Theorem mbfposb 25530

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 6851	. . . . . . . . 9 ⊢ Ⅎ𝑥((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦)
4	1, 2, 3	nfbr 5149	. . . . . . . 8 ⊢ Ⅎ𝑥0 ≤ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦)
5	4, 3, 1	nfif 4515	. . . . . . 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 6840	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) = ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))
8	7	breq2d 5114	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (0 ≤ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) ↔ 0 ≤ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)))
9	8, 7	ifbieq1d 4509	. . . . . . 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 5204	. . . . . 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 6961	. . . . . . . . . 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 5114	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (0 ≤ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ↔ 0 ≤ 𝐵))
17	16, 15	ifbieq1d 4509	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥), ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥), 0) = if(0 ≤ 𝐵, 𝐵, 0))
18	17	mpteq2dva 5195	. . . . . 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 7069	. . . . . . 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 7038	. . . . 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 5204	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 ↦ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦)) = (𝑥 ∈ 𝐴 ↦ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))
26	15	mpteq2dva 5195	. . . . . . . 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 25528	. . . 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 11393	. . . . . . . . 9 ⊢ Ⅎ𝑥-((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦)
33	1, 2, 32	nfbr 5149	. . . . . . . 8 ⊢ Ⅎ𝑥0 ≤ -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦)
34	33, 32, 1	nfif 4515	. . . . . . 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 11391	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) = -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))
37	36	breq2d 5114	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (0 ≤ -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) ↔ 0 ≤ -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)))
38	37, 36	ifbieq1d 4509	. . . . . . 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 5204	. . . . . 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 11391	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = -𝐵)
41	40	breq2d 5114	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (0 ≤ -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ↔ 0 ≤ -𝐵))
42	41, 40	ifbieq1d 4509	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥), -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥), 0) = if(0 ≤ -𝐵, -𝐵, 0))
43	42	mpteq2dva 5195	. . . . . 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 11581	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) ∧ 𝑦 ∈ 𝐴) → -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) ∈ ℝ)
47	23, 29	mbfneg 25527	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) → (𝑦 ∈ 𝐴 ↦ -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦)) ∈ MblFn)
48	46, 47	mbfpos 25528	. . . 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 7038	. . . . 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 25529	. . 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 4484   class class class wbr 5102   ↦ cmpt 5183  ⟶wf 6495  ‘cfv 6499  ℝcr 11043  0cc0 11044   ≤ cle 11185  -cneg 11382  MblFncmbf 25491

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 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-inf2 9570  ax-cnex 11100  ax-resscn 11101  ax-1cn 11102  ax-icn 11103  ax-addcl 11104  ax-addrcl 11105  ax-mulcl 11106  ax-mulrcl 11107  ax-mulcom 11108  ax-addass 11109  ax-mulass 11110  ax-distr 11111  ax-i2m1 11112  ax-1ne0 11113  ax-1rid 11114  ax-rnegex 11115  ax-rrecex 11116  ax-cnre 11117  ax-pre-lttri 11118  ax-pre-lttrn 11119  ax-pre-ltadd 11120  ax-pre-mulgt0 11121  ax-pre-sup 11122

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 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-int 4907  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-isom 6508  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-of 7633  df-om 7823  df-1st 7947  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-1o 8411  df-2o 8412  df-er 8648  df-map 8778  df-pm 8779  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-sup 9369  df-inf 9370  df-oi 9439  df-dju 9830  df-card 9868  df-pnf 11186  df-mnf 11187  df-xr 11188  df-ltxr 11189  df-le 11190  df-sub 11383  df-neg 11384  df-div 11812  df-nn 12163  df-2 12225  df-3 12226  df-n0 12419  df-z 12506  df-uz 12770  df-q 12884  df-rp 12928  df-xadd 13049  df-ioo 13286  df-ico 13288  df-icc 13289  df-fz 13445  df-fzo 13592  df-fl 13730  df-seq 13943  df-exp 14003  df-hash 14272  df-cj 15041  df-re 15042  df-im 15043  df-sqrt 15177  df-abs 15178  df-clim 15430  df-sum 15629  df-xmet 21233  df-met 21234  df-ovol 25341  df-vol 25342  df-mbf 25496

This theorem is referenced by:  iblre  25671