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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 ‘ 𝑀 )
mscl.d ⊢ 𝐷 = ( dist ‘ 𝑀 )
Assertion xmssym ( ( 𝑀 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 𝐷 𝐵 ) = ( 𝐵 𝐷 𝐴 ) )

Proof

Step Hyp Ref Expression
1 mscl.x ⊢ 𝑋 = ( Base ‘ 𝑀 )
2 mscl.d ⊢ 𝐷 = ( dist ‘ 𝑀 )
3 1 2 xmsxmet2 ⊢ ( 𝑀 ∈ ∞MetSp → ( 𝐷 ↾ ( 𝑋 × 𝑋 ) ) ∈ ( ∞Met ‘ 𝑋 ) )
4 xmetsym ⊢ ( ( ( 𝐷 ↾ ( 𝑋 × 𝑋 ) ) ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 ( 𝐷 ↾ ( 𝑋 × 𝑋 ) ) 𝐵 ) = ( 𝐵 ( 𝐷 ↾ ( 𝑋 × 𝑋 ) ) 𝐴 ) )
5 3 4 syl3an1 ⊢ ( ( 𝑀 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 ( 𝐷 ↾ ( 𝑋 × 𝑋 ) ) 𝐵 ) = ( 𝐵 ( 𝐷 ↾ ( 𝑋 × 𝑋 ) ) 𝐴 ) )
6 simp2 ⊢ ( ( 𝑀 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → 𝐴 ∈ 𝑋 )
7 simp3 ⊢ ( ( 𝑀 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → 𝐵 ∈ 𝑋 )
8 6 7 ovresd ⊢ ( ( 𝑀 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 ( 𝐷 ↾ ( 𝑋 × 𝑋 ) ) 𝐵 ) = ( 𝐴 𝐷 𝐵 ) )
9 7 6 ovresd ⊢ ( ( 𝑀 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐵 ( 𝐷 ↾ ( 𝑋 × 𝑋 ) ) 𝐴 ) = ( 𝐵 𝐷 𝐴 ) )
10 5 8 9 3eqtr3d ⊢ ( ( 𝑀 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 𝐷 𝐵 ) = ( 𝐵 𝐷 𝐴 ) )