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Definition df-cgrg 28387
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 28684 and expanded in iscgra 28685. This does not mean our system is weaker; dfcgrg2 28739 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 28396, cgr3simp2 28397, cgr3simp3 28398, cgrcgra 28697, and permutation laws such as cgr3swap12 28399 and dfcgrg2 28739.

Ideally, we would define this for functions of any set, but we will use words (see df-word 14501) 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 𝑎𝑗 ∈ dom 𝑎((𝑎𝑖)(dist‘𝑔)(𝑎𝑗)) = ((𝑏𝑖)(dist‘𝑔)(𝑏𝑗))))})
Distinct variable group:   𝑎,𝑏,𝑔,𝑖,𝑗

Detailed syntax breakdown of Definition df-cgrg
StepHypRef Expression
1 ccgrg 28386 . 2 class cgrG
2 vg . . 3 setvar 𝑔
3 cvv 3461 . . 3 class V
4 va . . . . . . . 8 setvar 𝑎
54cv 1532 . . . . . . 7 class 𝑎
62cv 1532 . . . . . . . . 9 class 𝑔
7 cbs 17183 . . . . . . . . 9 class Base
86, 7cfv 6549 . . . . . . . 8 class (Base‘𝑔)
9 cr 11139 . . . . . . . 8 class
10 cpm 8846 . . . . . . . 8 class pm
118, 9, 10co 7419 . . . . . . 7 class ((Base‘𝑔) ↑pm ℝ)
125, 11wcel 2098 . . . . . 6 wff 𝑎 ∈ ((Base‘𝑔) ↑pm ℝ)
13 vb . . . . . . . 8 setvar 𝑏
1413cv 1532 . . . . . . 7 class 𝑏
1514, 11wcel 2098 . . . . . 6 wff 𝑏 ∈ ((Base‘𝑔) ↑pm ℝ)
1612, 15wa 394 . . . . 5 wff (𝑎 ∈ ((Base‘𝑔) ↑pm ℝ) ∧ 𝑏 ∈ ((Base‘𝑔) ↑pm ℝ))
175cdm 5678 . . . . . . 7 class dom 𝑎
1814cdm 5678 . . . . . . 7 class dom 𝑏
1917, 18wceq 1533 . . . . . 6 wff dom 𝑎 = dom 𝑏
20 vi . . . . . . . . . . . 12 setvar 𝑖
2120cv 1532 . . . . . . . . . . 11 class 𝑖
2221, 5cfv 6549 . . . . . . . . . 10 class (𝑎𝑖)
23 vj . . . . . . . . . . . 12 setvar 𝑗
2423cv 1532 . . . . . . . . . . 11 class 𝑗
2524, 5cfv 6549 . . . . . . . . . 10 class (𝑎𝑗)
26 cds 17245 . . . . . . . . . . 11 class dist
276, 26cfv 6549 . . . . . . . . . 10 class (dist‘𝑔)
2822, 25, 27co 7419 . . . . . . . . 9 class ((𝑎𝑖)(dist‘𝑔)(𝑎𝑗))
2921, 14cfv 6549 . . . . . . . . . 10 class (𝑏𝑖)
3024, 14cfv 6549 . . . . . . . . . 10 class (𝑏𝑗)
3129, 30, 27co 7419 . . . . . . . . 9 class ((𝑏𝑖)(dist‘𝑔)(𝑏𝑗))
3228, 31wceq 1533 . . . . . . . 8 wff ((𝑎𝑖)(dist‘𝑔)(𝑎𝑗)) = ((𝑏𝑖)(dist‘𝑔)(𝑏𝑗))
3332, 23, 17wral 3050 . . . . . . 7 wff 𝑗 ∈ dom 𝑎((𝑎𝑖)(dist‘𝑔)(𝑎𝑗)) = ((𝑏𝑖)(dist‘𝑔)(𝑏𝑗))
3433, 20, 17wral 3050 . . . . . 6 wff 𝑖 ∈ dom 𝑎𝑗 ∈ dom 𝑎((𝑎𝑖)(dist‘𝑔)(𝑎𝑗)) = ((𝑏𝑖)(dist‘𝑔)(𝑏𝑗))
3519, 34wa 394 . . . . 5 wff (dom 𝑎 = dom 𝑏 ∧ ∀𝑖 ∈ dom 𝑎𝑗 ∈ dom 𝑎((𝑎𝑖)(dist‘𝑔)(𝑎𝑗)) = ((𝑏𝑖)(dist‘𝑔)(𝑏𝑗)))
3616, 35wa 394 . . . 4 wff ((𝑎 ∈ ((Base‘𝑔) ↑pm ℝ) ∧ 𝑏 ∈ ((Base‘𝑔) ↑pm ℝ)) ∧ (dom 𝑎 = dom 𝑏 ∧ ∀𝑖 ∈ dom 𝑎𝑗 ∈ dom 𝑎((𝑎𝑖)(dist‘𝑔)(𝑎𝑗)) = ((𝑏𝑖)(dist‘𝑔)(𝑏𝑗))))
3736, 4, 13copab 5211 . . 3 class {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ((Base‘𝑔) ↑pm ℝ) ∧ 𝑏 ∈ ((Base‘𝑔) ↑pm ℝ)) ∧ (dom 𝑎 = dom 𝑏 ∧ ∀𝑖 ∈ dom 𝑎𝑗 ∈ dom 𝑎((𝑎𝑖)(dist‘𝑔)(𝑎𝑗)) = ((𝑏𝑖)(dist‘𝑔)(𝑏𝑗))))}
382, 3, 37cmpt 5232 . 2 class (𝑔 ∈ V ↦ {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ((Base‘𝑔) ↑pm ℝ) ∧ 𝑏 ∈ ((Base‘𝑔) ↑pm ℝ)) ∧ (dom 𝑎 = dom 𝑏 ∧ ∀𝑖 ∈ dom 𝑎𝑗 ∈ dom 𝑎((𝑎𝑖)(dist‘𝑔)(𝑎𝑗)) = ((𝑏𝑖)(dist‘𝑔)(𝑏𝑗))))})
391, 38wceq 1533 1 wff cgrG = (𝑔 ∈ V ↦ {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ((Base‘𝑔) ↑pm ℝ) ∧ 𝑏 ∈ ((Base‘𝑔) ↑pm ℝ)) ∧ (dom 𝑎 = dom 𝑏 ∧ ∀𝑖 ∈ dom 𝑎𝑗 ∈ dom 𝑎((𝑎𝑖)(dist‘𝑔)(𝑎𝑗)) = ((𝑏𝑖)(dist‘𝑔)(𝑏𝑗))))})
Colors of variables: wff setvar class
This definition is referenced by:  iscgrg  28388  ercgrg  28393
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