Theorem itgitg1 24411

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 24279	. . . 4 ⊢ (𝐹 ∈ dom ∫1 → 𝐹:ℝ⟶ℝ)
2	1	ffvelrnda 6853	. . 3 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) ∈ ℝ)
3	1	feqmptd 6735	. . . 4 ⊢ (𝐹 ∈ dom ∫1 → 𝐹 = (𝑥 ∈ ℝ ↦ (𝐹‘𝑥)))
4		i1fibl 24410	. . . 4 ⊢ (𝐹 ∈ dom ∫1 → 𝐹 ∈ 𝐿1)
5	3, 4	eqeltrrd 2916	. . 3 ⊢ (𝐹 ∈ dom ∫1 → (𝑥 ∈ ℝ ↦ (𝐹‘𝑥)) ∈ 𝐿1)
6	2, 5	itgreval 24399	. 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 10645	. . . . . . 7 ⊢ 0 ∈ ℝ
8		ifcl 4513	. . . . . . 7 ⊢ (((𝐹‘𝑥) ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0) ∈ ℝ)
9	2, 7, 8	sylancl 588	. . . . . 6 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝑥 ∈ ℝ) → if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0) ∈ ℝ)
10		max1 12581	. . . . . . 7 ⊢ ((0 ∈ ℝ ∧ (𝐹‘𝑥) ∈ ℝ) → 0 ≤ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0))
11	7, 2, 10	sylancr 589	. . . . . 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 2916	. . . . . . . 8 ⊢ (𝐹 ∈ dom ∫1 → (𝑥 ∈ ℝ ↦ (𝐹‘𝑥)) ∈ dom ∫1)
14	13	i1fposd 24310	. . . . . . 7 ⊢ (𝐹 ∈ dom ∫1 → (𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0)) ∈ dom ∫1)
15		i1fibl 24410	. . . . . . 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 24409	. . . . 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 3184	. . . . . . . 8 ⊢ (𝐹 ∈ dom ∫1 → ∀𝑥 ∈ ℝ 0 ≤ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0))
19		reex 10630	. . . . . . . . . 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 5616	. . . . . . . . . 10 ⊢ (ℝ × {0}) = (𝑥 ∈ ℝ ↦ 0)
23	22	a1i 11	. . . . . . . . 9 ⊢ (𝐹 ∈ dom ∫1 → (ℝ × {0}) = (𝑥 ∈ ℝ ↦ 0))
24		eqidd 2824	. . . . . . . . 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 7429	. . . . . . . 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 259	. . . . . . 7 ⊢ (𝐹 ∈ dom ∫1 → (ℝ × {0}) ∘r ≤ (𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0)))
27		ax-resscn 10596	. . . . . . . . 9 ⊢ ℝ ⊆ ℂ
28	27	a1i 11	. . . . . . . 8 ⊢ (𝐹 ∈ dom ∫1 → ℝ ⊆ ℂ)
29	9	fmpttd 6881	. . . . . . . . 9 ⊢ (𝐹 ∈ dom ∫1 → (𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0)):ℝ⟶ℝ)
30	29	ffnd 6517	. . . . . . . 8 ⊢ (𝐹 ∈ dom ∫1 → (𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0)) Fn ℝ)
31	28, 30	0pledm 24276	. . . . . . 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 259	. . . . . 6 ⊢ (𝐹 ∈ dom ∫1 → 0𝑝 ∘r ≤ (𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0)))
33		itg2itg1 24339	. . . . . 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 586	. . . . 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 2858	. . . 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 11069	. . . . . . 7 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝑥 ∈ ℝ) → -(𝐹‘𝑥) ∈ ℝ)
37		ifcl 4513	. . . . . . 7 ⊢ ((-(𝐹‘𝑥) ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0) ∈ ℝ)
38	36, 7, 37	sylancl 588	. . . . . 6 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝑥 ∈ ℝ) → if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0) ∈ ℝ)
39		max1 12581	. . . . . . 7 ⊢ ((0 ∈ ℝ ∧ -(𝐹‘𝑥) ∈ ℝ) → 0 ≤ if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0))
40	7, 36, 39	sylancr 589	. . . . . 6 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝑥 ∈ ℝ) → 0 ≤ if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0))
41		neg1rr 11755	. . . . . . . . . . . 12 ⊢ -1 ∈ ℝ
42	41	a1i 11	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝑥 ∈ ℝ) → -1 ∈ ℝ)
43		fconstmpt 5616	. . . . . . . . . . . 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 7428	. . . . . . . . . 10 ⊢ (𝐹 ∈ dom ∫1 → ((ℝ × {-1}) ∘f · 𝐹) = (𝑥 ∈ ℝ ↦ (-1 · (𝐹‘𝑥))))
46	2	recnd 10671	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) ∈ ℂ)
47	46	mulm1d 11094	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝑥 ∈ ℝ) → (-1 · (𝐹‘𝑥)) = -(𝐹‘𝑥))
48	47	mpteq2dva 5163	. . . . . . . . . 10 ⊢ (𝐹 ∈ dom ∫1 → (𝑥 ∈ ℝ ↦ (-1 · (𝐹‘𝑥))) = (𝑥 ∈ ℝ ↦ -(𝐹‘𝑥)))
49	45, 48	eqtrd 2858	. . . . . . . . 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 24306	. . . . . . . . 9 ⊢ (𝐹 ∈ dom ∫1 → ((ℝ × {-1}) ∘f · 𝐹) ∈ dom ∫1)
52	49, 51	eqeltrrd 2916	. . . . . . . 8 ⊢ (𝐹 ∈ dom ∫1 → (𝑥 ∈ ℝ ↦ -(𝐹‘𝑥)) ∈ dom ∫1)
53	52	i1fposd 24310	. . . . . . 7 ⊢ (𝐹 ∈ dom ∫1 → (𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0)) ∈ dom ∫1)
54		i1fibl 24410	. . . . . . 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 24409	. . . . 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 3184	. . . . . . . 8 ⊢ (𝐹 ∈ dom ∫1 → ∀𝑥 ∈ ℝ 0 ≤ if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0))
58		eqidd 2824	. . . . . . . . 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 7429	. . . . . . . 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 259	. . . . . . 7 ⊢ (𝐹 ∈ dom ∫1 → (ℝ × {0}) ∘r ≤ (𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0)))
61	38	fmpttd 6881	. . . . . . . . 9 ⊢ (𝐹 ∈ dom ∫1 → (𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0)):ℝ⟶ℝ)
62	61	ffnd 6517	. . . . . . . 8 ⊢ (𝐹 ∈ dom ∫1 → (𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0)) Fn ℝ)
63	28, 62	0pledm 24276	. . . . . . 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 259	. . . . . 6 ⊢ (𝐹 ∈ dom ∫1 → 0𝑝 ∘r ≤ (𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0)))
65		itg2itg1 24339	. . . . . 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 586	. . . . 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 2858	. . . 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 7176	. . 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 24312	. . . 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 586	. . 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 2861	. 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 12592	. . . . . 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 5163	. . . 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 7428	. . . 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 2868	. . 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 6676	. 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 2862	1 ⊢ (𝐹 ∈ dom ∫1 → ∫ℝ(𝐹‘𝑥) d𝑥 = (∫1‘𝐹))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∀wral 3140  Vcvv 3496   ⊆ wss 3938  ifcif 4469  {csn 4569   class class class wbr 5068   ↦ cmpt 5148   × cxp 5555  dom cdm 5557  ‘cfv 6357  (class class class)co 7158   ∘_f cof 7409   ∘_r cofr 7410  ℂcc 10537  ℝcr 10538  0cc0 10539  1c1 10540   · cmul 10544   ≤ cle 10678   − cmin 10872  -cneg 10873  ∫₁citg1 24218  ∫₂citg2 24219  𝐿¹cibl 24220  ∫citg 24221  0_𝑝c0p 24272

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463  ax-inf2 9106  ax-cnex 10595  ax-resscn 10596  ax-1cn 10597  ax-icn 10598  ax-addcl 10599  ax-addrcl 10600  ax-mulcl 10601  ax-mulrcl 10602  ax-mulcom 10603  ax-addass 10604  ax-mulass 10605  ax-distr 10606  ax-i2m1 10607  ax-1ne0 10608  ax-1rid 10609  ax-rnegex 10610  ax-rrecex 10611  ax-cnre 10612  ax-pre-lttri 10613  ax-pre-lttrn 10614  ax-pre-ltadd 10615  ax-pre-mulgt0 10616  ax-pre-sup 10617  ax-addf 10618

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-fal 1550  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-nel 3126  df-ral 3145  df-rex 3146  df-reu 3147  df-rmo 3148  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-int 4879  df-iun 4923  df-disj 5034  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-se 5517  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-isom 6366  df-riota 7116  df-ov 7161  df-oprab 7162  df-mpo 7163  df-of 7411  df-ofr 7412  df-om 7583  df-1st 7691  df-2nd 7692  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-1o 8104  df-2o 8105  df-oadd 8108  df-er 8291  df-map 8410  df-pm 8411  df-en 8512  df-dom 8513  df-sdom 8514  df-fin 8515  df-fi 8877  df-sup 8908  df-inf 8909  df-oi 8976  df-dju 9332  df-card 9370  df-pnf 10679  df-mnf 10680  df-xr 10681  df-ltxr 10682  df-le 10683  df-sub 10874  df-neg 10875  df-div 11300  df-nn 11641  df-2 11703  df-3 11704  df-4 11705  df-n0 11901  df-z 11985  df-uz 12247  df-q 12352  df-rp 12393  df-xneg 12510  df-xadd 12511  df-xmul 12512  df-ioo 12745  df-ico 12747  df-icc 12748  df-fz 12896  df-fzo 13037  df-fl 13165  df-mod 13241  df-seq 13373  df-exp 13433  df-hash 13694  df-cj 14460  df-re 14461  df-im 14462  df-sqrt 14596  df-abs 14597  df-clim 14847  df-sum 15045  df-rest 16698  df-topgen 16719  df-psmet 20539  df-xmet 20540  df-met 20541  df-bl 20542  df-mopn 20543  df-top 21504  df-topon 21521  df-bases 21556  df-cmp 21997  df-ovol 24067  df-vol 24068  df-mbf 24222  df-itg1 24223  df-itg2 24224  df-ibl 24225  df-itg 24226  df-0p 24273

This theorem is referenced by: (None)