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	⊢ ( ⊤ → ( Base ‘ 𝐻 ) = ( Base ‘ 𝐺 ) )
6		eqid	⊢ ( +g ‘ 𝐻 ) = ( +g ‘ 𝐻 )
7	1 6	ttgplusg	⊢ ( +g ‘ 𝐻 ) = ( +g ‘ 𝐺 )
8	7	a1i	⊢ ( ⊤ → ( +g ‘ 𝐻 ) = ( +g ‘ 𝐺 ) )
9	5 8	grpsubpropd	⊢ ( ⊤ → ( -g ‘ 𝐻 ) = ( -g ‘ 𝐺 ) )
10	9	mptru	⊢ ( -g ‘ 𝐻 ) = ( -g ‘ 𝐺 )
11	2 10	eqtri	⊢ − = ( -g ‘ 𝐺 )