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Theorem istrkgc 27104
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 483 . . . . . 6 ((𝑝 = 𝑃𝑑 = ) → 𝑝 = 𝑃)
43eqcomd 2742 . . . . 5 ((𝑝 = 𝑃𝑑 = ) → 𝑃 = 𝑝)
54adantr 481 . . . . . 6 (((𝑝 = 𝑃𝑑 = ) ∧ 𝑥𝑃) → 𝑃 = 𝑝)
6 simpllr 773 . . . . . . . . 9 ((((𝑝 = 𝑃𝑑 = ) ∧ 𝑥𝑃) ∧ 𝑦𝑃) → 𝑑 = )
76eqcomd 2742 . . . . . . . 8 ((((𝑝 = 𝑃𝑑 = ) ∧ 𝑥𝑃) ∧ 𝑦𝑃) → = 𝑑)
87oveqd 7354 . . . . . . 7 ((((𝑝 = 𝑃𝑑 = ) ∧ 𝑥𝑃) ∧ 𝑦𝑃) → (𝑥 𝑦) = (𝑥𝑑𝑦))
97oveqd 7354 . . . . . . 7 ((((𝑝 = 𝑃𝑑 = ) ∧ 𝑥𝑃) ∧ 𝑦𝑃) → (𝑦 𝑥) = (𝑦𝑑𝑥))
108, 9eqeq12d 2752 . . . . . 6 ((((𝑝 = 𝑃𝑑 = ) ∧ 𝑥𝑃) ∧ 𝑦𝑃) → ((𝑥 𝑦) = (𝑦 𝑥) ↔ (𝑥𝑑𝑦) = (𝑦𝑑𝑥)))
115, 10raleqbidva 3317 . . . . 5 (((𝑝 = 𝑃𝑑 = ) ∧ 𝑥𝑃) → (∀𝑦𝑃 (𝑥 𝑦) = (𝑦 𝑥) ↔ ∀𝑦𝑝 (𝑥𝑑𝑦) = (𝑦𝑑𝑥)))
124, 11raleqbidva 3317 . . . 4 ((𝑝 = 𝑃𝑑 = ) → (∀𝑥𝑃𝑦𝑃 (𝑥 𝑦) = (𝑦 𝑥) ↔ ∀𝑥𝑝𝑦𝑝 (𝑥𝑑𝑦) = (𝑦𝑑𝑥)))
135adantr 481 . . . . . . 7 ((((𝑝 = 𝑃𝑑 = ) ∧ 𝑥𝑃) ∧ 𝑦𝑃) → 𝑃 = 𝑝)
147oveqdr 7365 . . . . . . . . 9 (((((𝑝 = 𝑃𝑑 = ) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑧𝑃) → (𝑥 𝑦) = (𝑥𝑑𝑦))
157oveqdr 7365 . . . . . . . . 9 (((((𝑝 = 𝑃𝑑 = ) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑧𝑃) → (𝑧 𝑧) = (𝑧𝑑𝑧))
1614, 15eqeq12d 2752 . . . . . . . 8 (((((𝑝 = 𝑃𝑑 = ) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑧𝑃) → ((𝑥 𝑦) = (𝑧 𝑧) ↔ (𝑥𝑑𝑦) = (𝑧𝑑𝑧)))
1716imbi1d 341 . . . . . . 7 (((((𝑝 = 𝑃𝑑 = ) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑧𝑃) → (((𝑥 𝑦) = (𝑧 𝑧) → 𝑥 = 𝑦) ↔ ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦)))
1813, 17raleqbidva 3317 . . . . . 6 ((((𝑝 = 𝑃𝑑 = ) ∧ 𝑥𝑃) ∧ 𝑦𝑃) → (∀𝑧𝑃 ((𝑥 𝑦) = (𝑧 𝑧) → 𝑥 = 𝑦) ↔ ∀𝑧𝑝 ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦)))
195, 18raleqbidva 3317 . . . . 5 (((𝑝 = 𝑃𝑑 = ) ∧ 𝑥𝑃) → (∀𝑦𝑃𝑧𝑃 ((𝑥 𝑦) = (𝑧 𝑧) → 𝑥 = 𝑦) ↔ ∀𝑦𝑝𝑧𝑝 ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦)))
204, 19raleqbidva 3317 . . . 4 ((𝑝 = 𝑃𝑑 = ) → (∀𝑥𝑃𝑦𝑃𝑧𝑃 ((𝑥 𝑦) = (𝑧 𝑧) → 𝑥 = 𝑦) ↔ ∀𝑥𝑝𝑦𝑝𝑧𝑝 ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦)))
2112, 20anbi12d 631 . . 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 16959 . 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 27098 . 2 TarskiGC = {𝑓[(Base‘𝑓) / 𝑝][(dist‘𝑓) / 𝑑](∀𝑥𝑝𝑦𝑝 (𝑥𝑑𝑦) = (𝑦𝑑𝑥) ∧ ∀𝑥𝑝𝑦𝑝𝑧𝑝 ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦))}
2422, 23elab4g 3624 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 205  wa 396   = wceq 1540  wcel 2105  wral 3061  Vcvv 3441  [wsbc 3727  cfv 6479  (class class class)co 7337  Basecbs 17009  distcds 17068  TarskiGCcstrkgc 27078  Itvcitv 27083
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707  ax-nul 5250
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2941  df-ral 3062  df-rab 3404  df-v 3443  df-sbc 3728  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-br 5093  df-iota 6431  df-fv 6487  df-ov 7340  df-trkgc 27098
This theorem is referenced by:  axtgcgrrflx  27112  axtgcgrid  27113  f1otrg  27521  xmstrkgc  27542  eengtrkg  27643
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