Theorem dscmet 23184

Description: The discrete metric on any set 𝑋. Definition 1.1-8 of [Kreyszig] p. 8. (Contributed by FL, 12-Oct-2006.)

Hypothesis

Ref	Expression
dscmet.1	⊢ 𝐷 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ if(𝑥 = 𝑦, 0, 1))

Assertion

Ref	Expression
dscmet	⊢ (𝑋 ∈ 𝑉 → 𝐷 ∈ (Met‘𝑋))

Distinct variable group:   𝑥,𝑦,𝑋

Allowed substitution hints:   𝐷(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem dscmet

Dummy variables 𝑣 𝑢 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0re 10645	. . . . . 6 ⊢ 0 ∈ ℝ
2		1re 10643	. . . . . 6 ⊢ 1 ∈ ℝ
3	1, 2	ifcli 4515	. . . . 5 ⊢ if(𝑥 = 𝑦, 0, 1) ∈ ℝ
4	3	rgen2w 3153	. . . 4 ⊢ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 if(𝑥 = 𝑦, 0, 1) ∈ ℝ
5		dscmet.1	. . . . 5 ⊢ 𝐷 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ if(𝑥 = 𝑦, 0, 1))
6	5	fmpo 7768	. . . 4 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 if(𝑥 = 𝑦, 0, 1) ∈ ℝ ↔ 𝐷:(𝑋 × 𝑋)⟶ℝ)
7	4, 6	mpbi 232	. . 3 ⊢ 𝐷:(𝑋 × 𝑋)⟶ℝ
8		equequ1 2032	. . . . . . . . 9 ⊢ (𝑥 = 𝑤 → (𝑥 = 𝑦 ↔ 𝑤 = 𝑦))
9	8	ifbid 4491	. . . . . . . 8 ⊢ (𝑥 = 𝑤 → if(𝑥 = 𝑦, 0, 1) = if(𝑤 = 𝑦, 0, 1))
10		equequ2 2033	. . . . . . . . 9 ⊢ (𝑦 = 𝑣 → (𝑤 = 𝑦 ↔ 𝑤 = 𝑣))
11	10	ifbid 4491	. . . . . . . 8 ⊢ (𝑦 = 𝑣 → if(𝑤 = 𝑦, 0, 1) = if(𝑤 = 𝑣, 0, 1))
12		0nn0 11915	. . . . . . . . . 10 ⊢ 0 ∈ ℕ0
13		1nn0 11916	. . . . . . . . . 10 ⊢ 1 ∈ ℕ0
14	12, 13	ifcli 4515	. . . . . . . . 9 ⊢ if(𝑤 = 𝑣, 0, 1) ∈ ℕ0
15	14	elexi 3515	. . . . . . . 8 ⊢ if(𝑤 = 𝑣, 0, 1) ∈ V
16	9, 11, 5, 15	ovmpo 7312	. . . . . . 7 ⊢ ((𝑤 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋) → (𝑤𝐷𝑣) = if(𝑤 = 𝑣, 0, 1))
17	16	eqeq1d 2825	. . . . . 6 ⊢ ((𝑤 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋) → ((𝑤𝐷𝑣) = 0 ↔ if(𝑤 = 𝑣, 0, 1) = 0))
18		iffalse 4478	. . . . . . . . . 10 ⊢ (¬ 𝑤 = 𝑣 → if(𝑤 = 𝑣, 0, 1) = 1)
19		ax-1ne0 10608	. . . . . . . . . . 11 ⊢ 1 ≠ 0
20	19	a1i 11	. . . . . . . . . 10 ⊢ (¬ 𝑤 = 𝑣 → 1 ≠ 0)
21	18, 20	eqnetrd 3085	. . . . . . . . 9 ⊢ (¬ 𝑤 = 𝑣 → if(𝑤 = 𝑣, 0, 1) ≠ 0)
22	21	neneqd 3023	. . . . . . . 8 ⊢ (¬ 𝑤 = 𝑣 → ¬ if(𝑤 = 𝑣, 0, 1) = 0)
23	22	con4i 114	. . . . . . 7 ⊢ (if(𝑤 = 𝑣, 0, 1) = 0 → 𝑤 = 𝑣)
24		iftrue 4475	. . . . . . 7 ⊢ (𝑤 = 𝑣 → if(𝑤 = 𝑣, 0, 1) = 0)
25	23, 24	impbii 211	. . . . . 6 ⊢ (if(𝑤 = 𝑣, 0, 1) = 0 ↔ 𝑤 = 𝑣)
26	17, 25	syl6bb 289	. . . . 5 ⊢ ((𝑤 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋) → ((𝑤𝐷𝑣) = 0 ↔ 𝑤 = 𝑣))
27	12, 13	ifcli 4515	. . . . . . . . . . 11 ⊢ if(𝑢 = 𝑤, 0, 1) ∈ ℕ0
28	12, 13	ifcli 4515	. . . . . . . . . . 11 ⊢ if(𝑢 = 𝑣, 0, 1) ∈ ℕ0
29	27, 28	nn0addcli 11937	. . . . . . . . . 10 ⊢ (if(𝑢 = 𝑤, 0, 1) + if(𝑢 = 𝑣, 0, 1)) ∈ ℕ0
30		elnn0 11902	. . . . . . . . . 10 ⊢ ((if(𝑢 = 𝑤, 0, 1) + if(𝑢 = 𝑣, 0, 1)) ∈ ℕ0 ↔ ((if(𝑢 = 𝑤, 0, 1) + if(𝑢 = 𝑣, 0, 1)) ∈ ℕ ∨ (if(𝑢 = 𝑤, 0, 1) + if(𝑢 = 𝑣, 0, 1)) = 0))
31	29, 30	mpbi 232	. . . . . . . . 9 ⊢ ((if(𝑢 = 𝑤, 0, 1) + if(𝑢 = 𝑣, 0, 1)) ∈ ℕ ∨ (if(𝑢 = 𝑤, 0, 1) + if(𝑢 = 𝑣, 0, 1)) = 0)
32		breq1 5071	. . . . . . . . . . . 12 ⊢ (0 = if(𝑤 = 𝑣, 0, 1) → (0 ≤ 1 ↔ if(𝑤 = 𝑣, 0, 1) ≤ 1))
33		breq1 5071	. . . . . . . . . . . 12 ⊢ (1 = if(𝑤 = 𝑣, 0, 1) → (1 ≤ 1 ↔ if(𝑤 = 𝑣, 0, 1) ≤ 1))
34		0le1 11165	. . . . . . . . . . . 12 ⊢ 0 ≤ 1
35	2	leidi 11176	. . . . . . . . . . . 12 ⊢ 1 ≤ 1
36	32, 33, 34, 35	keephyp 4538	. . . . . . . . . . 11 ⊢ if(𝑤 = 𝑣, 0, 1) ≤ 1
37		nnge1 11668	. . . . . . . . . . 11 ⊢ ((if(𝑢 = 𝑤, 0, 1) + if(𝑢 = 𝑣, 0, 1)) ∈ ℕ → 1 ≤ (if(𝑢 = 𝑤, 0, 1) + if(𝑢 = 𝑣, 0, 1)))
38	14	nn0rei 11911	. . . . . . . . . . . 12 ⊢ if(𝑤 = 𝑣, 0, 1) ∈ ℝ
39	29	nn0rei 11911	. . . . . . . . . . . 12 ⊢ (if(𝑢 = 𝑤, 0, 1) + if(𝑢 = 𝑣, 0, 1)) ∈ ℝ
40	38, 2, 39	letri 10771	. . . . . . . . . . 11 ⊢ ((if(𝑤 = 𝑣, 0, 1) ≤ 1 ∧ 1 ≤ (if(𝑢 = 𝑤, 0, 1) + if(𝑢 = 𝑣, 0, 1))) → if(𝑤 = 𝑣, 0, 1) ≤ (if(𝑢 = 𝑤, 0, 1) + if(𝑢 = 𝑣, 0, 1)))
41	36, 37, 40	sylancr 589	. . . . . . . . . 10 ⊢ ((if(𝑢 = 𝑤, 0, 1) + if(𝑢 = 𝑣, 0, 1)) ∈ ℕ → if(𝑤 = 𝑣, 0, 1) ≤ (if(𝑢 = 𝑤, 0, 1) + if(𝑢 = 𝑣, 0, 1)))
42	27	nn0ge0i 11927	. . . . . . . . . . . . 13 ⊢ 0 ≤ if(𝑢 = 𝑤, 0, 1)
43	28	nn0ge0i 11927	. . . . . . . . . . . . 13 ⊢ 0 ≤ if(𝑢 = 𝑣, 0, 1)
44	27	nn0rei 11911	. . . . . . . . . . . . . 14 ⊢ if(𝑢 = 𝑤, 0, 1) ∈ ℝ
45	28	nn0rei 11911	. . . . . . . . . . . . . 14 ⊢ if(𝑢 = 𝑣, 0, 1) ∈ ℝ
46	44, 45	add20i 11185	. . . . . . . . . . . . 13 ⊢ ((0 ≤ if(𝑢 = 𝑤, 0, 1) ∧ 0 ≤ if(𝑢 = 𝑣, 0, 1)) → ((if(𝑢 = 𝑤, 0, 1) + if(𝑢 = 𝑣, 0, 1)) = 0 ↔ (if(𝑢 = 𝑤, 0, 1) = 0 ∧ if(𝑢 = 𝑣, 0, 1) = 0)))
47	42, 43, 46	mp2an 690	. . . . . . . . . . . 12 ⊢ ((if(𝑢 = 𝑤, 0, 1) + if(𝑢 = 𝑣, 0, 1)) = 0 ↔ (if(𝑢 = 𝑤, 0, 1) = 0 ∧ if(𝑢 = 𝑣, 0, 1) = 0))
48		equequ2 2033	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑣 = 𝑤 → (𝑢 = 𝑣 ↔ 𝑢 = 𝑤))
49	48	ifbid 4491	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 = 𝑤 → if(𝑢 = 𝑣, 0, 1) = if(𝑢 = 𝑤, 0, 1))
50	49	eqeq1d 2825	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 = 𝑤 → (if(𝑢 = 𝑣, 0, 1) = 0 ↔ if(𝑢 = 𝑤, 0, 1) = 0))
51	50, 48	bibi12d 348	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 = 𝑤 → ((if(𝑢 = 𝑣, 0, 1) = 0 ↔ 𝑢 = 𝑣) ↔ (if(𝑢 = 𝑤, 0, 1) = 0 ↔ 𝑢 = 𝑤)))
52		equequ1 2032	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤 = 𝑢 → (𝑤 = 𝑣 ↔ 𝑢 = 𝑣))
53	52	ifbid 4491	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 = 𝑢 → if(𝑤 = 𝑣, 0, 1) = if(𝑢 = 𝑣, 0, 1))
54	53	eqeq1d 2825	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = 𝑢 → (if(𝑤 = 𝑣, 0, 1) = 0 ↔ if(𝑢 = 𝑣, 0, 1) = 0))
55	54, 52	bibi12d 348	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = 𝑢 → ((if(𝑤 = 𝑣, 0, 1) = 0 ↔ 𝑤 = 𝑣) ↔ (if(𝑢 = 𝑣, 0, 1) = 0 ↔ 𝑢 = 𝑣)))
56	55, 25	chvarvv 2005	. . . . . . . . . . . . . . . 16 ⊢ (if(𝑢 = 𝑣, 0, 1) = 0 ↔ 𝑢 = 𝑣)
57	51, 56	chvarvv 2005	. . . . . . . . . . . . . . 15 ⊢ (if(𝑢 = 𝑤, 0, 1) = 0 ↔ 𝑢 = 𝑤)
58		eqtr2 2844	. . . . . . . . . . . . . . 15 ⊢ ((𝑢 = 𝑤 ∧ 𝑢 = 𝑣) → 𝑤 = 𝑣)
59	57, 56, 58	syl2anb 599	. . . . . . . . . . . . . 14 ⊢ ((if(𝑢 = 𝑤, 0, 1) = 0 ∧ if(𝑢 = 𝑣, 0, 1) = 0) → 𝑤 = 𝑣)
60	59	iftrued 4477	. . . . . . . . . . . . 13 ⊢ ((if(𝑢 = 𝑤, 0, 1) = 0 ∧ if(𝑢 = 𝑣, 0, 1) = 0) → if(𝑤 = 𝑣, 0, 1) = 0)
61	1	leidi 11176	. . . . . . . . . . . . 13 ⊢ 0 ≤ 0
62	60, 61	eqbrtrdi 5107	. . . . . . . . . . . 12 ⊢ ((if(𝑢 = 𝑤, 0, 1) = 0 ∧ if(𝑢 = 𝑣, 0, 1) = 0) → if(𝑤 = 𝑣, 0, 1) ≤ 0)
63	47, 62	sylbi 219	. . . . . . . . . . 11 ⊢ ((if(𝑢 = 𝑤, 0, 1) + if(𝑢 = 𝑣, 0, 1)) = 0 → if(𝑤 = 𝑣, 0, 1) ≤ 0)
64		id 22	. . . . . . . . . . 11 ⊢ ((if(𝑢 = 𝑤, 0, 1) + if(𝑢 = 𝑣, 0, 1)) = 0 → (if(𝑢 = 𝑤, 0, 1) + if(𝑢 = 𝑣, 0, 1)) = 0)
65	63, 64	breqtrrd 5096	. . . . . . . . . 10 ⊢ ((if(𝑢 = 𝑤, 0, 1) + if(𝑢 = 𝑣, 0, 1)) = 0 → if(𝑤 = 𝑣, 0, 1) ≤ (if(𝑢 = 𝑤, 0, 1) + if(𝑢 = 𝑣, 0, 1)))
66	41, 65	jaoi 853	. . . . . . . . 9 ⊢ (((if(𝑢 = 𝑤, 0, 1) + if(𝑢 = 𝑣, 0, 1)) ∈ ℕ ∨ (if(𝑢 = 𝑤, 0, 1) + if(𝑢 = 𝑣, 0, 1)) = 0) → if(𝑤 = 𝑣, 0, 1) ≤ (if(𝑢 = 𝑤, 0, 1) + if(𝑢 = 𝑣, 0, 1)))
67	31, 66	mp1i 13	. . . . . . . 8 ⊢ ((𝑢 ∈ 𝑋 ∧ (𝑤 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → if(𝑤 = 𝑣, 0, 1) ≤ (if(𝑢 = 𝑤, 0, 1) + if(𝑢 = 𝑣, 0, 1)))
68	16	adantl 484	. . . . . . . 8 ⊢ ((𝑢 ∈ 𝑋 ∧ (𝑤 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → (𝑤𝐷𝑣) = if(𝑤 = 𝑣, 0, 1))
69		eqeq12 2837	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑤) → (𝑥 = 𝑦 ↔ 𝑢 = 𝑤))
70	69	ifbid 4491	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑤) → if(𝑥 = 𝑦, 0, 1) = if(𝑢 = 𝑤, 0, 1))
71	27	elexi 3515	. . . . . . . . . . 11 ⊢ if(𝑢 = 𝑤, 0, 1) ∈ V
72	70, 5, 71	ovmpoa 7307	. . . . . . . . . 10 ⊢ ((𝑢 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) → (𝑢𝐷𝑤) = if(𝑢 = 𝑤, 0, 1))
73	72	adantrr 715	. . . . . . . . 9 ⊢ ((𝑢 ∈ 𝑋 ∧ (𝑤 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → (𝑢𝐷𝑤) = if(𝑢 = 𝑤, 0, 1))
74		eqeq12 2837	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (𝑥 = 𝑦 ↔ 𝑢 = 𝑣))
75	74	ifbid 4491	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → if(𝑥 = 𝑦, 0, 1) = if(𝑢 = 𝑣, 0, 1))
76	28	elexi 3515	. . . . . . . . . . 11 ⊢ if(𝑢 = 𝑣, 0, 1) ∈ V
77	75, 5, 76	ovmpoa 7307	. . . . . . . . . 10 ⊢ ((𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋) → (𝑢𝐷𝑣) = if(𝑢 = 𝑣, 0, 1))
78	77	adantrl 714	. . . . . . . . 9 ⊢ ((𝑢 ∈ 𝑋 ∧ (𝑤 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → (𝑢𝐷𝑣) = if(𝑢 = 𝑣, 0, 1))
79	73, 78	oveq12d 7176	. . . . . . . 8 ⊢ ((𝑢 ∈ 𝑋 ∧ (𝑤 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → ((𝑢𝐷𝑤) + (𝑢𝐷𝑣)) = (if(𝑢 = 𝑤, 0, 1) + if(𝑢 = 𝑣, 0, 1)))
80	67, 68, 79	3brtr4d 5100	. . . . . . 7 ⊢ ((𝑢 ∈ 𝑋 ∧ (𝑤 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → (𝑤𝐷𝑣) ≤ ((𝑢𝐷𝑤) + (𝑢𝐷𝑣)))
81	80	expcom 416	. . . . . 6 ⊢ ((𝑤 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋) → (𝑢 ∈ 𝑋 → (𝑤𝐷𝑣) ≤ ((𝑢𝐷𝑤) + (𝑢𝐷𝑣))))
82	81	ralrimiv 3183	. . . . 5 ⊢ ((𝑤 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋) → ∀𝑢 ∈ 𝑋 (𝑤𝐷𝑣) ≤ ((𝑢𝐷𝑤) + (𝑢𝐷𝑣)))
83	26, 82	jca 514	. . . 4 ⊢ ((𝑤 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋) → (((𝑤𝐷𝑣) = 0 ↔ 𝑤 = 𝑣) ∧ ∀𝑢 ∈ 𝑋 (𝑤𝐷𝑣) ≤ ((𝑢𝐷𝑤) + (𝑢𝐷𝑣))))
84	83	rgen2 3205	. . 3 ⊢ ∀𝑤 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (((𝑤𝐷𝑣) = 0 ↔ 𝑤 = 𝑣) ∧ ∀𝑢 ∈ 𝑋 (𝑤𝐷𝑣) ≤ ((𝑢𝐷𝑤) + (𝑢𝐷𝑣)))
85	7, 84	pm3.2i 473	. 2 ⊢ (𝐷:(𝑋 × 𝑋)⟶ℝ ∧ ∀𝑤 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (((𝑤𝐷𝑣) = 0 ↔ 𝑤 = 𝑣) ∧ ∀𝑢 ∈ 𝑋 (𝑤𝐷𝑣) ≤ ((𝑢𝐷𝑤) + (𝑢𝐷𝑣))))
86		ismet 22935	. 2 ⊢ (𝑋 ∈ 𝑉 → (𝐷 ∈ (Met‘𝑋) ↔ (𝐷:(𝑋 × 𝑋)⟶ℝ ∧ ∀𝑤 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (((𝑤𝐷𝑣) = 0 ↔ 𝑤 = 𝑣) ∧ ∀𝑢 ∈ 𝑋 (𝑤𝐷𝑣) ≤ ((𝑢𝐷𝑤) + (𝑢𝐷𝑣))))))
87	85, 86	mpbiri 260	1 ⊢ (𝑋 ∈ 𝑉 → 𝐷 ∈ (Met‘𝑋))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∨ wo 843   = wceq 1537   ∈ wcel 2114   ≠ wne 3018  ∀wral 3140  ifcif 4469   class class class wbr 5068   × cxp 5555  ⟶wf 6353  ‘cfv 6357  (class class class)co 7158   ∈ cmpo 7160  ℝcr 10538  0cc0 10539  1c1 10540   + caddc 10542   ≤ cle 10678  ℕcn 11640  ℕ₀cn0 11900  Metcmet 20533

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463  ax-cnex 10595  ax-resscn 10596  ax-1cn 10597  ax-icn 10598  ax-addcl 10599  ax-addrcl 10600  ax-mulcl 10601  ax-mulrcl 10602  ax-mulcom 10603  ax-addass 10604  ax-mulass 10605  ax-distr 10606  ax-i2m1 10607  ax-1ne0 10608  ax-1rid 10609  ax-rnegex 10610  ax-rrecex 10611  ax-cnre 10612  ax-pre-lttri 10613  ax-pre-lttrn 10614  ax-pre-ltadd 10615  ax-pre-mulgt0 10616

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-nel 3126  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-riota 7116  df-ov 7161  df-oprab 7162  df-mpo 7163  df-om 7583  df-1st 7691  df-2nd 7692  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-er 8291  df-map 8410  df-en 8512  df-dom 8513  df-sdom 8514  df-pnf 10679  df-mnf 10680  df-xr 10681  df-ltxr 10682  df-le 10683  df-sub 10874  df-neg 10875  df-nn 11641  df-n0 11901  df-met 20541

This theorem is referenced by:  dscopn  23185