Definition df-cgrg 27751

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 28048 and expanded in iscgra 28049. This does not mean our system is weaker; dfcgrg2 28103 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 27760, cgr3simp2 27761, cgr3simp3 27762, cgrcgra 28061, and permutation laws such as cgr3swap12 27763 and dfcgrg2 28103.

Ideally, we would define this for functions of any set, but we will use words (see df-word 14461) 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 27750	. 2 class cgrG
2		vg	. . 3 setvar 𝑔
3		cvv 3474	. . 3 class V
4		va	. . . . . . . 8 setvar 𝑎
5	4	cv 1540	. . . . . . 7 class 𝑎
6	2	cv 1540	. . . . . . . . 9 class 𝑔
7		cbs 17140	. . . . . . . . 9 class Base
8	6, 7	cfv 6540	. . . . . . . 8 class (Base‘𝑔)
9		cr 11105	. . . . . . . 8 class ℝ
10		cpm 8817	. . . . . . . 8 class ↑pm
11	8, 9, 10	co 7405	. . . . . . 7 class ((Base‘𝑔) ↑pm ℝ)
12	5, 11	wcel 2106	. . . . . 6 wff 𝑎 ∈ ((Base‘𝑔) ↑pm ℝ)
13		vb	. . . . . . . 8 setvar 𝑏
14	13	cv 1540	. . . . . . 7 class 𝑏
15	14, 11	wcel 2106	. . . . . 6 wff 𝑏 ∈ ((Base‘𝑔) ↑pm ℝ)
16	12, 15	wa 396	. . . . 5 wff (𝑎 ∈ ((Base‘𝑔) ↑pm ℝ) ∧ 𝑏 ∈ ((Base‘𝑔) ↑pm ℝ))
17	5	cdm 5675	. . . . . . 7 class dom 𝑎
18	14	cdm 5675	. . . . . . 7 class dom 𝑏
19	17, 18	wceq 1541	. . . . . 6 wff dom 𝑎 = dom 𝑏
20		vi	. . . . . . . . . . . 12 setvar 𝑖
21	20	cv 1540	. . . . . . . . . . 11 class 𝑖
22	21, 5	cfv 6540	. . . . . . . . . 10 class (𝑎‘𝑖)
23		vj	. . . . . . . . . . . 12 setvar 𝑗
24	23	cv 1540	. . . . . . . . . . 11 class 𝑗
25	24, 5	cfv 6540	. . . . . . . . . 10 class (𝑎‘𝑗)
26		cds 17202	. . . . . . . . . . 11 class dist
27	6, 26	cfv 6540	. . . . . . . . . 10 class (dist‘𝑔)
28	22, 25, 27	co 7405	. . . . . . . . 9 class ((𝑎‘𝑖)(dist‘𝑔)(𝑎‘𝑗))
29	21, 14	cfv 6540	. . . . . . . . . 10 class (𝑏‘𝑖)
30	24, 14	cfv 6540	. . . . . . . . . 10 class (𝑏‘𝑗)
31	29, 30, 27	co 7405	. . . . . . . . 9 class ((𝑏‘𝑖)(dist‘𝑔)(𝑏‘𝑗))
32	28, 31	wceq 1541	. . . . . . . 8 wff ((𝑎‘𝑖)(dist‘𝑔)(𝑎‘𝑗)) = ((𝑏‘𝑖)(dist‘𝑔)(𝑏‘𝑗))
33	32, 23, 17	wral 3061	. . . . . . 7 wff ∀𝑗 ∈ dom 𝑎((𝑎‘𝑖)(dist‘𝑔)(𝑎‘𝑗)) = ((𝑏‘𝑖)(dist‘𝑔)(𝑏‘𝑗))
34	33, 20, 17	wral 3061	. . . . . 6 wff ∀𝑖 ∈ dom 𝑎∀𝑗 ∈ dom 𝑎((𝑎‘𝑖)(dist‘𝑔)(𝑎‘𝑗)) = ((𝑏‘𝑖)(dist‘𝑔)(𝑏‘𝑗))
35	19, 34	wa 396	. . . . 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 396	. . . 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 5209	. . 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 5230	. 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 1541	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  27752  ercgrg  27757