Theorem tgcgrcoml

Description: Congruence commutes on the LHS. Variant of Theorem 2.5 of Schwabhauser p. 27, but in a convenient form for a common case. (Contributed by David A. Wheeler, 29-Jun-2020)

	Ref	Expression
Hypotheses	tkgeom.p	|- P = ( Base ` G )
	tkgeom.d	|- .- = ( dist ` G )
	tkgeom.i	|- I = ( Itv ` G )
	tkgeom.g	|- ( ph -> G e. TarskiG )
	tgcgrcomr.a	|- ( ph -> A e. P )
	tgcgrcomr.b	|- ( ph -> B e. P )
	tgcgrcomr.c	|- ( ph -> C e. P )
	tgcgrcomr.d	|- ( ph -> D e. P )
	tgcgrcomr.6	|- ( ph -> ( A .- B ) = ( C .- D ) )
Assertion	tgcgrcoml	|- ( ph -> ( B .- A ) = ( C .- D ) )

Proof

Step	Hyp	Ref	Expression
1		tkgeom.p	|- P = ( Base ` G )
2		tkgeom.d	|- .- = ( dist ` G )
3		tkgeom.i	|- I = ( Itv ` G )
4		tkgeom.g	|- ( ph -> G e. TarskiG )
5		tgcgrcomr.a	|- ( ph -> A e. P )
6		tgcgrcomr.b	|- ( ph -> B e. P )
7		tgcgrcomr.c	|- ( ph -> C e. P )
8		tgcgrcomr.d	|- ( ph -> D e. P )
9		tgcgrcomr.6	|- ( ph -> ( A .- B ) = ( C .- D ) )
10	1 2 3 4 5 6	axtgcgrrflx	|- ( ph -> ( A .- B ) = ( B .- A ) )
11	10 9	eqtr3d	|- ( ph -> ( B .- A ) = ( C .- D ) )