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Definition df-trkgc 27963
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 2754, so it is not listed in this definition. (Contributed by Thierry Arnoux, 24-Aug-2017.)
Assertion
Ref Expression
df-trkgc TarskiGC = {𝑓 ∣ [(Baseβ€˜π‘“) / 𝑝][(distβ€˜π‘“) / 𝑑](βˆ€π‘₯ ∈ 𝑝 βˆ€π‘¦ ∈ 𝑝 (π‘₯𝑑𝑦) = (𝑦𝑑π‘₯) ∧ βˆ€π‘₯ ∈ 𝑝 βˆ€π‘¦ ∈ 𝑝 βˆ€π‘§ ∈ 𝑝 ((π‘₯𝑑𝑦) = (𝑧𝑑𝑧) β†’ π‘₯ = 𝑦))}
Distinct variable group:   𝑓,𝑑,𝑝,π‘₯,𝑦,𝑧

Detailed syntax breakdown of Definition df-trkgc
StepHypRef Expression
1 cstrkgc 27943 . 2 class TarskiGC
2 vx . . . . . . . . . . 11 setvar π‘₯
32cv 1539 . . . . . . . . . 10 class π‘₯
4 vy . . . . . . . . . . 11 setvar 𝑦
54cv 1539 . . . . . . . . . 10 class 𝑦
6 vd . . . . . . . . . . 11 setvar 𝑑
76cv 1539 . . . . . . . . . 10 class 𝑑
83, 5, 7co 7412 . . . . . . . . 9 class (π‘₯𝑑𝑦)
95, 3, 7co 7412 . . . . . . . . 9 class (𝑦𝑑π‘₯)
108, 9wceq 1540 . . . . . . . 8 wff (π‘₯𝑑𝑦) = (𝑦𝑑π‘₯)
11 vp . . . . . . . . 9 setvar 𝑝
1211cv 1539 . . . . . . . 8 class 𝑝
1310, 4, 12wral 3060 . . . . . . 7 wff βˆ€π‘¦ ∈ 𝑝 (π‘₯𝑑𝑦) = (𝑦𝑑π‘₯)
1413, 2, 12wral 3060 . . . . . 6 wff βˆ€π‘₯ ∈ 𝑝 βˆ€π‘¦ ∈ 𝑝 (π‘₯𝑑𝑦) = (𝑦𝑑π‘₯)
15 vz . . . . . . . . . . . . 13 setvar 𝑧
1615cv 1539 . . . . . . . . . . . 12 class 𝑧
1716, 16, 7co 7412 . . . . . . . . . . 11 class (𝑧𝑑𝑧)
188, 17wceq 1540 . . . . . . . . . 10 wff (π‘₯𝑑𝑦) = (𝑧𝑑𝑧)
192, 4weq 1965 . . . . . . . . . 10 wff π‘₯ = 𝑦
2018, 19wi 4 . . . . . . . . 9 wff ((π‘₯𝑑𝑦) = (𝑧𝑑𝑧) β†’ π‘₯ = 𝑦)
2120, 15, 12wral 3060 . . . . . . . 8 wff βˆ€π‘§ ∈ 𝑝 ((π‘₯𝑑𝑦) = (𝑧𝑑𝑧) β†’ π‘₯ = 𝑦)
2221, 4, 12wral 3060 . . . . . . 7 wff βˆ€π‘¦ ∈ 𝑝 βˆ€π‘§ ∈ 𝑝 ((π‘₯𝑑𝑦) = (𝑧𝑑𝑧) β†’ π‘₯ = 𝑦)
2322, 2, 12wral 3060 . . . . . 6 wff βˆ€π‘₯ ∈ 𝑝 βˆ€π‘¦ ∈ 𝑝 βˆ€π‘§ ∈ 𝑝 ((π‘₯𝑑𝑦) = (𝑧𝑑𝑧) β†’ π‘₯ = 𝑦)
2414, 23wa 395 . . . . 5 wff (βˆ€π‘₯ ∈ 𝑝 βˆ€π‘¦ ∈ 𝑝 (π‘₯𝑑𝑦) = (𝑦𝑑π‘₯) ∧ βˆ€π‘₯ ∈ 𝑝 βˆ€π‘¦ ∈ 𝑝 βˆ€π‘§ ∈ 𝑝 ((π‘₯𝑑𝑦) = (𝑧𝑑𝑧) β†’ π‘₯ = 𝑦))
25 vf . . . . . . 7 setvar 𝑓
2625cv 1539 . . . . . 6 class 𝑓
27 cds 17211 . . . . . 6 class dist
2826, 27cfv 6544 . . . . 5 class (distβ€˜π‘“)
2924, 6, 28wsbc 3778 . . . 4 wff [(distβ€˜π‘“) / 𝑑](βˆ€π‘₯ ∈ 𝑝 βˆ€π‘¦ ∈ 𝑝 (π‘₯𝑑𝑦) = (𝑦𝑑π‘₯) ∧ βˆ€π‘₯ ∈ 𝑝 βˆ€π‘¦ ∈ 𝑝 βˆ€π‘§ ∈ 𝑝 ((π‘₯𝑑𝑦) = (𝑧𝑑𝑧) β†’ π‘₯ = 𝑦))
30 cbs 17149 . . . . 5 class Base
3126, 30cfv 6544 . . . 4 class (Baseβ€˜π‘“)
3229, 11, 31wsbc 3778 . . 3 wff [(Baseβ€˜π‘“) / 𝑝][(distβ€˜π‘“) / 𝑑](βˆ€π‘₯ ∈ 𝑝 βˆ€π‘¦ ∈ 𝑝 (π‘₯𝑑𝑦) = (𝑦𝑑π‘₯) ∧ βˆ€π‘₯ ∈ 𝑝 βˆ€π‘¦ ∈ 𝑝 βˆ€π‘§ ∈ 𝑝 ((π‘₯𝑑𝑦) = (𝑧𝑑𝑧) β†’ π‘₯ = 𝑦))
3332, 25cab 2708 . 2 class {𝑓 ∣ [(Baseβ€˜π‘“) / 𝑝][(distβ€˜π‘“) / 𝑑](βˆ€π‘₯ ∈ 𝑝 βˆ€π‘¦ ∈ 𝑝 (π‘₯𝑑𝑦) = (𝑦𝑑π‘₯) ∧ βˆ€π‘₯ ∈ 𝑝 βˆ€π‘¦ ∈ 𝑝 βˆ€π‘§ ∈ 𝑝 ((π‘₯𝑑𝑦) = (𝑧𝑑𝑧) β†’ π‘₯ = 𝑦))}
341, 33wceq 1540 1 wff TarskiGC = {𝑓 ∣ [(Baseβ€˜π‘“) / 𝑝][(distβ€˜π‘“) / 𝑑](βˆ€π‘₯ ∈ 𝑝 βˆ€π‘¦ ∈ 𝑝 (π‘₯𝑑𝑦) = (𝑦𝑑π‘₯) ∧ βˆ€π‘₯ ∈ 𝑝 βˆ€π‘¦ ∈ 𝑝 βˆ€π‘§ ∈ 𝑝 ((π‘₯𝑑𝑦) = (𝑧𝑑𝑧) β†’ π‘₯ = 𝑦))}
Colors of variables: wff setvar class
This definition is referenced by:  istrkgc  27969
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