Definition df-cgrg 26776

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 27073 and expanded in iscgra 27074. This does not mean our system is weaker; dfcgrg2 27128 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 26785, cgr3simp2 26786, cgr3simp3 26787, cgrcgra 27086, and permutation laws such as cgr3swap12 26788 and dfcgrg2 27128.

Ideally, we would define this for functions of any set, but we will use words (see df-word 14146) 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 26775	. 2 class cgrG
2		vg	. . 3 setvar 𝑔
3		cvv 3422	. . 3 class V
4		va	. . . . . . . 8 setvar 𝑎
5	4	cv 1538	. . . . . . 7 class 𝑎
6	2	cv 1538	. . . . . . . . 9 class 𝑔
7		cbs 16840	. . . . . . . . 9 class Base
8	6, 7	cfv 6418	. . . . . . . 8 class (Base‘𝑔)
9		cr 10801	. . . . . . . 8 class ℝ
10		cpm 8574	. . . . . . . 8 class ↑pm
11	8, 9, 10	co 7255	. . . . . . 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 1538	. . . . . . 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 5580	. . . . . . 7 class dom 𝑎
18	14	cdm 5580	. . . . . . 7 class dom 𝑏
19	17, 18	wceq 1539	. . . . . 6 wff dom 𝑎 = dom 𝑏
20		vi	. . . . . . . . . . . 12 setvar 𝑖
21	20	cv 1538	. . . . . . . . . . 11 class 𝑖
22	21, 5	cfv 6418	. . . . . . . . . 10 class (𝑎‘𝑖)
23		vj	. . . . . . . . . . . 12 setvar 𝑗
24	23	cv 1538	. . . . . . . . . . 11 class 𝑗
25	24, 5	cfv 6418	. . . . . . . . . 10 class (𝑎‘𝑗)
26		cds 16897	. . . . . . . . . . 11 class dist
27	6, 26	cfv 6418	. . . . . . . . . 10 class (dist‘𝑔)
28	22, 25, 27	co 7255	. . . . . . . . 9 class ((𝑎‘𝑖)(dist‘𝑔)(𝑎‘𝑗))
29	21, 14	cfv 6418	. . . . . . . . . 10 class (𝑏‘𝑖)
30	24, 14	cfv 6418	. . . . . . . . . 10 class (𝑏‘𝑗)
31	29, 30, 27	co 7255	. . . . . . . . 9 class ((𝑏‘𝑖)(dist‘𝑔)(𝑏‘𝑗))
32	28, 31	wceq 1539	. . . . . . . 8 wff ((𝑎‘𝑖)(dist‘𝑔)(𝑎‘𝑗)) = ((𝑏‘𝑖)(dist‘𝑔)(𝑏‘𝑗))
33	32, 23, 17	wral 3063	. . . . . . 7 wff ∀𝑗 ∈ dom 𝑎((𝑎‘𝑖)(dist‘𝑔)(𝑎‘𝑗)) = ((𝑏‘𝑖)(dist‘𝑔)(𝑏‘𝑗))
34	33, 20, 17	wral 3063	. . . . . 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 5132	. . 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 5153	. 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 1539	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  26777  ercgrg  26782