Theorem ragtrivb

Description: Trivial right angle. Theorem 8.5 of Schwabhauser p. 58. (Contributed by Thierry Arnoux, 25-Aug-2019)

	Ref	Expression
Hypotheses	israg.p	|- P = ( Base ` G )
	israg.d	|- .- = ( dist ` G )
	israg.i	|- I = ( Itv ` G )
	israg.l	|- L = ( LineG ` G )
	israg.s	|- S = ( pInvG ` G )
	israg.g	|- ( ph -> G e. TarskiG )
	israg.a	|- ( ph -> A e. P )
	israg.b	|- ( ph -> B e. P )
	israg.c	|- ( ph -> C e. P )
Assertion	ragtrivb	|- ( ph -> <" A B B "> e. ( raG ` G ) )

Proof

Step	Hyp	Ref	Expression
1		israg.p	|- P = ( Base ` G )
2		israg.d	|- .- = ( dist ` G )
3		israg.i	|- I = ( Itv ` G )
4		israg.l	|- L = ( LineG ` G )
5		israg.s	|- S = ( pInvG ` G )
6		israg.g	|- ( ph -> G e. TarskiG )
7		israg.a	|- ( ph -> A e. P )
8		israg.b	|- ( ph -> B e. P )
9		israg.c	|- ( ph -> C e. P )
10		eqid	|- ( S ` B ) = ( S ` B )
11	1 2 3 4 5 6 8 10	mircinv	|- ( ph -> ( ( S ` B ) ` B ) = B )
12	11	oveq2d	|- ( ph -> ( A .- ( ( S ` B ) ` B ) ) = ( A .- B ) )
13	12	eqcomd	|- ( ph -> ( A .- B ) = ( A .- ( ( S ` B ) ` B ) ) )
14	1 2 3 4 5 6 7 8 8	israg	|- ( ph -> ( <" A B B "> e. ( raG ` G ) <-> ( A .- B ) = ( A .- ( ( S ` B ) ` B ) ) ) )
15	13 14	mpbird	|- ( ph -> <" A B B "> e. ( raG ` G ) )