Definition df-cgrg 28667

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 28964 and expanded in iscgra 28965. This does not mean our system is weaker; dfcgrg2 29019 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 28676, cgr3simp2 28677, cgr3simp3 28678, cgrcgra 28977, and permutation laws such as cgr3swap12 28679 and dfcgrg2 29019.

Ideally, we would define this for functions of any set, but we will use words (see df-word 14520) 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 28666	. 2 class cgrG
2		vg	. . 3 setvar 𝑔
3		cvv 3453	. . 3 class V
4		va	. . . . . . . 8 setvar 𝑎
5	4	cv 1558	. . . . . . 7 class 𝑎
6	2	cv 1558	. . . . . . . . 9 class 𝑔
7		cbs 17235	. . . . . . . . 9 class Base
8	6, 7	cfv 6515	. . . . . . . 8 class (Base‘𝑔)
9		cr 11065	. . . . . . . 8 class ℝ
10		cpm 8802	. . . . . . . 8 class ↑pm
11	8, 9, 10	co 7390	. . . . . . 7 class ((Base‘𝑔) ↑pm ℝ)
12	5, 11	wcel 2141	. . . . . 6 wff 𝑎 ∈ ((Base‘𝑔) ↑pm ℝ)
13		vb	. . . . . . . 8 setvar 𝑏
14	13	cv 1558	. . . . . . 7 class 𝑏
15	14, 11	wcel 2141	. . . . . 6 wff 𝑏 ∈ ((Base‘𝑔) ↑pm ℝ)
16	12, 15	wa 399	. . . . 5 wff (𝑎 ∈ ((Base‘𝑔) ↑pm ℝ) ∧ 𝑏 ∈ ((Base‘𝑔) ↑pm ℝ))
17	5	cdm 5643	. . . . . . 7 class dom 𝑎
18	14	cdm 5643	. . . . . . 7 class dom 𝑏
19	17, 18	wceq 1559	. . . . . 6 wff dom 𝑎 = dom 𝑏
20		vi	. . . . . . . . . . . 12 setvar 𝑖
21	20	cv 1558	. . . . . . . . . . 11 class 𝑖
22	21, 5	cfv 6515	. . . . . . . . . 10 class (𝑎‘𝑖)
23		vj	. . . . . . . . . . . 12 setvar 𝑗
24	23	cv 1558	. . . . . . . . . . 11 class 𝑗
25	24, 5	cfv 6515	. . . . . . . . . 10 class (𝑎‘𝑗)
26		cds 17285	. . . . . . . . . . 11 class dist
27	6, 26	cfv 6515	. . . . . . . . . 10 class (dist‘𝑔)
28	22, 25, 27	co 7390	. . . . . . . . 9 class ((𝑎‘𝑖)(dist‘𝑔)(𝑎‘𝑗))
29	21, 14	cfv 6515	. . . . . . . . . 10 class (𝑏‘𝑖)
30	24, 14	cfv 6515	. . . . . . . . . 10 class (𝑏‘𝑗)
31	29, 30, 27	co 7390	. . . . . . . . 9 class ((𝑏‘𝑖)(dist‘𝑔)(𝑏‘𝑗))
32	28, 31	wceq 1559	. . . . . . . 8 wff ((𝑎‘𝑖)(dist‘𝑔)(𝑎‘𝑗)) = ((𝑏‘𝑖)(dist‘𝑔)(𝑏‘𝑗))
33	32, 23, 17	wral 3075	. . . . . . 7 wff ∀𝑗 ∈ dom 𝑎((𝑎‘𝑖)(dist‘𝑔)(𝑎‘𝑗)) = ((𝑏‘𝑖)(dist‘𝑔)(𝑏‘𝑗))
34	33, 20, 17	wral 3075	. . . . . 6 wff ∀𝑖 ∈ dom 𝑎∀𝑗 ∈ dom 𝑎((𝑎‘𝑖)(dist‘𝑔)(𝑎‘𝑗)) = ((𝑏‘𝑖)(dist‘𝑔)(𝑏‘𝑗))
35	19, 34	wa 399	. . . . 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 399	. . . 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 5159	. . 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 5178	. 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 1559	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  28668  ercgrg  28673