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Theorem mirf 28668
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 28663 . . 3 (𝜑 → (𝑆𝐴) = (𝑦𝑃 ↦ (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦)))))
123, 11eqtrid 2789 . 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 28664 . . 3 ((𝜑𝑥𝑃) → (𝑀𝑥) = (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝑥) ∧ 𝐴 ∈ (𝑧𝐼𝑥))))
174, 5, 6, 13, 15, 14mirreu3 28662 . . . 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 2841 . 2 ((𝜑𝑥𝑃) → (𝑀𝑥) ∈ 𝑃)
212, 12, 20fmpt2d 7144 1 (𝜑𝑀:𝑃𝑃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  ∃!wreu 3378  Vcvv 3480  cmpt 5225  wf 6557  cfv 6561  crio 7387  (class class class)co 7431  Basecbs 17247  distcds 17306  TarskiGcstrkg 28435  Itvcitv 28441  LineGclng 28442  pInvGcmir 28660
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-trkgc 28456  df-trkgb 28457  df-trkgcb 28458  df-trkg 28461  df-mir 28661
This theorem is referenced by:  mircl  28669  mirf1o  28677  mirbtwni  28679  mirbtwnb  28680  mirauto  28692  miduniq2  28695  krippenlem  28698
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