Theorem ragtriva

Description: Trivial right angle. Theorem 8.8 of Schwabhauser p. 58. (Contributed by Thierry Arnoux, 3-Sep-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 )
	ragtriva.1	|- ( ph -> <" A B A "> e. ( raG ` G ) )
Assertion	ragtriva	|- ( ph -> A = B )

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		ragtriva.1	|- ( ph -> <" A B A "> e. ( raG ` G ) )
11	1 2 3 4 5 6 8 7 9	ragtrivb	|- ( ph -> <" B A A "> e. ( raG ` G ) )
12	1 2 3 4 5 6 8 7 7 11	ragcom	|- ( ph -> <" A A B "> e. ( raG ` G ) )
13	1 2 3 4 5 6 7 7 8 12 10	ragflat	|- ( ph -> A = B )