Theorem mircgrs

Description: Point inversion preserves congruence. Theorem 7.16 of Schwabhauser p. 51. (Contributed by Thierry Arnoux, 30-Jul-2019)

	Ref	Expression
Hypotheses	mirval.p	|- P = ( Base ` G )
	mirval.d	|- .- = ( dist ` G )
	mirval.i	|- I = ( Itv ` G )
	mirval.l	|- L = ( LineG ` G )
	mirval.s	|- S = ( pInvG ` G )
	mirval.g	|- ( ph -> G e. TarskiG )
	mirval.a	|- ( ph -> A e. P )
	mirfv.m	|- M = ( S ` A )
	miriso.1	|- ( ph -> X e. P )
	miriso.2	|- ( ph -> Y e. P )
	mircgrs.z	|- ( ph -> Z e. P )
	mircgrs.t	|- ( ph -> T e. P )
	mircgrs.e	|- ( ph -> ( X .- Y ) = ( Z .- T ) )
Assertion	mircgrs	|- ( ph -> ( ( M ` X ) .- ( M ` Y ) ) = ( ( M ` Z ) .- ( M ` T ) ) )

Proof

Step	Hyp	Ref	Expression
1		mirval.p	|- P = ( Base ` G )
2		mirval.d	|- .- = ( dist ` G )
3		mirval.i	|- I = ( Itv ` G )
4		mirval.l	|- L = ( LineG ` G )
5		mirval.s	|- S = ( pInvG ` G )
6		mirval.g	|- ( ph -> G e. TarskiG )
7		mirval.a	|- ( ph -> A e. P )
8		mirfv.m	|- M = ( S ` A )
9		miriso.1	|- ( ph -> X e. P )
10		miriso.2	|- ( ph -> Y e. P )
11		mircgrs.z	|- ( ph -> Z e. P )
12		mircgrs.t	|- ( ph -> T e. P )
13		mircgrs.e	|- ( ph -> ( X .- Y ) = ( Z .- T ) )
14	1 2 3 4 5 6 7 8 9 10	miriso	|- ( ph -> ( ( M ` X ) .- ( M ` Y ) ) = ( X .- Y ) )
15	1 2 3 4 5 6 7 8 11 12	miriso	|- ( ph -> ( ( M ` Z ) .- ( M ` T ) ) = ( Z .- T ) )
16	13 14 15	3eqtr4d	|- ( ph -> ( ( M ` X ) .- ( M ` Y ) ) = ( ( M ` Z ) .- ( M ` T ) ) )