Metamath Proof Explorer
Description: An element of a basis is a vector. (Contributed by Mario Carneiro, 24-Jun-2014)
|
|
Ref |
Expression |
|
Hypotheses |
lbsss.v |
⊢ 𝑉 = ( Base ‘ 𝑊 ) |
|
|
lbsss.j |
⊢ 𝐽 = ( LBasis ‘ 𝑊 ) |
|
Assertion |
lbsel |
⊢ ( ( 𝐵 ∈ 𝐽 ∧ 𝐸 ∈ 𝐵 ) → 𝐸 ∈ 𝑉 ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
lbsss.v |
⊢ 𝑉 = ( Base ‘ 𝑊 ) |
2 |
|
lbsss.j |
⊢ 𝐽 = ( LBasis ‘ 𝑊 ) |
3 |
1 2
|
lbsss |
⊢ ( 𝐵 ∈ 𝐽 → 𝐵 ⊆ 𝑉 ) |
4 |
3
|
sselda |
⊢ ( ( 𝐵 ∈ 𝐽 ∧ 𝐸 ∈ 𝐵 ) → 𝐸 ∈ 𝑉 ) |