fourierdlem107 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem107 46748

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 46733 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 7402	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 − 𝑋) + 𝑇) = ((𝐴 − 𝑋) + (𝐵 − 𝐴))
3		fourierdlem107.a	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐴 ∈ ℝ)
4	3	recnd 11204	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐴 ∈ ℂ)
5		fourierdlem107.x	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑋 ∈ ℝ⁺)
6	5	rpred 13031	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑋 ∈ ℝ)
7	6	recnd 11204	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑋 ∈ ℂ)
8		fourierdlem107.b	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐵 ∈ ℝ)
9	8	recnd 11204	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐵 ∈ ℂ)
10	4, 7, 9, 4	subadd4b 45823	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝐴 − 𝑋) + (𝐵 − 𝐴)) = ((𝐴 − 𝐴) + (𝐵 − 𝑋)))
11	2, 10	eqtrid 2808	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝐴 − 𝑋) + 𝑇) = ((𝐴 − 𝐴) + (𝐵 − 𝑋)))
12	4	subidd 11524	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐴 − 𝐴) = 0)
13	12	oveq1d 7406	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝐴 − 𝐴) + (𝐵 − 𝑋)) = (0 + (𝐵 − 𝑋)))
14	8, 6	resubcld 11609	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐵 − 𝑋) ∈ ℝ)
15	14	recnd 11204	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐵 − 𝑋) ∈ ℂ)
16	15	addlidd 11378	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (0 + (𝐵 − 𝑋)) = (𝐵 − 𝑋))
17	11, 13, 16	3eqtrd 2800	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝐴 − 𝑋) + 𝑇) = (𝐵 − 𝑋))
18	1	oveq2i 7402	. . . . . . . . . . . . . . 15 ⊢ (𝐴 + 𝑇) = (𝐴 + (𝐵 − 𝐴))
19	4, 9	pncan3d 11539	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐴 + (𝐵 − 𝐴)) = 𝐵)
20	18, 19	eqtrid 2808	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐴 + 𝑇) = 𝐵)
21	17, 20	oveq12d 7409	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((𝐴 − 𝑋) + 𝑇)[,](𝐴 + 𝑇)) = ((𝐵 − 𝑋)[,]𝐵))
22	21	eqcomd 2767	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐵 − 𝑋)[,]𝐵) = (((𝐴 − 𝑋) + 𝑇)[,](𝐴 + 𝑇)))
23	22	itgeq1d 46492	. . . . . . . . . . 11 ⊢ (𝜑 → ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 = ∫(((𝐴 − 𝑋) + 𝑇)[,](𝐴 + 𝑇))(𝐹‘𝑥) d𝑥)
24	3, 6	resubcld 11609	. . . . . . . . . . . 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 6862	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 = 𝑗 → (𝑝‘𝑖) = (𝑝‘𝑗))
27		oveq1 7398	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 = 𝑗 → (𝑖 + 1) = (𝑗 + 1))
28	27	fveq2d 6866	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 = 𝑗 → (𝑝‘(𝑖 + 1)) = (𝑝‘(𝑗 + 1)))
29	26, 28	breq12d 5110	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 = 𝑗 → ((𝑝‘𝑖) < (𝑝‘(𝑖 + 1)) ↔ (𝑝‘𝑗) < (𝑝‘(𝑗 + 1))))
30	29	cbvralvw 3239	. . . . . . . . . . . . . . . . 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 639	. . . . . . . . . . . . . . 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 3420	. . . . . . . . . . . . . 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 5192	. . . . . . . . . . . . 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 2784	. . . . . . . . . . . 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 13063	. . . . . . . . . . . . . . 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 46695	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑁 ∈ ℕ ∧ 𝑆 ∈ (𝑂‘𝑁)) ∧ 𝑆 Isom < , < ((0...𝑁), 𝐻)))
44	43	simpld 498	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑁 ∈ ℕ ∧ 𝑆 ∈ (𝑂‘𝑁)))
45	44	simpld 498	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℕ)
46	8, 3	resubcld 11609	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ)
47	1, 46	eqeltrid 2865	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑇 ∈ ℝ)
48	44	simprd 499	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑆 ∈ (𝑂‘𝑁))
49	24	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 − 𝑋)[,]𝐴)) → (𝐴 − 𝑋) ∈ ℝ)
50	3	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 − 𝑋)[,]𝐴)) → 𝐴 ∈ ℝ)
51		simpr 488	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 − 𝑋)[,]𝐴)) → 𝑥 ∈ ((𝐴 − 𝑋)[,]𝐴))
52		eliccre 46042	. . . . . . . . . . . . . 14 ⊢ (((𝐴 − 𝑋) ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ 𝑥 ∈ ((𝐴 − 𝑋)[,]𝐴)) → 𝑥 ∈ ℝ)
53	49, 50, 51, 52	syl3anc 1389	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 − 𝑋)[,]𝐴)) → 𝑥 ∈ ℝ)
54		fourierdlem107.fper	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥))
55	53, 54	syldan 600	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 − 𝑋)[,]𝐴)) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥))
56		fveq2 6862	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑗 → (𝑆‘𝑖) = (𝑆‘𝑗))
57	56	oveq1d 7406	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑗 → ((𝑆‘𝑖) + 𝑇) = ((𝑆‘𝑗) + 𝑇))
58	57	cbvmptv 5201	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (0...𝑁) ↦ ((𝑆‘𝑖) + 𝑇)) = (𝑗 ∈ (0...𝑁) ↦ ((𝑆‘𝑗) + 𝑇))
59		eqid 2761	. . . . . . . . . . . 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 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝑀 ∈ ℕ)
62	38	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝑄 ∈ (𝑃‘𝑀))
63	60	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝐹:ℝ⟶ℂ)
64	54	adantlr 725	. . . . . . . . . . . . 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 725	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
67	24	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝐴 − 𝑋) ∈ ℝ)
68	67	rexrd 11226	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝐴 − 𝑋) ∈ ℝ^*)
69		pnfxr 11230	. . . . . . . . . . . . . . 15 ⊢ +∞ ∈ ℝ^*
70	69	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → +∞ ∈ ℝ^*)
71	3	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝐴 ∈ ℝ)
72	39	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝐴 − 𝑋) < 𝐴)
73	3	ltpnfd 13117	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐴 < +∞)
74	73	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝐴 < +∞)
75	68, 70, 71, 72, 74	eliood 46035	. . . . . . . . . . . . 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 488	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝑗 ∈ (0..^𝑁))
79		eqid 2761	. . . . . . . . . . . . 13 ⊢ ((𝑆‘(𝑗 + 1)) − (𝐸‘(𝑆‘(𝑗 + 1)))) = ((𝑆‘(𝑗 + 1)) − (𝐸‘(𝑆‘(𝑗 + 1))))
80		eqid 2761	. . . . . . . . . . . . 13 ⊢ (𝐹 ↾ ((𝑍‘(𝐸‘(𝑆‘𝑗)))(,)(𝐸‘(𝑆‘(𝑗 + 1))))) = (𝐹 ↾ ((𝑍‘(𝐸‘(𝑆‘𝑗)))(,)(𝐸‘(𝑆‘(𝑗 + 1)))))
81		eqid 2761	. . . . . . . . . . . . 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 46731	. . . . . . . . . . . 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 725	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)))
86		eqid 2761	. . . . . . . . . . . . 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 46730	. . . . . . . . . . . 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 725	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1))))
90		eqid 2761	. . . . . . . . . . . . 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 46732	. . . . . . . . . . . 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 46733	. . . . . . . . . . 11 ⊢ (𝜑 → ∫(((𝐴 − 𝑋) + 𝑇)[,](𝐴 + 𝑇))(𝐹‘𝑥) d𝑥 = ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥)
93	23, 92	eqtrd 2796	. . . . . . . . . 10 ⊢ (𝜑 → ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 = ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥)
94	60	adantr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐵 − 𝑋)[,]𝐵)) → 𝐹:ℝ⟶ℂ)
95	14	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐵 − 𝑋)[,]𝐵)) → (𝐵 − 𝑋) ∈ ℝ)
96	8	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐵 − 𝑋)[,]𝐵)) → 𝐵 ∈ ℝ)
97		simpr 488	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐵 − 𝑋)[,]𝐵)) → 𝑥 ∈ ((𝐵 − 𝑋)[,]𝐵))
98		eliccre 46042	. . . . . . . . . . . . 13 ⊢ (((𝐵 − 𝑋) ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑥 ∈ ((𝐵 − 𝑋)[,]𝐵)) → 𝑥 ∈ ℝ)
99	95, 96, 97, 98	syl3anc 1389	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐵 − 𝑋)[,]𝐵)) → 𝑥 ∈ ℝ)
100	94, 99	ffvelcdmd 7061	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐵 − 𝑋)[,]𝐵)) → (𝐹‘𝑥) ∈ ℂ)
101	14	rexrd 11226	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐵 − 𝑋) ∈ ℝ^*)
102	69	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → +∞ ∈ ℝ^*)
103	8, 5	ltsubrpd 13063	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐵 − 𝑋) < 𝐵)
104	8	ltpnfd 13117	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 < +∞)
105	101, 102, 8, 103, 104	eliood 46035	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ∈ ((𝐵 − 𝑋)(,)+∞))
106	36, 1, 37, 38, 60, 54, 65, 84, 88, 14, 105	fourierdlem105 46746	. . . . . . . . . . 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 2862	. . . . . . . . 9 ⊢ (𝜑 → ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 ∈ ℂ)
109	108	subidd 11524	. . . . . . . 8 ⊢ (𝜑 → (∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥) = 0)
110	109	eqcomd 2767	. . . . . . 7 ⊢ (𝜑 → 0 = (∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥))
111	110	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → 0 = (∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥))
112	24	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (𝐴 − 𝑋) ∈ ℝ)
113	3	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → 𝐴 ∈ ℝ)
114	14	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (𝐵 − 𝑋) ∈ ℝ)
115	36, 37, 38	fourierdlem11 46653	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵))
116	115	simp3d 1156	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 < 𝐵)
117	3, 8, 116	ltled 11325	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
118	117	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → 𝐴 ≤ 𝐵)
119	3, 8, 6	lesub1d 11788	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ (𝐴 − 𝑋) ≤ (𝐵 − 𝑋)))
120	119	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (𝐴 ≤ 𝐵 ↔ (𝐴 − 𝑋) ≤ (𝐵 − 𝑋)))
121	118, 120	mpbid 234	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (𝐴 − 𝑋) ≤ (𝐵 − 𝑋))
122	8	adantr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → 𝐵 ∈ ℝ)
123	6	adantr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → 𝑋 ∈ ℝ)
124		simpr 488	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → 𝑇 < 𝑋)
125	1, 124	eqbrtrrid 5133	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (𝐵 − 𝐴) < 𝑋)
126	122, 113, 123, 125	ltsub23d 11786	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (𝐵 − 𝑋) < 𝐴)
127	114, 113, 126	ltled 11325	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (𝐵 − 𝑋) ≤ 𝐴)
128	112, 113, 114, 121, 127	eliccd 46041	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (𝐵 − 𝑋) ∈ ((𝐴 − 𝑋)[,]𝐴))
129	60	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 − 𝑋)[,]𝐴)) → 𝐹:ℝ⟶ℂ)
130	129, 53	ffvelcdmd 7061	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 − 𝑋)[,]𝐴)) → (𝐹‘𝑥) ∈ ℂ)
131	130	adantlr 725	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑇 < 𝑋) ∧ 𝑥 ∈ ((𝐴 − 𝑋)[,]𝐴)) → (𝐹‘𝑥) ∈ ℂ)
132	24	rexrd 11226	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 − 𝑋) ∈ ℝ^*)
133	3, 8, 6, 116	ltsub1dd 11793	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 − 𝑋) < (𝐵 − 𝑋))
134	14	ltpnfd 13117	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐵 − 𝑋) < +∞)
135	132, 102, 14, 133, 134	eliood 46035	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵 − 𝑋) ∈ ((𝐴 − 𝑋)(,)+∞))
136	36, 1, 37, 38, 60, 54, 65, 84, 88, 24, 135	fourierdlem105 46746	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋)) ↦ (𝐹‘𝑥)) ∈ 𝐿¹)
137	136	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (𝑥 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋)) ↦ (𝐹‘𝑥)) ∈ 𝐿¹)
138	37	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → 𝑀 ∈ ℕ)
139	38	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → 𝑄 ∈ (𝑃‘𝑀))
140	60	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → 𝐹:ℝ⟶ℂ)
141	54	adantlr 725	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑇 < 𝑋) ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥))
142	65	adantlr 725	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑇 < 𝑋) ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
143	84	adantlr 725	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑇 < 𝑋) ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)))
144	88	adantlr 725	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑇 < 𝑋) ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1))))
145	101	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (𝐵 − 𝑋) ∈ ℝ^*)
146	69	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → +∞ ∈ ℝ^*)
147	113	ltpnfd 13117	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → 𝐴 < +∞)
148	145, 146, 113, 126, 147	eliood 46035	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → 𝐴 ∈ ((𝐵 − 𝑋)(,)+∞))
149	36, 1, 138, 139, 140, 141, 142, 143, 144, 114, 148	fourierdlem105 46746	. . . . . . . 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 7406	. . . . . 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 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → 𝐹:ℝ⟶ℂ)
153	24	adantr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → (𝐴 − 𝑋) ∈ ℝ)
154	14	adantr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → (𝐵 − 𝑋) ∈ ℝ)
155		simpr 488	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → 𝑥 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋)))
156		eliccre 46042	. . . . . . . . . . 11 ⊢ (((𝐴 − 𝑋) ∈ ℝ ∧ (𝐵 − 𝑋) ∈ ℝ ∧ 𝑥 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → 𝑥 ∈ ℝ)
157	153, 154, 155, 156	syl3anc 1389	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → 𝑥 ∈ ℝ)
158	152, 157	ffvelcdmd 7061	. . . . . . . . 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 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → ∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 ∈ ℂ)
161	60	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐵 − 𝑋)[,]𝐴)) → 𝐹:ℝ⟶ℂ)
162	14	adantr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐵 − 𝑋)[,]𝐴)) → (𝐵 − 𝑋) ∈ ℝ)
163	3	adantr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐵 − 𝑋)[,]𝐴)) → 𝐴 ∈ ℝ)
164		simpr 488	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐵 − 𝑋)[,]𝐴)) → 𝑥 ∈ ((𝐵 − 𝑋)[,]𝐴))
165		eliccre 46042	. . . . . . . . . . 11 ⊢ (((𝐵 − 𝑋) ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ 𝑥 ∈ ((𝐵 − 𝑋)[,]𝐴)) → 𝑥 ∈ ℝ)
166	162, 163, 164, 165	syl3anc 1389	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐵 − 𝑋)[,]𝐴)) → 𝑥 ∈ ℝ)
167	161, 166	ffvelcdmd 7061	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐵 − 𝑋)[,]𝐴)) → (𝐹‘𝑥) ∈ ℂ)
168	167	adantlr 725	. . . . . . . 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 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 ∈ ℂ)
171	160, 169, 170	addsubassd 11556	. . . . . 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 2800	. . . . 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 7407	. . . 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 11525	. . . 4 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 − 0) = ∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥)
175	159	subidd 11524	. . . . . . 7 ⊢ (𝜑 → (∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 − ∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥) = 0)
176	175	oveq1d 7406	. . . . . 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 484	. . . . 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 11536	. . . . . 6 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥) ∈ ℂ)
179	160, 160, 178	subsub4d 11567	. . . . 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 11411	. . . . . 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 11552	. . . . . 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 2810	. . . . 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 2804	. . . 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 2804	. . 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 11524	. . . . . . . 8 ⊢ (𝜑 → (∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 − ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥) = 0)
186	185	eqcomd 2767	. . . . . . 7 ⊢ (𝜑 → 0 = (∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 − ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥))
187	186	oveq2d 7407	. . . . . 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 484	. . . . 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 11377	. . . . 5 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 + 0) = ∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥)
190	114, 122, 113, 127, 118	eliccd 46041	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → 𝐴 ∈ ((𝐵 − 𝑋)[,]𝐵))
191	100	adantlr 725	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑇 < 𝑋) ∧ 𝑥 ∈ ((𝐵 − 𝑋)[,]𝐵)) → (𝐹‘𝑥) ∈ ℂ)
192	3, 8	iccssred 13432	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
193	60, 192	feqresmpt 6931	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹 ↾ (𝐴[,]𝐵)) = (𝑥 ∈ (𝐴[,]𝐵) ↦ (𝐹‘𝑥)))
194	60, 192	fssresd 6726	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹 ↾ (𝐴[,]𝐵)):(𝐴[,]𝐵)⟶ℂ)
195		ioossicc 13431	. . . . . . . . . . . . . . 15 ⊢ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))
196	3	rexrd 11226	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
197	196	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐴 ∈ ℝ^*)
198	8	rexrd 11226	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
199	198	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐵 ∈ ℝ^*)
200	36, 37, 38	fourierdlem15 46657	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶(𝐴[,]𝐵))
201	200	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶(𝐴[,]𝐵))
202		simpr 488	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0..^𝑀))
203	197, 199, 201, 202	fourierdlem8 46650	. . . . . . . . . . . . . . 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 5990	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ (𝐴[,]𝐵)) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))))
206	205, 65	eqeltrd 2861	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝐹 ↾ (𝐴[,]𝐵)) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
207	205	eqcomd 2767	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝐹 ↾ (𝐴[,]𝐵)) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))))
208	207	oveq1d 7406	. . . . . . . . . . . . 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 2863	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ (((𝐹 ↾ (𝐴[,]𝐵)) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)))
210	207	oveq1d 7406	. . . . . . . . . . . . 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 2863	. . . . . . . . . . . 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 46710	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹 ↾ (𝐴[,]𝐵)) ∈ 𝐿¹)
213	193, 212	eqeltrrd 2862	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ (𝐹‘𝑥)) ∈ 𝐿¹)
214	213	adantr 484	. . . . . . . . 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 7407	. . . . . . 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 7407	. . . . . 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 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 ∈ ℂ)
219	215, 218	eqeltrrd 2862	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 + ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥) ∈ ℂ)
220	169, 218, 219	addsub12d 11559	. . . . . 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 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐹:ℝ⟶ℂ)
222	3	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐴 ∈ ℝ)
223	8	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐵 ∈ ℝ)
224		simpr 488	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ∈ (𝐴[,]𝐵))
225		eliccre 46042	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ∈ ℝ)
226	222, 223, 224, 225	syl3anc 1389	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ∈ ℝ)
227	221, 226	ffvelcdmd 7061	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℂ)
228	227, 213	itgcl 25834	. . . . . . . . . . 11 ⊢ (𝜑 → ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥 ∈ ℂ)
229	228	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥 ∈ ℂ)
230	169, 169, 229	subsub4d 11567	. . . . . . . . 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 2767	. . . . . . . 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 7407	. . . . . . 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 11524	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → (∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥) = 0)
234	233	oveq1d 7406	. . . . . . . . 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 11411	. . . . . . . . 9 ⊢ -∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥 = (0 − ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)
236	234, 235	eqtr4di 2814	. . . . . . . 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 7407	. . . . . . 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 11542	. . . . . . 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 2800	. . . . . 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 2800	. . . . 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 2804	. . . 4 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → ∫((𝐵 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 = (∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 − ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥))
242	241	oveq2d 7407	. . 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 11564	. . . . 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 7407	. . . . . . 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 2796	. . . . . 6 ⊢ (𝜑 → (∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥 − ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥) = 0)
246	245	oveq1d 7406	. . . . 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 11378	. . . . 5 ⊢ (𝜑 → (0 + ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥) = ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)
248	243, 246, 247	3eqtrd 2800	. . . 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 484	. . 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 2800	. 2 ⊢ ((𝜑 ∧ 𝑇 < 𝑋) → ∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 = ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)
251	24	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → (𝐴 − 𝑋) ∈ ℝ)
252	14	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → (𝐵 − 𝑋) ∈ ℝ)
253	3	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → 𝐴 ∈ ℝ)
254	24, 3, 39	ltled 11325	. . . . . . 7 ⊢ (𝜑 → (𝐴 − 𝑋) ≤ 𝐴)
255	254	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → (𝐴 − 𝑋) ≤ 𝐴)
256	6	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → 𝑋 ∈ ℝ)
257	8	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → 𝐵 ∈ ℝ)
258		id 22	. . . . . . . . 9 ⊢ (𝑋 ≤ 𝑇 → 𝑋 ≤ 𝑇)
259	258, 1	breqtrdi 5138	. . . . . . . 8 ⊢ (𝑋 ≤ 𝑇 → 𝑋 ≤ (𝐵 − 𝐴))
260	259	adantl 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → 𝑋 ≤ (𝐵 − 𝐴))
261	256, 257, 253, 260	lesubd 11785	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → 𝐴 ≤ (𝐵 − 𝑋))
262	251, 252, 253, 255, 261	eliccd 46041	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → 𝐴 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋)))
263	158	adantlr 725	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ≤ 𝑇) ∧ 𝑥 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → (𝐹‘𝑥) ∈ ℂ)
264	132, 102, 3, 39, 73	eliood 46035	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ((𝐴 − 𝑋)(,)+∞))
265	36, 1, 37, 38, 60, 54, 65, 84, 88, 24, 264	fourierdlem105 46746	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ((𝐴 − 𝑋)[,]𝐴) ↦ (𝐹‘𝑥)) ∈ 𝐿¹)
266	265	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → (𝑥 ∈ ((𝐴 − 𝑋)[,]𝐴) ↦ (𝐹‘𝑥)) ∈ 𝐿¹)
267	3	leidd 11747	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ≤ 𝐴)
268	5	rpge0d 13035	. . . . . . . . 9 ⊢ (𝜑 → 0 ≤ 𝑋)
269	8, 6	subge02d 11773	. . . . . . . . 9 ⊢ (𝜑 → (0 ≤ 𝑋 ↔ (𝐵 − 𝑋) ≤ 𝐵))
270	268, 269	mpbid 234	. . . . . . . 8 ⊢ (𝜑 → (𝐵 − 𝑋) ≤ 𝐵)
271		iccss 13412	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐴 ≤ 𝐴 ∧ (𝐵 − 𝑋) ≤ 𝐵)) → (𝐴[,](𝐵 − 𝑋)) ⊆ (𝐴[,]𝐵))
272	3, 8, 267, 270, 271	syl22anc 849	. . . . . . 7 ⊢ (𝜑 → (𝐴[,](𝐵 − 𝑋)) ⊆ (𝐴[,]𝐵))
273		iccmbl 25616	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ (𝐵 − 𝑋) ∈ ℝ) → (𝐴[,](𝐵 − 𝑋)) ∈ dom vol)
274	3, 14, 273	syl2anc 593	. . . . . . 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 484	. . . . 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 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → 0 ≤ 𝑋)
279	269	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → (0 ≤ 𝑋 ↔ (𝐵 − 𝑋) ≤ 𝐵))
280	278, 279	mpbid 234	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → (𝐵 − 𝑋) ≤ 𝐵)
281	253, 257, 252, 261, 280	eliccd 46041	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → (𝐵 − 𝑋) ∈ (𝐴[,]𝐵))
282	227	adantlr 725	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑋 ≤ 𝑇) ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℂ)
283	8	leidd 11747	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ≤ 𝐵)
284	283	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → 𝐵 ≤ 𝐵)
285		iccss 13412	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐴 ≤ (𝐵 − 𝑋) ∧ 𝐵 ≤ 𝐵)) → ((𝐵 − 𝑋)[,]𝐵) ⊆ (𝐴[,]𝐵))
286	253, 257, 261, 284, 285	syl22anc 849	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → ((𝐵 − 𝑋)[,]𝐵) ⊆ (𝐴[,]𝐵))
287		iccmbl 25616	. . . . . . . . . . 11 ⊢ (((𝐵 − 𝑋) ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐵 − 𝑋)[,]𝐵) ∈ dom vol)
288	14, 8, 287	syl2anc 593	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐵 − 𝑋)[,]𝐵) ∈ dom vol)
289	288	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → ((𝐵 − 𝑋)[,]𝐵) ∈ dom vol)
290	213	adantr 484	. . . . . . . . 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 7406	. . . . . 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 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,](𝐵 − 𝑋))) → 𝐹:ℝ⟶ℂ)
295	3	adantr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,](𝐵 − 𝑋))) → 𝐴 ∈ ℝ)
296	14	adantr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,](𝐵 − 𝑋))) → (𝐵 − 𝑋) ∈ ℝ)
297		simpr 488	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,](𝐵 − 𝑋))) → 𝑥 ∈ (𝐴[,](𝐵 − 𝑋)))
298		eliccre 46042	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ (𝐵 − 𝑋) ∈ ℝ ∧ 𝑥 ∈ (𝐴[,](𝐵 − 𝑋))) → 𝑥 ∈ ℝ)
299	295, 296, 297, 298	syl3anc 1389	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,](𝐵 − 𝑋))) → 𝑥 ∈ ℝ)
300	294, 299	ffvelcdmd 7061	. . . . . . . . 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 11556	. . . . . . 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 484	. . . . . 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 7407	. . . . . . . 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 11377	. . . . . . . 8 ⊢ (𝜑 → (∫(𝐴[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 + 0) = ∫(𝐴[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥)
306	304, 305	eqtrd 2796	. . . . . . 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 484	. . . . . 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 2801	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → ∫(𝐴[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 = (∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥 − ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥))
309	308	oveq2d 7407	. . . 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 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 = ∫((𝐴 − 𝑋)[,]𝐴)(𝐹‘𝑥) d𝑥)
311	107	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → ∫((𝐵 − 𝑋)[,]𝐵)(𝐹‘𝑥) d𝑥 ∈ ℂ)
312	310, 311	eqeltrrd 2862	. . . . . 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 11559	. . . . 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 11556	. . . . 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 2799	. . . 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 2800	. . 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 7407	. . 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 11537	. . 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 2800	. 2 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑇) → ∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 = ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)
321	250, 320, 47, 6	ltlecasei 11285	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 208 ∧ wa 399 = wceq 1559 ∈ wcel 2141 ∀wral 3075 ∃wrex 3085 {crab 3413 ∪ cun 3900 ⊆ wss 3902 ifcif 4477 {cpr 4581 class class class wbr 5097 ↦ cmpt 5178 dom cdm 5643 ran crn 5644 ↾ cres 5645 ℩cio 6470 ⟶wf 6512 ‘cfv 6516 Isom wiso 6517 (class class class)co 7391 ↑_m cmap 8802 supcsup 9380 ℂcc 11065 ℝcr 11066 0cc0 11067 1c1 11068 + caddc 11070 · cmul 11072 +∞cpnf 11207 ℝ^*cxr 11209 < clt 11210 ≤ cle 11211 − cmin 11408 -cneg 11409 / cdiv 11838 ℕcn 12204 ℤcz 12562 ℝ⁺crp 12987 (,)cioo 13343 (,]cioc 13344 [,]cicc 13346 ...cfz 13506 ..^cfzo 13653 ⌊cfl 13794 ♯chash 14337 –cn→ccncf 24926 volcvol 25513 𝐿¹cibl 25667 ∫citg 25668 lim_ℂ climc 25912

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5224 ax-sep 5243 ax-nul 5253 ax-pow 5319 ax-pr 5387 ax-un 7713 ax-inf2 9590 ax-cc 10386 ax-cnex 11123 ax-resscn 11124 ax-1cn 11125 ax-icn 11126 ax-addcl 11127 ax-addrcl 11128 ax-mulcl 11129 ax-mulrcl 11130 ax-mulcom 11131 ax-addass 11132 ax-mulass 11133 ax-distr 11134 ax-i2m1 11135 ax-1ne0 11136 ax-1rid 11137 ax-rnegex 11138 ax-rrecex 11139 ax-cnre 11140 ax-pre-lttri 11141 ax-pre-lttrn 11142 ax-pre-ltadd 11143 ax-pre-mulgt0 11144 ax-pre-sup 11145 ax-addf 11146

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-symdif 4203 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-uni 4863 df-int 4903 df-iun 4948 df-iin 4949 df-disj 5065 df-br 5098 df-opab 5160 df-mpt 5179 df-tr 5205 df-id 5538 df-eprel 5543 df-po 5551 df-so 5552 df-fr 5596 df-se 5597 df-we 5598 df-xp 5649 df-rel 5650 df-cnv 5651 df-co 5652 df-dm 5653 df-rn 5654 df-res 5655 df-ima 5656 df-pred 6283 df-ord 6344 df-on 6345 df-lim 6346 df-suc 6347 df-iota 6472 df-fun 6518 df-fn 6519 df-f 6520 df-f1 6521 df-fo 6522 df-f1o 6523 df-fv 6524 df-isom 6525 df-riota 7348 df-ov 7394 df-oprab 7395 df-mpo 7396 df-of 7655 df-ofr 7656 df-om 7842 df-1st 7965 df-2nd 7966 df-supp 8135 df-frecs 8256 df-wrecs 8287 df-recs 8336 df-rdg 8375 df-1o 8431 df-2o 8432 df-oadd 8435 df-omul 8436 df-er 8672 df-map 8804 df-pm 8805 df-ixp 8874 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-fsupp 9302 df-fi 9351 df-sup 9382 df-inf 9383 df-oi 9452 df-dju 9853 df-card 9891 df-acn 9894 df-pnf 11212 df-mnf 11213 df-xr 11214 df-ltxr 11215 df-le 11216 df-sub 11410 df-neg 11411 df-div 11839 df-nn 12205 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12476 df-xnn0 12549 df-z 12563 df-dec 12683 df-uz 12834 df-q 12944 df-rp 12988 df-xneg 13108 df-xadd 13109 df-xmul 13110 df-ioo 13347 df-ioc 13348 df-ico 13349 df-icc 13350 df-fz 13507 df-fzo 13654 df-fl 13796 df-mod 13874 df-seq 14009 df-exp 14069 df-hash 14338 df-cj 15117 df-re 15118 df-im 15119 df-sqrt 15253 df-abs 15254 df-limsup 15489 df-clim 15506 df-rlim 15507 df-sum 15705 df-struct 17174 df-sets 17191 df-slot 17209 df-ndx 17221 df-base 17237 df-ress 17258 df-plusg 17290 df-mulr 17291 df-starv 17292 df-sca 17293 df-vsca 17294 df-ip 17295 df-tset 17296 df-ple 17297 df-ds 17299 df-unif 17300 df-hom 17301 df-cco 17302 df-rest 17442 df-topn 17443 df-0g 17461 df-gsum 17462 df-topgen 17463 df-pt 17464 df-prds 17467 df-xrs 17523 df-qtop 17528 df-imas 17529 df-xps 17531 df-mre 17605 df-mrc 17606 df-acs 17608 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-submnd 18809 df-mulg 19101 df-cntz 19348 df-cmn 19813 df-psmet 21404 df-xmet 21405 df-met 21406 df-bl 21407 df-mopn 21408 df-fbas 21409 df-fg 21410 df-cnfld 21413 df-top 22942 df-topon 22959 df-topsp 22981 df-bases 22994 df-cld 23067 df-ntr 23068 df-cls 23069 df-nei 23146 df-lp 23184 df-perf 23185 df-cn 23275 df-cnp 23276 df-haus 23363 df-cmp 23435 df-tx 23610 df-hmeo 23803 df-fil 23894 df-fm 23986 df-flim 23987 df-flf 23988 df-xms 24368 df-ms 24369 df-tms 24370 df-cncf 24928 df-ovol 25514 df-vol 25515 df-mbf 25669 df-itg1 25670 df-itg2 25671 df-ibl 25672 df-itg 25673 df-0p 25720 df-ditg 25897 df-limc 25916 df-dv 25917

This theorem is referenced by: fourierdlem108 46749

W3C validator