Theorem mircl

Description: Closure of the point inversion function. (Contributed by Thierry Arnoux, 20-Oct-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 )
	mircl.x	|- ( ph -> X e. P )
Assertion	mircl	|- ( ph -> ( M ` X ) e. P )

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		mircl.x	|- ( ph -> X e. P )
10	1 2 3 4 5 6 7 8	mirf	|- ( ph -> M : P --> P )
11	10 9	ffvelcdmd	|- ( ph -> ( M ` X ) e. P )