itgiccshift - Mathbox for Glauco Siliprandi

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Theorem itgiccshift 46795

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 13088	. . . . 5 ⊢ (𝜑 → 𝑇 ∈ ℝ)
5		itgiccshift.aleb	. . . . 5 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
6	1, 2, 4, 5	leadd1dd 11855	. . . 4 ⊢ (𝜑 → (𝐴 + 𝑇) ≤ (𝐵 + 𝑇))
7	6	ditgpos 26085	. . 3 ⊢ (𝜑 → ⨜[(𝐴 + 𝑇) → (𝐵 + 𝑇)](𝐺‘𝑥) d𝑥 = ∫((𝐴 + 𝑇)(,)(𝐵 + 𝑇))(𝐺‘𝑥) d𝑥)
8	1, 4	readdcld 11265	. . . 4 ⊢ (𝜑 → (𝐴 + 𝑇) ∈ ℝ)
9	2, 4	readdcld 11265	. . . 4 ⊢ (𝜑 → (𝐵 + 𝑇) ∈ ℝ)
10		itgiccshift.f	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ))
11		cncff 25122	. . . . . . . . 9 ⊢ (𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ) → 𝐹:(𝐴[,]𝐵)⟶ℂ)
12	10, 11	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝐹:(𝐴[,]𝐵)⟶ℂ)
13	12	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝐹:(𝐴[,]𝐵)⟶ℂ)
14	1	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝐴 ∈ ℝ)
15	2	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝐵 ∈ ℝ)
16	8	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝐴 + 𝑇) ∈ ℝ)
17	9	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝐵 + 𝑇) ∈ ℝ)
18		simpr 490	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇)))
19		eliccre 46322	. . . . . . . . . 10 ⊢ (((𝐴 + 𝑇) ∈ ℝ ∧ (𝐵 + 𝑇) ∈ ℝ ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝑥 ∈ ℝ)
20	16, 17, 18, 19	syl3anc 1398	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝑥 ∈ ℝ)
21	4	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝑇 ∈ ℝ)
22	20, 21	resubcld 11669	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝑥 − 𝑇) ∈ ℝ)
23	1	recnd 11264	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ ℂ)
24	4	recnd 11264	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑇 ∈ ℂ)
25	23, 24	pncand 11597	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐴 + 𝑇) − 𝑇) = 𝐴)
26	25	eqcomd 2768	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 = ((𝐴 + 𝑇) − 𝑇))
27	26	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝐴 = ((𝐴 + 𝑇) − 𝑇))
28		elicc2 13466	. . . . . . . . . . . . 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 596	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇)) ↔ (𝑥 ∈ ℝ ∧ (𝐴 + 𝑇) ≤ 𝑥 ∧ 𝑥 ≤ (𝐵 + 𝑇))))
30	18, 29	mpbid 235	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝑥 ∈ ℝ ∧ (𝐴 + 𝑇) ≤ 𝑥 ∧ 𝑥 ≤ (𝐵 + 𝑇)))
31	30	simp2d 1161	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝐴 + 𝑇) ≤ 𝑥)
32	16, 20, 21, 31	lesub1dd 11857	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → ((𝐴 + 𝑇) − 𝑇) ≤ (𝑥 − 𝑇))
33	27, 32	eqbrtrd 5131	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝐴 ≤ (𝑥 − 𝑇))
34	30	simp3d 1162	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝑥 ≤ (𝐵 + 𝑇))
35	20, 17, 21, 34	lesub1dd 11857	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝑥 − 𝑇) ≤ ((𝐵 + 𝑇) − 𝑇))
36	2	recnd 11264	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ ℂ)
37	36, 24	pncand 11597	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐵 + 𝑇) − 𝑇) = 𝐵)
38	37	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → ((𝐵 + 𝑇) − 𝑇) = 𝐵)
39	35, 38	breqtrd 5135	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝑥 − 𝑇) ≤ 𝐵)
40	14, 15, 22, 33, 39	eliccd 46321	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝑥 − 𝑇) ∈ (𝐴[,]𝐵))
41	13, 40	ffvelcdmd 7081	. . . . . 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 7110	. . . . 5 ⊢ (𝜑 → 𝐺:((𝐴 + 𝑇)[,](𝐵 + 𝑇))⟶ℂ)
44	43	ffvelcdmda 7080	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝐺‘𝑥) ∈ ℂ)
45	8, 9, 44	itgioo 26045	. . 3 ⊢ (𝜑 → ∫((𝐴 + 𝑇)(,)(𝐵 + 𝑇))(𝐺‘𝑥) d𝑥 = ∫((𝐴 + 𝑇)[,](𝐵 + 𝑇))(𝐺‘𝑥) d𝑥)
46	7, 45	eqtr2d 2798	. 2 ⊢ (𝜑 → ∫((𝐴 + 𝑇)[,](𝐵 + 𝑇))(𝐺‘𝑥) d𝑥 = ⨜[(𝐴 + 𝑇) → (𝐵 + 𝑇)](𝐺‘𝑥) d𝑥)
47		eqid 2762	. . . 4 ⊢ (𝑦 ∈ ℂ ↦ (𝑦 + 𝑇)) = (𝑦 ∈ ℂ ↦ (𝑦 + 𝑇))
48	47	addccncf 25146	. . . . 5 ⊢ (𝑇 ∈ ℂ → (𝑦 ∈ ℂ ↦ (𝑦 + 𝑇)) ∈ (ℂ–cn→ℂ))
49	24, 48	syl 18	. . . 4 ⊢ (𝜑 → (𝑦 ∈ ℂ ↦ (𝑦 + 𝑇)) ∈ (ℂ–cn→ℂ))
50	1, 2	iccssred 13489	. . . . 5 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
51		ax-resscn 11184	. . . . 5 ⊢ ℝ ⊆ ℂ
52	50, 51	sstrdi 3946	. . . 4 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℂ)
53	8, 9	iccssred 13489	. . . . 5 ⊢ (𝜑 → ((𝐴 + 𝑇)[,](𝐵 + 𝑇)) ⊆ ℝ)
54	53, 51	sstrdi 3946	. . . 4 ⊢ (𝜑 → ((𝐴 + 𝑇)[,](𝐵 + 𝑇)) ⊆ ℂ)
55	8	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝐴 + 𝑇) ∈ ℝ)
56	9	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝐵 + 𝑇) ∈ ℝ)
57	50	sselda 3934	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → 𝑦 ∈ ℝ)
58	4	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → 𝑇 ∈ ℝ)
59	57, 58	readdcld 11265	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝑦 + 𝑇) ∈ ℝ)
60	1	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → 𝐴 ∈ ℝ)
61		simpr 490	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → 𝑦 ∈ (𝐴[,]𝐵))
62	2	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → 𝐵 ∈ ℝ)
63		elicc2 13466	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝑦 ∈ (𝐴[,]𝐵) ↔ (𝑦 ∈ ℝ ∧ 𝐴 ≤ 𝑦 ∧ 𝑦 ≤ 𝐵)))
64	60, 62, 63	syl2anc 596	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝑦 ∈ (𝐴[,]𝐵) ↔ (𝑦 ∈ ℝ ∧ 𝐴 ≤ 𝑦 ∧ 𝑦 ≤ 𝐵)))
65	61, 64	mpbid 235	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝑦 ∈ ℝ ∧ 𝐴 ≤ 𝑦 ∧ 𝑦 ≤ 𝐵))
66	65	simp2d 1161	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → 𝐴 ≤ 𝑦)
67	60, 57, 58, 66	leadd1dd 11855	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝐴 + 𝑇) ≤ (𝑦 + 𝑇))
68	65	simp3d 1162	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → 𝑦 ≤ 𝐵)
69	57, 62, 58, 68	leadd1dd 11855	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝑦 + 𝑇) ≤ (𝐵 + 𝑇))
70	55, 56, 59, 67, 69	eliccd 46321	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝑦 + 𝑇) ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇)))
71	47, 49, 52, 54, 70	cncfmptssg 46686	. . 3 ⊢ (𝜑 → (𝑦 ∈ (𝐴[,]𝐵) ↦ (𝑦 + 𝑇)) ∈ ((𝐴[,]𝐵)–cn→((𝐴 + 𝑇)[,](𝐵 + 𝑇))))
72		fvoveq1 7439	. . . . . . . 8 ⊢ (𝑥 = 𝑤 → (𝐹‘(𝑥 − 𝑇)) = (𝐹‘(𝑤 − 𝑇)))
73	72	cbvmptv 5213	. . . . . . 7 ⊢ (𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇)) ↦ (𝐹‘(𝑥 − 𝑇))) = (𝑤 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇)) ↦ (𝐹‘(𝑤 − 𝑇)))
74	1, 2, 4	iccshift 46335	. . . . . . . 8 ⊢ (𝜑 → ((𝐴 + 𝑇)[,](𝐵 + 𝑇)) = {𝑥 ∈ ℂ ∣ ∃𝑦 ∈ (𝐴[,]𝐵)𝑥 = (𝑦 + 𝑇)})
75	74	mpteq1d 5199	. . . . . . 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 2809	. . . . . 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 2809	. . . . 5 ⊢ (𝜑 → 𝐺 = (𝑤 ∈ {𝑥 ∈ ℂ ∣ ∃𝑦 ∈ (𝐴[,]𝐵)𝑥 = (𝑦 + 𝑇)} ↦ (𝐹‘(𝑤 − 𝑇))))
78		eqeq1 2766	. . . . . . . . . 10 ⊢ (𝑤 = 𝑥 → (𝑤 = (𝑧 + 𝑇) ↔ 𝑥 = (𝑧 + 𝑇)))
79	78	rexbidv 3188	. . . . . . . . 9 ⊢ (𝑤 = 𝑥 → (∃𝑧 ∈ (𝐴[,]𝐵)𝑤 = (𝑧 + 𝑇) ↔ ∃𝑧 ∈ (𝐴[,]𝐵)𝑥 = (𝑧 + 𝑇)))
80		oveq1 7423	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑦 → (𝑧 + 𝑇) = (𝑦 + 𝑇))
81	80	eqeq2d 2773	. . . . . . . . . 10 ⊢ (𝑧 = 𝑦 → (𝑥 = (𝑧 + 𝑇) ↔ 𝑥 = (𝑦 + 𝑇)))
82	81	cbvrexvw 3243	. . . . . . . . 9 ⊢ (∃𝑧 ∈ (𝐴[,]𝐵)𝑥 = (𝑧 + 𝑇) ↔ ∃𝑦 ∈ (𝐴[,]𝐵)𝑥 = (𝑦 + 𝑇))
83	79, 82	bitrdi 290	. . . . . . . 8 ⊢ (𝑤 = 𝑥 → (∃𝑧 ∈ (𝐴[,]𝐵)𝑤 = (𝑧 + 𝑇) ↔ ∃𝑦 ∈ (𝐴[,]𝐵)𝑥 = (𝑦 + 𝑇)))
84	83	cbvrabv 3424	. . . . . . 7 ⊢ {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ (𝐴[,]𝐵)𝑤 = (𝑧 + 𝑇)} = {𝑥 ∈ ℂ ∣ ∃𝑦 ∈ (𝐴[,]𝐵)𝑥 = (𝑦 + 𝑇)}
85	84	eqcomi 2771	. . . . . 6 ⊢ {𝑥 ∈ ℂ ∣ ∃𝑦 ∈ (𝐴[,]𝐵)𝑥 = (𝑦 + 𝑇)} = {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ (𝐴[,]𝐵)𝑤 = (𝑧 + 𝑇)}
86		eqid 2762	. . . . . 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 46689	. . . . 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 2862	. . . 4 ⊢ (𝜑 → 𝐺 ∈ ({𝑥 ∈ ℂ ∣ ∃𝑦 ∈ (𝐴[,]𝐵)𝑥 = (𝑦 + 𝑇)}–cn→ℂ))
89	43	feqmptd 6950	. . . 4 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇)) ↦ (𝐺‘𝑥)))
90	74	eqcomd 2768	. . . . 5 ⊢ (𝜑 → {𝑥 ∈ ℂ ∣ ∃𝑦 ∈ (𝐴[,]𝐵)𝑥 = (𝑦 + 𝑇)} = ((𝐴 + 𝑇)[,](𝐵 + 𝑇)))
91	90	oveq1d 7431	. . . 4 ⊢ (𝜑 → ({𝑥 ∈ ℂ ∣ ∃𝑦 ∈ (𝐴[,]𝐵)𝑥 = (𝑦 + 𝑇)}–cn→ℂ) = (((𝐴 + 𝑇)[,](𝐵 + 𝑇))–cn→ℂ))
92	88, 89, 91	3eltr3d 2876	. . 3 ⊢ (𝜑 → (𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇)) ↦ (𝐺‘𝑥)) ∈ (((𝐴 + 𝑇)[,](𝐵 + 𝑇))–cn→ℂ))
93		ioosscn 13463	. . . . . 6 ⊢ (𝐴(,)𝐵) ⊆ ℂ
94	93	a1i 11	. . . . 5 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℂ)
95		1cnd 11229	. . . . 5 ⊢ (𝜑 → 1 ∈ ℂ)
96		ssid 3956	. . . . . 6 ⊢ ℂ ⊆ ℂ
97	96	a1i 11	. . . . 5 ⊢ (𝜑 → ℂ ⊆ ℂ)
98	94, 95, 97	constcncfg 46687	. . . 4 ⊢ (𝜑 → (𝑦 ∈ (𝐴(,)𝐵) ↦ 1) ∈ ((𝐴(,)𝐵)–cn→ℂ))
99		fconstmpt 5721	. . . . 5 ⊢ ((𝐴(,)𝐵) × {1}) = (𝑦 ∈ (𝐴(,)𝐵) ↦ 1)
100		ioombl 25794	. . . . . . 7 ⊢ (𝐴(,)𝐵) ∈ dom vol
101	100	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝐴(,)𝐵) ∈ dom vol)
102		ioovolcl 25799	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (vol‘(𝐴(,)𝐵)) ∈ ℝ)
103	1, 2, 102	syl2anc 596	. . . . . 6 ⊢ (𝜑 → (vol‘(𝐴(,)𝐵)) ∈ ℝ)
104		iblconst 26047	. . . . . 6 ⊢ (((𝐴(,)𝐵) ∈ dom vol ∧ (vol‘(𝐴(,)𝐵)) ∈ ℝ ∧ 1 ∈ ℂ) → ((𝐴(,)𝐵) × {1}) ∈ 𝐿¹)
105	101, 103, 95, 104	syl3anc 1398	. . . . 5 ⊢ (𝜑 → ((𝐴(,)𝐵) × {1}) ∈ 𝐿¹)
106	99, 105	eqeltrrid 2867	. . . 4 ⊢ (𝜑 → (𝑦 ∈ (𝐴(,)𝐵) ↦ 1) ∈ 𝐿¹)
107	98, 106	elind 4149	. . 3 ⊢ (𝜑 → (𝑦 ∈ (𝐴(,)𝐵) ↦ 1) ∈ (((𝐴(,)𝐵)–cn→ℂ) ∩ 𝐿¹))
108	50	resmptd 6040	. . . . . . 7 ⊢ (𝜑 → ((𝑦 ∈ ℝ ↦ (𝑦 + 𝑇)) ↾ (𝐴[,]𝐵)) = (𝑦 ∈ (𝐴[,]𝐵) ↦ (𝑦 + 𝑇)))
109	108	eqcomd 2768	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ (𝐴[,]𝐵) ↦ (𝑦 + 𝑇)) = ((𝑦 ∈ ℝ ↦ (𝑦 + 𝑇)) ↾ (𝐴[,]𝐵)))
110	109	oveq2d 7432	. . . . 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 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝑇 ∈ ℂ)
114	112, 113	addcld 11255	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝑦 + 𝑇) ∈ ℂ)
115	114	fmpttd 7111	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ ℝ ↦ (𝑦 + 𝑇)):ℝ⟶ℂ)
116		ssid 3956	. . . . . . 7 ⊢ ℝ ⊆ ℝ
117	116	a1i 11	. . . . . 6 ⊢ (𝜑 → ℝ ⊆ ℝ)
118		eqid 2762	. . . . . . 7 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
119		tgioo4 25032	. . . . . . 7 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
120	118, 119	dvres 26140	. . . . . 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 852	. . . . 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 2797	. . . 4 ⊢ (𝜑 → (ℝ D (𝑦 ∈ (𝐴[,]𝐵) ↦ (𝑦 + 𝑇))) = ((ℝ D (𝑦 ∈ ℝ ↦ (𝑦 + 𝑇))) ↾ ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵))))
123		iccntr 25049	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
124	1, 2, 123	syl2anc 596	. . . . 5 ⊢ (𝜑 → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
125	124	reseq2d 5976	. . . 4 ⊢ (𝜑 → ((ℝ D (𝑦 ∈ ℝ ↦ (𝑦 + 𝑇))) ↾ ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵))) = ((ℝ D (𝑦 ∈ ℝ ↦ (𝑦 + 𝑇))) ↾ (𝐴(,)𝐵)))
126		reelprrecn 11219	. . . . . . . 8 ⊢ ℝ ∈ {ℝ, ℂ}
127	126	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
128		1cnd 11229	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 1 ∈ ℂ)
129	127	dvmptid 26186	. . . . . . 7 ⊢ (𝜑 → (ℝ D (𝑦 ∈ ℝ ↦ 𝑦)) = (𝑦 ∈ ℝ ↦ 1))
130		0cnd 11226	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 0 ∈ ℂ)
131	127, 24	dvmptc 26187	. . . . . . 7 ⊢ (𝜑 → (ℝ D (𝑦 ∈ ℝ ↦ 𝑇)) = (𝑦 ∈ ℝ ↦ 0))
132	127, 112, 128, 129, 113, 130, 131	dvmptadd 26189	. . . . . 6 ⊢ (𝜑 → (ℝ D (𝑦 ∈ ℝ ↦ (𝑦 + 𝑇))) = (𝑦 ∈ ℝ ↦ (1 + 0)))
133	132	reseq1d 5975	. . . . 5 ⊢ (𝜑 → ((ℝ D (𝑦 ∈ ℝ ↦ (𝑦 + 𝑇))) ↾ (𝐴(,)𝐵)) = ((𝑦 ∈ ℝ ↦ (1 + 0)) ↾ (𝐴(,)𝐵)))
134		ioossre 13462	. . . . . . 7 ⊢ (𝐴(,)𝐵) ⊆ ℝ
135	134	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℝ)
136	135	resmptd 6040	. . . . 5 ⊢ (𝜑 → ((𝑦 ∈ ℝ ↦ (1 + 0)) ↾ (𝐴(,)𝐵)) = (𝑦 ∈ (𝐴(,)𝐵) ↦ (1 + 0)))
137		1p0e1 12390	. . . . . . 7 ⊢ (1 + 0) = 1
138	137	mpteq2i 5205	. . . . . 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 2801	. . . 4 ⊢ (𝜑 → ((ℝ D (𝑦 ∈ ℝ ↦ (𝑦 + 𝑇))) ↾ (𝐴(,)𝐵)) = (𝑦 ∈ (𝐴(,)𝐵) ↦ 1))
141	122, 125, 140	3eqtrd 2801	. . 3 ⊢ (𝜑 → (ℝ D (𝑦 ∈ (𝐴[,]𝐵) ↦ (𝑦 + 𝑇))) = (𝑦 ∈ (𝐴(,)𝐵) ↦ 1))
142		fveq2 6882	. . 3 ⊢ (𝑥 = (𝑦 + 𝑇) → (𝐺‘𝑥) = (𝐺‘(𝑦 + 𝑇)))
143		oveq1 7423	. . 3 ⊢ (𝑦 = 𝐴 → (𝑦 + 𝑇) = (𝐴 + 𝑇))
144		oveq1 7423	. . 3 ⊢ (𝑦 = 𝐵 → (𝑦 + 𝑇) = (𝐵 + 𝑇))
145	1, 2, 5, 71, 92, 107, 141, 142, 143, 144, 8, 9	itgsubsticc 46791	. 2 ⊢ (𝜑 → ⨜[(𝐴 + 𝑇) → (𝐵 + 𝑇)](𝐺‘𝑥) d𝑥 = ⨜[𝐴 → 𝐵]((𝐺‘(𝑦 + 𝑇)) · 1) d𝑦)
146	5	ditgpos 26085	. . 3 ⊢ (𝜑 → ⨜[𝐴 → 𝐵]((𝐺‘(𝑦 + 𝑇)) · 1) d𝑦 = ∫(𝐴(,)𝐵)((𝐺‘(𝑦 + 𝑇)) · 1) d𝑦)
147	43	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → 𝐺:((𝐴 + 𝑇)[,](𝐵 + 𝑇))⟶ℂ)
148	147, 70	ffvelcdmd 7081	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝐺‘(𝑦 + 𝑇)) ∈ ℂ)
149		1cnd 11229	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → 1 ∈ ℂ)
150	148, 149	mulcld 11256	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → ((𝐺‘(𝑦 + 𝑇)) · 1) ∈ ℂ)
151	1, 2, 150	itgioo 26045	. . 3 ⊢ (𝜑 → ∫(𝐴(,)𝐵)((𝐺‘(𝑦 + 𝑇)) · 1) d𝑦 = ∫(𝐴[,]𝐵)((𝐺‘(𝑦 + 𝑇)) · 1) d𝑦)
152		fvoveq1 7439	. . . . . 6 ⊢ (𝑦 = 𝑥 → (𝐺‘(𝑦 + 𝑇)) = (𝐺‘(𝑥 + 𝑇)))
153	152	oveq1d 7431	. . . . 5 ⊢ (𝑦 = 𝑥 → ((𝐺‘(𝑦 + 𝑇)) · 1) = ((𝐺‘(𝑥 + 𝑇)) · 1))
154	153	cbvitgv 26006	. . . 4 ⊢ ∫(𝐴[,]𝐵)((𝐺‘(𝑦 + 𝑇)) · 1) d𝑦 = ∫(𝐴[,]𝐵)((𝐺‘(𝑥 + 𝑇)) · 1) d𝑥
155	43	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐺:((𝐴 + 𝑇)[,](𝐵 + 𝑇))⟶ℂ)
156	8	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐴 + 𝑇) ∈ ℝ)
157	9	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐵 + 𝑇) ∈ ℝ)
158	50	sselda 3934	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ∈ ℝ)
159	4	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑇 ∈ ℝ)
160	158, 159	readdcld 11265	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝑥 + 𝑇) ∈ ℝ)
161	1	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐴 ∈ ℝ)
162	1	rexrd 11286	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
163	162	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐴 ∈ ℝ^*)
164	2	rexrd 11286	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
165	164	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐵 ∈ ℝ^*)
166		simpr 490	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ∈ (𝐴[,]𝐵))
167		iccgelb 13457	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐴 ≤ 𝑥)
168	163, 165, 166, 167	syl3anc 1398	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐴 ≤ 𝑥)
169	161, 158, 159, 168	leadd1dd 11855	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐴 + 𝑇) ≤ (𝑥 + 𝑇))
170	2	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐵 ∈ ℝ)
171		iccleub 13456	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ≤ 𝐵)
172	163, 165, 166, 171	syl3anc 1398	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ≤ 𝐵)
173	158, 170, 159, 172	leadd1dd 11855	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝑥 + 𝑇) ≤ (𝐵 + 𝑇))
174	156, 157, 160, 169, 173	eliccd 46321	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝑥 + 𝑇) ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇)))
175	155, 174	ffvelcdmd 7081	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐺‘(𝑥 + 𝑇)) ∈ ℂ)
176	175	mulridd 11253	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → ((𝐺‘(𝑥 + 𝑇)) · 1) = (𝐺‘(𝑥 + 𝑇)))
177	42, 73	eqtri 2785	. . . . . . . 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 7439	. . . . . . . 8 ⊢ (𝑤 = (𝑥 + 𝑇) → (𝐹‘(𝑤 − 𝑇)) = (𝐹‘((𝑥 + 𝑇) − 𝑇)))
180	158	recnd 11264	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ∈ ℂ)
181	24	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑇 ∈ ℂ)
182	180, 181	pncand 11597	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → ((𝑥 + 𝑇) − 𝑇) = 𝑥)
183	182	fveq2d 6886	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘((𝑥 + 𝑇) − 𝑇)) = (𝐹‘𝑥))
184	179, 183	sylan9eqr 2819	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ 𝑤 = (𝑥 + 𝑇)) → (𝐹‘(𝑤 − 𝑇)) = (𝐹‘𝑥))
185	12	ffvelcdmda 7080	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℂ)
186	178, 184, 174, 185	fvmptd 6998	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐺‘(𝑥 + 𝑇)) = (𝐹‘𝑥))
187	176, 186	eqtrd 2797	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → ((𝐺‘(𝑥 + 𝑇)) · 1) = (𝐹‘𝑥))
188	187	itgeq2dv 26011	. . . 4 ⊢ (𝜑 → ∫(𝐴[,]𝐵)((𝐺‘(𝑥 + 𝑇)) · 1) d𝑥 = ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)
189	154, 188	eqtrid 2809	. . 3 ⊢ (𝜑 → ∫(𝐴[,]𝐵)((𝐺‘(𝑦 + 𝑇)) · 1) d𝑦 = ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)
190	146, 151, 189	3eqtrd 2801	. 2 ⊢ (𝜑 → ⨜[𝐴 → 𝐵]((𝐺‘(𝑦 + 𝑇)) · 1) d𝑦 = ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)
191	46, 145, 190	3eqtrd 2801	1 ⊢ (𝜑 → ∫((𝐴 + 𝑇)[,](𝐵 + 𝑇))(𝐺‘𝑥) d𝑥 = ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ∃wrex 3088 {crab 3414 ⊆ wss 3902 {csn 4587 {cpr 4589 class class class wbr 5107 ↦ cmpt 5190 × cxp 5657 dom cdm 5659 ran crn 5660 ↾ cres 5661 ⟶wf 6533 ‘cfv 6537 (class class class)co 7416 ℂcc 11125 ℝcr 11126 0cc0 11127 1c1 11128 + caddc 11130 · cmul 11132 ℝ^*cxr 11269 ≤ cle 11271 − cmin 11468 ℝ⁺crp 13044 (,)cioo 13400 [,]cicc 13403 TopOpenctopn 17510 topGenctg 17526 ℂ_fldccnfld 21586 intcnt 23243 –cn→ccncf 25105 volcvol 25692 𝐿¹cibl 25846 ∫citg 25847 ⨜cdit 26075 D cdv 26092

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7739 ax-inf2 9623 ax-cc 10440 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204 ax-pre-sup 11205 ax-addf 11206

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3367 df-reu 3368 df-rab 3415 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 4202 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4871 df-int 4911 df-iun 4956 df-iin 4957 df-disj 5075 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-se 5613 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7681 df-ofr 7682 df-om 7866 df-1st 7989 df-2nd 7990 df-supp 8162 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-1o 8458 df-2o 8459 df-oadd 8462 df-omul 8463 df-er 8699 df-map 8831 df-pm 8832 df-ixp 8908 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-fsupp 9335 df-fi 9384 df-sup 9415 df-inf 9416 df-oi 9485 df-dju 9909 df-card 9947 df-acn 9950 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11470 df-neg 11471 df-div 11899 df-nn 12261 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12532 df-z 12619 df-dec 12740 df-uz 12891 df-q 13001 df-rp 13045 df-xneg 13165 df-xadd 13166 df-xmul 13167 df-ioo 13404 df-ioc 13405 df-ico 13406 df-icc 13407 df-fz 13564 df-fzo 13712 df-fl 13855 df-mod 13933 df-seq 14068 df-exp 14128 df-hash 14397 df-cj 15188 df-re 15189 df-im 15190 df-sqrt 15324 df-abs 15325 df-limsup 15560 df-clim 15577 df-rlim 15578 df-sum 15776 df-struct 17243 df-sets 17260 df-slot 17278 df-ndx 17290 df-base 17306 df-ress 17327 df-plusg 17359 df-mulr 17360 df-starv 17361 df-sca 17362 df-vsca 17363 df-ip 17364 df-tset 17365 df-ple 17366 df-ds 17368 df-unif 17369 df-hom 17370 df-cco 17371 df-rest 17511 df-topn 17512 df-0g 17530 df-gsum 17531 df-topgen 17532 df-pt 17533 df-prds 17536 df-xrs 17592 df-qtop 17597 df-imas 17598 df-xps 17600 df-mre 17674 df-mrc 17675 df-acs 17677 df-mgm 18734 df-sgrp 18823 df-mnd 18839 df-submnd 18893 df-mulg 19192 df-cntz 19445 df-cmn 19910 df-psmet 21578 df-xmet 21579 df-met 21580 df-bl 21581 df-mopn 21582 df-fbas 21583 df-fg 21584 df-cnfld 21587 df-top 23120 df-topon 23137 df-topsp 23159 df-bases 23172 df-cld 23245 df-ntr 23246 df-cls 23247 df-nei 23324 df-lp 23362 df-perf 23363 df-cn 23453 df-cnp 23454 df-haus 23541 df-cmp 23613 df-tx 23789 df-hmeo 23982 df-fil 24073 df-fm 24165 df-flim 24166 df-flf 24167 df-xms 24547 df-ms 24548 df-tms 24549 df-cncf 25107 df-ovol 25693 df-vol 25694 df-mbf 25848 df-itg1 25849 df-itg2 25850 df-ibl 25851 df-itg 25852 df-0p 25899 df-ditg 26076 df-limc 26095 df-dv 26096

This theorem is used by: fourierdlem81 47002

W3C validator