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Theorem iblss2 26034

Description: Change the domain of an integrability predicate. (Contributed by Mario Carneiro, 13-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)

**Hypotheses**
Ref	Expression
iblss2.1	⊢ (𝜑 → 𝐴 ⊆ 𝐵)
iblss2.2	⊢ (𝜑 → 𝐵 ∈ dom vol)
iblss2.3	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑉)
iblss2.4	⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ 𝐴)) → 𝐶 = 0)
iblss2.5	⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝐿¹)

**Assertion**
Ref	Expression
iblss2	⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ 𝐶) ∈ 𝐿¹)

Distinct variable groups: 𝑥,𝐴 𝑥,𝐵 𝜑,𝑥 𝑥,𝑉 Allowed substitution hint: 𝐶(𝑥)

Proof of Theorem iblss2 Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iblss2.1	. . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
2		iblss2.2	. . 3 ⊢ (𝜑 → 𝐵 ∈ dom vol)
3		iblss2.3	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑉)
4		iblss2.4	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ 𝐴)) → 𝐶 = 0)
5		iblss2.5	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝐿¹)
6		iblmbf 25996	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝐿¹ → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ MblFn)
7	5, 6	syl 18	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ MblFn)
8	1, 2, 3, 4, 7	mbfss 25875	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ 𝐶) ∈ MblFn)
9	1	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → 𝐴 ⊆ 𝐵)
10	9	sselda 3931	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐵)
11	10	iftrued 4490	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → if(𝑥 ∈ 𝐵, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0) = if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0))
12		iftrue 4488	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0) = if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0))
13	12	adantl 487	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → if(𝑥 ∈ 𝐴, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0) = if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0))
14	11, 13	eqtr4d 2798	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → if(𝑥 ∈ 𝐵, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0) = if(𝑥 ∈ 𝐴, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0))
15		ifid 4523	. . . . . . . . 9 ⊢ if(𝑥 ∈ 𝐵, 0, 0) = 0
16		simplll 787	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ ¬ 𝑥 ∈ 𝐴) ∧ 𝑥 ∈ 𝐵) → 𝜑)
17		simpr 490	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ ¬ 𝑥 ∈ 𝐴) ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
18		simplr 781	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ ¬ 𝑥 ∈ 𝐴) ∧ 𝑥 ∈ 𝐵) → ¬ 𝑥 ∈ 𝐴)
19	17, 18	eldifd 3910	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ ¬ 𝑥 ∈ 𝐴) ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ (𝐵 ∖ 𝐴))
20	16, 19, 4	syl2anc 596	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ ¬ 𝑥 ∈ 𝐴) ∧ 𝑥 ∈ 𝐵) → 𝐶 = 0)
21	20	oveq1d 7429	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ ¬ 𝑥 ∈ 𝐴) ∧ 𝑥 ∈ 𝐵) → (𝐶 / (i↑𝑘)) = (0 / (i↑𝑘)))
22		simpllr 788	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ ¬ 𝑥 ∈ 𝐴) ∧ 𝑥 ∈ 𝐵) → 𝑘 ∈ (0...3))
23		elfzelz 13579	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (0...3) → 𝑘 ∈ ℤ)
24		ax-icn 11184	. . . . . . . . . . . . . . . . 17 ⊢ i ∈ ℂ
25		ine0 11674	. . . . . . . . . . . . . . . . 17 ⊢ i ≠ 0
26		expclz 14149	. . . . . . . . . . . . . . . . . 18 ⊢ ((i ∈ ℂ ∧ i ≠ 0 ∧ 𝑘 ∈ ℤ) → (i↑𝑘) ∈ ℂ)
27		expne0i 14159	. . . . . . . . . . . . . . . . . 18 ⊢ ((i ∈ ℂ ∧ i ≠ 0 ∧ 𝑘 ∈ ℤ) → (i↑𝑘) ≠ 0)
28	26, 27	div0d 12015	. . . . . . . . . . . . . . . . 17 ⊢ ((i ∈ ℂ ∧ i ≠ 0 ∧ 𝑘 ∈ ℤ) → (0 / (i↑𝑘)) = 0)
29	24, 25, 28	mp3an12 1480	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ ℤ → (0 / (i↑𝑘)) = 0)
30	22, 23, 29	3syl 19	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ ¬ 𝑥 ∈ 𝐴) ∧ 𝑥 ∈ 𝐵) → (0 / (i↑𝑘)) = 0)
31	21, 30	eqtrd 2795	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ ¬ 𝑥 ∈ 𝐴) ∧ 𝑥 ∈ 𝐵) → (𝐶 / (i↑𝑘)) = 0)
32	31	fveq2d 6883	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ ¬ 𝑥 ∈ 𝐴) ∧ 𝑥 ∈ 𝐵) → (ℜ‘(𝐶 / (i↑𝑘))) = (ℜ‘0))
33		re0 15240	. . . . . . . . . . . . 13 ⊢ (ℜ‘0) = 0
34	32, 33	eqtrdi 2811	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ ¬ 𝑥 ∈ 𝐴) ∧ 𝑥 ∈ 𝐵) → (ℜ‘(𝐶 / (i↑𝑘))) = 0)
35	34	ifeq1d 4502	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ ¬ 𝑥 ∈ 𝐴) ∧ 𝑥 ∈ 𝐵) → if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0) = if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), 0, 0))
36		ifid 4523	. . . . . . . . . . 11 ⊢ if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), 0, 0) = 0
37	35, 36	eqtrdi 2811	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ ¬ 𝑥 ∈ 𝐴) ∧ 𝑥 ∈ 𝐵) → if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0) = 0)
38	37	ifeq1da 4514	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ ¬ 𝑥 ∈ 𝐴) → if(𝑥 ∈ 𝐵, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0) = if(𝑥 ∈ 𝐵, 0, 0))
39		iffalse 4491	. . . . . . . . . 10 ⊢ (¬ 𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0) = 0)
40	39	adantl 487	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ ¬ 𝑥 ∈ 𝐴) → if(𝑥 ∈ 𝐴, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0) = 0)
41	15, 38, 40	3eqtr4a 2821	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ ¬ 𝑥 ∈ 𝐴) → if(𝑥 ∈ 𝐵, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0) = if(𝑥 ∈ 𝐴, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0))
42	14, 41	pm2.61dan 825	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → if(𝑥 ∈ 𝐵, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0) = if(𝑥 ∈ 𝐴, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0))
43		ifan 4536	. . . . . . 7 ⊢ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0) = if(𝑥 ∈ 𝐵, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0)
44		ifan 4536	. . . . . . 7 ⊢ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0) = if(𝑥 ∈ 𝐴, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0)
45	42, 43, 44	3eqtr4g 2820	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0) = if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))
46	45	mpteq2dv 5199	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)) = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)))
47	46	fveq2d 6883	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) = (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))))
48		eqidd 2761	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)) = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)))
49		eqidd 2761	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘(𝐶 / (i↑𝑘))) = (ℜ‘(𝐶 / (i↑𝑘))))
50	48, 49, 5, 3	iblitg 25997	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℤ) → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) ∈ ℝ)
51	23, 50	sylan2 605	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) ∈ ℝ)
52	47, 51	eqeltrd 2860	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) ∈ ℝ)
53	52	ralrimiva 3154	. 2 ⊢ (𝜑 → ∀𝑘 ∈ (0...3)(∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) ∈ ℝ)
54		eqidd 2761	. . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)) = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)))
55		eqidd 2761	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (ℜ‘(𝐶 / (i↑𝑘))) = (ℜ‘(𝐶 / (i↑𝑘))))
56		elun 4100	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∪ (𝐵 ∖ 𝐴)) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ (𝐵 ∖ 𝐴)))
57		undif2 4431	. . . . . . . 8 ⊢ (𝐴 ∪ (𝐵 ∖ 𝐴)) = (𝐴 ∪ 𝐵)
58		ssequn1 4132	. . . . . . . . 9 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∪ 𝐵) = 𝐵)
59	1, 58	sylib 221	. . . . . . . 8 ⊢ (𝜑 → (𝐴 ∪ 𝐵) = 𝐵)
60	57, 59	eqtrid 2807	. . . . . . 7 ⊢ (𝜑 → (𝐴 ∪ (𝐵 ∖ 𝐴)) = 𝐵)
61	60	eleq2d 2846	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∪ (𝐵 ∖ 𝐴)) ↔ 𝑥 ∈ 𝐵))
62	56, 61	bitr3id 288	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ (𝐵 ∖ 𝐴)) ↔ 𝑥 ∈ 𝐵))
63	62	biimpar 483	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ (𝐵 ∖ 𝐴)))
64	7, 3	mbfmptcl 25865	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℂ)
65		0cn 11223	. . . . . 6 ⊢ 0 ∈ ℂ
66	4, 65	eqeltrdi 2868	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ 𝐴)) → 𝐶 ∈ ℂ)
67	64, 66	jaodan 972	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ (𝐵 ∖ 𝐴))) → 𝐶 ∈ ℂ)
68	63, 67	syldan 603	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ ℂ)
69	54, 55, 68	isibl2 25995	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ↦ 𝐶) ∈ 𝐿¹ ↔ ((𝑥 ∈ 𝐵 ↦ 𝐶) ∈ MblFn ∧ ∀𝑘 ∈ (0...3)(∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) ∈ ℝ)))
70	8, 53, 69	mpbir2and 726	1 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ 𝐶) ∈ 𝐿¹)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 ∨ wo 861 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ≠ wne 2955 ∀wral 3076 ∖ cdif 3896 ∪ cun 3897 ⊆ wss 3899 ifcif 4482 class class class wbr 5103 ↦ cmpt 5186 dom cdm 5655 ‘cfv 6533 (class class class)co 7414 ℂcc 11123 ℝcr 11124 0cc0 11125 ici 11127 ≤ cle 11269 / cdiv 11896 3c3 12321 ℤcz 12616 ...cfz 13562 ↑cexp 14126 ℜcre 15185 volcvol 25692 MblFncmbf 25843 ∫₂citg2 25845 𝐿¹cibl 25846

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7737 ax-inf2 9621 ax-cnex 11181 ax-resscn 11182 ax-1cn 11183 ax-icn 11184 ax-addcl 11185 ax-addrcl 11186 ax-mulcl 11187 ax-mulrcl 11188 ax-mulcom 11189 ax-addass 11190 ax-mulass 11191 ax-distr 11192 ax-i2m1 11193 ax-1ne0 11194 ax-1rid 11195 ax-rnegex 11196 ax-rrecex 11197 ax-cnre 11198 ax-pre-lttri 11199 ax-pre-lttrn 11200 ax-pre-ltadd 11201 ax-pre-mulgt0 11202 ax-pre-sup 11203

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7371 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7679 df-om 7864 df-1st 7987 df-2nd 7988 df-frecs 8281 df-wrecs 8312 df-recs 8361 df-rdg 8400 df-1o 8456 df-2o 8457 df-er 8697 df-map 8829 df-pm 8830 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-sup 9413 df-inf 9414 df-oi 9483 df-dju 9907 df-card 9945 df-pnf 11270 df-mnf 11271 df-xr 11272 df-ltxr 11273 df-le 11274 df-sub 11468 df-neg 11469 df-div 11897 df-nn 12259 df-2 12328 df-3 12329 df-4 12330 df-n0 12530 df-z 12617 df-uz 12889 df-q 12999 df-rp 13044 df-xadd 13165 df-ioo 13403 df-ico 13405 df-icc 13406 df-fz 13563 df-fzo 13711 df-fl 13854 df-mod 13932 df-seq 14067 df-exp 14127 df-hash 14396 df-cj 15187 df-re 15188 df-im 15189 df-sqrt 15323 df-abs 15324 df-clim 15576 df-sum 15775 df-xmet 21579 df-met 21580 df-ovol 25693 df-vol 25694 df-mbf 25848 df-ibl 25851

This theorem is used by: itgss3 26043 itgless 26045 ftc1anclem5 38447 ftc1anclem6 38448 areacirc 38463 arearect 44057 areaquad 44058

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