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Definition df-met 19509
Description: Define the (proper) class of all metrics. (A metric space is the metric's base set paired with the metric; see df-ms 21883. However, we will often also call the metric itself a "metric space".) Equivalent to Definition 14-1.1 of [Gleason] p. 223. The 4 properties in Gleason's definition are shown by met0 21905, metgt0 21921, metsym 21912, and mettri 21914. (Contributed by NM, 25-Aug-2006.)
Assertion
Ref Expression
df-met Met = (𝑥 ∈ V ↦ {𝑑 ∈ (ℝ ↑𝑚 (𝑥 × 𝑥)) ∣ ∀𝑦𝑥𝑧𝑥 (((𝑦𝑑𝑧) = 0 ↔ 𝑦 = 𝑧) ∧ ∀𝑤𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) + (𝑤𝑑𝑧)))})
Distinct variable group:   𝑥,𝑑,𝑦,𝑧,𝑤

Detailed syntax breakdown of Definition df-met
StepHypRef Expression
1 cme 19501 . 2 class Met
2 vx . . 3 setvar 𝑥
3 cvv 3172 . . 3 class V
4 vy . . . . . . . . . . 11 setvar 𝑦
54cv 1473 . . . . . . . . . 10 class 𝑦
6 vz . . . . . . . . . . 11 setvar 𝑧
76cv 1473 . . . . . . . . . 10 class 𝑧
8 vd . . . . . . . . . . 11 setvar 𝑑
98cv 1473 . . . . . . . . . 10 class 𝑑
105, 7, 9co 6526 . . . . . . . . 9 class (𝑦𝑑𝑧)
11 cc0 9792 . . . . . . . . 9 class 0
1210, 11wceq 1474 . . . . . . . 8 wff (𝑦𝑑𝑧) = 0
134, 6weq 1860 . . . . . . . 8 wff 𝑦 = 𝑧
1412, 13wb 194 . . . . . . 7 wff ((𝑦𝑑𝑧) = 0 ↔ 𝑦 = 𝑧)
15 vw . . . . . . . . . . . 12 setvar 𝑤
1615cv 1473 . . . . . . . . . . 11 class 𝑤
1716, 5, 9co 6526 . . . . . . . . . 10 class (𝑤𝑑𝑦)
1816, 7, 9co 6526 . . . . . . . . . 10 class (𝑤𝑑𝑧)
19 caddc 9795 . . . . . . . . . 10 class +
2017, 18, 19co 6526 . . . . . . . . 9 class ((𝑤𝑑𝑦) + (𝑤𝑑𝑧))
21 cle 9931 . . . . . . . . 9 class
2210, 20, 21wbr 4577 . . . . . . . 8 wff (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) + (𝑤𝑑𝑧))
232cv 1473 . . . . . . . 8 class 𝑥
2422, 15, 23wral 2895 . . . . . . 7 wff 𝑤𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) + (𝑤𝑑𝑧))
2514, 24wa 382 . . . . . 6 wff (((𝑦𝑑𝑧) = 0 ↔ 𝑦 = 𝑧) ∧ ∀𝑤𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) + (𝑤𝑑𝑧)))
2625, 6, 23wral 2895 . . . . 5 wff 𝑧𝑥 (((𝑦𝑑𝑧) = 0 ↔ 𝑦 = 𝑧) ∧ ∀𝑤𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) + (𝑤𝑑𝑧)))
2726, 4, 23wral 2895 . . . 4 wff 𝑦𝑥𝑧𝑥 (((𝑦𝑑𝑧) = 0 ↔ 𝑦 = 𝑧) ∧ ∀𝑤𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) + (𝑤𝑑𝑧)))
28 cr 9791 . . . . 5 class
2923, 23cxp 5025 . . . . 5 class (𝑥 × 𝑥)
30 cmap 7721 . . . . 5 class 𝑚
3128, 29, 30co 6526 . . . 4 class (ℝ ↑𝑚 (𝑥 × 𝑥))
3227, 8, 31crab 2899 . . 3 class {𝑑 ∈ (ℝ ↑𝑚 (𝑥 × 𝑥)) ∣ ∀𝑦𝑥𝑧𝑥 (((𝑦𝑑𝑧) = 0 ↔ 𝑦 = 𝑧) ∧ ∀𝑤𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) + (𝑤𝑑𝑧)))}
332, 3, 32cmpt 4637 . 2 class (𝑥 ∈ V ↦ {𝑑 ∈ (ℝ ↑𝑚 (𝑥 × 𝑥)) ∣ ∀𝑦𝑥𝑧𝑥 (((𝑦𝑑𝑧) = 0 ↔ 𝑦 = 𝑧) ∧ ∀𝑤𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) + (𝑤𝑑𝑧)))})
341, 33wceq 1474 1 wff Met = (𝑥 ∈ V ↦ {𝑑 ∈ (ℝ ↑𝑚 (𝑥 × 𝑥)) ∣ ∀𝑦𝑥𝑧𝑥 (((𝑦𝑑𝑧) = 0 ↔ 𝑦 = 𝑧) ∧ ∀𝑤𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) + (𝑤𝑑𝑧)))})
Colors of variables: wff setvar class
This definition is referenced by:  ismet  21885
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