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Theorem i1fibl 25316

Description: A simple function is integrable. (Contributed by Mario Carneiro, 6-Aug-2014.)

**Assertion**
Ref	Expression
i1fibl	⊢ (𝐹 ∈ dom ∫₁ → 𝐹 ∈ 𝐿¹)

Proof of Theorem i1fibl Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		i1ff 25184	. . 3 ⊢ (𝐹 ∈ dom ∫₁ → 𝐹:ℝ⟶ℝ)
2	1	feqmptd 6957	. 2 ⊢ (𝐹 ∈ dom ∫₁ → 𝐹 = (𝑥 ∈ ℝ ↦ (𝐹‘𝑥)))
3		i1fmbf 25183	. . . 4 ⊢ (𝐹 ∈ dom ∫₁ → 𝐹 ∈ MblFn)
4	2, 3	eqeltrrd 2834	. . 3 ⊢ (𝐹 ∈ dom ∫₁ → (𝑥 ∈ ℝ ↦ (𝐹‘𝑥)) ∈ MblFn)
5		simpr 485	. . . . . . . . 9 ⊢ ((𝐹 ∈ dom ∫₁ ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℝ)
6	5	biantrurd 533	. . . . . . . 8 ⊢ ((𝐹 ∈ dom ∫₁ ∧ 𝑥 ∈ ℝ) → (0 ≤ (𝐹‘𝑥) ↔ (𝑥 ∈ ℝ ∧ 0 ≤ (𝐹‘𝑥))))
7	6	ifbid 4550	. . . . . . 7 ⊢ ((𝐹 ∈ dom ∫₁ ∧ 𝑥 ∈ ℝ) → if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0) = if((𝑥 ∈ ℝ ∧ 0 ≤ (𝐹‘𝑥)), (𝐹‘𝑥), 0))
8	7	mpteq2dva 5247	. . . . . 6 ⊢ (𝐹 ∈ dom ∫₁ → (𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0)) = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ ℝ ∧ 0 ≤ (𝐹‘𝑥)), (𝐹‘𝑥), 0)))
9	8	fveq2d 6892	. . . . 5 ⊢ (𝐹 ∈ dom ∫₁ → (∫₂‘(𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0))) = (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ ℝ ∧ 0 ≤ (𝐹‘𝑥)), (𝐹‘𝑥), 0))))
10		eqid 2732	. . . . . . 7 ⊢ (𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0)) = (𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0))
11	10	i1fpos 25215	. . . . . 6 ⊢ (𝐹 ∈ dom ∫₁ → (𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0)) ∈ dom ∫₁)
12		0re 11212	. . . . . . . . . 10 ⊢ 0 ∈ ℝ
13	1	ffvelcdmda 7083	. . . . . . . . . 10 ⊢ ((𝐹 ∈ dom ∫₁ ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) ∈ ℝ)
14		max1 13160	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ ∧ (𝐹‘𝑥) ∈ ℝ) → 0 ≤ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0))
15	12, 13, 14	sylancr 587	. . . . . . . . 9 ⊢ ((𝐹 ∈ dom ∫₁ ∧ 𝑥 ∈ ℝ) → 0 ≤ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0))
16	15	ralrimiva 3146	. . . . . . . 8 ⊢ (𝐹 ∈ dom ∫₁ → ∀𝑥 ∈ ℝ 0 ≤ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0))
17		reex 11197	. . . . . . . . . 10 ⊢ ℝ ∈ V
18	17	a1i 11	. . . . . . . . 9 ⊢ (𝐹 ∈ dom ∫₁ → ℝ ∈ V)
19	12	a1i 11	. . . . . . . . 9 ⊢ ((𝐹 ∈ dom ∫₁ ∧ 𝑥 ∈ ℝ) → 0 ∈ ℝ)
20		fvex 6901	. . . . . . . . . . 11 ⊢ (𝐹‘𝑥) ∈ V
21		c0ex 11204	. . . . . . . . . . 11 ⊢ 0 ∈ V
22	20, 21	ifex 4577	. . . . . . . . . 10 ⊢ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0) ∈ V
23	22	a1i 11	. . . . . . . . 9 ⊢ ((𝐹 ∈ dom ∫₁ ∧ 𝑥 ∈ ℝ) → if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0) ∈ V)
24		fconstmpt 5736	. . . . . . . . . 10 ⊢ (ℝ × {0}) = (𝑥 ∈ ℝ ↦ 0)
25	24	a1i 11	. . . . . . . . 9 ⊢ (𝐹 ∈ dom ∫₁ → (ℝ × {0}) = (𝑥 ∈ ℝ ↦ 0))
26		eqidd 2733	. . . . . . . . 9 ⊢ (𝐹 ∈ dom ∫₁ → (𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0)) = (𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0)))
27	18, 19, 23, 25, 26	ofrfval2 7687	. . . . . . . 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)))
28	16, 27	mpbird 256	. . . . . . 7 ⊢ (𝐹 ∈ dom ∫₁ → (ℝ × {0}) ∘_r ≤ (𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0)))
29		ax-resscn 11163	. . . . . . . . 9 ⊢ ℝ ⊆ ℂ
30	29	a1i 11	. . . . . . . 8 ⊢ (𝐹 ∈ dom ∫₁ → ℝ ⊆ ℂ)
31	22, 10	fnmpti 6690	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0)) Fn ℝ
32	31	a1i 11	. . . . . . . 8 ⊢ (𝐹 ∈ dom ∫₁ → (𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0)) Fn ℝ)
33	30, 32	0pledm 25181	. . . . . . 7 ⊢ (𝐹 ∈ dom ∫₁ → (0_𝑝 ∘_r ≤ (𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0)) ↔ (ℝ × {0}) ∘_r ≤ (𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0))))
34	28, 33	mpbird 256	. . . . . 6 ⊢ (𝐹 ∈ dom ∫₁ → 0_𝑝 ∘_r ≤ (𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0)))
35		itg2itg1 25245	. . . . . 6 ⊢ (((𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0)) ∈ dom ∫₁ ∧ 0_𝑝 ∘_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))))
36	11, 34, 35	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))))
37	9, 36	eqtr3d 2774	. . . 4 ⊢ (𝐹 ∈ dom ∫₁ → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ ℝ ∧ 0 ≤ (𝐹‘𝑥)), (𝐹‘𝑥), 0))) = (∫₁‘(𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0))))
38		itg1cl 25193	. . . . 5 ⊢ ((𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0)) ∈ dom ∫₁ → (∫₁‘(𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0))) ∈ ℝ)
39	11, 38	syl 17	. . . 4 ⊢ (𝐹 ∈ dom ∫₁ → (∫₁‘(𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0))) ∈ ℝ)
40	37, 39	eqeltrd 2833	. . 3 ⊢ (𝐹 ∈ dom ∫₁ → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ ℝ ∧ 0 ≤ (𝐹‘𝑥)), (𝐹‘𝑥), 0))) ∈ ℝ)
41	5	biantrurd 533	. . . . . . . 8 ⊢ ((𝐹 ∈ dom ∫₁ ∧ 𝑥 ∈ ℝ) → (0 ≤ -(𝐹‘𝑥) ↔ (𝑥 ∈ ℝ ∧ 0 ≤ -(𝐹‘𝑥))))
42	41	ifbid 4550	. . . . . . 7 ⊢ ((𝐹 ∈ dom ∫₁ ∧ 𝑥 ∈ ℝ) → if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0) = if((𝑥 ∈ ℝ ∧ 0 ≤ -(𝐹‘𝑥)), -(𝐹‘𝑥), 0))
43	42	mpteq2dva 5247	. . . . . 6 ⊢ (𝐹 ∈ dom ∫₁ → (𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0)) = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ ℝ ∧ 0 ≤ -(𝐹‘𝑥)), -(𝐹‘𝑥), 0)))
44	43	fveq2d 6892	. . . . 5 ⊢ (𝐹 ∈ dom ∫₁ → (∫₂‘(𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0))) = (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ ℝ ∧ 0 ≤ -(𝐹‘𝑥)), -(𝐹‘𝑥), 0))))
45		neg1rr 12323	. . . . . . . . . . 11 ⊢ -1 ∈ ℝ
46	45	a1i 11	. . . . . . . . . 10 ⊢ ((𝐹 ∈ dom ∫₁ ∧ 𝑥 ∈ ℝ) → -1 ∈ ℝ)
47		fconstmpt 5736	. . . . . . . . . . 11 ⊢ (ℝ × {-1}) = (𝑥 ∈ ℝ ↦ -1)
48	47	a1i 11	. . . . . . . . . 10 ⊢ (𝐹 ∈ dom ∫₁ → (ℝ × {-1}) = (𝑥 ∈ ℝ ↦ -1))
49	18, 46, 13, 48, 2	offval2 7686	. . . . . . . . 9 ⊢ (𝐹 ∈ dom ∫₁ → ((ℝ × {-1}) ∘_f · 𝐹) = (𝑥 ∈ ℝ ↦ (-1 · (𝐹‘𝑥))))
50	13	recnd 11238	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ dom ∫₁ ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) ∈ ℂ)
51	50	mulm1d 11662	. . . . . . . . . 10 ⊢ ((𝐹 ∈ dom ∫₁ ∧ 𝑥 ∈ ℝ) → (-1 · (𝐹‘𝑥)) = -(𝐹‘𝑥))
52	51	mpteq2dva 5247	. . . . . . . . 9 ⊢ (𝐹 ∈ dom ∫₁ → (𝑥 ∈ ℝ ↦ (-1 · (𝐹‘𝑥))) = (𝑥 ∈ ℝ ↦ -(𝐹‘𝑥)))
53	49, 52	eqtrd 2772	. . . . . . . 8 ⊢ (𝐹 ∈ dom ∫₁ → ((ℝ × {-1}) ∘_f · 𝐹) = (𝑥 ∈ ℝ ↦ -(𝐹‘𝑥)))
54		id 22	. . . . . . . . 9 ⊢ (𝐹 ∈ dom ∫₁ → 𝐹 ∈ dom ∫₁)
55	45	a1i 11	. . . . . . . . 9 ⊢ (𝐹 ∈ dom ∫₁ → -1 ∈ ℝ)
56	54, 55	i1fmulc 25212	. . . . . . . 8 ⊢ (𝐹 ∈ dom ∫₁ → ((ℝ × {-1}) ∘_f · 𝐹) ∈ dom ∫₁)
57	53, 56	eqeltrrd 2834	. . . . . . 7 ⊢ (𝐹 ∈ dom ∫₁ → (𝑥 ∈ ℝ ↦ -(𝐹‘𝑥)) ∈ dom ∫₁)
58	57	i1fposd 25216	. . . . . 6 ⊢ (𝐹 ∈ dom ∫₁ → (𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0)) ∈ dom ∫₁)
59	13	renegcld 11637	. . . . . . . . . 10 ⊢ ((𝐹 ∈ dom ∫₁ ∧ 𝑥 ∈ ℝ) → -(𝐹‘𝑥) ∈ ℝ)
60		max1 13160	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ ∧ -(𝐹‘𝑥) ∈ ℝ) → 0 ≤ if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0))
61	12, 59, 60	sylancr 587	. . . . . . . . 9 ⊢ ((𝐹 ∈ dom ∫₁ ∧ 𝑥 ∈ ℝ) → 0 ≤ if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0))
62	61	ralrimiva 3146	. . . . . . . 8 ⊢ (𝐹 ∈ dom ∫₁ → ∀𝑥 ∈ ℝ 0 ≤ if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0))
63		negex 11454	. . . . . . . . . . 11 ⊢ -(𝐹‘𝑥) ∈ V
64	63, 21	ifex 4577	. . . . . . . . . 10 ⊢ if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0) ∈ V
65	64	a1i 11	. . . . . . . . 9 ⊢ ((𝐹 ∈ dom ∫₁ ∧ 𝑥 ∈ ℝ) → if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0) ∈ V)
66		eqidd 2733	. . . . . . . . 9 ⊢ (𝐹 ∈ dom ∫₁ → (𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0)) = (𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0)))
67	18, 19, 65, 25, 66	ofrfval2 7687	. . . . . . . 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)))
68	62, 67	mpbird 256	. . . . . . 7 ⊢ (𝐹 ∈ dom ∫₁ → (ℝ × {0}) ∘_r ≤ (𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0)))
69		eqid 2732	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0)) = (𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0))
70	64, 69	fnmpti 6690	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0)) Fn ℝ
71	70	a1i 11	. . . . . . . 8 ⊢ (𝐹 ∈ dom ∫₁ → (𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0)) Fn ℝ)
72	30, 71	0pledm 25181	. . . . . . 7 ⊢ (𝐹 ∈ dom ∫₁ → (0_𝑝 ∘_r ≤ (𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0)) ↔ (ℝ × {0}) ∘_r ≤ (𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0))))
73	68, 72	mpbird 256	. . . . . 6 ⊢ (𝐹 ∈ dom ∫₁ → 0_𝑝 ∘_r ≤ (𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0)))
74		itg2itg1 25245	. . . . . 6 ⊢ (((𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0)) ∈ dom ∫₁ ∧ 0_𝑝 ∘_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))))
75	58, 73, 74	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))))
76	44, 75	eqtr3d 2774	. . . 4 ⊢ (𝐹 ∈ dom ∫₁ → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ ℝ ∧ 0 ≤ -(𝐹‘𝑥)), -(𝐹‘𝑥), 0))) = (∫₁‘(𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0))))
77		itg1cl 25193	. . . . 5 ⊢ ((𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0)) ∈ dom ∫₁ → (∫₁‘(𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0))) ∈ ℝ)
78	58, 77	syl 17	. . . 4 ⊢ (𝐹 ∈ dom ∫₁ → (∫₁‘(𝑥 ∈ ℝ ↦ if(0 ≤ -(𝐹‘𝑥), -(𝐹‘𝑥), 0))) ∈ ℝ)
79	76, 78	eqeltrd 2833	. . 3 ⊢ (𝐹 ∈ dom ∫₁ → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ ℝ ∧ 0 ≤ -(𝐹‘𝑥)), -(𝐹‘𝑥), 0))) ∈ ℝ)
80	13	iblrelem 25299	. . 3 ⊢ (𝐹 ∈ dom ∫₁ → ((𝑥 ∈ ℝ ↦ (𝐹‘𝑥)) ∈ 𝐿¹ ↔ ((𝑥 ∈ ℝ ↦ (𝐹‘𝑥)) ∈ MblFn ∧ (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ ℝ ∧ 0 ≤ (𝐹‘𝑥)), (𝐹‘𝑥), 0))) ∈ ℝ ∧ (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ ℝ ∧ 0 ≤ -(𝐹‘𝑥)), -(𝐹‘𝑥), 0))) ∈ ℝ)))
81	4, 40, 79, 80	mpbir3and 1342	. 2 ⊢ (𝐹 ∈ dom ∫₁ → (𝑥 ∈ ℝ ↦ (𝐹‘𝑥)) ∈ 𝐿¹)
82	2, 81	eqeltrd 2833	1 ⊢ (𝐹 ∈ dom ∫₁ → 𝐹 ∈ 𝐿¹)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3061 Vcvv 3474 ⊆ wss 3947 ifcif 4527 {csn 4627 class class class wbr 5147 ↦ cmpt 5230 × cxp 5673 dom cdm 5675 Fn wfn 6535 ‘cfv 6540 (class class class)co 7405 ∘_f cof 7664 ∘_r cofr 7665 ℂcc 11104 ℝcr 11105 0cc0 11106 1c1 11107 · cmul 11111 ≤ cle 11245 -cneg 11441 MblFncmbf 25122 ∫₁citg1 25123 ∫₂citg2 25124 𝐿¹cibl 25125 0_𝑝c0p 25177

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-inf2 9632 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-disj 5113 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7666 df-ofr 7667 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-er 8699 df-map 8818 df-pm 8819 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fi 9402 df-sup 9433 df-inf 9434 df-oi 9501 df-dju 9892 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-n0 12469 df-z 12555 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-ioo 13324 df-ico 13326 df-icc 13327 df-fz 13481 df-fzo 13624 df-fl 13753 df-seq 13963 df-exp 14024 df-hash 14287 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-clim 15428 df-sum 15629 df-rest 17364 df-topgen 17385 df-psmet 20928 df-xmet 20929 df-met 20930 df-bl 20931 df-mopn 20932 df-top 22387 df-topon 22404 df-bases 22440 df-cmp 22882 df-ovol 24972 df-vol 24973 df-mbf 25127 df-itg1 25128 df-itg2 25129 df-ibl 25130 df-0p 25178

This theorem is referenced by: itgitg1 25317 ftc1anclem4 36552

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