Metamath Proof Explorer


Theorem xmsxmet2

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 ) )