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Theorem iblss2 24093

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 24055	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝐿¹ → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ MblFn)
7	5, 6	syl 17	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ MblFn)
8	1, 2, 3, 4, 7	mbfss 23934	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ 𝐶) ∈ MblFn)
9	1	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → 𝐴 ⊆ 𝐵)
10	9	sselda 3895	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐵)
11	10	iftrued 4395	. . . . . . . . 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 4393	. . . . . . . . . 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 482	. . . . . . . . 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 2836	. . . . . . . 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 4426	. . . . . . . . 9 ⊢ if(𝑥 ∈ 𝐵, 0, 0) = 0
16		simplll 771	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ ¬ 𝑥 ∈ 𝐴) ∧ 𝑥 ∈ 𝐵) → 𝜑)
17		simpr 485	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ ¬ 𝑥 ∈ 𝐴) ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
18		simplr 765	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ ¬ 𝑥 ∈ 𝐴) ∧ 𝑥 ∈ 𝐵) → ¬ 𝑥 ∈ 𝐴)
19	17, 18	eldifd 3876	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ ¬ 𝑥 ∈ 𝐴) ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ (𝐵 ∖ 𝐴))
20	16, 19, 4	syl2anc 584	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ ¬ 𝑥 ∈ 𝐴) ∧ 𝑥 ∈ 𝐵) → 𝐶 = 0)
21	20	oveq1d 7038	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ ¬ 𝑥 ∈ 𝐴) ∧ 𝑥 ∈ 𝐵) → (𝐶 / (i↑𝑘)) = (0 / (i↑𝑘)))
22		simpllr 772	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ ¬ 𝑥 ∈ 𝐴) ∧ 𝑥 ∈ 𝐵) → 𝑘 ∈ (0...3))
23		elfzelz 12762	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (0...3) → 𝑘 ∈ ℤ)
24		ax-icn 10449	. . . . . . . . . . . . . . . . 17 ⊢ i ∈ ℂ
25		ine0 10929	. . . . . . . . . . . . . . . . 17 ⊢ i ≠ 0
26		expclz 13308	. . . . . . . . . . . . . . . . . 18 ⊢ ((i ∈ ℂ ∧ i ≠ 0 ∧ 𝑘 ∈ ℤ) → (i↑𝑘) ∈ ℂ)
27		expne0i 13315	. . . . . . . . . . . . . . . . . 18 ⊢ ((i ∈ ℂ ∧ i ≠ 0 ∧ 𝑘 ∈ ℤ) → (i↑𝑘) ≠ 0)
28	26, 27	div0d 11269	. . . . . . . . . . . . . . . . 17 ⊢ ((i ∈ ℂ ∧ i ≠ 0 ∧ 𝑘 ∈ ℤ) → (0 / (i↑𝑘)) = 0)
29	24, 25, 28	mp3an12 1443	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ ℤ → (0 / (i↑𝑘)) = 0)
30	22, 23, 29	3syl 18	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ ¬ 𝑥 ∈ 𝐴) ∧ 𝑥 ∈ 𝐵) → (0 / (i↑𝑘)) = 0)
31	21, 30	eqtrd 2833	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ ¬ 𝑥 ∈ 𝐴) ∧ 𝑥 ∈ 𝐵) → (𝐶 / (i↑𝑘)) = 0)
32	31	fveq2d 6549	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ ¬ 𝑥 ∈ 𝐴) ∧ 𝑥 ∈ 𝐵) → (ℜ‘(𝐶 / (i↑𝑘))) = (ℜ‘0))
33		re0 14349	. . . . . . . . . . . . 13 ⊢ (ℜ‘0) = 0
34	32, 33	syl6eq 2849	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ ¬ 𝑥 ∈ 𝐴) ∧ 𝑥 ∈ 𝐵) → (ℜ‘(𝐶 / (i↑𝑘))) = 0)
35	34	ifeq1d 4405	. . . . . . . . . . 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 4426	. . . . . . . . . . 11 ⊢ if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), 0, 0) = 0
37	35, 36	syl6eq 2849	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ ¬ 𝑥 ∈ 𝐴) ∧ 𝑥 ∈ 𝐵) → if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0) = 0)
38	37	ifeq1da 4417	. . . . . . . . 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 4396	. . . . . . . . . 10 ⊢ (¬ 𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0) = 0)
40	39	adantl 482	. . . . . . . . 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 2859	. . . . . . . 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 809	. . . . . . 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 4438	. . . . . . 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 4438	. . . . . . 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 2858	. . . . . 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 5063	. . . . 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 6549	. . . 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 2798	. . . . . 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 2798	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘(𝐶 / (i↑𝑘))) = (ℜ‘(𝐶 / (i↑𝑘))))
50	48, 49, 5, 3	iblitg 24056	. . . . 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 592	. . . 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 2885	. . 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 3151	. 2 ⊢ (𝜑 → ∀𝑘 ∈ (0...3)(∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) ∈ ℝ)
54		eqidd 2798	. . 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 2798	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (ℜ‘(𝐶 / (i↑𝑘))) = (ℜ‘(𝐶 / (i↑𝑘))))
56		elun 4052	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∪ (𝐵 ∖ 𝐴)) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ (𝐵 ∖ 𝐴)))
57		undif2 4345	. . . . . . . 8 ⊢ (𝐴 ∪ (𝐵 ∖ 𝐴)) = (𝐴 ∪ 𝐵)
58		ssequn1 4083	. . . . . . . . 9 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∪ 𝐵) = 𝐵)
59	1, 58	sylib 219	. . . . . . . 8 ⊢ (𝜑 → (𝐴 ∪ 𝐵) = 𝐵)
60	57, 59	syl5eq 2845	. . . . . . 7 ⊢ (𝜑 → (𝐴 ∪ (𝐵 ∖ 𝐴)) = 𝐵)
61	60	eleq2d 2870	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∪ (𝐵 ∖ 𝐴)) ↔ 𝑥 ∈ 𝐵))
62	56, 61	syl5bbr 286	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ (𝐵 ∖ 𝐴)) ↔ 𝑥 ∈ 𝐵))
63	62	biimpar 478	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ (𝐵 ∖ 𝐴)))
64	7, 3	mbfmptcl 23924	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℂ)
65		0cn 10486	. . . . . 6 ⊢ 0 ∈ ℂ
66	4, 65	syl6eqel 2893	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ 𝐴)) → 𝐶 ∈ ℂ)
67	64, 66	jaodan 952	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ (𝐵 ∖ 𝐴))) → 𝐶 ∈ ℂ)
68	63, 67	syldan 591	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ ℂ)
69	54, 55, 68	isibl2 24054	. 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 709	1 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ 𝐶) ∈ 𝐿¹)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∨ wo 842 ∧ w3a 1080 = wceq 1525 ∈ wcel 2083 ≠ wne 2986 ∀wral 3107 ∖ cdif 3862 ∪ cun 3863 ⊆ wss 3865 ifcif 4387 class class class wbr 4968 ↦ cmpt 5047 dom cdm 5450 ‘cfv 6232 (class class class)co 7023 ℂcc 10388 ℝcr 10389 0cc0 10390 ici 10392 ≤ cle 10529 / cdiv 11151 3c3 11547 ℤcz 11835 ...cfz 12746 ↑cexp 13283 ℜcre 14294 volcvol 23751 MblFncmbf 23902 ∫₂citg2 23904 𝐿¹cibl 23905

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-rep 5088 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 ax-un 7326 ax-inf2 8957 ax-cnex 10446 ax-resscn 10447 ax-1cn 10448 ax-icn 10449 ax-addcl 10450 ax-addrcl 10451 ax-mulcl 10452 ax-mulrcl 10453 ax-mulcom 10454 ax-addass 10455 ax-mulass 10456 ax-distr 10457 ax-i2m1 10458 ax-1ne0 10459 ax-1rid 10460 ax-rnegex 10461 ax-rrecex 10462 ax-cnre 10463 ax-pre-lttri 10464 ax-pre-lttrn 10465 ax-pre-ltadd 10466 ax-pre-mulgt0 10467 ax-pre-sup 10468

This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1528 df-fal 1538 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-nel 3093 df-ral 3112 df-rex 3113 df-reu 3114 df-rmo 3115 df-rab 3116 df-v 3442 df-sbc 3712 df-csb 3818 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-pss 3882 df-nul 4218 df-if 4388 df-pw 4461 df-sn 4479 df-pr 4481 df-tp 4483 df-op 4485 df-uni 4752 df-int 4789 df-iun 4833 df-br 4969 df-opab 5031 df-mpt 5048 df-tr 5071 df-id 5355 df-eprel 5360 df-po 5369 df-so 5370 df-fr 5409 df-se 5410 df-we 5411 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-pred 6030 df-ord 6076 df-on 6077 df-lim 6078 df-suc 6079 df-iota 6196 df-fun 6234 df-fn 6235 df-f 6236 df-f1 6237 df-fo 6238 df-f1o 6239 df-fv 6240 df-isom 6241 df-riota 6984 df-ov 7026 df-oprab 7027 df-mpo 7028 df-of 7274 df-om 7444 df-1st 7552 df-2nd 7553 df-wrecs 7805 df-recs 7867 df-rdg 7905 df-1o 7960 df-2o 7961 df-oadd 7964 df-er 8146 df-map 8265 df-pm 8266 df-en 8365 df-dom 8366 df-sdom 8367 df-fin 8368 df-sup 8759 df-inf 8760 df-oi 8827 df-dju 9183 df-card 9221 df-pnf 10530 df-mnf 10531 df-xr 10532 df-ltxr 10533 df-le 10534 df-sub 10725 df-neg 10726 df-div 11152 df-nn 11493 df-2 11554 df-3 11555 df-4 11556 df-n0 11752 df-z 11836 df-uz 12098 df-q 12202 df-rp 12244 df-xadd 12362 df-ioo 12596 df-ico 12598 df-icc 12599 df-fz 12747 df-fzo 12888 df-fl 13016 df-mod 13092 df-seq 13224 df-exp 13284 df-hash 13545 df-cj 14296 df-re 14297 df-im 14298 df-sqrt 14432 df-abs 14433 df-clim 14683 df-sum 14881 df-xmet 20224 df-met 20225 df-ovol 23752 df-vol 23753 df-mbf 23907 df-ibl 23910

This theorem is referenced by: itgss3 24102 itgless 24104 ftc1anclem5 34523 ftc1anclem6 34524 areacirc 34539 arearect 39328 areaquad 39329

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