Theorem itgitg1 25307

Description: Transfer an integral using ∫₁ to an equivalent integral using ∫. (Contributed by Mario Carneiro, 6-Aug-2014.)

Assertion

Ref	Expression
itgitg1	⊢ (𝐹 ∈ dom ∫1 → ∫ℝ(𝐹‘𝑥) d𝑥 = (∫1‘𝐹))

Distinct variable group:   𝑥,𝐹

Proof of Theorem itgitg1

Step	Hyp	Ref	Expression
1		i1ff 25174	. . . 4 ⊢ (𝐹 ∈ dom ∫1 → 𝐹:ℝ⟶ℝ)
2	1	ffvelcdmda 7081	. . 3 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) ∈ ℝ)
3	1	feqmptd 6955	. . . 4 ⊢ (𝐹 ∈ dom ∫1 → 𝐹 = (𝑥 ∈ ℝ ↦ (𝐹‘𝑥)))
4		i1fibl 25306	. . . 4 ⊢ (𝐹 ∈ dom ∫1 → 𝐹 ∈ 𝐿1)
5	3, 4	eqeltrrd 2835	. . 3 ⊢ (𝐹 ∈ dom ∫1 → (𝑥 ∈ ℝ ↦ (𝐹‘𝑥)) ∈ 𝐿1)
6	2, 5	itgreval 25295	. 2 ⊢ (𝐹 ∈ dom ∫1 → ∫ℝ(𝐹‘𝑥) d𝑥 = (∫ℝif(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0) d𝑥 − ∫ℝif(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0) d𝑥))
7		0re 11211	. . . . . . 7 ⊢ 0 ∈ ℝ
8		ifcl 4571	. . . . . . 7 ⊢ (((𝐹‘𝑥) ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0) ∈ ℝ)
9	2, 7, 8	sylancl 587	. . . . . 6 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝑥 ∈ ℝ) → if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0) ∈ ℝ)
10		max1 13159	. . . . . . 7 ⊢ ((0 ∈ ℝ ∧ (𝐹‘𝑥) ∈ ℝ) → 0 ≤ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0))
11	7, 2, 10	sylancr 588	. . . . . 6 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝑥 ∈ ℝ) → 0 ≤ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0))
12		id 22	. . . . . . . . 9 ⊢ (𝐹 ∈ dom ∫1 → 𝐹 ∈ dom ∫1)
13	3, 12	eqeltrrd 2835	. . . . . . . 8 ⊢ (𝐹 ∈ dom ∫1 → (𝑥 ∈ ℝ ↦ (𝐹‘𝑥)) ∈ dom ∫1)
14	13	i1fposd 25206	. . . . . . 7 ⊢ (𝐹 ∈ dom ∫1 → (𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0)) ∈ dom ∫1)
15		i1fibl 25306	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0)) ∈ dom ∫1 → (𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0)) ∈ 𝐿1)
16	14, 15	syl 17	. . . . . 6 ⊢ (𝐹 ∈ dom ∫1 → (𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0)) ∈ 𝐿1)
17	9, 11, 16	itgitg2 25305	. . . . 5 ⊢ (𝐹 ∈ dom ∫1 → ∫ℝif(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0) d𝑥 = (∫2‘(𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0))))
18	11	ralrimiva 3147	. . . . . . . 8 ⊢ (𝐹 ∈ dom ∫1 → ∀𝑥 ∈ ℝ 0 ≤ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0))
19		reex 11196	. . . . . . . . . 10 ⊢ ℝ ∈ V
20	19	a1i 11	. . . . . . . . 9 ⊢ (𝐹 ∈ dom ∫1 → ℝ ∈ V)
21	7	a1i 11	. . . . . . . . 9 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝑥 ∈ ℝ) → 0 ∈ ℝ)
22		fconstmpt 5735	. . . . . . . . . 10 ⊢ (ℝ × {0}) = (𝑥 ∈ ℝ ↦ 0)
23	22	a1i 11	. . . . . . . . 9 ⊢ (𝐹 ∈ dom ∫1 → (ℝ × {0}) = (𝑥 ∈ ℝ ↦ 0))
24		eqidd 2734	. . . . . . . . 9 ⊢ (𝐹 ∈ dom ∫1 → (𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0)) = (𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0)))
25	20, 21, 9, 23, 24	ofrfval2 7685	. . . . . . . 8 ⊢ (𝐹 ∈ dom ∫1 → ((ℝ × {0}) ∘r ≤ (𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0)) ↔ ∀𝑥 ∈ ℝ 0 ≤ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0)))
26	18, 25	mpbird 257	. . . . . . 7 ⊢ (𝐹 ∈ dom ∫1 → (ℝ × {0}) ∘r ≤ (𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0)))
27		ax-resscn 11162	. . . . . . . . 9 ⊢ ℝ ⊆ ℂ
28	27	a1i 11	. . . . . . . 8 ⊢ (𝐹 ∈ dom ∫1 → ℝ ⊆ ℂ)
29	9	fmpttd 7109	. . . . . . . . 9 ⊢ (𝐹 ∈ dom ∫1 → (𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0)):ℝ⟶ℝ)
30	29	ffnd 6714	. . . . . . . 8 ⊢ (𝐹 ∈ dom ∫1 → (𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0)) Fn ℝ)
31	28, 30	0pledm 25171	. . . . . . 7 ⊢ (𝐹 ∈ dom ∫1 → (0𝑝 ∘r ≤ (𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0)) ↔ (ℝ × {0}) ∘r ≤ (𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0))))
32	26, 31	mpbird 257	. . . . . 6 ⊢ (𝐹 ∈ dom ∫1 → 0𝑝 ∘r ≤ (𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0)))
33		itg2itg1 25235	. . . . . 6 ⊢ (((𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0)) ∈ dom ∫1 ∧ 0𝑝 ∘r ≤ (𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0))) → (∫2‘(𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0))) = (∫1‘(𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0))))
34	14, 32, 33	syl2anc 585	. . . . 5 ⊢ (𝐹 ∈ dom ∫1 → (∫2‘(𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0))) = (∫1‘(𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0))))
35	17, 34	eqtrd 2773	. . . 4 ⊢ (𝐹 ∈ dom ∫1 → ∫ℝif(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0) d𝑥 = (∫1‘(𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0))))
36	2	renegcld 11636	. . . . . . 7 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝑥 ∈ ℝ) → -(𝐹‘𝑥) ∈ ℝ)
37		ifcl 4571	. . . . . . 7 ⊢ ((-(𝐹‘𝑥) ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0) ∈ ℝ)
38	36, 7, 37	sylancl 587	. . . . . 6 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝑥 ∈ ℝ) → if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0) ∈ ℝ)
39		max1 13159	. . . . . . 7 ⊢ ((0 ∈ ℝ ∧ -(𝐹‘𝑥) ∈ ℝ) → 0 ≤ if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0))
40	7, 36, 39	sylancr 588	. . . . . 6 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝑥 ∈ ℝ) → 0 ≤ if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0))
41		neg1rr 12322	. . . . . . . . . . . 12 ⊢ -1 ∈ ℝ
42	41	a1i 11	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝑥 ∈ ℝ) → -1 ∈ ℝ)
43		fconstmpt 5735	. . . . . . . . . . . 12 ⊢ (ℝ × {-1}) = (𝑥 ∈ ℝ ↦ -1)
44	43	a1i 11	. . . . . . . . . . 11 ⊢ (𝐹 ∈ dom ∫1 → (ℝ × {-1}) = (𝑥 ∈ ℝ ↦ -1))
45	20, 42, 2, 44, 3	offval2 7684	. . . . . . . . . 10 ⊢ (𝐹 ∈ dom ∫1 → ((ℝ × {-1}) ∘f · 𝐹) = (𝑥 ∈ ℝ ↦ (-1 · (𝐹‘𝑥))))
46	2	recnd 11237	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) ∈ ℂ)
47	46	mulm1d 11661	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝑥 ∈ ℝ) → (-1 · (𝐹‘𝑥)) = -(𝐹‘𝑥))
48	47	mpteq2dva 5246	. . . . . . . . . 10 ⊢ (𝐹 ∈ dom ∫1 → (𝑥 ∈ ℝ ↦ (-1 · (𝐹‘𝑥))) = (𝑥 ∈ ℝ ↦ -(𝐹‘𝑥)))
49	45, 48	eqtrd 2773	. . . . . . . . 9 ⊢ (𝐹 ∈ dom ∫1 → ((ℝ × {-1}) ∘f · 𝐹) = (𝑥 ∈ ℝ ↦ -(𝐹‘𝑥)))
50	41	a1i 11	. . . . . . . . . 10 ⊢ (𝐹 ∈ dom ∫1 → -1 ∈ ℝ)
51	12, 50	i1fmulc 25202	. . . . . . . . 9 ⊢ (𝐹 ∈ dom ∫1 → ((ℝ × {-1}) ∘f · 𝐹) ∈ dom ∫1)
52	49, 51	eqeltrrd 2835	. . . . . . . 8 ⊢ (𝐹 ∈ dom ∫1 → (𝑥 ∈ ℝ ↦ -(𝐹‘𝑥)) ∈ dom ∫1)
53	52	i1fposd 25206	. . . . . . 7 ⊢ (𝐹 ∈ dom ∫1 → (𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0)) ∈ dom ∫1)
54		i1fibl 25306	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0)) ∈ dom ∫1 → (𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0)) ∈ 𝐿1)
55	53, 54	syl 17	. . . . . 6 ⊢ (𝐹 ∈ dom ∫1 → (𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0)) ∈ 𝐿1)
56	38, 40, 55	itgitg2 25305	. . . . 5 ⊢ (𝐹 ∈ dom ∫1 → ∫ℝif(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0) d𝑥 = (∫2‘(𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0))))
57	40	ralrimiva 3147	. . . . . . . 8 ⊢ (𝐹 ∈ dom ∫1 → ∀𝑥 ∈ ℝ 0 ≤ if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0))
58		eqidd 2734	. . . . . . . . 9 ⊢ (𝐹 ∈ dom ∫1 → (𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0)) = (𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0)))
59	20, 21, 38, 23, 58	ofrfval2 7685	. . . . . . . 8 ⊢ (𝐹 ∈ dom ∫1 → ((ℝ × {0}) ∘r ≤ (𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0)) ↔ ∀𝑥 ∈ ℝ 0 ≤ if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0)))
60	57, 59	mpbird 257	. . . . . . 7 ⊢ (𝐹 ∈ dom ∫1 → (ℝ × {0}) ∘r ≤ (𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0)))
61	38	fmpttd 7109	. . . . . . . . 9 ⊢ (𝐹 ∈ dom ∫1 → (𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0)):ℝ⟶ℝ)
62	61	ffnd 6714	. . . . . . . 8 ⊢ (𝐹 ∈ dom ∫1 → (𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0)) Fn ℝ)
63	28, 62	0pledm 25171	. . . . . . 7 ⊢ (𝐹 ∈ dom ∫1 → (0𝑝 ∘r ≤ (𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0)) ↔ (ℝ × {0}) ∘r ≤ (𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0))))
64	60, 63	mpbird 257	. . . . . 6 ⊢ (𝐹 ∈ dom ∫1 → 0𝑝 ∘r ≤ (𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0)))
65		itg2itg1 25235	. . . . . 6 ⊢ (((𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0)) ∈ dom ∫1 ∧ 0𝑝 ∘r ≤ (𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0))) → (∫2‘(𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0))) = (∫1‘(𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0))))
66	53, 64, 65	syl2anc 585	. . . . 5 ⊢ (𝐹 ∈ dom ∫1 → (∫2‘(𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0))) = (∫1‘(𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0))))
67	56, 66	eqtrd 2773	. . . 4 ⊢ (𝐹 ∈ dom ∫1 → ∫ℝif(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0) d𝑥 = (∫1‘(𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0))))
68	35, 67	oveq12d 7421	. . 3 ⊢ (𝐹 ∈ dom ∫1 → (∫ℝif(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0) d𝑥 − ∫ℝif(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0) d𝑥) = ((∫1‘(𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0))) − (∫1‘(𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0)))))
69		itg1sub 25208	. . . 4 ⊢ (((𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0)) ∈ dom ∫1 ∧ (𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0)) ∈ dom ∫1) → (∫1‘((𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0)) ∘f − (𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0)))) = ((∫1‘(𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0))) − (∫1‘(𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0)))))
70	14, 53, 69	syl2anc 585	. . 3 ⊢ (𝐹 ∈ dom ∫1 → (∫1‘((𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0)) ∘f − (𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0)))) = ((∫1‘(𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0))) − (∫1‘(𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0)))))
71	68, 70	eqtr4d 2776	. 2 ⊢ (𝐹 ∈ dom ∫1 → (∫ℝif(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0) d𝑥 − ∫ℝif(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0) d𝑥) = (∫1‘((𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0)) ∘f − (𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0)))))
72		max0sub 13170	. . . . . 6 ⊢ ((𝐹‘𝑥) ∈ ℝ → (if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0) − if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0)) = (𝐹‘𝑥))
73	2, 72	syl 17	. . . . 5 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝑥 ∈ ℝ) → (if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0) − if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0)) = (𝐹‘𝑥))
74	73	mpteq2dva 5246	. . . 4 ⊢ (𝐹 ∈ dom ∫1 → (𝑥 ∈ ℝ ↦ (if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0) − if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0))) = (𝑥 ∈ ℝ ↦ (𝐹‘𝑥)))
75	20, 9, 38, 24, 58	offval2 7684	. . . 4 ⊢ (𝐹 ∈ dom ∫1 → ((𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0)) ∘f − (𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0))) = (𝑥 ∈ ℝ ↦ (if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0) − if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0))))
76	74, 75, 3	3eqtr4d 2783	. . 3 ⊢ (𝐹 ∈ dom ∫1 → ((𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0)) ∘f − (𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0))) = 𝐹)
77	76	fveq2d 6891	. 2 ⊢ (𝐹 ∈ dom ∫1 → (∫1‘((𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0)) ∘f − (𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0)))) = (∫1‘𝐹))
78	6, 71, 77	3eqtrd 2777	1 ⊢ (𝐹 ∈ dom ∫1 → ∫ℝ(𝐹‘𝑥) d𝑥 = (∫1‘𝐹))

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  ∀wral 3062  Vcvv 3475   ⊆ wss 3946  ifcif 4526  {csn 4626   class class class wbr 5146   ↦ cmpt 5229   × cxp 5672  dom cdm 5674  ‘cfv 6539  (class class class)co 7403   ∘_f cof 7662   ∘_r cofr 7663  ℂcc 11103  ℝcr 11104  0cc0 11105  1c1 11106   · cmul 11110   ≤ cle 11244   − cmin 11439  -cneg 11440  ∫₁citg1 25113  ∫₂citg2 25114  𝐿¹cibl 25115  ∫citg 25116  0_𝑝c0p 25167

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 5283  ax-sep 5297  ax-nul 5304  ax-pow 5361  ax-pr 5425  ax-un 7719  ax-inf2 9631  ax-cnex 11161  ax-resscn 11162  ax-1cn 11163  ax-icn 11164  ax-addcl 11165  ax-addrcl 11166  ax-mulcl 11167  ax-mulrcl 11168  ax-mulcom 11169  ax-addass 11170  ax-mulass 11171  ax-distr 11172  ax-i2m1 11173  ax-1ne0 11174  ax-1rid 11175  ax-rnegex 11176  ax-rrecex 11177  ax-cnre 11178  ax-pre-lttri 11179  ax-pre-lttrn 11180  ax-pre-ltadd 11181  ax-pre-mulgt0 11182  ax-pre-sup 11183  ax-addf 11184

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 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3965  df-nul 4321  df-if 4527  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4907  df-int 4949  df-iun 4997  df-disj 5112  df-br 5147  df-opab 5209  df-mpt 5230  df-tr 5264  df-id 5572  df-eprel 5578  df-po 5586  df-so 5587  df-fr 5629  df-se 5630  df-we 5631  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-pred 6296  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6491  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-isom 6548  df-riota 7359  df-ov 7406  df-oprab 7407  df-mpo 7408  df-of 7664  df-ofr 7665  df-om 7850  df-1st 7969  df-2nd 7970  df-frecs 8260  df-wrecs 8291  df-recs 8365  df-rdg 8404  df-1o 8460  df-2o 8461  df-er 8698  df-map 8817  df-pm 8818  df-en 8935  df-dom 8936  df-sdom 8937  df-fin 8938  df-fi 9401  df-sup 9432  df-inf 9433  df-oi 9500  df-dju 9891  df-card 9929  df-pnf 11245  df-mnf 11246  df-xr 11247  df-ltxr 11248  df-le 11249  df-sub 11441  df-neg 11442  df-div 11867  df-nn 12208  df-2 12270  df-3 12271  df-4 12272  df-n0 12468  df-z 12554  df-uz 12818  df-q 12928  df-rp 12970  df-xneg 13087  df-xadd 13088  df-xmul 13089  df-ioo 13323  df-ico 13325  df-icc 13326  df-fz 13480  df-fzo 13623  df-fl 13752  df-mod 13830  df-seq 13962  df-exp 14023  df-hash 14286  df-cj 15041  df-re 15042  df-im 15043  df-sqrt 15177  df-abs 15178  df-clim 15427  df-sum 15628  df-rest 17363  df-topgen 17384  df-psmet 20920  df-xmet 20921  df-met 20922  df-bl 20923  df-mopn 20924  df-top 22377  df-topon 22394  df-bases 22430  df-cmp 22872  df-ovol 24962  df-vol 24963  df-mbf 25117  df-itg1 25118  df-itg2 25119  df-ibl 25120  df-itg 25121  df-0p 25168

This theorem is referenced by: (None)