Theorem grpsubadd0sub

Description: Subtraction expressed as addition of the difference of the identity element and the subtrahend. (Contributed by AV, 9-Nov-2019)

	Ref	Expression
Hypotheses	grpsubid.b	|- B = ( Base ` G )
	grpsubid.o	|- .0. = ( 0g ` G )
	grpsubid.m	|- .- = ( -g ` G )
	grpsubadd0sub.p	|- .+ = ( +g ` G )
Assertion	grpsubadd0sub	|- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( X .- Y ) = ( X .+ ( .0. .- Y ) ) )

Proof

Step	Hyp	Ref	Expression
1		grpsubid.b	|- B = ( Base ` G )
2		grpsubid.o	|- .0. = ( 0g ` G )
3		grpsubid.m	|- .- = ( -g ` G )
4		grpsubadd0sub.p	|- .+ = ( +g ` G )
5		eqid	|- ( invg ` G ) = ( invg ` G )
6	1 4 5 3	grpsubval	|- ( ( X e. B /\ Y e. B ) -> ( X .- Y ) = ( X .+ ( ( invg ` G ) ` Y ) ) )
7	6	3adant1	|- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( X .- Y ) = ( X .+ ( ( invg ` G ) ` Y ) ) )
8	1 3 5 2	grpinvval2	|- ( ( G e. Grp /\ Y e. B ) -> ( ( invg ` G ) ` Y ) = ( .0. .- Y ) )
9	8	3adant2	|- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( ( invg ` G ) ` Y ) = ( .0. .- Y ) )
10	9	oveq2d	|- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( X .+ ( ( invg ` G ) ` Y ) ) = ( X .+ ( .0. .- Y ) ) )
11	7 10	eqtrd	|- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( X .- Y ) = ( X .+ ( .0. .- Y ) ) )