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Theorem mirreu 26377
Description: Any point has a unique antecedent through point inversion. Theorem 7.8 of [Schwabhauser] p. 50. (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)
mirmir.b (𝜑𝐵𝑃)
Assertion
Ref Expression
mirreu (𝜑 → ∃!𝑎𝑃 (𝑀𝑎) = 𝐵)
Distinct variable groups:   𝐵,𝑎   𝑀,𝑎   𝑃,𝑎   𝜑,𝑎
Allowed substitution hints:   𝐴(𝑎)   𝑆(𝑎)   𝐺(𝑎)   𝐼(𝑎)   𝐿(𝑎)   (𝑎)

Proof of Theorem mirreu
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 mirmir.b . . 3 (𝜑𝐵𝑃)
101, 2, 3, 4, 5, 6, 7, 8, 9mircl 26374 . 2 (𝜑 → (𝑀𝐵) ∈ 𝑃)
111, 2, 3, 4, 5, 6, 7, 8, 9mirmir 26375 . 2 (𝜑 → (𝑀‘(𝑀𝐵)) = 𝐵)
126ad2antrr 722 . . . . . 6 (((𝜑𝑎𝑃) ∧ (𝑀𝑎) = 𝐵) → 𝐺 ∈ TarskiG)
137ad2antrr 722 . . . . . 6 (((𝜑𝑎𝑃) ∧ (𝑀𝑎) = 𝐵) → 𝐴𝑃)
14 simplr 765 . . . . . 6 (((𝜑𝑎𝑃) ∧ (𝑀𝑎) = 𝐵) → 𝑎𝑃)
151, 2, 3, 4, 5, 12, 13, 8, 14mirmir 26375 . . . . 5 (((𝜑𝑎𝑃) ∧ (𝑀𝑎) = 𝐵) → (𝑀‘(𝑀𝑎)) = 𝑎)
16 simpr 485 . . . . . 6 (((𝜑𝑎𝑃) ∧ (𝑀𝑎) = 𝐵) → (𝑀𝑎) = 𝐵)
1716fveq2d 6667 . . . . 5 (((𝜑𝑎𝑃) ∧ (𝑀𝑎) = 𝐵) → (𝑀‘(𝑀𝑎)) = (𝑀𝐵))
1815, 17eqtr3d 2855 . . . 4 (((𝜑𝑎𝑃) ∧ (𝑀𝑎) = 𝐵) → 𝑎 = (𝑀𝐵))
1918ex 413 . . 3 ((𝜑𝑎𝑃) → ((𝑀𝑎) = 𝐵𝑎 = (𝑀𝐵)))
2019ralrimiva 3179 . 2 (𝜑 → ∀𝑎𝑃 ((𝑀𝑎) = 𝐵𝑎 = (𝑀𝐵)))
21 fveqeq2 6672 . . 3 (𝑎 = (𝑀𝐵) → ((𝑀𝑎) = 𝐵 ↔ (𝑀‘(𝑀𝐵)) = 𝐵))
2221eqreu 3717 . 2 (((𝑀𝐵) ∈ 𝑃 ∧ (𝑀‘(𝑀𝐵)) = 𝐵 ∧ ∀𝑎𝑃 ((𝑀𝑎) = 𝐵𝑎 = (𝑀𝐵))) → ∃!𝑎𝑃 (𝑀𝑎) = 𝐵)
2310, 11, 20, 22syl3anc 1363 1 (𝜑 → ∃!𝑎𝑃 (𝑀𝑎) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  wral 3135  ∃!wreu 3137  cfv 6348  Basecbs 16471  distcds 16562  TarskiGcstrkg 26143  Itvcitv 26149  LineGclng 26150  pInvGcmir 26365
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-trkgc 26161  df-trkgb 26162  df-trkgcb 26163  df-trkg 26166  df-mir 26366
This theorem is referenced by: (None)
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