Theorem itgitg1 25730

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 25597	. . . 4 ⊢ (𝐹 ∈ dom ∫1 → 𝐹:ℝ⟶ℝ)
2	1	ffvelcdmda 7012	. . 3 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) ∈ ℝ)
3	1	feqmptd 6885	. . . 4 ⊢ (𝐹 ∈ dom ∫1 → 𝐹 = (𝑥 ∈ ℝ ↦ (𝐹‘𝑥)))
4		i1fibl 25729	. . . 4 ⊢ (𝐹 ∈ dom ∫1 → 𝐹 ∈ 𝐿1)
5	3, 4	eqeltrrd 2830	. . 3 ⊢ (𝐹 ∈ dom ∫1 → (𝑥 ∈ ℝ ↦ (𝐹‘𝑥)) ∈ 𝐿1)
6	2, 5	itgreval 25718	. 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 11106	. . . . . . 7 ⊢ 0 ∈ ℝ
8		ifcl 4519	. . . . . . 7 ⊢ (((𝐹‘𝑥) ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0) ∈ ℝ)
9	2, 7, 8	sylancl 586	. . . . . 6 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝑥 ∈ ℝ) → if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0) ∈ ℝ)
10		max1 13076	. . . . . . 7 ⊢ ((0 ∈ ℝ ∧ (𝐹‘𝑥) ∈ ℝ) → 0 ≤ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0))
11	7, 2, 10	sylancr 587	. . . . . 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 2830	. . . . . . . 8 ⊢ (𝐹 ∈ dom ∫1 → (𝑥 ∈ ℝ ↦ (𝐹‘𝑥)) ∈ dom ∫1)
14	13	i1fposd 25628	. . . . . . 7 ⊢ (𝐹 ∈ dom ∫1 → (𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0)) ∈ dom ∫1)
15		i1fibl 25729	. . . . . . 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 25728	. . . . 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 3122	. . . . . . . 8 ⊢ (𝐹 ∈ dom ∫1 → ∀𝑥 ∈ ℝ 0 ≤ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0))
19		reex 11089	. . . . . . . . . 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 5676	. . . . . . . . . 10 ⊢ (ℝ × {0}) = (𝑥 ∈ ℝ ↦ 0)
23	22	a1i 11	. . . . . . . . 9 ⊢ (𝐹 ∈ dom ∫1 → (ℝ × {0}) = (𝑥 ∈ ℝ ↦ 0))
24		eqidd 2731	. . . . . . . . 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 7626	. . . . . . . 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 11055	. . . . . . . . 9 ⊢ ℝ ⊆ ℂ
28	27	a1i 11	. . . . . . . 8 ⊢ (𝐹 ∈ dom ∫1 → ℝ ⊆ ℂ)
29	9	fmpttd 7043	. . . . . . . . 9 ⊢ (𝐹 ∈ dom ∫1 → (𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0)):ℝ⟶ℝ)
30	29	ffnd 6648	. . . . . . . 8 ⊢ (𝐹 ∈ dom ∫1 → (𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0)) Fn ℝ)
31	28, 30	0pledm 25594	. . . . . . 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 25657	. . . . . 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 584	. . . . 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 2765	. . . 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 11536	. . . . . . 7 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝑥 ∈ ℝ) → -(𝐹‘𝑥) ∈ ℝ)
37		ifcl 4519	. . . . . . 7 ⊢ ((-(𝐹‘𝑥) ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0) ∈ ℝ)
38	36, 7, 37	sylancl 586	. . . . . 6 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝑥 ∈ ℝ) → if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0) ∈ ℝ)
39		max1 13076	. . . . . . 7 ⊢ ((0 ∈ ℝ ∧ -(𝐹‘𝑥) ∈ ℝ) → 0 ≤ if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0))
40	7, 36, 39	sylancr 587	. . . . . 6 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝑥 ∈ ℝ) → 0 ≤ if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0))
41		neg1rr 12103	. . . . . . . . . . . 12 ⊢ -1 ∈ ℝ
42	41	a1i 11	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝑥 ∈ ℝ) → -1 ∈ ℝ)
43		fconstmpt 5676	. . . . . . . . . . . 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 7625	. . . . . . . . . 10 ⊢ (𝐹 ∈ dom ∫1 → ((ℝ × {-1}) ∘f · 𝐹) = (𝑥 ∈ ℝ ↦ (-1 · (𝐹‘𝑥))))
46	2	recnd 11132	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) ∈ ℂ)
47	46	mulm1d 11561	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝑥 ∈ ℝ) → (-1 · (𝐹‘𝑥)) = -(𝐹‘𝑥))
48	47	mpteq2dva 5182	. . . . . . . . . 10 ⊢ (𝐹 ∈ dom ∫1 → (𝑥 ∈ ℝ ↦ (-1 · (𝐹‘𝑥))) = (𝑥 ∈ ℝ ↦ -(𝐹‘𝑥)))
49	45, 48	eqtrd 2765	. . . . . . . . 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 25624	. . . . . . . . 9 ⊢ (𝐹 ∈ dom ∫1 → ((ℝ × {-1}) ∘f · 𝐹) ∈ dom ∫1)
52	49, 51	eqeltrrd 2830	. . . . . . . 8 ⊢ (𝐹 ∈ dom ∫1 → (𝑥 ∈ ℝ ↦ -(𝐹‘𝑥)) ∈ dom ∫1)
53	52	i1fposd 25628	. . . . . . 7 ⊢ (𝐹 ∈ dom ∫1 → (𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0)) ∈ dom ∫1)
54		i1fibl 25729	. . . . . . 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 25728	. . . . 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 3122	. . . . . . . 8 ⊢ (𝐹 ∈ dom ∫1 → ∀𝑥 ∈ ℝ 0 ≤ if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0))
58		eqidd 2731	. . . . . . . . 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 7626	. . . . . . . 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 7043	. . . . . . . . 9 ⊢ (𝐹 ∈ dom ∫1 → (𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0)):ℝ⟶ℝ)
62	61	ffnd 6648	. . . . . . . 8 ⊢ (𝐹 ∈ dom ∫1 → (𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0)) Fn ℝ)
63	28, 62	0pledm 25594	. . . . . . 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 25657	. . . . . 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 584	. . . . 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 2765	. . . 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 7359	. . 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 25630	. . . 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 584	. . 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 2768	. 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 13087	. . . . . 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 5182	. . . 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 7625	. . . 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 2775	. . 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 6821	. 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 2769	1 ⊢ (𝐹 ∈ dom ∫1 → ∫ℝ(𝐹‘𝑥) d𝑥 = (∫1‘𝐹))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2110  ∀wral 3045  Vcvv 3434   ⊆ wss 3900  ifcif 4473  {csn 4574   class class class wbr 5089   ↦ cmpt 5170   × cxp 5612  dom cdm 5614  ‘cfv 6477  (class class class)co 7341   ∘_f cof 7603   ∘_r cofr 7604  ℂcc 10996  ℝcr 10997  0cc0 10998  1c1 10999   · cmul 11003   ≤ cle 11139   − cmin 11336  -cneg 11337  ∫₁citg1 25536  ∫₂citg2 25537  𝐿¹cibl 25538  ∫citg 25539  0_𝑝c0p 25590

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7663  ax-inf2 9526  ax-cnex 11054  ax-resscn 11055  ax-1cn 11056  ax-icn 11057  ax-addcl 11058  ax-addrcl 11059  ax-mulcl 11060  ax-mulrcl 11061  ax-mulcom 11062  ax-addass 11063  ax-mulass 11064  ax-distr 11065  ax-i2m1 11066  ax-1ne0 11067  ax-1rid 11068  ax-rnegex 11069  ax-rrecex 11070  ax-cnre 11071  ax-pre-lttri 11072  ax-pre-lttrn 11073  ax-pre-ltadd 11074  ax-pre-mulgt0 11075  ax-pre-sup 11076  ax-addf 11077

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3344  df-reu 3345  df-rab 3394  df-v 3436  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4282  df-if 4474  df-pw 4550  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-int 4896  df-iun 4941  df-disj 5057  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-se 5568  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6244  df-ord 6305  df-on 6306  df-lim 6307  df-suc 6308  df-iota 6433  df-fun 6479  df-fn 6480  df-f 6481  df-f1 6482  df-fo 6483  df-f1o 6484  df-fv 6485  df-isom 6486  df-riota 7298  df-ov 7344  df-oprab 7345  df-mpo 7346  df-of 7605  df-ofr 7606  df-om 7792  df-1st 7916  df-2nd 7917  df-frecs 8206  df-wrecs 8237  df-recs 8286  df-rdg 8324  df-1o 8380  df-2o 8381  df-er 8617  df-map 8747  df-pm 8748  df-en 8865  df-dom 8866  df-sdom 8867  df-fin 8868  df-fi 9290  df-sup 9321  df-inf 9322  df-oi 9391  df-dju 9786  df-card 9824  df-pnf 11140  df-mnf 11141  df-xr 11142  df-ltxr 11143  df-le 11144  df-sub 11338  df-neg 11339  df-div 11767  df-nn 12118  df-2 12180  df-3 12181  df-4 12182  df-n0 12374  df-z 12461  df-uz 12725  df-q 12839  df-rp 12883  df-xneg 13003  df-xadd 13004  df-xmul 13005  df-ioo 13241  df-ico 13243  df-icc 13244  df-fz 13400  df-fzo 13547  df-fl 13688  df-mod 13766  df-seq 13901  df-exp 13961  df-hash 14230  df-cj 14998  df-re 14999  df-im 15000  df-sqrt 15134  df-abs 15135  df-clim 15387  df-sum 15586  df-rest 17318  df-topgen 17339  df-psmet 21276  df-xmet 21277  df-met 21278  df-bl 21279  df-mopn 21280  df-top 22802  df-topon 22819  df-bases 22854  df-cmp 23295  df-ovol 25385  df-vol 25386  df-mbf 25540  df-itg1 25541  df-itg2 25542  df-ibl 25543  df-itg 25544  df-0p 25591

This theorem is referenced by: (None)