0ome - Mathbox for Glauco Siliprandi

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Theorem 0ome 46566

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 2731	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝒫 𝑋 ↦ 0) = (𝑦 ∈ 𝒫 𝑋 ↦ 0)
2		0e0iccpnf 13356	. . . . . . . . . 10 ⊢ 0 ∈ (0[,]+∞)
3	2	a1i 11	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝒫 𝑋 → 0 ∈ (0[,]+∞))
4	1, 3	fmpti 7045	. . . . . . . 8 ⊢ (𝑦 ∈ 𝒫 𝑋 ↦ 0):𝒫 𝑋⟶(0[,]+∞)
5		0ome.2	. . . . . . . . . . 11 ⊢ 𝑂 = (𝑥 ∈ 𝒫 𝑋 ↦ 0)
6		eqidd 2732	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → 0 = 0)
7	6	cbvmptv 5195	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝒫 𝑋 ↦ 0) = (𝑦 ∈ 𝒫 𝑋 ↦ 0)
8	5, 7	eqtri 2754	. . . . . . . . . 10 ⊢ 𝑂 = (𝑦 ∈ 𝒫 𝑋 ↦ 0)
9	8	feq1i 6642	. . . . . . . . 9 ⊢ (𝑂:dom 𝑂⟶(0[,]+∞) ↔ (𝑦 ∈ 𝒫 𝑋 ↦ 0):dom 𝑂⟶(0[,]+∞))
10	8	dmeqi 5844	. . . . . . . . . . 11 ⊢ dom 𝑂 = dom (𝑦 ∈ 𝒫 𝑋 ↦ 0)
11		0re 11111	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℝ
12	11	rgenw 3051	. . . . . . . . . . . 12 ⊢ ∀𝑦 ∈ 𝒫 𝑋0 ∈ ℝ
13		dmmptg 6189	. . . . . . . . . . . 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 2754	. . . . . . . . . 10 ⊢ dom 𝑂 = 𝒫 𝑋
16	15	feq2i 6643	. . . . . . . . 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 5391	. . . . . . . . . 10 ⊢ ∪ 𝒫 𝑋 = 𝑋
20	19	pweqi 4566	. . . . . . . . 9 ⊢ 𝒫 ∪ 𝒫 𝑋 = 𝒫 𝑋
21	20	eqcomi 2740	. . . . . . . 8 ⊢ 𝒫 𝑋 = 𝒫 ∪ 𝒫 𝑋
22	15	eqcomi 2740	. . . . . . . . . 10 ⊢ 𝒫 𝑋 = dom 𝑂
23	22	unieqi 4871	. . . . . . . . 9 ⊢ ∪ 𝒫 𝑋 = ∪ dom 𝑂
24	23	pweqi 4566	. . . . . . . 8 ⊢ 𝒫 ∪ 𝒫 𝑋 = 𝒫 ∪ dom 𝑂
25	15, 21, 24	3eqtri 2758	. . . . . . 7 ⊢ dom 𝑂 = 𝒫 ∪ dom 𝑂
26	18, 25	pm3.2i 470	. . . . . 6 ⊢ (𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 ∪ dom 𝑂)
27		0elpw 5294	. . . . . . 7 ⊢ ∅ ∈ 𝒫 𝑋
28		eqidd 2732	. . . . . . . 8 ⊢ (𝑦 = ∅ → 0 = 0)
29	11	elexi 3459	. . . . . . . 8 ⊢ 0 ∈ V
30	28, 8, 29	fvmpt 6929	. . . . . . 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 11648	. . . . . . . . 9 ⊢ 0 ≤ 0
34	33	a1i 11	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝒫 ∪ dom 𝑂 ∧ 𝑧 ∈ 𝒫 𝑦) → 0 ≤ 0)
35		eqidd 2732	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → 0 = 0)
36		elpwi 4557	. . . . . . . . . . . . 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 2755	. . . . . . . . . . . . . . . 16 ⊢ 𝒫 ∪ dom 𝑂 = 𝒫 𝑋
40	39	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ 𝒫 ∪ dom 𝑂 → 𝒫 ∪ dom 𝑂 = 𝒫 𝑋)
41	38, 40	eleqtrd 2833	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ 𝒫 ∪ dom 𝑂 → 𝑦 ∈ 𝒫 𝑋)
42		elpwi 4557	. . . . . . . . . . . . . 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 3945	. . . . . . . . . . 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 4553	. . . . . . . . . . . 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 6952	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝒫 ∪ dom 𝑂 ∧ 𝑧 ∈ 𝒫 𝑦) → (𝑂‘𝑧) = 0)
52	8	fvmpt2 6940	. . . . . . . . . . 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 5104	. . . . . . . 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 3124	. . . . . 6 ⊢ (𝑦 ∈ 𝒫 ∪ dom 𝑂 → ∀𝑧 ∈ 𝒫 𝑦(𝑂‘𝑧) ≤ (𝑂‘𝑦))
58	57	rgen 3049	. . . . 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 5195	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝒫 𝑋 ↦ 0) = (𝑧 ∈ 𝒫 𝑋 ↦ 0)
62	8, 61	eqtri 2754	. . . . . . . . . 10 ⊢ 𝑂 = (𝑧 ∈ 𝒫 𝑋 ↦ 0)
63	62	a1i 11	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝒫 dom 𝑂 → 𝑂 = (𝑧 ∈ 𝒫 𝑋 ↦ 0))
64		eqidd 2732	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝒫 dom 𝑂 ∧ 𝑧 = ∪ 𝑦) → 0 = 0)
65		id 22	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝒫 dom 𝑂 → 𝑦 ∈ 𝒫 dom 𝑂)
66	15	pweqi 4566	. . . . . . . . . . . . . 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 2833	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝒫 dom 𝑂 → 𝑦 ∈ 𝒫 𝒫 𝑋)
69		elpwi 4557	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝒫 𝒫 𝑋 → 𝑦 ⊆ 𝒫 𝑋)
70	68, 69	syl 17	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝒫 dom 𝑂 → 𝑦 ⊆ 𝒫 𝑋)
71		sspwuni 5048	. . . . . . . . . . 11 ⊢ (𝑦 ⊆ 𝒫 𝑋 ↔ ∪ 𝑦 ⊆ 𝑋)
72	70, 71	sylib 218	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝒫 dom 𝑂 → ∪ 𝑦 ⊆ 𝑋)
73		vuniex 7672	. . . . . . . . . . . 12 ⊢ ∪ 𝑦 ∈ V
74	73	a1i 11	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝒫 dom 𝑂 → ∪ 𝑦 ∈ V)
75		elpwg 4553	. . . . . . . . . . 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 6936	. . . . . . . 8 ⊢ (𝑦 ∈ 𝒫 dom 𝑂 → (𝑂‘∪ 𝑦) = 0)
80	63	reseq1d 5927	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝒫 dom 𝑂 → (𝑂 ↾ 𝑦) = ((𝑧 ∈ 𝒫 𝑋 ↦ 0) ↾ 𝑦))
81		resmpt 5986	. . . . . . . . . . . 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 2766	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝒫 dom 𝑂 → (𝑂 ↾ 𝑦) = (𝑧 ∈ 𝑦 ↦ 0))
84	83	fveq2d 6826	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝒫 dom 𝑂 → (Σ^{^}‘(𝑂 ↾ 𝑦)) = (Σ^{^}‘(𝑧 ∈ 𝑦 ↦ 0)))
85		nfv 1915	. . . . . . . . . 10 ⊢ Ⅎ𝑧 𝑦 ∈ 𝒫 dom 𝑂
86	85, 65	sge0z 46412	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝒫 dom 𝑂 → (Σ^{^}‘(𝑧 ∈ 𝑦 ↦ 0)) = 0)
87	84, 86	eqtrd 2766	. . . . . . . 8 ⊢ (𝑦 ∈ 𝒫 dom 𝑂 → (Σ^{^}‘(𝑂 ↾ 𝑦)) = 0)
88	79, 87	breq12d 5104	. . . . . . 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 3049	. . . 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 5317	. . . . 5 ⊢ (𝜑 → 𝒫 𝑋 ∈ V)
97		mptexg 7155	. . . . 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 2831	. . 3 ⊢ (𝜑 → 𝑂 ∈ V)
100		isome 46531	. . 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 1541 ∈ wcel 2111 ∀wral 3047 Vcvv 3436 ⊆ wss 3902 ∅c0 4283 𝒫 cpw 4550 ∪ cuni 4859 class class class wbr 5091 ↦ cmpt 5172 dom cdm 5616 ↾ cres 5618 ⟶wf 6477 ‘cfv 6481 (class class class)co 7346 ωcom 7796 ≼ cdom 8867 ℝcr 11002 0cc0 11003 +∞cpnf 11140 ≤ cle 11144 [,]cicc 13245 Σ^{^}csumge0 46399 OutMeascome 46526

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5217 ax-sep 5234 ax-nul 5244 ax-pow 5303 ax-pr 5370 ax-un 7668 ax-inf2 9531 ax-cnex 11059 ax-resscn 11060 ax-1cn 11061 ax-icn 11062 ax-addcl 11063 ax-addrcl 11064 ax-mulcl 11065 ax-mulrcl 11066 ax-mulcom 11067 ax-addass 11068 ax-mulass 11069 ax-distr 11070 ax-i2m1 11071 ax-1ne0 11072 ax-1rid 11073 ax-rnegex 11074 ax-rrecex 11075 ax-cnre 11076 ax-pre-lttri 11077 ax-pre-lttrn 11078 ax-pre-ltadd 11079 ax-pre-mulgt0 11080 ax-pre-sup 11081

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4476 df-pw 4552 df-sn 4577 df-pr 4579 df-op 4583 df-uni 4860 df-int 4898 df-iun 4943 df-br 5092 df-opab 5154 df-mpt 5173 df-tr 5199 df-id 5511 df-eprel 5516 df-po 5524 df-so 5525 df-fr 5569 df-se 5570 df-we 5571 df-xp 5622 df-rel 5623 df-cnv 5624 df-co 5625 df-dm 5626 df-rn 5627 df-res 5628 df-ima 5629 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-isom 6490 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-om 7797 df-1st 7921 df-2nd 7922 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-er 8622 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-sup 9326 df-oi 9396 df-card 9829 df-pnf 11145 df-mnf 11146 df-xr 11147 df-ltxr 11148 df-le 11149 df-sub 11343 df-neg 11344 df-div 11772 df-nn 12123 df-2 12185 df-3 12186 df-n0 12379 df-z 12466 df-uz 12730 df-rp 12888 df-ico 13248 df-icc 13249 df-fz 13405 df-fzo 13552 df-seq 13906 df-exp 13966 df-hash 14235 df-cj 15003 df-re 15004 df-im 15005 df-sqrt 15139 df-abs 15140 df-clim 15392 df-sum 15591 df-sumge0 46400 df-ome 46527

This theorem is referenced by: (None)

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