Theorem ragcom

Description: Commutative rule for right angles. Theorem 8.2 of Schwabhauser p. 57. (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 )
	ragcom.1	|- ( ph -> <" A B C "> e. ( raG ` G ) )
Assertion	ragcom	|- ( ph -> <" C B A "> 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		ragcom.1	|- ( ph -> <" A B C "> e. ( raG ` G ) )
11		eqid	|- ( S ` B ) = ( S ` B )
12	1 2 3 4 5 6 8 11 9	mircl	|- ( ph -> ( ( S ` B ) ` C ) e. P )
13	1 2 3 4 5 6 7 8 9	israg	|- ( ph -> ( <" A B C "> e. ( raG ` G ) <-> ( A .- C ) = ( A .- ( ( S ` B ) ` C ) ) ) )
14	10 13	mpbid	|- ( ph -> ( A .- C ) = ( A .- ( ( S ` B ) ` C ) ) )
15	1 2 3 6 7 9 7 12 14	tgcgrcomlr	|- ( ph -> ( C .- A ) = ( ( ( S ` B ) ` C ) .- A ) )
16	1 2 3 4 5 6 8 11 12 7	miriso	|- ( ph -> ( ( ( S ` B ) ` ( ( S ` B ) ` C ) ) .- ( ( S ` B ) ` A ) ) = ( ( ( S ` B ) ` C ) .- A ) )
17	1 2 3 4 5 6 8 11 9	mirmir	|- ( ph -> ( ( S ` B ) ` ( ( S ` B ) ` C ) ) = C )
18	17	oveq1d	|- ( ph -> ( ( ( S ` B ) ` ( ( S ` B ) ` C ) ) .- ( ( S ` B ) ` A ) ) = ( C .- ( ( S ` B ) ` A ) ) )
19	15 16 18	3eqtr2d	|- ( ph -> ( C .- A ) = ( C .- ( ( S ` B ) ` A ) ) )
20	1 2 3 4 5 6 9 8 7	israg	|- ( ph -> ( <" C B A "> e. ( raG ` G ) <-> ( C .- A ) = ( C .- ( ( S ` B ) ` A ) ) ) )
21	19 20	mpbird	|- ( ph -> <" C B A "> e. ( raG ` G ) )