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Theorem ismir 25354
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)
ismir.1 (𝜑𝐶𝑃)
ismir.2 (𝜑 → (𝐴 𝐶) = (𝐴 𝐵))
ismir.3 (𝜑𝐴 ∈ (𝐶𝐼𝐵))
Assertion
Ref Expression
ismir (𝜑𝐶 = (𝑀𝐵))

Proof of Theorem ismir
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 mirval.p . . 3 𝑃 = (Base‘𝐺)
2 mirval.d . . 3 = (dist‘𝐺)
3 mirval.i . . 3 𝐼 = (Itv‘𝐺)
4 mirval.l . . 3 𝐿 = (LineG‘𝐺)
5 mirval.s . . 3 𝑆 = (pInvG‘𝐺)
6 mirval.g . . 3 (𝜑𝐺 ∈ TarskiG)
7 mirval.a . . 3 (𝜑𝐴𝑃)
8 mirfv.m . . 3 𝑀 = (𝑆𝐴)
9 mirfv.b . . 3 (𝜑𝐵𝑃)
101, 2, 3, 4, 5, 6, 7, 8, 9mirfv 25351 . 2 (𝜑 → (𝑀𝐵) = (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))))
11 ismir.2 . . 3 (𝜑 → (𝐴 𝐶) = (𝐴 𝐵))
12 ismir.3 . . 3 (𝜑𝐴 ∈ (𝐶𝐼𝐵))
13 ismir.1 . . . 4 (𝜑𝐶𝑃)
141, 2, 3, 6, 9, 7mirreu3 25349 . . . 4 (𝜑 → ∃!𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵)))
15 oveq2 6557 . . . . . . 7 (𝑧 = 𝐶 → (𝐴 𝑧) = (𝐴 𝐶))
1615eqeq1d 2612 . . . . . 6 (𝑧 = 𝐶 → ((𝐴 𝑧) = (𝐴 𝐵) ↔ (𝐴 𝐶) = (𝐴 𝐵)))
17 oveq1 6556 . . . . . . 7 (𝑧 = 𝐶 → (𝑧𝐼𝐵) = (𝐶𝐼𝐵))
1817eleq2d 2673 . . . . . 6 (𝑧 = 𝐶 → (𝐴 ∈ (𝑧𝐼𝐵) ↔ 𝐴 ∈ (𝐶𝐼𝐵)))
1916, 18anbi12d 743 . . . . 5 (𝑧 = 𝐶 → (((𝐴 𝑧) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵)) ↔ ((𝐴 𝐶) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝐶𝐼𝐵))))
2019riota2 6533 . . . 4 ((𝐶𝑃 ∧ ∃!𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))) → (((𝐴 𝐶) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝐶𝐼𝐵)) ↔ (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))) = 𝐶))
2113, 14, 20syl2anc 691 . . 3 (𝜑 → (((𝐴 𝐶) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝐶𝐼𝐵)) ↔ (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))) = 𝐶))
2211, 12, 21mpbi2and 958 . 2 (𝜑 → (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))) = 𝐶)
2310, 22eqtr2d 2645 1 (𝜑𝐶 = (𝑀𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977  ∃!wreu 2898  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  mireq  25360  mirinv  25361  miriso  25365  mirmir2  25369  mirauto  25379  colmid  25383  krippenlem  25385  midexlem  25387  mideulem2  25426  opphllem  25427  midcom  25474  trgcopyeulem  25497
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