Theorem tcphsub

Description: The subtraction operation of a subcomplex pre-Hilbert space augmented with norm. (Contributed by Thierry Arnoux, 30-Jun-2019)

	Ref	Expression
Hypotheses	tcphval.n	|- G = ( toCPreHil ` W )
	tcphsub.v	|- .- = ( -g ` W )
Assertion	tcphsub	|- .- = ( -g ` G )

Proof

Step	Hyp	Ref	Expression
1		tcphval.n	|- G = ( toCPreHil ` W )
2		tcphsub.v	|- .- = ( -g ` W )
3		eqid	|- ( Base ` W ) = ( Base ` W )
4	1 3	tcphbas	|- ( Base ` W ) = ( Base ` G )
5	4	a1i	|- ( T. -> ( Base ` W ) = ( Base ` G ) )
6		eqid	|- ( +g ` W ) = ( +g ` W )
7	1 6	tchplusg	|- ( +g ` W ) = ( +g ` G )
8	7	a1i	|- ( T. -> ( +g ` W ) = ( +g ` G ) )
9	5 8	grpsubpropd	|- ( T. -> ( -g ` W ) = ( -g ` G ) )
10	9	mptru	|- ( -g ` W ) = ( -g ` G )
11	2 10	eqtri	|- .- = ( -g ` G )