Definition df-trkgc 28516

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 2756, 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](∀𝑥 ∈ 𝑝 ∀𝑦 ∈ 𝑝 (𝑥𝑑𝑦) = (𝑦𝑑𝑥) ∧ ∀𝑥 ∈ 𝑝 ∀𝑦 ∈ 𝑝 ∀𝑧 ∈ 𝑝 ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦))}

Distinct variable group:   𝑓,𝑑,𝑝,𝑥,𝑦,𝑧

Detailed syntax breakdown of Definition df-trkgc

Step	Hyp	Ref	Expression
1		cstrkgc 28496	. 2 class TarskiGC
2		vx	. . . . . . . . . . 11 setvar 𝑥
3	2	cv 1541	. . . . . . . . . 10 class 𝑥
4		vy	. . . . . . . . . . 11 setvar 𝑦
5	4	cv 1541	. . . . . . . . . 10 class 𝑦
6		vd	. . . . . . . . . . 11 setvar 𝑑
7	6	cv 1541	. . . . . . . . . 10 class 𝑑
8	3, 5, 7	co 7367	. . . . . . . . 9 class (𝑥𝑑𝑦)
9	5, 3, 7	co 7367	. . . . . . . . 9 class (𝑦𝑑𝑥)
10	8, 9	wceq 1542	. . . . . . . 8 wff (𝑥𝑑𝑦) = (𝑦𝑑𝑥)
11		vp	. . . . . . . . 9 setvar 𝑝
12	11	cv 1541	. . . . . . . 8 class 𝑝
13	10, 4, 12	wral 3051	. . . . . . 7 wff ∀𝑦 ∈ 𝑝 (𝑥𝑑𝑦) = (𝑦𝑑𝑥)
14	13, 2, 12	wral 3051	. . . . . 6 wff ∀𝑥 ∈ 𝑝 ∀𝑦 ∈ 𝑝 (𝑥𝑑𝑦) = (𝑦𝑑𝑥)
15		vz	. . . . . . . . . . . . 13 setvar 𝑧
16	15	cv 1541	. . . . . . . . . . . 12 class 𝑧
17	16, 16, 7	co 7367	. . . . . . . . . . 11 class (𝑧𝑑𝑧)
18	8, 17	wceq 1542	. . . . . . . . . 10 wff (𝑥𝑑𝑦) = (𝑧𝑑𝑧)
19	2, 4	weq 1964	. . . . . . . . . 10 wff 𝑥 = 𝑦
20	18, 19	wi 4	. . . . . . . . 9 wff ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦)
21	20, 15, 12	wral 3051	. . . . . . . 8 wff ∀𝑧 ∈ 𝑝 ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦)
22	21, 4, 12	wral 3051	. . . . . . 7 wff ∀𝑦 ∈ 𝑝 ∀𝑧 ∈ 𝑝 ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦)
23	22, 2, 12	wral 3051	. . . . . 6 wff ∀𝑥 ∈ 𝑝 ∀𝑦 ∈ 𝑝 ∀𝑧 ∈ 𝑝 ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦)
24	14, 23	wa 395	. . . . 5 wff (∀𝑥 ∈ 𝑝 ∀𝑦 ∈ 𝑝 (𝑥𝑑𝑦) = (𝑦𝑑𝑥) ∧ ∀𝑥 ∈ 𝑝 ∀𝑦 ∈ 𝑝 ∀𝑧 ∈ 𝑝 ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦))
25		vf	. . . . . . 7 setvar 𝑓
26	25	cv 1541	. . . . . 6 class 𝑓
27		cds 17229	. . . . . 6 class dist
28	26, 27	cfv 6498	. . . . 5 class (dist‘𝑓)
29	24, 6, 28	wsbc 3728	. . . 4 wff [(dist‘𝑓) / 𝑑](∀𝑥 ∈ 𝑝 ∀𝑦 ∈ 𝑝 (𝑥𝑑𝑦) = (𝑦𝑑𝑥) ∧ ∀𝑥 ∈ 𝑝 ∀𝑦 ∈ 𝑝 ∀𝑧 ∈ 𝑝 ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦))
30		cbs 17179	. . . . 5 class Base
31	26, 30	cfv 6498	. . . 4 class (Base‘𝑓)
32	29, 11, 31	wsbc 3728	. . 3 wff [(Base‘𝑓) / 𝑝][(dist‘𝑓) / 𝑑](∀𝑥 ∈ 𝑝 ∀𝑦 ∈ 𝑝 (𝑥𝑑𝑦) = (𝑦𝑑𝑥) ∧ ∀𝑥 ∈ 𝑝 ∀𝑦 ∈ 𝑝 ∀𝑧 ∈ 𝑝 ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦))
33	32, 25	cab 2714	. 2 class {𝑓 ∣ [(Base‘𝑓) / 𝑝][(dist‘𝑓) / 𝑑](∀𝑥 ∈ 𝑝 ∀𝑦 ∈ 𝑝 (𝑥𝑑𝑦) = (𝑦𝑑𝑥) ∧ ∀𝑥 ∈ 𝑝 ∀𝑦 ∈ 𝑝 ∀𝑧 ∈ 𝑝 ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦))}
34	1, 33	wceq 1542	1 wff TarskiGC = {𝑓 ∣ [(Base‘𝑓) / 𝑝][(dist‘𝑓) / 𝑑](∀𝑥 ∈ 𝑝 ∀𝑦 ∈ 𝑝 (𝑥𝑑𝑦) = (𝑦𝑑𝑥) ∧ ∀𝑥 ∈ 𝑝 ∀𝑦 ∈ 𝑝 ∀𝑧 ∈ 𝑝 ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦))}

This definition is referenced by:  istrkgc  28522