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Theorem mirmir 23749
Description: The point inversion function is an involution. Theorem 7.7 of [Schwabhauser] p. 49. (Contributed by Thierry Arnoux, 3-Jun-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
)
mirmir.b  |-  ( ph  ->  B  e.  P )
Assertion
Ref Expression
mirmir  |-  ( ph  ->  ( M `  ( M `  B )
)  =  B )

Proof of Theorem mirmir
StepHypRef Expression
1 mirval.p . . 3  |-  P  =  ( Base `  G
)
2 mirval.d . . 3  |-  .-  =  ( dist `  G )
3 mirval.i . . 3  |-  I  =  (Itv `  G )
4 mirval.l . . 3  |-  L  =  (LineG `  G )
5 mirval.s . . 3  |-  S  =  (pInvG `  G )
6 mirval.g . . 3  |-  ( ph  ->  G  e. TarskiG )
7 mirval.a . . 3  |-  ( ph  ->  A  e.  P )
8 mirfv.m . . 3  |-  M  =  ( S `  A
)
9 mirmir.b . . . 4  |-  ( ph  ->  B  e.  P )
101, 2, 3, 4, 5, 6, 7, 8, 9mircl 23748 . . 3  |-  ( ph  ->  ( M `  B
)  e.  P )
111, 2, 3, 4, 5, 6, 7, 8, 9mircgr 23744 . . . 4  |-  ( ph  ->  ( A  .-  ( M `  B )
)  =  ( A 
.-  B ) )
1211eqcomd 2468 . . 3  |-  ( ph  ->  ( A  .-  B
)  =  ( A 
.-  ( M `  B ) ) )
131, 2, 3, 4, 5, 6, 7, 8, 9mirbtwn 23745 . . . 4  |-  ( ph  ->  A  e.  ( ( M `  B ) I B ) )
141, 2, 3, 6, 10, 7, 9, 13tgbtwncom 23600 . . 3  |-  ( ph  ->  A  e.  ( B I ( M `  B ) ) )
151, 2, 3, 4, 5, 6, 7, 8, 10, 9, 12, 14ismir 23746 . 2  |-  ( ph  ->  B  =  ( M `
 ( M `  B ) ) )
1615eqcomd 2468 1  |-  ( ph  ->  ( M `  ( M `  B )
)  =  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1374    e. wcel 1762   ` cfv 5579  (class class class)co 6275   Basecbs 14479   distcds 14553  TarskiGcstrkg 23546  Itvcitv 23553  LineGclng 23554  pInvGcmir 23739
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-trkgc 23565  df-trkgb 23566  df-trkgcb 23567  df-trkg 23571  df-mir 23740
This theorem is referenced by:  mircom  23750  mirreu  23751  mireq  23752  mirf1o  23755  mirbtwnb  23758  miduniq  23763  miduniq2  23765  ragcom  23776  ragmir  23778  colperpexlem1  23802  colperpexlem2  23803
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