Definition df-cgrg 28537

Description: Define the relation of congruence between shapes. Definition 4.4 of [Schwabhauser] p. 35. A "shape" is a finite sequence of points, and a triangle can be represented as a shape with three points. Two shapes are congruent if all corresponding segments between all corresponding points are congruent.

Many systems of geometry define triangle congruence as requiring both segment congruence and angle congruence. Such systems, such as Hilbert's axiomatic system, typically have a primitive notion of angle congruence in addition to segment congruence. Here, angle congruence is instead a derived notion, defined later in df-cgra 28834 and expanded in iscgra 28835. This does not mean our system is weaker; dfcgrg2 28889 proves that these two definitions are equivalent, and using the Tarski definition instead (given in [Schwabhauser] p. 35) is simpler. Once two triangles are proven congruent as defined here, you can use various theorems to prove that corresponding parts of congruent triangles are congruent (CPCTC). For example, see cgr3simp1 28546, cgr3simp2 28547, cgr3simp3 28548, cgrcgra 28847, and permutation laws such as cgr3swap12 28549 and dfcgrg2 28889.

Ideally, we would define this for functions of any set, but we will use words (see df-word 14563) in most cases.

(Contributed by Thierry Arnoux, 3-Apr-2019.)

Assertion

Ref	Expression
df-cgrg	⊢ cgrG = (𝑔 ∈ V ↦ {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ((Base‘𝑔) ↑pm ℝ) ∧ 𝑏 ∈ ((Base‘𝑔) ↑pm ℝ)) ∧ (dom 𝑎 = dom 𝑏 ∧ ∀𝑖 ∈ dom 𝑎∀𝑗 ∈ dom 𝑎((𝑎‘𝑖)(dist‘𝑔)(𝑎‘𝑗)) = ((𝑏‘𝑖)(dist‘𝑔)(𝑏‘𝑗))))})

Distinct variable group:   𝑎,𝑏,𝑔,𝑖,𝑗

Detailed syntax breakdown of Definition df-cgrg

Step	Hyp	Ref	Expression
1		ccgrg 28536	. 2 class cgrG
2		vg	. . 3 setvar 𝑔
3		cvv 3488	. . 3 class V
4		va	. . . . . . . 8 setvar 𝑎
5	4	cv 1536	. . . . . . 7 class 𝑎
6	2	cv 1536	. . . . . . . . 9 class 𝑔
7		cbs 17258	. . . . . . . . 9 class Base
8	6, 7	cfv 6573	. . . . . . . 8 class (Base‘𝑔)
9		cr 11183	. . . . . . . 8 class ℝ
10		cpm 8885	. . . . . . . 8 class ↑pm
11	8, 9, 10	co 7448	. . . . . . 7 class ((Base‘𝑔) ↑pm ℝ)
12	5, 11	wcel 2108	. . . . . 6 wff 𝑎 ∈ ((Base‘𝑔) ↑pm ℝ)
13		vb	. . . . . . . 8 setvar 𝑏
14	13	cv 1536	. . . . . . 7 class 𝑏
15	14, 11	wcel 2108	. . . . . 6 wff 𝑏 ∈ ((Base‘𝑔) ↑pm ℝ)
16	12, 15	wa 395	. . . . 5 wff (𝑎 ∈ ((Base‘𝑔) ↑pm ℝ) ∧ 𝑏 ∈ ((Base‘𝑔) ↑pm ℝ))
17	5	cdm 5700	. . . . . . 7 class dom 𝑎
18	14	cdm 5700	. . . . . . 7 class dom 𝑏
19	17, 18	wceq 1537	. . . . . 6 wff dom 𝑎 = dom 𝑏
20		vi	. . . . . . . . . . . 12 setvar 𝑖
21	20	cv 1536	. . . . . . . . . . 11 class 𝑖
22	21, 5	cfv 6573	. . . . . . . . . 10 class (𝑎‘𝑖)
23		vj	. . . . . . . . . . . 12 setvar 𝑗
24	23	cv 1536	. . . . . . . . . . 11 class 𝑗
25	24, 5	cfv 6573	. . . . . . . . . 10 class (𝑎‘𝑗)
26		cds 17320	. . . . . . . . . . 11 class dist
27	6, 26	cfv 6573	. . . . . . . . . 10 class (dist‘𝑔)
28	22, 25, 27	co 7448	. . . . . . . . 9 class ((𝑎‘𝑖)(dist‘𝑔)(𝑎‘𝑗))
29	21, 14	cfv 6573	. . . . . . . . . 10 class (𝑏‘𝑖)
30	24, 14	cfv 6573	. . . . . . . . . 10 class (𝑏‘𝑗)
31	29, 30, 27	co 7448	. . . . . . . . 9 class ((𝑏‘𝑖)(dist‘𝑔)(𝑏‘𝑗))
32	28, 31	wceq 1537	. . . . . . . 8 wff ((𝑎‘𝑖)(dist‘𝑔)(𝑎‘𝑗)) = ((𝑏‘𝑖)(dist‘𝑔)(𝑏‘𝑗))
33	32, 23, 17	wral 3067	. . . . . . 7 wff ∀𝑗 ∈ dom 𝑎((𝑎‘𝑖)(dist‘𝑔)(𝑎‘𝑗)) = ((𝑏‘𝑖)(dist‘𝑔)(𝑏‘𝑗))
34	33, 20, 17	wral 3067	. . . . . 6 wff ∀𝑖 ∈ dom 𝑎∀𝑗 ∈ dom 𝑎((𝑎‘𝑖)(dist‘𝑔)(𝑎‘𝑗)) = ((𝑏‘𝑖)(dist‘𝑔)(𝑏‘𝑗))
35	19, 34	wa 395	. . . . 5 wff (dom 𝑎 = dom 𝑏 ∧ ∀𝑖 ∈ dom 𝑎∀𝑗 ∈ dom 𝑎((𝑎‘𝑖)(dist‘𝑔)(𝑎‘𝑗)) = ((𝑏‘𝑖)(dist‘𝑔)(𝑏‘𝑗)))
36	16, 35	wa 395	. . . 4 wff ((𝑎 ∈ ((Base‘𝑔) ↑pm ℝ) ∧ 𝑏 ∈ ((Base‘𝑔) ↑pm ℝ)) ∧ (dom 𝑎 = dom 𝑏 ∧ ∀𝑖 ∈ dom 𝑎∀𝑗 ∈ dom 𝑎((𝑎‘𝑖)(dist‘𝑔)(𝑎‘𝑗)) = ((𝑏‘𝑖)(dist‘𝑔)(𝑏‘𝑗))))
37	36, 4, 13	copab 5228	. . 3 class {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ((Base‘𝑔) ↑pm ℝ) ∧ 𝑏 ∈ ((Base‘𝑔) ↑pm ℝ)) ∧ (dom 𝑎 = dom 𝑏 ∧ ∀𝑖 ∈ dom 𝑎∀𝑗 ∈ dom 𝑎((𝑎‘𝑖)(dist‘𝑔)(𝑎‘𝑗)) = ((𝑏‘𝑖)(dist‘𝑔)(𝑏‘𝑗))))}
38	2, 3, 37	cmpt 5249	. 2 class (𝑔 ∈ V ↦ {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ((Base‘𝑔) ↑pm ℝ) ∧ 𝑏 ∈ ((Base‘𝑔) ↑pm ℝ)) ∧ (dom 𝑎 = dom 𝑏 ∧ ∀𝑖 ∈ dom 𝑎∀𝑗 ∈ dom 𝑎((𝑎‘𝑖)(dist‘𝑔)(𝑎‘𝑗)) = ((𝑏‘𝑖)(dist‘𝑔)(𝑏‘𝑗))))})
39	1, 38	wceq 1537	1 wff cgrG = (𝑔 ∈ V ↦ {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ((Base‘𝑔) ↑pm ℝ) ∧ 𝑏 ∈ ((Base‘𝑔) ↑pm ℝ)) ∧ (dom 𝑎 = dom 𝑏 ∧ ∀𝑖 ∈ dom 𝑎∀𝑗 ∈ dom 𝑎((𝑎‘𝑖)(dist‘𝑔)(𝑎‘𝑗)) = ((𝑏‘𝑖)(dist‘𝑔)(𝑏‘𝑗))))})

This definition is referenced by:  iscgrg  28538  ercgrg  28543