Metamath Proof Explorer


Definition df-lsp

Description: Define span of a set of vectors of a left module or left vector space. (Contributed by NM, 8-Dec-2013)

Ref Expression
Assertion df-lsp LSpan = ( 𝑤 ∈ V ↦ ( 𝑠 ∈ 𝒫 ( Base ‘ 𝑤 ) ↦ { 𝑡 ∈ ( LSubSp ‘ 𝑤 ) ∣ 𝑠𝑡 } ) )

Detailed syntax breakdown

Step Hyp Ref Expression
0 clspn LSpan
1 vw 𝑤
2 cvv V
3 vs 𝑠
4 cbs Base
5 1 cv 𝑤
6 5 4 cfv ( Base ‘ 𝑤 )
7 6 cpw 𝒫 ( Base ‘ 𝑤 )
8 vt 𝑡
9 clss LSubSp
10 5 9 cfv ( LSubSp ‘ 𝑤 )
11 3 cv 𝑠
12 8 cv 𝑡
13 11 12 wss 𝑠𝑡
14 13 8 10 crab { 𝑡 ∈ ( LSubSp ‘ 𝑤 ) ∣ 𝑠𝑡 }
15 14 cint { 𝑡 ∈ ( LSubSp ‘ 𝑤 ) ∣ 𝑠𝑡 }
16 3 7 15 cmpt ( 𝑠 ∈ 𝒫 ( Base ‘ 𝑤 ) ↦ { 𝑡 ∈ ( LSubSp ‘ 𝑤 ) ∣ 𝑠𝑡 } )
17 1 2 16 cmpt ( 𝑤 ∈ V ↦ ( 𝑠 ∈ 𝒫 ( Base ‘ 𝑤 ) ↦ { 𝑡 ∈ ( LSubSp ‘ 𝑤 ) ∣ 𝑠𝑡 } ) )
18 0 17 wceq LSpan = ( 𝑤 ∈ V ↦ ( 𝑠 ∈ 𝒫 ( Base ‘ 𝑤 ) ↦ { 𝑡 ∈ ( LSubSp ‘ 𝑤 ) ∣ 𝑠𝑡 } ) )