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Theorem istrkgc 25644
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 474 . . . . . 6 ((𝑝 = 𝑃𝑑 = ) → 𝑝 = 𝑃)
43eqcomd 2771 . . . . 5 ((𝑝 = 𝑃𝑑 = ) → 𝑃 = 𝑝)
54adantr 472 . . . . . 6 (((𝑝 = 𝑃𝑑 = ) ∧ 𝑥𝑃) → 𝑃 = 𝑝)
6 simpllr 793 . . . . . . . . 9 ((((𝑝 = 𝑃𝑑 = ) ∧ 𝑥𝑃) ∧ 𝑦𝑃) → 𝑑 = )
76eqcomd 2771 . . . . . . . 8 ((((𝑝 = 𝑃𝑑 = ) ∧ 𝑥𝑃) ∧ 𝑦𝑃) → = 𝑑)
87oveqd 6859 . . . . . . 7 ((((𝑝 = 𝑃𝑑 = ) ∧ 𝑥𝑃) ∧ 𝑦𝑃) → (𝑥 𝑦) = (𝑥𝑑𝑦))
97oveqd 6859 . . . . . . 7 ((((𝑝 = 𝑃𝑑 = ) ∧ 𝑥𝑃) ∧ 𝑦𝑃) → (𝑦 𝑥) = (𝑦𝑑𝑥))
108, 9eqeq12d 2780 . . . . . 6 ((((𝑝 = 𝑃𝑑 = ) ∧ 𝑥𝑃) ∧ 𝑦𝑃) → ((𝑥 𝑦) = (𝑦 𝑥) ↔ (𝑥𝑑𝑦) = (𝑦𝑑𝑥)))
115, 10raleqbidva 3302 . . . . 5 (((𝑝 = 𝑃𝑑 = ) ∧ 𝑥𝑃) → (∀𝑦𝑃 (𝑥 𝑦) = (𝑦 𝑥) ↔ ∀𝑦𝑝 (𝑥𝑑𝑦) = (𝑦𝑑𝑥)))
124, 11raleqbidva 3302 . . . 4 ((𝑝 = 𝑃𝑑 = ) → (∀𝑥𝑃𝑦𝑃 (𝑥 𝑦) = (𝑦 𝑥) ↔ ∀𝑥𝑝𝑦𝑝 (𝑥𝑑𝑦) = (𝑦𝑑𝑥)))
135adantr 472 . . . . . . 7 ((((𝑝 = 𝑃𝑑 = ) ∧ 𝑥𝑃) ∧ 𝑦𝑃) → 𝑃 = 𝑝)
147oveqdr 6870 . . . . . . . . 9 (((((𝑝 = 𝑃𝑑 = ) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑧𝑃) → (𝑥 𝑦) = (𝑥𝑑𝑦))
157oveqdr 6870 . . . . . . . . 9 (((((𝑝 = 𝑃𝑑 = ) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑧𝑃) → (𝑧 𝑧) = (𝑧𝑑𝑧))
1614, 15eqeq12d 2780 . . . . . . . 8 (((((𝑝 = 𝑃𝑑 = ) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑧𝑃) → ((𝑥 𝑦) = (𝑧 𝑧) ↔ (𝑥𝑑𝑦) = (𝑧𝑑𝑧)))
1716imbi1d 332 . . . . . . 7 (((((𝑝 = 𝑃𝑑 = ) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ 𝑧𝑃) → (((𝑥 𝑦) = (𝑧 𝑧) → 𝑥 = 𝑦) ↔ ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦)))
1813, 17raleqbidva 3302 . . . . . 6 ((((𝑝 = 𝑃𝑑 = ) ∧ 𝑥𝑃) ∧ 𝑦𝑃) → (∀𝑧𝑃 ((𝑥 𝑦) = (𝑧 𝑧) → 𝑥 = 𝑦) ↔ ∀𝑧𝑝 ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦)))
195, 18raleqbidva 3302 . . . . 5 (((𝑝 = 𝑃𝑑 = ) ∧ 𝑥𝑃) → (∀𝑦𝑃𝑧𝑃 ((𝑥 𝑦) = (𝑧 𝑧) → 𝑥 = 𝑦) ↔ ∀𝑦𝑝𝑧𝑝 ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦)))
204, 19raleqbidva 3302 . . . 4 ((𝑝 = 𝑃𝑑 = ) → (∀𝑥𝑃𝑦𝑃𝑧𝑃 ((𝑥 𝑦) = (𝑧 𝑧) → 𝑥 = 𝑦) ↔ ∀𝑥𝑝𝑦𝑝𝑧𝑝 ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦)))
2112, 20anbi12d 624 . . 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 16188 . 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 25638 . 2 TarskiGC = {𝑓[(Base‘𝑓) / 𝑝][(dist‘𝑓) / 𝑑](∀𝑥𝑝𝑦𝑝 (𝑥𝑑𝑦) = (𝑦𝑑𝑥) ∧ ∀𝑥𝑝𝑦𝑝𝑧𝑝 ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦))}
2422, 23elab4g 3510 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 197  wa 384   = wceq 1652  wcel 2155  wral 3055  Vcvv 3350  [wsbc 3596  cfv 6068  (class class class)co 6842  Basecbs 16130  distcds 16223  TarskiGCcstrkgc 25621  Itvcitv 25626
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-nul 4949
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-iota 6031  df-fv 6076  df-ov 6845  df-trkgc 25638
This theorem is referenced by:  axtgcgrrflx  25652  axtgcgrid  25653  f1otrg  26042  xmstrkgc  26057  eengtrkg  26156
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