fourierdlem109 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem109 44931

Description: The integral of a piecewise continuous periodic function 𝐹 is unchanged if the domain is shifted by any value 𝑋. This lemma generalizes fourierdlem92 44914 where the integral was shifted by the exact period. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Hypotheses**
Ref	Expression
fourierdlem109.a	⊢ (𝜑 → 𝐴 ∈ ℝ)
fourierdlem109.b	⊢ (𝜑 → 𝐵 ∈ ℝ)
fourierdlem109.t	⊢ 𝑇 = (𝐵 − 𝐴)
fourierdlem109.x	⊢ (𝜑 → 𝑋 ∈ ℝ)
fourierdlem109.p	⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
fourierdlem109.m	⊢ (𝜑 → 𝑀 ∈ ℕ)
fourierdlem109.q	⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀))
fourierdlem109.f	⊢ (𝜑 → 𝐹:ℝ⟶ℂ)
fourierdlem109.fper	⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥))
fourierdlem109.fcn	⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
fourierdlem109.r	⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)))
fourierdlem109.l	⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1))))
fourierdlem109.o	⊢ 𝑂 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = (𝐴 − 𝑋) ∧ (𝑝‘𝑚) = (𝐵 − 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
fourierdlem109.h	⊢ 𝐻 = ({(𝐴 − 𝑋), (𝐵 − 𝑋)} ∪ {𝑥 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄})
fourierdlem109.n	⊢ 𝑁 = ((♯‘𝐻) − 1)
fourierdlem109.16	⊢ 𝑆 = (℩𝑓𝑓 Isom < , < ((0...𝑁), 𝐻))
fourierdlem109.17	⊢ 𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)))
fourierdlem109.18	⊢ 𝐽 = (𝑦 ∈ (𝐴(,]𝐵) ↦ if(𝑦 = 𝐵, 𝐴, 𝑦))
fourierdlem109.19	⊢ 𝐼 = (𝑥 ∈ ℝ ↦ sup({𝑗 ∈ (0..^𝑀) ∣ (𝑄‘𝑗) ≤ (𝐽‘(𝐸‘𝑥))}, ℝ, < ))

**Assertion**
Ref	Expression
fourierdlem109	⊢ (𝜑 → ∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 = ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)

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

**Proof of Theorem fourierdlem109**
Step	Hyp	Ref	Expression
1		fourierdlem109.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ)
2	1	adantr 482	. . 3 ⊢ ((𝜑 ∧ 0 < 𝑋) → 𝐴 ∈ ℝ)
3		fourierdlem109.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ ℝ)
4	3	adantr 482	. . 3 ⊢ ((𝜑 ∧ 0 < 𝑋) → 𝐵 ∈ ℝ)
5		fourierdlem109.t	. . 3 ⊢ 𝑇 = (𝐵 − 𝐴)
6		fourierdlem109.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ ℝ)
7	6	adantr 482	. . . 4 ⊢ ((𝜑 ∧ 0 < 𝑋) → 𝑋 ∈ ℝ)
8		simpr 486	. . . 4 ⊢ ((𝜑 ∧ 0 < 𝑋) → 0 < 𝑋)
9	7, 8	elrpd 13013	. . 3 ⊢ ((𝜑 ∧ 0 < 𝑋) → 𝑋 ∈ ℝ⁺)
10		fourierdlem109.p	. . 3 ⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
11		fourierdlem109.m	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℕ)
12	11	adantr 482	. . 3 ⊢ ((𝜑 ∧ 0 < 𝑋) → 𝑀 ∈ ℕ)
13		fourierdlem109.q	. . . 4 ⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀))
14	13	adantr 482	. . 3 ⊢ ((𝜑 ∧ 0 < 𝑋) → 𝑄 ∈ (𝑃‘𝑀))
15		fourierdlem109.f	. . . 4 ⊢ (𝜑 → 𝐹:ℝ⟶ℂ)
16	15	adantr 482	. . 3 ⊢ ((𝜑 ∧ 0 < 𝑋) → 𝐹:ℝ⟶ℂ)
17		fourierdlem109.fper	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥))
18	17	adantlr 714	. . 3 ⊢ (((𝜑 ∧ 0 < 𝑋) ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥))
19		fourierdlem109.fcn	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
20	19	adantlr 714	. . 3 ⊢ (((𝜑 ∧ 0 < 𝑋) ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
21		fourierdlem109.r	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)))
22	21	adantlr 714	. . 3 ⊢ (((𝜑 ∧ 0 < 𝑋) ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)))
23		fourierdlem109.l	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1))))
24	23	adantlr 714	. . 3 ⊢ (((𝜑 ∧ 0 < 𝑋) ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1))))
25	2, 4, 5, 9, 10, 12, 14, 16, 18, 20, 22, 24	fourierdlem108 44930	. 2 ⊢ ((𝜑 ∧ 0 < 𝑋) → ∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 = ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)
26		oveq2 7417	. . . . . . 7 ⊢ (𝑋 = 0 → (𝐴 − 𝑋) = (𝐴 − 0))
27	1	recnd 11242	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℂ)
28	27	subid1d 11560	. . . . . . 7 ⊢ (𝜑 → (𝐴 − 0) = 𝐴)
29	26, 28	sylan9eqr 2795	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = 0) → (𝐴 − 𝑋) = 𝐴)
30		oveq2 7417	. . . . . . 7 ⊢ (𝑋 = 0 → (𝐵 − 𝑋) = (𝐵 − 0))
31	3	recnd 11242	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℂ)
32	31	subid1d 11560	. . . . . . 7 ⊢ (𝜑 → (𝐵 − 0) = 𝐵)
33	30, 32	sylan9eqr 2795	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = 0) → (𝐵 − 𝑋) = 𝐵)
34	29, 33	oveq12d 7427	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 = 0) → ((𝐴 − 𝑋)[,](𝐵 − 𝑋)) = (𝐴[,]𝐵))
35	34	itgeq1d 44673	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = 0) → ∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 = ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)
36	35	adantlr 714	. . 3 ⊢ (((𝜑 ∧ ¬ 0 < 𝑋) ∧ 𝑋 = 0) → ∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 = ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)
37		simpll 766	. . . 4 ⊢ (((𝜑 ∧ ¬ 0 < 𝑋) ∧ ¬ 𝑋 = 0) → 𝜑)
38	37, 6	syl 17	. . . . 5 ⊢ (((𝜑 ∧ ¬ 0 < 𝑋) ∧ ¬ 𝑋 = 0) → 𝑋 ∈ ℝ)
39		0red 11217	. . . . 5 ⊢ (((𝜑 ∧ ¬ 0 < 𝑋) ∧ ¬ 𝑋 = 0) → 0 ∈ ℝ)
40		simpr 486	. . . . . 6 ⊢ (((𝜑 ∧ ¬ 0 < 𝑋) ∧ ¬ 𝑋 = 0) → ¬ 𝑋 = 0)
41	40	neqned 2948	. . . . 5 ⊢ (((𝜑 ∧ ¬ 0 < 𝑋) ∧ ¬ 𝑋 = 0) → 𝑋 ≠ 0)
42		simplr 768	. . . . 5 ⊢ (((𝜑 ∧ ¬ 0 < 𝑋) ∧ ¬ 𝑋 = 0) → ¬ 0 < 𝑋)
43	38, 39, 41, 42	lttri5d 44009	. . . 4 ⊢ (((𝜑 ∧ ¬ 0 < 𝑋) ∧ ¬ 𝑋 = 0) → 𝑋 < 0)
44	6	recnd 11242	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋 ∈ ℂ)
45	27, 44	subcld 11571	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 − 𝑋) ∈ ℂ)
46	45, 44	subnegd 11578	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐴 − 𝑋) − -𝑋) = ((𝐴 − 𝑋) + 𝑋))
47	27, 44	npcand 11575	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐴 − 𝑋) + 𝑋) = 𝐴)
48	46, 47	eqtrd 2773	. . . . . . . . 9 ⊢ (𝜑 → ((𝐴 − 𝑋) − -𝑋) = 𝐴)
49	31, 44	subcld 11571	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐵 − 𝑋) ∈ ℂ)
50	49, 44	subnegd 11578	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐵 − 𝑋) − -𝑋) = ((𝐵 − 𝑋) + 𝑋))
51	31, 44	npcand 11575	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐵 − 𝑋) + 𝑋) = 𝐵)
52	50, 51	eqtrd 2773	. . . . . . . . 9 ⊢ (𝜑 → ((𝐵 − 𝑋) − -𝑋) = 𝐵)
53	48, 52	oveq12d 7427	. . . . . . . 8 ⊢ (𝜑 → (((𝐴 − 𝑋) − -𝑋)[,]((𝐵 − 𝑋) − -𝑋)) = (𝐴[,]𝐵))
54	53	eqcomd 2739	. . . . . . 7 ⊢ (𝜑 → (𝐴[,]𝐵) = (((𝐴 − 𝑋) − -𝑋)[,]((𝐵 − 𝑋) − -𝑋)))
55	54	itgeq1d 44673	. . . . . 6 ⊢ (𝜑 → ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥 = ∫(((𝐴 − 𝑋) − -𝑋)[,]((𝐵 − 𝑋) − -𝑋))(𝐹‘𝑥) d𝑥)
56	55	adantr 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 < 0) → ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥 = ∫(((𝐴 − 𝑋) − -𝑋)[,]((𝐵 − 𝑋) − -𝑋))(𝐹‘𝑥) d𝑥)
57	1, 6	resubcld 11642	. . . . . . 7 ⊢ (𝜑 → (𝐴 − 𝑋) ∈ ℝ)
58	57	adantr 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 < 0) → (𝐴 − 𝑋) ∈ ℝ)
59	3, 6	resubcld 11642	. . . . . . 7 ⊢ (𝜑 → (𝐵 − 𝑋) ∈ ℝ)
60	59	adantr 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 < 0) → (𝐵 − 𝑋) ∈ ℝ)
61		eqid 2733	. . . . . 6 ⊢ ((𝐵 − 𝑋) − (𝐴 − 𝑋)) = ((𝐵 − 𝑋) − (𝐴 − 𝑋))
62	6	renegcld 11641	. . . . . . . 8 ⊢ (𝜑 → -𝑋 ∈ ℝ)
63	62	adantr 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 < 0) → -𝑋 ∈ ℝ)
64	6	lt0neg1d 11783	. . . . . . . 8 ⊢ (𝜑 → (𝑋 < 0 ↔ 0 < -𝑋))
65	64	biimpa 478	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 < 0) → 0 < -𝑋)
66	63, 65	elrpd 13013	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 < 0) → -𝑋 ∈ ℝ⁺)
67		fourierdlem109.o	. . . . . . 7 ⊢ 𝑂 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = (𝐴 − 𝑋) ∧ (𝑝‘𝑚) = (𝐵 − 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
68		fveq2 6892	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑗 → (𝑝‘𝑖) = (𝑝‘𝑗))
69		oveq1 7416	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑗 → (𝑖 + 1) = (𝑗 + 1))
70	69	fveq2d 6896	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑗 → (𝑝‘(𝑖 + 1)) = (𝑝‘(𝑗 + 1)))
71	68, 70	breq12d 5162	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑗 → ((𝑝‘𝑖) < (𝑝‘(𝑖 + 1)) ↔ (𝑝‘𝑗) < (𝑝‘(𝑗 + 1))))
72	71	cbvralvw 3235	. . . . . . . . . . 11 ⊢ (∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)) ↔ ∀𝑗 ∈ (0..^𝑚)(𝑝‘𝑗) < (𝑝‘(𝑗 + 1)))
73	72	anbi2i 624	. . . . . . . . . 10 ⊢ ((((𝑝‘0) = (𝐴 − 𝑋) ∧ (𝑝‘𝑚) = (𝐵 − 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1))) ↔ (((𝑝‘0) = (𝐴 − 𝑋) ∧ (𝑝‘𝑚) = (𝐵 − 𝑋)) ∧ ∀𝑗 ∈ (0..^𝑚)(𝑝‘𝑗) < (𝑝‘(𝑗 + 1))))
74	73	a1i 11	. . . . . . . . 9 ⊢ (𝑝 ∈ (ℝ ↑_m (0...𝑚)) → ((((𝑝‘0) = (𝐴 − 𝑋) ∧ (𝑝‘𝑚) = (𝐵 − 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1))) ↔ (((𝑝‘0) = (𝐴 − 𝑋) ∧ (𝑝‘𝑚) = (𝐵 − 𝑋)) ∧ ∀𝑗 ∈ (0..^𝑚)(𝑝‘𝑗) < (𝑝‘(𝑗 + 1)))))
75	74	rabbiia 3437	. . . . . . . 8 ⊢ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = (𝐴 − 𝑋) ∧ (𝑝‘𝑚) = (𝐵 − 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))} = {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = (𝐴 − 𝑋) ∧ (𝑝‘𝑚) = (𝐵 − 𝑋)) ∧ ∀𝑗 ∈ (0..^𝑚)(𝑝‘𝑗) < (𝑝‘(𝑗 + 1)))}
76	75	mpteq2i 5254	. . . . . . 7 ⊢ (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = (𝐴 − 𝑋) ∧ (𝑝‘𝑚) = (𝐵 − 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = (𝐴 − 𝑋) ∧ (𝑝‘𝑚) = (𝐵 − 𝑋)) ∧ ∀𝑗 ∈ (0..^𝑚)(𝑝‘𝑗) < (𝑝‘(𝑗 + 1)))})
77	67, 76	eqtri 2761	. . . . . 6 ⊢ 𝑂 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = (𝐴 − 𝑋) ∧ (𝑝‘𝑚) = (𝐵 − 𝑋)) ∧ ∀𝑗 ∈ (0..^𝑚)(𝑝‘𝑗) < (𝑝‘(𝑗 + 1)))})
78	10, 11, 13	fourierdlem11 44834	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵))
79	78	simp3d 1145	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 < 𝐵)
80	1, 3, 6, 79	ltsub1dd 11826	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 − 𝑋) < (𝐵 − 𝑋))
81		fourierdlem109.h	. . . . . . . . . 10 ⊢ 𝐻 = ({(𝐴 − 𝑋), (𝐵 − 𝑋)} ∪ {𝑥 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄})
82		fourierdlem109.n	. . . . . . . . . 10 ⊢ 𝑁 = ((♯‘𝐻) − 1)
83		fourierdlem109.16	. . . . . . . . . 10 ⊢ 𝑆 = (℩𝑓𝑓 Isom < , < ((0...𝑁), 𝐻))
84	5, 10, 11, 13, 57, 59, 80, 67, 81, 82, 83	fourierdlem54 44876	. . . . . . . . 9 ⊢ (𝜑 → ((𝑁 ∈ ℕ ∧ 𝑆 ∈ (𝑂‘𝑁)) ∧ 𝑆 Isom < , < ((0...𝑁), 𝐻)))
85	84	simpld 496	. . . . . . . 8 ⊢ (𝜑 → (𝑁 ∈ ℕ ∧ 𝑆 ∈ (𝑂‘𝑁)))
86	85	simpld 496	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ)
87	86	adantr 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 < 0) → 𝑁 ∈ ℕ)
88	85	simprd 497	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ (𝑂‘𝑁))
89	88	adantr 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 < 0) → 𝑆 ∈ (𝑂‘𝑁))
90	15	adantr 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 < 0) → 𝐹:ℝ⟶ℂ)
91	31, 27, 44	nnncan2d 11606	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐵 − 𝑋) − (𝐴 − 𝑋)) = (𝐵 − 𝐴))
92	91, 5	eqtr4di 2791	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐵 − 𝑋) − (𝐴 − 𝑋)) = 𝑇)
93	92	oveq2d 7425	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 + ((𝐵 − 𝑋) − (𝐴 − 𝑋))) = (𝑥 + 𝑇))
94	93	adantr 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑥 + ((𝐵 − 𝑋) − (𝐴 − 𝑋))) = (𝑥 + 𝑇))
95	94	fveq2d 6896	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + ((𝐵 − 𝑋) − (𝐴 − 𝑋)))) = (𝐹‘(𝑥 + 𝑇)))
96	95, 17	eqtrd 2773	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + ((𝐵 − 𝑋) − (𝐴 − 𝑋)))) = (𝐹‘𝑥))
97	96	adantlr 714	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 < 0) ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + ((𝐵 − 𝑋) − (𝐴 − 𝑋)))) = (𝐹‘𝑥))
98	11	adantr 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝑀 ∈ ℕ)
99	13	adantr 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝑄 ∈ (𝑃‘𝑀))
100	15	adantr 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝐹:ℝ⟶ℂ)
101	17	adantlr 714	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥))
102	19	adantlr 714	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
103	57	adantr 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝐴 − 𝑋) ∈ ℝ)
104	57	rexrd 11264	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 − 𝑋) ∈ ℝ^*)
105		pnfxr 11268	. . . . . . . . . . 11 ⊢ +∞ ∈ ℝ^*
106	105	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → +∞ ∈ ℝ^*)
107	59	ltpnfd 13101	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵 − 𝑋) < +∞)
108	104, 106, 59, 80, 107	eliood 44211	. . . . . . . . 9 ⊢ (𝜑 → (𝐵 − 𝑋) ∈ ((𝐴 − 𝑋)(,)+∞))
109	108	adantr 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝐵 − 𝑋) ∈ ((𝐴 − 𝑋)(,)+∞))
110		oveq1 7416	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (𝑥 + (𝑘 · 𝑇)) = (𝑦 + (𝑘 · 𝑇)))
111	110	eleq1d 2819	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → ((𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄))
112	111	rexbidv 3179	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄))
113	112	cbvrabv 3443	. . . . . . . . . 10 ⊢ {𝑥 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄} = {𝑦 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄}
114	113	uneq2i 4161	. . . . . . . . 9 ⊢ ({(𝐴 − 𝑋), (𝐵 − 𝑋)} ∪ {𝑥 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄}) = ({(𝐴 − 𝑋), (𝐵 − 𝑋)} ∪ {𝑦 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})
115	81, 114	eqtri 2761	. . . . . . . 8 ⊢ 𝐻 = ({(𝐴 − 𝑋), (𝐵 − 𝑋)} ∪ {𝑦 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})
116		fourierdlem109.17	. . . . . . . 8 ⊢ 𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)))
117		fourierdlem109.18	. . . . . . . 8 ⊢ 𝐽 = (𝑦 ∈ (𝐴(,]𝐵) ↦ if(𝑦 = 𝐵, 𝐴, 𝑦))
118		simpr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝑗 ∈ (0..^𝑁))
119		eqid 2733	. . . . . . . 8 ⊢ ((𝑆‘(𝑗 + 1)) − (𝐸‘(𝑆‘(𝑗 + 1)))) = ((𝑆‘(𝑗 + 1)) − (𝐸‘(𝑆‘(𝑗 + 1))))
120		eqid 2733	. . . . . . . 8 ⊢ (𝐹 ↾ ((𝐽‘(𝐸‘(𝑆‘𝑗)))(,)(𝐸‘(𝑆‘(𝑗 + 1))))) = (𝐹 ↾ ((𝐽‘(𝐸‘(𝑆‘𝑗)))(,)(𝐸‘(𝑆‘(𝑗 + 1)))))
121		eqid 2733	. . . . . . . 8 ⊢ (𝑦 ∈ (((𝐽‘(𝐸‘(𝑆‘𝑗))) + ((𝑆‘(𝑗 + 1)) − (𝐸‘(𝑆‘(𝑗 + 1)))))(,)((𝐸‘(𝑆‘(𝑗 + 1))) + ((𝑆‘(𝑗 + 1)) − (𝐸‘(𝑆‘(𝑗 + 1)))))) ↦ ((𝐹 ↾ ((𝐽‘(𝐸‘(𝑆‘𝑗)))(,)(𝐸‘(𝑆‘(𝑗 + 1)))))‘(𝑦 − ((𝑆‘(𝑗 + 1)) − (𝐸‘(𝑆‘(𝑗 + 1))))))) = (𝑦 ∈ (((𝐽‘(𝐸‘(𝑆‘𝑗))) + ((𝑆‘(𝑗 + 1)) − (𝐸‘(𝑆‘(𝑗 + 1)))))(,)((𝐸‘(𝑆‘(𝑗 + 1))) + ((𝑆‘(𝑗 + 1)) − (𝐸‘(𝑆‘(𝑗 + 1)))))) ↦ ((𝐹 ↾ ((𝐽‘(𝐸‘(𝑆‘𝑗)))(,)(𝐸‘(𝑆‘(𝑗 + 1)))))‘(𝑦 − ((𝑆‘(𝑗 + 1)) − (𝐸‘(𝑆‘(𝑗 + 1)))))))
122		fourierdlem109.19	. . . . . . . . 9 ⊢ 𝐼 = (𝑥 ∈ ℝ ↦ sup({𝑗 ∈ (0..^𝑀) ∣ (𝑄‘𝑗) ≤ (𝐽‘(𝐸‘𝑥))}, ℝ, < ))
123		fveq2 6892	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑖 → (𝑄‘𝑗) = (𝑄‘𝑖))
124	123	breq1d 5159	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑖 → ((𝑄‘𝑗) ≤ (𝐽‘(𝐸‘𝑥)) ↔ (𝑄‘𝑖) ≤ (𝐽‘(𝐸‘𝑥))))
125	124	cbvrabv 3443	. . . . . . . . . . 11 ⊢ {𝑗 ∈ (0..^𝑀) ∣ (𝑄‘𝑗) ≤ (𝐽‘(𝐸‘𝑥))} = {𝑖 ∈ (0..^𝑀) ∣ (𝑄‘𝑖) ≤ (𝐽‘(𝐸‘𝑥))}
126	125	supeq1i 9442	. . . . . . . . . 10 ⊢ sup({𝑗 ∈ (0..^𝑀) ∣ (𝑄‘𝑗) ≤ (𝐽‘(𝐸‘𝑥))}, ℝ, < ) = sup({𝑖 ∈ (0..^𝑀) ∣ (𝑄‘𝑖) ≤ (𝐽‘(𝐸‘𝑥))}, ℝ, < )
127	126	mpteq2i 5254	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ ↦ sup({𝑗 ∈ (0..^𝑀) ∣ (𝑄‘𝑗) ≤ (𝐽‘(𝐸‘𝑥))}, ℝ, < )) = (𝑥 ∈ ℝ ↦ sup({𝑖 ∈ (0..^𝑀) ∣ (𝑄‘𝑖) ≤ (𝐽‘(𝐸‘𝑥))}, ℝ, < ))
128	122, 127	eqtri 2761	. . . . . . . 8 ⊢ 𝐼 = (𝑥 ∈ ℝ ↦ sup({𝑖 ∈ (0..^𝑀) ∣ (𝑄‘𝑖) ≤ (𝐽‘(𝐸‘𝑥))}, ℝ, < ))
129	10, 5, 98, 99, 100, 101, 102, 103, 109, 67, 115, 82, 83, 116, 117, 118, 119, 120, 121, 128	fourierdlem90 44912	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝐹 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) ∈ (((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))–cn→ℂ))
130	129	adantlr 714	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 < 0) ∧ 𝑗 ∈ (0..^𝑁)) → (𝐹 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) ∈ (((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))–cn→ℂ))
131	21	adantlr 714	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)))
132		eqid 2733	. . . . . . . 8 ⊢ (𝑖 ∈ (0..^𝑀) ↦ 𝑅) = (𝑖 ∈ (0..^𝑀) ↦ 𝑅)
133	10, 5, 98, 99, 100, 101, 102, 131, 103, 109, 67, 115, 82, 83, 116, 117, 118, 119, 128, 132	fourierdlem89 44911	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → if((𝐽‘(𝐸‘(𝑆‘𝑗))) = (𝑄‘(𝐼‘(𝑆‘𝑗))), ((𝑖 ∈ (0..^𝑀) ↦ 𝑅)‘(𝐼‘(𝑆‘𝑗))), (𝐹‘(𝐽‘(𝐸‘(𝑆‘𝑗))))) ∈ ((𝐹 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) lim_ℂ (𝑆‘𝑗)))
134	133	adantlr 714	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 < 0) ∧ 𝑗 ∈ (0..^𝑁)) → if((𝐽‘(𝐸‘(𝑆‘𝑗))) = (𝑄‘(𝐼‘(𝑆‘𝑗))), ((𝑖 ∈ (0..^𝑀) ↦ 𝑅)‘(𝐼‘(𝑆‘𝑗))), (𝐹‘(𝐽‘(𝐸‘(𝑆‘𝑗))))) ∈ ((𝐹 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) lim_ℂ (𝑆‘𝑗)))
135	23	adantlr 714	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1))))
136		eqid 2733	. . . . . . . 8 ⊢ (𝑖 ∈ (0..^𝑀) ↦ 𝐿) = (𝑖 ∈ (0..^𝑀) ↦ 𝐿)
137	10, 5, 98, 99, 100, 101, 102, 135, 103, 109, 67, 115, 82, 83, 116, 117, 118, 119, 128, 136	fourierdlem91 44913	. . . . . . 7 ⊢ ((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))))
138	137	adantlr 714	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 < 0) ∧ 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))))
139	58, 60, 61, 66, 77, 87, 89, 90, 97, 130, 134, 138	fourierdlem108 44930	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 < 0) → ∫(((𝐴 − 𝑋) − -𝑋)[,]((𝐵 − 𝑋) − -𝑋))(𝐹‘𝑥) d𝑥 = ∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥)
140	56, 139	eqtr2d 2774	. . . 4 ⊢ ((𝜑 ∧ 𝑋 < 0) → ∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 = ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)
141	37, 43, 140	syl2anc 585	. . 3 ⊢ (((𝜑 ∧ ¬ 0 < 𝑋) ∧ ¬ 𝑋 = 0) → ∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 = ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)
142	36, 141	pm2.61dan 812	. 2 ⊢ ((𝜑 ∧ ¬ 0 < 𝑋) → ∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 = ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)
143	25, 142	pm2.61dan 812	1 ⊢ (𝜑 → ∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 = ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3062 ∃wrex 3071 {crab 3433 ∪ cun 3947 ifcif 4529 {cpr 4631 class class class wbr 5149 ↦ cmpt 5232 ran crn 5678 ↾ cres 5679 ℩cio 6494 ⟶wf 6540 ‘cfv 6544 Isom wiso 6545 (class class class)co 7409 ↑_m cmap 8820 supcsup 9435 ℂcc 11108 ℝcr 11109 0cc0 11110 1c1 11111 + caddc 11113 · cmul 11115 +∞cpnf 11245 ℝ^*cxr 11247 < clt 11248 ≤ cle 11249 − cmin 11444 -cneg 11445 / cdiv 11871 ℕcn 12212 ℤcz 12558 (,)cioo 13324 (,]cioc 13325 [,]cicc 13327 ...cfz 13484 ..^cfzo 13627 ⌊cfl 13755 ♯chash 14290 –cn→ccncf 24392 ∫citg 25135 lim_ℂ climc 25379

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-inf2 9636 ax-cc 10430 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188 ax-addf 11189 ax-mulf 11190

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-symdif 4243 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-disj 5115 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-of 7670 df-ofr 7671 df-om 7856 df-1st 7975 df-2nd 7976 df-supp 8147 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-2o 8467 df-oadd 8470 df-omul 8471 df-er 8703 df-map 8822 df-pm 8823 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9362 df-fi 9406 df-sup 9437 df-inf 9438 df-oi 9505 df-dju 9896 df-card 9934 df-acn 9937 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-9 12282 df-n0 12473 df-xnn0 12545 df-z 12559 df-dec 12678 df-uz 12823 df-q 12933 df-rp 12975 df-xneg 13092 df-xadd 13093 df-xmul 13094 df-ioo 13328 df-ioc 13329 df-ico 13330 df-icc 13331 df-fz 13485 df-fzo 13628 df-fl 13757 df-mod 13835 df-seq 13967 df-exp 14028 df-hash 14291 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-limsup 15415 df-clim 15432 df-rlim 15433 df-sum 15633 df-struct 17080 df-sets 17097 df-slot 17115 df-ndx 17127 df-base 17145 df-ress 17174 df-plusg 17210 df-mulr 17211 df-starv 17212 df-sca 17213 df-vsca 17214 df-ip 17215 df-tset 17216 df-ple 17217 df-ds 17219 df-unif 17220 df-hom 17221 df-cco 17222 df-rest 17368 df-topn 17369 df-0g 17387 df-gsum 17388 df-topgen 17389 df-pt 17390 df-prds 17393 df-xrs 17448 df-qtop 17453 df-imas 17454 df-xps 17456 df-mre 17530 df-mrc 17531 df-acs 17533 df-mgm 18561 df-sgrp 18610 df-mnd 18626 df-submnd 18672 df-mulg 18951 df-cntz 19181 df-cmn 19650 df-psmet 20936 df-xmet 20937 df-met 20938 df-bl 20939 df-mopn 20940 df-fbas 20941 df-fg 20942 df-cnfld 20945 df-top 22396 df-topon 22413 df-topsp 22435 df-bases 22449 df-cld 22523 df-ntr 22524 df-cls 22525 df-nei 22602 df-lp 22640 df-perf 22641 df-cn 22731 df-cnp 22732 df-haus 22819 df-cmp 22891 df-tx 23066 df-hmeo 23259 df-fil 23350 df-fm 23442 df-flim 23443 df-flf 23444 df-xms 23826 df-ms 23827 df-tms 23828 df-cncf 24394 df-ovol 24981 df-vol 24982 df-mbf 25136 df-itg1 25137 df-itg2 25138 df-ibl 25139 df-itg 25140 df-0p 25187 df-ditg 25364 df-limc 25383 df-dv 25384

This theorem is referenced by: fourierdlem110 44932

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