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Theorem mirreu 26462
 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 26459 . 2 (𝜑 → (𝑀𝐵) ∈ 𝑃)
111, 2, 3, 4, 5, 6, 7, 8, 9mirmir 26460 . 2 (𝜑 → (𝑀‘(𝑀𝐵)) = 𝐵)
126ad2antrr 725 . . . . . 6 (((𝜑𝑎𝑃) ∧ (𝑀𝑎) = 𝐵) → 𝐺 ∈ TarskiG)
137ad2antrr 725 . . . . . 6 (((𝜑𝑎𝑃) ∧ (𝑀𝑎) = 𝐵) → 𝐴𝑃)
14 simplr 768 . . . . . 6 (((𝜑𝑎𝑃) ∧ (𝑀𝑎) = 𝐵) → 𝑎𝑃)
151, 2, 3, 4, 5, 12, 13, 8, 14mirmir 26460 . . . . 5 (((𝜑𝑎𝑃) ∧ (𝑀𝑎) = 𝐵) → (𝑀‘(𝑀𝑎)) = 𝑎)
16 simpr 488 . . . . . 6 (((𝜑𝑎𝑃) ∧ (𝑀𝑎) = 𝐵) → (𝑀𝑎) = 𝐵)
1716fveq2d 6653 . . . . 5 (((𝜑𝑎𝑃) ∧ (𝑀𝑎) = 𝐵) → (𝑀‘(𝑀𝑎)) = (𝑀𝐵))
1815, 17eqtr3d 2838 . . . 4 (((𝜑𝑎𝑃) ∧ (𝑀𝑎) = 𝐵) → 𝑎 = (𝑀𝐵))
1918ex 416 . . 3 ((𝜑𝑎𝑃) → ((𝑀𝑎) = 𝐵𝑎 = (𝑀𝐵)))
2019ralrimiva 3152 . 2 (𝜑 → ∀𝑎𝑃 ((𝑀𝑎) = 𝐵𝑎 = (𝑀𝐵)))
21 fveqeq2 6658 . . 3 (𝑎 = (𝑀𝐵) → ((𝑀𝑎) = 𝐵 ↔ (𝑀‘(𝑀𝐵)) = 𝐵))
2221eqreu 3671 . 2 (((𝑀𝐵) ∈ 𝑃 ∧ (𝑀‘(𝑀𝐵)) = 𝐵 ∧ ∀𝑎𝑃 ((𝑀𝑎) = 𝐵𝑎 = (𝑀𝐵))) → ∃!𝑎𝑃 (𝑀𝑎) = 𝐵)
2310, 11, 20, 22syl3anc 1368 1 (𝜑 → ∃!𝑎𝑃 (𝑀𝑎) = 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2112  ∀wral 3109  ∃!wreu 3111  ‘cfv 6328  Basecbs 16479  distcds 16570  TarskiGcstrkg 26228  Itvcitv 26234  LineGclng 26235  pInvGcmir 26450 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-trkgc 26246  df-trkgb 26247  df-trkgcb 26248  df-trkg 26251  df-mir 26451 This theorem is referenced by: (None)
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