Description: Vector subtraction on a subspace is a restriction of vector subtraction on the parent space. (Contributed by NM, 28-Jan-2008) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | sspm.y | ⊢ 𝑌 = ( BaseSet ‘ 𝑊 ) | |
| sspm.m | ⊢ 𝑀 = ( −𝑣 ‘ 𝑈 ) | ||
| sspm.l | ⊢ 𝐿 = ( −𝑣 ‘ 𝑊 ) | ||
| sspm.h | ⊢ 𝐻 = ( SubSp ‘ 𝑈 ) | ||
| Assertion | sspm | ⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) → 𝐿 = ( 𝑀 ↾ ( 𝑌 × 𝑌 ) ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | sspm.y | ⊢ 𝑌 = ( BaseSet ‘ 𝑊 ) | |
| 2 | sspm.m | ⊢ 𝑀 = ( −𝑣 ‘ 𝑈 ) | |
| 3 | sspm.l | ⊢ 𝐿 = ( −𝑣 ‘ 𝑊 ) | |
| 4 | sspm.h | ⊢ 𝐻 = ( SubSp ‘ 𝑈 ) | |
| 5 | 1 2 3 4 | sspmval | ⊢ ( ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌 ) ) → ( 𝑥 𝐿 𝑦 ) = ( 𝑥 𝑀 𝑦 ) ) |
| 6 | 1 3 | nvmf | ⊢ ( 𝑊 ∈ NrmCVec → 𝐿 : ( 𝑌 × 𝑌 ) ⟶ 𝑌 ) |
| 7 | eqid | ⊢ ( BaseSet ‘ 𝑈 ) = ( BaseSet ‘ 𝑈 ) | |
| 8 | 7 2 | nvmf | ⊢ ( 𝑈 ∈ NrmCVec → 𝑀 : ( ( BaseSet ‘ 𝑈 ) × ( BaseSet ‘ 𝑈 ) ) ⟶ ( BaseSet ‘ 𝑈 ) ) |
| 9 | 1 4 5 6 8 | sspmlem | ⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) → 𝐿 = ( 𝑀 ↾ ( 𝑌 × 𝑌 ) ) ) |