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Theorem ismir 25272
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 25269 . 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 25267 . . . 4 (𝜑 → ∃!𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵)))
15 oveq2 6535 . . . . . . 7 (𝑧 = 𝐶 → (𝐴 𝑧) = (𝐴 𝐶))
1615eqeq1d 2611 . . . . . 6 (𝑧 = 𝐶 → ((𝐴 𝑧) = (𝐴 𝐵) ↔ (𝐴 𝐶) = (𝐴 𝐵)))
17 oveq1 6534 . . . . . . 7 (𝑧 = 𝐶 → (𝑧𝐼𝐵) = (𝐶𝐼𝐵))
1817eleq2d 2672 . . . . . 6 (𝑧 = 𝐶 → (𝐴 ∈ (𝑧𝐼𝐵) ↔ 𝐴 ∈ (𝐶𝐼𝐵)))
1916, 18anbi12d 742 . . . . 5 (𝑧 = 𝐶 → (((𝐴 𝑧) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵)) ↔ ((𝐴 𝐶) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝐶𝐼𝐵))))
2019riota2 6511 . . . 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 690 . . 3 (𝜑 → (((𝐴 𝐶) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝐶𝐼𝐵)) ↔ (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))) = 𝐶))
2211, 12, 21mpbi2and 957 . 2 (𝜑 → (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))) = 𝐶)
2310, 22eqtr2d 2644 1 (𝜑𝐶 = (𝑀𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wcel 1976  ∃!wreu 2897  cfv 5790  crio 6488  (class class class)co 6527  Basecbs 15641  distcds 15723  TarskiGcstrkg 25046  Itvcitv 25052  LineGclng 25053  pInvGcmir 25265
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-trkgc 25064  df-trkgb 25065  df-trkgcb 25066  df-trkg 25069  df-mir 25266
This theorem is referenced by:  mirmir  25275  mireq  25278  mirinv  25279  miriso  25283  mirmir2  25287  mirauto  25297  colmid  25301  krippenlem  25303  midexlem  25305  mideulem2  25344  opphllem  25345  midcom  25392  trgcopyeulem  25415
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