Theorem mirmir2

Description: Point inversion of a point inversion through another point. (Contributed by Thierry Arnoux, 3-Nov-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 )
Assertion	mirmir2	|- ( ph -> ( M ` ( ( S ` Y ) ` X ) ) = ( ( S ` ( M ` Y ) ) ` ( M ` X ) ) )

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	1 2 3 4 5 6 7 8 10	mircl	|- ( ph -> ( M ` Y ) e. P )
12		eqid	|- ( S ` ( M ` Y ) ) = ( S ` ( M ` Y ) )
13	1 2 3 4 5 6 7 8 9	mircl	|- ( ph -> ( M ` X ) e. P )
14		eqid	|- ( S ` Y ) = ( S ` Y )
15	1 2 3 4 5 6 10 14 9	mircl	|- ( ph -> ( ( S ` Y ) ` X ) e. P )
16	1 2 3 4 5 6 7 8 15	mircl	|- ( ph -> ( M ` ( ( S ` Y ) ` X ) ) e. P )
17	1 2 3 4 5 6 10 14 9	mircgr	|- ( ph -> ( Y .- ( ( S ` Y ) ` X ) ) = ( Y .- X ) )
18	1 2 3 4 5 6 7 8 10 15 10 9 17	mircgrs	|- ( ph -> ( ( M ` Y ) .- ( M ` ( ( S ` Y ) ` X ) ) ) = ( ( M ` Y ) .- ( M ` X ) ) )
19	1 2 3 4 5 6 10 14 9	mirbtwn	|- ( ph -> Y e. ( ( ( S ` Y ) ` X ) I X ) )
20	1 2 3 4 5 6 7 8 15 10 9 19	mirbtwni	|- ( ph -> ( M ` Y ) e. ( ( M ` ( ( S ` Y ) ` X ) ) I ( M ` X ) ) )
21	1 2 3 4 5 6 11 12 13 16 18 20	ismir	|- ( ph -> ( M ` ( ( S ` Y ) ` X ) ) = ( ( S ` ( M ` Y ) ) ` ( M ` X ) ) )