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Mirrors  >  Home  >  MPE Home  >  Th. List  >  mirinv Structured version   Visualization version   GIF version

Theorem mirinv 25456
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 481 . . . 4 ((𝜑 ∧ (𝑀𝐵) = 𝐵) → 𝐺 ∈ TarskiG)
6 mirinv.b . . . . 5 (𝜑𝐵𝑃)
76adantr 481 . . . 4 ((𝜑 ∧ (𝑀𝐵) = 𝐵) → 𝐵𝑃)
8 mirval.a . . . . 5 (𝜑𝐴𝑃)
98adantr 481 . . . 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 25448 . . . . 5 ((𝜑 ∧ (𝑀𝐵) = 𝐵) → 𝐴 ∈ ((𝑀𝐵)𝐼𝐵))
14 simpr 477 . . . . . 6 ((𝜑 ∧ (𝑀𝐵) = 𝐵) → (𝑀𝐵) = 𝐵)
1514oveq1d 6620 . . . . 5 ((𝜑 ∧ (𝑀𝐵) = 𝐵) → ((𝑀𝐵)𝐼𝐵) = (𝐵𝐼𝐵))
1613, 15eleqtrd 2706 . . . 4 ((𝜑 ∧ (𝑀𝐵) = 𝐵) → 𝐴 ∈ (𝐵𝐼𝐵))
171, 2, 3, 5, 7, 9, 16axtgbtwnid 25260 . . 3 ((𝜑 ∧ (𝑀𝐵) = 𝐵) → 𝐵 = 𝐴)
1817eqcomd 2632 . 2 ((𝜑 ∧ (𝑀𝐵) = 𝐵) → 𝐴 = 𝐵)
194adantr 481 . . . 4 ((𝜑𝐴 = 𝐵) → 𝐺 ∈ TarskiG)
208adantr 481 . . . 4 ((𝜑𝐴 = 𝐵) → 𝐴𝑃)
216adantr 481 . . . 4 ((𝜑𝐴 = 𝐵) → 𝐵𝑃)
22 eqidd 2627 . . . 4 ((𝜑𝐴 = 𝐵) → (𝐴 𝐵) = (𝐴 𝐵))
23 simpr 477 . . . . 5 ((𝜑𝐴 = 𝐵) → 𝐴 = 𝐵)
241, 2, 3, 19, 21, 21tgbtwntriv1 25281 . . . . 5 ((𝜑𝐴 = 𝐵) → 𝐵 ∈ (𝐵𝐼𝐵))
2523, 24eqeltrd 2704 . . . 4 ((𝜑𝐴 = 𝐵) → 𝐴 ∈ (𝐵𝐼𝐵))
261, 2, 3, 10, 11, 19, 20, 12, 21, 21, 22, 25ismir 25449 . . 3 ((𝜑𝐴 = 𝐵) → 𝐵 = (𝑀𝐵))
2726eqcomd 2632 . 2 ((𝜑𝐴 = 𝐵) → (𝑀𝐵) = 𝐵)
2818, 27impbida 876 1 (𝜑 → ((𝑀𝐵) = 𝐵𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1992  cfv 5850  (class class class)co 6605  Basecbs 15776  distcds 15866  TarskiGcstrkg 25224  Itvcitv 25230  LineGclng 25231  pInvGcmir 25442
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-riota 6566  df-ov 6608  df-trkgc 25242  df-trkgb 25243  df-trkgcb 25244  df-trkg 25247  df-mir 25443
This theorem is referenced by:  mirne  25457  mircinv  25458  mirln2  25467  miduniq  25475  miduniq2  25477  krippenlem  25480  ragflat2  25493  footex  25508  colperpexlem2  25518  colperpexlem3  25519  opphllem6  25539  lmimid  25581  hypcgrlem2  25587
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