Theorem lmieq

Description: Equality deduction for line mirroring. Theorem 10.7 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 )
	lmieq.c	|- ( ph -> B e. P )
	lmieq.d	|- ( ph -> ( M ` A ) = ( M ` B ) )
Assertion	lmieq	|- ( ph -> A = B )

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		lmieq.c	|- ( ph -> B e. P )
11		lmieq.d	|- ( ph -> ( M ` A ) = ( M ` B ) )
12		fveqeq2	|- ( b = A -> ( ( M ` b ) = ( M ` B ) <-> ( M ` A ) = ( M ` B ) ) )
13		fveqeq2	|- ( b = B -> ( ( M ` b ) = ( M ` B ) <-> ( M ` B ) = ( M ` B ) ) )
14	1 2 3 4 5 6 7 8 10	lmicl	|- ( ph -> ( M ` B ) e. P )
15	1 2 3 4 5 6 7 8 14	lmireu	|- ( ph -> E! b e. P ( M ` b ) = ( M ` B ) )
16		eqidd	|- ( ph -> ( M ` B ) = ( M ` B ) )
17	12 13 15 9 10 11 16	reu2eqd	|- ( ph -> A = B )