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Theorem istrkgc 25153
 Description: Property of being a Tarski geometry - congruence part. (Contributed by Thierry Arnoux, 14-Mar-2019.)
Hypotheses
Ref Expression
istrkg.p 𝑃 = (Base‘𝐺)
istrkg.d = (dist‘𝐺)
istrkg.i 𝐼 = (Itv‘𝐺)
Assertion
Ref Expression
istrkgc (𝐺 ∈ TarskiGC ↔ (𝐺 ∈ V ∧ (∀𝑥𝑃𝑦𝑃 (𝑥 𝑦) = (𝑦 𝑥) ∧ ∀𝑥𝑃𝑦𝑃𝑧𝑃 ((𝑥 𝑦) = (𝑧 𝑧) → 𝑥 = 𝑦))))
Distinct variable groups:   𝑥,𝑦,𝑧,𝐼   𝑥,𝑃,𝑦,𝑧   𝑥, ,𝑦,𝑧
Allowed substitution hints:   𝐺(𝑥,𝑦,𝑧)

Proof of Theorem istrkgc
Dummy variables 𝑓 𝑑 𝑝 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 istrkg.p . . 3 𝑃 = (Base‘𝐺)
2 istrkg.d . . 3 = (dist‘𝐺)
3 simpl 472 . . . . . 6 ((𝑝 = 𝑃𝑑 = ) → 𝑝 = 𝑃)
43eqcomd 2616 . . . . 5 ((𝑝 = 𝑃𝑑 = ) → 𝑃 = 𝑝)
54adantr 480 . . . . . 6 (((𝑝 = 𝑃𝑑 = ) ∧ 𝑥𝑃) → 𝑃 = 𝑝)
6 simpllr 795 . . . . . . . . 9 ((((𝑝 = 𝑃𝑑 = ) ∧ 𝑥𝑃) ∧ 𝑦𝑃) → 𝑑 = )
76eqcomd 2616 . . . . . . . 8 ((((𝑝 = 𝑃𝑑 = ) ∧ 𝑥𝑃) ∧ 𝑦𝑃) → = 𝑑)
87oveqd 6566 . . . . . . 7 ((((𝑝 = 𝑃𝑑 = ) ∧ 𝑥𝑃) ∧ 𝑦𝑃) → (𝑥 𝑦) = (𝑥𝑑𝑦))
97oveqd 6566 . . . . . . 7 ((((𝑝 = 𝑃𝑑 = ) ∧ 𝑥𝑃) ∧ 𝑦𝑃) → (𝑦 𝑥) = (𝑦𝑑𝑥))
108, 9eqeq12d 2625 . . . . . 6 ((((𝑝 = 𝑃𝑑 = ) ∧ 𝑥𝑃) ∧ 𝑦𝑃) → ((𝑥 𝑦) = (𝑦 𝑥) ↔ (𝑥𝑑𝑦) = (𝑦𝑑𝑥)))
115, 10raleqbidva 3131 . . . . 5 (((𝑝 = 𝑃𝑑 = ) ∧ 𝑥𝑃) → (∀𝑦𝑃 (𝑥 𝑦) = (𝑦 𝑥) ↔ ∀𝑦𝑝 (𝑥𝑑𝑦) = (𝑦𝑑𝑥)))
124, 11raleqbidva 3131 . . . 4 ((𝑝 = 𝑃𝑑 = ) → (∀𝑥𝑃𝑦𝑃 (𝑥 𝑦) = (𝑦 𝑥) ↔ ∀𝑥𝑝𝑦𝑝 (𝑥𝑑𝑦) = (𝑦𝑑𝑥)))
135adantr 480 . . . . . . 7 ((((𝑝 = 𝑃𝑑 = ) ∧ 𝑥𝑃) ∧ 𝑦𝑃) → 𝑃 = 𝑝)
147oveqdr 6573 . . . . . . . . 9 (((((𝑝 = 𝑃𝑑 = ) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑧𝑃) → (𝑥 𝑦) = (𝑥𝑑𝑦))
157oveqdr 6573 . . . . . . . . 9 (((((𝑝 = 𝑃𝑑 = ) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑧𝑃) → (𝑧 𝑧) = (𝑧𝑑𝑧))
1614, 15eqeq12d 2625 . . . . . . . 8 (((((𝑝 = 𝑃𝑑 = ) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑧𝑃) → ((𝑥 𝑦) = (𝑧 𝑧) ↔ (𝑥𝑑𝑦) = (𝑧𝑑𝑧)))
1716imbi1d 330 . . . . . . 7 (((((𝑝 = 𝑃𝑑 = ) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑧𝑃) → (((𝑥 𝑦) = (𝑧 𝑧) → 𝑥 = 𝑦) ↔ ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦)))
1813, 17raleqbidva 3131 . . . . . 6 ((((𝑝 = 𝑃𝑑 = ) ∧ 𝑥𝑃) ∧ 𝑦𝑃) → (∀𝑧𝑃 ((𝑥 𝑦) = (𝑧 𝑧) → 𝑥 = 𝑦) ↔ ∀𝑧𝑝 ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦)))
195, 18raleqbidva 3131 . . . . 5 (((𝑝 = 𝑃𝑑 = ) ∧ 𝑥𝑃) → (∀𝑦𝑃𝑧𝑃 ((𝑥 𝑦) = (𝑧 𝑧) → 𝑥 = 𝑦) ↔ ∀𝑦𝑝𝑧𝑝 ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦)))
204, 19raleqbidva 3131 . . . 4 ((𝑝 = 𝑃𝑑 = ) → (∀𝑥𝑃𝑦𝑃𝑧𝑃 ((𝑥 𝑦) = (𝑧 𝑧) → 𝑥 = 𝑦) ↔ ∀𝑥𝑝𝑦𝑝𝑧𝑝 ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦)))
2112, 20anbi12d 743 . . 3 ((𝑝 = 𝑃𝑑 = ) → ((∀𝑥𝑃𝑦𝑃 (𝑥 𝑦) = (𝑦 𝑥) ∧ ∀𝑥𝑃𝑦𝑃𝑧𝑃 ((𝑥 𝑦) = (𝑧 𝑧) → 𝑥 = 𝑦)) ↔ (∀𝑥𝑝𝑦𝑝 (𝑥𝑑𝑦) = (𝑦𝑑𝑥) ∧ ∀𝑥𝑝𝑦𝑝𝑧𝑝 ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦))))
221, 2, 21sbcie2s 15744 . 2 (𝑓 = 𝐺 → ([(Base‘𝑓) / 𝑝][(dist‘𝑓) / 𝑑](∀𝑥𝑝𝑦𝑝 (𝑥𝑑𝑦) = (𝑦𝑑𝑥) ∧ ∀𝑥𝑝𝑦𝑝𝑧𝑝 ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦)) ↔ (∀𝑥𝑃𝑦𝑃 (𝑥 𝑦) = (𝑦 𝑥) ∧ ∀𝑥𝑃𝑦𝑃𝑧𝑃 ((𝑥 𝑦) = (𝑧 𝑧) → 𝑥 = 𝑦))))
23 df-trkgc 25147 . 2 TarskiGC = {𝑓[(Base‘𝑓) / 𝑝][(dist‘𝑓) / 𝑑](∀𝑥𝑝𝑦𝑝 (𝑥𝑑𝑦) = (𝑦𝑑𝑥) ∧ ∀𝑥𝑝𝑦𝑝𝑧𝑝 ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦))}
2422, 23elab4g 3324 1 (𝐺 ∈ TarskiGC ↔ (𝐺 ∈ V ∧ (∀𝑥𝑃𝑦𝑃 (𝑥 𝑦) = (𝑦 𝑥) ∧ ∀𝑥𝑃𝑦𝑃𝑧𝑃 ((𝑥 𝑦) = (𝑧 𝑧) → 𝑥 = 𝑦))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173  [wsbc 3402  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  distcds 15777  TarskiGCcstrkgc 25130  Itvcitv 25135 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ov 6552  df-trkgc 25147 This theorem is referenced by:  axtgcgrrflx  25161  axtgcgrid  25162  f1otrg  25551  xmstrkgc  25566  eengtrkg  25665
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