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 ) )