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Theorem mirinv 25782
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 466 . . . 4 ((𝜑 ∧ (𝑀𝐵) = 𝐵) → 𝐺 ∈ TarskiG)
6 mirinv.b . . . . 5 (𝜑𝐵𝑃)
76adantr 466 . . . 4 ((𝜑 ∧ (𝑀𝐵) = 𝐵) → 𝐵𝑃)
8 mirval.a . . . . 5 (𝜑𝐴𝑃)
98adantr 466 . . . 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 25774 . . . . 5 ((𝜑 ∧ (𝑀𝐵) = 𝐵) → 𝐴 ∈ ((𝑀𝐵)𝐼𝐵))
14 simpr 471 . . . . . 6 ((𝜑 ∧ (𝑀𝐵) = 𝐵) → (𝑀𝐵) = 𝐵)
1514oveq1d 6811 . . . . 5 ((𝜑 ∧ (𝑀𝐵) = 𝐵) → ((𝑀𝐵)𝐼𝐵) = (𝐵𝐼𝐵))
1613, 15eleqtrd 2852 . . . 4 ((𝜑 ∧ (𝑀𝐵) = 𝐵) → 𝐴 ∈ (𝐵𝐼𝐵))
171, 2, 3, 5, 7, 9, 16axtgbtwnid 25586 . . 3 ((𝜑 ∧ (𝑀𝐵) = 𝐵) → 𝐵 = 𝐴)
1817eqcomd 2777 . 2 ((𝜑 ∧ (𝑀𝐵) = 𝐵) → 𝐴 = 𝐵)
194adantr 466 . . . 4 ((𝜑𝐴 = 𝐵) → 𝐺 ∈ TarskiG)
208adantr 466 . . . 4 ((𝜑𝐴 = 𝐵) → 𝐴𝑃)
216adantr 466 . . . 4 ((𝜑𝐴 = 𝐵) → 𝐵𝑃)
22 eqidd 2772 . . . 4 ((𝜑𝐴 = 𝐵) → (𝐴 𝐵) = (𝐴 𝐵))
23 simpr 471 . . . . 5 ((𝜑𝐴 = 𝐵) → 𝐴 = 𝐵)
241, 2, 3, 19, 21, 21tgbtwntriv1 25607 . . . . 5 ((𝜑𝐴 = 𝐵) → 𝐵 ∈ (𝐵𝐼𝐵))
2523, 24eqeltrd 2850 . . . 4 ((𝜑𝐴 = 𝐵) → 𝐴 ∈ (𝐵𝐼𝐵))
261, 2, 3, 10, 11, 19, 20, 12, 21, 21, 22, 25ismir 25775 . . 3 ((𝜑𝐴 = 𝐵) → 𝐵 = (𝑀𝐵))
2726eqcomd 2777 . 2 ((𝜑𝐴 = 𝐵) → (𝑀𝐵) = 𝐵)
2818, 27impbida 802 1 (𝜑 → ((𝑀𝐵) = 𝐵𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  cfv 6030  (class class class)co 6796  Basecbs 16064  distcds 16158  TarskiGcstrkg 25550  Itvcitv 25556  LineGclng 25557  pInvGcmir 25768
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-riota 6757  df-ov 6799  df-trkgc 25568  df-trkgb 25569  df-trkgcb 25570  df-trkg 25573  df-mir 25769
This theorem is referenced by:  mirne  25783  mircinv  25784  mirln2  25793  miduniq  25801  miduniq2  25803  krippenlem  25806  ragflat2  25819  footex  25834  colperpexlem2  25844  colperpexlem3  25845  opphllem6  25865  lmimid  25907  hypcgrlem2  25913
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