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Theorem mirbtwn 25909
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
mirbtwn (𝜑𝐴 ∈ ((𝑀𝐵)𝐼𝐵))

Proof of Theorem mirbtwn
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 mirval.p . . . . 5 𝑃 = (Base‘𝐺)
2 mirval.d . . . . 5 = (dist‘𝐺)
3 mirval.i . . . . 5 𝐼 = (Itv‘𝐺)
4 mirval.l . . . . 5 𝐿 = (LineG‘𝐺)
5 mirval.s . . . . 5 𝑆 = (pInvG‘𝐺)
6 mirval.g . . . . 5 (𝜑𝐺 ∈ TarskiG)
7 mirval.a . . . . 5 (𝜑𝐴𝑃)
8 mirfv.m . . . . 5 𝑀 = (𝑆𝐴)
9 mirfv.b . . . . 5 (𝜑𝐵𝑃)
101, 2, 3, 4, 5, 6, 7, 8, 9mirfv 25907 . . . 4 (𝜑 → (𝑀𝐵) = (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))))
111, 2, 3, 6, 9, 7mirreu3 25905 . . . . 5 (𝜑 → ∃!𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵)))
12 riotacl2 6852 . . . . 5 (∃!𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵)) → (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))) ∈ {𝑧𝑃 ∣ ((𝐴 𝑧) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))})
1311, 12syl 17 . . . 4 (𝜑 → (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))) ∈ {𝑧𝑃 ∣ ((𝐴 𝑧) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))})
1410, 13eqeltrd 2878 . . 3 (𝜑 → (𝑀𝐵) ∈ {𝑧𝑃 ∣ ((𝐴 𝑧) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))})
15 oveq2 6886 . . . . . 6 (𝑧 = (𝑀𝐵) → (𝐴 𝑧) = (𝐴 (𝑀𝐵)))
1615eqeq1d 2801 . . . . 5 (𝑧 = (𝑀𝐵) → ((𝐴 𝑧) = (𝐴 𝐵) ↔ (𝐴 (𝑀𝐵)) = (𝐴 𝐵)))
17 oveq1 6885 . . . . . 6 (𝑧 = (𝑀𝐵) → (𝑧𝐼𝐵) = ((𝑀𝐵)𝐼𝐵))
1817eleq2d 2864 . . . . 5 (𝑧 = (𝑀𝐵) → (𝐴 ∈ (𝑧𝐼𝐵) ↔ 𝐴 ∈ ((𝑀𝐵)𝐼𝐵)))
1916, 18anbi12d 625 . . . 4 (𝑧 = (𝑀𝐵) → (((𝐴 𝑧) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵)) ↔ ((𝐴 (𝑀𝐵)) = (𝐴 𝐵) ∧ 𝐴 ∈ ((𝑀𝐵)𝐼𝐵))))
2019elrab 3556 . . 3 ((𝑀𝐵) ∈ {𝑧𝑃 ∣ ((𝐴 𝑧) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))} ↔ ((𝑀𝐵) ∈ 𝑃 ∧ ((𝐴 (𝑀𝐵)) = (𝐴 𝐵) ∧ 𝐴 ∈ ((𝑀𝐵)𝐼𝐵))))
2114, 20sylib 210 . 2 (𝜑 → ((𝑀𝐵) ∈ 𝑃 ∧ ((𝐴 (𝑀𝐵)) = (𝐴 𝐵) ∧ 𝐴 ∈ ((𝑀𝐵)𝐼𝐵))))
2221simprrd 791 1 (𝜑𝐴 ∈ ((𝑀𝐵)𝐼𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385   = wceq 1653  wcel 2157  ∃!wreu 3091  {crab 3093  cfv 6101  crio 6838  (class class class)co 6878  Basecbs 16184  distcds 16276  TarskiGcstrkg 25681  Itvcitv 25687  LineGclng 25688  pInvGcmir 25903
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rmo 3097  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-riota 6839  df-ov 6881  df-trkgc 25699  df-trkgb 25700  df-trkgcb 25701  df-trkg 25704  df-mir 25904
This theorem is referenced by:  mirmir  25913  mirinv  25917  miriso  25921  mirmir2  25925  mirln  25927  mirln2  25928  mirconn  25929  mirhl2  25932  mircgrextend  25933  mirtrcgr  25934  mirauto  25935  miduniq  25936  krippenlem  25941  ragflat  25955  ragcgr  25958  footex  25969  colperpexlem1  25978  colperpexlem3  25980  mideulem2  25982  opphllem  25983  opphllem1  25995  opphllem2  25996  opphllem4  25998  colhp  26018  midbtwn  26027  lmieu  26032  lmiisolem  26044
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