MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-cgrg Structured version   Visualization version   GIF version

Definition df-cgrg 28533
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 28830 and expanded in iscgra 28831. This does not mean our system is weaker; dfcgrg2 28885 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 28542, cgr3simp2 28543, cgr3simp3 28544, cgrcgra 28843, and permutation laws such as cgr3swap12 28545 and dfcgrg2 28885.

Ideally, we would define this for functions of any set, but we will use words (see df-word 14549) 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 28532 . 2 class cgrG
2 vg . . 3 setvar 𝑔
3 cvv 3477 . . 3 class V
4 va . . . . . . . 8 setvar 𝑎
54cv 1535 . . . . . . 7 class 𝑎
62cv 1535 . . . . . . . . 9 class 𝑔
7 cbs 17244 . . . . . . . . 9 class Base
86, 7cfv 6562 . . . . . . . 8 class (Base‘𝑔)
9 cr 11151 . . . . . . . 8 class
10 cpm 8865 . . . . . . . 8 class pm
118, 9, 10co 7430 . . . . . . 7 class ((Base‘𝑔) ↑pm ℝ)
125, 11wcel 2105 . . . . . 6 wff 𝑎 ∈ ((Base‘𝑔) ↑pm ℝ)
13 vb . . . . . . . 8 setvar 𝑏
1413cv 1535 . . . . . . 7 class 𝑏
1514, 11wcel 2105 . . . . . 6 wff 𝑏 ∈ ((Base‘𝑔) ↑pm ℝ)
1612, 15wa 395 . . . . 5 wff (𝑎 ∈ ((Base‘𝑔) ↑pm ℝ) ∧ 𝑏 ∈ ((Base‘𝑔) ↑pm ℝ))
175cdm 5688 . . . . . . 7 class dom 𝑎
1814cdm 5688 . . . . . . 7 class dom 𝑏
1917, 18wceq 1536 . . . . . 6 wff dom 𝑎 = dom 𝑏
20 vi . . . . . . . . . . . 12 setvar 𝑖
2120cv 1535 . . . . . . . . . . 11 class 𝑖
2221, 5cfv 6562 . . . . . . . . . 10 class (𝑎𝑖)
23 vj . . . . . . . . . . . 12 setvar 𝑗
2423cv 1535 . . . . . . . . . . 11 class 𝑗
2524, 5cfv 6562 . . . . . . . . . 10 class (𝑎𝑗)
26 cds 17306 . . . . . . . . . . 11 class dist
276, 26cfv 6562 . . . . . . . . . 10 class (dist‘𝑔)
2822, 25, 27co 7430 . . . . . . . . 9 class ((𝑎𝑖)(dist‘𝑔)(𝑎𝑗))
2921, 14cfv 6562 . . . . . . . . . 10 class (𝑏𝑖)
3024, 14cfv 6562 . . . . . . . . . 10 class (𝑏𝑗)
3129, 30, 27co 7430 . . . . . . . . 9 class ((𝑏𝑖)(dist‘𝑔)(𝑏𝑗))
3228, 31wceq 1536 . . . . . . . 8 wff ((𝑎𝑖)(dist‘𝑔)(𝑎𝑗)) = ((𝑏𝑖)(dist‘𝑔)(𝑏𝑗))
3332, 23, 17wral 3058 . . . . . . 7 wff 𝑗 ∈ dom 𝑎((𝑎𝑖)(dist‘𝑔)(𝑎𝑗)) = ((𝑏𝑖)(dist‘𝑔)(𝑏𝑗))
3433, 20, 17wral 3058 . . . . . 6 wff 𝑖 ∈ dom 𝑎𝑗 ∈ dom 𝑎((𝑎𝑖)(dist‘𝑔)(𝑎𝑗)) = ((𝑏𝑖)(dist‘𝑔)(𝑏𝑗))
3519, 34wa 395 . . . . 5 wff (dom 𝑎 = dom 𝑏 ∧ ∀𝑖 ∈ dom 𝑎𝑗 ∈ dom 𝑎((𝑎𝑖)(dist‘𝑔)(𝑎𝑗)) = ((𝑏𝑖)(dist‘𝑔)(𝑏𝑗)))
3616, 35wa 395 . . . 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 5209 . . 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 5230 . 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 1536 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  28534  ercgrg  28539
  Copyright terms: Public domain W3C validator