Definition df-xmet 12157

Description: Define the set of all extended metrics on a given base set. The definition is similar to df-met 12158, but we also allow the metric to take on the value +∞. (Contributed by Mario Carneiro, 20-Aug-2015.)

Assertion

Ref	Expression
df-xmet	⊢ ∞Met = (𝑥 ∈ V ↦ {𝑑 ∈ (ℝ* ↑𝑚 (𝑥 × 𝑥)) ∣ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (((𝑦𝑑𝑧) = 0 ↔ 𝑦 = 𝑧) ∧ ∀𝑤 ∈ 𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) +𝑒 (𝑤𝑑𝑧)))})

Distinct variable group:   𝑥,𝑑,𝑦,𝑧,𝑤

Detailed syntax breakdown of Definition df-xmet

Step	Hyp	Ref	Expression
1		cxmet 12149	. 2 class ∞Met
2		vx	. . 3 setvar 𝑥
3		cvv 2686	. . 3 class V
4		vy	. . . . . . . . . . 11 setvar 𝑦
5	4	cv 1330	. . . . . . . . . 10 class 𝑦
6		vz	. . . . . . . . . . 11 setvar 𝑧
7	6	cv 1330	. . . . . . . . . 10 class 𝑧
8		vd	. . . . . . . . . . 11 setvar 𝑑
9	8	cv 1330	. . . . . . . . . 10 class 𝑑
10	5, 7, 9	co 5774	. . . . . . . . 9 class (𝑦𝑑𝑧)
11		cc0 7620	. . . . . . . . 9 class 0
12	10, 11	wceq 1331	. . . . . . . 8 wff (𝑦𝑑𝑧) = 0
13	4, 6	weq 1479	. . . . . . . 8 wff 𝑦 = 𝑧
14	12, 13	wb 104	. . . . . . 7 wff ((𝑦𝑑𝑧) = 0 ↔ 𝑦 = 𝑧)
15		vw	. . . . . . . . . . . 12 setvar 𝑤
16	15	cv 1330	. . . . . . . . . . 11 class 𝑤
17	16, 5, 9	co 5774	. . . . . . . . . 10 class (𝑤𝑑𝑦)
18	16, 7, 9	co 5774	. . . . . . . . . 10 class (𝑤𝑑𝑧)
19		cxad 9557	. . . . . . . . . 10 class +𝑒
20	17, 18, 19	co 5774	. . . . . . . . 9 class ((𝑤𝑑𝑦) +𝑒 (𝑤𝑑𝑧))
21		cle 7801	. . . . . . . . 9 class ≤
22	10, 20, 21	wbr 3929	. . . . . . . 8 wff (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) +𝑒 (𝑤𝑑𝑧))
23	2	cv 1330	. . . . . . . 8 class 𝑥
24	22, 15, 23	wral 2416	. . . . . . 7 wff ∀𝑤 ∈ 𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) +𝑒 (𝑤𝑑𝑧))
25	14, 24	wa 103	. . . . . 6 wff (((𝑦𝑑𝑧) = 0 ↔ 𝑦 = 𝑧) ∧ ∀𝑤 ∈ 𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) +𝑒 (𝑤𝑑𝑧)))
26	25, 6, 23	wral 2416	. . . . 5 wff ∀𝑧 ∈ 𝑥 (((𝑦𝑑𝑧) = 0 ↔ 𝑦 = 𝑧) ∧ ∀𝑤 ∈ 𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) +𝑒 (𝑤𝑑𝑧)))
27	26, 4, 23	wral 2416	. . . 4 wff ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (((𝑦𝑑𝑧) = 0 ↔ 𝑦 = 𝑧) ∧ ∀𝑤 ∈ 𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) +𝑒 (𝑤𝑑𝑧)))
28		cxr 7799	. . . . 5 class ℝ*
29	23, 23	cxp 4537	. . . . 5 class (𝑥 × 𝑥)
30		cmap 6542	. . . . 5 class ↑𝑚
31	28, 29, 30	co 5774	. . . 4 class (ℝ* ↑𝑚 (𝑥 × 𝑥))
32	27, 8, 31	crab 2420	. . 3 class {𝑑 ∈ (ℝ* ↑𝑚 (𝑥 × 𝑥)) ∣ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (((𝑦𝑑𝑧) = 0 ↔ 𝑦 = 𝑧) ∧ ∀𝑤 ∈ 𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) +𝑒 (𝑤𝑑𝑧)))}
33	2, 3, 32	cmpt 3989	. 2 class (𝑥 ∈ V ↦ {𝑑 ∈ (ℝ* ↑𝑚 (𝑥 × 𝑥)) ∣ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (((𝑦𝑑𝑧) = 0 ↔ 𝑦 = 𝑧) ∧ ∀𝑤 ∈ 𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) +𝑒 (𝑤𝑑𝑧)))})
34	1, 33	wceq 1331	1 wff ∞Met = (𝑥 ∈ V ↦ {𝑑 ∈ (ℝ* ↑𝑚 (𝑥 × 𝑥)) ∣ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (((𝑦𝑑𝑧) = 0 ↔ 𝑦 = 𝑧) ∧ ∀𝑤 ∈ 𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) +𝑒 (𝑤𝑑𝑧)))})

This definition is referenced by:  xmetrel  12512  isxmet  12514  xmetf  12519  xmetunirn  12527