Metamath Proof Explorer


Theorem xmseq0

Description: The distance between two points in an extended metric space is zero iff the two points are identical. (Contributed by Mario Carneiro, 2-Oct-2015)

Ref Expression
Hypotheses mscl.x 𝑋 = ( Base ‘ 𝑀 )
mscl.d 𝐷 = ( dist ‘ 𝑀 )
Assertion xmseq0 ( ( 𝑀 ∈ ∞MetSp ∧ 𝐴𝑋𝐵𝑋 ) → ( ( 𝐴 𝐷 𝐵 ) = 0 ↔ 𝐴 = 𝐵 ) )

Proof

Step Hyp Ref Expression
1 mscl.x 𝑋 = ( Base ‘ 𝑀 )
2 mscl.d 𝐷 = ( dist ‘ 𝑀 )
3 ovres ( ( 𝐴𝑋𝐵𝑋 ) → ( 𝐴 ( 𝐷 ↾ ( 𝑋 × 𝑋 ) ) 𝐵 ) = ( 𝐴 𝐷 𝐵 ) )
4 3 3adant1 ( ( 𝑀 ∈ ∞MetSp ∧ 𝐴𝑋𝐵𝑋 ) → ( 𝐴 ( 𝐷 ↾ ( 𝑋 × 𝑋 ) ) 𝐵 ) = ( 𝐴 𝐷 𝐵 ) )
5 4 eqeq1d ( ( 𝑀 ∈ ∞MetSp ∧ 𝐴𝑋𝐵𝑋 ) → ( ( 𝐴 ( 𝐷 ↾ ( 𝑋 × 𝑋 ) ) 𝐵 ) = 0 ↔ ( 𝐴 𝐷 𝐵 ) = 0 ) )
6 1 2 xmsxmet2 ( 𝑀 ∈ ∞MetSp → ( 𝐷 ↾ ( 𝑋 × 𝑋 ) ) ∈ ( ∞Met ‘ 𝑋 ) )
7 xmeteq0 ( ( ( 𝐷 ↾ ( 𝑋 × 𝑋 ) ) ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐴𝑋𝐵𝑋 ) → ( ( 𝐴 ( 𝐷 ↾ ( 𝑋 × 𝑋 ) ) 𝐵 ) = 0 ↔ 𝐴 = 𝐵 ) )
8 6 7 syl3an1 ( ( 𝑀 ∈ ∞MetSp ∧ 𝐴𝑋𝐵𝑋 ) → ( ( 𝐴 ( 𝐷 ↾ ( 𝑋 × 𝑋 ) ) 𝐵 ) = 0 ↔ 𝐴 = 𝐵 ) )
9 5 8 bitr3d ( ( 𝑀 ∈ ∞MetSp ∧ 𝐴𝑋𝐵𝑋 ) → ( ( 𝐴 𝐷 𝐵 ) = 0 ↔ 𝐴 = 𝐵 ) )