Theorem dscmet 22789

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 10380	. . . . . 6 ⊢ 0 ∈ ℝ
2		1re 10378	. . . . . 6 ⊢ 1 ∈ ℝ
3	1, 2	ifcli 4353	. . . . 5 ⊢ if(𝑥 = 𝑦, 0, 1) ∈ ℝ
4	3	rgen2w 3107	. . . 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 7519	. . . 4 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 if(𝑥 = 𝑦, 0, 1) ∈ ℝ ↔ 𝐷:(𝑋 × 𝑋)⟶ℝ)
7	4, 6	mpbi 222	. . 3 ⊢ 𝐷:(𝑋 × 𝑋)⟶ℝ
8		equequ1 2072	. . . . . . . . 9 ⊢ (𝑥 = 𝑤 → (𝑥 = 𝑦 ↔ 𝑤 = 𝑦))
9	8	ifbid 4329	. . . . . . . 8 ⊢ (𝑥 = 𝑤 → if(𝑥 = 𝑦, 0, 1) = if(𝑤 = 𝑦, 0, 1))
10		equequ2 2073	. . . . . . . . 9 ⊢ (𝑦 = 𝑣 → (𝑤 = 𝑦 ↔ 𝑤 = 𝑣))
11	10	ifbid 4329	. . . . . . . 8 ⊢ (𝑦 = 𝑣 → if(𝑤 = 𝑦, 0, 1) = if(𝑤 = 𝑣, 0, 1))
12		0nn0 11663	. . . . . . . . . 10 ⊢ 0 ∈ ℕ0
13		1nn0 11664	. . . . . . . . . 10 ⊢ 1 ∈ ℕ0
14	12, 13	ifcli 4353	. . . . . . . . 9 ⊢ if(𝑤 = 𝑣, 0, 1) ∈ ℕ0
15	14	elexi 3415	. . . . . . . 8 ⊢ if(𝑤 = 𝑣, 0, 1) ∈ V
16	9, 11, 5, 15	ovmpt2 7075	. . . . . . 7 ⊢ ((𝑤 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋) → (𝑤𝐷𝑣) = if(𝑤 = 𝑣, 0, 1))
17	16	eqeq1d 2780	. . . . . 6 ⊢ ((𝑤 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋) → ((𝑤𝐷𝑣) = 0 ↔ if(𝑤 = 𝑣, 0, 1) = 0))
18		iffalse 4316	. . . . . . . . . 10 ⊢ (¬ 𝑤 = 𝑣 → if(𝑤 = 𝑣, 0, 1) = 1)
19		ax-1ne0 10343	. . . . . . . . . . 11 ⊢ 1 ≠ 0
20	19	a1i 11	. . . . . . . . . 10 ⊢ (¬ 𝑤 = 𝑣 → 1 ≠ 0)
21	18, 20	eqnetrd 3036	. . . . . . . . 9 ⊢ (¬ 𝑤 = 𝑣 → if(𝑤 = 𝑣, 0, 1) ≠ 0)
22	21	neneqd 2974	. . . . . . . 8 ⊢ (¬ 𝑤 = 𝑣 → ¬ if(𝑤 = 𝑣, 0, 1) = 0)
23	22	con4i 114	. . . . . . 7 ⊢ (if(𝑤 = 𝑣, 0, 1) = 0 → 𝑤 = 𝑣)
24		iftrue 4313	. . . . . . 7 ⊢ (𝑤 = 𝑣 → if(𝑤 = 𝑣, 0, 1) = 0)
25	23, 24	impbii 201	. . . . . 6 ⊢ (if(𝑤 = 𝑣, 0, 1) = 0 ↔ 𝑤 = 𝑣)
26	17, 25	syl6bb 279	. . . . 5 ⊢ ((𝑤 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋) → ((𝑤𝐷𝑣) = 0 ↔ 𝑤 = 𝑣))
27	12, 13	ifcli 4353	. . . . . . . . . . 11 ⊢ if(𝑢 = 𝑤, 0, 1) ∈ ℕ0
28	12, 13	ifcli 4353	. . . . . . . . . . 11 ⊢ if(𝑢 = 𝑣, 0, 1) ∈ ℕ0
29	27, 28	nn0addcli 11685	. . . . . . . . . 10 ⊢ (if(𝑢 = 𝑤, 0, 1) + if(𝑢 = 𝑣, 0, 1)) ∈ ℕ0
30		elnn0 11648	. . . . . . . . . 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 222	. . . . . . . . 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 4891	. . . . . . . . . . . 12 ⊢ (0 = if(𝑤 = 𝑣, 0, 1) → (0 ≤ 1 ↔ if(𝑤 = 𝑣, 0, 1) ≤ 1))
33		breq1 4891	. . . . . . . . . . . 12 ⊢ (1 = if(𝑤 = 𝑣, 0, 1) → (1 ≤ 1 ↔ if(𝑤 = 𝑣, 0, 1) ≤ 1))
34		0le1 10900	. . . . . . . . . . . 12 ⊢ 0 ≤ 1
35	2	leidi 10911	. . . . . . . . . . . 12 ⊢ 1 ≤ 1
36	32, 33, 34, 35	keephyp 4376	. . . . . . . . . . 11 ⊢ if(𝑤 = 𝑣, 0, 1) ≤ 1
37		nnge1 11408	. . . . . . . . . . 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 11658	. . . . . . . . . . . 12 ⊢ if(𝑤 = 𝑣, 0, 1) ∈ ℝ
39	29	nn0rei 11658	. . . . . . . . . . . 12 ⊢ (if(𝑢 = 𝑤, 0, 1) + if(𝑢 = 𝑣, 0, 1)) ∈ ℝ
40	38, 2, 39	letri 10507	. . . . . . . . . . 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 581	. . . . . . . . . 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 11675	. . . . . . . . . . . . 13 ⊢ 0 ≤ if(𝑢 = 𝑤, 0, 1)
43	28	nn0ge0i 11675	. . . . . . . . . . . . 13 ⊢ 0 ≤ if(𝑢 = 𝑣, 0, 1)
44	27	nn0rei 11658	. . . . . . . . . . . . . 14 ⊢ if(𝑢 = 𝑤, 0, 1) ∈ ℝ
45	28	nn0rei 11658	. . . . . . . . . . . . . 14 ⊢ if(𝑢 = 𝑣, 0, 1) ∈ ℝ
46	44, 45	add20i 10920	. . . . . . . . . . . . 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 682	. . . . . . . . . . . 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 2073	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑣 = 𝑤 → (𝑢 = 𝑣 ↔ 𝑢 = 𝑤))
49	48	ifbid 4329	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 = 𝑤 → if(𝑢 = 𝑣, 0, 1) = if(𝑢 = 𝑤, 0, 1))
50	49	eqeq1d 2780	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 = 𝑤 → (if(𝑢 = 𝑣, 0, 1) = 0 ↔ if(𝑢 = 𝑤, 0, 1) = 0))
51	50, 48	bibi12d 337	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 = 𝑤 → ((if(𝑢 = 𝑣, 0, 1) = 0 ↔ 𝑢 = 𝑣) ↔ (if(𝑢 = 𝑤, 0, 1) = 0 ↔ 𝑢 = 𝑤)))
52		equequ1 2072	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤 = 𝑢 → (𝑤 = 𝑣 ↔ 𝑢 = 𝑣))
53	52	ifbid 4329	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 = 𝑢 → if(𝑤 = 𝑣, 0, 1) = if(𝑢 = 𝑣, 0, 1))
54	53	eqeq1d 2780	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = 𝑢 → (if(𝑤 = 𝑣, 0, 1) = 0 ↔ if(𝑢 = 𝑣, 0, 1) = 0))
55	54, 52	bibi12d 337	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = 𝑢 → ((if(𝑤 = 𝑣, 0, 1) = 0 ↔ 𝑤 = 𝑣) ↔ (if(𝑢 = 𝑣, 0, 1) = 0 ↔ 𝑢 = 𝑣)))
56	55, 25	chvarv 2361	. . . . . . . . . . . . . . . 16 ⊢ (if(𝑢 = 𝑣, 0, 1) = 0 ↔ 𝑢 = 𝑣)
57	51, 56	chvarv 2361	. . . . . . . . . . . . . . 15 ⊢ (if(𝑢 = 𝑤, 0, 1) = 0 ↔ 𝑢 = 𝑤)
58		eqtr2 2800	. . . . . . . . . . . . . . 15 ⊢ ((𝑢 = 𝑤 ∧ 𝑢 = 𝑣) → 𝑤 = 𝑣)
59	57, 56, 58	syl2anb 591	. . . . . . . . . . . . . 14 ⊢ ((if(𝑢 = 𝑤, 0, 1) = 0 ∧ if(𝑢 = 𝑣, 0, 1) = 0) → 𝑤 = 𝑣)
60	59	iftrued 4315	. . . . . . . . . . . . 13 ⊢ ((if(𝑢 = 𝑤, 0, 1) = 0 ∧ if(𝑢 = 𝑣, 0, 1) = 0) → if(𝑤 = 𝑣, 0, 1) = 0)
61	1	leidi 10911	. . . . . . . . . . . . 13 ⊢ 0 ≤ 0
62	60, 61	syl6eqbr 4927	. . . . . . . . . . . 12 ⊢ ((if(𝑢 = 𝑤, 0, 1) = 0 ∧ if(𝑢 = 𝑣, 0, 1) = 0) → if(𝑤 = 𝑣, 0, 1) ≤ 0)
63	47, 62	sylbi 209	. . . . . . . . . . 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 4916	. . . . . . . . . 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 846	. . . . . . . . 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 475	. . . . . . . 8 ⊢ ((𝑢 ∈ 𝑋 ∧ (𝑤 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → (𝑤𝐷𝑣) = if(𝑤 = 𝑣, 0, 1))
69		eqeq12 2791	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑤) → (𝑥 = 𝑦 ↔ 𝑢 = 𝑤))
70	69	ifbid 4329	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑤) → if(𝑥 = 𝑦, 0, 1) = if(𝑢 = 𝑤, 0, 1))
71	27	elexi 3415	. . . . . . . . . . 11 ⊢ if(𝑢 = 𝑤, 0, 1) ∈ V
72	70, 5, 71	ovmpt2a 7070	. . . . . . . . . 10 ⊢ ((𝑢 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) → (𝑢𝐷𝑤) = if(𝑢 = 𝑤, 0, 1))
73	72	adantrr 707	. . . . . . . . 9 ⊢ ((𝑢 ∈ 𝑋 ∧ (𝑤 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → (𝑢𝐷𝑤) = if(𝑢 = 𝑤, 0, 1))
74		eqeq12 2791	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (𝑥 = 𝑦 ↔ 𝑢 = 𝑣))
75	74	ifbid 4329	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → if(𝑥 = 𝑦, 0, 1) = if(𝑢 = 𝑣, 0, 1))
76	28	elexi 3415	. . . . . . . . . . 11 ⊢ if(𝑢 = 𝑣, 0, 1) ∈ V
77	75, 5, 76	ovmpt2a 7070	. . . . . . . . . 10 ⊢ ((𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋) → (𝑢𝐷𝑣) = if(𝑢 = 𝑣, 0, 1))
78	77	adantrl 706	. . . . . . . . 9 ⊢ ((𝑢 ∈ 𝑋 ∧ (𝑤 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → (𝑢𝐷𝑣) = if(𝑢 = 𝑣, 0, 1))
79	73, 78	oveq12d 6942	. . . . . . . 8 ⊢ ((𝑢 ∈ 𝑋 ∧ (𝑤 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → ((𝑢𝐷𝑤) + (𝑢𝐷𝑣)) = (if(𝑢 = 𝑤, 0, 1) + if(𝑢 = 𝑣, 0, 1)))
80	67, 68, 79	3brtr4d 4920	. . . . . . 7 ⊢ ((𝑢 ∈ 𝑋 ∧ (𝑤 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → (𝑤𝐷𝑣) ≤ ((𝑢𝐷𝑤) + (𝑢𝐷𝑣)))
81	80	expcom 404	. . . . . 6 ⊢ ((𝑤 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋) → (𝑢 ∈ 𝑋 → (𝑤𝐷𝑣) ≤ ((𝑢𝐷𝑤) + (𝑢𝐷𝑣))))
82	81	ralrimiv 3147	. . . . 5 ⊢ ((𝑤 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋) → ∀𝑢 ∈ 𝑋 (𝑤𝐷𝑣) ≤ ((𝑢𝐷𝑤) + (𝑢𝐷𝑣)))
83	26, 82	jca 507	. . . 4 ⊢ ((𝑤 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋) → (((𝑤𝐷𝑣) = 0 ↔ 𝑤 = 𝑣) ∧ ∀𝑢 ∈ 𝑋 (𝑤𝐷𝑣) ≤ ((𝑢𝐷𝑤) + (𝑢𝐷𝑣))))
84	83	rgen2a 3159	. . 3 ⊢ ∀𝑤 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (((𝑤𝐷𝑣) = 0 ↔ 𝑤 = 𝑣) ∧ ∀𝑢 ∈ 𝑋 (𝑤𝐷𝑣) ≤ ((𝑢𝐷𝑤) + (𝑢𝐷𝑣)))
85	7, 84	pm3.2i 464	. 2 ⊢ (𝐷:(𝑋 × 𝑋)⟶ℝ ∧ ∀𝑤 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (((𝑤𝐷𝑣) = 0 ↔ 𝑤 = 𝑣) ∧ ∀𝑢 ∈ 𝑋 (𝑤𝐷𝑣) ≤ ((𝑢𝐷𝑤) + (𝑢𝐷𝑣))))
86		ismet 22540	. 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 250	1 ⊢ (𝑋 ∈ 𝑉 → 𝐷 ∈ (Met‘𝑋))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 386   ∨ wo 836   = wceq 1601   ∈ wcel 2107   ≠ wne 2969  ∀wral 3090  ifcif 4307   class class class wbr 4888   × cxp 5355  ⟶wf 6133  ‘cfv 6137  (class class class)co 6924   ↦ cmpt2 6926  ℝcr 10273  0cc0 10274  1c1 10275   + caddc 10277   ≤ cle 10414  ℕcn 11378  ℕ₀cn0 11646  Metcmet 20132

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228  ax-cnex 10330  ax-resscn 10331  ax-1cn 10332  ax-icn 10333  ax-addcl 10334  ax-addrcl 10335  ax-mulcl 10336  ax-mulrcl 10337  ax-mulcom 10338  ax-addass 10339  ax-mulass 10340  ax-distr 10341  ax-i2m1 10342  ax-1ne0 10343  ax-1rid 10344  ax-rnegex 10345  ax-rrecex 10346  ax-cnre 10347  ax-pre-lttri 10348  ax-pre-lttrn 10349  ax-pre-ltadd 10350  ax-pre-mulgt0 10351

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4674  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-tr 4990  df-id 5263  df-eprel 5268  df-po 5276  df-so 5277  df-fr 5316  df-we 5318  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-pred 5935  df-ord 5981  df-on 5982  df-lim 5983  df-suc 5984  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-riota 6885  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-om 7346  df-1st 7447  df-2nd 7448  df-wrecs 7691  df-recs 7753  df-rdg 7791  df-er 8028  df-map 8144  df-en 8244  df-dom 8245  df-sdom 8246  df-pnf 10415  df-mnf 10416  df-xr 10417  df-ltxr 10418  df-le 10419  df-sub 10610  df-neg 10611  df-nn 11379  df-n0 11647  df-met 20140

This theorem is referenced by:  dscopn  22790