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Theorem dscmet 24080

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 11215	. . . . . 6 ⊢ 0 ∈ ℝ
2		1re 11213	. . . . . 6 ⊢ 1 ∈ ℝ
3	1, 2	ifcli 4575	. . . . 5 ⊢ if(𝑥 = 𝑦, 0, 1) ∈ ℝ
4	3	rgen2w 3066	. . . 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 8053	. . . 4 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 if(𝑥 = 𝑦, 0, 1) ∈ ℝ ↔ 𝐷:(𝑋 × 𝑋)⟶ℝ)
7	4, 6	mpbi 229	. . 3 ⊢ 𝐷:(𝑋 × 𝑋)⟶ℝ
8		equequ1 2028	. . . . . . . . 9 ⊢ (𝑥 = 𝑤 → (𝑥 = 𝑦 ↔ 𝑤 = 𝑦))
9	8	ifbid 4551	. . . . . . . 8 ⊢ (𝑥 = 𝑤 → if(𝑥 = 𝑦, 0, 1) = if(𝑤 = 𝑦, 0, 1))
10		equequ2 2029	. . . . . . . . 9 ⊢ (𝑦 = 𝑣 → (𝑤 = 𝑦 ↔ 𝑤 = 𝑣))
11	10	ifbid 4551	. . . . . . . 8 ⊢ (𝑦 = 𝑣 → if(𝑤 = 𝑦, 0, 1) = if(𝑤 = 𝑣, 0, 1))
12		0nn0 12486	. . . . . . . . . 10 ⊢ 0 ∈ ℕ₀
13		1nn0 12487	. . . . . . . . . 10 ⊢ 1 ∈ ℕ₀
14	12, 13	ifcli 4575	. . . . . . . . 9 ⊢ if(𝑤 = 𝑣, 0, 1) ∈ ℕ₀
15	14	elexi 3493	. . . . . . . 8 ⊢ if(𝑤 = 𝑣, 0, 1) ∈ V
16	9, 11, 5, 15	ovmpo 7567	. . . . . . 7 ⊢ ((𝑤 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋) → (𝑤𝐷𝑣) = if(𝑤 = 𝑣, 0, 1))
17	16	eqeq1d 2734	. . . . . 6 ⊢ ((𝑤 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋) → ((𝑤𝐷𝑣) = 0 ↔ if(𝑤 = 𝑣, 0, 1) = 0))
18		iffalse 4537	. . . . . . . . . 10 ⊢ (¬ 𝑤 = 𝑣 → if(𝑤 = 𝑣, 0, 1) = 1)
19		ax-1ne0 11178	. . . . . . . . . . 11 ⊢ 1 ≠ 0
20	19	a1i 11	. . . . . . . . . 10 ⊢ (¬ 𝑤 = 𝑣 → 1 ≠ 0)
21	18, 20	eqnetrd 3008	. . . . . . . . 9 ⊢ (¬ 𝑤 = 𝑣 → if(𝑤 = 𝑣, 0, 1) ≠ 0)
22	21	neneqd 2945	. . . . . . . 8 ⊢ (¬ 𝑤 = 𝑣 → ¬ if(𝑤 = 𝑣, 0, 1) = 0)
23	22	con4i 114	. . . . . . 7 ⊢ (if(𝑤 = 𝑣, 0, 1) = 0 → 𝑤 = 𝑣)
24		iftrue 4534	. . . . . . 7 ⊢ (𝑤 = 𝑣 → if(𝑤 = 𝑣, 0, 1) = 0)
25	23, 24	impbii 208	. . . . . 6 ⊢ (if(𝑤 = 𝑣, 0, 1) = 0 ↔ 𝑤 = 𝑣)
26	17, 25	bitrdi 286	. . . . 5 ⊢ ((𝑤 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋) → ((𝑤𝐷𝑣) = 0 ↔ 𝑤 = 𝑣))
27	12, 13	ifcli 4575	. . . . . . . . . . 11 ⊢ if(𝑢 = 𝑤, 0, 1) ∈ ℕ₀
28	12, 13	ifcli 4575	. . . . . . . . . . 11 ⊢ if(𝑢 = 𝑣, 0, 1) ∈ ℕ₀
29	27, 28	nn0addcli 12508	. . . . . . . . . 10 ⊢ (if(𝑢 = 𝑤, 0, 1) + if(𝑢 = 𝑣, 0, 1)) ∈ ℕ₀
30		elnn0 12473	. . . . . . . . . 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 229	. . . . . . . . 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 5151	. . . . . . . . . . . 12 ⊢ (0 = if(𝑤 = 𝑣, 0, 1) → (0 ≤ 1 ↔ if(𝑤 = 𝑣, 0, 1) ≤ 1))
33		breq1 5151	. . . . . . . . . . . 12 ⊢ (1 = if(𝑤 = 𝑣, 0, 1) → (1 ≤ 1 ↔ if(𝑤 = 𝑣, 0, 1) ≤ 1))
34		0le1 11736	. . . . . . . . . . . 12 ⊢ 0 ≤ 1
35	2	leidi 11747	. . . . . . . . . . . 12 ⊢ 1 ≤ 1
36	32, 33, 34, 35	keephyp 4599	. . . . . . . . . . 11 ⊢ if(𝑤 = 𝑣, 0, 1) ≤ 1
37		nnge1 12239	. . . . . . . . . . 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 12482	. . . . . . . . . . . 12 ⊢ if(𝑤 = 𝑣, 0, 1) ∈ ℝ
39	29	nn0rei 12482	. . . . . . . . . . . 12 ⊢ (if(𝑢 = 𝑤, 0, 1) + if(𝑢 = 𝑣, 0, 1)) ∈ ℝ
40	38, 2, 39	letri 11342	. . . . . . . . . . 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 587	. . . . . . . . . 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 12498	. . . . . . . . . . . . 13 ⊢ 0 ≤ if(𝑢 = 𝑤, 0, 1)
43	28	nn0ge0i 12498	. . . . . . . . . . . . 13 ⊢ 0 ≤ if(𝑢 = 𝑣, 0, 1)
44	27	nn0rei 12482	. . . . . . . . . . . . . 14 ⊢ if(𝑢 = 𝑤, 0, 1) ∈ ℝ
45	28	nn0rei 12482	. . . . . . . . . . . . . 14 ⊢ if(𝑢 = 𝑣, 0, 1) ∈ ℝ
46	44, 45	add20i 11756	. . . . . . . . . . . . 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 2029	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑣 = 𝑤 → (𝑢 = 𝑣 ↔ 𝑢 = 𝑤))
49	48	ifbid 4551	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 = 𝑤 → if(𝑢 = 𝑣, 0, 1) = if(𝑢 = 𝑤, 0, 1))
50	49	eqeq1d 2734	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 = 𝑤 → (if(𝑢 = 𝑣, 0, 1) = 0 ↔ if(𝑢 = 𝑤, 0, 1) = 0))
51	50, 48	bibi12d 345	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 = 𝑤 → ((if(𝑢 = 𝑣, 0, 1) = 0 ↔ 𝑢 = 𝑣) ↔ (if(𝑢 = 𝑤, 0, 1) = 0 ↔ 𝑢 = 𝑤)))
52		equequ1 2028	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤 = 𝑢 → (𝑤 = 𝑣 ↔ 𝑢 = 𝑣))
53	52	ifbid 4551	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 = 𝑢 → if(𝑤 = 𝑣, 0, 1) = if(𝑢 = 𝑣, 0, 1))
54	53	eqeq1d 2734	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = 𝑢 → (if(𝑤 = 𝑣, 0, 1) = 0 ↔ if(𝑢 = 𝑣, 0, 1) = 0))
55	54, 52	bibi12d 345	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = 𝑢 → ((if(𝑤 = 𝑣, 0, 1) = 0 ↔ 𝑤 = 𝑣) ↔ (if(𝑢 = 𝑣, 0, 1) = 0 ↔ 𝑢 = 𝑣)))
56	55, 25	chvarvv 2002	. . . . . . . . . . . . . . . 16 ⊢ (if(𝑢 = 𝑣, 0, 1) = 0 ↔ 𝑢 = 𝑣)
57	51, 56	chvarvv 2002	. . . . . . . . . . . . . . 15 ⊢ (if(𝑢 = 𝑤, 0, 1) = 0 ↔ 𝑢 = 𝑤)
58		eqtr2 2756	. . . . . . . . . . . . . . 15 ⊢ ((𝑢 = 𝑤 ∧ 𝑢 = 𝑣) → 𝑤 = 𝑣)
59	57, 56, 58	syl2anb 598	. . . . . . . . . . . . . 14 ⊢ ((if(𝑢 = 𝑤, 0, 1) = 0 ∧ if(𝑢 = 𝑣, 0, 1) = 0) → 𝑤 = 𝑣)
60	59	iftrued 4536	. . . . . . . . . . . . 13 ⊢ ((if(𝑢 = 𝑤, 0, 1) = 0 ∧ if(𝑢 = 𝑣, 0, 1) = 0) → if(𝑤 = 𝑣, 0, 1) = 0)
61	1	leidi 11747	. . . . . . . . . . . . 13 ⊢ 0 ≤ 0
62	60, 61	eqbrtrdi 5187	. . . . . . . . . . . 12 ⊢ ((if(𝑢 = 𝑤, 0, 1) = 0 ∧ if(𝑢 = 𝑣, 0, 1) = 0) → if(𝑤 = 𝑣, 0, 1) ≤ 0)
63	47, 62	sylbi 216	. . . . . . . . . . 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 5176	. . . . . . . . . 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 855	. . . . . . . . 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 2749	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑤) → (𝑥 = 𝑦 ↔ 𝑢 = 𝑤))
70	69	ifbid 4551	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑤) → if(𝑥 = 𝑦, 0, 1) = if(𝑢 = 𝑤, 0, 1))
71	27	elexi 3493	. . . . . . . . . . 11 ⊢ if(𝑢 = 𝑤, 0, 1) ∈ V
72	70, 5, 71	ovmpoa 7562	. . . . . . . . . 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 2749	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (𝑥 = 𝑦 ↔ 𝑢 = 𝑣))
75	74	ifbid 4551	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → if(𝑥 = 𝑦, 0, 1) = if(𝑢 = 𝑣, 0, 1))
76	28	elexi 3493	. . . . . . . . . . 11 ⊢ if(𝑢 = 𝑣, 0, 1) ∈ V
77	75, 5, 76	ovmpoa 7562	. . . . . . . . . 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 7426	. . . . . . . 8 ⊢ ((𝑢 ∈ 𝑋 ∧ (𝑤 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → ((𝑢𝐷𝑤) + (𝑢𝐷𝑣)) = (if(𝑢 = 𝑤, 0, 1) + if(𝑢 = 𝑣, 0, 1)))
80	67, 68, 79	3brtr4d 5180	. . . . . . 7 ⊢ ((𝑢 ∈ 𝑋 ∧ (𝑤 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → (𝑤𝐷𝑣) ≤ ((𝑢𝐷𝑤) + (𝑢𝐷𝑣)))
81	80	expcom 414	. . . . . 6 ⊢ ((𝑤 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋) → (𝑢 ∈ 𝑋 → (𝑤𝐷𝑣) ≤ ((𝑢𝐷𝑤) + (𝑢𝐷𝑣))))
82	81	ralrimiv 3145	. . . . 5 ⊢ ((𝑤 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋) → ∀𝑢 ∈ 𝑋 (𝑤𝐷𝑣) ≤ ((𝑢𝐷𝑤) + (𝑢𝐷𝑣)))
83	26, 82	jca 512	. . . 4 ⊢ ((𝑤 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋) → (((𝑤𝐷𝑣) = 0 ↔ 𝑤 = 𝑣) ∧ ∀𝑢 ∈ 𝑋 (𝑤𝐷𝑣) ≤ ((𝑢𝐷𝑤) + (𝑢𝐷𝑣))))
84	83	rgen2 3197	. . 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 23828	. 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 257	1 ⊢ (𝑋 ∈ 𝑉 → 𝐷 ∈ (Met‘𝑋))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∨ wo 845 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ∀wral 3061 ifcif 4528 class class class wbr 5148 × cxp 5674 ⟶wf 6539 ‘cfv 6543 (class class class)co 7408 ∈ cmpo 7410 ℝcr 11108 0cc0 11109 1c1 11110 + caddc 11112 ≤ cle 11248 ℕcn 12211 ℕ₀cn0 12471 Metcmet 20929

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7855 df-1st 7974 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-er 8702 df-map 8821 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-n0 12472 df-met 20937

This theorem is referenced by: dscopn 24081

W3C validator