Description: Vector addition on a subspace in terms of vector addition on the parent space. (Contributed by NM, 28-Jan-2008) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | sspg.y | ⊢ 𝑌 = ( BaseSet ‘ 𝑊 ) | |
| sspg.g | ⊢ 𝐺 = ( +𝑣 ‘ 𝑈 ) | ||
| sspg.f | ⊢ 𝐹 = ( +𝑣 ‘ 𝑊 ) | ||
| sspg.h | ⊢ 𝐻 = ( SubSp ‘ 𝑈 ) | ||
| Assertion | sspgval | ⊢ ( ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐴 ∈ 𝑌 ∧ 𝐵 ∈ 𝑌 ) ) → ( 𝐴 𝐹 𝐵 ) = ( 𝐴 𝐺 𝐵 ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | sspg.y | ⊢ 𝑌 = ( BaseSet ‘ 𝑊 ) | |
| 2 | sspg.g | ⊢ 𝐺 = ( +𝑣 ‘ 𝑈 ) | |
| 3 | sspg.f | ⊢ 𝐹 = ( +𝑣 ‘ 𝑊 ) | |
| 4 | sspg.h | ⊢ 𝐻 = ( SubSp ‘ 𝑈 ) | |
| 5 | 1 2 3 4 | sspg | ⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) → 𝐹 = ( 𝐺 ↾ ( 𝑌 × 𝑌 ) ) ) |
| 6 | 5 | oveqd | ⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) → ( 𝐴 𝐹 𝐵 ) = ( 𝐴 ( 𝐺 ↾ ( 𝑌 × 𝑌 ) ) 𝐵 ) ) |
| 7 | ovres | ⊢ ( ( 𝐴 ∈ 𝑌 ∧ 𝐵 ∈ 𝑌 ) → ( 𝐴 ( 𝐺 ↾ ( 𝑌 × 𝑌 ) ) 𝐵 ) = ( 𝐴 𝐺 𝐵 ) ) | |
| 8 | 6 7 | sylan9eq | ⊢ ( ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐴 ∈ 𝑌 ∧ 𝐵 ∈ 𝑌 ) ) → ( 𝐴 𝐹 𝐵 ) = ( 𝐴 𝐺 𝐵 ) ) |