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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)

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