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Metamath Proof Explorer
Description: The distance function, suitably truncated, is an extended metric on
X . (Contributed by Mario Carneiro, 2-Oct-2015)
|
|
Ref |
Expression |
|
Hypotheses |
mscl.x |
⊢ 𝑋 = ( Base ‘ 𝑀 ) |
|
|
mscl.d |
⊢ 𝐷 = ( dist ‘ 𝑀 ) |
|
Assertion |
xmsxmet2 |
⊢ ( 𝑀 ∈ ∞MetSp → ( 𝐷 ↾ ( 𝑋 × 𝑋 ) ) ∈ ( ∞Met ‘ 𝑋 ) ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
mscl.x |
⊢ 𝑋 = ( Base ‘ 𝑀 ) |
| 2 |
|
mscl.d |
⊢ 𝐷 = ( dist ‘ 𝑀 ) |
| 3 |
2
|
reseq1i |
⊢ ( 𝐷 ↾ ( 𝑋 × 𝑋 ) ) = ( ( dist ‘ 𝑀 ) ↾ ( 𝑋 × 𝑋 ) ) |
| 4 |
1 3
|
xmsxmet |
⊢ ( 𝑀 ∈ ∞MetSp → ( 𝐷 ↾ ( 𝑋 × 𝑋 ) ) ∈ ( ∞Met ‘ 𝑋 ) ) |