Metamath Proof Explorer


Theorem ttgds

Description: The metric of a subcomplex Hilbert space augmented with betweenness. (Contributed by Thierry Arnoux, 25-Mar-2019) (Revised by AV, 29-Oct-2024)

Ref Expression
Hypotheses ttgval.n 𝐺 = ( toTG ‘ 𝐻 )
ttgds.1 𝐷 = ( dist ‘ 𝐻 )
Assertion ttgds 𝐷 = ( dist ‘ 𝐺 )

Proof

Step Hyp Ref Expression
1 ttgval.n 𝐺 = ( toTG ‘ 𝐻 )
2 ttgds.1 𝐷 = ( dist ‘ 𝐻 )
3 dsid dist = Slot ( dist ‘ ndx )
4 slotslnbpsd ( ( ( LineG ‘ ndx ) ≠ ( Base ‘ ndx ) ∧ ( LineG ‘ ndx ) ≠ ( +g ‘ ndx ) ) ∧ ( ( LineG ‘ ndx ) ≠ ( ·𝑠 ‘ ndx ) ∧ ( LineG ‘ ndx ) ≠ ( dist ‘ ndx ) ) )
5 simprr ( ( ( ( LineG ‘ ndx ) ≠ ( Base ‘ ndx ) ∧ ( LineG ‘ ndx ) ≠ ( +g ‘ ndx ) ) ∧ ( ( LineG ‘ ndx ) ≠ ( ·𝑠 ‘ ndx ) ∧ ( LineG ‘ ndx ) ≠ ( dist ‘ ndx ) ) ) → ( LineG ‘ ndx ) ≠ ( dist ‘ ndx ) )
6 4 5 ax-mp ( LineG ‘ ndx ) ≠ ( dist ‘ ndx )
7 6 necomi ( dist ‘ ndx ) ≠ ( LineG ‘ ndx )
8 slotsinbpsd ( ( ( Itv ‘ ndx ) ≠ ( Base ‘ ndx ) ∧ ( Itv ‘ ndx ) ≠ ( +g ‘ ndx ) ) ∧ ( ( Itv ‘ ndx ) ≠ ( ·𝑠 ‘ ndx ) ∧ ( Itv ‘ ndx ) ≠ ( dist ‘ ndx ) ) )
9 simprr ( ( ( ( Itv ‘ ndx ) ≠ ( Base ‘ ndx ) ∧ ( Itv ‘ ndx ) ≠ ( +g ‘ ndx ) ) ∧ ( ( Itv ‘ ndx ) ≠ ( ·𝑠 ‘ ndx ) ∧ ( Itv ‘ ndx ) ≠ ( dist ‘ ndx ) ) ) → ( Itv ‘ ndx ) ≠ ( dist ‘ ndx ) )
10 8 9 ax-mp ( Itv ‘ ndx ) ≠ ( dist ‘ ndx )
11 10 necomi ( dist ‘ ndx ) ≠ ( Itv ‘ ndx )
12 1 3 7 11 ttglem ( dist ‘ 𝐻 ) = ( dist ‘ 𝐺 )
13 2 12 eqtri 𝐷 = ( dist ‘ 𝐺 )