Theorem trgcgr

Description: Triangle congruence. (Contributed by Thierry Arnoux, 27-Apr-2019)

	Ref	Expression
Hypotheses	trgcgrg.p	|- P = ( Base ` G )
	trgcgrg.m	|- .- = ( dist ` G )
	trgcgrg.r	|- .~ = ( cgrG ` G )
	trgcgrg.g	|- ( ph -> G e. TarskiG )
	trgcgrg.a	|- ( ph -> A e. P )
	trgcgrg.b	|- ( ph -> B e. P )
	trgcgrg.c	|- ( ph -> C e. P )
	trgcgrg.d	|- ( ph -> D e. P )
	trgcgrg.e	|- ( ph -> E e. P )
	trgcgrg.f	|- ( ph -> F e. P )
	trgcgr.1	|- ( ph -> ( A .- B ) = ( D .- E ) )
	trgcgr.2	|- ( ph -> ( B .- C ) = ( E .- F ) )
	trgcgr.3	|- ( ph -> ( C .- A ) = ( F .- D ) )
Assertion	trgcgr	|- ( ph -> <" A B C "> .~ <" D E F "> )

Proof

Step	Hyp	Ref	Expression
1		trgcgrg.p	|- P = ( Base ` G )
2		trgcgrg.m	|- .- = ( dist ` G )
3		trgcgrg.r	|- .~ = ( cgrG ` G )
4		trgcgrg.g	|- ( ph -> G e. TarskiG )
5		trgcgrg.a	|- ( ph -> A e. P )
6		trgcgrg.b	|- ( ph -> B e. P )
7		trgcgrg.c	|- ( ph -> C e. P )
8		trgcgrg.d	|- ( ph -> D e. P )
9		trgcgrg.e	|- ( ph -> E e. P )
10		trgcgrg.f	|- ( ph -> F e. P )
11		trgcgr.1	|- ( ph -> ( A .- B ) = ( D .- E ) )
12		trgcgr.2	|- ( ph -> ( B .- C ) = ( E .- F ) )
13		trgcgr.3	|- ( ph -> ( C .- A ) = ( F .- D ) )
14	1 2 3 4 5 6 7 8 9 10	trgcgrg	|- ( ph -> ( <" A B C "> .~ <" D E F "> <-> ( ( A .- B ) = ( D .- E ) /\ ( B .- C ) = ( E .- F ) /\ ( C .- A ) = ( F .- D ) ) ) )
15	11 12 13 14	mpbir3and	|- ( ph -> <" A B C "> .~ <" D E F "> )