Metamath Proof Explorer

Theorem xmssym

Description: The distance function in an extended metric space is symmetric. (Contributed by Mario Carneiro, 2-Oct-2015)

Ref Expression
Hypotheses mscl.x ⊒ 𝑋 = ( Base β€˜ 𝑀 )
mscl.d ⊒ 𝐷 = ( dist β€˜ 𝑀 )
Assertion xmssym ( ( 𝑀 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ) β†’ ( 𝐴 𝐷 𝐡 ) = ( 𝐡 𝐷 𝐴 ) )

Proof

Step Hyp Ref Expression
1 mscl.x ⊒ 𝑋 = ( Base β€˜ 𝑀 )
2 mscl.d ⊒ 𝐷 = ( dist β€˜ 𝑀 )
3 1 2 xmsxmet2 ⊒ ( 𝑀 ∈ ∞MetSp β†’ ( 𝐷 β†Ύ ( 𝑋 Γ— 𝑋 ) ) ∈ ( ∞Met β€˜ 𝑋 ) )
4 xmetsym ⊒ ( ( ( 𝐷 β†Ύ ( 𝑋 Γ— 𝑋 ) ) ∈ ( ∞Met β€˜ 𝑋 ) ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ) β†’ ( 𝐴 ( 𝐷 β†Ύ ( 𝑋 Γ— 𝑋 ) ) 𝐡 ) = ( 𝐡 ( 𝐷 β†Ύ ( 𝑋 Γ— 𝑋 ) ) 𝐴 ) )
5 3 4 syl3an1 ⊒ ( ( 𝑀 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ) β†’ ( 𝐴 ( 𝐷 β†Ύ ( 𝑋 Γ— 𝑋 ) ) 𝐡 ) = ( 𝐡 ( 𝐷 β†Ύ ( 𝑋 Γ— 𝑋 ) ) 𝐴 ) )
6 simp2 ⊒ ( ( 𝑀 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ) β†’ 𝐴 ∈ 𝑋 )
7 simp3 ⊒ ( ( 𝑀 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ) β†’ 𝐡 ∈ 𝑋 )
8 6 7 ovresd ⊒ ( ( 𝑀 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ) β†’ ( 𝐴 ( 𝐷 β†Ύ ( 𝑋 Γ— 𝑋 ) ) 𝐡 ) = ( 𝐴 𝐷 𝐡 ) )
9 7 6 ovresd ⊒ ( ( 𝑀 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ) β†’ ( 𝐡 ( 𝐷 β†Ύ ( 𝑋 Γ— 𝑋 ) ) 𝐴 ) = ( 𝐡 𝐷 𝐴 ) )
10 5 8 9 3eqtr3d ⊒ ( ( 𝑀 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ) β†’ ( 𝐴 𝐷 𝐡 ) = ( 𝐡 𝐷 𝐴 ) )