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Definition df-trkgc 26162
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 2841, 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 26145 . 2 class TarskiGC
2 vx . . . . . . . . . . 11 setvar 𝑥
32cv 1527 . . . . . . . . . 10 class 𝑥
4 vy . . . . . . . . . . 11 setvar 𝑦
54cv 1527 . . . . . . . . . 10 class 𝑦
6 vd . . . . . . . . . . 11 setvar 𝑑
76cv 1527 . . . . . . . . . 10 class 𝑑
83, 5, 7co 7145 . . . . . . . . 9 class (𝑥𝑑𝑦)
95, 3, 7co 7145 . . . . . . . . 9 class (𝑦𝑑𝑥)
108, 9wceq 1528 . . . . . . . 8 wff (𝑥𝑑𝑦) = (𝑦𝑑𝑥)
11 vp . . . . . . . . 9 setvar 𝑝
1211cv 1527 . . . . . . . 8 class 𝑝
1310, 4, 12wral 3138 . . . . . . 7 wff 𝑦𝑝 (𝑥𝑑𝑦) = (𝑦𝑑𝑥)
1413, 2, 12wral 3138 . . . . . 6 wff 𝑥𝑝𝑦𝑝 (𝑥𝑑𝑦) = (𝑦𝑑𝑥)
15 vz . . . . . . . . . . . . 13 setvar 𝑧
1615cv 1527 . . . . . . . . . . . 12 class 𝑧
1716, 16, 7co 7145 . . . . . . . . . . 11 class (𝑧𝑑𝑧)
188, 17wceq 1528 . . . . . . . . . 10 wff (𝑥𝑑𝑦) = (𝑧𝑑𝑧)
192, 4weq 1955 . . . . . . . . . 10 wff 𝑥 = 𝑦
2018, 19wi 4 . . . . . . . . 9 wff ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦)
2120, 15, 12wral 3138 . . . . . . . 8 wff 𝑧𝑝 ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦)
2221, 4, 12wral 3138 . . . . . . 7 wff 𝑦𝑝𝑧𝑝 ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦)
2322, 2, 12wral 3138 . . . . . 6 wff 𝑥𝑝𝑦𝑝𝑧𝑝 ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦)
2414, 23wa 396 . . . . 5 wff (∀𝑥𝑝𝑦𝑝 (𝑥𝑑𝑦) = (𝑦𝑑𝑥) ∧ ∀𝑥𝑝𝑦𝑝𝑧𝑝 ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦))
25 vf . . . . . . 7 setvar 𝑓
2625cv 1527 . . . . . 6 class 𝑓
27 cds 16564 . . . . . 6 class dist
2826, 27cfv 6349 . . . . 5 class (dist‘𝑓)
2924, 6, 28wsbc 3771 . . . 4 wff [(dist‘𝑓) / 𝑑](∀𝑥𝑝𝑦𝑝 (𝑥𝑑𝑦) = (𝑦𝑑𝑥) ∧ ∀𝑥𝑝𝑦𝑝𝑧𝑝 ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦))
30 cbs 16473 . . . . 5 class Base
3126, 30cfv 6349 . . . 4 class (Base‘𝑓)
3229, 11, 31wsbc 3771 . . 3 wff [(Base‘𝑓) / 𝑝][(dist‘𝑓) / 𝑑](∀𝑥𝑝𝑦𝑝 (𝑥𝑑𝑦) = (𝑦𝑑𝑥) ∧ ∀𝑥𝑝𝑦𝑝𝑧𝑝 ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦))
3332, 25cab 2799 . 2 class {𝑓[(Base‘𝑓) / 𝑝][(dist‘𝑓) / 𝑑](∀𝑥𝑝𝑦𝑝 (𝑥𝑑𝑦) = (𝑦𝑑𝑥) ∧ ∀𝑥𝑝𝑦𝑝𝑧𝑝 ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦))}
341, 33wceq 1528 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  26168
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