0ome - Mathbox for Glauco Siliprandi

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Theorem 0ome 46527

Description: The map that assigns 0 to every subset, is an outer measure. (Contributed by Glauco Siliprandi, 11-Oct-2020.)

**Hypotheses**
Ref	Expression
0ome.1	⊢ (𝜑 → 𝑋 ∈ 𝑉)
0ome.2	⊢ 𝑂 = (𝑥 ∈ 𝒫 𝑋 ↦ 0)

**Assertion**
Ref	Expression
0ome	⊢ (𝜑 → 𝑂 ∈ OutMeas)

Distinct variable group: 𝑥,𝑋 Allowed substitution hints: 𝜑(𝑥) 𝑂(𝑥) 𝑉(𝑥)

Proof of Theorem 0ome Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2729	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝒫 𝑋 ↦ 0) = (𝑦 ∈ 𝒫 𝑋 ↦ 0)
2		0e0iccpnf 13420	. . . . . . . . . 10 ⊢ 0 ∈ (0[,]+∞)
3	2	a1i 11	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝒫 𝑋 → 0 ∈ (0[,]+∞))
4	1, 3	fmpti 7084	. . . . . . . 8 ⊢ (𝑦 ∈ 𝒫 𝑋 ↦ 0):𝒫 𝑋⟶(0[,]+∞)
5		0ome.2	. . . . . . . . . . 11 ⊢ 𝑂 = (𝑥 ∈ 𝒫 𝑋 ↦ 0)
6		eqidd 2730	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → 0 = 0)
7	6	cbvmptv 5211	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝒫 𝑋 ↦ 0) = (𝑦 ∈ 𝒫 𝑋 ↦ 0)
8	5, 7	eqtri 2752	. . . . . . . . . 10 ⊢ 𝑂 = (𝑦 ∈ 𝒫 𝑋 ↦ 0)
9	8	feq1i 6679	. . . . . . . . 9 ⊢ (𝑂:dom 𝑂⟶(0[,]+∞) ↔ (𝑦 ∈ 𝒫 𝑋 ↦ 0):dom 𝑂⟶(0[,]+∞))
10	8	dmeqi 5868	. . . . . . . . . . 11 ⊢ dom 𝑂 = dom (𝑦 ∈ 𝒫 𝑋 ↦ 0)
11		0re 11176	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℝ
12	11	rgenw 3048	. . . . . . . . . . . 12 ⊢ ∀𝑦 ∈ 𝒫 𝑋0 ∈ ℝ
13		dmmptg 6215	. . . . . . . . . . . 12 ⊢ (∀𝑦 ∈ 𝒫 𝑋0 ∈ ℝ → dom (𝑦 ∈ 𝒫 𝑋 ↦ 0) = 𝒫 𝑋)
14	12, 13	ax-mp 5	. . . . . . . . . . 11 ⊢ dom (𝑦 ∈ 𝒫 𝑋 ↦ 0) = 𝒫 𝑋
15	10, 14	eqtri 2752	. . . . . . . . . 10 ⊢ dom 𝑂 = 𝒫 𝑋
16	15	feq2i 6680	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝒫 𝑋 ↦ 0):dom 𝑂⟶(0[,]+∞) ↔ (𝑦 ∈ 𝒫 𝑋 ↦ 0):𝒫 𝑋⟶(0[,]+∞))
17	9, 16	bitri 275	. . . . . . . 8 ⊢ (𝑂:dom 𝑂⟶(0[,]+∞) ↔ (𝑦 ∈ 𝒫 𝑋 ↦ 0):𝒫 𝑋⟶(0[,]+∞))
18	4, 17	mpbir 231	. . . . . . 7 ⊢ 𝑂:dom 𝑂⟶(0[,]+∞)
19		unipw 5410	. . . . . . . . . 10 ⊢ ∪ 𝒫 𝑋 = 𝑋
20	19	pweqi 4579	. . . . . . . . 9 ⊢ 𝒫 ∪ 𝒫 𝑋 = 𝒫 𝑋
21	20	eqcomi 2738	. . . . . . . 8 ⊢ 𝒫 𝑋 = 𝒫 ∪ 𝒫 𝑋
22	15	eqcomi 2738	. . . . . . . . . 10 ⊢ 𝒫 𝑋 = dom 𝑂
23	22	unieqi 4883	. . . . . . . . 9 ⊢ ∪ 𝒫 𝑋 = ∪ dom 𝑂
24	23	pweqi 4579	. . . . . . . 8 ⊢ 𝒫 ∪ 𝒫 𝑋 = 𝒫 ∪ dom 𝑂
25	15, 21, 24	3eqtri 2756	. . . . . . 7 ⊢ dom 𝑂 = 𝒫 ∪ dom 𝑂
26	18, 25	pm3.2i 470	. . . . . 6 ⊢ (𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 ∪ dom 𝑂)
27		0elpw 5311	. . . . . . 7 ⊢ ∅ ∈ 𝒫 𝑋
28		eqidd 2730	. . . . . . . 8 ⊢ (𝑦 = ∅ → 0 = 0)
29	11	elexi 3470	. . . . . . . 8 ⊢ 0 ∈ V
30	28, 8, 29	fvmpt 6968	. . . . . . 7 ⊢ (∅ ∈ 𝒫 𝑋 → (𝑂‘∅) = 0)
31	27, 30	ax-mp 5	. . . . . 6 ⊢ (𝑂‘∅) = 0
32	26, 31	pm3.2i 470	. . . . 5 ⊢ ((𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 ∪ dom 𝑂) ∧ (𝑂‘∅) = 0)
33	11	leidi 11712	. . . . . . . . 9 ⊢ 0 ≤ 0
34	33	a1i 11	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝒫 ∪ dom 𝑂 ∧ 𝑧 ∈ 𝒫 𝑦) → 0 ≤ 0)
35		eqidd 2730	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → 0 = 0)
36		elpwi 4570	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ 𝒫 𝑦 → 𝑧 ⊆ 𝑦)
37	36	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ 𝒫 ∪ dom 𝑂 ∧ 𝑧 ∈ 𝒫 𝑦) → 𝑧 ⊆ 𝑦)
38		id 22	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ 𝒫 ∪ dom 𝑂 → 𝑦 ∈ 𝒫 ∪ dom 𝑂)
39	21, 24	eqtr2i 2753	. . . . . . . . . . . . . . . 16 ⊢ 𝒫 ∪ dom 𝑂 = 𝒫 𝑋
40	39	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ 𝒫 ∪ dom 𝑂 → 𝒫 ∪ dom 𝑂 = 𝒫 𝑋)
41	38, 40	eleqtrd 2830	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ 𝒫 ∪ dom 𝑂 → 𝑦 ∈ 𝒫 𝑋)
42		elpwi 4570	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ 𝒫 𝑋 → 𝑦 ⊆ 𝑋)
43	41, 42	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝒫 ∪ dom 𝑂 → 𝑦 ⊆ 𝑋)
44	43	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ 𝒫 ∪ dom 𝑂 ∧ 𝑧 ∈ 𝒫 𝑦) → 𝑦 ⊆ 𝑋)
45	37, 44	sstrd 3957	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ 𝒫 ∪ dom 𝑂 ∧ 𝑧 ∈ 𝒫 𝑦) → 𝑧 ⊆ 𝑋)
46		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ 𝒫 ∪ dom 𝑂 ∧ 𝑧 ∈ 𝒫 𝑦) → 𝑧 ∈ 𝒫 𝑦)
47		elpwg 4566	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ 𝒫 𝑦 → (𝑧 ∈ 𝒫 𝑋 ↔ 𝑧 ⊆ 𝑋))
48	46, 47	syl 17	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ 𝒫 ∪ dom 𝑂 ∧ 𝑧 ∈ 𝒫 𝑦) → (𝑧 ∈ 𝒫 𝑋 ↔ 𝑧 ⊆ 𝑋))
49	45, 48	mpbird 257	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝒫 ∪ dom 𝑂 ∧ 𝑧 ∈ 𝒫 𝑦) → 𝑧 ∈ 𝒫 𝑋)
50	11	a1i 11	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝒫 ∪ dom 𝑂 ∧ 𝑧 ∈ 𝒫 𝑦) → 0 ∈ ℝ)
51	8, 35, 49, 50	fvmptd3 6991	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝒫 ∪ dom 𝑂 ∧ 𝑧 ∈ 𝒫 𝑦) → (𝑂‘𝑧) = 0)
52	8	fvmpt2 6979	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ 𝒫 𝑋 ∧ 0 ∈ ℝ) → (𝑂‘𝑦) = 0)
53	41, 11, 52	sylancl 586	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝒫 ∪ dom 𝑂 → (𝑂‘𝑦) = 0)
54	53	adantr 480	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝒫 ∪ dom 𝑂 ∧ 𝑧 ∈ 𝒫 𝑦) → (𝑂‘𝑦) = 0)
55	51, 54	breq12d 5120	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝒫 ∪ dom 𝑂 ∧ 𝑧 ∈ 𝒫 𝑦) → ((𝑂‘𝑧) ≤ (𝑂‘𝑦) ↔ 0 ≤ 0))
56	34, 55	mpbird 257	. . . . . . 7 ⊢ ((𝑦 ∈ 𝒫 ∪ dom 𝑂 ∧ 𝑧 ∈ 𝒫 𝑦) → (𝑂‘𝑧) ≤ (𝑂‘𝑦))
57	56	ralrimiva 3125	. . . . . 6 ⊢ (𝑦 ∈ 𝒫 ∪ dom 𝑂 → ∀𝑧 ∈ 𝒫 𝑦(𝑂‘𝑧) ≤ (𝑂‘𝑦))
58	57	rgen 3046	. . . . 5 ⊢ ∀𝑦 ∈ 𝒫 ∪ dom 𝑂∀𝑧 ∈ 𝒫 𝑦(𝑂‘𝑧) ≤ (𝑂‘𝑦)
59	32, 58	pm3.2i 470	. . . 4 ⊢ (((𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 ∪ dom 𝑂) ∧ (𝑂‘∅) = 0) ∧ ∀𝑦 ∈ 𝒫 ∪ dom 𝑂∀𝑧 ∈ 𝒫 𝑦(𝑂‘𝑧) ≤ (𝑂‘𝑦))
60	33	a1i 11	. . . . . . 7 ⊢ (𝑦 ∈ 𝒫 dom 𝑂 → 0 ≤ 0)
61	35	cbvmptv 5211	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝒫 𝑋 ↦ 0) = (𝑧 ∈ 𝒫 𝑋 ↦ 0)
62	8, 61	eqtri 2752	. . . . . . . . . 10 ⊢ 𝑂 = (𝑧 ∈ 𝒫 𝑋 ↦ 0)
63	62	a1i 11	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝒫 dom 𝑂 → 𝑂 = (𝑧 ∈ 𝒫 𝑋 ↦ 0))
64		eqidd 2730	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝒫 dom 𝑂 ∧ 𝑧 = ∪ 𝑦) → 0 = 0)
65		id 22	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝒫 dom 𝑂 → 𝑦 ∈ 𝒫 dom 𝑂)
66	15	pweqi 4579	. . . . . . . . . . . . . 14 ⊢ 𝒫 dom 𝑂 = 𝒫 𝒫 𝑋
67	66	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝒫 dom 𝑂 → 𝒫 dom 𝑂 = 𝒫 𝒫 𝑋)
68	65, 67	eleqtrd 2830	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝒫 dom 𝑂 → 𝑦 ∈ 𝒫 𝒫 𝑋)
69		elpwi 4570	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝒫 𝒫 𝑋 → 𝑦 ⊆ 𝒫 𝑋)
70	68, 69	syl 17	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝒫 dom 𝑂 → 𝑦 ⊆ 𝒫 𝑋)
71		sspwuni 5064	. . . . . . . . . . 11 ⊢ (𝑦 ⊆ 𝒫 𝑋 ↔ ∪ 𝑦 ⊆ 𝑋)
72	70, 71	sylib 218	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝒫 dom 𝑂 → ∪ 𝑦 ⊆ 𝑋)
73		vuniex 7715	. . . . . . . . . . . 12 ⊢ ∪ 𝑦 ∈ V
74	73	a1i 11	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝒫 dom 𝑂 → ∪ 𝑦 ∈ V)
75		elpwg 4566	. . . . . . . . . . 11 ⊢ (∪ 𝑦 ∈ V → (∪ 𝑦 ∈ 𝒫 𝑋 ↔ ∪ 𝑦 ⊆ 𝑋))
76	74, 75	syl 17	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝒫 dom 𝑂 → (∪ 𝑦 ∈ 𝒫 𝑋 ↔ ∪ 𝑦 ⊆ 𝑋))
77	72, 76	mpbird 257	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝒫 dom 𝑂 → ∪ 𝑦 ∈ 𝒫 𝑋)
78	11	a1i 11	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝒫 dom 𝑂 → 0 ∈ ℝ)
79	63, 64, 77, 78	fvmptd 6975	. . . . . . . 8 ⊢ (𝑦 ∈ 𝒫 dom 𝑂 → (𝑂‘∪ 𝑦) = 0)
80	63	reseq1d 5949	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝒫 dom 𝑂 → (𝑂 ↾ 𝑦) = ((𝑧 ∈ 𝒫 𝑋 ↦ 0) ↾ 𝑦))
81		resmpt 6008	. . . . . . . . . . . 12 ⊢ (𝑦 ⊆ 𝒫 𝑋 → ((𝑧 ∈ 𝒫 𝑋 ↦ 0) ↾ 𝑦) = (𝑧 ∈ 𝑦 ↦ 0))
82	70, 81	syl 17	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝒫 dom 𝑂 → ((𝑧 ∈ 𝒫 𝑋 ↦ 0) ↾ 𝑦) = (𝑧 ∈ 𝑦 ↦ 0))
83	80, 82	eqtrd 2764	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝒫 dom 𝑂 → (𝑂 ↾ 𝑦) = (𝑧 ∈ 𝑦 ↦ 0))
84	83	fveq2d 6862	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝒫 dom 𝑂 → (Σ^{^}‘(𝑂 ↾ 𝑦)) = (Σ^{^}‘(𝑧 ∈ 𝑦 ↦ 0)))
85		nfv 1914	. . . . . . . . . 10 ⊢ Ⅎ𝑧 𝑦 ∈ 𝒫 dom 𝑂
86	85, 65	sge0z 46373	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝒫 dom 𝑂 → (Σ^{^}‘(𝑧 ∈ 𝑦 ↦ 0)) = 0)
87	84, 86	eqtrd 2764	. . . . . . . 8 ⊢ (𝑦 ∈ 𝒫 dom 𝑂 → (Σ^{^}‘(𝑂 ↾ 𝑦)) = 0)
88	79, 87	breq12d 5120	. . . . . . 7 ⊢ (𝑦 ∈ 𝒫 dom 𝑂 → ((𝑂‘∪ 𝑦) ≤ (Σ^{^}‘(𝑂 ↾ 𝑦)) ↔ 0 ≤ 0))
89	60, 88	mpbird 257	. . . . . 6 ⊢ (𝑦 ∈ 𝒫 dom 𝑂 → (𝑂‘∪ 𝑦) ≤ (Σ^{^}‘(𝑂 ↾ 𝑦)))
90	89	a1d 25	. . . . 5 ⊢ (𝑦 ∈ 𝒫 dom 𝑂 → (𝑦 ≼ ω → (𝑂‘∪ 𝑦) ≤ (Σ^{^}‘(𝑂 ↾ 𝑦))))
91	90	rgen 3046	. . . 4 ⊢ ∀𝑦 ∈ 𝒫 dom 𝑂(𝑦 ≼ ω → (𝑂‘∪ 𝑦) ≤ (Σ^{^}‘(𝑂 ↾ 𝑦)))
92	59, 91	pm3.2i 470	. . 3 ⊢ ((((𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 ∪ dom 𝑂) ∧ (𝑂‘∅) = 0) ∧ ∀𝑦 ∈ 𝒫 ∪ dom 𝑂∀𝑧 ∈ 𝒫 𝑦(𝑂‘𝑧) ≤ (𝑂‘𝑦)) ∧ ∀𝑦 ∈ 𝒫 dom 𝑂(𝑦 ≼ ω → (𝑂‘∪ 𝑦) ≤ (Σ^{^}‘(𝑂 ↾ 𝑦))))
93	92	a1i 11	. 2 ⊢ (𝜑 → ((((𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 ∪ dom 𝑂) ∧ (𝑂‘∅) = 0) ∧ ∀𝑦 ∈ 𝒫 ∪ dom 𝑂∀𝑧 ∈ 𝒫 𝑦(𝑂‘𝑧) ≤ (𝑂‘𝑦)) ∧ ∀𝑦 ∈ 𝒫 dom 𝑂(𝑦 ≼ ω → (𝑂‘∪ 𝑦) ≤ (Σ^{^}‘(𝑂 ↾ 𝑦)))))
94	8	a1i 11	. . . 4 ⊢ (𝜑 → 𝑂 = (𝑦 ∈ 𝒫 𝑋 ↦ 0))
95		0ome.1	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
96	95	pwexd 5334	. . . . 5 ⊢ (𝜑 → 𝒫 𝑋 ∈ V)
97		mptexg 7195	. . . . 5 ⊢ (𝒫 𝑋 ∈ V → (𝑦 ∈ 𝒫 𝑋 ↦ 0) ∈ V)
98	96, 97	syl 17	. . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝒫 𝑋 ↦ 0) ∈ V)
99	94, 98	eqeltrd 2828	. . 3 ⊢ (𝜑 → 𝑂 ∈ V)
100		isome 46492	. . 3 ⊢ (𝑂 ∈ V → (𝑂 ∈ OutMeas ↔ ((((𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 ∪ dom 𝑂) ∧ (𝑂‘∅) = 0) ∧ ∀𝑦 ∈ 𝒫 ∪ dom 𝑂∀𝑧 ∈ 𝒫 𝑦(𝑂‘𝑧) ≤ (𝑂‘𝑦)) ∧ ∀𝑦 ∈ 𝒫 dom 𝑂(𝑦 ≼ ω → (𝑂‘∪ 𝑦) ≤ (Σ^{^}‘(𝑂 ↾ 𝑦))))))
101	99, 100	syl 17	. 2 ⊢ (𝜑 → (𝑂 ∈ OutMeas ↔ ((((𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 ∪ dom 𝑂) ∧ (𝑂‘∅) = 0) ∧ ∀𝑦 ∈ 𝒫 ∪ dom 𝑂∀𝑧 ∈ 𝒫 𝑦(𝑂‘𝑧) ≤ (𝑂‘𝑦)) ∧ ∀𝑦 ∈ 𝒫 dom 𝑂(𝑦 ≼ ω → (𝑂‘∪ 𝑦) ≤ (Σ^{^}‘(𝑂 ↾ 𝑦))))))
102	93, 101	mpbird 257	1 ⊢ (𝜑 → 𝑂 ∈ OutMeas)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ∀wral 3044 Vcvv 3447 ⊆ wss 3914 ∅c0 4296 𝒫 cpw 4563 ∪ cuni 4871 class class class wbr 5107 ↦ cmpt 5188 dom cdm 5638 ↾ cres 5640 ⟶wf 6507 ‘cfv 6511 (class class class)co 7387 ωcom 7842 ≼ cdom 8916 ℝcr 11067 0cc0 11068 +∞cpnf 11205 ≤ cle 11209 [,]cicc 13309 Σ^{^}csumge0 46360 OutMeascome 46487

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 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-inf2 9594 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146

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 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-se 5592 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-isom 6520 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-om 7843 df-1st 7968 df-2nd 7969 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-er 8671 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-sup 9393 df-oi 9463 df-card 9892 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-2 12249 df-3 12250 df-n0 12443 df-z 12530 df-uz 12794 df-rp 12952 df-ico 13312 df-icc 13313 df-fz 13469 df-fzo 13616 df-seq 13967 df-exp 14027 df-hash 14296 df-cj 15065 df-re 15066 df-im 15067 df-sqrt 15201 df-abs 15202 df-clim 15454 df-sum 15653 df-sumge0 46361 df-ome 46488

This theorem is referenced by: (None)

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