . . . . . 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 45751	. . . . . . . . . . . . 13 ⊢ (((𝐵 − 𝑋) ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑥 ∈ ((𝐵 − 𝑋)[,]𝐵)) → 𝑥 ∈ ℝ)
99	95, 96, 97, 98	syl3anc 1373	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐵 − 𝑋)[,]𝐵)) → 𝑥 ∈ ℝ)
100	94, 99	ffvelcdmd 7030	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐵 − 𝑋)[,]𝐵)) → (𝐹‘𝑥) ∈ ℂ)
101	14	rexrd 11182	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐵 − 𝑋) ∈ ℝ^*)
102	69	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → +∞ ∈ ℝ^*)
103	8, 5	ltsubrpd 12981	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐵 − 𝑋) < 𝐵)
104	8	ltpnfd 13035	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 < +∞)
105	101, 102, 8, 103, 104	eliood 45744	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ∈ ((𝐵 − 𝑋)(,)+∞))
106	36, 1, 37, 38, 60, 54, 65, 84, 88, 14, 105	fourierdlem105 46455	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ ((𝐵 − 𝑋)[,]𝐵) ↦ (𝐹‘𝑥)) ∈ 𝐿¹)
107	100, 106	itgcl 25741	. . . . . . . . . 10 ⊢ (𝜑 → ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 ∈ ℂ)
108	93, 107	eqeltrrd 2837	. . . . . . . . 9 ⊢ (𝜑 → ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 ∈ ℂ)
109	108	subidd 11480	. . . . . . . 8 ⊢ (𝜑 → (∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥) = 0)
110	109	eqcomd 2742	. . . . . . 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 46362	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵))
116	115	simp3d 1144	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 < 𝐵)
117	3, 8, 116	ltled 11281	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
118	117	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → 𝐴 ≤ 𝐵)
119	3, 8, 6	lesub1d 11744	. . . . . . . . . . 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 5134	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (𝐵 − 𝐴) < 𝑋)
126	122, 113, 123, 125	ltsub23d 11742	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (𝐵 − 𝑋) < 𝐴)
127	114, 113, 126	ltled 11281	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (𝐵 − 𝑋) ≤ 𝐴)
128	112, 113, 114, 121, 127	eliccd 45750	. . . . . . . 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 7030	. . . . . . . . 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 11182	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 − 𝑋) ∈ ℝ^*)
133	3, 8, 6, 116	ltsub1dd 11749	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 − 𝑋) < (𝐵 − 𝑋))
134	14	ltpnfd 13035	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐵 − 𝑋) < +∞)
135	132, 102, 14, 133, 134	eliood 45744	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵 − 𝑋) ∈ ((𝐴 − 𝑋)(,)+∞))
136	36, 1, 37, 38, 60, 54, 65, 84, 88, 24, 135	fourierdlem105 46455	. . . . . . . . 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 13035	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → 𝐴 < +∞)
148	145, 146, 113, 126, 147	eliood 45744	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → 𝐴 ∈ ((𝐵 − 𝑋)(,)+∞))
149	36, 1, 138, 139, 140, 141, 142, 143, 144, 114, 148	fourierdlem105 46455	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (𝑥 ∈ ((𝐵 − 𝑋)[,]𝐴) ↦ (𝐹‘𝑥)) ∈ 𝐿¹)
150	112, 113, 128, 131, 137, 149	itgspliticc 25794	. . . . . . 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 7373	. . . . . 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 45751	. . . . . . . . . . 11 ⊢ (((𝐴 − 𝑋) ∈ ℝ ∧ (𝐵 − 𝑋) ∈ ℝ ∧ 𝑥 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → 𝑥 ∈ ℝ)
157	153, 154, 155, 156	syl3anc 1373	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → 𝑥 ∈ ℝ)
158	152, 157	ffvelcdmd 7030	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → (𝐹‘𝑥) ∈ ℂ)
159	158, 136	itgcl 25741	. . . . . . . 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 45751	. . . . . . . . . . 11 ⊢ (((𝐵 − 𝑋) ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ 𝑥 ∈ ((𝐵 − 𝑋)[,]𝐴)) → 𝑥 ∈ ℝ)
166	162, 163, 164, 165	syl3anc 1373	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐵 − 𝑋)[,]𝐴)) → 𝑥 ∈ ℝ)
167	161, 166	ffvelcdmd 7030	. . . . . . . . 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 25741	. . . . . . 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 11512	. . . . . 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 2775	. . . . 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 7374	. . . 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 11481	. . . 4 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 − 0) = ∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥)
175	159	subidd 11480	. . . . . . 7 ⊢ (𝜑 → (∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 − ∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥) = 0)
176	175	oveq1d 7373	. . . . . 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 11492	. . . . . 6 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥) ∈ ℂ)
179	160, 160, 178	subsub4d 11523	. . . . 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 11367	. . . . . 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 11508	. . . . . 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 2785	. . . . 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 2779	. . . 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 2779	. . 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 11480	. . . . . . . 8 ⊢ (𝜑 → (∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 − ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥) = 0)
186	185	eqcomd 2742	. . . . . . 7 ⊢ (𝜑 → 0 = (∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 − ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥))
187	186	oveq2d 7374	. . . . . 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 11333	. . . . 5 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 + 0) = ∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥)
190	114, 122, 113, 127, 118	eliccd 45750	. . . . . . . . 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 13350	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
193	60, 192	feqresmpt 6903	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹 ↾ (𝐴[,]𝐵)) = (𝑥 ∈ (𝐴[,]𝐵) ↦ (𝐹‘𝑥)))
194	60, 192	fssresd 6701	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹 ↾ (𝐴[,]𝐵)):(𝐴[,]𝐵)⟶ℂ)
195		ioossicc 13349	. . . . . . . . . . . . . . 15 ⊢ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))
196	3	rexrd 11182	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
197	196	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐴 ∈ ℝ^*)
198	8	rexrd 11182	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
199	198	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐵 ∈ ℝ^*)
200	36, 37, 38	fourierdlem15 46366	. . . . . . . . . . . . . . . . 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 46359	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))) ⊆ (𝐴[,]𝐵))
204	195, 203	sstrid 3945	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ (𝐴[,]𝐵))
205	204	resabs1d 5967	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ (𝐴[,]𝐵)) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))))
206	205, 65	eqeltrd 2836	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ (𝐴[,]𝐵)) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
207	205	eqcomd 2742	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝐹 ↾ (𝐴[,]𝐵)) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))))
208	207	oveq1d 7373	. . . . . . . . . . . . 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 2838	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ (((𝐹 ↾ (𝐴[,]𝐵)) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)))
210	207	oveq1d 7373	. . . . . . . . . . . . 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 2838	. . . . . . . . . . . 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 46419	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹 ↾ (𝐴[,]𝐵)) ∈ 𝐿¹)
213	193, 212	eqeltrrd 2837	. . . . . . . . . 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 25794	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 = (∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 + ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥))
216	215	oveq2d 7374	. . . . . . 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 7374	. . . . . 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 2837	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 + ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥) ∈ ℂ)
220	169, 218, 219	addsub12d 11515	. . . . . 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 45751	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ∈ ℝ)
226	222, 223, 224, 225	syl3anc 1373	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ∈ ℝ)
227	221, 226	ffvelcdmd 7030	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℂ)
228	227, 213	itgcl 25741	. . . . . . . . . . 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 11523	. . . . . . . . 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 2742	. . . . . . . 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 7374	. . . . . . 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 11480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥) = 0)
234	233	oveq1d 7373	. . . . . . . . 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 11367	. . . . . . . . 9 ⊢ -∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥 = (0 − ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)
236	234, 235	eqtr4di 2789	. . . . . . . 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 7374	. . . . . . 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 11498	. . . . . . 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 2775	. . . . . 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 2775	. . . . 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 2779	. . . 4 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → ∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 = (∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 − ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥))
242	241	oveq2d 7374	. . 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 11520	. . . . 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 7374	. . . . . . 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 2771	. . . . . 6 ⊢ (𝜑 → (∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥) = 0)
246	245	oveq1d 7373	. . . . 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 11334	. . . . 5 ⊢ (𝜑 → (0 + ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥) = ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)
248	243, 246, 247	3eqtrd 2775	. . . 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 2775	. 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 11281	. . . . . . 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 5139	. . . . . . . 8 ⊢ (𝑋 ≤ 𝑇 → 𝑋 ≤ (𝐵 − 𝐴))
260	259	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → 𝑋 ≤ (𝐵 − 𝐴))
261	256, 257, 253, 260	lesubd 11741	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → 𝐴 ≤ (𝐵 − 𝑋))
262	251, 252, 253, 255, 261	eliccd 45750	. . . . 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 45744	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ((𝐴 − 𝑋)(,)+∞))
265	36, 1, 37, 38, 60, 54, 65, 84, 88, 24, 264	fourierdlem105 46455	. . . . . 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 11703	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ≤ 𝐴)
268	5	rpge0d 12953	. . . . . . . . 9 ⊢ (𝜑 → 0 ≤ 𝑋)
269	8, 6	subge02d 11729	. . . . . . . . 9 ⊢ (𝜑 → (0 ≤ 𝑋 ↔ (𝐵 − 𝑋) ≤ 𝐵))
270	268, 269	mpbid 232	. . . . . . . 8 ⊢ (𝜑 → (𝐵 − 𝑋) ≤ 𝐵)
271		iccss 13330	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐴 ≤ 𝐴 ∧ (𝐵 − 𝑋) ≤ 𝐵)) → (𝐴[,](𝐵 − 𝑋)) ⊆ (𝐴[,]𝐵))
272	3, 8, 267, 270, 271	syl22anc 838	. . . . . . 7 ⊢ (𝜑 → (𝐴[,](𝐵 − 𝑋)) ⊆ (𝐴[,]𝐵))
273		iccmbl 25523	. . . . . . . 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 25762	. . . . . 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 25794	. . . 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 45750	. . . . . . . 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 11703	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ≤ 𝐵)
284	283	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → 𝐵 ≤ 𝐵)
285		iccss 13330	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐴 ≤ (𝐵 − 𝑋) ∧ 𝐵 ≤ 𝐵)) → ((𝐵 − 𝑋)[,]𝐵) ⊆ (𝐴[,]𝐵))
286	253, 257, 261, 284, 285	syl22anc 838	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → ((𝐵 − 𝑋)[,]𝐵) ⊆ (𝐴[,]𝐵))
287		iccmbl 25523	. . . . . . . . . . 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 25762	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → (𝑥 ∈ ((𝐵 − 𝑋)[,]𝐵) ↦ (𝐹‘𝑥)) ∈ 𝐿¹)
292	253, 257, 281, 282, 276, 291	itgspliticc 25794	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥 = (∫(𝐴[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 + ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥))
293	292	oveq1d 7373	. . . . . 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 45751	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ (𝐵 − 𝑋) ∈ ℝ ∧ 𝑥 ∈ (𝐴[,](𝐵 − 𝑋))) → 𝑥 ∈ ℝ)
299	295, 296, 297, 298	syl3anc 1373	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,](𝐵 − 𝑋))) → 𝑥 ∈ ℝ)
300	294, 299	ffvelcdmd 7030	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,](𝐵 − 𝑋))) → (𝐹‘𝑥) ∈ ℂ)
301	300, 275	itgcl 25741	. . . . . . . 8 ⊢ (𝜑 → ∫(𝐴[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 ∈ ℂ)
302	301, 107, 107	addsubassd 11512	. . . . . . 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 7374	. . . . . . . 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 11333	. . . . . . . 8 ⊢ (𝜑 → (∫(𝐴[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 + 0) = ∫(𝐴[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥)
306	304, 305	eqtrd 2771	. . . . . . 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 2776	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → ∫(𝐴[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 = (∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥 − ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥))
309	308	oveq2d 7374	. . . 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 2837	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 ∈ ℂ)
313	282, 290	itgcl 25741	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥 ∈ ℂ)
314	312, 313, 311	addsub12d 11515	. . . . 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 11512	. . . . 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 2774	. . . 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 2775	. . 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 7374	. . 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 11493	. . 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 2775	. 2 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → ∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 = ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)
321	250, 320, 47, 6	ltlecasei 11241	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 1541 ∈ wcel 2113 ∀wral 3051 ∃wrex 3060 {crab 3399 ∪ cun 3899 ⊆ wss 3901 ifcif 4479 {cpr 4582 class class class wbr 5098 ↦ cmpt 5179 dom cdm 5624 ran crn 5625 ↾ cres 5626 ℩cio 6446 ⟶wf 6488 ‘cfv 6492 Isom wiso 6493 (class class class)co 7358 ↑_m cmap 8763 supcsup 9343 ℂcc 11024 ℝcr 11025 0cc0 11026 1c1 11027 + caddc 11029 · cmul 11031 +∞cpnf 11163 ℝ^*cxr 11165 < clt 11166 ≤ cle 11167 − cmin 11364 -cneg 11365 / cdiv 11794 ℕcn 12145 ℤcz 12488 ℝ⁺crp 12905 (,)cioo 13261 (,]cioc 13262 [,]cicc 13264 ...cfz 13423 ..^cfzo 13570 ⌊cfl 13710 ♯chash 14253 –cn→ccncf 24825 volcvol 25420 𝐿¹cibl 25574 ∫citg 25575 lim_ℂ climc 25819

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-inf2 9550 ax-cc 10345 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103 ax-pre-sup 11104 ax-addf 11105

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-symdif 4205 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-tp 4585 df-op 4587 df-uni 4864 df-int 4903 df-iun 4948 df-iin 4949 df-disj 5066 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-ofr 7623 df-om 7809 df-1st 7933 df-2nd 7934 df-supp 8103 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-2o 8398 df-oadd 8401 df-omul 8402 df-er 8635 df-map 8765 df-pm 8766 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9265 df-fi 9314 df-sup 9345 df-inf 9346 df-oi 9415 df-dju 9813 df-card 9851 df-acn 9854 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-div 11795 df-nn 12146 df-2 12208 df-3 12209 df-4 12210 df-5 12211 df-6 12212 df-7 12213 df-8 12214 df-9 12215 df-n0 12402 df-xnn0 12475 df-z 12489 df-dec 12608 df-uz 12752 df-q 12862 df-rp 12906 df-xneg 13026 df-xadd 13027 df-xmul 13028 df-ioo 13265 df-ioc 13266 df-ico 13267 df-icc 13268 df-fz 13424 df-fzo 13571 df-fl 13712 df-mod 13790 df-seq 13925 df-exp 13985 df-hash 14254 df-cj 15022 df-re 15023 df-im 15024 df-sqrt 15158 df-abs 15159 df-limsup 15394 df-clim 15411 df-rlim 15412 df-sum 15610 df-struct 17074 df-sets 17091 df-slot 17109 df-ndx 17121 df-base 17137 df-ress 17158 df-plusg 17190 df-mulr 17191 df-starv 17192 df-sca 17193 df-vsca 17194 df-ip 17195 df-tset 17196 df-ple 17197 df-ds 17199 df-unif 17200 df-hom 17201 df-cco 17202 df-rest 17342 df-topn 17343 df-0g 17361 df-gsum 17362 df-topgen 17363 df-pt 17364 df-prds 17367 df-xrs 17423 df-qtop 17428 df-imas 17429 df-xps 17431 df-mre 17505 df-mrc 17506 df-acs 17508 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-submnd 18709 df-mulg 18998 df-cntz 19246 df-cmn 19711 df-psmet 21301 df-xmet 21302 df-met 21303 df-bl 21304 df-mopn 21305 df-fbas 21306 df-fg 21307 df-cnfld 21310 df-top 22838 df-topon 22855 df-topsp 22877 df-bases 22890 df-cld 22963 df-ntr 22964 df-cls 22965 df-nei 23042 df-lp 23080 df-perf 23081 df-cn 23171 df-cnp 23172 df-haus 23259 df-cmp 23331 df-tx 23506 df-hmeo 23699 df-fil 23790 df-fm 23882 df-flim 23883 df-flf 23884 df-xms 24264 df-ms 24265 df-tms 24266 df-cncf 24827 df-ovol 25421 df-vol 25422 df-mbf 25576 df-itg1 25577 df-itg2 25578 df-ibl 25579 df-itg 25580 df-0p 25627 df-ditg 25804 df-limc 25823 df-dv 25824

This theorem is referenced by: fourierdlem108 46458

W3C validator