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Theorem mirmir 26376
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 (𝜑𝐺 ∈ TarskiG)
mirval.a (𝜑𝐴𝑃)
mirfv.m 𝑀 = (𝑆𝐴)
mirmir.b (𝜑𝐵𝑃)
Assertion
Ref Expression
mirmir (𝜑 → (𝑀‘(𝑀𝐵)) = 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 (𝜑𝐺 ∈ TarskiG)
7 mirval.a . . 3 (𝜑𝐴𝑃)
8 mirfv.m . . 3 𝑀 = (𝑆𝐴)
9 mirmir.b . . . 4 (𝜑𝐵𝑃)
101, 2, 3, 4, 5, 6, 7, 8, 9mircl 26375 . . 3 (𝜑 → (𝑀𝐵) ∈ 𝑃)
111, 2, 3, 4, 5, 6, 7, 8, 9mircgr 26371 . . . 4 (𝜑 → (𝐴 (𝑀𝐵)) = (𝐴 𝐵))
1211eqcomd 2827 . . 3 (𝜑 → (𝐴 𝐵) = (𝐴 (𝑀𝐵)))
131, 2, 3, 4, 5, 6, 7, 8, 9mirbtwn 26372 . . . 4 (𝜑𝐴 ∈ ((𝑀𝐵)𝐼𝐵))
141, 2, 3, 6, 10, 7, 9, 13tgbtwncom 26202 . . 3 (𝜑𝐴 ∈ (𝐵𝐼(𝑀𝐵)))
151, 2, 3, 4, 5, 6, 7, 8, 10, 9, 12, 14ismir 26373 . 2 (𝜑𝐵 = (𝑀‘(𝑀𝐵)))
1615eqcomd 2827 1 (𝜑 → (𝑀‘(𝑀𝐵)) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wcel 2105  cfv 6349  (class class class)co 7145  Basecbs 16473  distcds 16564  TarskiGcstrkg 26144  Itvcitv 26150  LineGclng 26151  pInvGcmir 26366
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pr 5321
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3497  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4833  df-iun 4914  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-riota 7103  df-ov 7148  df-trkgc 26162  df-trkgb 26163  df-trkgcb 26164  df-trkg 26167  df-mir 26367
This theorem is referenced by:  mircom  26377  mirreu  26378  mireq  26379  mirne  26381  mirf1o  26383  mirbtwnb  26386  miduniq2  26401  ragcom  26412  ragmir  26414  colperpexlem1  26444  colperpexlem2  26445  opphllem2  26462  opphllem3  26463  opphllem4  26464  opphllem6  26466  opphl  26468  colhp  26484  sacgr  26545
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