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Theorem List for Metamath Proof Explorer - 26201-26300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
15.2.2  Betweenness
 
Theoremtgbtwntriv2 26201 Betweenness always holds for the second endpoint. Theorem 3.1 of [Schwabhauser] p. 30. (Contributed by Thierry Arnoux, 15-Mar-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)       (𝜑𝐵 ∈ (𝐴𝐼𝐵))
 
Theoremtgbtwncom 26202 Betweenness commutes. Theorem 3.2 of [Schwabhauser] p. 30. (Contributed by Thierry Arnoux, 15-Mar-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐵 ∈ (𝐴𝐼𝐶))       (𝜑𝐵 ∈ (𝐶𝐼𝐴))
 
Theoremtgbtwncomb 26203 Betweenness commutes, biconditional version. (Contributed by Thierry Arnoux, 3-Apr-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)       (𝜑 → (𝐵 ∈ (𝐴𝐼𝐶) ↔ 𝐵 ∈ (𝐶𝐼𝐴)))
 
Theoremtgbtwnne 26204 Betweenness and inequality. (Contributed by Thierry Arnoux, 1-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐵 ∈ (𝐴𝐼𝐶))    &   (𝜑𝐵𝐴)       (𝜑𝐴𝐶)
 
Theoremtgbtwntriv1 26205 Betweenness always holds for the first endpoint. Theorem 3.3 of [Schwabhauser] p. 30. (Contributed by Thierry Arnoux, 15-Mar-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)       (𝜑𝐴 ∈ (𝐴𝐼𝐵))
 
Theoremtgbtwnswapid 26206 If you can swap the first two arguments of a betweenness statement, then those arguments are identical. Theorem 3.4 of [Schwabhauser] p. 30. (Contributed by Thierry Arnoux, 16-Mar-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐴 ∈ (𝐵𝐼𝐶))    &   (𝜑𝐵 ∈ (𝐴𝐼𝐶))       (𝜑𝐴 = 𝐵)
 
Theoremtgbtwnintr 26207 Inner transitivity law for betweenness. Left-hand side of Theorem 3.5 of [Schwabhauser] p. 30. (Contributed by Thierry Arnoux, 18-Mar-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐴 ∈ (𝐵𝐼𝐷))    &   (𝜑𝐵 ∈ (𝐶𝐼𝐷))       (𝜑𝐵 ∈ (𝐴𝐼𝐶))
 
Theoremtgbtwnexch3 26208 Exchange the first endpoint in betweenness. Left-hand side of Theorem 3.6 of [Schwabhauser] p. 30. (Contributed by Thierry Arnoux, 18-Mar-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐵 ∈ (𝐴𝐼𝐶))    &   (𝜑𝐶 ∈ (𝐴𝐼𝐷))       (𝜑𝐶 ∈ (𝐵𝐼𝐷))
 
Theoremtgbtwnouttr2 26209 Outer transitivity law for betweenness. Left-hand side of Theorem 3.7 of [Schwabhauser] p. 30. (Contributed by Thierry Arnoux, 18-Mar-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐵𝐶)    &   (𝜑𝐵 ∈ (𝐴𝐼𝐶))    &   (𝜑𝐶 ∈ (𝐵𝐼𝐷))       (𝜑𝐶 ∈ (𝐴𝐼𝐷))
 
Theoremtgbtwnexch2 26210 Exchange the outer point of two betweenness statements. Right-hand side of Theorem 3.5 of [Schwabhauser] p. 30. (Contributed by Thierry Arnoux, 23-Mar-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐵 ∈ (𝐴𝐼𝐷))    &   (𝜑𝐶 ∈ (𝐵𝐼𝐷))       (𝜑𝐶 ∈ (𝐴𝐼𝐷))
 
Theoremtgbtwnouttr 26211 Outer transitivity law for betweenness. Right-hand side of Theorem 3.7 of [Schwabhauser] p. 30. (Contributed by Thierry Arnoux, 23-Mar-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐵𝐶)    &   (𝜑𝐵 ∈ (𝐴𝐼𝐶))    &   (𝜑𝐶 ∈ (𝐵𝐼𝐷))       (𝜑𝐵 ∈ (𝐴𝐼𝐷))
 
Theoremtgbtwnexch 26212 Outer transitivity law for betweenness. Right-hand side of Theorem 3.6 of [Schwabhauser] p. 30. (Contributed by Thierry Arnoux, 23-Mar-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐵 ∈ (𝐴𝐼𝐶))    &   (𝜑𝐶 ∈ (𝐴𝐼𝐷))       (𝜑𝐵 ∈ (𝐴𝐼𝐷))
 
Theoremtgtrisegint 26213* A line segment between two sides of a triange intersects a segment crossing from the remaining side to the opposite vertex. Theorem 3.17 of [Schwabhauser] p. 33. (Contributed by Thierry Arnoux, 23-Mar-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑𝐵 ∈ (𝐴𝐼𝐶))    &   (𝜑𝐸 ∈ (𝐷𝐼𝐶))    &   (𝜑𝐹 ∈ (𝐴𝐼𝐷))       (𝜑 → ∃𝑞𝑃 (𝑞 ∈ (𝐹𝐼𝐶) ∧ 𝑞 ∈ (𝐵𝐼𝐸)))
 
15.2.3  Dimension
 
Theoremtglowdim1 26214* Lower dimension axiom for one dimension. In dimension at least 1, there are at least two distinct points. The condition "the space is of dimension 1 or more" is written here as 2 ≤ (♯‘𝑃) to avoid a new definition, but a different convention could be chosen. (Contributed by Thierry Arnoux, 23-Mar-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑 → 2 ≤ (♯‘𝑃))       (𝜑 → ∃𝑥𝑃𝑦𝑃 𝑥𝑦)
 
Theoremtglowdim1i 26215* Lower dimension axiom for one dimension. (Contributed by Thierry Arnoux, 28-May-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑 → 2 ≤ (♯‘𝑃))    &   (𝜑𝑋𝑃)       (𝜑 → ∃𝑦𝑃 𝑋𝑦)
 
Theoremtgldimor 26216 Excluded-middle like statement allowing to treat dimension zero as a special case. (Contributed by Thierry Arnoux, 11-Apr-2019.)
𝑃 = (𝐸𝐹)    &   (𝜑𝐴𝑃)       (𝜑 → ((♯‘𝑃) = 1 ∨ 2 ≤ (♯‘𝑃)))
 
Theoremtgldim0eq 26217 In dimension zero, any two points are equal. (Contributed by Thierry Arnoux, 11-Apr-2019.)
𝑃 = (𝐸𝐹)    &   (𝜑 → (♯‘𝑃) = 1)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)       (𝜑𝐴 = 𝐵)
 
Theoremtgldim0itv 26218 In dimension zero, any two points are equal. (Contributed by Thierry Arnoux, 12-Apr-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑 → (♯‘𝑃) = 1)       (𝜑𝐴 ∈ (𝐵𝐼𝐶))
 
Theoremtgldim0cgr 26219 In dimension zero, any two pairs of points are congruent. (Contributed by Thierry Arnoux, 12-Apr-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑 → (♯‘𝑃) = 1)    &   (𝜑𝐷𝑃)       (𝜑 → (𝐴 𝐵) = (𝐶 𝐷))
 
Theoremtgbtwndiff 26220* 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 26221 In geometries of dimension less 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 26222 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 26223 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 26224 Declare the constant for the congruence between shapes relation.
class cgrG
 
Definitiondf-cgrg 26225* 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 26522 and expanded in iscgra 26523. This does not mean our system is weaker; dfcgrg2 26577 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 26234, cgr3simp2 26235, cgr3simp3 26236, cgrcgra 26535, and permutation laws such as cgr3swap12 26237 and dfcgrg2 26577.

Ideally, we would define this for functions of any set, but we will use words (see df-word 13852) 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 26226* 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 26227* 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 26228* 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 26229 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 26230 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 26231 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 26232* 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 26233 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 26234 Deduce segment congruence from a triangle congruence. This is a portion of the theorem that corresponding parts of congruent triangles are congruent (CPCTC), focusing on a specific segment. (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 26235 Deduce segment congruence from a triangle congruence. This is a portion of CPCTC, focusing on a specific segment. (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 26236 Deduce segment congruence from a triangle congruence. This is a portion of CPCTC, focusing on a specific segment. (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 26237 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 26238 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 26239 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 26240 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 26241 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 26242 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 26243 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 26244 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 26245 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 26246 Declare the constant for the isometry builder.
class Ismt
 
Definitiondf-ismt 26247* Define the set of isometries between two structures. Definition 4.8 of [Schwabhauser] p. 36. See isismt 26248. (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 26248* Property of being an isometry. Compare with isismty 34962. (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 26249* 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 26250 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 26251 The identity is a motion. (Contributed by Thierry Arnoux, 15-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   (𝜑𝐺𝑉)       (𝜑 → ( I ↾ 𝑃) ∈ (𝐺Ismt𝐺))
 
Theoremmotf1o 26252 Motions are bijections. (Contributed by Thierry Arnoux, 15-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   (𝜑𝐺𝑉)    &   (𝜑𝐹 ∈ (𝐺Ismt𝐺))       (𝜑𝐹:𝑃1-1-onto𝑃)
 
Theoremmotcl 26253 Closure of motions. (Contributed by Thierry Arnoux, 15-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   (𝜑𝐺𝑉)    &   (𝜑𝐹 ∈ (𝐺Ismt𝐺))    &   (𝜑𝐴𝑃)       (𝜑 → (𝐹𝐴) ∈ 𝑃)
 
Theoremmotco 26254 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 26255 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 26256* 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 26257* 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 26258* 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 26259 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 26260* 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 26261 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 26262 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 26263 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 26264 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 26265* 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 26266 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 26267 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 26268 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 26269 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 26270 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 26271 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 26272 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 26273 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 26274 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 26275 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 26276 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 26277 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 26278 In geometries of dimension less than two, then any three 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 26279 If there are three non-colinear points, then the dimension is at least two. Converse of tglowdim2l 26364. (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 26280 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 26281* 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 26282 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 26283 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 26284 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 26285 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 26286 Lemma for tgbtwnconn1 26289. (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 26287 Lemma for tgbtwnconn1 26289. (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 26288 Lemma for tgbtwnconn1 26289. (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 26289 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 26290 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 26291 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 26292 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 26293 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 26294 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 26295 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 26296 Less-than relation for geometric congruences.
class ≤G
 
Definitiondf-leg 26297* Define the less-than relationship between geometric distance congruence classes. See legval 26298. (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 26298* 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 26299* 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 26300* 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))))
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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 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44804
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