Theorem grpnpncan0

Description: Cancellation law for group subtraction ( npncan2 analog). (Contributed by AV, 24-Nov-2019)

	Ref	Expression
Hypotheses	grpsubadd.b	|- B = ( Base ` G )
	grpsubadd.p	|- .+ = ( +g ` G )
	grpsubadd.m	|- .- = ( -g ` G )
	grpnpncan0.0	|- .0. = ( 0g ` G )
Assertion	grpnpncan0	|- ( ( G e. Grp /\ ( X e. B /\ Y e. B ) ) -> ( ( X .- Y ) .+ ( Y .- X ) ) = .0. )

Proof

Step	Hyp	Ref	Expression
1		grpsubadd.b	|- B = ( Base ` G )
2		grpsubadd.p	|- .+ = ( +g ` G )
3		grpsubadd.m	|- .- = ( -g ` G )
4		grpnpncan0.0	|- .0. = ( 0g ` G )
5		simpl	|- ( ( G e. Grp /\ ( X e. B /\ Y e. B ) ) -> G e. Grp )
6		simprl	|- ( ( G e. Grp /\ ( X e. B /\ Y e. B ) ) -> X e. B )
7		simprr	|- ( ( G e. Grp /\ ( X e. B /\ Y e. B ) ) -> Y e. B )
8	1 2 3	grpnpncan	|- ( ( G e. Grp /\ ( X e. B /\ Y e. B /\ X e. B ) ) -> ( ( X .- Y ) .+ ( Y .- X ) ) = ( X .- X ) )
9	5 6 7 6 8	syl13anc	|- ( ( G e. Grp /\ ( X e. B /\ Y e. B ) ) -> ( ( X .- Y ) .+ ( Y .- X ) ) = ( X .- X ) )
10	1 4 3	grpsubid	|- ( ( G e. Grp /\ X e. B ) -> ( X .- X ) = .0. )
11	10	adantrr	|- ( ( G e. Grp /\ ( X e. B /\ Y e. B ) ) -> ( X .- X ) = .0. )
12	9 11	eqtrd	|- ( ( G e. Grp /\ ( X e. B /\ Y e. B ) ) -> ( ( X .- Y ) .+ ( Y .- X ) ) = .0. )