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Theorem mirf 28683
Description: Point inversion as function. (Contributed by Thierry Arnoux, 30-May-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)
Assertion
Ref Expression
mirf (𝜑𝑀:𝑃𝑃)

Proof of Theorem mirf
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 riotaex 7392 . . 3 (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦))) ∈ V
21a1i 11 . 2 ((𝜑𝑦𝑃) → (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦))) ∈ V)
3 mirfv.m . . 3 𝑀 = (𝑆𝐴)
4 mirval.p . . . 4 𝑃 = (Base‘𝐺)
5 mirval.d . . . 4 = (dist‘𝐺)
6 mirval.i . . . 4 𝐼 = (Itv‘𝐺)
7 mirval.l . . . 4 𝐿 = (LineG‘𝐺)
8 mirval.s . . . 4 𝑆 = (pInvG‘𝐺)
9 mirval.g . . . 4 (𝜑𝐺 ∈ TarskiG)
10 mirval.a . . . 4 (𝜑𝐴𝑃)
114, 5, 6, 7, 8, 9, 10mirval 28678 . . 3 (𝜑 → (𝑆𝐴) = (𝑦𝑃 ↦ (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦)))))
123, 11eqtrid 2787 . 2 (𝜑𝑀 = (𝑦𝑃 ↦ (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦)))))
139adantr 480 . . . 4 ((𝜑𝑥𝑃) → 𝐺 ∈ TarskiG)
1410adantr 480 . . . 4 ((𝜑𝑥𝑃) → 𝐴𝑃)
15 simpr 484 . . . 4 ((𝜑𝑥𝑃) → 𝑥𝑃)
164, 5, 6, 7, 8, 13, 14, 3, 15mirfv 28679 . . 3 ((𝜑𝑥𝑃) → (𝑀𝑥) = (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝑥) ∧ 𝐴 ∈ (𝑧𝐼𝑥))))
174, 5, 6, 13, 15, 14mirreu3 28677 . . . 4 ((𝜑𝑥𝑃) → ∃!𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝑥) ∧ 𝐴 ∈ (𝑧𝐼𝑥)))
18 riotacl 7405 . . . 4 (∃!𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝑥) ∧ 𝐴 ∈ (𝑧𝐼𝑥)) → (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝑥) ∧ 𝐴 ∈ (𝑧𝐼𝑥))) ∈ 𝑃)
1917, 18syl 17 . . 3 ((𝜑𝑥𝑃) → (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝑥) ∧ 𝐴 ∈ (𝑧𝐼𝑥))) ∈ 𝑃)
2016, 19eqeltrd 2839 . 2 ((𝜑𝑥𝑃) → (𝑀𝑥) ∈ 𝑃)
212, 12, 20fmpt2d 7144 1 (𝜑𝑀:𝑃𝑃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  ∃!wreu 3376  Vcvv 3478  cmpt 5231  wf 6559  cfv 6563  crio 7387  (class class class)co 7431  Basecbs 17245  distcds 17307  TarskiGcstrkg 28450  Itvcitv 28456  LineGclng 28457  pInvGcmir 28675
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-trkgc 28471  df-trkgb 28472  df-trkgcb 28473  df-trkg 28476  df-mir 28676
This theorem is referenced by:  mircl  28684  mirf1o  28692  mirbtwni  28694  mirbtwnb  28695  mirauto  28707  miduniq2  28710  krippenlem  28713
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