Theorem miduniq1

Description: Uniqueness of the middle point, expressed with point inversion. Theorem 7.18 of Schwabhauser p. 52. (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 )
	miduniq1.a	|- ( ph -> A e. P )
	miduniq1.b	|- ( ph -> B e. P )
	miduniq1.x	|- ( ph -> X e. P )
	miduniq1.e	|- ( ph -> ( ( S ` A ) ` X ) = ( ( S ` B ) ` X ) )
Assertion	miduniq1	|- ( ph -> A = 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		miduniq1.a	|- ( ph -> A e. P )
8		miduniq1.b	|- ( ph -> B e. P )
9		miduniq1.x	|- ( ph -> X e. P )
10		miduniq1.e	|- ( ph -> ( ( S ` A ) ` X ) = ( ( S ` B ) ` X ) )
11		eqid	|- ( S ` A ) = ( S ` A )
12	1 2 3 4 5 6 7 11 9	mircl	|- ( ph -> ( ( S ` A ) ` X ) e. P )
13		eqidd	|- ( ph -> ( ( S ` A ) ` X ) = ( ( S ` A ) ` X ) )
14	10	eqcomd	|- ( ph -> ( ( S ` B ) ` X ) = ( ( S ` A ) ` X ) )
15	1 2 3 4 5 6 7 8 9 12 13 14	miduniq	|- ( ph -> A = B )