Theorem ragflat2

Description: Deduce equality from two right angles. Theorem 8.6 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 )
	ragflat2.d	|- ( ph -> D e. P )
	ragflat2.1	|- ( ph -> <" A B C "> e. ( raG ` G ) )
	ragflat2.2	|- ( ph -> <" D B C "> e. ( raG ` G ) )
	ragflat2.3	|- ( ph -> C e. ( A I D ) )
Assertion	ragflat2	|- ( ph -> B = C )

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		ragflat2.d	|- ( ph -> D e. P )
11		ragflat2.1	|- ( ph -> <" A B C "> e. ( raG ` G ) )
12		ragflat2.2	|- ( ph -> <" D B C "> e. ( raG ` G ) )
13		ragflat2.3	|- ( ph -> C e. ( A I 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	1 2 3 4 5 6 7 8 9	israg	|- ( ph -> ( <" A B C "> e. ( raG ` G ) <-> ( A .- C ) = ( A .- ( ( S ` B ) ` C ) ) ) )
18	11 17	mpbid	|- ( ph -> ( A .- C ) = ( A .- ( ( S ` B ) ` C ) ) )
19	1 2 3 4 5 6 10 8 9	israg	|- ( ph -> ( <" D B C "> e. ( raG ` G ) <-> ( D .- C ) = ( D .- ( ( S ` B ) ` C ) ) ) )
20	12 19	mpbid	|- ( ph -> ( D .- C ) = ( D .- ( ( S ` B ) ` C ) ) )
21	1 4 3 6 7 10 9 14 16 7 2 13 18 20	tgidinside	|- ( ph -> C = ( ( S ` B ) ` C ) )
22	21	eqcomd	|- ( ph -> ( ( S ` B ) ` C ) = C )
23	1 2 3 4 5 6 8 15 9	mirinv	|- ( ph -> ( ( ( S ` B ) ` C ) = C <-> B = C ) )
24	22 23	mpbid	|- ( ph -> B = C )