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Theorem istrkgc 27396
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 774 . . . . . . . . 9 ((((𝑝 = 𝑃𝑑 = ) ∧ 𝑥𝑃) ∧ 𝑦𝑃) → 𝑑 = )
76eqcomd 2742 . . . . . . . 8 ((((𝑝 = 𝑃𝑑 = ) ∧ 𝑥𝑃) ∧ 𝑦𝑃) → = 𝑑)
87oveqd 7374 . . . . . . 7 ((((𝑝 = 𝑃𝑑 = ) ∧ 𝑥𝑃) ∧ 𝑦𝑃) → (𝑥 𝑦) = (𝑥𝑑𝑦))
97oveqd 7374 . . . . . . 7 ((((𝑝 = 𝑃𝑑 = ) ∧ 𝑥𝑃) ∧ 𝑦𝑃) → (𝑦 𝑥) = (𝑦𝑑𝑥))
108, 9eqeq12d 2752 . . . . . 6 ((((𝑝 = 𝑃𝑑 = ) ∧ 𝑥𝑃) ∧ 𝑦𝑃) → ((𝑥 𝑦) = (𝑦 𝑥) ↔ (𝑥𝑑𝑦) = (𝑦𝑑𝑥)))
115, 10raleqbidva 3321 . . . . 5 (((𝑝 = 𝑃𝑑 = ) ∧ 𝑥𝑃) → (∀𝑦𝑃 (𝑥 𝑦) = (𝑦 𝑥) ↔ ∀𝑦𝑝 (𝑥𝑑𝑦) = (𝑦𝑑𝑥)))
124, 11raleqbidva 3321 . . . 4 ((𝑝 = 𝑃𝑑 = ) → (∀𝑥𝑃𝑦𝑃 (𝑥 𝑦) = (𝑦 𝑥) ↔ ∀𝑥𝑝𝑦𝑝 (𝑥𝑑𝑦) = (𝑦𝑑𝑥)))
135adantr 481 . . . . . . 7 ((((𝑝 = 𝑃𝑑 = ) ∧ 𝑥𝑃) ∧ 𝑦𝑃) → 𝑃 = 𝑝)
147oveqdr 7385 . . . . . . . . 9 (((((𝑝 = 𝑃𝑑 = ) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑧𝑃) → (𝑥 𝑦) = (𝑥𝑑𝑦))
157oveqdr 7385 . . . . . . . . 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 3321 . . . . . 6 ((((𝑝 = 𝑃𝑑 = ) ∧ 𝑥𝑃) ∧ 𝑦𝑃) → (∀𝑧𝑃 ((𝑥 𝑦) = (𝑧 𝑧) → 𝑥 = 𝑦) ↔ ∀𝑧𝑝 ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦)))
195, 18raleqbidva 3321 . . . . 5 (((𝑝 = 𝑃𝑑 = ) ∧ 𝑥𝑃) → (∀𝑦𝑃𝑧𝑃 ((𝑥 𝑦) = (𝑧 𝑧) → 𝑥 = 𝑦) ↔ ∀𝑦𝑝𝑧𝑝 ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦)))
204, 19raleqbidva 3321 . . . 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 17033 . 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 27390 . 2 TarskiGC = {𝑓[(Base‘𝑓) / 𝑝][(dist‘𝑓) / 𝑑](∀𝑥𝑝𝑦𝑝 (𝑥𝑑𝑦) = (𝑦𝑑𝑥) ∧ ∀𝑥𝑝𝑦𝑝𝑧𝑝 ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦))}
2422, 23elab4g 3635 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 1541  wcel 2106  wral 3064  Vcvv 3445  [wsbc 3739  cfv 6496  (class class class)co 7357  Basecbs 17083  distcds 17142  TarskiGCcstrkgc 27370  Itvcitv 27375
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707  ax-nul 5263
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2944  df-ral 3065  df-rab 3408  df-v 3447  df-sbc 3740  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-iota 6448  df-fv 6504  df-ov 7360  df-trkgc 27390
This theorem is referenced by:  axtgcgrrflx  27404  axtgcgrid  27405  f1otrg  27813  xmstrkgc  27834  eengtrkg  27935
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