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Theorem mircgr 25352
 Description: Property of the image by the point inversion function. Definition 7.5 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)
mirfv.b (𝜑𝐵𝑃)
Assertion
Ref Expression
mircgr (𝜑 → (𝐴 (𝑀𝐵)) = (𝐴 𝐵))

Proof of Theorem mircgr
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 mirval.p . . . . . 6 𝑃 = (Base‘𝐺)
2 mirval.d . . . . . 6 = (dist‘𝐺)
3 mirval.i . . . . . 6 𝐼 = (Itv‘𝐺)
4 mirval.l . . . . . 6 𝐿 = (LineG‘𝐺)
5 mirval.s . . . . . 6 𝑆 = (pInvG‘𝐺)
6 mirval.g . . . . . 6 (𝜑𝐺 ∈ TarskiG)
7 mirval.a . . . . . 6 (𝜑𝐴𝑃)
8 mirfv.m . . . . . 6 𝑀 = (𝑆𝐴)
9 mirfv.b . . . . . 6 (𝜑𝐵𝑃)
101, 2, 3, 4, 5, 6, 7, 8, 9mirfv 25351 . . . . 5 (𝜑 → (𝑀𝐵) = (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))))
111, 2, 3, 6, 9, 7mirreu3 25349 . . . . . 6 (𝜑 → ∃!𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵)))
12 riotacl2 6524 . . . . . 6 (∃!𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵)) → (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))) ∈ {𝑧𝑃 ∣ ((𝐴 𝑧) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))})
1311, 12syl 17 . . . . 5 (𝜑 → (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))) ∈ {𝑧𝑃 ∣ ((𝐴 𝑧) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))})
1410, 13eqeltrd 2688 . . . 4 (𝜑 → (𝑀𝐵) ∈ {𝑧𝑃 ∣ ((𝐴 𝑧) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))})
15 oveq2 6557 . . . . . . 7 (𝑧 = (𝑀𝐵) → (𝐴 𝑧) = (𝐴 (𝑀𝐵)))
1615eqeq1d 2612 . . . . . 6 (𝑧 = (𝑀𝐵) → ((𝐴 𝑧) = (𝐴 𝐵) ↔ (𝐴 (𝑀𝐵)) = (𝐴 𝐵)))
17 oveq1 6556 . . . . . . 7 (𝑧 = (𝑀𝐵) → (𝑧𝐼𝐵) = ((𝑀𝐵)𝐼𝐵))
1817eleq2d 2673 . . . . . 6 (𝑧 = (𝑀𝐵) → (𝐴 ∈ (𝑧𝐼𝐵) ↔ 𝐴 ∈ ((𝑀𝐵)𝐼𝐵)))
1916, 18anbi12d 743 . . . . 5 (𝑧 = (𝑀𝐵) → (((𝐴 𝑧) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵)) ↔ ((𝐴 (𝑀𝐵)) = (𝐴 𝐵) ∧ 𝐴 ∈ ((𝑀𝐵)𝐼𝐵))))
2019elrab 3331 . . . 4 ((𝑀𝐵) ∈ {𝑧𝑃 ∣ ((𝐴 𝑧) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))} ↔ ((𝑀𝐵) ∈ 𝑃 ∧ ((𝐴 (𝑀𝐵)) = (𝐴 𝐵) ∧ 𝐴 ∈ ((𝑀𝐵)𝐼𝐵))))
2114, 20sylib 207 . . 3 (𝜑 → ((𝑀𝐵) ∈ 𝑃 ∧ ((𝐴 (𝑀𝐵)) = (𝐴 𝐵) ∧ 𝐴 ∈ ((𝑀𝐵)𝐼𝐵))))
2221simprd 478 . 2 (𝜑 → ((𝐴 (𝑀𝐵)) = (𝐴 𝐵) ∧ 𝐴 ∈ ((𝑀𝐵)𝐼𝐵)))
2322simpld 474 1 (𝜑 → (𝐴 (𝑀𝐵)) = (𝐴 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∃!wreu 2898  {crab 2900  ‘cfv 5804  ℩crio 6510  (class class class)co 6549  Basecbs 15695  distcds 15777  TarskiGcstrkg 25129  Itvcitv 25135  LineGclng 25136  pInvGcmir 25347 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-trkgc 25147  df-trkgb 25148  df-trkgcb 25149  df-trkg 25152  df-mir 25348 This theorem is referenced by:  mirmir  25357  miriso  25365  mirmir2  25369  mircgrextend  25377  mirtrcgr  25378  mirauto  25379  miduniq  25380  krippenlem  25385  ragcol  25394  ragflat  25399  ragcgr  25402  footex  25413  colperpexlem1  25422  colperpexlem3  25424  mideulem2  25426  opphllem  25427  midcgr  25472  lmiisolem  25488  sacgr  25522
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