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Theorem mirinv 23919
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
StepHypRef 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 )
54adantr 465 . . . 4  |-  ( (
ph  /\  ( M `  B )  =  B )  ->  G  e. TarskiG )
6 mirinv.b . . . . 5  |-  ( ph  ->  B  e.  P )
76adantr 465 . . . 4  |-  ( (
ph  /\  ( M `  B )  =  B )  ->  B  e.  P )
8 mirval.a . . . . 5  |-  ( ph  ->  A  e.  P )
98adantr 465 . . . 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
)
131, 2, 3, 10, 11, 5, 9, 12, 7mirbtwn 23911 . . . . 5  |-  ( (
ph  /\  ( M `  B )  =  B )  ->  A  e.  ( ( M `  B ) I B ) )
14 simpr 461 . . . . . 6  |-  ( (
ph  /\  ( M `  B )  =  B )  ->  ( M `  B )  =  B )
1514oveq1d 6296 . . . . 5  |-  ( (
ph  /\  ( M `  B )  =  B )  ->  ( ( M `  B )
I B )  =  ( B I B ) )
1613, 15eleqtrd 2533 . . . 4  |-  ( (
ph  /\  ( M `  B )  =  B )  ->  A  e.  ( B I B ) )
171, 2, 3, 5, 7, 9, 16axtgbtwnid 23735 . . 3  |-  ( (
ph  /\  ( M `  B )  =  B )  ->  B  =  A )
1817eqcomd 2451 . 2  |-  ( (
ph  /\  ( M `  B )  =  B )  ->  A  =  B )
194adantr 465 . . . 4  |-  ( (
ph  /\  A  =  B )  ->  G  e. TarskiG )
208adantr 465 . . . 4  |-  ( (
ph  /\  A  =  B )  ->  A  e.  P )
216adantr 465 . . . 4  |-  ( (
ph  /\  A  =  B )  ->  B  e.  P )
22 eqidd 2444 . . . 4  |-  ( (
ph  /\  A  =  B )  ->  ( A  .-  B )  =  ( A  .-  B
) )
23 simpr 461 . . . . 5  |-  ( (
ph  /\  A  =  B )  ->  A  =  B )
241, 2, 3, 19, 21, 21tgbtwntriv1 23754 . . . . 5  |-  ( (
ph  /\  A  =  B )  ->  B  e.  ( B I B ) )
2523, 24eqeltrd 2531 . . . 4  |-  ( (
ph  /\  A  =  B )  ->  A  e.  ( B I B ) )
261, 2, 3, 10, 11, 19, 20, 12, 21, 21, 22, 25ismir 23912 . . 3  |-  ( (
ph  /\  A  =  B )  ->  B  =  ( M `  B ) )
2726eqcomd 2451 . 2  |-  ( (
ph  /\  A  =  B )  ->  ( M `  B )  =  B )
2818, 27impbida 832 1  |-  ( ph  ->  ( ( M `  B )  =  B  <-> 
A  =  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1383    e. wcel 1804   ` cfv 5578  (class class class)co 6281   Basecbs 14509   distcds 14583  TarskiGcstrkg 23697  Itvcitv 23704  LineGclng 23705  pInvGcmir 23905
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-trkgc 23716  df-trkgb 23717  df-trkgcb 23718  df-trkg 23722  df-mir 23906
This theorem is referenced by:  mircinv  23920  mirln2  23929  miduniq  23934  miduniq2  23936  krippenlem  23939  ragflat2  23952  footex  23967  colperpexlem2  23977  colperpexlem3  23978  opphllem6  23996  lmimid  24031  hypcgrlem2  24037
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