Definition df-span 31295

Description: Define the linear span of a subset of Hilbert space. Definition of span in [Schechter] p. 276. See spanval 31319 for its value. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.)

Assertion

Ref	Expression
df-span	⊢ span = (𝑥 ∈ 𝒫 ℋ ↦ ∩ {𝑦 ∈ Sℋ ∣ 𝑥 ⊆ 𝑦})

Distinct variable group:   𝑥,𝑦

Detailed syntax breakdown of Definition df-span

Step	Hyp	Ref	Expression
1		cspn 30918	. 2 class span
2		vx	. . 3 setvar 𝑥
3		chba 30905	. . . 4 class ℋ
4	3	cpw 4580	. . 3 class 𝒫 ℋ
5	2	cv 1539	. . . . . 6 class 𝑥
6		vy	. . . . . . 7 setvar 𝑦
7	6	cv 1539	. . . . . 6 class 𝑦
8	5, 7	wss 3931	. . . . 5 wff 𝑥 ⊆ 𝑦
9		csh 30914	. . . . 5 class Sℋ
10	8, 6, 9	crab 3420	. . . 4 class {𝑦 ∈ Sℋ ∣ 𝑥 ⊆ 𝑦}
11	10	cint 4927	. . 3 class ∩ {𝑦 ∈ Sℋ ∣ 𝑥 ⊆ 𝑦}
12	2, 4, 11	cmpt 5206	. 2 class (𝑥 ∈ 𝒫 ℋ ↦ ∩ {𝑦 ∈ Sℋ ∣ 𝑥 ⊆ 𝑦})
13	1, 12	wceq 1540	1 wff span = (𝑥 ∈ 𝒫 ℋ ↦ ∩ {𝑦 ∈ Sℋ ∣ 𝑥 ⊆ 𝑦})

This definition is referenced by:  spanval  31319