itgiccshift - Mathbox for Glauco Siliprandi

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Theorem itgiccshift 46260

Description: The integral of a function, 𝐹 stays the same if its closed interval domain is shifted by 𝑇. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Hypotheses**
Ref	Expression
itgiccshift.a	⊢ (𝜑 → 𝐴 ∈ ℝ)
itgiccshift.b	⊢ (𝜑 → 𝐵 ∈ ℝ)
itgiccshift.aleb	⊢ (𝜑 → 𝐴 ≤ 𝐵)
itgiccshift.f	⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ))
itgiccshift.t	⊢ (𝜑 → 𝑇 ∈ ℝ⁺)
itgiccshift.g	⊢ 𝐺 = (𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇)) ↦ (𝐹‘(𝑥 − 𝑇)))

**Assertion**
Ref	Expression
itgiccshift	⊢ (𝜑 → ∫((𝐴 + 𝑇)[,](𝐵 + 𝑇))(𝐺‘𝑥) d𝑥 = ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)

Distinct variable groups: 𝑥,𝐴 𝑥,𝐵 𝑥,𝐹 𝑥,𝐺 𝑥,𝑇 𝜑,𝑥

Proof of Theorem itgiccshift Dummy variables 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		itgiccshift.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ)
2		itgiccshift.b	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ)
3		itgiccshift.t	. . . . . 6 ⊢ (𝜑 → 𝑇 ∈ ℝ⁺)
4	3	rpred 12953	. . . . 5 ⊢ (𝜑 → 𝑇 ∈ ℝ)
5		itgiccshift.aleb	. . . . 5 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
6	1, 2, 4, 5	leadd1dd 11755	. . . 4 ⊢ (𝜑 → (𝐴 + 𝑇) ≤ (𝐵 + 𝑇))
7	6	ditgpos 25817	. . 3 ⊢ (𝜑 → ⨜[(𝐴 + 𝑇) → (𝐵 + 𝑇)](𝐺‘𝑥) d𝑥 = ∫((𝐴 + 𝑇)(,)(𝐵 + 𝑇))(𝐺‘𝑥) d𝑥)
8	1, 4	readdcld 11165	. . . 4 ⊢ (𝜑 → (𝐴 + 𝑇) ∈ ℝ)
9	2, 4	readdcld 11165	. . . 4 ⊢ (𝜑 → (𝐵 + 𝑇) ∈ ℝ)
10		itgiccshift.f	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ))
11		cncff 24846	. . . . . . . . 9 ⊢ (𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ) → 𝐹:(𝐴[,]𝐵)⟶ℂ)
12	10, 11	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐹:(𝐴[,]𝐵)⟶ℂ)
13	12	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝐹:(𝐴[,]𝐵)⟶ℂ)
14	1	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝐴 ∈ ℝ)
15	2	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝐵 ∈ ℝ)
16	8	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝐴 + 𝑇) ∈ ℝ)
17	9	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝐵 + 𝑇) ∈ ℝ)
18		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇)))
19		eliccre 45787	. . . . . . . . . 10 ⊢ (((𝐴 + 𝑇) ∈ ℝ ∧ (𝐵 + 𝑇) ∈ ℝ ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝑥 ∈ ℝ)
20	16, 17, 18, 19	syl3anc 1374	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝑥 ∈ ℝ)
21	4	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝑇 ∈ ℝ)
22	20, 21	resubcld 11569	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝑥 − 𝑇) ∈ ℝ)
23	1	recnd 11164	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ ℂ)
24	4	recnd 11164	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑇 ∈ ℂ)
25	23, 24	pncand 11497	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐴 + 𝑇) − 𝑇) = 𝐴)
26	25	eqcomd 2743	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 = ((𝐴 + 𝑇) − 𝑇))
27	26	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝐴 = ((𝐴 + 𝑇) − 𝑇))
28		elicc2 13331	. . . . . . . . . . . . 13 ⊢ (((𝐴 + 𝑇) ∈ ℝ ∧ (𝐵 + 𝑇) ∈ ℝ) → (𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇)) ↔ (𝑥 ∈ ℝ ∧ (𝐴 + 𝑇) ≤ 𝑥 ∧ 𝑥 ≤ (𝐵 + 𝑇))))
29	16, 17, 28	syl2anc 585	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇)) ↔ (𝑥 ∈ ℝ ∧ (𝐴 + 𝑇) ≤ 𝑥 ∧ 𝑥 ≤ (𝐵 + 𝑇))))
30	18, 29	mpbid 232	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝑥 ∈ ℝ ∧ (𝐴 + 𝑇) ≤ 𝑥 ∧ 𝑥 ≤ (𝐵 + 𝑇)))
31	30	simp2d 1144	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝐴 + 𝑇) ≤ 𝑥)
32	16, 20, 21, 31	lesub1dd 11757	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → ((𝐴 + 𝑇) − 𝑇) ≤ (𝑥 − 𝑇))
33	27, 32	eqbrtrd 5121	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝐴 ≤ (𝑥 − 𝑇))
34	30	simp3d 1145	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝑥 ≤ (𝐵 + 𝑇))
35	20, 17, 21, 34	lesub1dd 11757	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝑥 − 𝑇) ≤ ((𝐵 + 𝑇) − 𝑇))
36	2	recnd 11164	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ ℂ)
37	36, 24	pncand 11497	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐵 + 𝑇) − 𝑇) = 𝐵)
38	37	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → ((𝐵 + 𝑇) − 𝑇) = 𝐵)
39	35, 38	breqtrd 5125	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝑥 − 𝑇) ≤ 𝐵)
40	14, 15, 22, 33, 39	eliccd 45786	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝑥 − 𝑇) ∈ (𝐴[,]𝐵))
41	13, 40	ffvelcdmd 7032	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝐹‘(𝑥 − 𝑇)) ∈ ℂ)
42		itgiccshift.g	. . . . . 6 ⊢ 𝐺 = (𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇)) ↦ (𝐹‘(𝑥 − 𝑇)))
43	41, 42	fmptd 7061	. . . . 5 ⊢ (𝜑 → 𝐺:((𝐴 + 𝑇)[,](𝐵 + 𝑇))⟶ℂ)
44	43	ffvelcdmda 7031	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝐺‘𝑥) ∈ ℂ)
45	8, 9, 44	itgioo 25777	. . 3 ⊢ (𝜑 → ∫((𝐴 + 𝑇)(,)(𝐵 + 𝑇))(𝐺‘𝑥) d𝑥 = ∫((𝐴 + 𝑇)[,](𝐵 + 𝑇))(𝐺‘𝑥) d𝑥)
46	7, 45	eqtr2d 2773	. 2 ⊢ (𝜑 → ∫((𝐴 + 𝑇)[,](𝐵 + 𝑇))(𝐺‘𝑥) d𝑥 = ⨜[(𝐴 + 𝑇) → (𝐵 + 𝑇)](𝐺‘𝑥) d𝑥)
47		eqid 2737	. . . 4 ⊢ (𝑦 ∈ ℂ ↦ (𝑦 + 𝑇)) = (𝑦 ∈ ℂ ↦ (𝑦 + 𝑇))
48	47	addccncf 24870	. . . . 5 ⊢ (𝑇 ∈ ℂ → (𝑦 ∈ ℂ ↦ (𝑦 + 𝑇)) ∈ (ℂ–cn→ℂ))
49	24, 48	syl 17	. . . 4 ⊢ (𝜑 → (𝑦 ∈ ℂ ↦ (𝑦 + 𝑇)) ∈ (ℂ–cn→ℂ))
50	1, 2	iccssred 13354	. . . . 5 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
51		ax-resscn 11087	. . . . 5 ⊢ ℝ ⊆ ℂ
52	50, 51	sstrdi 3947	. . . 4 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℂ)
53	8, 9	iccssred 13354	. . . . 5 ⊢ (𝜑 → ((𝐴 + 𝑇)[,](𝐵 + 𝑇)) ⊆ ℝ)
54	53, 51	sstrdi 3947	. . . 4 ⊢ (𝜑 → ((𝐴 + 𝑇)[,](𝐵 + 𝑇)) ⊆ ℂ)
55	8	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝐴 + 𝑇) ∈ ℝ)
56	9	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝐵 + 𝑇) ∈ ℝ)
57	50	sselda 3934	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → 𝑦 ∈ ℝ)
58	4	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → 𝑇 ∈ ℝ)
59	57, 58	readdcld 11165	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝑦 + 𝑇) ∈ ℝ)
60	1	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → 𝐴 ∈ ℝ)
61		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → 𝑦 ∈ (𝐴[,]𝐵))
62	2	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → 𝐵 ∈ ℝ)
63		elicc2 13331	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝑦 ∈ (𝐴[,]𝐵) ↔ (𝑦 ∈ ℝ ∧ 𝐴 ≤ 𝑦 ∧ 𝑦 ≤ 𝐵)))
64	60, 62, 63	syl2anc 585	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝑦 ∈ (𝐴[,]𝐵) ↔ (𝑦 ∈ ℝ ∧ 𝐴 ≤ 𝑦 ∧ 𝑦 ≤ 𝐵)))
65	61, 64	mpbid 232	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝑦 ∈ ℝ ∧ 𝐴 ≤ 𝑦 ∧ 𝑦 ≤ 𝐵))
66	65	simp2d 1144	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → 𝐴 ≤ 𝑦)
67	60, 57, 58, 66	leadd1dd 11755	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝐴 + 𝑇) ≤ (𝑦 + 𝑇))
68	65	simp3d 1145	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → 𝑦 ≤ 𝐵)
69	57, 62, 58, 68	leadd1dd 11755	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝑦 + 𝑇) ≤ (𝐵 + 𝑇))
70	55, 56, 59, 67, 69	eliccd 45786	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝑦 + 𝑇) ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇)))
71	47, 49, 52, 54, 70	cncfmptssg 46151	. . 3 ⊢ (𝜑 → (𝑦 ∈ (𝐴[,]𝐵) ↦ (𝑦 + 𝑇)) ∈ ((𝐴[,]𝐵)–cn→((𝐴 + 𝑇)[,](𝐵 + 𝑇))))
72		fvoveq1 7383	. . . . . . . 8 ⊢ (𝑥 = 𝑤 → (𝐹‘(𝑥 − 𝑇)) = (𝐹‘(𝑤 − 𝑇)))
73	72	cbvmptv 5203	. . . . . . 7 ⊢ (𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇)) ↦ (𝐹‘(𝑥 − 𝑇))) = (𝑤 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇)) ↦ (𝐹‘(𝑤 − 𝑇)))
74	1, 2, 4	iccshift 45800	. . . . . . . 8 ⊢ (𝜑 → ((𝐴 + 𝑇)[,](𝐵 + 𝑇)) = {𝑥 ∈ ℂ ∣ ∃𝑦 ∈ (𝐴[,]𝐵)𝑥 = (𝑦 + 𝑇)})
75	74	mpteq1d 5189	. . . . . . 7 ⊢ (𝜑 → (𝑤 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇)) ↦ (𝐹‘(𝑤 − 𝑇))) = (𝑤 ∈ {𝑥 ∈ ℂ ∣ ∃𝑦 ∈ (𝐴[,]𝐵)𝑥 = (𝑦 + 𝑇)} ↦ (𝐹‘(𝑤 − 𝑇))))
76	73, 75	eqtrid 2784	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇)) ↦ (𝐹‘(𝑥 − 𝑇))) = (𝑤 ∈ {𝑥 ∈ ℂ ∣ ∃𝑦 ∈ (𝐴[,]𝐵)𝑥 = (𝑦 + 𝑇)} ↦ (𝐹‘(𝑤 − 𝑇))))
77	42, 76	eqtrid 2784	. . . . 5 ⊢ (𝜑 → 𝐺 = (𝑤 ∈ {𝑥 ∈ ℂ ∣ ∃𝑦 ∈ (𝐴[,]𝐵)𝑥 = (𝑦 + 𝑇)} ↦ (𝐹‘(𝑤 − 𝑇))))
78		eqeq1 2741	. . . . . . . . . 10 ⊢ (𝑤 = 𝑥 → (𝑤 = (𝑧 + 𝑇) ↔ 𝑥 = (𝑧 + 𝑇)))
79	78	rexbidv 3161	. . . . . . . . 9 ⊢ (𝑤 = 𝑥 → (∃𝑧 ∈ (𝐴[,]𝐵)𝑤 = (𝑧 + 𝑇) ↔ ∃𝑧 ∈ (𝐴[,]𝐵)𝑥 = (𝑧 + 𝑇)))
80		oveq1 7367	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑦 → (𝑧 + 𝑇) = (𝑦 + 𝑇))
81	80	eqeq2d 2748	. . . . . . . . . 10 ⊢ (𝑧 = 𝑦 → (𝑥 = (𝑧 + 𝑇) ↔ 𝑥 = (𝑦 + 𝑇)))
82	81	cbvrexvw 3216	. . . . . . . . 9 ⊢ (∃𝑧 ∈ (𝐴[,]𝐵)𝑥 = (𝑧 + 𝑇) ↔ ∃𝑦 ∈ (𝐴[,]𝐵)𝑥 = (𝑦 + 𝑇))
83	79, 82	bitrdi 287	. . . . . . . 8 ⊢ (𝑤 = 𝑥 → (∃𝑧 ∈ (𝐴[,]𝐵)𝑤 = (𝑧 + 𝑇) ↔ ∃𝑦 ∈ (𝐴[,]𝐵)𝑥 = (𝑦 + 𝑇)))
84	83	cbvrabv 3410	. . . . . . 7 ⊢ {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ (𝐴[,]𝐵)𝑤 = (𝑧 + 𝑇)} = {𝑥 ∈ ℂ ∣ ∃𝑦 ∈ (𝐴[,]𝐵)𝑥 = (𝑦 + 𝑇)}
85	84	eqcomi 2746	. . . . . 6 ⊢ {𝑥 ∈ ℂ ∣ ∃𝑦 ∈ (𝐴[,]𝐵)𝑥 = (𝑦 + 𝑇)} = {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ (𝐴[,]𝐵)𝑤 = (𝑧 + 𝑇)}
86		eqid 2737	. . . . . 6 ⊢ (𝑤 ∈ {𝑥 ∈ ℂ ∣ ∃𝑦 ∈ (𝐴[,]𝐵)𝑥 = (𝑦 + 𝑇)} ↦ (𝐹‘(𝑤 − 𝑇))) = (𝑤 ∈ {𝑥 ∈ ℂ ∣ ∃𝑦 ∈ (𝐴[,]𝐵)𝑥 = (𝑦 + 𝑇)} ↦ (𝐹‘(𝑤 − 𝑇)))
87	52, 24, 85, 10, 86	cncfshift 46154	. . . . 5 ⊢ (𝜑 → (𝑤 ∈ {𝑥 ∈ ℂ ∣ ∃𝑦 ∈ (𝐴[,]𝐵)𝑥 = (𝑦 + 𝑇)} ↦ (𝐹‘(𝑤 − 𝑇))) ∈ ({𝑥 ∈ ℂ ∣ ∃𝑦 ∈ (𝐴[,]𝐵)𝑥 = (𝑦 + 𝑇)}–cn→ℂ))
88	77, 87	eqeltrd 2837	. . . 4 ⊢ (𝜑 → 𝐺 ∈ ({𝑥 ∈ ℂ ∣ ∃𝑦 ∈ (𝐴[,]𝐵)𝑥 = (𝑦 + 𝑇)}–cn→ℂ))
89	43	feqmptd 6903	. . . 4 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇)) ↦ (𝐺‘𝑥)))
90	74	eqcomd 2743	. . . . 5 ⊢ (𝜑 → {𝑥 ∈ ℂ ∣ ∃𝑦 ∈ (𝐴[,]𝐵)𝑥 = (𝑦 + 𝑇)} = ((𝐴 + 𝑇)[,](𝐵 + 𝑇)))
91	90	oveq1d 7375	. . . 4 ⊢ (𝜑 → ({𝑥 ∈ ℂ ∣ ∃𝑦 ∈ (𝐴[,]𝐵)𝑥 = (𝑦 + 𝑇)}–cn→ℂ) = (((𝐴 + 𝑇)[,](𝐵 + 𝑇))–cn→ℂ))
92	88, 89, 91	3eltr3d 2851	. . 3 ⊢ (𝜑 → (𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇)) ↦ (𝐺‘𝑥)) ∈ (((𝐴 + 𝑇)[,](𝐵 + 𝑇))–cn→ℂ))
93		ioosscn 13328	. . . . . 6 ⊢ (𝐴(,)𝐵) ⊆ ℂ
94	93	a1i 11	. . . . 5 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℂ)
95		1cnd 11131	. . . . 5 ⊢ (𝜑 → 1 ∈ ℂ)
96		ssid 3957	. . . . . 6 ⊢ ℂ ⊆ ℂ
97	96	a1i 11	. . . . 5 ⊢ (𝜑 → ℂ ⊆ ℂ)
98	94, 95, 97	constcncfg 46152	. . . 4 ⊢ (𝜑 → (𝑦 ∈ (𝐴(,)𝐵) ↦ 1) ∈ ((𝐴(,)𝐵)–cn→ℂ))
99		fconstmpt 5687	. . . . 5 ⊢ ((𝐴(,)𝐵) × {1}) = (𝑦 ∈ (𝐴(,)𝐵) ↦ 1)
100		ioombl 25526	. . . . . . 7 ⊢ (𝐴(,)𝐵) ∈ dom vol
101	100	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝐴(,)𝐵) ∈ dom vol)
102		ioovolcl 25531	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (vol‘(𝐴(,)𝐵)) ∈ ℝ)
103	1, 2, 102	syl2anc 585	. . . . . 6 ⊢ (𝜑 → (vol‘(𝐴(,)𝐵)) ∈ ℝ)
104		iblconst 25779	. . . . . 6 ⊢ (((𝐴(,)𝐵) ∈ dom vol ∧ (vol‘(𝐴(,)𝐵)) ∈ ℝ ∧ 1 ∈ ℂ) → ((𝐴(,)𝐵) × {1}) ∈ 𝐿¹)
105	101, 103, 95, 104	syl3anc 1374	. . . . 5 ⊢ (𝜑 → ((𝐴(,)𝐵) × {1}) ∈ 𝐿¹)
106	99, 105	eqeltrrid 2842	. . . 4 ⊢ (𝜑 → (𝑦 ∈ (𝐴(,)𝐵) ↦ 1) ∈ 𝐿¹)
107	98, 106	elind 4153	. . 3 ⊢ (𝜑 → (𝑦 ∈ (𝐴(,)𝐵) ↦ 1) ∈ (((𝐴(,)𝐵)–cn→ℂ) ∩ 𝐿¹))
108	50	resmptd 6000	. . . . . . 7 ⊢ (𝜑 → ((𝑦 ∈ ℝ ↦ (𝑦 + 𝑇)) ↾ (𝐴[,]𝐵)) = (𝑦 ∈ (𝐴[,]𝐵) ↦ (𝑦 + 𝑇)))
109	108	eqcomd 2743	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ (𝐴[,]𝐵) ↦ (𝑦 + 𝑇)) = ((𝑦 ∈ ℝ ↦ (𝑦 + 𝑇)) ↾ (𝐴[,]𝐵)))
110	109	oveq2d 7376	. . . . 5 ⊢ (𝜑 → (ℝ D (𝑦 ∈ (𝐴[,]𝐵) ↦ (𝑦 + 𝑇))) = (ℝ D ((𝑦 ∈ ℝ ↦ (𝑦 + 𝑇)) ↾ (𝐴[,]𝐵))))
111	51	a1i 11	. . . . . 6 ⊢ (𝜑 → ℝ ⊆ ℂ)
112	111	sselda 3934	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℂ)
113	24	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝑇 ∈ ℂ)
114	112, 113	addcld 11155	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝑦 + 𝑇) ∈ ℂ)
115	114	fmpttd 7062	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ ℝ ↦ (𝑦 + 𝑇)):ℝ⟶ℂ)
116		ssid 3957	. . . . . . 7 ⊢ ℝ ⊆ ℝ
117	116	a1i 11	. . . . . 6 ⊢ (𝜑 → ℝ ⊆ ℝ)
118		eqid 2737	. . . . . . 7 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
119		tgioo4 24753	. . . . . . 7 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
120	118, 119	dvres 25872	. . . . . 6 ⊢ (((ℝ ⊆ ℂ ∧ (𝑦 ∈ ℝ ↦ (𝑦 + 𝑇)):ℝ⟶ℂ) ∧ (ℝ ⊆ ℝ ∧ (𝐴[,]𝐵) ⊆ ℝ)) → (ℝ D ((𝑦 ∈ ℝ ↦ (𝑦 + 𝑇)) ↾ (𝐴[,]𝐵))) = ((ℝ D (𝑦 ∈ ℝ ↦ (𝑦 + 𝑇))) ↾ ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵))))
121	111, 115, 117, 50, 120	syl22anc 839	. . . . 5 ⊢ (𝜑 → (ℝ D ((𝑦 ∈ ℝ ↦ (𝑦 + 𝑇)) ↾ (𝐴[,]𝐵))) = ((ℝ D (𝑦 ∈ ℝ ↦ (𝑦 + 𝑇))) ↾ ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵))))
122	110, 121	eqtrd 2772	. . . 4 ⊢ (𝜑 → (ℝ D (𝑦 ∈ (𝐴[,]𝐵) ↦ (𝑦 + 𝑇))) = ((ℝ D (𝑦 ∈ ℝ ↦ (𝑦 + 𝑇))) ↾ ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵))))
123		iccntr 24770	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
124	1, 2, 123	syl2anc 585	. . . . 5 ⊢ (𝜑 → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
125	124	reseq2d 5939	. . . 4 ⊢ (𝜑 → ((ℝ D (𝑦 ∈ ℝ ↦ (𝑦 + 𝑇))) ↾ ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵))) = ((ℝ D (𝑦 ∈ ℝ ↦ (𝑦 + 𝑇))) ↾ (𝐴(,)𝐵)))
126		reelprrecn 11122	. . . . . . . 8 ⊢ ℝ ∈ {ℝ, ℂ}
127	126	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
128		1cnd 11131	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 1 ∈ ℂ)
129	127	dvmptid 25921	. . . . . . 7 ⊢ (𝜑 → (ℝ D (𝑦 ∈ ℝ ↦ 𝑦)) = (𝑦 ∈ ℝ ↦ 1))
130		0cnd 11129	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 0 ∈ ℂ)
131	127, 24	dvmptc 25922	. . . . . . 7 ⊢ (𝜑 → (ℝ D (𝑦 ∈ ℝ ↦ 𝑇)) = (𝑦 ∈ ℝ ↦ 0))
132	127, 112, 128, 129, 113, 130, 131	dvmptadd 25924	. . . . . 6 ⊢ (𝜑 → (ℝ D (𝑦 ∈ ℝ ↦ (𝑦 + 𝑇))) = (𝑦 ∈ ℝ ↦ (1 + 0)))
133	132	reseq1d 5938	. . . . 5 ⊢ (𝜑 → ((ℝ D (𝑦 ∈ ℝ ↦ (𝑦 + 𝑇))) ↾ (𝐴(,)𝐵)) = ((𝑦 ∈ ℝ ↦ (1 + 0)) ↾ (𝐴(,)𝐵)))
134		ioossre 13327	. . . . . . 7 ⊢ (𝐴(,)𝐵) ⊆ ℝ
135	134	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℝ)
136	135	resmptd 6000	. . . . 5 ⊢ (𝜑 → ((𝑦 ∈ ℝ ↦ (1 + 0)) ↾ (𝐴(,)𝐵)) = (𝑦 ∈ (𝐴(,)𝐵) ↦ (1 + 0)))
137		1p0e1 12268	. . . . . . 7 ⊢ (1 + 0) = 1
138	137	mpteq2i 5195	. . . . . 6 ⊢ (𝑦 ∈ (𝐴(,)𝐵) ↦ (1 + 0)) = (𝑦 ∈ (𝐴(,)𝐵) ↦ 1)
139	138	a1i 11	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ (𝐴(,)𝐵) ↦ (1 + 0)) = (𝑦 ∈ (𝐴(,)𝐵) ↦ 1))
140	133, 136, 139	3eqtrd 2776	. . . 4 ⊢ (𝜑 → ((ℝ D (𝑦 ∈ ℝ ↦ (𝑦 + 𝑇))) ↾ (𝐴(,)𝐵)) = (𝑦 ∈ (𝐴(,)𝐵) ↦ 1))
141	122, 125, 140	3eqtrd 2776	. . 3 ⊢ (𝜑 → (ℝ D (𝑦 ∈ (𝐴[,]𝐵) ↦ (𝑦 + 𝑇))) = (𝑦 ∈ (𝐴(,)𝐵) ↦ 1))
142		fveq2 6835	. . 3 ⊢ (𝑥 = (𝑦 + 𝑇) → (𝐺‘𝑥) = (𝐺‘(𝑦 + 𝑇)))
143		oveq1 7367	. . 3 ⊢ (𝑦 = 𝐴 → (𝑦 + 𝑇) = (𝐴 + 𝑇))
144		oveq1 7367	. . 3 ⊢ (𝑦 = 𝐵 → (𝑦 + 𝑇) = (𝐵 + 𝑇))
145	1, 2, 5, 71, 92, 107, 141, 142, 143, 144, 8, 9	itgsubsticc 46256	. 2 ⊢ (𝜑 → ⨜[(𝐴 + 𝑇) → (𝐵 + 𝑇)](𝐺‘𝑥) d𝑥 = ⨜[𝐴 → 𝐵]((𝐺‘(𝑦 + 𝑇)) · 1) d𝑦)
146	5	ditgpos 25817	. . 3 ⊢ (𝜑 → ⨜[𝐴 → 𝐵]((𝐺‘(𝑦 + 𝑇)) · 1) d𝑦 = ∫(𝐴(,)𝐵)((𝐺‘(𝑦 + 𝑇)) · 1) d𝑦)
147	43	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → 𝐺:((𝐴 + 𝑇)[,](𝐵 + 𝑇))⟶ℂ)
148	147, 70	ffvelcdmd 7032	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝐺‘(𝑦 + 𝑇)) ∈ ℂ)
149		1cnd 11131	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → 1 ∈ ℂ)
150	148, 149	mulcld 11156	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → ((𝐺‘(𝑦 + 𝑇)) · 1) ∈ ℂ)
151	1, 2, 150	itgioo 25777	. . 3 ⊢ (𝜑 → ∫(𝐴(,)𝐵)((𝐺‘(𝑦 + 𝑇)) · 1) d𝑦 = ∫(𝐴[,]𝐵)((𝐺‘(𝑦 + 𝑇)) · 1) d𝑦)
152		fvoveq1 7383	. . . . . 6 ⊢ (𝑦 = 𝑥 → (𝐺‘(𝑦 + 𝑇)) = (𝐺‘(𝑥 + 𝑇)))
153	152	oveq1d 7375	. . . . 5 ⊢ (𝑦 = 𝑥 → ((𝐺‘(𝑦 + 𝑇)) · 1) = ((𝐺‘(𝑥 + 𝑇)) · 1))
154	153	cbvitgv 25738	. . . 4 ⊢ ∫(𝐴[,]𝐵)((𝐺‘(𝑦 + 𝑇)) · 1) d𝑦 = ∫(𝐴[,]𝐵)((𝐺‘(𝑥 + 𝑇)) · 1) d𝑥
155	43	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐺:((𝐴 + 𝑇)[,](𝐵 + 𝑇))⟶ℂ)
156	8	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐴 + 𝑇) ∈ ℝ)
157	9	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐵 + 𝑇) ∈ ℝ)
158	50	sselda 3934	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ∈ ℝ)
159	4	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑇 ∈ ℝ)
160	158, 159	readdcld 11165	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝑥 + 𝑇) ∈ ℝ)
161	1	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐴 ∈ ℝ)
162	1	rexrd 11186	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
163	162	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐴 ∈ ℝ^*)
164	2	rexrd 11186	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
165	164	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐵 ∈ ℝ^*)
166		simpr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ∈ (𝐴[,]𝐵))
167		iccgelb 13322	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐴 ≤ 𝑥)
168	163, 165, 166, 167	syl3anc 1374	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐴 ≤ 𝑥)
169	161, 158, 159, 168	leadd1dd 11755	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐴 + 𝑇) ≤ (𝑥 + 𝑇))
170	2	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐵 ∈ ℝ)
171		iccleub 13321	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ≤ 𝐵)
172	163, 165, 166, 171	syl3anc 1374	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ≤ 𝐵)
173	158, 170, 159, 172	leadd1dd 11755	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝑥 + 𝑇) ≤ (𝐵 + 𝑇))
174	156, 157, 160, 169, 173	eliccd 45786	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝑥 + 𝑇) ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇)))
175	155, 174	ffvelcdmd 7032	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐺‘(𝑥 + 𝑇)) ∈ ℂ)
176	175	mulridd 11153	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → ((𝐺‘(𝑥 + 𝑇)) · 1) = (𝐺‘(𝑥 + 𝑇)))
177	42, 73	eqtri 2760	. . . . . . . 8 ⊢ 𝐺 = (𝑤 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇)) ↦ (𝐹‘(𝑤 − 𝑇)))
178	177	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐺 = (𝑤 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇)) ↦ (𝐹‘(𝑤 − 𝑇))))
179		fvoveq1 7383	. . . . . . . 8 ⊢ (𝑤 = (𝑥 + 𝑇) → (𝐹‘(𝑤 − 𝑇)) = (𝐹‘((𝑥 + 𝑇) − 𝑇)))
180	158	recnd 11164	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ∈ ℂ)
181	24	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑇 ∈ ℂ)
182	180, 181	pncand 11497	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → ((𝑥 + 𝑇) − 𝑇) = 𝑥)
183	182	fveq2d 6839	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘((𝑥 + 𝑇) − 𝑇)) = (𝐹‘𝑥))
184	179, 183	sylan9eqr 2794	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ 𝑤 = (𝑥 + 𝑇)) → (𝐹‘(𝑤 − 𝑇)) = (𝐹‘𝑥))
185	12	ffvelcdmda 7031	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℂ)
186	178, 184, 174, 185	fvmptd 6950	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐺‘(𝑥 + 𝑇)) = (𝐹‘𝑥))
187	176, 186	eqtrd 2772	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → ((𝐺‘(𝑥 + 𝑇)) · 1) = (𝐹‘𝑥))
188	187	itgeq2dv 25743	. . . 4 ⊢ (𝜑 → ∫(𝐴[,]𝐵)((𝐺‘(𝑥 + 𝑇)) · 1) d𝑥 = ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)
189	154, 188	eqtrid 2784	. . 3 ⊢ (𝜑 → ∫(𝐴[,]𝐵)((𝐺‘(𝑦 + 𝑇)) · 1) d𝑦 = ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)
190	146, 151, 189	3eqtrd 2776	. 2 ⊢ (𝜑 → ⨜[𝐴 → 𝐵]((𝐺‘(𝑦 + 𝑇)) · 1) d𝑦 = ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)
191	46, 145, 190	3eqtrd 2776	1 ⊢ (𝜑 → ∫((𝐴 + 𝑇)[,](𝐵 + 𝑇))(𝐺‘𝑥) d𝑥 = ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ∃wrex 3061 {crab 3400 ⊆ wss 3902 {csn 4581 {cpr 4583 class class class wbr 5099 ↦ cmpt 5180 × cxp 5623 dom cdm 5625 ran crn 5626 ↾ cres 5627 ⟶wf 6489 ‘cfv 6493 (class class class)co 7360 ℂcc 11028 ℝcr 11029 0cc0 11030 1c1 11031 + caddc 11033 · cmul 11035 ℝ^*cxr 11169 ≤ cle 11171 − cmin 11368 ℝ⁺crp 12909 (,)cioo 13265 [,]cicc 13268 TopOpenctopn 17345 topGenctg 17361 ℂ_fldccnfld 21313 intcnt 22965 –cn→ccncf 24829 volcvol 25424 𝐿¹cibl 25578 ∫citg 25579 ⨜cdit 25807 D cdv 25824

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5225 ax-sep 5242 ax-nul 5252 ax-pow 5311 ax-pr 5378 ax-un 7682 ax-inf2 9554 ax-cc 10349 ax-cnex 11086 ax-resscn 11087 ax-1cn 11088 ax-icn 11089 ax-addcl 11090 ax-addrcl 11091 ax-mulcl 11092 ax-mulrcl 11093 ax-mulcom 11094 ax-addass 11095 ax-mulass 11096 ax-distr 11097 ax-i2m1 11098 ax-1ne0 11099 ax-1rid 11100 ax-rnegex 11101 ax-rrecex 11102 ax-cnre 11103 ax-pre-lttri 11104 ax-pre-lttrn 11105 ax-pre-ltadd 11106 ax-pre-mulgt0 11107 ax-pre-sup 11108 ax-addf 11109

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3351 df-reu 3352 df-rab 3401 df-v 3443 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-symdif 4206 df-nul 4287 df-if 4481 df-pw 4557 df-sn 4582 df-pr 4584 df-tp 4586 df-op 4588 df-uni 4865 df-int 4904 df-iun 4949 df-iin 4950 df-disj 5067 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-se 5579 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-of 7624 df-ofr 7625 df-om 7811 df-1st 7935 df-2nd 7936 df-supp 8105 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-2o 8400 df-oadd 8403 df-omul 8404 df-er 8637 df-map 8769 df-pm 8770 df-ixp 8840 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-fsupp 9269 df-fi 9318 df-sup 9349 df-inf 9350 df-oi 9419 df-dju 9817 df-card 9855 df-acn 9858 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12150 df-2 12212 df-3 12213 df-4 12214 df-5 12215 df-6 12216 df-7 12217 df-8 12218 df-9 12219 df-n0 12406 df-z 12493 df-dec 12612 df-uz 12756 df-q 12866 df-rp 12910 df-xneg 13030 df-xadd 13031 df-xmul 13032 df-ioo 13269 df-ioc 13270 df-ico 13271 df-icc 13272 df-fz 13428 df-fzo 13575 df-fl 13716 df-mod 13794 df-seq 13929 df-exp 13989 df-hash 14258 df-cj 15026 df-re 15027 df-im 15028 df-sqrt 15162 df-abs 15163 df-limsup 15398 df-clim 15415 df-rlim 15416 df-sum 15614 df-struct 17078 df-sets 17095 df-slot 17113 df-ndx 17125 df-base 17141 df-ress 17162 df-plusg 17194 df-mulr 17195 df-starv 17196 df-sca 17197 df-vsca 17198 df-ip 17199 df-tset 17200 df-ple 17201 df-ds 17203 df-unif 17204 df-hom 17205 df-cco 17206 df-rest 17346 df-topn 17347 df-0g 17365 df-gsum 17366 df-topgen 17367 df-pt 17368 df-prds 17371 df-xrs 17427 df-qtop 17432 df-imas 17433 df-xps 17435 df-mre 17509 df-mrc 17510 df-acs 17512 df-mgm 18569 df-sgrp 18648 df-mnd 18664 df-submnd 18713 df-mulg 19002 df-cntz 19250 df-cmn 19715 df-psmet 21305 df-xmet 21306 df-met 21307 df-bl 21308 df-mopn 21309 df-fbas 21310 df-fg 21311 df-cnfld 21314 df-top 22842 df-topon 22859 df-topsp 22881 df-bases 22894 df-cld 22967 df-ntr 22968 df-cls 22969 df-nei 23046 df-lp 23084 df-perf 23085 df-cn 23175 df-cnp 23176 df-haus 23263 df-cmp 23335 df-tx 23510 df-hmeo 23703 df-fil 23794 df-fm 23886 df-flim 23887 df-flf 23888 df-xms 24268 df-ms 24269 df-tms 24270 df-cncf 24831 df-ovol 25425 df-vol 25426 df-mbf 25580 df-itg1 25581 df-itg2 25582 df-ibl 25583 df-itg 25584 df-0p 25631 df-ditg 25808 df-limc 25827 df-dv 25828

This theorem is referenced by: fourierdlem81 46467

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