Theorem lmicinv

Description: The mirroring line is an invariant. (Contributed by Thierry Arnoux, 8-Aug-2020)

	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 )
	lmicinv.1	|- ( ph -> A e. D )
Assertion	lmicinv	|- ( ph -> ( 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		lmicinv.1	|- ( ph -> A e. D )
11	1 2 3 4 5 6 7 8 9	lmiinv	|- ( ph -> ( ( M ` A ) = A <-> A e. D ) )
12	10 11	mpbird	|- ( ph -> ( M ` A ) = A )