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Theorem istrkgc 25248
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 473 . . . . . 6 ((𝑝 = 𝑃𝑑 = ) → 𝑝 = 𝑃)
43eqcomd 2632 . . . . 5 ((𝑝 = 𝑃𝑑 = ) → 𝑃 = 𝑝)
54adantr 481 . . . . . 6 (((𝑝 = 𝑃𝑑 = ) ∧ 𝑥𝑃) → 𝑃 = 𝑝)
6 simpllr 798 . . . . . . . . 9 ((((𝑝 = 𝑃𝑑 = ) ∧ 𝑥𝑃) ∧ 𝑦𝑃) → 𝑑 = )
76eqcomd 2632 . . . . . . . 8 ((((𝑝 = 𝑃𝑑 = ) ∧ 𝑥𝑃) ∧ 𝑦𝑃) → = 𝑑)
87oveqd 6622 . . . . . . 7 ((((𝑝 = 𝑃𝑑 = ) ∧ 𝑥𝑃) ∧ 𝑦𝑃) → (𝑥 𝑦) = (𝑥𝑑𝑦))
97oveqd 6622 . . . . . . 7 ((((𝑝 = 𝑃𝑑 = ) ∧ 𝑥𝑃) ∧ 𝑦𝑃) → (𝑦 𝑥) = (𝑦𝑑𝑥))
108, 9eqeq12d 2641 . . . . . 6 ((((𝑝 = 𝑃𝑑 = ) ∧ 𝑥𝑃) ∧ 𝑦𝑃) → ((𝑥 𝑦) = (𝑦 𝑥) ↔ (𝑥𝑑𝑦) = (𝑦𝑑𝑥)))
115, 10raleqbidva 3148 . . . . 5 (((𝑝 = 𝑃𝑑 = ) ∧ 𝑥𝑃) → (∀𝑦𝑃 (𝑥 𝑦) = (𝑦 𝑥) ↔ ∀𝑦𝑝 (𝑥𝑑𝑦) = (𝑦𝑑𝑥)))
124, 11raleqbidva 3148 . . . 4 ((𝑝 = 𝑃𝑑 = ) → (∀𝑥𝑃𝑦𝑃 (𝑥 𝑦) = (𝑦 𝑥) ↔ ∀𝑥𝑝𝑦𝑝 (𝑥𝑑𝑦) = (𝑦𝑑𝑥)))
135adantr 481 . . . . . . 7 ((((𝑝 = 𝑃𝑑 = ) ∧ 𝑥𝑃) ∧ 𝑦𝑃) → 𝑃 = 𝑝)
147oveqdr 6629 . . . . . . . . 9 (((((𝑝 = 𝑃𝑑 = ) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑧𝑃) → (𝑥 𝑦) = (𝑥𝑑𝑦))
157oveqdr 6629 . . . . . . . . 9 (((((𝑝 = 𝑃𝑑 = ) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑧𝑃) → (𝑧 𝑧) = (𝑧𝑑𝑧))
1614, 15eqeq12d 2641 . . . . . . . 8 (((((𝑝 = 𝑃𝑑 = ) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑧𝑃) → ((𝑥 𝑦) = (𝑧 𝑧) ↔ (𝑥𝑑𝑦) = (𝑧𝑑𝑧)))
1716imbi1d 331 . . . . . . 7 (((((𝑝 = 𝑃𝑑 = ) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑧𝑃) → (((𝑥 𝑦) = (𝑧 𝑧) → 𝑥 = 𝑦) ↔ ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦)))
1813, 17raleqbidva 3148 . . . . . 6 ((((𝑝 = 𝑃𝑑 = ) ∧ 𝑥𝑃) ∧ 𝑦𝑃) → (∀𝑧𝑃 ((𝑥 𝑦) = (𝑧 𝑧) → 𝑥 = 𝑦) ↔ ∀𝑧𝑝 ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦)))
195, 18raleqbidva 3148 . . . . 5 (((𝑝 = 𝑃𝑑 = ) ∧ 𝑥𝑃) → (∀𝑦𝑃𝑧𝑃 ((𝑥 𝑦) = (𝑧 𝑧) → 𝑥 = 𝑦) ↔ ∀𝑦𝑝𝑧𝑝 ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦)))
204, 19raleqbidva 3148 . . . 4 ((𝑝 = 𝑃𝑑 = ) → (∀𝑥𝑃𝑦𝑃𝑧𝑃 ((𝑥 𝑦) = (𝑧 𝑧) → 𝑥 = 𝑦) ↔ ∀𝑥𝑝𝑦𝑝𝑧𝑝 ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦)))
2112, 20anbi12d 746 . . 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 15832 . 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 25242 . 2 TarskiGC = {𝑓[(Base‘𝑓) / 𝑝][(dist‘𝑓) / 𝑑](∀𝑥𝑝𝑦𝑝 (𝑥𝑑𝑦) = (𝑦𝑑𝑥) ∧ ∀𝑥𝑝𝑦𝑝𝑧𝑝 ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦))}
2422, 23elab4g 3343 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 196  wa 384   = wceq 1480  wcel 1992  wral 2912  Vcvv 3191  [wsbc 3422  cfv 5850  (class class class)co 6605  Basecbs 15776  distcds 15866  TarskiGCcstrkgc 25225  Itvcitv 25230
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 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-nul 4754
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 1883  df-eu 2478  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-iota 5813  df-fv 5858  df-ov 6608  df-trkgc 25242
This theorem is referenced by:  axtgcgrrflx  25256  axtgcgrid  25257  f1otrg  25646  xmstrkgc  25661  eengtrkg  25760
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