Theorem mbfposb 25571

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 2891	. . . . . . . . 9 ⊢ Ⅎ𝑥0
2		nfcv 2891	. . . . . . . . 9 ⊢ Ⅎ𝑥 ≤
3		nffvmpt1 6837	. . . . . . . . 9 ⊢ Ⅎ𝑥((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦)
4	1, 2, 3	nfbr 5142	. . . . . . . 8 ⊢ Ⅎ𝑥0 ≤ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦)
5	4, 3, 1	nfif 4509	. . . . . . 7 ⊢ Ⅎ𝑥if(0 ≤ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦), ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦), 0)
6		nfcv 2891	. . . . . . 7 ⊢ Ⅎ𝑦if(0 ≤ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥), ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥), 0)
7		fveq2 6826	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) = ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))
8	7	breq2d 5107	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (0 ≤ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) ↔ 0 ≤ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)))
9	8, 7	ifbieq1d 4503	. . . . . . 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 5197	. . . . . 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 2729	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
14	13	fvmpt2 6945	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ ℝ) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
15	11, 12, 14	syl2anc 584	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
16	15	breq2d 5107	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (0 ≤ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ↔ 0 ≤ 𝐵))
17	16, 15	ifbieq1d 4503	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥), ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥), 0) = if(0 ≤ 𝐵, 𝐵, 0))
18	17	mpteq2dva 5188	. . . . . 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	eqtrid 2776	. . . . 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 7053	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℝ)
22	21	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℝ)
23	22	ffvelcdmda 7022	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) ∧ 𝑦 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) ∈ ℝ)
24		nfcv 2891	. . . . . . . . 9 ⊢ Ⅎ𝑦((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)
25	3, 24, 7	cbvmpt 5197	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 ↦ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦)) = (𝑥 ∈ 𝐴 ↦ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))
26	15	mpteq2dva 5188	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)) = (𝑥 ∈ 𝐴 ↦ 𝐵))
27	25, 26	eqtrid 2776	. . . . . . 7 ⊢ (𝜑 → (𝑦 ∈ 𝐴 ↦ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦)) = (𝑥 ∈ 𝐴 ↦ 𝐵))
28	27	eleq1d 2813	. . . . . 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 25569	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) → (𝑦 ∈ 𝐴 ↦ if(0 ≤ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦), ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦), 0)) ∈ MblFn)
31	20, 30	eqeltrrd 2829	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) → (𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn)
32	3	nfneg 11378	. . . . . . . . 9 ⊢ Ⅎ𝑥-((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦)
33	1, 2, 32	nfbr 5142	. . . . . . . 8 ⊢ Ⅎ𝑥0 ≤ -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦)
34	33, 32, 1	nfif 4509	. . . . . . 7 ⊢ Ⅎ𝑥if(0 ≤ -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦), -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦), 0)
35		nfcv 2891	. . . . . . 7 ⊢ Ⅎ𝑦if(0 ≤ -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥), -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥), 0)
36	7	negeqd 11376	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) = -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))
37	36	breq2d 5107	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (0 ≤ -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) ↔ 0 ≤ -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)))
38	37, 36	ifbieq1d 4503	. . . . . . 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 5197	. . . . . 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 11376	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = -𝐵)
41	40	breq2d 5107	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (0 ≤ -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ↔ 0 ≤ -𝐵))
42	41, 40	ifbieq1d 4503	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥), -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥), 0) = if(0 ≤ -𝐵, -𝐵, 0))
43	42	mpteq2dva 5188	. . . . . 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	eqtrid 2776	. . . . 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 11566	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) ∧ 𝑦 ∈ 𝐴) → -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) ∈ ℝ)
47	23, 29	mbfneg 25568	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) → (𝑦 ∈ 𝐴 ↦ -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦)) ∈ MblFn)
48	46, 47	mbfpos 25569	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) → (𝑦 ∈ 𝐴 ↦ if(0 ≤ -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦), -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦), 0)) ∈ MblFn)
49	45, 48	eqeltrrd 2829	. . 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	ffvelcdmda 7022	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) ∈ ℝ)
53	52	adantlr 715	. . . 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 770	. . . . 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 2828	. . . 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 772	. . . . 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 2828	. . . 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 25570	. . 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 2829	. 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 800	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 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ifcif 4478   class class class wbr 5095   ↦ cmpt 5176  ⟶wf 6482  ‘cfv 6486  ℝcr 11027  0cc0 11028   ≤ cle 11169  -cneg 11367  MblFncmbf 25532

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675  ax-inf2 9556  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105  ax-pre-sup 11106

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3345  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-int 4900  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-se 5577  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-isom 6495  df-riota 7310  df-ov 7356  df-oprab 7357  df-mpo 7358  df-of 7617  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-2o 8396  df-er 8632  df-map 8762  df-pm 8763  df-en 8880  df-dom 8881  df-sdom 8882  df-fin 8883  df-sup 9351  df-inf 9352  df-oi 9421  df-dju 9816  df-card 9854  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11368  df-neg 11369  df-div 11797  df-nn 12148  df-2 12210  df-3 12211  df-n0 12404  df-z 12491  df-uz 12755  df-q 12869  df-rp 12913  df-xadd 13034  df-ioo 13271  df-ico 13273  df-icc 13274  df-fz 13430  df-fzo 13577  df-fl 13715  df-seq 13928  df-exp 13988  df-hash 14257  df-cj 15025  df-re 15026  df-im 15027  df-sqrt 15161  df-abs 15162  df-clim 15414  df-sum 15613  df-xmet 21273  df-met 21274  df-ovol 25382  df-vol 25383  df-mbf 25537

This theorem is referenced by:  iblre  25712