Theorem ragcol

Description: The right angle property is independent of the choice of point on one side. Theorem 8.3 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 )
	ragcol.d	|- ( ph -> D e. P )
	ragcol.1	|- ( ph -> <" A B C "> e. ( raG ` G ) )
	ragcol.2	|- ( ph -> A =/= B )
	ragcol.3	|- ( ph -> ( A e. ( B L D ) \/ B = D ) )
Assertion	ragcol	|- ( ph -> <" D B C "> 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		ragcol.d	|- ( ph -> D e. P )
11		ragcol.1	|- ( ph -> <" A B C "> e. ( raG ` G ) )
12		ragcol.2	|- ( ph -> A =/= B )
13		ragcol.3	|- ( ph -> ( A e. ( B L D ) \/ B = D ) )
14		eqid	|- ( cgrG ` G ) = ( cgrG ` G )
15		eqid	|- ( S ` B ) = ( S ` B )
16	1 2 3 4 5 6 8 15 9	mircl	|- ( ph -> ( ( S ` B ) ` C ) e. P )
17	12	necomd	|- ( ph -> B =/= A )
18	1 2 3 4 5 6 8 15 9	mircgr	|- ( ph -> ( B .- ( ( S ` B ) ` C ) ) = ( B .- C ) )
19	18	eqcomd	|- ( ph -> ( B .- C ) = ( B .- ( ( S ` B ) ` C ) ) )
20	1 2 3 4 5 6 7 8 9	israg	|- ( ph -> ( <" A B C "> e. ( raG ` G ) <-> ( A .- C ) = ( A .- ( ( S ` B ) ` C ) ) ) )
21	11 20	mpbid	|- ( ph -> ( A .- C ) = ( A .- ( ( S ` B ) ` C ) ) )
22	1 4 3 6 8 7 10 14 9 16 2 17 13 19 21	lncgr	|- ( ph -> ( D .- C ) = ( D .- ( ( S ` B ) ` C ) ) )
23	1 2 3 4 5 6 10 8 9	israg	|- ( ph -> ( <" D B C "> e. ( raG ` G ) <-> ( D .- C ) = ( D .- ( ( S ` B ) ` C ) ) ) )
24	22 23	mpbird	|- ( ph -> <" D B C "> e. ( raG ` G ) )