Theorem mbfposb 24722

Description: A function is measurable iff its positive and negative parts are measurable. (Contributed by Mario Carneiro, 11-Aug-2014.)

Hypothesis

Ref	Expression
mbfpos.1	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)

Assertion

Ref	Expression
mbfposb	⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn ↔ ((𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn ∧ (𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ MblFn)))

Distinct variable groups:   𝑥,𝐴   𝜑,𝑥

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem mbfposb

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfcv 2906	. . . . . . . . 9 ⊢ Ⅎ𝑥0
2		nfcv 2906	. . . . . . . . 9 ⊢ Ⅎ𝑥 ≤
3		nffvmpt1 6767	. . . . . . . . 9 ⊢ Ⅎ𝑥((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦)
4	1, 2, 3	nfbr 5117	. . . . . . . 8 ⊢ Ⅎ𝑥0 ≤ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦)
5	4, 3, 1	nfif 4486	. . . . . . 7 ⊢ Ⅎ𝑥if(0 ≤ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦), ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦), 0)
6		nfcv 2906	. . . . . . 7 ⊢ Ⅎ𝑦if(0 ≤ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥), ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥), 0)
7		fveq2 6756	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) = ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))
8	7	breq2d 5082	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (0 ≤ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) ↔ 0 ≤ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)))
9	8, 7	ifbieq1d 4480	. . . . . . 7 ⊢ (𝑦 = 𝑥 → if(0 ≤ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦), ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦), 0) = if(0 ≤ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥), ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥), 0))
10	5, 6, 9	cbvmpt 5181	. . . . . 6 ⊢ (𝑦 ∈ 𝐴 ↦ if(0 ≤ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦), ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦), 0)) = (𝑥 ∈ 𝐴 ↦ if(0 ≤ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥), ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥), 0))
11		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
12		mbfpos.1	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)
13		eqid 2738	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
14	13	fvmpt2 6868	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ ℝ) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
15	11, 12, 14	syl2anc 583	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
16	15	breq2d 5082	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (0 ≤ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ↔ 0 ≤ 𝐵))
17	16, 15	ifbieq1d 4480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥), ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥), 0) = if(0 ≤ 𝐵, 𝐵, 0))
18	17	mpteq2dva 5170	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(0 ≤ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥), ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥), 0)) = (𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)))
19	10, 18	syl5eq 2791	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ 𝐴 ↦ if(0 ≤ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦), ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦), 0)) = (𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)))
20	19	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) → (𝑦 ∈ 𝐴 ↦ if(0 ≤ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦), ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦), 0)) = (𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)))
21	12	fmpttd 6971	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℝ)
22	21	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℝ)
23	22	ffvelrnda 6943	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) ∧ 𝑦 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) ∈ ℝ)
24		nfcv 2906	. . . . . . . . 9 ⊢ Ⅎ𝑦((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)
25	3, 24, 7	cbvmpt 5181	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 ↦ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦)) = (𝑥 ∈ 𝐴 ↦ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))
26	15	mpteq2dva 5170	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)) = (𝑥 ∈ 𝐴 ↦ 𝐵))
27	25, 26	syl5eq 2791	. . . . . . 7 ⊢ (𝜑 → (𝑦 ∈ 𝐴 ↦ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦)) = (𝑥 ∈ 𝐴 ↦ 𝐵))
28	27	eleq1d 2823	. . . . . 6 ⊢ (𝜑 → ((𝑦 ∈ 𝐴 ↦ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦)) ∈ MblFn ↔ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn))
29	28	biimpar 477	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) → (𝑦 ∈ 𝐴 ↦ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦)) ∈ MblFn)
30	23, 29	mbfpos 24720	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) → (𝑦 ∈ 𝐴 ↦ if(0 ≤ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦), ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦), 0)) ∈ MblFn)
31	20, 30	eqeltrrd 2840	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) → (𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn)
32	3	nfneg 11147	. . . . . . . . 9 ⊢ Ⅎ𝑥-((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦)
33	1, 2, 32	nfbr 5117	. . . . . . . 8 ⊢ Ⅎ𝑥0 ≤ -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦)
34	33, 32, 1	nfif 4486	. . . . . . 7 ⊢ Ⅎ𝑥if(0 ≤ -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦), -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦), 0)
35		nfcv 2906	. . . . . . 7 ⊢ Ⅎ𝑦if(0 ≤ -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥), -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥), 0)
36	7	negeqd 11145	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) = -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))
37	36	breq2d 5082	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (0 ≤ -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) ↔ 0 ≤ -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)))
38	37, 36	ifbieq1d 4480	. . . . . . 7 ⊢ (𝑦 = 𝑥 → if(0 ≤ -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦), -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦), 0) = if(0 ≤ -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥), -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥), 0))
39	34, 35, 38	cbvmpt 5181	. . . . . 6 ⊢ (𝑦 ∈ 𝐴 ↦ if(0 ≤ -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦), -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦), 0)) = (𝑥 ∈ 𝐴 ↦ if(0 ≤ -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥), -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥), 0))
40	15	negeqd 11145	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = -𝐵)
41	40	breq2d 5082	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (0 ≤ -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ↔ 0 ≤ -𝐵))
42	41, 40	ifbieq1d 4480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥), -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥), 0) = if(0 ≤ -𝐵, -𝐵, 0))
43	42	mpteq2dva 5170	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(0 ≤ -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥), -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥), 0)) = (𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)))
44	39, 43	syl5eq 2791	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ 𝐴 ↦ if(0 ≤ -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦), -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦), 0)) = (𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)))
45	44	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) → (𝑦 ∈ 𝐴 ↦ if(0 ≤ -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦), -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦), 0)) = (𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)))
46	23	renegcld 11332	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) ∧ 𝑦 ∈ 𝐴) → -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) ∈ ℝ)
47	23, 29	mbfneg 24719	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) → (𝑦 ∈ 𝐴 ↦ -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦)) ∈ MblFn)
48	46, 47	mbfpos 24720	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) → (𝑦 ∈ 𝐴 ↦ if(0 ≤ -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦), -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦), 0)) ∈ MblFn)
49	45, 48	eqeltrrd 2840	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) → (𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ MblFn)
50	31, 49	jca 511	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) → ((𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn ∧ (𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ MblFn))
51	27	adantr 480	. . 3 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn ∧ (𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ MblFn)) → (𝑦 ∈ 𝐴 ↦ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦)) = (𝑥 ∈ 𝐴 ↦ 𝐵))
52	21	ffvelrnda 6943	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) ∈ ℝ)
53	52	adantlr 711	. . . 4 ⊢ (((𝜑 ∧ ((𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn ∧ (𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ MblFn)) ∧ 𝑦 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) ∈ ℝ)
54	19	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn ∧ (𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ MblFn)) → (𝑦 ∈ 𝐴 ↦ if(0 ≤ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦), ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦), 0)) = (𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)))
55		simprl 767	. . . . 5 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn ∧ (𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ MblFn)) → (𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn)
56	54, 55	eqeltrd 2839	. . . 4 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn ∧ (𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ MblFn)) → (𝑦 ∈ 𝐴 ↦ if(0 ≤ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦), ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦), 0)) ∈ MblFn)
57	44	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn ∧ (𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ MblFn)) → (𝑦 ∈ 𝐴 ↦ if(0 ≤ -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦), -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦), 0)) = (𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)))
58		simprr 769	. . . . 5 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn ∧ (𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ MblFn)) → (𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ MblFn)
59	57, 58	eqeltrd 2839	. . . 4 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn ∧ (𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ MblFn)) → (𝑦 ∈ 𝐴 ↦ if(0 ≤ -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦), -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦), 0)) ∈ MblFn)
60	53, 56, 59	mbfposr 24721	. . 3 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn ∧ (𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ MblFn)) → (𝑦 ∈ 𝐴 ↦ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦)) ∈ MblFn)
61	51, 60	eqeltrrd 2840	. 2 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn ∧ (𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ MblFn)) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn)
62	50, 61	impbida 797	1 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn ↔ ((𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn ∧ (𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ MblFn)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ifcif 4456   class class class wbr 5070   ↦ cmpt 5153  ⟶wf 6414  ‘cfv 6418  ℝcr 10801  0cc0 10802   ≤ cle 10941  -cneg 11136  MblFncmbf 24683

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-inf2 9329  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879  ax-pre-sup 10880

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-se 5536  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-isom 6427  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-of 7511  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-2o 8268  df-er 8456  df-map 8575  df-pm 8576  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-sup 9131  df-inf 9132  df-oi 9199  df-dju 9590  df-card 9628  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-div 11563  df-nn 11904  df-2 11966  df-3 11967  df-n0 12164  df-z 12250  df-uz 12512  df-q 12618  df-rp 12660  df-xadd 12778  df-ioo 13012  df-ico 13014  df-icc 13015  df-fz 13169  df-fzo 13312  df-fl 13440  df-seq 13650  df-exp 13711  df-hash 13973  df-cj 14738  df-re 14739  df-im 14740  df-sqrt 14874  df-abs 14875  df-clim 15125  df-sum 15326  df-xmet 20503  df-met 20504  df-ovol 24533  df-vol 24534  df-mbf 24688

This theorem is referenced by:  iblre  24863