0ome - Mathbox for Glauco Siliprandi

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Theorem 0ome 45790

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 2724	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝒫 𝑋 ↦ 0) = (𝑦 ∈ 𝒫 𝑋 ↦ 0)
2		0e0iccpnf 13437	. . . . . . . . . 10 ⊢ 0 ∈ (0[,]+∞)
3	2	a1i 11	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝒫 𝑋 → 0 ∈ (0[,]+∞))
4	1, 3	fmpti 7104	. . . . . . . 8 ⊢ (𝑦 ∈ 𝒫 𝑋 ↦ 0):𝒫 𝑋⟶(0[,]+∞)
5		0ome.2	. . . . . . . . . . 11 ⊢ 𝑂 = (𝑥 ∈ 𝒫 𝑋 ↦ 0)
6		eqidd 2725	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → 0 = 0)
7	6	cbvmptv 5252	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝒫 𝑋 ↦ 0) = (𝑦 ∈ 𝒫 𝑋 ↦ 0)
8	5, 7	eqtri 2752	. . . . . . . . . 10 ⊢ 𝑂 = (𝑦 ∈ 𝒫 𝑋 ↦ 0)
9	8	feq1i 6699	. . . . . . . . 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 11215	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℝ
12	11	rgenw 3057	. . . . . . . . . . . 12 ⊢ ∀𝑦 ∈ 𝒫 𝑋0 ∈ ℝ
13		dmmptg 6232	. . . . . . . . . . . 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 6700	. . . . . . . . 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 230	. . . . . . 7 ⊢ 𝑂:dom 𝑂⟶(0[,]+∞)
19		unipw 5441	. . . . . . . . . 10 ⊢ ∪ 𝒫 𝑋 = 𝑋
20	19	pweqi 4611	. . . . . . . . 9 ⊢ 𝒫 ∪ 𝒫 𝑋 = 𝒫 𝑋
21	20	eqcomi 2733	. . . . . . . 8 ⊢ 𝒫 𝑋 = 𝒫 ∪ 𝒫 𝑋
22	15	eqcomi 2733	. . . . . . . . . 10 ⊢ 𝒫 𝑋 = dom 𝑂
23	22	unieqi 4912	. . . . . . . . 9 ⊢ ∪ 𝒫 𝑋 = ∪ dom 𝑂
24	23	pweqi 4611	. . . . . . . 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 5345	. . . . . . 7 ⊢ ∅ ∈ 𝒫 𝑋
28		eqidd 2725	. . . . . . . 8 ⊢ (𝑦 = ∅ → 0 = 0)
29	11	elexi 3486	. . . . . . . 8 ⊢ 0 ∈ V
30	28, 8, 29	fvmpt 6989	. . . . . . 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 11747	. . . . . . . . 9 ⊢ 0 ≤ 0
34	33	a1i 11	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝒫 ∪ dom 𝑂 ∧ 𝑧 ∈ 𝒫 𝑦) → 0 ≤ 0)
35		eqidd 2725	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → 0 = 0)
36		elpwi 4602	. . . . . . . . . . . . 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 2827	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ 𝒫 ∪ dom 𝑂 → 𝑦 ∈ 𝒫 𝑋)
42		elpwi 4602	. . . . . . . . . . . . . 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 3985	. . . . . . . . . . 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 4598	. . . . . . . . . . . 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 7012	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝒫 ∪ dom 𝑂 ∧ 𝑧 ∈ 𝒫 𝑦) → (𝑂‘𝑧) = 0)
52	8	fvmpt2 7000	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ 𝒫 𝑋 ∧ 0 ∈ ℝ) → (𝑂‘𝑦) = 0)
53	41, 11, 52	sylancl 585	. . . . . . . . . 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 5152	. . . . . . . 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 3138	. . . . . 6 ⊢ (𝑦 ∈ 𝒫 ∪ dom 𝑂 → ∀𝑧 ∈ 𝒫 𝑦(𝑂‘𝑧) ≤ (𝑂‘𝑦))
58	57	rgen 3055	. . . . 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 5252	. . . . . . . . . . 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 2725	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝒫 dom 𝑂 ∧ 𝑧 = ∪ 𝑦) → 0 = 0)
65		id 22	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝒫 dom 𝑂 → 𝑦 ∈ 𝒫 dom 𝑂)
66	15	pweqi 4611	. . . . . . . . . . . . . 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 2827	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝒫 dom 𝑂 → 𝑦 ∈ 𝒫 𝒫 𝑋)
69		elpwi 4602	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝒫 𝒫 𝑋 → 𝑦 ⊆ 𝒫 𝑋)
70	68, 69	syl 17	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝒫 dom 𝑂 → 𝑦 ⊆ 𝒫 𝑋)
71		sspwuni 5094	. . . . . . . . . . 11 ⊢ (𝑦 ⊆ 𝒫 𝑋 ↔ ∪ 𝑦 ⊆ 𝑋)
72	70, 71	sylib 217	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝒫 dom 𝑂 → ∪ 𝑦 ⊆ 𝑋)
73		vuniex 7723	. . . . . . . . . . . 12 ⊢ ∪ 𝑦 ∈ V
74	73	a1i 11	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝒫 dom 𝑂 → ∪ 𝑦 ∈ V)
75		elpwg 4598	. . . . . . . . . . 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 6996	. . . . . . . 8 ⊢ (𝑦 ∈ 𝒫 dom 𝑂 → (𝑂‘∪ 𝑦) = 0)
80	63	reseq1d 5971	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝒫 dom 𝑂 → (𝑂 ↾ 𝑦) = ((𝑧 ∈ 𝒫 𝑋 ↦ 0) ↾ 𝑦))
81		resmpt 6028	. . . . . . . . . . . 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 6886	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝒫 dom 𝑂 → (Σ^{^}‘(𝑂 ↾ 𝑦)) = (Σ^{^}‘(𝑧 ∈ 𝑦 ↦ 0)))
85		nfv 1909	. . . . . . . . . 10 ⊢ Ⅎ𝑧 𝑦 ∈ 𝒫 dom 𝑂
86	85, 65	sge0z 45636	. . . . . . . . 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 5152	. . . . . . 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 3055	. . . 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 5368	. . . . 5 ⊢ (𝜑 → 𝒫 𝑋 ∈ V)
97		mptexg 7215	. . . . 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 2825	. . 3 ⊢ (𝜑 → 𝑂 ∈ V)
100		isome 45755	. . 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 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∀wral 3053 Vcvv 3466 ⊆ wss 3941 ∅c0 4315 𝒫 cpw 4595 ∪ cuni 4900 class class class wbr 5139 ↦ cmpt 5222 dom cdm 5667 ↾ cres 5669 ⟶wf 6530 ‘cfv 6534 (class class class)co 7402 ωcom 7849 ≼ cdom 8934 ℝcr 11106 0cc0 11107 +∞cpnf 11244 ≤ cle 11248 [,]cicc 13328 Σ^{^}csumge0 45623 OutMeascome 45750

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-inf2 9633 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-pre-sup 11185

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-se 5623 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-isom 6543 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-sup 9434 df-oi 9502 df-card 9931 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-div 11871 df-nn 12212 df-2 12274 df-3 12275 df-n0 12472 df-z 12558 df-uz 12822 df-rp 12976 df-ico 13331 df-icc 13332 df-fz 13486 df-fzo 13629 df-seq 13968 df-exp 14029 df-hash 14292 df-cj 15048 df-re 15049 df-im 15050 df-sqrt 15184 df-abs 15185 df-clim 15434 df-sum 15635 df-sumge0 45624 df-ome 45751

This theorem is referenced by: (None)

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