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Theorem mirreu 25459
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 25456 . 2 (𝜑 → (𝑀𝐵) ∈ 𝑃)
111, 2, 3, 4, 5, 6, 7, 8, 9mirmir 25457 . 2 (𝜑 → (𝑀‘(𝑀𝐵)) = 𝐵)
126ad2antrr 761 . . . . . 6 (((𝜑𝑎𝑃) ∧ (𝑀𝑎) = 𝐵) → 𝐺 ∈ TarskiG)
137ad2antrr 761 . . . . . 6 (((𝜑𝑎𝑃) ∧ (𝑀𝑎) = 𝐵) → 𝐴𝑃)
14 simplr 791 . . . . . 6 (((𝜑𝑎𝑃) ∧ (𝑀𝑎) = 𝐵) → 𝑎𝑃)
151, 2, 3, 4, 5, 12, 13, 8, 14mirmir 25457 . . . . 5 (((𝜑𝑎𝑃) ∧ (𝑀𝑎) = 𝐵) → (𝑀‘(𝑀𝑎)) = 𝑎)
16 simpr 477 . . . . . 6 (((𝜑𝑎𝑃) ∧ (𝑀𝑎) = 𝐵) → (𝑀𝑎) = 𝐵)
1716fveq2d 6152 . . . . 5 (((𝜑𝑎𝑃) ∧ (𝑀𝑎) = 𝐵) → (𝑀‘(𝑀𝑎)) = (𝑀𝐵))
1815, 17eqtr3d 2657 . . . 4 (((𝜑𝑎𝑃) ∧ (𝑀𝑎) = 𝐵) → 𝑎 = (𝑀𝐵))
1918ex 450 . . 3 ((𝜑𝑎𝑃) → ((𝑀𝑎) = 𝐵𝑎 = (𝑀𝐵)))
2019ralrimiva 2960 . 2 (𝜑 → ∀𝑎𝑃 ((𝑀𝑎) = 𝐵𝑎 = (𝑀𝐵)))
21 fveq2 6148 . . . 4 (𝑎 = (𝑀𝐵) → (𝑀𝑎) = (𝑀‘(𝑀𝐵)))
2221eqeq1d 2623 . . 3 (𝑎 = (𝑀𝐵) → ((𝑀𝑎) = 𝐵 ↔ (𝑀‘(𝑀𝐵)) = 𝐵))
2322eqreu 3380 . 2 (((𝑀𝐵) ∈ 𝑃 ∧ (𝑀‘(𝑀𝐵)) = 𝐵 ∧ ∀𝑎𝑃 ((𝑀𝑎) = 𝐵𝑎 = (𝑀𝐵))) → ∃!𝑎𝑃 (𝑀𝑎) = 𝐵)
2410, 11, 20, 23syl3anc 1323 1 (𝜑 → ∃!𝑎𝑃 (𝑀𝑎) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wral 2907  ∃!wreu 2909  cfv 5847  Basecbs 15781  distcds 15871  TarskiGcstrkg 25229  Itvcitv 25235  LineGclng 25236  pInvGcmir 25447
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-trkgc 25247  df-trkgb 25248  df-trkgcb 25249  df-trkg 25252  df-mir 25448
This theorem is referenced by: (None)
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