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 β€˜ 𝑀 )
mscl.d ⊒ 𝐷 = ( dist β€˜ 𝑀 )
Assertion xmseq0 ( ( 𝑀 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ) β†’ ( ( 𝐴 𝐷 𝐡 ) = 0 ↔ 𝐴 = 𝐡 ) )

Proof

Step Hyp Ref Expression
1 mscl.x ⊒ 𝑋 = ( Base β€˜ 𝑀 )
2 mscl.d ⊒ 𝐷 = ( dist β€˜ 𝑀 )
3 ovres ⊒ ( ( 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ) β†’ ( 𝐴 ( 𝐷 β†Ύ ( 𝑋 Γ— 𝑋 ) ) 𝐡 ) = ( 𝐴 𝐷 𝐡 ) )
4 3 3adant1 ⊒ ( ( 𝑀 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ) β†’ ( 𝐴 ( 𝐷 β†Ύ ( 𝑋 Γ— 𝑋 ) ) 𝐡 ) = ( 𝐴 𝐷 𝐡 ) )
5 4 eqeq1d ⊒ ( ( 𝑀 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ) β†’ ( ( 𝐴 ( 𝐷 β†Ύ ( 𝑋 Γ— 𝑋 ) ) 𝐡 ) = 0 ↔ ( 𝐴 𝐷 𝐡 ) = 0 ) )
6 1 2 xmsxmet2 ⊒ ( 𝑀 ∈ ∞MetSp β†’ ( 𝐷 β†Ύ ( 𝑋 Γ— 𝑋 ) ) ∈ ( ∞Met β€˜ 𝑋 ) )
7 xmeteq0 ⊒ ( ( ( 𝐷 β†Ύ ( 𝑋 Γ— 𝑋 ) ) ∈ ( ∞Met β€˜ 𝑋 ) ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ) β†’ ( ( 𝐴 ( 𝐷 β†Ύ ( 𝑋 Γ— 𝑋 ) ) 𝐡 ) = 0 ↔ 𝐴 = 𝐡 ) )
8 6 7 syl3an1 ⊒ ( ( 𝑀 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ) β†’ ( ( 𝐴 ( 𝐷 β†Ύ ( 𝑋 Γ— 𝑋 ) ) 𝐡 ) = 0 ↔ 𝐴 = 𝐡 ) )
9 5 8 bitr3d ⊒ ( ( 𝑀 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ) β†’ ( ( 𝐴 𝐷 𝐡 ) = 0 ↔ 𝐴 = 𝐡 ) )