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Definition df-trkgc 25392
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 2670, 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 25375 . 2 class TarskiGC
2 vx . . . . . . . . . . 11 setvar 𝑥
32cv 1522 . . . . . . . . . 10 class 𝑥
4 vy . . . . . . . . . . 11 setvar 𝑦
54cv 1522 . . . . . . . . . 10 class 𝑦
6 vd . . . . . . . . . . 11 setvar 𝑑
76cv 1522 . . . . . . . . . 10 class 𝑑
83, 5, 7co 6690 . . . . . . . . 9 class (𝑥𝑑𝑦)
95, 3, 7co 6690 . . . . . . . . 9 class (𝑦𝑑𝑥)
108, 9wceq 1523 . . . . . . . 8 wff (𝑥𝑑𝑦) = (𝑦𝑑𝑥)
11 vp . . . . . . . . 9 setvar 𝑝
1211cv 1522 . . . . . . . 8 class 𝑝
1310, 4, 12wral 2941 . . . . . . 7 wff 𝑦𝑝 (𝑥𝑑𝑦) = (𝑦𝑑𝑥)
1413, 2, 12wral 2941 . . . . . 6 wff 𝑥𝑝𝑦𝑝 (𝑥𝑑𝑦) = (𝑦𝑑𝑥)
15 vz . . . . . . . . . . . . 13 setvar 𝑧
1615cv 1522 . . . . . . . . . . . 12 class 𝑧
1716, 16, 7co 6690 . . . . . . . . . . 11 class (𝑧𝑑𝑧)
188, 17wceq 1523 . . . . . . . . . 10 wff (𝑥𝑑𝑦) = (𝑧𝑑𝑧)
192, 4weq 1931 . . . . . . . . . 10 wff 𝑥 = 𝑦
2018, 19wi 4 . . . . . . . . 9 wff ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦)
2120, 15, 12wral 2941 . . . . . . . 8 wff 𝑧𝑝 ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦)
2221, 4, 12wral 2941 . . . . . . 7 wff 𝑦𝑝𝑧𝑝 ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦)
2322, 2, 12wral 2941 . . . . . 6 wff 𝑥𝑝𝑦𝑝𝑧𝑝 ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦)
2414, 23wa 383 . . . . 5 wff (∀𝑥𝑝𝑦𝑝 (𝑥𝑑𝑦) = (𝑦𝑑𝑥) ∧ ∀𝑥𝑝𝑦𝑝𝑧𝑝 ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦))
25 vf . . . . . . 7 setvar 𝑓
2625cv 1522 . . . . . 6 class 𝑓
27 cds 15997 . . . . . 6 class dist
2826, 27cfv 5926 . . . . 5 class (dist‘𝑓)
2924, 6, 28wsbc 3468 . . . 4 wff [(dist‘𝑓) / 𝑑](∀𝑥𝑝𝑦𝑝 (𝑥𝑑𝑦) = (𝑦𝑑𝑥) ∧ ∀𝑥𝑝𝑦𝑝𝑧𝑝 ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦))
30 cbs 15904 . . . . 5 class Base
3126, 30cfv 5926 . . . 4 class (Base‘𝑓)
3229, 11, 31wsbc 3468 . . 3 wff [(Base‘𝑓) / 𝑝][(dist‘𝑓) / 𝑑](∀𝑥𝑝𝑦𝑝 (𝑥𝑑𝑦) = (𝑦𝑑𝑥) ∧ ∀𝑥𝑝𝑦𝑝𝑧𝑝 ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦))
3332, 25cab 2637 . 2 class {𝑓[(Base‘𝑓) / 𝑝][(dist‘𝑓) / 𝑑](∀𝑥𝑝𝑦𝑝 (𝑥𝑑𝑦) = (𝑦𝑑𝑥) ∧ ∀𝑥𝑝𝑦𝑝𝑧𝑝 ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦))}
341, 33wceq 1523 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  25398
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