itgiccshift - Mathbox for Glauco Siliprandi

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Theorem itgiccshift 46408

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 12986	. . . . 5 ⊢ (𝜑 → 𝑇 ∈ ℝ)
5		itgiccshift.aleb	. . . . 5 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
6	1, 2, 4, 5	leadd1dd 11764	. . . 4 ⊢ (𝜑 → (𝐴 + 𝑇) ≤ (𝐵 + 𝑇))
7	6	ditgpos 25823	. . 3 ⊢ (𝜑 → ⨜[(𝐴 + 𝑇) → (𝐵 + 𝑇)](𝐺‘𝑥) d𝑥 = ∫((𝐴 + 𝑇)(,)(𝐵 + 𝑇))(𝐺‘𝑥) d𝑥)
8	1, 4	readdcld 11174	. . . 4 ⊢ (𝜑 → (𝐴 + 𝑇) ∈ ℝ)
9	2, 4	readdcld 11174	. . . 4 ⊢ (𝜑 → (𝐵 + 𝑇) ∈ ℝ)
10		itgiccshift.f	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ))
11		cncff 24860	. . . . . . . . 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 45935	. . . . . . . . . 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 11578	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝑥 − 𝑇) ∈ ℝ)
23	1	recnd 11173	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ ℂ)
24	4	recnd 11173	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑇 ∈ ℂ)
25	23, 24	pncand 11506	. . . . . . . . . . 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 13364	. . . . . . . . . . . . 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 11766	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → ((𝐴 + 𝑇) − 𝑇) ≤ (𝑥 − 𝑇))
33	27, 32	eqbrtrd 5108	. . . . . . . 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 11766	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝑥 − 𝑇) ≤ ((𝐵 + 𝑇) − 𝑇))
36	2	recnd 11173	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ ℂ)
37	36, 24	pncand 11506	. . . . . . . . . 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 5112	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝑥 − 𝑇) ≤ 𝐵)
40	14, 15, 22, 33, 39	eliccd 45934	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝑥 − 𝑇) ∈ (𝐴[,]𝐵))
41	13, 40	ffvelcdmd 7038	. . . . . 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 7067	. . . . 5 ⊢ (𝜑 → 𝐺:((𝐴 + 𝑇)[,](𝐵 + 𝑇))⟶ℂ)
44	43	ffvelcdmda 7037	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝐺‘𝑥) ∈ ℂ)
45	8, 9, 44	itgioo 25783	. . 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 24884	. . . . 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 13387	. . . . 5 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
51		ax-resscn 11095	. . . . 5 ⊢ ℝ ⊆ ℂ
52	50, 51	sstrdi 3935	. . . 4 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℂ)
53	8, 9	iccssred 13387	. . . . 5 ⊢ (𝜑 → ((𝐴 + 𝑇)[,](𝐵 + 𝑇)) ⊆ ℝ)
54	53, 51	sstrdi 3935	. . . 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 3922	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → 𝑦 ∈ ℝ)
58	4	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → 𝑇 ∈ ℝ)
59	57, 58	readdcld 11174	. . . . 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 13364	. . . . . . . . 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 11764	. . . . 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 11764	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝑦 + 𝑇) ≤ (𝐵 + 𝑇))
70	55, 56, 59, 67, 69	eliccd 45934	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝑦 + 𝑇) ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇)))
71	47, 49, 52, 54, 70	cncfmptssg 46299	. . 3 ⊢ (𝜑 → (𝑦 ∈ (𝐴[,]𝐵) ↦ (𝑦 + 𝑇)) ∈ ((𝐴[,]𝐵)–cn→((𝐴 + 𝑇)[,](𝐵 + 𝑇))))
72		fvoveq1 7390	. . . . . . . 8 ⊢ (𝑥 = 𝑤 → (𝐹‘(𝑥 − 𝑇)) = (𝐹‘(𝑤 − 𝑇)))
73	72	cbvmptv 5190	. . . . . . 7 ⊢ (𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇)) ↦ (𝐹‘(𝑥 − 𝑇))) = (𝑤 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇)) ↦ (𝐹‘(𝑤 − 𝑇)))
74	1, 2, 4	iccshift 45948	. . . . . . . 8 ⊢ (𝜑 → ((𝐴 + 𝑇)[,](𝐵 + 𝑇)) = {𝑥 ∈ ℂ ∣ ∃𝑦 ∈ (𝐴[,]𝐵)𝑥 = (𝑦 + 𝑇)})
75	74	mpteq1d 5176	. . . . . . 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 3162	. . . . . . . . 9 ⊢ (𝑤 = 𝑥 → (∃𝑧 ∈ (𝐴[,]𝐵)𝑤 = (𝑧 + 𝑇) ↔ ∃𝑧 ∈ (𝐴[,]𝐵)𝑥 = (𝑧 + 𝑇)))
80		oveq1 7374	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑦 → (𝑧 + 𝑇) = (𝑦 + 𝑇))
81	80	eqeq2d 2748	. . . . . . . . . 10 ⊢ (𝑧 = 𝑦 → (𝑥 = (𝑧 + 𝑇) ↔ 𝑥 = (𝑦 + 𝑇)))
82	81	cbvrexvw 3217	. . . . . . . . 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 3400	. . . . . . 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 46302	. . . . 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 6909	. . . 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 7382	. . . 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 13361	. . . . . 6 ⊢ (𝐴(,)𝐵) ⊆ ℂ
94	93	a1i 11	. . . . 5 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℂ)
95		1cnd 11139	. . . . 5 ⊢ (𝜑 → 1 ∈ ℂ)
96		ssid 3945	. . . . . 6 ⊢ ℂ ⊆ ℂ
97	96	a1i 11	. . . . 5 ⊢ (𝜑 → ℂ ⊆ ℂ)
98	94, 95, 97	constcncfg 46300	. . . 4 ⊢ (𝜑 → (𝑦 ∈ (𝐴(,)𝐵) ↦ 1) ∈ ((𝐴(,)𝐵)–cn→ℂ))
99		fconstmpt 5693	. . . . 5 ⊢ ((𝐴(,)𝐵) × {1}) = (𝑦 ∈ (𝐴(,)𝐵) ↦ 1)
100		ioombl 25532	. . . . . . 7 ⊢ (𝐴(,)𝐵) ∈ dom vol
101	100	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝐴(,)𝐵) ∈ dom vol)
102		ioovolcl 25537	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (vol‘(𝐴(,)𝐵)) ∈ ℝ)
103	1, 2, 102	syl2anc 585	. . . . . 6 ⊢ (𝜑 → (vol‘(𝐴(,)𝐵)) ∈ ℝ)
104		iblconst 25785	. . . . . 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 4141	. . 3 ⊢ (𝜑 → (𝑦 ∈ (𝐴(,)𝐵) ↦ 1) ∈ (((𝐴(,)𝐵)–cn→ℂ) ∩ 𝐿¹))
108	50	resmptd 6006	. . . . . . 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 7383	. . . . 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 3922	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℂ)
113	24	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝑇 ∈ ℂ)
114	112, 113	addcld 11164	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝑦 + 𝑇) ∈ ℂ)
115	114	fmpttd 7068	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ ℝ ↦ (𝑦 + 𝑇)):ℝ⟶ℂ)
116		ssid 3945	. . . . . . 7 ⊢ ℝ ⊆ ℝ
117	116	a1i 11	. . . . . 6 ⊢ (𝜑 → ℝ ⊆ ℝ)
118		eqid 2737	. . . . . . 7 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
119		tgioo4 24770	. . . . . . 7 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
120	118, 119	dvres 25878	. . . . . 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 24787	. . . . . 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 5945	. . . 4 ⊢ (𝜑 → ((ℝ D (𝑦 ∈ ℝ ↦ (𝑦 + 𝑇))) ↾ ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵))) = ((ℝ D (𝑦 ∈ ℝ ↦ (𝑦 + 𝑇))) ↾ (𝐴(,)𝐵)))
126		reelprrecn 11130	. . . . . . . 8 ⊢ ℝ ∈ {ℝ, ℂ}
127	126	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
128		1cnd 11139	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 1 ∈ ℂ)
129	127	dvmptid 25924	. . . . . . 7 ⊢ (𝜑 → (ℝ D (𝑦 ∈ ℝ ↦ 𝑦)) = (𝑦 ∈ ℝ ↦ 1))
130		0cnd 11137	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 0 ∈ ℂ)
131	127, 24	dvmptc 25925	. . . . . . 7 ⊢ (𝜑 → (ℝ D (𝑦 ∈ ℝ ↦ 𝑇)) = (𝑦 ∈ ℝ ↦ 0))
132	127, 112, 128, 129, 113, 130, 131	dvmptadd 25927	. . . . . 6 ⊢ (𝜑 → (ℝ D (𝑦 ∈ ℝ ↦ (𝑦 + 𝑇))) = (𝑦 ∈ ℝ ↦ (1 + 0)))
133	132	reseq1d 5944	. . . . 5 ⊢ (𝜑 → ((ℝ D (𝑦 ∈ ℝ ↦ (𝑦 + 𝑇))) ↾ (𝐴(,)𝐵)) = ((𝑦 ∈ ℝ ↦ (1 + 0)) ↾ (𝐴(,)𝐵)))
134		ioossre 13360	. . . . . . 7 ⊢ (𝐴(,)𝐵) ⊆ ℝ
135	134	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℝ)
136	135	resmptd 6006	. . . . 5 ⊢ (𝜑 → ((𝑦 ∈ ℝ ↦ (1 + 0)) ↾ (𝐴(,)𝐵)) = (𝑦 ∈ (𝐴(,)𝐵) ↦ (1 + 0)))
137		1p0e1 12300	. . . . . . 7 ⊢ (1 + 0) = 1
138	137	mpteq2i 5182	. . . . . 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 6841	. . 3 ⊢ (𝑥 = (𝑦 + 𝑇) → (𝐺‘𝑥) = (𝐺‘(𝑦 + 𝑇)))
143		oveq1 7374	. . 3 ⊢ (𝑦 = 𝐴 → (𝑦 + 𝑇) = (𝐴 + 𝑇))
144		oveq1 7374	. . 3 ⊢ (𝑦 = 𝐵 → (𝑦 + 𝑇) = (𝐵 + 𝑇))
145	1, 2, 5, 71, 92, 107, 141, 142, 143, 144, 8, 9	itgsubsticc 46404	. 2 ⊢ (𝜑 → ⨜[(𝐴 + 𝑇) → (𝐵 + 𝑇)](𝐺‘𝑥) d𝑥 = ⨜[𝐴 → 𝐵]((𝐺‘(𝑦 + 𝑇)) · 1) d𝑦)
146	5	ditgpos 25823	. . 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 7038	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝐺‘(𝑦 + 𝑇)) ∈ ℂ)
149		1cnd 11139	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → 1 ∈ ℂ)
150	148, 149	mulcld 11165	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → ((𝐺‘(𝑦 + 𝑇)) · 1) ∈ ℂ)
151	1, 2, 150	itgioo 25783	. . 3 ⊢ (𝜑 → ∫(𝐴(,)𝐵)((𝐺‘(𝑦 + 𝑇)) · 1) d𝑦 = ∫(𝐴[,]𝐵)((𝐺‘(𝑦 + 𝑇)) · 1) d𝑦)
152		fvoveq1 7390	. . . . . 6 ⊢ (𝑦 = 𝑥 → (𝐺‘(𝑦 + 𝑇)) = (𝐺‘(𝑥 + 𝑇)))
153	152	oveq1d 7382	. . . . 5 ⊢ (𝑦 = 𝑥 → ((𝐺‘(𝑦 + 𝑇)) · 1) = ((𝐺‘(𝑥 + 𝑇)) · 1))
154	153	cbvitgv 25744	. . . 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 3922	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ∈ ℝ)
159	4	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑇 ∈ ℝ)
160	158, 159	readdcld 11174	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝑥 + 𝑇) ∈ ℝ)
161	1	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐴 ∈ ℝ)
162	1	rexrd 11195	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
163	162	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐴 ∈ ℝ^*)
164	2	rexrd 11195	. . . . . . . . . . . 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 13355	. . . . . . . . . . 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 11764	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐴 + 𝑇) ≤ (𝑥 + 𝑇))
170	2	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐵 ∈ ℝ)
171		iccleub 13354	. . . . . . . . . . 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 11764	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝑥 + 𝑇) ≤ (𝐵 + 𝑇))
174	156, 157, 160, 169, 173	eliccd 45934	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝑥 + 𝑇) ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇)))
175	155, 174	ffvelcdmd 7038	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐺‘(𝑥 + 𝑇)) ∈ ℂ)
176	175	mulridd 11162	. . . . . 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 7390	. . . . . . . 8 ⊢ (𝑤 = (𝑥 + 𝑇) → (𝐹‘(𝑤 − 𝑇)) = (𝐹‘((𝑥 + 𝑇) − 𝑇)))
180	158	recnd 11173	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ∈ ℂ)
181	24	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑇 ∈ ℂ)
182	180, 181	pncand 11506	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → ((𝑥 + 𝑇) − 𝑇) = 𝑥)
183	182	fveq2d 6845	. . . . . . . 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 7037	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℂ)
186	178, 184, 174, 185	fvmptd 6956	. . . . . 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 25749	. . . 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 3062 {crab 3390 ⊆ wss 3890 {csn 4568 {cpr 4570 class class class wbr 5086 ↦ cmpt 5167 × cxp 5629 dom cdm 5631 ran crn 5632 ↾ cres 5633 ⟶wf 6495 ‘cfv 6499 (class class class)co 7367 ℂcc 11036 ℝcr 11037 0cc0 11038 1c1 11039 + caddc 11041 · cmul 11043 ℝ^*cxr 11178 ≤ cle 11180 − cmin 11377 ℝ⁺crp 12942 (,)cioo 13298 [,]cicc 13301 TopOpenctopn 17384 topGenctg 17400 ℂ_fldccnfld 21352 intcnt 22982 –cn→ccncf 24843 volcvol 25430 𝐿¹cibl 25584 ∫citg 25585 ⨜cdit 25813 D cdv 25830

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 5213 ax-sep 5232 ax-nul 5242 ax-pow 5308 ax-pr 5376 ax-un 7689 ax-inf2 9562 ax-cc 10357 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 ax-addf 11117

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 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-symdif 4194 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-disj 5054 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6266 df-ord 6327 df-on 6328 df-lim 6329 df-suc 6330 df-iota 6455 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-isom 6508 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-of 7631 df-ofr 7632 df-om 7818 df-1st 7942 df-2nd 7943 df-supp 8111 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-2o 8406 df-oadd 8409 df-omul 8410 df-er 8643 df-map 8775 df-pm 8776 df-ixp 8846 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-fsupp 9275 df-fi 9324 df-sup 9355 df-inf 9356 df-oi 9425 df-dju 9825 df-card 9863 df-acn 9866 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-7 12249 df-8 12250 df-9 12251 df-n0 12438 df-z 12525 df-dec 12645 df-uz 12789 df-q 12899 df-rp 12943 df-xneg 13063 df-xadd 13064 df-xmul 13065 df-ioo 13302 df-ioc 13303 df-ico 13304 df-icc 13305 df-fz 13462 df-fzo 13609 df-fl 13751 df-mod 13829 df-seq 13964 df-exp 14024 df-hash 14293 df-cj 15061 df-re 15062 df-im 15063 df-sqrt 15197 df-abs 15198 df-limsup 15433 df-clim 15450 df-rlim 15451 df-sum 15649 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-hom 17244 df-cco 17245 df-rest 17385 df-topn 17386 df-0g 17404 df-gsum 17405 df-topgen 17406 df-pt 17407 df-prds 17410 df-xrs 17466 df-qtop 17471 df-imas 17472 df-xps 17474 df-mre 17548 df-mrc 17549 df-acs 17551 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-submnd 18752 df-mulg 19044 df-cntz 19292 df-cmn 19757 df-psmet 21344 df-xmet 21345 df-met 21346 df-bl 21347 df-mopn 21348 df-fbas 21349 df-fg 21350 df-cnfld 21353 df-top 22859 df-topon 22876 df-topsp 22898 df-bases 22911 df-cld 22984 df-ntr 22985 df-cls 22986 df-nei 23063 df-lp 23101 df-perf 23102 df-cn 23192 df-cnp 23193 df-haus 23280 df-cmp 23352 df-tx 23527 df-hmeo 23720 df-fil 23811 df-fm 23903 df-flim 23904 df-flf 23905 df-xms 24285 df-ms 24286 df-tms 24287 df-cncf 24845 df-ovol 25431 df-vol 25432 df-mbf 25586 df-itg1 25587 df-itg2 25588 df-ibl 25589 df-itg 25590 df-0p 25637 df-ditg 25814 df-limc 25833 df-dv 25834

This theorem is referenced by: fourierdlem81 46615

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