Theorem trgcgrcom

Description: Commutative law for three-place congruence. (Contributed by Thierry Arnoux, 27-Apr-2019)

	Ref	Expression
Hypotheses	tgcgrxfr.p	|- P = ( Base ` G )
	tgcgrxfr.m	|- .- = ( dist ` G )
	tgcgrxfr.i	|- I = ( Itv ` G )
	tgcgrxfr.r	|- .~ = ( cgrG ` G )
	tgcgrxfr.g	|- ( ph -> G e. TarskiG )
	tgbtwnxfr.a	|- ( ph -> A e. P )
	tgbtwnxfr.b	|- ( ph -> B e. P )
	tgbtwnxfr.c	|- ( ph -> C e. P )
	tgbtwnxfr.d	|- ( ph -> D e. P )
	tgbtwnxfr.e	|- ( ph -> E e. P )
	tgbtwnxfr.f	|- ( ph -> F e. P )
	tgbtwnxfr.2	|- ( ph -> <" A B C "> .~ <" D E F "> )
Assertion	trgcgrcom	|- ( ph -> <" D E F "> .~ <" A B C "> )

Proof

Step	Hyp	Ref	Expression
1		tgcgrxfr.p	|- P = ( Base ` G )
2		tgcgrxfr.m	|- .- = ( dist ` G )
3		tgcgrxfr.i	|- I = ( Itv ` G )
4		tgcgrxfr.r	|- .~ = ( cgrG ` G )
5		tgcgrxfr.g	|- ( ph -> G e. TarskiG )
6		tgbtwnxfr.a	|- ( ph -> A e. P )
7		tgbtwnxfr.b	|- ( ph -> B e. P )
8		tgbtwnxfr.c	|- ( ph -> C e. P )
9		tgbtwnxfr.d	|- ( ph -> D e. P )
10		tgbtwnxfr.e	|- ( ph -> E e. P )
11		tgbtwnxfr.f	|- ( ph -> F e. P )
12		tgbtwnxfr.2	|- ( ph -> <" A B C "> .~ <" D E F "> )
13	1 2 3 4 5 6 7 8 9 10 11 12	cgr3simp1	|- ( ph -> ( A .- B ) = ( D .- E ) )
14	13	eqcomd	|- ( ph -> ( D .- E ) = ( A .- B ) )
15	1 2 3 4 5 6 7 8 9 10 11 12	cgr3simp2	|- ( ph -> ( B .- C ) = ( E .- F ) )
16	15	eqcomd	|- ( ph -> ( E .- F ) = ( B .- C ) )
17	1 2 3 4 5 6 7 8 9 10 11 12	cgr3simp3	|- ( ph -> ( C .- A ) = ( F .- D ) )
18	17	eqcomd	|- ( ph -> ( F .- D ) = ( C .- A ) )
19	1 2 4 5 9 10 11 6 7 8 14 16 18	trgcgr	|- ( ph -> <" D E F "> .~ <" A B C "> )