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Definition df-trkgc 25064
Description: Define the class of geometries fulfilling the congruence axioms of reflexivity, identity and transitivity. These are axioms A1 to A3 of [Schwabhauser] p. 10. With our distance based notation for congruence, transitivity of congruence boils down to transitivity of equality and is already given by eqtr 2628, so it is not listed in this definition. (Contributed by Thierry Arnoux, 24-Aug-2017.)
Assertion
Ref Expression
df-trkgc TarskiGC = {𝑓[(Base‘𝑓) / 𝑝][(dist‘𝑓) / 𝑑](∀𝑥𝑝𝑦𝑝 (𝑥𝑑𝑦) = (𝑦𝑑𝑥) ∧ ∀𝑥𝑝𝑦𝑝𝑧𝑝 ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦))}
Distinct variable group:   𝑓,𝑑,𝑝,𝑥,𝑦,𝑧

Detailed syntax breakdown of Definition df-trkgc
StepHypRef Expression
1 cstrkgc 25047 . 2 class TarskiGC
2 vx . . . . . . . . . . 11 setvar 𝑥
32cv 1473 . . . . . . . . . 10 class 𝑥
4 vy . . . . . . . . . . 11 setvar 𝑦
54cv 1473 . . . . . . . . . 10 class 𝑦
6 vd . . . . . . . . . . 11 setvar 𝑑
76cv 1473 . . . . . . . . . 10 class 𝑑
83, 5, 7co 6527 . . . . . . . . 9 class (𝑥𝑑𝑦)
95, 3, 7co 6527 . . . . . . . . 9 class (𝑦𝑑𝑥)
108, 9wceq 1474 . . . . . . . 8 wff (𝑥𝑑𝑦) = (𝑦𝑑𝑥)
11 vp . . . . . . . . 9 setvar 𝑝
1211cv 1473 . . . . . . . 8 class 𝑝
1310, 4, 12wral 2895 . . . . . . 7 wff 𝑦𝑝 (𝑥𝑑𝑦) = (𝑦𝑑𝑥)
1413, 2, 12wral 2895 . . . . . 6 wff 𝑥𝑝𝑦𝑝 (𝑥𝑑𝑦) = (𝑦𝑑𝑥)
15 vz . . . . . . . . . . . . 13 setvar 𝑧
1615cv 1473 . . . . . . . . . . . 12 class 𝑧
1716, 16, 7co 6527 . . . . . . . . . . 11 class (𝑧𝑑𝑧)
188, 17wceq 1474 . . . . . . . . . 10 wff (𝑥𝑑𝑦) = (𝑧𝑑𝑧)
192, 4weq 1860 . . . . . . . . . 10 wff 𝑥 = 𝑦
2018, 19wi 4 . . . . . . . . 9 wff ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦)
2120, 15, 12wral 2895 . . . . . . . 8 wff 𝑧𝑝 ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦)
2221, 4, 12wral 2895 . . . . . . 7 wff 𝑦𝑝𝑧𝑝 ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦)
2322, 2, 12wral 2895 . . . . . 6 wff 𝑥𝑝𝑦𝑝𝑧𝑝 ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦)
2414, 23wa 382 . . . . 5 wff (∀𝑥𝑝𝑦𝑝 (𝑥𝑑𝑦) = (𝑦𝑑𝑥) ∧ ∀𝑥𝑝𝑦𝑝𝑧𝑝 ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦))
25 vf . . . . . . 7 setvar 𝑓
2625cv 1473 . . . . . 6 class 𝑓
27 cds 15723 . . . . . 6 class dist
2826, 27cfv 5790 . . . . 5 class (dist‘𝑓)
2924, 6, 28wsbc 3401 . . . 4 wff [(dist‘𝑓) / 𝑑](∀𝑥𝑝𝑦𝑝 (𝑥𝑑𝑦) = (𝑦𝑑𝑥) ∧ ∀𝑥𝑝𝑦𝑝𝑧𝑝 ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦))
30 cbs 15641 . . . . 5 class Base
3126, 30cfv 5790 . . . 4 class (Base‘𝑓)
3229, 11, 31wsbc 3401 . . 3 wff [(Base‘𝑓) / 𝑝][(dist‘𝑓) / 𝑑](∀𝑥𝑝𝑦𝑝 (𝑥𝑑𝑦) = (𝑦𝑑𝑥) ∧ ∀𝑥𝑝𝑦𝑝𝑧𝑝 ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦))
3332, 25cab 2595 . 2 class {𝑓[(Base‘𝑓) / 𝑝][(dist‘𝑓) / 𝑑](∀𝑥𝑝𝑦𝑝 (𝑥𝑑𝑦) = (𝑦𝑑𝑥) ∧ ∀𝑥𝑝𝑦𝑝𝑧𝑝 ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦))}
341, 33wceq 1474 1 wff TarskiGC = {𝑓[(Base‘𝑓) / 𝑝][(dist‘𝑓) / 𝑑](∀𝑥𝑝𝑦𝑝 (𝑥𝑑𝑦) = (𝑦𝑑𝑥) ∧ ∀𝑥𝑝𝑦𝑝𝑧𝑝 ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦))}
Colors of variables: wff setvar class
This definition is referenced by:  istrkgc  25070
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