Theorem mirinv 25561

Description: The only invariant point of a point inversion Theorem 7.3 of [Schwabhauser] p. 49, Theorem 7.10 of [Schwabhauser] p. 50. (Contributed by Thierry Arnoux, 30-Jul-2019.)

Hypotheses

Ref	Expression
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 )

Assertion

Ref	Expression
mirinv	|- ( ph -> ( ( M ` B ) = B <-> A = B ) )

Proof of Theorem mirinv

Step	Hyp	Ref	Expression
1		mirval.p	. . . 4 |- P = ( Base ` G )
2		mirval.d	. . . 4 |- .- = ( dist ` G )
3		mirval.i	. . . 4 |- I = (Itv ` G )
4		mirval.g	. . . . 5 |- ( ph -> G e. TarskiG )
5	4	adantr 481	. . . 4 |- ( ( ph /\ ( M ` B ) = B ) -> G e. TarskiG )
6		mirinv.b	. . . . 5 |- ( ph -> B e. P )
7	6	adantr 481	. . . 4 |- ( ( ph /\ ( M ` B ) = B ) -> B e. P )
8		mirval.a	. . . . 5 |- ( ph -> A e. P )
9	8	adantr 481	. . . 4 |- ( ( ph /\ ( M ` B ) = B ) -> A e. P )
10		mirval.l	. . . . . 6 |- L = (LineG ` G )
11		mirval.s	. . . . . 6 |- S = (pInvG ` G )
12		mirfv.m	. . . . . 6 |- M = ( S ` A )
13	1, 2, 3, 10, 11, 5, 9, 12, 7	mirbtwn 25553	. . . . 5 |- ( ( ph /\ ( M ` B ) = B ) -> A e. ( ( M ` B ) I B ) )
14		simpr 477	. . . . . 6 |- ( ( ph /\ ( M ` B ) = B ) -> ( M ` B ) = B )
15	14	oveq1d 6665	. . . . 5 |- ( ( ph /\ ( M ` B ) = B ) -> ( ( M ` B ) I B ) = ( B I B ) )
16	13, 15	eleqtrd 2703	. . . 4 |- ( ( ph /\ ( M ` B ) = B ) -> A e. ( B I B ) )
17	1, 2, 3, 5, 7, 9, 16	axtgbtwnid 25365	. . 3 |- ( ( ph /\ ( M ` B ) = B ) -> B = A )
18	17	eqcomd 2628	. 2 |- ( ( ph /\ ( M ` B ) = B ) -> A = B )
19	4	adantr 481	. . . 4 |- ( ( ph /\ A = B ) -> G e. TarskiG )
20	8	adantr 481	. . . 4 |- ( ( ph /\ A = B ) -> A e. P )
21	6	adantr 481	. . . 4 |- ( ( ph /\ A = B ) -> B e. P )
22		eqidd 2623	. . . 4 |- ( ( ph /\ A = B ) -> ( A .- B ) = ( A .- B ) )
23		simpr 477	. . . . 5 |- ( ( ph /\ A = B ) -> A = B )
24	1, 2, 3, 19, 21, 21	tgbtwntriv1 25386	. . . . 5 |- ( ( ph /\ A = B ) -> B e. ( B I B ) )
25	23, 24	eqeltrd 2701	. . . 4 |- ( ( ph /\ A = B ) -> A e. ( B I B ) )
26	1, 2, 3, 10, 11, 19, 20, 12, 21, 21, 22, 25	ismir 25554	. . 3 |- ( ( ph /\ A = B ) -> B = ( M ` B ) )
27	26	eqcomd 2628	. 2 |- ( ( ph /\ A = B ) -> ( M ` B ) = B )
28	18, 27	impbida 877	1 |- ( ph -> ( ( M ` B ) = B <-> A = B ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   ` cfv 5888  (class class class)co 6650   Basecbs 15857   distcds 15950  TarskiGcstrkg 25329  Itvcitv 25335  LineGclng 25336  pInvGcmir 25547

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-trkgc 25347  df-trkgb 25348  df-trkgcb 25349  df-trkg 25352  df-mir 25548

This theorem is referenced by:  mirne  25562  mircinv  25563  mirln2  25572  miduniq  25580  miduniq2  25582  krippenlem  25585  ragflat2  25598  footex  25613  colperpexlem2  25623  colperpexlem3  25624  opphllem6  25644  lmimid  25686  hypcgrlem2  25692