Step |
Hyp |
Ref |
Expression |
1 |
|
obsss.v |
⊢ 𝑉 = ( Base ‘ 𝑊 ) |
2 |
|
eqid |
⊢ ( ·𝑖 ‘ 𝑊 ) = ( ·𝑖 ‘ 𝑊 ) |
3 |
|
eqid |
⊢ ( Scalar ‘ 𝑊 ) = ( Scalar ‘ 𝑊 ) |
4 |
|
eqid |
⊢ ( 1r ‘ ( Scalar ‘ 𝑊 ) ) = ( 1r ‘ ( Scalar ‘ 𝑊 ) ) |
5 |
|
eqid |
⊢ ( 0g ‘ ( Scalar ‘ 𝑊 ) ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) |
6 |
|
eqid |
⊢ ( ocv ‘ 𝑊 ) = ( ocv ‘ 𝑊 ) |
7 |
|
eqid |
⊢ ( 0g ‘ 𝑊 ) = ( 0g ‘ 𝑊 ) |
8 |
1 2 3 4 5 6 7
|
isobs |
⊢ ( 𝐵 ∈ ( OBasis ‘ 𝑊 ) ↔ ( 𝑊 ∈ PreHil ∧ 𝐵 ⊆ 𝑉 ∧ ( ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 ( ·𝑖 ‘ 𝑊 ) 𝑦 ) = if ( 𝑥 = 𝑦 , ( 1r ‘ ( Scalar ‘ 𝑊 ) ) , ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) ∧ ( ( ocv ‘ 𝑊 ) ‘ 𝐵 ) = { ( 0g ‘ 𝑊 ) } ) ) ) |
9 |
8
|
simp2bi |
⊢ ( 𝐵 ∈ ( OBasis ‘ 𝑊 ) → 𝐵 ⊆ 𝑉 ) |