Description: Scalar multiplication on a subspace in terms of scalar multiplication on the parent space. (Contributed by NM, 28-Jan-2008) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | ssps.y | ⊢ 𝑌 = ( BaseSet ‘ 𝑊 ) | |
| ssps.s | ⊢ 𝑆 = ( ·𝑠OLD ‘ 𝑈 ) | ||
| ssps.r | ⊢ 𝑅 = ( ·𝑠OLD ‘ 𝑊 ) | ||
| ssps.h | ⊢ 𝐻 = ( SubSp ‘ 𝑈 ) | ||
| Assertion | sspsval | ⊢ ( ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑌 ) ) → ( 𝐴 𝑅 𝐵 ) = ( 𝐴 𝑆 𝐵 ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ssps.y | ⊢ 𝑌 = ( BaseSet ‘ 𝑊 ) | |
| 2 | ssps.s | ⊢ 𝑆 = ( ·𝑠OLD ‘ 𝑈 ) | |
| 3 | ssps.r | ⊢ 𝑅 = ( ·𝑠OLD ‘ 𝑊 ) | |
| 4 | ssps.h | ⊢ 𝐻 = ( SubSp ‘ 𝑈 ) | |
| 5 | 1 2 3 4 | ssps | ⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) → 𝑅 = ( 𝑆 ↾ ( ℂ × 𝑌 ) ) ) |
| 6 | 5 | oveqd | ⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) → ( 𝐴 𝑅 𝐵 ) = ( 𝐴 ( 𝑆 ↾ ( ℂ × 𝑌 ) ) 𝐵 ) ) |
| 7 | ovres | ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑌 ) → ( 𝐴 ( 𝑆 ↾ ( ℂ × 𝑌 ) ) 𝐵 ) = ( 𝐴 𝑆 𝐵 ) ) | |
| 8 | 6 7 | sylan9eq | ⊢ ( ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑌 ) ) → ( 𝐴 𝑅 𝐵 ) = ( 𝐴 𝑆 𝐵 ) ) |