HomeHome Metamath Proof Explorer
Theorem List (p. 253 of 424)
< Previous  Next >
Bad symbols? Try the
GIF version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-27159)
  Hilbert Space Explorer  Hilbert Space Explorer
(27160-28684)
  Users' Mathboxes  Users' Mathboxes
(28685-42360)
 

Theorem List for Metamath Proof Explorer - 25201-25300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremtgbtwndiff 25201* There is always a 𝑐 distinct from 𝐵 such that 𝐵 lies between 𝐴 and 𝑐. Theorem 3.14 of [Schwabhauser] p. 32. The condition "the space is of dimension 1 or more" is written here as 2 ≤ (#‘𝑃) for simplicity. (Contributed by Thierry Arnoux, 23-Mar-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑 → 2 ≤ (#‘𝑃))       (𝜑 → ∃𝑐𝑃 (𝐵 ∈ (𝐴𝐼𝑐) ∧ 𝐵𝑐))
 
Theoremtgdim01 25202 In geometries of dimension lower than 2, all points are colinear. (Contributed by Thierry Arnoux, 27-Aug-2019.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺𝑉)    &   (𝜑 → ¬ 𝐺DimTarskiG≥2)    &   (𝜑𝑋𝑃)    &   (𝜑𝑌𝑃)    &   (𝜑𝑍𝑃)       (𝜑 → (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)))
 
15.2.4  Betweenness and Congruence
 
Theoremtgifscgr 25203 Inner five segment congruence. Take two triangles, 𝐴𝐷𝐶 and 𝐸𝐻𝐾, with 𝐵 between 𝐴 and 𝐶 and 𝐹 between 𝐸 and 𝐾. If the other components of the triangles are congruent, then so are 𝐵𝐷 and 𝐹𝐻. Theorem 4.2 of [Schwabhauser] p. 34. (Contributed by Thierry Arnoux, 24-Mar-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑𝐾𝑃)    &   (𝜑𝐻𝑃)    &   (𝜑𝐵 ∈ (𝐴𝐼𝐶))    &   (𝜑𝐹 ∈ (𝐸𝐼𝐾))    &   (𝜑 → (𝐴 𝐶) = (𝐸 𝐾))    &   (𝜑 → (𝐵 𝐶) = (𝐹 𝐾))    &   (𝜑 → (𝐴 𝐷) = (𝐸 𝐻))    &   (𝜑 → (𝐶 𝐷) = (𝐾 𝐻))       (𝜑 → (𝐵 𝐷) = (𝐹 𝐻))
 
Theoremtgcgrsub 25204 Removing identical parts from the end of a line segment preserves congruence. Theorem 4.3 of [Schwabhauser] p. 35. (Contributed by Thierry Arnoux, 3-Apr-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑𝐵 ∈ (𝐴𝐼𝐶))    &   (𝜑𝐸 ∈ (𝐷𝐼𝐹))    &   (𝜑 → (𝐴 𝐶) = (𝐷 𝐹))    &   (𝜑 → (𝐵 𝐶) = (𝐸 𝐹))       (𝜑 → (𝐴 𝐵) = (𝐷 𝐸))
 
15.2.5  Congruence of a series of points
 
Syntaxccgrg 25205 Declare the constant for the congruence between shapes relation.
class cgrG
 
Definitiondf-cgrg 25206* Define the relation congruence bewteen shapes. Definition 4.4 of [Schwabhauser] p. 35. Ideally, we would define this for functions of any set, but we will used words (functions over ) in most cases. (Contributed by Thierry Arnoux, 3-Apr-2019.)
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))))})
 
Theoremiscgrg 25207* The congruence property for sequences of points. (Contributed by Thierry Arnoux, 3-Apr-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &    = (cgrG‘𝐺)       (𝐺𝑉 → (𝐴 𝐵 ↔ ((𝐴 ∈ (𝑃pm ℝ) ∧ 𝐵 ∈ (𝑃pm ℝ)) ∧ (dom 𝐴 = dom 𝐵 ∧ ∀𝑖 ∈ dom 𝐴𝑗 ∈ dom 𝐴((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗))))))
 
Theoremiscgrgd 25208* The property for two sequences 𝐴 and 𝐵 of points to be congruent. (Contributed by Thierry Arnoux, 3-Apr-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &    = (cgrG‘𝐺)    &   (𝜑𝐺𝑉)    &   (𝜑𝐷 ⊆ ℝ)    &   (𝜑𝐴:𝐷𝑃)    &   (𝜑𝐵:𝐷𝑃)       (𝜑 → (𝐴 𝐵 ↔ ∀𝑖 ∈ dom 𝐴𝑗 ∈ dom 𝐴((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗))))
 
Theoremiscgrglt 25209* The property for two sequences 𝐴 and 𝐵 of points to be congruent, where the congruence is only required for indices verifying a less-than relation. (Contributed by Thierry Arnoux, 7-Oct-2020.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &    = (cgrG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐷 ⊆ ℝ)    &   (𝜑𝐴:𝐷𝑃)    &   (𝜑𝐵:𝐷𝑃)       (𝜑 → (𝐴 𝐵 ↔ ∀𝑖 ∈ dom 𝐴𝑗 ∈ dom 𝐴(𝑖 < 𝑗 → ((𝐴𝑖) (𝐴𝑗)) = ((𝐵𝑖) (𝐵𝑗)))))
 
Theoremtrgcgrg 25210 The property for two triangles to be congruent to each other. (Contributed by Thierry Arnoux, 3-Apr-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &    = (cgrG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)       (𝜑 → (⟨“𝐴𝐵𝐶”⟩ ⟨“𝐷𝐸𝐹”⟩ ↔ ((𝐴 𝐵) = (𝐷 𝐸) ∧ (𝐵 𝐶) = (𝐸 𝐹) ∧ (𝐶 𝐴) = (𝐹 𝐷))))
 
Theoremtrgcgr 25211 Triangle congruence. (Contributed by Thierry Arnoux, 27-Apr-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &    = (cgrG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑 → (𝐴 𝐵) = (𝐷 𝐸))    &   (𝜑 → (𝐵 𝐶) = (𝐸 𝐹))    &   (𝜑 → (𝐶 𝐴) = (𝐹 𝐷))       (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ⟨“𝐷𝐸𝐹”⟩)
 
Theoremercgrg 25212 The shape congruence relation is an equivalence relation. Statement 4.4 of [Schwabhauser] p. 35. (Contributed by Thierry Arnoux, 9-Apr-2019.)
𝑃 = (Base‘𝐺)       (𝐺 ∈ TarskiG → (cgrG‘𝐺) Er (𝑃pm ℝ))
 
Theoremtgcgrxfr 25213* A line segment can be divided at the same place as a congruent line segment is divided. Theorem 4.5 of [Schwabhauser] p. 35. (Contributed by Thierry Arnoux, 9-Apr-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &    = (cgrG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑𝐵 ∈ (𝐴𝐼𝐶))    &   (𝜑 → (𝐴 𝐶) = (𝐷 𝐹))       (𝜑 → ∃𝑒𝑃 (𝑒 ∈ (𝐷𝐼𝐹) ∧ ⟨“𝐴𝐵𝐶”⟩ ⟨“𝐷𝑒𝐹”⟩))
 
Theoremcgr3id 25214 Reflexivity law for three-place congruence. (Contributed by Thierry Arnoux, 28-Apr-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &    = (cgrG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)       (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ⟨“𝐴𝐵𝐶”⟩)
 
Theoremcgr3simp1 25215 Deduce segment congruence from a triangle congruence. (Contributed by Thierry Arnoux, 27-Apr-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &    = (cgrG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ⟨“𝐷𝐸𝐹”⟩)       (𝜑 → (𝐴 𝐵) = (𝐷 𝐸))
 
Theoremcgr3simp2 25216 Deduce segment congruence from a triangle congruence. (Contributed by Thierry Arnoux, 27-Apr-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &    = (cgrG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ⟨“𝐷𝐸𝐹”⟩)       (𝜑 → (𝐵 𝐶) = (𝐸 𝐹))
 
Theoremcgr3simp3 25217 Deduce segment congruence from a triangle congruence. (Contributed by Thierry Arnoux, 27-Apr-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &    = (cgrG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ⟨“𝐷𝐸𝐹”⟩)       (𝜑 → (𝐶 𝐴) = (𝐹 𝐷))
 
Theoremcgr3swap12 25218 Permutation law for three-place congruence. (Contributed by Thierry Arnoux, 27-Apr-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &    = (cgrG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ⟨“𝐷𝐸𝐹”⟩)       (𝜑 → ⟨“𝐵𝐴𝐶”⟩ ⟨“𝐸𝐷𝐹”⟩)
 
Theoremcgr3swap23 25219 Permutation law for three-place congruence. (Contributed by Thierry Arnoux, 27-Apr-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &    = (cgrG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ⟨“𝐷𝐸𝐹”⟩)       (𝜑 → ⟨“𝐴𝐶𝐵”⟩ ⟨“𝐷𝐹𝐸”⟩)
 
Theoremcgr3swap13 25220 Permutation law for three-place congruence. (Contributed by Thierry Arnoux, 3-Oct-2020.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &    = (cgrG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ⟨“𝐷𝐸𝐹”⟩)       (𝜑 → ⟨“𝐶𝐵𝐴”⟩ ⟨“𝐹𝐸𝐷”⟩)
 
Theoremcgr3rotr 25221 Permutation law for three-place congruence. (Contributed by Thierry Arnoux, 1-Aug-2020.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &    = (cgrG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ⟨“𝐷𝐸𝐹”⟩)       (𝜑 → ⟨“𝐶𝐴𝐵”⟩ ⟨“𝐹𝐷𝐸”⟩)
 
Theoremcgr3rotl 25222 Permutation law for three-place congruence. (Contributed by Thierry Arnoux, 1-Aug-2020.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &    = (cgrG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ⟨“𝐷𝐸𝐹”⟩)       (𝜑 → ⟨“𝐵𝐶𝐴”⟩ ⟨“𝐸𝐹𝐷”⟩)
 
Theoremtrgcgrcom 25223 Commutative law for three-place congruence. (Contributed by Thierry Arnoux, 27-Apr-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &    = (cgrG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ⟨“𝐷𝐸𝐹”⟩)       (𝜑 → ⟨“𝐷𝐸𝐹”⟩ ⟨“𝐴𝐵𝐶”⟩)
 
Theoremcgr3tr 25224 Transitivity law for three-place congruence. (Contributed by Thierry Arnoux, 27-Apr-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &    = (cgrG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ⟨“𝐷𝐸𝐹”⟩)    &   (𝜑𝐽𝑃)    &   (𝜑𝐾𝑃)    &   (𝜑𝐿𝑃)    &   (𝜑 → ⟨“𝐷𝐸𝐹”⟩ ⟨“𝐽𝐾𝐿”⟩)       (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ⟨“𝐽𝐾𝐿”⟩)
 
Theoremtgbtwnxfr 25225 A condition for extending betweenness to a new set of points based on congruence with another set of points. Theorem 4.6 of [Schwabhauser] p. 36. (Contributed by Thierry Arnoux, 27-Apr-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &    = (cgrG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ⟨“𝐷𝐸𝐹”⟩)    &   (𝜑𝐵 ∈ (𝐴𝐼𝐶))       (𝜑𝐸 ∈ (𝐷𝐼𝐹))
 
Theoremtgcgr4 25226 Two quadrilaterals to be congruent to each other if one triangle formed by their vertices is, and the additional points are equidistant too. (Contributed by Thierry Arnoux, 8-Oct-2020.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &    = (cgrG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝑊𝑃)    &   (𝜑𝑋𝑃)    &   (𝜑𝑌𝑃)    &   (𝜑𝑍𝑃)       (𝜑 → (⟨“𝐴𝐵𝐶𝐷”⟩ ⟨“𝑊𝑋𝑌𝑍”⟩ ↔ (⟨“𝐴𝐵𝐶”⟩ ⟨“𝑊𝑋𝑌”⟩ ∧ ((𝐴 𝐷) = (𝑊 𝑍) ∧ (𝐵 𝐷) = (𝑋 𝑍) ∧ (𝐶 𝐷) = (𝑌 𝑍)))))
 
15.2.6  Motions
 
Syntaxcismt 25227 Declare the constant for the isometry builder.
class Ismt
 
Definitiondf-ismt 25228* Define the set of isometries between two structures. Definition 4.8 of [Schwabhauser] p. 36. See isismt 25229. (Contributed by Thierry Arnoux, 13-Dec-2019.)
Ismt = (𝑔 ∈ V, ∈ V ↦ {𝑓 ∣ (𝑓:(Base‘𝑔)–1-1-onto→(Base‘) ∧ ∀𝑎 ∈ (Base‘𝑔)∀𝑏 ∈ (Base‘𝑔)((𝑓𝑎)(dist‘)(𝑓𝑏)) = (𝑎(dist‘𝑔)𝑏))})
 
Theoremisismt 25229* Property of being an isometry. Compare with isismty 32770. (Contributed by Thierry Arnoux, 13-Dec-2019.)
𝐵 = (Base‘𝐺)    &   𝑃 = (Base‘𝐻)    &   𝐷 = (dist‘𝐺)    &    = (dist‘𝐻)       ((𝐺𝑉𝐻𝑊) → (𝐹 ∈ (𝐺Ismt𝐻) ↔ (𝐹:𝐵1-1-onto𝑃 ∧ ∀𝑎𝐵𝑏𝐵 ((𝐹𝑎) (𝐹𝑏)) = (𝑎𝐷𝑏))))
 
Theoremismot 25230* Property of being an isometry mapping to the same space. In geometry, this is also called a motion. (Contributed by Thierry Arnoux, 15-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)       (𝐺𝑉 → (𝐹 ∈ (𝐺Ismt𝐺) ↔ (𝐹:𝑃1-1-onto𝑃 ∧ ∀𝑎𝑃𝑏𝑃 ((𝐹𝑎) (𝐹𝑏)) = (𝑎 𝑏))))
 
Theoremmotcgr 25231 Property of a motion: distances are preserved. (Contributed by Thierry Arnoux, 15-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   (𝜑𝐺𝑉)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐹 ∈ (𝐺Ismt𝐺))       (𝜑 → ((𝐹𝐴) (𝐹𝐵)) = (𝐴 𝐵))
 
Theoremidmot 25232 The identity is a motion. (Contributed by Thierry Arnoux, 15-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   (𝜑𝐺𝑉)       (𝜑 → ( I ↾ 𝑃) ∈ (𝐺Ismt𝐺))
 
Theoremmotf1o 25233 Motions are bijections. (Contributed by Thierry Arnoux, 15-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   (𝜑𝐺𝑉)    &   (𝜑𝐹 ∈ (𝐺Ismt𝐺))       (𝜑𝐹:𝑃1-1-onto𝑃)
 
Theoremmotcl 25234 Closure of motions. (Contributed by Thierry Arnoux, 15-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   (𝜑𝐺𝑉)    &   (𝜑𝐹 ∈ (𝐺Ismt𝐺))    &   (𝜑𝐴𝑃)       (𝜑 → (𝐹𝐴) ∈ 𝑃)
 
Theoremmotco 25235 The composition of two motions is a motion. (Contributed by Thierry Arnoux, 15-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   (𝜑𝐺𝑉)    &   (𝜑𝐹 ∈ (𝐺Ismt𝐺))    &   (𝜑𝐻 ∈ (𝐺Ismt𝐺))       (𝜑 → (𝐹𝐻) ∈ (𝐺Ismt𝐺))
 
Theoremcnvmot 25236 The converse of a motion is a motion. (Contributed by Thierry Arnoux, 15-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   (𝜑𝐺𝑉)    &   (𝜑𝐹 ∈ (𝐺Ismt𝐺))       (𝜑𝐹 ∈ (𝐺Ismt𝐺))
 
Theoremmotplusg 25237* The operation for motions is their composition. (Contributed by Thierry Arnoux, 15-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   (𝜑𝐺𝑉)    &   𝐼 = {⟨(Base‘ndx), (𝐺Ismt𝐺)⟩, ⟨(+g‘ndx), (𝑓 ∈ (𝐺Ismt𝐺), 𝑔 ∈ (𝐺Ismt𝐺) ↦ (𝑓𝑔))⟩}    &   (𝜑𝐹 ∈ (𝐺Ismt𝐺))    &   (𝜑𝐻 ∈ (𝐺Ismt𝐺))       (𝜑 → (𝐹(+g𝐼)𝐻) = (𝐹𝐻))
 
Theoremmotgrp 25238* The motions of a geometry form a group with respect to function composition, called the Isometry group. (Contributed by Thierry Arnoux, 15-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   (𝜑𝐺𝑉)    &   𝐼 = {⟨(Base‘ndx), (𝐺Ismt𝐺)⟩, ⟨(+g‘ndx), (𝑓 ∈ (𝐺Ismt𝐺), 𝑔 ∈ (𝐺Ismt𝐺) ↦ (𝑓𝑔))⟩}       (𝜑𝐼 ∈ Grp)
 
Theoremmotcgrg 25239* Property of a motion: distances are preserved. (Contributed by Thierry Arnoux, 15-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   (𝜑𝐺𝑉)    &   𝐼 = {⟨(Base‘ndx), (𝐺Ismt𝐺)⟩, ⟨(+g‘ndx), (𝑓 ∈ (𝐺Ismt𝐺), 𝑔 ∈ (𝐺Ismt𝐺) ↦ (𝑓𝑔))⟩}    &    = (cgrG‘𝐺)    &   (𝜑𝑇 ∈ Word 𝑃)    &   (𝜑𝐹 ∈ (𝐺Ismt𝐺))       (𝜑 → (𝐹𝑇) 𝑇)
 
Theoremmotcgr3 25240 Property of a motion: distances are preserved, special case of triangles. (Contributed by Thierry Arnoux, 15-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &    = (cgrG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷 = (𝐻𝐴))    &   (𝜑𝐸 = (𝐻𝐵))    &   (𝜑𝐹 = (𝐻𝐶))    &   (𝜑𝐻 ∈ (𝐺Ismt𝐺))       (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ⟨“𝐷𝐸𝐹”⟩)
 
15.2.7  Colinearity
 
Theoremtglng 25241* Lines of a Tarski Geometry. This relates to both Definition 4.10 of [Schwabhauser] p. 36. and Definition 6.14 of [Schwabhauser] p. 45. (Contributed by Thierry Arnoux, 28-Mar-2019.)
𝑃 = (Base‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝐼 = (Itv‘𝐺)       (𝐺 ∈ TarskiG → 𝐿 = (𝑥𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}))
 
Theoremtglnfn 25242 Lines as functions. (Contributed by Thierry Arnoux, 25-May-2019.)
𝑃 = (Base‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝐼 = (Itv‘𝐺)       (𝐺 ∈ TarskiG → 𝐿 Fn ((𝑃 × 𝑃) ∖ I ))
 
Theoremtglnunirn 25243 Lines are sets of points. (Contributed by Thierry Arnoux, 25-May-2019.)
𝑃 = (Base‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝐼 = (Itv‘𝐺)       (𝐺 ∈ TarskiG → ran 𝐿𝑃)
 
Theoremtglnpt 25244 Lines are sets of points. (Contributed by Thierry Arnoux, 17-Oct-2019.)
𝑃 = (Base‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴 ∈ ran 𝐿)    &   (𝜑𝑋𝐴)       (𝜑𝑋𝑃)
 
Theoremtglngne 25245 It takes two different points to form a line. (Contributed by Thierry Arnoux, 6-Aug-2019.)
𝑃 = (Base‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝑋𝑃)    &   (𝜑𝑌𝑃)    &   (𝜑𝑍 ∈ (𝑋𝐿𝑌))       (𝜑𝑋𝑌)
 
Theoremtglngval 25246* The line going through points 𝑋 and 𝑌. (Contributed by Thierry Arnoux, 28-Mar-2019.)
𝑃 = (Base‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝑋𝑃)    &   (𝜑𝑌𝑃)    &   (𝜑𝑋𝑌)       (𝜑 → (𝑋𝐿𝑌) = {𝑧𝑃 ∣ (𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧))})
 
Theoremtglnssp 25247 Lines are subset of the geometry base set. That is, lines are sets of points. (Contributed by Thierry Arnoux, 17-May-2019.)
𝑃 = (Base‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝑋𝑃)    &   (𝜑𝑌𝑃)    &   (𝜑𝑋𝑌)       (𝜑 → (𝑋𝐿𝑌) ⊆ 𝑃)
 
Theoremtgellng 25248 Property of lying on the line going through points 𝑋 and 𝑌. Definition 4.10 of [Schwabhauser] p. 36. We choose the notation 𝑍 ∈ (𝑋(LineG‘𝐺)𝑌) instead of "colinear" because LineG is a common structure slot for other axiomatizations of geometry. (Contributed by Thierry Arnoux, 28-Mar-2019.)
𝑃 = (Base‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝑋𝑃)    &   (𝜑𝑌𝑃)    &   (𝜑𝑋𝑌)    &   (𝜑𝑍𝑃)       (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))))
 
Theoremtgcolg 25249 We choose the notation (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌) instead of "colinear" in order to avoid defining an additional symbol for colinearity because LineG is a common structure slot for other axiomatizations of geometry. (Contributed by Thierry Arnoux, 25-May-2019.)
𝑃 = (Base‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝑋𝑃)    &   (𝜑𝑌𝑃)    &   (𝜑𝑍𝑃)       (𝜑 → ((𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))))
 
Theorembtwncolg1 25250 Betweenness implies colinearity. (Contributed by Thierry Arnoux, 28-Mar-2019.)
𝑃 = (Base‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝑋𝑃)    &   (𝜑𝑌𝑃)    &   (𝜑𝑍𝑃)    &   (𝜑𝑍 ∈ (𝑋𝐼𝑌))       (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌))
 
Theorembtwncolg2 25251 Betweenness implies colinearity. (Contributed by Thierry Arnoux, 28-Mar-2019.)
𝑃 = (Base‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝑋𝑃)    &   (𝜑𝑌𝑃)    &   (𝜑𝑍𝑃)    &   (𝜑𝑋 ∈ (𝑍𝐼𝑌))       (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌))
 
Theorembtwncolg3 25252 Betweenness implies colinearity. (Contributed by Thierry Arnoux, 28-Mar-2019.)
𝑃 = (Base‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝑋𝑃)    &   (𝜑𝑌𝑃)    &   (𝜑𝑍𝑃)    &   (𝜑𝑌 ∈ (𝑋𝐼𝑍))       (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌))
 
Theoremcolcom 25253 Swapping the points defining a line keeps it unchanged. Part of Theorem 4.11 of [Schwabhauser] p. 34. (Contributed by Thierry Arnoux, 3-Apr-2019.)
𝑃 = (Base‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝑋𝑃)    &   (𝜑𝑌𝑃)    &   (𝜑𝑍𝑃)    &   (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌))       (𝜑 → (𝑍 ∈ (𝑌𝐿𝑋) ∨ 𝑌 = 𝑋))
 
Theoremcolrot1 25254 Rotating the points defining a line. Part of Theorem 4.11 of [Schwabhauser] p. 34. (Contributed by Thierry Arnoux, 3-Apr-2019.)
𝑃 = (Base‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝑋𝑃)    &   (𝜑𝑌𝑃)    &   (𝜑𝑍𝑃)    &   (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌))       (𝜑 → (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))
 
Theoremcolrot2 25255 Rotating the points defining a line. Part of Theorem 4.11 of [Schwabhauser] p. 34. (Contributed by Thierry Arnoux, 3-Apr-2019.)
𝑃 = (Base‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝑋𝑃)    &   (𝜑𝑌𝑃)    &   (𝜑𝑍𝑃)    &   (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌))       (𝜑 → (𝑌 ∈ (𝑍𝐿𝑋) ∨ 𝑍 = 𝑋))
 
Theoremncolcom 25256 Swapping non-colinear points. (Contributed by Thierry Arnoux, 19-Oct-2019.)
𝑃 = (Base‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝑋𝑃)    &   (𝜑𝑌𝑃)    &   (𝜑𝑍𝑃)    &   (𝜑 → ¬ (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌))       (𝜑 → ¬ (𝑍 ∈ (𝑌𝐿𝑋) ∨ 𝑌 = 𝑋))
 
Theoremncolrot1 25257 Rotating non-colinear points. (Contributed by Thierry Arnoux, 19-Oct-2019.)
𝑃 = (Base‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝑋𝑃)    &   (𝜑𝑌𝑃)    &   (𝜑𝑍𝑃)    &   (𝜑 → ¬ (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌))       (𝜑 → ¬ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))
 
Theoremncolrot2 25258 Rotating non-colinear points. (Contributed by Thierry Arnoux, 19-Oct-2019.)
𝑃 = (Base‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝑋𝑃)    &   (𝜑𝑌𝑃)    &   (𝜑𝑍𝑃)    &   (𝜑 → ¬ (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌))       (𝜑 → ¬ (𝑌 ∈ (𝑍𝐿𝑋) ∨ 𝑍 = 𝑋))
 
Theoremtgdim01ln 25259 In geometries of dimension lower than 2, any 3 points are colinear. (Contributed by Thierry Arnoux, 27-Aug-2019.)
𝑃 = (Base‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝑋𝑃)    &   (𝜑𝑌𝑃)    &   (𝜑𝑍𝑃)    &   (𝜑 → ¬ 𝐺DimTarskiG≥2)       (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌))
 
Theoremncoltgdim2 25260 If there are 3 non-colinear points, dimension must be 2 or more. tglowdim2l 25345 converse. (Contributed by Thierry Arnoux, 23-Feb-2020.)
𝑃 = (Base‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝑋𝑃)    &   (𝜑𝑌𝑃)    &   (𝜑𝑍𝑃)    &   (𝜑 → ¬ (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌))       (𝜑𝐺DimTarskiG≥2)
 
Theoremlnxfr 25261 Transfer law for colinearity. Theorem 4.13 of [Schwabhauser] p. 37. (Contributed by Thierry Arnoux, 27-Apr-2019.)
𝑃 = (Base‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝑋𝑃)    &   (𝜑𝑌𝑃)    &   (𝜑𝑍𝑃)    &    = (cgrG‘𝐺)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑 → (𝑌 ∈ (𝑋𝐿𝑍) ∨ 𝑋 = 𝑍))    &   (𝜑 → ⟨“𝑋𝑌𝑍”⟩ ⟨“𝐴𝐵𝐶”⟩)       (𝜑 → (𝐵 ∈ (𝐴𝐿𝐶) ∨ 𝐴 = 𝐶))
 
Theoremlnext 25262* Extend a line with a missing point. Theorem 4.14 of [Schwabhauser] p. 37. (Contributed by Thierry Arnoux, 27-Apr-2019.)
𝑃 = (Base‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝑋𝑃)    &   (𝜑𝑌𝑃)    &   (𝜑𝑍𝑃)    &    = (cgrG‘𝐺)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &    = (dist‘𝐺)    &   (𝜑 → (𝑌 ∈ (𝑋𝐿𝑍) ∨ 𝑋 = 𝑍))    &   (𝜑 → (𝑋 𝑌) = (𝐴 𝐵))       (𝜑 → ∃𝑐𝑃 ⟨“𝑋𝑌𝑍”⟩ ⟨“𝐴𝐵𝑐”⟩)
 
Theoremtgfscgr 25263 Congruence law for the general five segment configuration. Theorem 4.16 of [Schwabhauser] p. 37. (Contributed by Thierry Arnoux, 27-Apr-2019.)
𝑃 = (Base‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝑋𝑃)    &   (𝜑𝑌𝑃)    &   (𝜑𝑍𝑃)    &    = (cgrG‘𝐺)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &    = (dist‘𝐺)    &   (𝜑𝑇𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑 → (𝑌 ∈ (𝑋𝐿𝑍) ∨ 𝑋 = 𝑍))    &   (𝜑 → ⟨“𝑋𝑌𝑍”⟩ ⟨“𝐴𝐵𝐶”⟩)    &   (𝜑 → (𝑋 𝑇) = (𝐴 𝐷))    &   (𝜑 → (𝑌 𝑇) = (𝐵 𝐷))    &   (𝜑𝑋𝑌)       (𝜑 → (𝑍 𝑇) = (𝐶 𝐷))
 
Theoremlncgr 25264 Congruence rule for lines. Theorem 4.17 of [Schwabhauser] p. 37. (Contributed by Thierry Arnoux, 28-Apr-2019.)
𝑃 = (Base‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝑋𝑃)    &   (𝜑𝑌𝑃)    &   (𝜑𝑍𝑃)    &    = (cgrG‘𝐺)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &    = (dist‘𝐺)    &   (𝜑𝑋𝑌)    &   (𝜑 → (𝑌 ∈ (𝑋𝐿𝑍) ∨ 𝑋 = 𝑍))    &   (𝜑 → (𝑋 𝐴) = (𝑋 𝐵))    &   (𝜑 → (𝑌 𝐴) = (𝑌 𝐵))       (𝜑 → (𝑍 𝐴) = (𝑍 𝐵))
 
Theoremlnid 25265 Identity law for points on lines. Theorem 4.18 of [Schwabhauser] p. 38. (Contributed by Thierry Arnoux, 28-Apr-2019.)
𝑃 = (Base‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝑋𝑃)    &   (𝜑𝑌𝑃)    &   (𝜑𝑍𝑃)    &    = (cgrG‘𝐺)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &    = (dist‘𝐺)    &   (𝜑𝑋𝑌)    &   (𝜑 → (𝑌 ∈ (𝑋𝐿𝑍) ∨ 𝑋 = 𝑍))    &   (𝜑 → (𝑋 𝑍) = (𝑋 𝐴))    &   (𝜑 → (𝑌 𝑍) = (𝑌 𝐴))       (𝜑𝑍 = 𝐴)
 
Theoremtgidinside 25266 Law for finding a point inside a segment. Theorem 4.19 of [Schwabhauser] p. 38. (Contributed by Thierry Arnoux, 28-Apr-2019.)
𝑃 = (Base‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝑋𝑃)    &   (𝜑𝑌𝑃)    &   (𝜑𝑍𝑃)    &    = (cgrG‘𝐺)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &    = (dist‘𝐺)    &   (𝜑𝑍 ∈ (𝑋𝐼𝑌))    &   (𝜑 → (𝑋 𝑍) = (𝑋 𝐴))    &   (𝜑 → (𝑌 𝑍) = (𝑌 𝐴))       (𝜑𝑍 = 𝐴)
 
15.2.8  Connectivity of betweenness
 
Theoremtgbtwnconn1lem1 25267 Lemma for tgbtwnconn1 25270. (Contributed by Thierry Arnoux, 30-Apr-2019.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐴𝐵)    &   (𝜑𝐵 ∈ (𝐴𝐼𝐶))    &   (𝜑𝐵 ∈ (𝐴𝐼𝐷))    &    = (dist‘𝐺)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑𝐻𝑃)    &   (𝜑𝐽𝑃)    &   (𝜑𝐷 ∈ (𝐴𝐼𝐸))    &   (𝜑𝐶 ∈ (𝐴𝐼𝐹))    &   (𝜑𝐸 ∈ (𝐴𝐼𝐻))    &   (𝜑𝐹 ∈ (𝐴𝐼𝐽))    &   (𝜑 → (𝐸 𝐷) = (𝐶 𝐷))    &   (𝜑 → (𝐶 𝐹) = (𝐶 𝐷))    &   (𝜑 → (𝐸 𝐻) = (𝐵 𝐶))    &   (𝜑 → (𝐹 𝐽) = (𝐵 𝐷))       (𝜑𝐻 = 𝐽)
 
Theoremtgbtwnconn1lem2 25268 Lemma for tgbtwnconn1 25270. (Contributed by Thierry Arnoux, 30-Apr-2019.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐴𝐵)    &   (𝜑𝐵 ∈ (𝐴𝐼𝐶))    &   (𝜑𝐵 ∈ (𝐴𝐼𝐷))    &    = (dist‘𝐺)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑𝐻𝑃)    &   (𝜑𝐽𝑃)    &   (𝜑𝐷 ∈ (𝐴𝐼𝐸))    &   (𝜑𝐶 ∈ (𝐴𝐼𝐹))    &   (𝜑𝐸 ∈ (𝐴𝐼𝐻))    &   (𝜑𝐹 ∈ (𝐴𝐼𝐽))    &   (𝜑 → (𝐸 𝐷) = (𝐶 𝐷))    &   (𝜑 → (𝐶 𝐹) = (𝐶 𝐷))    &   (𝜑 → (𝐸 𝐻) = (𝐵 𝐶))    &   (𝜑 → (𝐹 𝐽) = (𝐵 𝐷))       (𝜑 → (𝐸 𝐹) = (𝐶 𝐷))
 
Theoremtgbtwnconn1lem3 25269 Lemma for tgbtwnconn1 25270. (Contributed by Thierry Arnoux, 30-Apr-2019.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐴𝐵)    &   (𝜑𝐵 ∈ (𝐴𝐼𝐶))    &   (𝜑𝐵 ∈ (𝐴𝐼𝐷))    &    = (dist‘𝐺)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑𝐻𝑃)    &   (𝜑𝐽𝑃)    &   (𝜑𝐷 ∈ (𝐴𝐼𝐸))    &   (𝜑𝐶 ∈ (𝐴𝐼𝐹))    &   (𝜑𝐸 ∈ (𝐴𝐼𝐻))    &   (𝜑𝐹 ∈ (𝐴𝐼𝐽))    &   (𝜑 → (𝐸 𝐷) = (𝐶 𝐷))    &   (𝜑 → (𝐶 𝐹) = (𝐶 𝐷))    &   (𝜑 → (𝐸 𝐻) = (𝐵 𝐶))    &   (𝜑 → (𝐹 𝐽) = (𝐵 𝐷))    &   (𝜑𝑋𝑃)    &   (𝜑𝑋 ∈ (𝐶𝐼𝐸))    &   (𝜑𝑋 ∈ (𝐷𝐼𝐹))    &   (𝜑𝐶𝐸)       (𝜑𝐷 = 𝐹)
 
Theoremtgbtwnconn1 25270 Connectivity law for betweenness. Theorem 5.1 of [Schwabhauser] p. 39-41. In earlier presentations of Tarski's axioms, this theorem appeared as an additional axiom. It was derived from the other axioms by Gupta, 1965. (Contributed by Thierry Arnoux, 30-Apr-2019.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐴𝐵)    &   (𝜑𝐵 ∈ (𝐴𝐼𝐶))    &   (𝜑𝐵 ∈ (𝐴𝐼𝐷))       (𝜑 → (𝐶 ∈ (𝐴𝐼𝐷) ∨ 𝐷 ∈ (𝐴𝐼𝐶)))
 
Theoremtgbtwnconn2 25271 Another connectivity law for betweenness. Theorem 5.2 of [Schwabhauser] p. 41. (Contributed by Thierry Arnoux, 17-May-2019.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐴𝐵)    &   (𝜑𝐵 ∈ (𝐴𝐼𝐶))    &   (𝜑𝐵 ∈ (𝐴𝐼𝐷))       (𝜑 → (𝐶 ∈ (𝐵𝐼𝐷) ∨ 𝐷 ∈ (𝐵𝐼𝐶)))
 
Theoremtgbtwnconn3 25272 Inner connectivity law for betweenness. Theorem 5.3 of [Schwabhauser] p. 41. (Contributed by Thierry Arnoux, 17-May-2019.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐵 ∈ (𝐴𝐼𝐷))    &   (𝜑𝐶 ∈ (𝐴𝐼𝐷))       (𝜑 → (𝐵 ∈ (𝐴𝐼𝐶) ∨ 𝐶 ∈ (𝐴𝐼𝐵)))
 
Theoremtgbtwnconnln3 25273 Derive colinearity from betweenness. (Contributed by Thierry Arnoux, 17-May-2019.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐵 ∈ (𝐴𝐼𝐷))    &   (𝜑𝐶 ∈ (𝐴𝐼𝐷))    &   𝐿 = (LineG‘𝐺)       (𝜑 → (𝐵 ∈ (𝐴𝐿𝐶) ∨ 𝐴 = 𝐶))
 
Theoremtgbtwnconn22 25274 Double connectivity law for betweenness. (Contributed by Thierry Arnoux, 1-Dec-2019.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐴𝐵)    &   (𝜑𝐶𝐵)    &   (𝜑𝐵 ∈ (𝐴𝐼𝐶))    &   (𝜑𝐵 ∈ (𝐴𝐼𝐷))    &   (𝜑𝐵 ∈ (𝐶𝐼𝐸))       (𝜑𝐵 ∈ (𝐷𝐼𝐸))
 
Theoremtgbtwnconnln1 25275 Derive colinearity from betweenness. (Contributed by Thierry Arnoux, 17-May-2019.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐴𝐵)    &   (𝜑𝐵 ∈ (𝐴𝐼𝐶))    &   (𝜑𝐵 ∈ (𝐴𝐼𝐷))       (𝜑 → (𝐴 ∈ (𝐶𝐿𝐷) ∨ 𝐶 = 𝐷))
 
Theoremtgbtwnconnln2 25276 Derive colinearity from betweenness. (Contributed by Thierry Arnoux, 17-May-2019.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐴𝐵)    &   (𝜑𝐵 ∈ (𝐴𝐼𝐶))    &   (𝜑𝐵 ∈ (𝐴𝐼𝐷))       (𝜑 → (𝐵 ∈ (𝐶𝐿𝐷) ∨ 𝐶 = 𝐷))
 
15.2.9  Less-than relation in geometric congruences
 
Syntaxcleg 25277 Less-than relation for geometric congruences.
class ≤G
 
Definitiondf-leg 25278* Define the less-than relationship between geometric distance congruence classes. See legval 25279. (Contributed by Thierry Arnoux, 21-Jun-2019.)
≤G = (𝑔 ∈ V ↦ {⟨𝑒, 𝑓⟩ ∣ [(Base‘𝑔) / 𝑝][(dist‘𝑔) / 𝑑][(Itv‘𝑔) / 𝑖]𝑥𝑝𝑦𝑝 (𝑓 = (𝑥𝑑𝑦) ∧ ∃𝑧𝑝 (𝑧 ∈ (𝑥𝑖𝑦) ∧ 𝑒 = (𝑥𝑑𝑧)))})
 
Theoremlegval 25279* Value of the less-than relationship. (Contributed by Thierry Arnoux, 21-Jun-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &    = (≤G‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)       (𝜑 = {⟨𝑒, 𝑓⟩ ∣ ∃𝑥𝑃𝑦𝑃 (𝑓 = (𝑥 𝑦) ∧ ∃𝑧𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 𝑧)))})
 
Theoremlegov 25280* Value of the less-than relationship. Definition 5.4 of [Schwabhauser] p. 41. (Contributed by Thierry Arnoux, 21-Jun-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &    = (≤G‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)       (𝜑 → ((𝐴 𝐵) (𝐶 𝐷) ↔ ∃𝑧𝑃 (𝑧 ∈ (𝐶𝐼𝐷) ∧ (𝐴 𝐵) = (𝐶 𝑧))))
 
Theoremlegov2 25281* An equivalent definition of the less-than relationship. Definition 5.5 of [Schwabhauser] p. 41. (Contributed by Thierry Arnoux, 21-Jun-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &    = (≤G‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)       (𝜑 → ((𝐴 𝐵) (𝐶 𝐷) ↔ ∃𝑥𝑃 (𝐵 ∈ (𝐴𝐼𝑥) ∧ (𝐴 𝑥) = (𝐶 𝐷))))
 
Theoremlegid 25282 Reflexivity of the less-than relationship. Proposition 5.7 of [Schwabhauser] p. 42. (Contributed by Thierry Arnoux, 27-Jun-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &    = (≤G‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)       (𝜑 → (𝐴 𝐵) (𝐴 𝐵))
 
Theorembtwnleg 25283 Betweenness implies less-than relation. (Contributed by Thierry Arnoux, 3-Jul-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &    = (≤G‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐵 ∈ (𝐴𝐼𝐶))       (𝜑 → (𝐴 𝐵) (𝐴 𝐶))
 
Theoremlegtrd 25284 Transitivity of the less-than relationship. Proposition 5.8 of [Schwabhauser] p. 42. (Contributed by Thierry Arnoux, 27-Jun-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &    = (≤G‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑 → (𝐴 𝐵) (𝐶 𝐷))    &   (𝜑 → (𝐶 𝐷) (𝐸 𝐹))       (𝜑 → (𝐴 𝐵) (𝐸 𝐹))
 
Theoremlegtri3 25285 Equality from the less-than relationship. Proposition 5.9 of [Schwabhauser] p. 42. (Contributed by Thierry Arnoux, 27-Jun-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &    = (≤G‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑 → (𝐴 𝐵) (𝐶 𝐷))    &   (𝜑 → (𝐶 𝐷) (𝐴 𝐵))       (𝜑 → (𝐴 𝐵) = (𝐶 𝐷))
 
Theoremlegtrid 25286 Trichotomy law for the less-than relationship. Proposition 5.10 of [Schwabhauser] p. 42. (Contributed by Thierry Arnoux, 27-Jun-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &    = (≤G‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)       (𝜑 → ((𝐴 𝐵) (𝐶 𝐷) ∨ (𝐶 𝐷) (𝐴 𝐵)))
 
Theoremleg0 25287 Degenerated (zero-length) segments are minimal. Proposition 5.11 of [Schwabhauser] p. 42. (Contributed by Thierry Arnoux, 27-Jun-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &    = (≤G‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)       (𝜑 → (𝐴 𝐴) (𝐶 𝐷))
 
Theoremlegeq 25288 Deduce equality from "less than" null segments. (Contributed by Thierry Arnoux, 12-Aug-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &    = (≤G‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑 → (𝐴 𝐵) (𝐶 𝐶))       (𝜑𝐴 = 𝐵)
 
Theoremlegbtwn 25289 Deduce betweenness from "less than" relation. Corresponds loosely to Proposition 6.13 of [Schwabhauser] p. 45. (Contributed by Thierry Arnoux, 25-Aug-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &    = (≤G‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑 → (𝐴 ∈ (𝐶𝐼𝐵) ∨ 𝐵 ∈ (𝐶𝐼𝐴)))    &   (𝜑 → (𝐶 𝐴) (𝐶 𝐵))       (𝜑𝐴 ∈ (𝐶𝐼𝐵))
 
Theoremtgcgrsub2 25290 Removing identical parts from the end of a line segment preserves congruence. In this version the order of points is not known. (Contributed by Thierry Arnoux, 3-Apr-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &    = (≤G‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑 → (𝐵 ∈ (𝐴𝐼𝐶) ∨ 𝐶 ∈ (𝐴𝐼𝐵)))    &   (𝜑 → (𝐸 ∈ (𝐷𝐼𝐹) ∨ 𝐹 ∈ (𝐷𝐼𝐸)))    &   (𝜑 → (𝐴 𝐵) = (𝐷 𝐸))    &   (𝜑 → (𝐴 𝐶) = (𝐷 𝐹))       (𝜑 → (𝐵 𝐶) = (𝐸 𝐹))
 
Theoremltgseg 25291* The set 𝐸 denotes the possible values of the congruence. (Contributed by Thierry Arnoux, 15-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &    = (≤G‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   𝐸 = ( “ (𝑃 × 𝑃))    &   (𝜑 → Fun )    &   (𝜑𝐴𝐸)       (𝜑 → ∃𝑥𝑃𝑦𝑃 𝐴 = (𝑥 𝑦))
 
Theoremltgov 25292 Strict "shorter than" geometric relation between segments. (Contributed by Thierry Arnoux, 15-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &    = (≤G‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   𝐸 = ( “ (𝑃 × 𝑃))    &   (𝜑 → Fun )    &    < = (( 𝐸) ∖ I )    &   (𝜑 → (𝑃 × 𝑃) ⊆ dom )    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)       (𝜑 → ((𝐴 𝐵) < (𝐶 𝐷) ↔ ((𝐴 𝐵) (𝐶 𝐷) ∧ (𝐴 𝐵) ≠ (𝐶 𝐷))))
 
Theoremlegov3 25293 An equivalent definition of the less-than relationship, from the strict relation. (Contributed by Thierry Arnoux, 15-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &    = (≤G‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   𝐸 = ( “ (𝑃 × 𝑃))    &   (𝜑 → Fun )    &    < = (( 𝐸) ∖ I )    &   (𝜑 → (𝑃 × 𝑃) ⊆ dom )    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)       (𝜑 → ((𝐴 𝐵) (𝐶 𝐷) ↔ ((𝐴 𝐵) < (𝐶 𝐷) ∨ (𝐴 𝐵) = (𝐶 𝐷))))
 
Theoremlegso 25294 The shorter-than relationship builds an order over pairs. Remark 5.13 of [Schwabhauser] p. 42. (Contributed by Thierry Arnoux, 27-Jun-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &    = (≤G‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   𝐸 = ( “ (𝑃 × 𝑃))    &   (𝜑 → Fun )    &    < = (( 𝐸) ∖ I )    &   (𝜑 → (𝑃 × 𝑃) ⊆ dom )       (𝜑< Or 𝐸)
 
15.2.10  Rays
 
Syntaxchlg 25295 Function producing the relation "belong to the same half-line".
class hlG
 
Definitiondf-hlg 25296* Define the function producting the relation "belong to the same half-line" (Contributed by Thierry Arnoux, 15-Aug-2020.)
hlG = (𝑔 ∈ V ↦ (𝑐 ∈ (Base‘𝑔) ↦ {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (Base‘𝑔) ∧ 𝑏 ∈ (Base‘𝑔)) ∧ (𝑎𝑐𝑏𝑐 ∧ (𝑎 ∈ (𝑐(Itv‘𝑔)𝑏) ∨ 𝑏 ∈ (𝑐(Itv‘𝑔)𝑎))))}))
 
Theoremishlg 25297 Rays : Definition 6.1 of [Schwabhauser] p. 43. With this definition, 𝐴(𝐾𝐶)𝐵 means that 𝐴 and 𝐵 are on the same ray with initial point 𝐶. This follows the same notation as Schwabhauser where rays are first defined as a relation. It is possible to recover the ray itself using e.g. ((𝐾𝐶) “ {𝐴}) (Contributed by Thierry Arnoux, 21-Dec-2019.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐾 = (hlG‘𝐺)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐺𝑉)       (𝜑 → (𝐴(𝐾𝐶)𝐵 ↔ (𝐴𝐶𝐵𝐶 ∧ (𝐴 ∈ (𝐶𝐼𝐵) ∨ 𝐵 ∈ (𝐶𝐼𝐴)))))
 
Theoremhlcomb 25298 The half-line relation commutes. Theorem 6.6 of [Schwabhauser] p. 44. (Contributed by Thierry Arnoux, 21-Feb-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐾 = (hlG‘𝐺)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐺𝑉)       (𝜑 → (𝐴(𝐾𝐶)𝐵𝐵(𝐾𝐶)𝐴))
 
Theoremhlcomd 25299 The half-line relation commutes. Theorem 6.6 of [Schwabhauser] p. 44. (Contributed by Thierry Arnoux, 21-Feb-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐾 = (hlG‘𝐺)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐺𝑉)    &   (𝜑𝐴(𝐾𝐶)𝐵)       (𝜑𝐵(𝐾𝐶)𝐴)
 
Theoremhlne1 25300 The half-line relation implies inequality. (Contributed by Thierry Arnoux, 22-Feb-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐾 = (hlG‘𝐺)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐺𝑉)    &   (𝜑𝐴(𝐾𝐶)𝐵)       (𝜑𝐴𝐶)
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42360
  Copyright terms: Public domain < Previous  Next >