Theorem mircom

Description: Variation on mirmir . (Contributed by Thierry Arnoux, 10-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 )
	mirmir.b	|- ( ph -> B e. P )
	mircom.1	|- ( ph -> ( M ` B ) = C )
Assertion	mircom	|- ( ph -> ( M ` C ) = 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		mircom.1	|- ( ph -> ( M ` B ) = C )
11	10	fveq2d	|- ( ph -> ( M ` ( M ` B ) ) = ( M ` C ) )
12	1 2 3 4 5 6 7 8 9	mirmir	|- ( ph -> ( M ` ( M ` B ) ) = B )
13	11 12	eqtr3d	|- ( ph -> ( M ` C ) = B )