Definition df-cgrg 28761

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 29100 and expanded in iscgra 29101. This does not mean our system is weaker; dfcgrg2 29161 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 28770, cgr3simp2 28771, cgr3simp3 28772, cgrcgra 29113, and permutation laws such as cgr3swap12 28773 and dfcgrg2 29161.

Ideally, we would define this for functions of any set, but we will use words (see df-word 14553) 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 28760	. 2 class cgrG
2		vg	. . 3 setvar 𝑔
3		cvv 3455	. . 3 class V
4		va	. . . . . . . 8 setvar 𝑎
5	4	cv 1569	. . . . . . 7 class 𝑎
6	2	cv 1569	. . . . . . . . 9 class 𝑔
7		cbs 17270	. . . . . . . . 9 class Base
8	6, 7	cfv 6538	. . . . . . . 8 class (Base‘𝑔)
9		cr 11100	. . . . . . . 8 class ℝ
10		cpm 8826	. . . . . . . 8 class ↑pm
11	8, 9, 10	co 7412	. . . . . . 7 class ((Base‘𝑔) ↑pm ℝ)
12	5, 11	wcel 2143	. . . . . 6 wff 𝑎 ∈ ((Base‘𝑔) ↑pm ℝ)
13		vb	. . . . . . . 8 setvar 𝑏
14	13	cv 1569	. . . . . . 7 class 𝑏
15	14, 11	wcel 2143	. . . . . 6 wff 𝑏 ∈ ((Base‘𝑔) ↑pm ℝ)
16	12, 15	wa 400	. . . . 5 wff (𝑎 ∈ ((Base‘𝑔) ↑pm ℝ) ∧ 𝑏 ∈ ((Base‘𝑔) ↑pm ℝ))
17	5	cdm 5663	. . . . . . 7 class dom 𝑎
18	14	cdm 5663	. . . . . . 7 class dom 𝑏
19	17, 18	wceq 1570	. . . . . 6 wff dom 𝑎 = dom 𝑏
20		vi	. . . . . . . . . . . 12 setvar 𝑖
21	20	cv 1569	. . . . . . . . . . 11 class 𝑖
22	21, 5	cfv 6538	. . . . . . . . . 10 class (𝑎‘𝑖)
23		vj	. . . . . . . . . . . 12 setvar 𝑗
24	23	cv 1569	. . . . . . . . . . 11 class 𝑗
25	24, 5	cfv 6538	. . . . . . . . . 10 class (𝑎‘𝑗)
26		cds 17320	. . . . . . . . . . 11 class dist
27	6, 26	cfv 6538	. . . . . . . . . 10 class (dist‘𝑔)
28	22, 25, 27	co 7412	. . . . . . . . 9 class ((𝑎‘𝑖)(dist‘𝑔)(𝑎‘𝑗))
29	21, 14	cfv 6538	. . . . . . . . . 10 class (𝑏‘𝑖)
30	24, 14	cfv 6538	. . . . . . . . . 10 class (𝑏‘𝑗)
31	29, 30, 27	co 7412	. . . . . . . . 9 class ((𝑏‘𝑖)(dist‘𝑔)(𝑏‘𝑗))
32	28, 31	wceq 1570	. . . . . . . 8 wff ((𝑎‘𝑖)(dist‘𝑔)(𝑎‘𝑗)) = ((𝑏‘𝑖)(dist‘𝑔)(𝑏‘𝑗))
33	32, 23, 17	wral 3079	. . . . . . 7 wff ∀𝑗 ∈ dom 𝑎((𝑎‘𝑖)(dist‘𝑔)(𝑎‘𝑗)) = ((𝑏‘𝑖)(dist‘𝑔)(𝑏‘𝑗))
34	33, 20, 17	wral 3079	. . . . . 6 wff ∀𝑖 ∈ dom 𝑎∀𝑗 ∈ dom 𝑎((𝑎‘𝑖)(dist‘𝑔)(𝑎‘𝑗)) = ((𝑏‘𝑖)(dist‘𝑔)(𝑏‘𝑗))
35	19, 34	wa 400	. . . . 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 400	. . . 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 5174	. . 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 5193	. 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 1570	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  28762  ercgrg  28767