Theorem lmilmi

Description: Line mirroring is an involution. Theorem 10.5 of Schwabhauser p. 89. (Contributed by Thierry Arnoux, 11-Dec-2019)

	Ref	Expression
Hypotheses	ismid.p	|- P = ( Base ` G )
	ismid.d	|- .- = ( dist ` G )
	ismid.i	|- I = ( Itv ` G )
	ismid.g	|- ( ph -> G e. TarskiG )
	ismid.1	|- ( ph -> G TarskiGDim>= 2 )
	lmif.m	|- M = ( ( lInvG ` G ) ` D )
	lmif.l	|- L = ( LineG ` G )
	lmif.d	|- ( ph -> D e. ran L )
	lmicl.1	|- ( ph -> A e. P )
Assertion	lmilmi	|- ( ph -> ( M ` ( M ` A ) ) = A )

Proof

Step	Hyp	Ref	Expression
1		ismid.p	|- P = ( Base ` G )
2		ismid.d	|- .- = ( dist ` G )
3		ismid.i	|- I = ( Itv ` G )
4		ismid.g	|- ( ph -> G e. TarskiG )
5		ismid.1	|- ( ph -> G TarskiGDim>= 2 )
6		lmif.m	|- M = ( ( lInvG ` G ) ` D )
7		lmif.l	|- L = ( LineG ` G )
8		lmif.d	|- ( ph -> D e. ran L )
9		lmicl.1	|- ( ph -> A e. P )
10	1 2 3 4 5 6 7 8 9	lmicl	|- ( ph -> ( M ` A ) e. P )
11		eqidd	|- ( ph -> ( M ` A ) = ( M ` A ) )
12	1 2 3 4 5 6 7 8 9 10 11	lmicom	|- ( ph -> ( M ` ( M ` A ) ) = A )