Theorem dscmet 22377

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 10040	. . . . . 6 ⊢ 0 ∈ ℝ
2		1re 10039	. . . . . 6 ⊢ 1 ∈ ℝ
3	1, 2	keepel 4155	. . . . 5 ⊢ if(𝑥 = 𝑦, 0, 1) ∈ ℝ
4	3	rgen2w 2925	. . . 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	fmpt2 7237	. . . 4 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 if(𝑥 = 𝑦, 0, 1) ∈ ℝ ↔ 𝐷:(𝑋 × 𝑋)⟶ℝ)
7	4, 6	mpbi 220	. . 3 ⊢ 𝐷:(𝑋 × 𝑋)⟶ℝ
8		equequ1 1952	. . . . . . . . 9 ⊢ (𝑥 = 𝑤 → (𝑥 = 𝑦 ↔ 𝑤 = 𝑦))
9	8	ifbid 4108	. . . . . . . 8 ⊢ (𝑥 = 𝑤 → if(𝑥 = 𝑦, 0, 1) = if(𝑤 = 𝑦, 0, 1))
10		equequ2 1953	. . . . . . . . 9 ⊢ (𝑦 = 𝑣 → (𝑤 = 𝑦 ↔ 𝑤 = 𝑣))
11	10	ifbid 4108	. . . . . . . 8 ⊢ (𝑦 = 𝑣 → if(𝑤 = 𝑦, 0, 1) = if(𝑤 = 𝑣, 0, 1))
12		0nn0 11307	. . . . . . . . . 10 ⊢ 0 ∈ ℕ0
13		1nn0 11308	. . . . . . . . . 10 ⊢ 1 ∈ ℕ0
14	12, 13	keepel 4155	. . . . . . . . 9 ⊢ if(𝑤 = 𝑣, 0, 1) ∈ ℕ0
15	14	elexi 3213	. . . . . . . 8 ⊢ if(𝑤 = 𝑣, 0, 1) ∈ V
16	9, 11, 5, 15	ovmpt2 6796	. . . . . . 7 ⊢ ((𝑤 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋) → (𝑤𝐷𝑣) = if(𝑤 = 𝑣, 0, 1))
17	16	eqeq1d 2624	. . . . . 6 ⊢ ((𝑤 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋) → ((𝑤𝐷𝑣) = 0 ↔ if(𝑤 = 𝑣, 0, 1) = 0))
18		iffalse 4095	. . . . . . . . . 10 ⊢ (¬ 𝑤 = 𝑣 → if(𝑤 = 𝑣, 0, 1) = 1)
19		ax-1ne0 10005	. . . . . . . . . . 11 ⊢ 1 ≠ 0
20	19	a1i 11	. . . . . . . . . 10 ⊢ (¬ 𝑤 = 𝑣 → 1 ≠ 0)
21	18, 20	eqnetrd 2861	. . . . . . . . 9 ⊢ (¬ 𝑤 = 𝑣 → if(𝑤 = 𝑣, 0, 1) ≠ 0)
22	21	neneqd 2799	. . . . . . . 8 ⊢ (¬ 𝑤 = 𝑣 → ¬ if(𝑤 = 𝑣, 0, 1) = 0)
23	22	con4i 113	. . . . . . 7 ⊢ (if(𝑤 = 𝑣, 0, 1) = 0 → 𝑤 = 𝑣)
24		iftrue 4092	. . . . . . 7 ⊢ (𝑤 = 𝑣 → if(𝑤 = 𝑣, 0, 1) = 0)
25	23, 24	impbii 199	. . . . . 6 ⊢ (if(𝑤 = 𝑣, 0, 1) = 0 ↔ 𝑤 = 𝑣)
26	17, 25	syl6bb 276	. . . . 5 ⊢ ((𝑤 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋) → ((𝑤𝐷𝑣) = 0 ↔ 𝑤 = 𝑣))
27	12, 13	keepel 4155	. . . . . . . . . . 11 ⊢ if(𝑢 = 𝑤, 0, 1) ∈ ℕ0
28	12, 13	keepel 4155	. . . . . . . . . . 11 ⊢ if(𝑢 = 𝑣, 0, 1) ∈ ℕ0
29	27, 28	nn0addcli 11330	. . . . . . . . . 10 ⊢ (if(𝑢 = 𝑤, 0, 1) + if(𝑢 = 𝑣, 0, 1)) ∈ ℕ0
30		elnn0 11294	. . . . . . . . . 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 220	. . . . . . . . 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 4656	. . . . . . . . . . . 12 ⊢ (0 = if(𝑤 = 𝑣, 0, 1) → (0 ≤ 1 ↔ if(𝑤 = 𝑣, 0, 1) ≤ 1))
33		breq1 4656	. . . . . . . . . . . 12 ⊢ (1 = if(𝑤 = 𝑣, 0, 1) → (1 ≤ 1 ↔ if(𝑤 = 𝑣, 0, 1) ≤ 1))
34		0le1 10551	. . . . . . . . . . . 12 ⊢ 0 ≤ 1
35	2	leidi 10562	. . . . . . . . . . . 12 ⊢ 1 ≤ 1
36	32, 33, 34, 35	keephyp 4152	. . . . . . . . . . 11 ⊢ if(𝑤 = 𝑣, 0, 1) ≤ 1
37		nnge1 11046	. . . . . . . . . . 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 11303	. . . . . . . . . . . 12 ⊢ if(𝑤 = 𝑣, 0, 1) ∈ ℝ
39	29	nn0rei 11303	. . . . . . . . . . . 12 ⊢ (if(𝑢 = 𝑤, 0, 1) + if(𝑢 = 𝑣, 0, 1)) ∈ ℝ
40	38, 2, 39	letri 10166	. . . . . . . . . . 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 695	. . . . . . . . . 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 11320	. . . . . . . . . . . . 13 ⊢ 0 ≤ if(𝑢 = 𝑤, 0, 1)
43	28	nn0ge0i 11320	. . . . . . . . . . . . 13 ⊢ 0 ≤ if(𝑢 = 𝑣, 0, 1)
44	27	nn0rei 11303	. . . . . . . . . . . . . 14 ⊢ if(𝑢 = 𝑤, 0, 1) ∈ ℝ
45	28	nn0rei 11303	. . . . . . . . . . . . . 14 ⊢ if(𝑢 = 𝑣, 0, 1) ∈ ℝ
46	44, 45	add20i 10571	. . . . . . . . . . . . 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 708	. . . . . . . . . . . 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 1953	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑣 = 𝑤 → (𝑢 = 𝑣 ↔ 𝑢 = 𝑤))
49	48	ifbid 4108	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 = 𝑤 → if(𝑢 = 𝑣, 0, 1) = if(𝑢 = 𝑤, 0, 1))
50	49	eqeq1d 2624	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 = 𝑤 → (if(𝑢 = 𝑣, 0, 1) = 0 ↔ if(𝑢 = 𝑤, 0, 1) = 0))
51	50, 48	bibi12d 335	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 = 𝑤 → ((if(𝑢 = 𝑣, 0, 1) = 0 ↔ 𝑢 = 𝑣) ↔ (if(𝑢 = 𝑤, 0, 1) = 0 ↔ 𝑢 = 𝑤)))
52		equequ1 1952	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤 = 𝑢 → (𝑤 = 𝑣 ↔ 𝑢 = 𝑣))
53	52	ifbid 4108	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 = 𝑢 → if(𝑤 = 𝑣, 0, 1) = if(𝑢 = 𝑣, 0, 1))
54	53	eqeq1d 2624	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = 𝑢 → (if(𝑤 = 𝑣, 0, 1) = 0 ↔ if(𝑢 = 𝑣, 0, 1) = 0))
55	54, 52	bibi12d 335	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = 𝑢 → ((if(𝑤 = 𝑣, 0, 1) = 0 ↔ 𝑤 = 𝑣) ↔ (if(𝑢 = 𝑣, 0, 1) = 0 ↔ 𝑢 = 𝑣)))
56	55, 25	chvarv 2263	. . . . . . . . . . . . . . . 16 ⊢ (if(𝑢 = 𝑣, 0, 1) = 0 ↔ 𝑢 = 𝑣)
57	51, 56	chvarv 2263	. . . . . . . . . . . . . . 15 ⊢ (if(𝑢 = 𝑤, 0, 1) = 0 ↔ 𝑢 = 𝑤)
58		eqtr2 2642	. . . . . . . . . . . . . . 15 ⊢ ((𝑢 = 𝑤 ∧ 𝑢 = 𝑣) → 𝑤 = 𝑣)
59	57, 56, 58	syl2anb 496	. . . . . . . . . . . . . 14 ⊢ ((if(𝑢 = 𝑤, 0, 1) = 0 ∧ if(𝑢 = 𝑣, 0, 1) = 0) → 𝑤 = 𝑣)
60	59	iftrued 4094	. . . . . . . . . . . . 13 ⊢ ((if(𝑢 = 𝑤, 0, 1) = 0 ∧ if(𝑢 = 𝑣, 0, 1) = 0) → if(𝑤 = 𝑣, 0, 1) = 0)
61	1	leidi 10562	. . . . . . . . . . . . 13 ⊢ 0 ≤ 0
62	60, 61	syl6eqbr 4692	. . . . . . . . . . . 12 ⊢ ((if(𝑢 = 𝑤, 0, 1) = 0 ∧ if(𝑢 = 𝑣, 0, 1) = 0) → if(𝑤 = 𝑣, 0, 1) ≤ 0)
63	47, 62	sylbi 207	. . . . . . . . . . 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 4681	. . . . . . . . . 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 394	. . . . . . . . 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 482	. . . . . . . 8 ⊢ ((𝑢 ∈ 𝑋 ∧ (𝑤 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → (𝑤𝐷𝑣) = if(𝑤 = 𝑣, 0, 1))
69		eqeq12 2635	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑤) → (𝑥 = 𝑦 ↔ 𝑢 = 𝑤))
70	69	ifbid 4108	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑤) → if(𝑥 = 𝑦, 0, 1) = if(𝑢 = 𝑤, 0, 1))
71	27	elexi 3213	. . . . . . . . . . 11 ⊢ if(𝑢 = 𝑤, 0, 1) ∈ V
72	70, 5, 71	ovmpt2a 6791	. . . . . . . . . 10 ⊢ ((𝑢 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) → (𝑢𝐷𝑤) = if(𝑢 = 𝑤, 0, 1))
73	72	adantrr 753	. . . . . . . . 9 ⊢ ((𝑢 ∈ 𝑋 ∧ (𝑤 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → (𝑢𝐷𝑤) = if(𝑢 = 𝑤, 0, 1))
74		eqeq12 2635	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (𝑥 = 𝑦 ↔ 𝑢 = 𝑣))
75	74	ifbid 4108	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → if(𝑥 = 𝑦, 0, 1) = if(𝑢 = 𝑣, 0, 1))
76	28	elexi 3213	. . . . . . . . . . 11 ⊢ if(𝑢 = 𝑣, 0, 1) ∈ V
77	75, 5, 76	ovmpt2a 6791	. . . . . . . . . 10 ⊢ ((𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋) → (𝑢𝐷𝑣) = if(𝑢 = 𝑣, 0, 1))
78	77	adantrl 752	. . . . . . . . 9 ⊢ ((𝑢 ∈ 𝑋 ∧ (𝑤 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → (𝑢𝐷𝑣) = if(𝑢 = 𝑣, 0, 1))
79	73, 78	oveq12d 6668	. . . . . . . 8 ⊢ ((𝑢 ∈ 𝑋 ∧ (𝑤 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → ((𝑢𝐷𝑤) + (𝑢𝐷𝑣)) = (if(𝑢 = 𝑤, 0, 1) + if(𝑢 = 𝑣, 0, 1)))
80	67, 68, 79	3brtr4d 4685	. . . . . . 7 ⊢ ((𝑢 ∈ 𝑋 ∧ (𝑤 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → (𝑤𝐷𝑣) ≤ ((𝑢𝐷𝑤) + (𝑢𝐷𝑣)))
81	80	expcom 451	. . . . . 6 ⊢ ((𝑤 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋) → (𝑢 ∈ 𝑋 → (𝑤𝐷𝑣) ≤ ((𝑢𝐷𝑤) + (𝑢𝐷𝑣))))
82	81	ralrimiv 2965	. . . . 5 ⊢ ((𝑤 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋) → ∀𝑢 ∈ 𝑋 (𝑤𝐷𝑣) ≤ ((𝑢𝐷𝑤) + (𝑢𝐷𝑣)))
83	26, 82	jca 554	. . . 4 ⊢ ((𝑤 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋) → (((𝑤𝐷𝑣) = 0 ↔ 𝑤 = 𝑣) ∧ ∀𝑢 ∈ 𝑋 (𝑤𝐷𝑣) ≤ ((𝑢𝐷𝑤) + (𝑢𝐷𝑣))))
84	83	rgen2a 2977	. . 3 ⊢ ∀𝑤 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (((𝑤𝐷𝑣) = 0 ↔ 𝑤 = 𝑣) ∧ ∀𝑢 ∈ 𝑋 (𝑤𝐷𝑣) ≤ ((𝑢𝐷𝑤) + (𝑢𝐷𝑣)))
85	7, 84	pm3.2i 471	. 2 ⊢ (𝐷:(𝑋 × 𝑋)⟶ℝ ∧ ∀𝑤 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (((𝑤𝐷𝑣) = 0 ↔ 𝑤 = 𝑣) ∧ ∀𝑢 ∈ 𝑋 (𝑤𝐷𝑣) ≤ ((𝑢𝐷𝑤) + (𝑢𝐷𝑣))))
86		ismet 22128	. 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 248	1 ⊢ (𝑋 ∈ 𝑉 → 𝐷 ∈ (Met‘𝑋))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  ifcif 4086   class class class wbr 4653   × cxp 5112  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650   ↦ cmpt2 6652  ℝcr 9935  0cc0 9936  1c1 9937   + caddc 9939   ≤ cle 10075  ℕcn 11020  ℕ₀cn0 11292  Metcme 19732

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-met 19740

This theorem is referenced by:  dscopn  22378