Theorem mirmot

Description: Point investion is a motion of the geometric space. Theorem 7.14 of Schwabhauser p. 51. (Contributed by Thierry Arnoux, 15-Dec-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 )
	mirmot.m	|- M = ( S ` A )
	mirmot.a	|- ( ph -> A e. P )
Assertion	mirmot	|- ( ph -> M e. ( G Ismt G ) )

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		mirmot.m	|- M = ( S ` A )
8		mirmot.a	|- ( ph -> A e. P )
9	1 2 3 4 5 6 8 7	mirf1o	|- ( ph -> M : P -1-1-onto-> P )
10	6	adantr	|- ( ( ph /\ ( a e. P /\ b e. P ) ) -> G e. TarskiG )
11	8	adantr	|- ( ( ph /\ ( a e. P /\ b e. P ) ) -> A e. P )
12		simprl	|- ( ( ph /\ ( a e. P /\ b e. P ) ) -> a e. P )
13		simprr	|- ( ( ph /\ ( a e. P /\ b e. P ) ) -> b e. P )
14	1 2 3 4 5 10 11 7 12 13	miriso	|- ( ( ph /\ ( a e. P /\ b e. P ) ) -> ( ( M ` a ) .- ( M ` b ) ) = ( a .- b ) )
15	14	ralrimivva	|- ( ph -> A. a e. P A. b e. P ( ( M ` a ) .- ( M ` b ) ) = ( a .- b ) )
16	1 2	ismot	|- ( G e. TarskiG -> ( M e. ( G Ismt G ) <-> ( M : P -1-1-onto-> P /\ A. a e. P A. b e. P ( ( M ` a ) .- ( M ` b ) ) = ( a .- b ) ) ) )
17	6 16	syl	|- ( ph -> ( M e. ( G Ismt G ) <-> ( M : P -1-1-onto-> P /\ A. a e. P A. b e. P ( ( M ` a ) .- ( M ` b ) ) = ( a .- b ) ) ) )
18	9 15 17	mpbir2and	|- ( ph -> M e. ( G Ismt G ) )