Theorem mbfposb 23226

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 2751	. . . . . . . . 9 ⊢ Ⅎ𝑥0
2		nfcv 2751	. . . . . . . . 9 ⊢ Ⅎ𝑥 ≤
3		nffvmpt1 6111	. . . . . . . . 9 ⊢ Ⅎ𝑥((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦)
4	1, 2, 3	nfbr 4629	. . . . . . . 8 ⊢ Ⅎ𝑥0 ≤ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦)
5	4, 3, 1	nfif 4065	. . . . . . 7 ⊢ Ⅎ𝑥if(0 ≤ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦), ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦), 0)
6		nfcv 2751	. . . . . . 7 ⊢ Ⅎ𝑦if(0 ≤ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥), ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥), 0)
7		fveq2 6103	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) = ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))
8	7	breq2d 4595	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (0 ≤ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) ↔ 0 ≤ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)))
9	8, 7	ifbieq1d 4059	. . . . . . 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 4677	. . . . . 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 476	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
12		mbfpos.1	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)
13		eqid 2610	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
14	13	fvmpt2 6200	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ ℝ) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
15	11, 12, 14	syl2anc 691	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
16	15	breq2d 4595	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (0 ≤ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ↔ 0 ≤ 𝐵))
17	16, 15	ifbieq1d 4059	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥), ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥), 0) = if(0 ≤ 𝐵, 𝐵, 0))
18	17	mpteq2dva 4672	. . . . . 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 2656	. . . . 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, 13	fmptd 6292	. . . . . . 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 6267	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) ∧ 𝑦 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) ∈ ℝ)
24		nfcv 2751	. . . . . . . . 9 ⊢ Ⅎ𝑦((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)
25	3, 24, 7	cbvmpt 4677	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 ↦ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦)) = (𝑥 ∈ 𝐴 ↦ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))
26	15	mpteq2dva 4672	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)) = (𝑥 ∈ 𝐴 ↦ 𝐵))
27	25, 26	syl5eq 2656	. . . . . . 7 ⊢ (𝜑 → (𝑦 ∈ 𝐴 ↦ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦)) = (𝑥 ∈ 𝐴 ↦ 𝐵))
28	27	eleq1d 2672	. . . . . 6 ⊢ (𝜑 → ((𝑦 ∈ 𝐴 ↦ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦)) ∈ MblFn ↔ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn))
29	28	biimpar 501	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) → (𝑦 ∈ 𝐴 ↦ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦)) ∈ MblFn)
30	23, 29	mbfpos 23224	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) → (𝑦 ∈ 𝐴 ↦ if(0 ≤ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦), ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦), 0)) ∈ MblFn)
31	20, 30	eqeltrrd 2689	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) → (𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn)
32	3	nfneg 10156	. . . . . . . . 9 ⊢ Ⅎ𝑥-((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦)
33	1, 2, 32	nfbr 4629	. . . . . . . 8 ⊢ Ⅎ𝑥0 ≤ -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦)
34	33, 32, 1	nfif 4065	. . . . . . 7 ⊢ Ⅎ𝑥if(0 ≤ -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦), -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦), 0)
35		nfcv 2751	. . . . . . 7 ⊢ Ⅎ𝑦if(0 ≤ -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥), -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥), 0)
36	7	negeqd 10154	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) = -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))
37	36	breq2d 4595	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (0 ≤ -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) ↔ 0 ≤ -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)))
38	37, 36	ifbieq1d 4059	. . . . . . 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 4677	. . . . . 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 10154	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = -𝐵)
41	40	breq2d 4595	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (0 ≤ -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ↔ 0 ≤ -𝐵))
42	41, 40	ifbieq1d 4059	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥), -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥), 0) = if(0 ≤ -𝐵, -𝐵, 0))
43	42	mpteq2dva 4672	. . . . . 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 2656	. . . . 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 10336	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) ∧ 𝑦 ∈ 𝐴) → -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) ∈ ℝ)
47	23, 29	mbfneg 23223	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) → (𝑦 ∈ 𝐴 ↦ -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦)) ∈ MblFn)
48	46, 47	mbfpos 23224	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) → (𝑦 ∈ 𝐴 ↦ if(0 ≤ -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦), -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦), 0)) ∈ MblFn)
49	45, 48	eqeltrrd 2689	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) → (𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ MblFn)
50	31, 49	jca 553	. 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 6267	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) ∈ ℝ)
53	52	adantlr 747	. . . 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 790	. . . . 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 2688	. . . 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 792	. . . . 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 2688	. . . 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 23225	. . 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 2689	. 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 873	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 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ifcif 4036   class class class wbr 4583   ↦ cmpt 4643  ⟶wf 5800  ‘cfv 5804  ℝcr 9814  0cc0 9815   ≤ cle 9954  -cneg 10146  MblFncmbf 23189

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-q 11665  df-rp 11709  df-xadd 11823  df-ioo 12050  df-ico 12052  df-icc 12053  df-fz 12198  df-fzo 12335  df-fl 12455  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-sum 14265  df-xmet 19560  df-met 19561  df-ovol 23040  df-vol 23041  df-mbf 23194

This theorem is referenced by:  iblre  23366