Metamath Proof Explorer


Theorem trkgdist

Description: The measure of a distance in a Tarski geometry. (Contributed by Thierry Arnoux, 24-Aug-2017)

Ref Expression
Hypothesis trkgstr.w 𝑊 = { ⟨ ( Base ‘ ndx ) , 𝑈 ⟩ , ⟨ ( dist ‘ ndx ) , 𝐷 ⟩ , ⟨ ( Itv ‘ ndx ) , 𝐼 ⟩ }
Assertion trkgdist ( 𝐷𝑉𝐷 = ( dist ‘ 𝑊 ) )

Proof

Step Hyp Ref Expression
1 trkgstr.w 𝑊 = { ⟨ ( Base ‘ ndx ) , 𝑈 ⟩ , ⟨ ( dist ‘ ndx ) , 𝐷 ⟩ , ⟨ ( Itv ‘ ndx ) , 𝐼 ⟩ }
2 1 trkgstr 𝑊 Struct ⟨ 1 , 1 6 ⟩
3 dsid dist = Slot ( dist ‘ ndx )
4 snsstp2 { ⟨ ( dist ‘ ndx ) , 𝐷 ⟩ } ⊆ { ⟨ ( Base ‘ ndx ) , 𝑈 ⟩ , ⟨ ( dist ‘ ndx ) , 𝐷 ⟩ , ⟨ ( Itv ‘ ndx ) , 𝐼 ⟩ }
5 4 1 sseqtrri { ⟨ ( dist ‘ ndx ) , 𝐷 ⟩ } ⊆ 𝑊
6 2 3 5 strfv ( 𝐷𝑉𝐷 = ( dist ‘ 𝑊 ) )