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 β€˜ 𝑋 ) β†’ tpos 𝐷 = 𝐷 )

Proof

Step Hyp Ref Expression
1 xmetsym ⊒ ( ( 𝐷 ∈ ( ∞Met β€˜ 𝑋 ) ∧ π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ) β†’ ( π‘₯ 𝐷 𝑦 ) = ( 𝑦 𝐷 π‘₯ ) )
2 1 3expb ⊒ ( ( 𝐷 ∈ ( ∞Met β€˜ 𝑋 ) ∧ ( π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ) ) β†’ ( π‘₯ 𝐷 𝑦 ) = ( 𝑦 𝐷 π‘₯ ) )
3 2 ralrimivva ⊒ ( 𝐷 ∈ ( ∞Met β€˜ 𝑋 ) β†’ βˆ€ π‘₯ ∈ 𝑋 βˆ€ 𝑦 ∈ 𝑋 ( π‘₯ 𝐷 𝑦 ) = ( 𝑦 𝐷 π‘₯ ) )
4 xmetf ⊒ ( 𝐷 ∈ ( ∞Met β€˜ 𝑋 ) β†’ 𝐷 : ( 𝑋 Γ— 𝑋 ) ⟢ ℝ* )
5 ffn ⊒ ( 𝐷 : ( 𝑋 Γ— 𝑋 ) ⟢ ℝ* β†’ 𝐷 Fn ( 𝑋 Γ— 𝑋 ) )
6 tpossym ⊒ ( 𝐷 Fn ( 𝑋 Γ— 𝑋 ) β†’ ( tpos 𝐷 = 𝐷 ↔ βˆ€ π‘₯ ∈ 𝑋 βˆ€ 𝑦 ∈ 𝑋 ( π‘₯ 𝐷 𝑦 ) = ( 𝑦 𝐷 π‘₯ ) ) )
7 4 5 6 3syl ⊒ ( 𝐷 ∈ ( ∞Met β€˜ 𝑋 ) β†’ ( tpos 𝐷 = 𝐷 ↔ βˆ€ π‘₯ ∈ 𝑋 βˆ€ 𝑦 ∈ 𝑋 ( π‘₯ 𝐷 𝑦 ) = ( 𝑦 𝐷 π‘₯ ) ) )
8 3 7 mpbird ⊒ ( 𝐷 ∈ ( ∞Met β€˜ 𝑋 ) β†’ tpos 𝐷 = 𝐷 )