Theorem mirne

Description: Mirror of non-center point cannot be the center point. (Contributed by Thierry Arnoux, 27-Sep-2020)

	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 )
	mirinv.b	|- ( ph -> B e. P )
	mirne.1	|- ( ph -> B =/= A )
Assertion	mirne	|- ( ph -> ( M ` B ) =/= A )

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		mirinv.b	|- ( ph -> B e. P )
10		mirne.1	|- ( ph -> B =/= A )
11		simpr	|- ( ( ph /\ ( M ` B ) = A ) -> ( M ` B ) = A )
12	11	fveq2d	|- ( ( ph /\ ( M ` B ) = A ) -> ( M ` ( M ` B ) ) = ( M ` A ) )
13	1 2 3 4 5 6 7 8 9	mirmir	|- ( ph -> ( M ` ( M ` B ) ) = B )
14	13	adantr	|- ( ( ph /\ ( M ` B ) = A ) -> ( M ` ( M ` B ) ) = B )
15		eqid	|- A = A
16	1 2 3 4 5 6 7 8 7	mirinv	|- ( ph -> ( ( M ` A ) = A <-> A = A ) )
17	15 16	mpbiri	|- ( ph -> ( M ` A ) = A )
18	17	adantr	|- ( ( ph /\ ( M ` B ) = A ) -> ( M ` A ) = A )
19	12 14 18	3eqtr3d	|- ( ( ph /\ ( M ` B ) = A ) -> B = A )
20	10	adantr	|- ( ( ph /\ ( M ` B ) = A ) -> B =/= A )
21	20	neneqd	|- ( ( ph /\ ( M ` B ) = A ) -> -. B = A )
22	19 21	pm2.65da	|- ( ph -> -. ( M ` B ) = A )
23	22	neqned	|- ( ph -> ( M ` B ) =/= A )