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Theorem mirmir 26431
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 26430 . . 3 (𝜑 → (𝑀𝐵) ∈ 𝑃)
111, 2, 3, 4, 5, 6, 7, 8, 9mircgr 26426 . . . 4 (𝜑 → (𝐴 (𝑀𝐵)) = (𝐴 𝐵))
1211eqcomd 2826 . . 3 (𝜑 → (𝐴 𝐵) = (𝐴 (𝑀𝐵)))
131, 2, 3, 4, 5, 6, 7, 8, 9mirbtwn 26427 . . . 4 (𝜑𝐴 ∈ ((𝑀𝐵)𝐼𝐵))
141, 2, 3, 6, 10, 7, 9, 13tgbtwncom 26257 . . 3 (𝜑𝐴 ∈ (𝐵𝐼(𝑀𝐵)))
151, 2, 3, 4, 5, 6, 7, 8, 10, 9, 12, 14ismir 26428 . 2 (𝜑𝐵 = (𝑀‘(𝑀𝐵)))
1615eqcomd 2826 1 (𝜑 → (𝑀‘(𝑀𝐵)) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2114  cfv 6327  (class class class)co 7129  Basecbs 16458  distcds 16549  TarskiGcstrkg 26199  Itvcitv 26205  LineGclng 26206  pInvGcmir 26421
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-rep 5162  ax-sep 5175  ax-nul 5182  ax-pr 5302
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-reu 3132  df-rmo 3133  df-rab 3134  df-v 3472  df-sbc 3749  df-csb 3857  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4811  df-iun 4893  df-br 5039  df-opab 5101  df-mpt 5119  df-id 5432  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-iota 6286  df-fun 6329  df-fn 6330  df-f 6331  df-f1 6332  df-fo 6333  df-f1o 6334  df-fv 6335  df-riota 7087  df-ov 7132  df-trkgc 26217  df-trkgb 26218  df-trkgcb 26219  df-trkg 26222  df-mir 26422
This theorem is referenced by:  mircom  26432  mirreu  26433  mireq  26434  mirne  26436  mirf1o  26438  mirbtwnb  26441  miduniq2  26456  ragcom  26467  ragmir  26469  colperpexlem1  26499  colperpexlem2  26500  opphllem2  26517  opphllem3  26518  opphllem4  26519  opphllem6  26521  opphl  26523  colhp  26539  sacgr  26600
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