0ome - Mathbox for Glauco Siliprandi

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Theorem 0ome 46516

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 2734	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝒫 𝑋 ↦ 0) = (𝑦 ∈ 𝒫 𝑋 ↦ 0)
2		0e0iccpnf 13481	. . . . . . . . . 10 ⊢ 0 ∈ (0[,]+∞)
3	2	a1i 11	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝒫 𝑋 → 0 ∈ (0[,]+∞))
4	1, 3	fmpti 7112	. . . . . . . 8 ⊢ (𝑦 ∈ 𝒫 𝑋 ↦ 0):𝒫 𝑋⟶(0[,]+∞)
5		0ome.2	. . . . . . . . . . 11 ⊢ 𝑂 = (𝑥 ∈ 𝒫 𝑋 ↦ 0)
6		eqidd 2735	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → 0 = 0)
7	6	cbvmptv 5235	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝒫 𝑋 ↦ 0) = (𝑦 ∈ 𝒫 𝑋 ↦ 0)
8	5, 7	eqtri 2757	. . . . . . . . . 10 ⊢ 𝑂 = (𝑦 ∈ 𝒫 𝑋 ↦ 0)
9	8	feq1i 6707	. . . . . . . . 9 ⊢ (𝑂:dom 𝑂⟶(0[,]+∞) ↔ (𝑦 ∈ 𝒫 𝑋 ↦ 0):dom 𝑂⟶(0[,]+∞))
10	8	dmeqi 5895	. . . . . . . . . . 11 ⊢ dom 𝑂 = dom (𝑦 ∈ 𝒫 𝑋 ↦ 0)
11		0re 11245	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℝ
12	11	rgenw 3054	. . . . . . . . . . . 12 ⊢ ∀𝑦 ∈ 𝒫 𝑋0 ∈ ℝ
13		dmmptg 6242	. . . . . . . . . . . 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 2757	. . . . . . . . . 10 ⊢ dom 𝑂 = 𝒫 𝑋
16	15	feq2i 6708	. . . . . . . . 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 5435	. . . . . . . . . 10 ⊢ ∪ 𝒫 𝑋 = 𝑋
20	19	pweqi 4596	. . . . . . . . 9 ⊢ 𝒫 ∪ 𝒫 𝑋 = 𝒫 𝑋
21	20	eqcomi 2743	. . . . . . . 8 ⊢ 𝒫 𝑋 = 𝒫 ∪ 𝒫 𝑋
22	15	eqcomi 2743	. . . . . . . . . 10 ⊢ 𝒫 𝑋 = dom 𝑂
23	22	unieqi 4899	. . . . . . . . 9 ⊢ ∪ 𝒫 𝑋 = ∪ dom 𝑂
24	23	pweqi 4596	. . . . . . . 8 ⊢ 𝒫 ∪ 𝒫 𝑋 = 𝒫 ∪ dom 𝑂
25	15, 21, 24	3eqtri 2761	. . . . . . 7 ⊢ dom 𝑂 = 𝒫 ∪ dom 𝑂
26	18, 25	pm3.2i 470	. . . . . 6 ⊢ (𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 ∪ dom 𝑂)
27		0elpw 5336	. . . . . . 7 ⊢ ∅ ∈ 𝒫 𝑋
28		eqidd 2735	. . . . . . . 8 ⊢ (𝑦 = ∅ → 0 = 0)
29	11	elexi 3486	. . . . . . . 8 ⊢ 0 ∈ V
30	28, 8, 29	fvmpt 6996	. . . . . . 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 11779	. . . . . . . . 9 ⊢ 0 ≤ 0
34	33	a1i 11	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝒫 ∪ dom 𝑂 ∧ 𝑧 ∈ 𝒫 𝑦) → 0 ≤ 0)
35		eqidd 2735	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → 0 = 0)
36		elpwi 4587	. . . . . . . . . . . . 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 2758	. . . . . . . . . . . . . . . 16 ⊢ 𝒫 ∪ dom 𝑂 = 𝒫 𝑋
40	39	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ 𝒫 ∪ dom 𝑂 → 𝒫 ∪ dom 𝑂 = 𝒫 𝑋)
41	38, 40	eleqtrd 2835	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ 𝒫 ∪ dom 𝑂 → 𝑦 ∈ 𝒫 𝑋)
42		elpwi 4587	. . . . . . . . . . . . . 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 3974	. . . . . . . . . . 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 4583	. . . . . . . . . . . 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 7019	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝒫 ∪ dom 𝑂 ∧ 𝑧 ∈ 𝒫 𝑦) → (𝑂‘𝑧) = 0)
52	8	fvmpt2 7007	. . . . . . . . . . 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 5136	. . . . . . . 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 3133	. . . . . 6 ⊢ (𝑦 ∈ 𝒫 ∪ dom 𝑂 → ∀𝑧 ∈ 𝒫 𝑦(𝑂‘𝑧) ≤ (𝑂‘𝑦))
58	57	rgen 3052	. . . . 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 5235	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝒫 𝑋 ↦ 0) = (𝑧 ∈ 𝒫 𝑋 ↦ 0)
62	8, 61	eqtri 2757	. . . . . . . . . 10 ⊢ 𝑂 = (𝑧 ∈ 𝒫 𝑋 ↦ 0)
63	62	a1i 11	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝒫 dom 𝑂 → 𝑂 = (𝑧 ∈ 𝒫 𝑋 ↦ 0))
64		eqidd 2735	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝒫 dom 𝑂 ∧ 𝑧 = ∪ 𝑦) → 0 = 0)
65		id 22	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝒫 dom 𝑂 → 𝑦 ∈ 𝒫 dom 𝑂)
66	15	pweqi 4596	. . . . . . . . . . . . . 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 2835	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝒫 dom 𝑂 → 𝑦 ∈ 𝒫 𝒫 𝑋)
69		elpwi 4587	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝒫 𝒫 𝑋 → 𝑦 ⊆ 𝒫 𝑋)
70	68, 69	syl 17	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝒫 dom 𝑂 → 𝑦 ⊆ 𝒫 𝑋)
71		sspwuni 5080	. . . . . . . . . . 11 ⊢ (𝑦 ⊆ 𝒫 𝑋 ↔ ∪ 𝑦 ⊆ 𝑋)
72	70, 71	sylib 218	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝒫 dom 𝑂 → ∪ 𝑦 ⊆ 𝑋)
73		vuniex 7741	. . . . . . . . . . . 12 ⊢ ∪ 𝑦 ∈ V
74	73	a1i 11	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝒫 dom 𝑂 → ∪ 𝑦 ∈ V)
75		elpwg 4583	. . . . . . . . . . 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 7003	. . . . . . . 8 ⊢ (𝑦 ∈ 𝒫 dom 𝑂 → (𝑂‘∪ 𝑦) = 0)
80	63	reseq1d 5976	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝒫 dom 𝑂 → (𝑂 ↾ 𝑦) = ((𝑧 ∈ 𝒫 𝑋 ↦ 0) ↾ 𝑦))
81		resmpt 6035	. . . . . . . . . . . 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 2769	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝒫 dom 𝑂 → (𝑂 ↾ 𝑦) = (𝑧 ∈ 𝑦 ↦ 0))
84	83	fveq2d 6890	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝒫 dom 𝑂 → (Σ^{^}‘(𝑂 ↾ 𝑦)) = (Σ^{^}‘(𝑧 ∈ 𝑦 ↦ 0)))
85		nfv 1913	. . . . . . . . . 10 ⊢ Ⅎ𝑧 𝑦 ∈ 𝒫 dom 𝑂
86	85, 65	sge0z 46362	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝒫 dom 𝑂 → (Σ^{^}‘(𝑧 ∈ 𝑦 ↦ 0)) = 0)
87	84, 86	eqtrd 2769	. . . . . . . 8 ⊢ (𝑦 ∈ 𝒫 dom 𝑂 → (Σ^{^}‘(𝑂 ↾ 𝑦)) = 0)
88	79, 87	breq12d 5136	. . . . . . 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 3052	. . . 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 5359	. . . . 5 ⊢ (𝜑 → 𝒫 𝑋 ∈ V)
97		mptexg 7223	. . . . 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 2833	. . 3 ⊢ (𝜑 → 𝑂 ∈ V)
100		isome 46481	. . 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 1539 ∈ wcel 2107 ∀wral 3050 Vcvv 3463 ⊆ wss 3931 ∅c0 4313 𝒫 cpw 4580 ∪ cuni 4887 class class class wbr 5123 ↦ cmpt 5205 dom cdm 5665 ↾ cres 5667 ⟶wf 6537 ‘cfv 6541 (class class class)co 7413 ωcom 7869 ≼ cdom 8965 ℝcr 11136 0cc0 11137 +∞cpnf 11274 ≤ cle 11278 [,]cicc 13372 Σ^{^}csumge0 46349 OutMeascome 46476

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2706 ax-rep 5259 ax-sep 5276 ax-nul 5286 ax-pow 5345 ax-pr 5412 ax-un 7737 ax-inf2 9663 ax-cnex 11193 ax-resscn 11194 ax-1cn 11195 ax-icn 11196 ax-addcl 11197 ax-addrcl 11198 ax-mulcl 11199 ax-mulrcl 11200 ax-mulcom 11201 ax-addass 11202 ax-mulass 11203 ax-distr 11204 ax-i2m1 11205 ax-1ne0 11206 ax-1rid 11207 ax-rnegex 11208 ax-rrecex 11209 ax-cnre 11210 ax-pre-lttri 11211 ax-pre-lttrn 11212 ax-pre-ltadd 11213 ax-pre-mulgt0 11214 ax-pre-sup 11215

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2808 df-nfc 2884 df-ne 2932 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4888 df-int 4927 df-iun 4973 df-br 5124 df-opab 5186 df-mpt 5206 df-tr 5240 df-id 5558 df-eprel 5564 df-po 5572 df-so 5573 df-fr 5617 df-se 5618 df-we 5619 df-xp 5671 df-rel 5672 df-cnv 5673 df-co 5674 df-dm 5675 df-rn 5676 df-res 5677 df-ima 5678 df-pred 6301 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-isom 6550 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7870 df-1st 7996 df-2nd 7997 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-1o 8488 df-er 8727 df-en 8968 df-dom 8969 df-sdom 8970 df-fin 8971 df-sup 9464 df-oi 9532 df-card 9961 df-pnf 11279 df-mnf 11280 df-xr 11281 df-ltxr 11282 df-le 11283 df-sub 11476 df-neg 11477 df-div 11903 df-nn 12249 df-2 12311 df-3 12312 df-n0 12510 df-z 12597 df-uz 12861 df-rp 13017 df-ico 13375 df-icc 13376 df-fz 13530 df-fzo 13677 df-seq 14025 df-exp 14085 df-hash 14353 df-cj 15121 df-re 15122 df-im 15123 df-sqrt 15257 df-abs 15258 df-clim 15507 df-sum 15706 df-sumge0 46350 df-ome 46477

This theorem is referenced by: (None)

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