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Definition df-cgrg 25866
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 26160 and expanded in iscgra 26161. This does not mean our system is weaker; dfcgrg2 26216 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 25875, cgr3simp2 25876, cgr3simp3 25877, cgrcgra 26173, and permutation laws such as cgr3swap12 25878 and dfcgrg2 26216.

Ideally, we would define this for functions of any set, but we will use words (see df-word 13604) 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 25865 . 2 class cgrG
2 vg . . 3 setvar 𝑔
3 cvv 3398 . . 3 class V
4 va . . . . . . . 8 setvar 𝑎
54cv 1600 . . . . . . 7 class 𝑎
62cv 1600 . . . . . . . . 9 class 𝑔
7 cbs 16259 . . . . . . . . 9 class Base
86, 7cfv 6137 . . . . . . . 8 class (Base‘𝑔)
9 cr 10273 . . . . . . . 8 class
10 cpm 8143 . . . . . . . 8 class pm
118, 9, 10co 6924 . . . . . . 7 class ((Base‘𝑔) ↑pm ℝ)
125, 11wcel 2107 . . . . . 6 wff 𝑎 ∈ ((Base‘𝑔) ↑pm ℝ)
13 vb . . . . . . . 8 setvar 𝑏
1413cv 1600 . . . . . . 7 class 𝑏
1514, 11wcel 2107 . . . . . 6 wff 𝑏 ∈ ((Base‘𝑔) ↑pm ℝ)
1612, 15wa 386 . . . . 5 wff (𝑎 ∈ ((Base‘𝑔) ↑pm ℝ) ∧ 𝑏 ∈ ((Base‘𝑔) ↑pm ℝ))
175cdm 5357 . . . . . . 7 class dom 𝑎
1814cdm 5357 . . . . . . 7 class dom 𝑏
1917, 18wceq 1601 . . . . . 6 wff dom 𝑎 = dom 𝑏
20 vi . . . . . . . . . . . 12 setvar 𝑖
2120cv 1600 . . . . . . . . . . 11 class 𝑖
2221, 5cfv 6137 . . . . . . . . . 10 class (𝑎𝑖)
23 vj . . . . . . . . . . . 12 setvar 𝑗
2423cv 1600 . . . . . . . . . . 11 class 𝑗
2524, 5cfv 6137 . . . . . . . . . 10 class (𝑎𝑗)
26 cds 16351 . . . . . . . . . . 11 class dist
276, 26cfv 6137 . . . . . . . . . 10 class (dist‘𝑔)
2822, 25, 27co 6924 . . . . . . . . 9 class ((𝑎𝑖)(dist‘𝑔)(𝑎𝑗))
2921, 14cfv 6137 . . . . . . . . . 10 class (𝑏𝑖)
3024, 14cfv 6137 . . . . . . . . . 10 class (𝑏𝑗)
3129, 30, 27co 6924 . . . . . . . . 9 class ((𝑏𝑖)(dist‘𝑔)(𝑏𝑗))
3228, 31wceq 1601 . . . . . . . 8 wff ((𝑎𝑖)(dist‘𝑔)(𝑎𝑗)) = ((𝑏𝑖)(dist‘𝑔)(𝑏𝑗))
3332, 23, 17wral 3090 . . . . . . 7 wff 𝑗 ∈ dom 𝑎((𝑎𝑖)(dist‘𝑔)(𝑎𝑗)) = ((𝑏𝑖)(dist‘𝑔)(𝑏𝑗))
3433, 20, 17wral 3090 . . . . . 6 wff 𝑖 ∈ dom 𝑎𝑗 ∈ dom 𝑎((𝑎𝑖)(dist‘𝑔)(𝑎𝑗)) = ((𝑏𝑖)(dist‘𝑔)(𝑏𝑗))
3519, 34wa 386 . . . . 5 wff (dom 𝑎 = dom 𝑏 ∧ ∀𝑖 ∈ dom 𝑎𝑗 ∈ dom 𝑎((𝑎𝑖)(dist‘𝑔)(𝑎𝑗)) = ((𝑏𝑖)(dist‘𝑔)(𝑏𝑗)))
3616, 35wa 386 . . . 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 4950 . . 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 4967 . 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 1601 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  25867  ercgrg  25872
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