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Theorem mirinv 28689
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 (𝜑𝐺 ∈ TarskiG)
mirval.a (𝜑𝐴𝑃)
mirfv.m 𝑀 = (𝑆𝐴)
mirinv.b (𝜑𝐵𝑃)
Assertion
Ref Expression
mirinv (𝜑 → ((𝑀𝐵) = 𝐵𝐴 = 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 (𝜑𝐺 ∈ TarskiG)
54adantr 480 . . . 4 ((𝜑 ∧ (𝑀𝐵) = 𝐵) → 𝐺 ∈ TarskiG)
6 mirinv.b . . . . 5 (𝜑𝐵𝑃)
76adantr 480 . . . 4 ((𝜑 ∧ (𝑀𝐵) = 𝐵) → 𝐵𝑃)
8 mirval.a . . . . 5 (𝜑𝐴𝑃)
98adantr 480 . . . 4 ((𝜑 ∧ (𝑀𝐵) = 𝐵) → 𝐴𝑃)
10 mirval.l . . . . . 6 𝐿 = (LineG‘𝐺)
11 mirval.s . . . . . 6 𝑆 = (pInvG‘𝐺)
12 mirfv.m . . . . . 6 𝑀 = (𝑆𝐴)
131, 2, 3, 10, 11, 5, 9, 12, 7mirbtwn 28681 . . . . 5 ((𝜑 ∧ (𝑀𝐵) = 𝐵) → 𝐴 ∈ ((𝑀𝐵)𝐼𝐵))
14 simpr 484 . . . . . 6 ((𝜑 ∧ (𝑀𝐵) = 𝐵) → (𝑀𝐵) = 𝐵)
1514oveq1d 7446 . . . . 5 ((𝜑 ∧ (𝑀𝐵) = 𝐵) → ((𝑀𝐵)𝐼𝐵) = (𝐵𝐼𝐵))
1613, 15eleqtrd 2841 . . . 4 ((𝜑 ∧ (𝑀𝐵) = 𝐵) → 𝐴 ∈ (𝐵𝐼𝐵))
171, 2, 3, 5, 7, 9, 16axtgbtwnid 28489 . . 3 ((𝜑 ∧ (𝑀𝐵) = 𝐵) → 𝐵 = 𝐴)
1817eqcomd 2741 . 2 ((𝜑 ∧ (𝑀𝐵) = 𝐵) → 𝐴 = 𝐵)
194adantr 480 . . . 4 ((𝜑𝐴 = 𝐵) → 𝐺 ∈ TarskiG)
208adantr 480 . . . 4 ((𝜑𝐴 = 𝐵) → 𝐴𝑃)
216adantr 480 . . . 4 ((𝜑𝐴 = 𝐵) → 𝐵𝑃)
22 eqidd 2736 . . . 4 ((𝜑𝐴 = 𝐵) → (𝐴 𝐵) = (𝐴 𝐵))
23 simpr 484 . . . . 5 ((𝜑𝐴 = 𝐵) → 𝐴 = 𝐵)
241, 2, 3, 19, 21, 21tgbtwntriv1 28514 . . . . 5 ((𝜑𝐴 = 𝐵) → 𝐵 ∈ (𝐵𝐼𝐵))
2523, 24eqeltrd 2839 . . . 4 ((𝜑𝐴 = 𝐵) → 𝐴 ∈ (𝐵𝐼𝐵))
261, 2, 3, 10, 11, 19, 20, 12, 21, 21, 22, 25ismir 28682 . . 3 ((𝜑𝐴 = 𝐵) → 𝐵 = (𝑀𝐵))
2726eqcomd 2741 . 2 ((𝜑𝐴 = 𝐵) → (𝑀𝐵) = 𝐵)
2818, 27impbida 801 1 (𝜑 → ((𝑀𝐵) = 𝐵𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  cfv 6563  (class class class)co 7431  Basecbs 17245  distcds 17307  TarskiGcstrkg 28450  Itvcitv 28456  LineGclng 28457  pInvGcmir 28675
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-trkgc 28471  df-trkgb 28472  df-trkgcb 28473  df-trkg 28476  df-mir 28676
This theorem is referenced by:  mirne  28690  mircinv  28691  mirln2  28700  miduniq  28708  miduniq2  28710  krippenlem  28713  ragflat2  28726  footexALT  28741  footexlem1  28742  footexlem2  28743  colperpexlem2  28754  colperpexlem3  28755  opphllem6  28775  lmimid  28817  hypcgrlem2  28823
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