Metamath Proof Explorer


Theorem ttgbas

Description: The base set 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 )
ttgbas.1
|- B = ( Base ` H )
Assertion ttgbas
|- B = ( Base ` G )

Proof

Step Hyp Ref Expression
1 ttgval.n
 |-  G = ( toTG ` H )
2 ttgbas.1
 |-  B = ( Base ` H )
3 baseid
 |-  Base = Slot ( Base ` ndx )
4 slotslnbpsd
 |-  ( ( ( LineG ` ndx ) =/= ( Base ` ndx ) /\ ( LineG ` ndx ) =/= ( +g ` ndx ) ) /\ ( ( LineG ` ndx ) =/= ( .s ` ndx ) /\ ( LineG ` ndx ) =/= ( dist ` ndx ) ) )
5 simpll
 |-  ( ( ( ( LineG ` ndx ) =/= ( Base ` ndx ) /\ ( LineG ` ndx ) =/= ( +g ` ndx ) ) /\ ( ( LineG ` ndx ) =/= ( .s ` ndx ) /\ ( LineG ` ndx ) =/= ( dist ` ndx ) ) ) -> ( LineG ` ndx ) =/= ( Base ` ndx ) )
6 4 5 ax-mp
 |-  ( LineG ` ndx ) =/= ( Base ` ndx )
7 6 necomi
 |-  ( Base ` ndx ) =/= ( LineG ` ndx )
8 slotsinbpsd
 |-  ( ( ( Itv ` ndx ) =/= ( Base ` ndx ) /\ ( Itv ` ndx ) =/= ( +g ` ndx ) ) /\ ( ( Itv ` ndx ) =/= ( .s ` ndx ) /\ ( Itv ` ndx ) =/= ( dist ` ndx ) ) )
9 simpll
 |-  ( ( ( ( Itv ` ndx ) =/= ( Base ` ndx ) /\ ( Itv ` ndx ) =/= ( +g ` ndx ) ) /\ ( ( Itv ` ndx ) =/= ( .s ` ndx ) /\ ( Itv ` ndx ) =/= ( dist ` ndx ) ) ) -> ( Itv ` ndx ) =/= ( Base ` ndx ) )
10 8 9 ax-mp
 |-  ( Itv ` ndx ) =/= ( Base ` ndx )
11 10 necomi
 |-  ( Base ` ndx ) =/= ( Itv ` ndx )
12 1 3 7 11 ttglem
 |-  ( Base ` H ) = ( Base ` G )
13 2 12 eqtri
 |-  B = ( Base ` G )