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Theorem mirmir 24719
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 24718 . . 3  |-  ( ph  ->  ( M `  B
)  e.  P )
111, 2, 3, 4, 5, 6, 7, 8, 9mircgr 24714 . . . 4  |-  ( ph  ->  ( A  .-  ( M `  B )
)  =  ( A 
.-  B ) )
1211eqcomd 2459 . . 3  |-  ( ph  ->  ( A  .-  B
)  =  ( A 
.-  ( M `  B ) ) )
131, 2, 3, 4, 5, 6, 7, 8, 9mirbtwn 24715 . . . 4  |-  ( ph  ->  A  e.  ( ( M `  B ) I B ) )
141, 2, 3, 6, 10, 7, 9, 13tgbtwncom 24544 . . 3  |-  ( ph  ->  A  e.  ( B I ( M `  B ) ) )
151, 2, 3, 4, 5, 6, 7, 8, 10, 9, 12, 14ismir 24716 . 2  |-  ( ph  ->  B  =  ( M `
 ( M `  B ) ) )
1615eqcomd 2459 1  |-  ( ph  ->  ( M `  ( M `  B )
)  =  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1446    e. wcel 1889   ` cfv 5585  (class class class)co 6295   Basecbs 15133   distcds 15211  TarskiGcstrkg 24490  Itvcitv 24496  LineGclng 24497  pInvGcmir 24709
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pr 4642
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-op 3977  df-uni 4202  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-id 4752  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-riota 6257  df-ov 6298  df-trkgc 24508  df-trkgb 24509  df-trkgcb 24510  df-trkg 24513  df-mir 24710
This theorem is referenced by:  mircom  24720  mirreu  24721  mireq  24722  mirne  24724  mirf1o  24726  mirbtwnb  24729  miduniq2  24744  ragcom  24755  ragmir  24757  colperpexlem1  24784  colperpexlem2  24785  opphllem2  24802  opphllem3  24803  opphllem4  24804  opphllem6  24806  opphl  24808  colhp  24824  sacgr  24884
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