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Theorem mirinv 26573
 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 484 . . . 4 ((𝜑 ∧ (𝑀𝐵) = 𝐵) → 𝐺 ∈ TarskiG)
6 mirinv.b . . . . 5 (𝜑𝐵𝑃)
76adantr 484 . . . 4 ((𝜑 ∧ (𝑀𝐵) = 𝐵) → 𝐵𝑃)
8 mirval.a . . . . 5 (𝜑𝐴𝑃)
98adantr 484 . . . 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 26565 . . . . 5 ((𝜑 ∧ (𝑀𝐵) = 𝐵) → 𝐴 ∈ ((𝑀𝐵)𝐼𝐵))
14 simpr 488 . . . . . 6 ((𝜑 ∧ (𝑀𝐵) = 𝐵) → (𝑀𝐵) = 𝐵)
1514oveq1d 7171 . . . . 5 ((𝜑 ∧ (𝑀𝐵) = 𝐵) → ((𝑀𝐵)𝐼𝐵) = (𝐵𝐼𝐵))
1613, 15eleqtrd 2854 . . . 4 ((𝜑 ∧ (𝑀𝐵) = 𝐵) → 𝐴 ∈ (𝐵𝐼𝐵))
171, 2, 3, 5, 7, 9, 16axtgbtwnid 26373 . . 3 ((𝜑 ∧ (𝑀𝐵) = 𝐵) → 𝐵 = 𝐴)
1817eqcomd 2764 . 2 ((𝜑 ∧ (𝑀𝐵) = 𝐵) → 𝐴 = 𝐵)
194adantr 484 . . . 4 ((𝜑𝐴 = 𝐵) → 𝐺 ∈ TarskiG)
208adantr 484 . . . 4 ((𝜑𝐴 = 𝐵) → 𝐴𝑃)
216adantr 484 . . . 4 ((𝜑𝐴 = 𝐵) → 𝐵𝑃)
22 eqidd 2759 . . . 4 ((𝜑𝐴 = 𝐵) → (𝐴 𝐵) = (𝐴 𝐵))
23 simpr 488 . . . . 5 ((𝜑𝐴 = 𝐵) → 𝐴 = 𝐵)
241, 2, 3, 19, 21, 21tgbtwntriv1 26398 . . . . 5 ((𝜑𝐴 = 𝐵) → 𝐵 ∈ (𝐵𝐼𝐵))
2523, 24eqeltrd 2852 . . . 4 ((𝜑𝐴 = 𝐵) → 𝐴 ∈ (𝐵𝐼𝐵))
261, 2, 3, 10, 11, 19, 20, 12, 21, 21, 22, 25ismir 26566 . . 3 ((𝜑𝐴 = 𝐵) → 𝐵 = (𝑀𝐵))
2726eqcomd 2764 . 2 ((𝜑𝐴 = 𝐵) → (𝑀𝐵) = 𝐵)
2818, 27impbida 800 1 (𝜑 → ((𝑀𝐵) = 𝐵𝐴 = 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ‘cfv 6340  (class class class)co 7156  Basecbs 16555  distcds 16646  TarskiGcstrkg 26337  Itvcitv 26343  LineGclng 26344  pInvGcmir 26559 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5160  ax-sep 5173  ax-nul 5180  ax-pr 5302 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-id 5434  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-riota 7114  df-ov 7159  df-trkgc 26355  df-trkgb 26356  df-trkgcb 26357  df-trkg 26360  df-mir 26560 This theorem is referenced by:  mirne  26574  mircinv  26575  mirln2  26584  miduniq  26592  miduniq2  26594  krippenlem  26597  ragflat2  26610  footexALT  26625  footexlem1  26626  footexlem2  26627  colperpexlem2  26638  colperpexlem3  26639  opphllem6  26659  lmimid  26701  hypcgrlem2  26707
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