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Theorem ismir 26453
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 26450 . 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 26448 . . . 4 (𝜑 → ∃!𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵)))
15 oveq2 7143 . . . . . . 7 (𝑧 = 𝐶 → (𝐴 𝑧) = (𝐴 𝐶))
1615eqeq1d 2800 . . . . . 6 (𝑧 = 𝐶 → ((𝐴 𝑧) = (𝐴 𝐵) ↔ (𝐴 𝐶) = (𝐴 𝐵)))
17 oveq1 7142 . . . . . . 7 (𝑧 = 𝐶 → (𝑧𝐼𝐵) = (𝐶𝐼𝐵))
1817eleq2d 2875 . . . . . 6 (𝑧 = 𝐶 → (𝐴 ∈ (𝑧𝐼𝐵) ↔ 𝐴 ∈ (𝐶𝐼𝐵)))
1916, 18anbi12d 633 . . . . 5 (𝑧 = 𝐶 → (((𝐴 𝑧) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵)) ↔ ((𝐴 𝐶) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝐶𝐼𝐵))))
2019riota2 7118 . . . 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 587 . . 3 (𝜑 → (((𝐴 𝐶) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝐶𝐼𝐵)) ↔ (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))) = 𝐶))
2211, 12, 21mpbi2and 711 . 2 (𝜑 → (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))) = 𝐶)
2310, 22eqtr2d 2834 1 (𝜑𝐶 = (𝑀𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2111  ∃!wreu 3108  cfv 6324  crio 7092  (class class class)co 7135  Basecbs 16475  distcds 16566  TarskiGcstrkg 26224  Itvcitv 26230  LineGclng 26231  pInvGcmir 26446
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 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pr 5295
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-trkgc 26242  df-trkgb 26243  df-trkgcb 26244  df-trkg 26247  df-mir 26447
This theorem is referenced by:  mirmir  26456  mireq  26459  mirinv  26460  miriso  26464  mirmir2  26468  mirauto  26478  colmid  26482  krippenlem  26484  midexlem  26486  mideulem2  26528  opphllem  26529  midcom  26576  trgcopyeulem  26599
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