Theorem mirmir

Description: The point inversion function is an involution. Theorem 7.7 of Schwabhauser p. 49. (Contributed by Thierry Arnoux, 3-Jun-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 )
	mirmir.b	|- ( ph -> B e. P )
Assertion	mirmir	|- ( ph -> ( M ` ( M ` B ) ) = B )

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		mirmir.b	|- ( ph -> B e. P )
10	1 2 3 4 5 6 7 8 9	mircl	|- ( ph -> ( M ` B ) e. P )
11	1 2 3 4 5 6 7 8 9	mircgr	|- ( ph -> ( A .- ( M ` B ) ) = ( A .- B ) )
12	11	eqcomd	|- ( ph -> ( A .- B ) = ( A .- ( M ` B ) ) )
13	1 2 3 4 5 6 7 8 9	mirbtwn	|- ( ph -> A e. ( ( M ` B ) I B ) )
14	1 2 3 6 10 7 9 13	tgbtwncom	|- ( ph -> A e. ( B I ( M ` B ) ) )
15	1 2 3 4 5 6 7 8 10 9 12 14	ismir	|- ( ph -> B = ( M ` ( M ` B ) ) )
16	15	eqcomd	|- ( ph -> ( M ` ( M ` B ) ) = B )