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Metamath Proof Explorer

Theorem xmettpos

Description: The distance function of an extended metric space is symmetric. (Contributed by Mario Carneiro, 20-Aug-2015)

Ref Expression
Assertion xmettpos ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → tpos 𝐷 = 𝐷 )

Proof

Step Hyp Ref Expression
1 xmetsym ⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ) → ( 𝑥 𝐷 𝑦 ) = ( 𝑦 𝐷 𝑥 ) )
2 1 3expb ⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ) ) → ( 𝑥 𝐷 𝑦 ) = ( 𝑦 𝐷 𝑥 ) )
3 2 ralrimivva ⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( 𝑥 𝐷 𝑦 ) = ( 𝑦 𝐷 𝑥 ) )
4 xmetf ⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → 𝐷 : ( 𝑋 × 𝑋 ) ⟶ ℝ* )
5 ffn ⊢ ( 𝐷 : ( 𝑋 × 𝑋 ) ⟶ ℝ* → 𝐷 Fn ( 𝑋 × 𝑋 ) )
6 tpossym ⊢ ( 𝐷 Fn ( 𝑋 × 𝑋 ) → ( tpos 𝐷 = 𝐷 ↔ ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( 𝑥 𝐷 𝑦 ) = ( 𝑦 𝐷 𝑥 ) ) )
7 4 5 6 3syl ⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → ( tpos 𝐷 = 𝐷 ↔ ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( 𝑥 𝐷 𝑦 ) = ( 𝑦 𝐷 𝑥 ) ) )
8 3 7 mpbird ⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → tpos 𝐷 = 𝐷 )