Theorem ttgsub

Description: The subtraction operation of a subcomplex Hilbert space augmented with betweenness. (Contributed by Thierry Arnoux, 25-Mar-2019)

	Ref	Expression
Hypotheses	ttgval.n	|- G = ( toTG ` H )
	ttgsub.1	|- .- = ( -g ` H )
Assertion	ttgsub	|- .- = ( -g ` G )

Proof

Step	Hyp	Ref	Expression
1		ttgval.n	|- G = ( toTG ` H )
2		ttgsub.1	|- .- = ( -g ` H )
3		eqid	|- ( Base ` H ) = ( Base ` H )
4	1 3	ttgbas	|- ( Base ` H ) = ( Base ` G )
5	4	a1i	|- ( T. -> ( Base ` H ) = ( Base ` G ) )
6		eqid	|- ( +g ` H ) = ( +g ` H )
7	1 6	ttgplusg	|- ( +g ` H ) = ( +g ` G )
8	7	a1i	|- ( T. -> ( +g ` H ) = ( +g ` G ) )
9	5 8	grpsubpropd	|- ( T. -> ( -g ` H ) = ( -g ` G ) )
10	9	mptru	|- ( -g ` H ) = ( -g ` G )
11	2 10	eqtri	|- .- = ( -g ` G )