Description: The induced metric on a subspace is a restriction of the induced metric on the parent space. (Contributed by NM, 1-Feb-2008) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | sspims.y | ⊢ 𝑌 = ( BaseSet ‘ 𝑊 ) | |
| sspims.d | ⊢ 𝐷 = ( IndMet ‘ 𝑈 ) | ||
| sspims.c | ⊢ 𝐶 = ( IndMet ‘ 𝑊 ) | ||
| sspims.h | ⊢ 𝐻 = ( SubSp ‘ 𝑈 ) | ||
| Assertion | sspims | ⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) → 𝐶 = ( 𝐷 ↾ ( 𝑌 × 𝑌 ) ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | sspims.y | ⊢ 𝑌 = ( BaseSet ‘ 𝑊 ) | |
| 2 | sspims.d | ⊢ 𝐷 = ( IndMet ‘ 𝑈 ) | |
| 3 | sspims.c | ⊢ 𝐶 = ( IndMet ‘ 𝑊 ) | |
| 4 | sspims.h | ⊢ 𝐻 = ( SubSp ‘ 𝑈 ) | |
| 5 | 1 2 3 4 | sspimsval | ⊢ ( ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌 ) ) → ( 𝑥 𝐶 𝑦 ) = ( 𝑥 𝐷 𝑦 ) ) |
| 6 | 1 3 | imsdf | ⊢ ( 𝑊 ∈ NrmCVec → 𝐶 : ( 𝑌 × 𝑌 ) ⟶ ℝ ) |
| 7 | eqid | ⊢ ( BaseSet ‘ 𝑈 ) = ( BaseSet ‘ 𝑈 ) | |
| 8 | 7 2 | imsdf | ⊢ ( 𝑈 ∈ NrmCVec → 𝐷 : ( ( BaseSet ‘ 𝑈 ) × ( BaseSet ‘ 𝑈 ) ) ⟶ ℝ ) |
| 9 | 1 4 5 6 8 | sspmlem | ⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) → 𝐶 = ( 𝐷 ↾ ( 𝑌 × 𝑌 ) ) ) |