Theorem dscmet 24804

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 11238	. . . . . 6 ⊢ 0 ∈ ℝ
2		1re 11236	. . . . . 6 ⊢ 1 ∈ ℝ
3	1, 2	ifcli 4533	. . . . 5 ⊢ if(𝑥 = 𝑦, 0, 1) ∈ ℝ
4	3	rgen2w 3083	. . . 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 8069	. . . 4 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 if(𝑥 = 𝑦, 0, 1) ∈ ℝ ↔ 𝐷:(𝑋 × 𝑋)⟶ℝ)
7	4, 6	mpbi 233	. . 3 ⊢ 𝐷:(𝑋 × 𝑋)⟶ℝ
8		equequ1 2058	. . . . . . . . 9 ⊢ (𝑥 = 𝑤 → (𝑥 = 𝑦 ↔ 𝑤 = 𝑦))
9	8	ifbid 4509	. . . . . . . 8 ⊢ (𝑥 = 𝑤 → if(𝑥 = 𝑦, 0, 1) = if(𝑤 = 𝑦, 0, 1))
10		equequ2 2059	. . . . . . . . 9 ⊢ (𝑦 = 𝑣 → (𝑤 = 𝑦 ↔ 𝑤 = 𝑣))
11	10	ifbid 4509	. . . . . . . 8 ⊢ (𝑦 = 𝑣 → if(𝑤 = 𝑦, 0, 1) = if(𝑤 = 𝑣, 0, 1))
12		0nn0 12547	. . . . . . . . . 10 ⊢ 0 ∈ ℕ0
13		1nn0 12548	. . . . . . . . . 10 ⊢ 1 ∈ ℕ0
14	12, 13	ifcli 4533	. . . . . . . . 9 ⊢ if(𝑤 = 𝑣, 0, 1) ∈ ℕ0
15	14	elexi 3475	. . . . . . . 8 ⊢ if(𝑤 = 𝑣, 0, 1) ∈ V
16	9, 11, 5, 15	ovmpo 7577	. . . . . . 7 ⊢ ((𝑤 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋) → (𝑤𝐷𝑣) = if(𝑤 = 𝑣, 0, 1))
17	16	eqeq1d 2764	. . . . . 6 ⊢ ((𝑤 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋) → ((𝑤𝐷𝑣) = 0 ↔ if(𝑤 = 𝑣, 0, 1) = 0))
18		iffalse 4494	. . . . . . . . . 10 ⊢ (¬ 𝑤 = 𝑣 → if(𝑤 = 𝑣, 0, 1) = 1)
19		ax-1ne0 11197	. . . . . . . . . . 11 ⊢ 1 ≠ 0
20	19	a1i 11	. . . . . . . . . 10 ⊢ (¬ 𝑤 = 𝑣 → 1 ≠ 0)
21	18, 20	eqnetrd 3024	. . . . . . . . 9 ⊢ (¬ 𝑤 = 𝑣 → if(𝑤 = 𝑣, 0, 1) ≠ 0)
22	21	neneqd 2962	. . . . . . . 8 ⊢ (¬ 𝑤 = 𝑣 → ¬ if(𝑤 = 𝑣, 0, 1) = 0)
23	22	con4i 115	. . . . . . 7 ⊢ (if(𝑤 = 𝑣, 0, 1) = 0 → 𝑤 = 𝑣)
24		iftrue 4491	. . . . . . 7 ⊢ (𝑤 = 𝑣 → if(𝑤 = 𝑣, 0, 1) = 0)
25	23, 24	impbii 212	. . . . . 6 ⊢ (if(𝑤 = 𝑣, 0, 1) = 0 ↔ 𝑤 = 𝑣)
26	17, 25	bitrdi 290	. . . . 5 ⊢ ((𝑤 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋) → ((𝑤𝐷𝑣) = 0 ↔ 𝑤 = 𝑣))
27	12, 13	ifcli 4533	. . . . . . . . . . 11 ⊢ if(𝑢 = 𝑤, 0, 1) ∈ ℕ0
28	12, 13	ifcli 4533	. . . . . . . . . . 11 ⊢ if(𝑢 = 𝑣, 0, 1) ∈ ℕ0
29	27, 28	nn0addcli 12569	. . . . . . . . . 10 ⊢ (if(𝑢 = 𝑤, 0, 1) + if(𝑢 = 𝑣, 0, 1)) ∈ ℕ0
30		elnn0 12534	. . . . . . . . . 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 233	. . . . . . . . 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 5110	. . . . . . . . . . . 12 ⊢ (0 = if(𝑤 = 𝑣, 0, 1) → (0 ≤ 1 ↔ if(𝑤 = 𝑣, 0, 1) ≤ 1))
33		breq1 5110	. . . . . . . . . . . 12 ⊢ (1 = if(𝑤 = 𝑣, 0, 1) → (1 ≤ 1 ↔ if(𝑤 = 𝑣, 0, 1) ≤ 1))
34		0le1 11765	. . . . . . . . . . . 12 ⊢ 0 ≤ 1
35	2	leidi 11776	. . . . . . . . . . . 12 ⊢ 1 ≤ 1
36	32, 33, 34, 35	keephyp 4557	. . . . . . . . . . 11 ⊢ if(𝑤 = 𝑣, 0, 1) ≤ 1
37		nnge1 12292	. . . . . . . . . . 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 12543	. . . . . . . . . . . 12 ⊢ if(𝑤 = 𝑣, 0, 1) ∈ ℝ
39	29	nn0rei 12543	. . . . . . . . . . . 12 ⊢ (if(𝑢 = 𝑤, 0, 1) + if(𝑢 = 𝑣, 0, 1)) ∈ ℝ
40	38, 2, 39	letri 11367	. . . . . . . . . . 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 599	. . . . . . . . . 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 12559	. . . . . . . . . . . . 13 ⊢ 0 ≤ if(𝑢 = 𝑤, 0, 1)
43	28	nn0ge0i 12559	. . . . . . . . . . . . 13 ⊢ 0 ≤ if(𝑢 = 𝑣, 0, 1)
44	27	nn0rei 12543	. . . . . . . . . . . . . 14 ⊢ if(𝑢 = 𝑤, 0, 1) ∈ ℝ
45	28	nn0rei 12543	. . . . . . . . . . . . . 14 ⊢ if(𝑢 = 𝑣, 0, 1) ∈ ℝ
46	44, 45	add20i 11785	. . . . . . . . . . . . 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 705	. . . . . . . . . . . 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 2059	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑣 = 𝑤 → (𝑢 = 𝑣 ↔ 𝑢 = 𝑤))
49	48	ifbid 4509	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 = 𝑤 → if(𝑢 = 𝑣, 0, 1) = if(𝑢 = 𝑤, 0, 1))
50	49	eqeq1d 2764	. . . . . . . . . . . . . . . . 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 2058	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤 = 𝑢 → (𝑤 = 𝑣 ↔ 𝑢 = 𝑣))
53	52	ifbid 4509	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 = 𝑢 → if(𝑤 = 𝑣, 0, 1) = if(𝑢 = 𝑣, 0, 1))
54	53	eqeq1d 2764	. . . . . . . . . . . . . . . . . 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 2022	. . . . . . . . . . . . . . . 16 ⊢ (if(𝑢 = 𝑣, 0, 1) = 0 ↔ 𝑢 = 𝑣)
57	51, 56	chvarvv 2022	. . . . . . . . . . . . . . 15 ⊢ (if(𝑢 = 𝑤, 0, 1) = 0 ↔ 𝑢 = 𝑤)
58		eqtr2 2783	. . . . . . . . . . . . . . 15 ⊢ ((𝑢 = 𝑤 ∧ 𝑢 = 𝑣) → 𝑤 = 𝑣)
59	57, 56, 58	syl2anb 610	. . . . . . . . . . . . . 14 ⊢ ((if(𝑢 = 𝑤, 0, 1) = 0 ∧ if(𝑢 = 𝑣, 0, 1) = 0) → 𝑤 = 𝑣)
60	59	iftrued 4493	. . . . . . . . . . . . 13 ⊢ ((if(𝑢 = 𝑤, 0, 1) = 0 ∧ if(𝑢 = 𝑣, 0, 1) = 0) → if(𝑤 = 𝑣, 0, 1) = 0)
61	1	leidi 11776	. . . . . . . . . . . . 13 ⊢ 0 ≤ 0
62	60, 61	eqbrtrdi 5148	. . . . . . . . . . . 12 ⊢ ((if(𝑢 = 𝑤, 0, 1) = 0 ∧ if(𝑢 = 𝑣, 0, 1) = 0) → if(𝑤 = 𝑣, 0, 1) ≤ 0)
63	47, 62	sylbi 220	. . . . . . . . . . 11 ⊢ ((if(𝑢 = 𝑤, 0, 1) + if(𝑢 = 𝑣, 0, 1)) = 0 → if(𝑤 = 𝑣, 0, 1) ≤ 0)
64		id 23	. . . . . . . . . . 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 5137	. . . . . . . . . 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 871	. . . . . . . . 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 14	. . . . . . . 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 487	. . . . . . . 8 ⊢ ((𝑢 ∈ 𝑋 ∧ (𝑤 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → (𝑤𝐷𝑣) = if(𝑤 = 𝑣, 0, 1))
69		eqeq12 2779	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑤) → (𝑥 = 𝑦 ↔ 𝑢 = 𝑤))
70	69	ifbid 4509	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑤) → if(𝑥 = 𝑦, 0, 1) = if(𝑢 = 𝑤, 0, 1))
71	27	elexi 3475	. . . . . . . . . . 11 ⊢ if(𝑢 = 𝑤, 0, 1) ∈ V
72	70, 5, 71	ovmpoa 7572	. . . . . . . . . 10 ⊢ ((𝑢 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) → (𝑢𝐷𝑤) = if(𝑢 = 𝑤, 0, 1))
73	72	adantrr 730	. . . . . . . . 9 ⊢ ((𝑢 ∈ 𝑋 ∧ (𝑤 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → (𝑢𝐷𝑤) = if(𝑢 = 𝑤, 0, 1))
74		eqeq12 2779	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (𝑥 = 𝑦 ↔ 𝑢 = 𝑣))
75	74	ifbid 4509	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → if(𝑥 = 𝑦, 0, 1) = if(𝑢 = 𝑣, 0, 1))
76	28	elexi 3475	. . . . . . . . . . 11 ⊢ if(𝑢 = 𝑣, 0, 1) ∈ V
77	75, 5, 76	ovmpoa 7572	. . . . . . . . . 10 ⊢ ((𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋) → (𝑢𝐷𝑣) = if(𝑢 = 𝑣, 0, 1))
78	77	adantrl 729	. . . . . . . . 9 ⊢ ((𝑢 ∈ 𝑋 ∧ (𝑤 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → (𝑢𝐷𝑣) = if(𝑢 = 𝑣, 0, 1))
79	73, 78	oveq12d 7435	. . . . . . . 8 ⊢ ((𝑢 ∈ 𝑋 ∧ (𝑤 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → ((𝑢𝐷𝑤) + (𝑢𝐷𝑣)) = (if(𝑢 = 𝑤, 0, 1) + if(𝑢 = 𝑣, 0, 1)))
80	67, 68, 79	3brtr4d 5141	. . . . . . 7 ⊢ ((𝑢 ∈ 𝑋 ∧ (𝑤 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → (𝑤𝐷𝑣) ≤ ((𝑢𝐷𝑤) + (𝑢𝐷𝑣)))
81	80	expcom 419	. . . . . 6 ⊢ ((𝑤 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋) → (𝑢 ∈ 𝑋 → (𝑤𝐷𝑣) ≤ ((𝑢𝐷𝑤) + (𝑢𝐷𝑣))))
82	81	ralrimiv 3155	. . . . 5 ⊢ ((𝑤 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋) → ∀𝑢 ∈ 𝑋 (𝑤𝐷𝑣) ≤ ((𝑢𝐷𝑤) + (𝑢𝐷𝑣)))
83	26, 82	jca 521	. . . 4 ⊢ ((𝑤 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋) → (((𝑤𝐷𝑣) = 0 ↔ 𝑤 = 𝑣) ∧ ∀𝑢 ∈ 𝑋 (𝑤𝐷𝑣) ≤ ((𝑢𝐷𝑤) + (𝑢𝐷𝑣))))
84	83	rgen2 3204	. . 3 ⊢ ∀𝑤 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (((𝑤𝐷𝑣) = 0 ↔ 𝑤 = 𝑣) ∧ ∀𝑢 ∈ 𝑋 (𝑤𝐷𝑣) ≤ ((𝑢𝐷𝑤) + (𝑢𝐷𝑣)))
85	7, 84	pm3.2i 476	. 2 ⊢ (𝐷:(𝑋 × 𝑋)⟶ℝ ∧ ∀𝑤 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (((𝑤𝐷𝑣) = 0 ↔ 𝑤 = 𝑣) ∧ ∀𝑢 ∈ 𝑋 (𝑤𝐷𝑣) ≤ ((𝑢𝐷𝑤) + (𝑢𝐷𝑣))))
86		ismet 24555	. 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 261	1 ⊢ (𝑋 ∈ 𝑉 → 𝐷 ∈ (Met‘𝑋))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2145   ≠ wne 2957  ∀wral 3078  ifcif 4485   class class class wbr 5107   × cxp 5657  ⟶wf 6533  ‘cfv 6537  (class class class)co 7417   ∈ cmpo 7419  ℝcr 11127  0cc0 11128  1c1 11129   + caddc 11131   ≤ cle 11272  ℕcn 12261  ℕ₀cn0 12532  Metcmet 21577

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740  ax-cnex 11184  ax-resscn 11185  ax-1cn 11186  ax-icn 11187  ax-addcl 11188  ax-addrcl 11189  ax-mulcl 11190  ax-mulrcl 11191  ax-mulcom 11192  ax-addass 11193  ax-mulass 11194  ax-distr 11195  ax-i2m1 11196  ax-1ne0 11197  ax-1rid 11198  ax-rnegex 11199  ax-rrecex 11200  ax-cnre 11201  ax-pre-lttri 11202  ax-pre-lttrn 11203  ax-pre-ltadd 11204  ax-pre-mulgt0 11205

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7374  df-ov 7420  df-oprab 7421  df-mpo 7422  df-om 7867  df-1st 7990  df-2nd 7991  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-er 8700  df-map 8832  df-en 8957  df-dom 8958  df-sdom 8959  df-pnf 11273  df-mnf 11274  df-xr 11275  df-ltxr 11276  df-le 11277  df-sub 11471  df-neg 11472  df-nn 12262  df-n0 12533  df-met 21585

This theorem is used by:  dscopn  24805