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Theorem List for Metamath Proof Explorer - 26301-26400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremlegid 26301 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 26302 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 26303 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 26304 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 26305 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 26306 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 26307 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 26308 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 26309 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 26310* 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 26311 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 26312 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 26313 The "shorter than" relation induces an order on 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 26314 Function producing the relation "belong to the same half-line".
class hlG
 
Definitiondf-hlg 26315* 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 26316 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 26317 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 26318 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 26319 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)
 
Theoremhlne2 26320 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)       (𝜑𝐵𝐶)
 
Theoremhlln 26321 The half-line relation implies colinearity, part of Theorem 6.4 of [Schwabhauser] p. 44. (Contributed by Thierry Arnoux, 22-Feb-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐾 = (hlG‘𝐺)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐺 ∈ TarskiG)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐴(𝐾𝐶)𝐵)       (𝜑𝐴 ∈ (𝐵𝐿𝐶))
 
Theoremhleqnid 26322 The endpoint does not belong to the half-line. (Contributed by Thierry Arnoux, 3-Mar-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐾 = (hlG‘𝐺)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐺 ∈ TarskiG)       (𝜑 → ¬ 𝐴(𝐾𝐴)𝐵)
 
Theoremhlid 26323 The half-line relation is reflexive. Theorem 6.5 of [Schwabhauser] p. 44. (Contributed by Thierry Arnoux, 21-Feb-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐾 = (hlG‘𝐺)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝐶)       (𝜑𝐴(𝐾𝐶)𝐴)
 
Theoremhltr 26324 The half-line relation is transitive. Theorem 6.7 of [Schwabhauser] p. 44. (Contributed by Thierry Arnoux, 23-Feb-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐾 = (hlG‘𝐺)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐷𝑃)    &   (𝜑𝐴(𝐾𝐷)𝐵)    &   (𝜑𝐵(𝐾𝐷)𝐶)       (𝜑𝐴(𝐾𝐷)𝐶)
 
Theoremhlbtwn 26325 Betweenness is a sufficient condition to swap half-lines. (Contributed by Thierry Arnoux, 21-Feb-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐾 = (hlG‘𝐺)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐷𝑃)    &   (𝜑𝐷 ∈ (𝐶𝐼𝐵))    &   (𝜑𝐵𝐶)    &   (𝜑𝐷𝐶)       (𝜑 → (𝐴(𝐾𝐶)𝐵𝐴(𝐾𝐶)𝐷))
 
Theorembtwnhl1 26326 Deduce half-line from betweenness. (Contributed by Thierry Arnoux, 4-Mar-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐾 = (hlG‘𝐺)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐷𝑃)    &   (𝜑𝐶 ∈ (𝐴𝐼𝐵))    &   (𝜑𝐴𝐵)    &   (𝜑𝐶𝐴)       (𝜑𝐶(𝐾𝐴)𝐵)
 
Theorembtwnhl2 26327 Deduce half-line from betweenness. (Contributed by Thierry Arnoux, 4-Mar-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐾 = (hlG‘𝐺)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐷𝑃)    &   (𝜑𝐶 ∈ (𝐴𝐼𝐵))    &   (𝜑𝐴𝐵)    &   (𝜑𝐶𝐵)       (𝜑𝐶(𝐾𝐵)𝐴)
 
Theorembtwnhl 26328 Swap betweenness for a half-line. (Contributed by Thierry Arnoux, 2-Mar-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐾 = (hlG‘𝐺)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐷𝑃)    &   (𝜑𝐴(𝐾𝐷)𝐵)    &   (𝜑𝐷 ∈ (𝐴𝐼𝐶))       (𝜑𝐷 ∈ (𝐵𝐼𝐶))
 
Theoremlnhl 26329 Either a point 𝐶 on the line AB is on the same side as 𝐴 or on the opposite side. (Contributed by Thierry Arnoux, 21-Sep-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐾 = (hlG‘𝐺)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐷𝑃)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐶 ∈ (𝐴𝐿𝐵))       (𝜑 → (𝐶(𝐾𝐵)𝐴𝐵 ∈ (𝐴𝐼𝐶)))
 
Theoremhlcgrex 26330* Construct a point on a half-line, at a given distance of its origin. (Contributed by Thierry Arnoux, 1-Aug-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐾 = (hlG‘𝐺)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐷𝑃)    &    = (dist‘𝐺)    &   (𝜑𝐷𝐴)    &   (𝜑𝐵𝐶)       (𝜑 → ∃𝑥𝑃 (𝑥(𝐾𝐴)𝐷 ∧ (𝐴 𝑥) = (𝐵 𝐶)))
 
Theoremhlcgreulem 26331 Lemma for hlcgreu 26332. (Contributed by Thierry Arnoux, 9-Aug-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐾 = (hlG‘𝐺)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐷𝑃)    &    = (dist‘𝐺)    &   (𝜑𝐷𝐴)    &   (𝜑𝐵𝐶)    &   (𝜑𝑋𝑃)    &   (𝜑𝑌𝑃)    &   (𝜑𝑋(𝐾𝐴)𝐷)    &   (𝜑𝑌(𝐾𝐴)𝐷)    &   (𝜑 → (𝐴 𝑋) = (𝐵 𝐶))    &   (𝜑 → (𝐴 𝑌) = (𝐵 𝐶))       (𝜑𝑋 = 𝑌)
 
Theoremhlcgreu 26332* The point constructed in hlcgrex 26330 is unique. Theorem 6.11 of [Schwabhauser] p. 44. (Contributed by Thierry Arnoux, 9-Aug-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐾 = (hlG‘𝐺)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐷𝑃)    &    = (dist‘𝐺)    &   (𝜑𝐷𝐴)    &   (𝜑𝐵𝐶)       (𝜑 → ∃!𝑥𝑃 (𝑥(𝐾𝐴)𝐷 ∧ (𝐴 𝑥) = (𝐵 𝐶)))
 
15.2.11  Lines
 
Theorembtwnlng1 26333 Betweenness implies colinearity. (Contributed by Thierry Arnoux, 28-Mar-2019.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝑋𝑃)    &   (𝜑𝑌𝑃)    &   (𝜑𝑍𝑃)    &   (𝜑𝑋𝑌)    &   (𝜑𝑍 ∈ (𝑋𝐼𝑌))       (𝜑𝑍 ∈ (𝑋𝐿𝑌))
 
Theorembtwnlng2 26334 Betweenness implies colinearity. (Contributed by Thierry Arnoux, 28-Mar-2019.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝑋𝑃)    &   (𝜑𝑌𝑃)    &   (𝜑𝑍𝑃)    &   (𝜑𝑋𝑌)    &   (𝜑𝑋 ∈ (𝑍𝐼𝑌))       (𝜑𝑍 ∈ (𝑋𝐿𝑌))
 
Theorembtwnlng3 26335 Betweenness implies colinearity. (Contributed by Thierry Arnoux, 28-Mar-2019.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝑋𝑃)    &   (𝜑𝑌𝑃)    &   (𝜑𝑍𝑃)    &   (𝜑𝑋𝑌)    &   (𝜑𝑌 ∈ (𝑋𝐼𝑍))       (𝜑𝑍 ∈ (𝑋𝐿𝑌))
 
Theoremlncom 26336 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)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝑋𝑃)    &   (𝜑𝑌𝑃)    &   (𝜑𝑍𝑃)    &   (𝜑𝑋𝑌)    &   (𝜑𝑍 ∈ (𝑌𝐿𝑋))       (𝜑𝑍 ∈ (𝑋𝐿𝑌))
 
Theoremlnrot1 26337 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)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝑋𝑃)    &   (𝜑𝑌𝑃)    &   (𝜑𝑍𝑃)    &   (𝜑𝑋𝑌)    &   (𝜑𝑌 ∈ (𝑍𝐿𝑋))    &   (𝜑𝑍𝑋)       (𝜑𝑍 ∈ (𝑋𝐿𝑌))
 
Theoremlnrot2 26338 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)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝑋𝑃)    &   (𝜑𝑌𝑃)    &   (𝜑𝑍𝑃)    &   (𝜑𝑋𝑌)    &   (𝜑𝑋 ∈ (𝑌𝐿𝑍))    &   (𝜑𝑌𝑍)       (𝜑𝑍 ∈ (𝑋𝐿𝑌))
 
Theoremncolne1 26339 Non-colinear points are different. (Contributed by Thierry Arnoux, 8-Aug-2019.)
𝐵 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &   (𝜑 → ¬ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))       (𝜑𝑋𝑌)
 
Theoremncolne2 26340 Non-colinear points are different. (Contributed by Thierry Arnoux, 8-Aug-2019.) TODO (NM): maybe ncolne2 26340 could be simplified out and deleted, replaced by ncolcom 26275.
𝐵 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &   (𝜑 → ¬ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))       (𝜑𝑋𝑍)
 
Theoremtgisline 26341* The property of being a proper line, generated by two distinct points. (Contributed by Thierry Arnoux, 25-May-2019.)
𝐵 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴 ∈ ran 𝐿)       (𝜑 → ∃𝑥𝐵𝑦𝐵 (𝐴 = (𝑥𝐿𝑦) ∧ 𝑥𝑦))
 
Theoremtglnne 26342 It takes two different points to form a line. (Contributed by Thierry Arnoux, 27-Nov-2019.)
𝐵 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑 → (𝑋𝐿𝑌) ∈ ran 𝐿)       (𝜑𝑋𝑌)
 
Theoremtglndim0 26343 There are no lines in dimension 0. (Contributed by Thierry Arnoux, 18-Oct-2019.)
𝐵 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑 → (♯‘𝐵) = 1)       (𝜑 → ¬ 𝐴 ∈ ran 𝐿)
 
Theoremtgelrnln 26344 The property of being a proper line, generated by two distinct points. (Contributed by Thierry Arnoux, 25-May-2019.)
𝐵 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑋𝑌)       (𝜑 → (𝑋𝐿𝑌) ∈ ran 𝐿)
 
Theoremtglineeltr 26345 Transitivity law for lines, one half of tglineelsb2 26346. (Contributed by Thierry Arnoux, 25-May-2019.)
𝐵 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝑃𝐵)    &   (𝜑𝑄𝐵)    &   (𝜑𝑃𝑄)    &   (𝜑𝑆𝐵)    &   (𝜑𝑆𝑃)    &   (𝜑𝑆 ∈ (𝑃𝐿𝑄))    &   (𝜑𝑅𝐵)    &   (𝜑𝑅 ∈ (𝑃𝐿𝑆))       (𝜑𝑅 ∈ (𝑃𝐿𝑄))
 
Theoremtglineelsb2 26346 If 𝑆 lies on PQ , then PQ = PS . Theorem 6.16 of [Schwabhauser] p. 45. (Contributed by Thierry Arnoux, 17-May-2019.)
𝐵 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝑃𝐵)    &   (𝜑𝑄𝐵)    &   (𝜑𝑃𝑄)    &   (𝜑𝑆𝐵)    &   (𝜑𝑆𝑃)    &   (𝜑𝑆 ∈ (𝑃𝐿𝑄))       (𝜑 → (𝑃𝐿𝑄) = (𝑃𝐿𝑆))
 
Theoremtglinerflx1 26347 Reflexivity law for line membership. Part of theorem 6.17 of [Schwabhauser] p. 45. (Contributed by Thierry Arnoux, 17-May-2019.)
𝐵 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝑃𝐵)    &   (𝜑𝑄𝐵)    &   (𝜑𝑃𝑄)       (𝜑𝑃 ∈ (𝑃𝐿𝑄))
 
Theoremtglinerflx2 26348 Reflexivity law for line membership. Part of theorem 6.17 of [Schwabhauser] p. 45. (Contributed by Thierry Arnoux, 17-May-2019.)
𝐵 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝑃𝐵)    &   (𝜑𝑄𝐵)    &   (𝜑𝑃𝑄)       (𝜑𝑄 ∈ (𝑃𝐿𝑄))
 
Theoremtglinecom 26349 Commutativity law for lines. Part of theorem 6.17 of [Schwabhauser] p. 45. (Contributed by Thierry Arnoux, 17-May-2019.)
𝐵 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝑃𝐵)    &   (𝜑𝑄𝐵)    &   (𝜑𝑃𝑄)       (𝜑 → (𝑃𝐿𝑄) = (𝑄𝐿𝑃))
 
Theoremtglinethru 26350 If 𝐴 is a line containing two distinct points 𝑃 and 𝑄, then 𝐴 is the line through 𝑃 and 𝑄. Theorem 6.18 of [Schwabhauser] p. 45. (Contributed by Thierry Arnoux, 25-May-2019.)
𝐵 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝑃𝐵)    &   (𝜑𝑄𝐵)    &   (𝜑𝑃𝑄)    &   (𝜑𝑃𝑄)    &   (𝜑𝐴 ∈ ran 𝐿)    &   (𝜑𝑃𝐴)    &   (𝜑𝑄𝐴)       (𝜑𝐴 = (𝑃𝐿𝑄))
 
Theoremtghilberti1 26351* There is a line through any two distinct points. Hilbert's axiom I.1 for geometry. (Contributed by Thierry Arnoux, 25-May-2019.)
𝐵 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝑃𝐵)    &   (𝜑𝑄𝐵)    &   (𝜑𝑃𝑄)       (𝜑 → ∃𝑥 ∈ ran 𝐿(𝑃𝑥𝑄𝑥))
 
Theoremtghilberti2 26352* There is at most one line through any two distinct points. Hilbert's axiom I.2 for geometry. (Contributed by Thierry Arnoux, 25-May-2019.)
𝐵 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝑃𝐵)    &   (𝜑𝑄𝐵)    &   (𝜑𝑃𝑄)       (𝜑 → ∃*𝑥 ∈ ran 𝐿(𝑃𝑥𝑄𝑥))
 
Theoremtglinethrueu 26353* There is a unique line going through any two distinct points. Theorem 6.19 of [Schwabhauser] p. 46. (Contributed by Thierry Arnoux, 25-May-2019.)
𝐵 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝑃𝐵)    &   (𝜑𝑄𝐵)    &   (𝜑𝑃𝑄)       (𝜑 → ∃!𝑥 ∈ ran 𝐿(𝑃𝑥𝑄𝑥))
 
Theoremtglnne0 26354 A line 𝐴 has at least one point. (Contributed by Thierry Arnoux, 4-Mar-2020.)
𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴 ∈ ran 𝐿)       (𝜑𝐴 ≠ ∅)
 
Theoremtglnpt2 26355* Find a second point on a line. (Contributed by Thierry Arnoux, 18-Oct-2019.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴 ∈ ran 𝐿)    &   (𝜑𝑋𝐴)       (𝜑 → ∃𝑦𝐴 𝑋𝑦)
 
Theoremtglineintmo 26356* Two distinct lines intersect in at most one point. Theorem 6.21 of [Schwabhauser] p. 46. (Contributed by Thierry Arnoux, 25-May-2019.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴 ∈ ran 𝐿)    &   (𝜑𝐵 ∈ ran 𝐿)    &   (𝜑𝐴𝐵)       (𝜑 → ∃*𝑥(𝑥𝐴𝑥𝐵))
 
Theoremtglineineq 26357 Two distinct lines intersect in at most one point, variation. Theorem 6.21 of [Schwabhauser] p. 46. (Contributed by Thierry Arnoux, 6-Aug-2019.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴 ∈ ran 𝐿)    &   (𝜑𝐵 ∈ ran 𝐿)    &   (𝜑𝐴𝐵)    &   (𝜑𝑋 ∈ (𝐴𝐵))    &   (𝜑𝑌 ∈ (𝐴𝐵))       (𝜑𝑋 = 𝑌)
 
Theoremtglineneq 26358 Given three non-colinear points, build two different lines. (Contributed by Thierry Arnoux, 6-Aug-2019.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑 → ¬ (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶))       (𝜑 → (𝐴𝐿𝐵) ≠ (𝐶𝐿𝐷))
 
Theoremtglineinteq 26359 Two distinct lines intersect in at most one point. Theorem 6.21 of [Schwabhauser] p. 46. (Contributed by Thierry Arnoux, 6-Aug-2019.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑 → ¬ (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶))    &   (𝜑𝑋 ∈ (𝐴𝐿𝐵))    &   (𝜑𝑌 ∈ (𝐴𝐿𝐵))    &   (𝜑𝑋 ∈ (𝐶𝐿𝐷))    &   (𝜑𝑌 ∈ (𝐶𝐿𝐷))       (𝜑𝑋 = 𝑌)
 
Theoremncolncol 26360 Deduce non-colinearity from non-colinearity and colinearity. (Contributed by Thierry Arnoux, 27-Aug-2019.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑 → ¬ (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶))    &   (𝜑𝐷 ∈ (𝐴𝐿𝐵))    &   (𝜑𝐷𝐵)       (𝜑 → ¬ (𝐷 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶))
 
Theoremcoltr 26361 A transitivity law for colinearity. (Contributed by Thierry Arnoux, 27-Nov-2019.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐴 ∈ (𝐵𝐿𝐶))    &   (𝜑 → (𝐵 ∈ (𝐶𝐿𝐷) ∨ 𝐶 = 𝐷))       (𝜑 → (𝐴 ∈ (𝐶𝐿𝐷) ∨ 𝐶 = 𝐷))
 
Theoremcoltr3 26362 A transitivity law for colinearity. (Contributed by Thierry Arnoux, 27-Nov-2019.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐴 ∈ (𝐵𝐿𝐶))    &   (𝜑𝐷 ∈ (𝐴𝐼𝐶))       (𝜑𝐷 ∈ (𝐵𝐿𝐶))
 
Theoremcolline 26363* Three points are colinear iff there is a line through all three of them. Theorem 6.23 of [Schwabhauser] p. 46. (Contributed by Thierry Arnoux, 28-May-2019.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝑋𝑃)    &   (𝜑𝑌𝑃)    &   (𝜑𝑍𝑃)    &   (𝜑 → 2 ≤ (♯‘𝑃))       (𝜑 → ((𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍) ↔ ∃𝑎 ∈ ran 𝐿(𝑋𝑎𝑌𝑎𝑍𝑎)))
 
Theoremtglowdim2l 26364* Reformulation of the lower dimension axiom for dimension two. There exist three non-colinear points. Theorem 6.24 of [Schwabhauser] p. 46. (Contributed by Thierry Arnoux, 30-May-2019.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐺DimTarskiG≥2)       (𝜑 → ∃𝑎𝑃𝑏𝑃𝑐𝑃 ¬ (𝑐 ∈ (𝑎𝐿𝑏) ∨ 𝑎 = 𝑏))
 
Theoremtglowdim2ln 26365* There is always one point outside of any line. Theorem 6.25 of [Schwabhauser] p. 46. (Contributed by Thierry Arnoux, 16-Nov-2019.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐺DimTarskiG≥2)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐴𝐵)       (𝜑 → ∃𝑐𝑃 ¬ 𝑐 ∈ (𝐴𝐿𝐵))
 
15.2.12  Point inversions
 
Syntaxcmir 26366 Declare the constant for the point inversion function.
class pInvG
 
Definitiondf-mir 26367* Define the point inversion ("mirror") function. Definition 7.5 of [Schwabhauser] p. 49. See mirval 26369 and ismir 26373. (Contributed by Thierry Arnoux, 30-May-2019.)
pInvG = (𝑔 ∈ V ↦ (𝑚 ∈ (Base‘𝑔) ↦ (𝑎 ∈ (Base‘𝑔) ↦ (𝑏 ∈ (Base‘𝑔)((𝑚(dist‘𝑔)𝑏) = (𝑚(dist‘𝑔)𝑎) ∧ 𝑚 ∈ (𝑏(Itv‘𝑔)𝑎))))))
 
Theoremmirreu3 26368* Existential uniqueness of the mirror point. Theorem 7.8 of [Schwabhauser] p. 49. (Contributed by Thierry Arnoux, 30-May-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝑀𝑃)       (𝜑 → ∃!𝑏𝑃 ((𝑀 𝑏) = (𝑀 𝐴) ∧ 𝑀 ∈ (𝑏𝐼𝐴)))
 
Theoremmirval 26369* Value of the point inversion function 𝑆. Definition 7.5 of [Schwabhauser] p. 49. (Contributed by Thierry Arnoux, 30-May-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝑆 = (pInvG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)       (𝜑 → (𝑆𝐴) = (𝑦𝑃 ↦ (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦)))))
 
Theoremmirfv 26370* Value of the point inversion function 𝑀. Definition 7.5 of [Schwabhauser] p. 49. (Contributed by Thierry Arnoux, 30-May-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝑆 = (pInvG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   𝑀 = (𝑆𝐴)    &   (𝜑𝐵𝑃)       (𝜑 → (𝑀𝐵) = (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))))
 
Theoremmircgr 26371 Property of the image by the point inversion function. Definition 7.5 of [Schwabhauser] p. 49. (Contributed by Thierry Arnoux, 3-Jun-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝑆 = (pInvG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   𝑀 = (𝑆𝐴)    &   (𝜑𝐵𝑃)       (𝜑 → (𝐴 (𝑀𝐵)) = (𝐴 𝐵))
 
Theoremmirbtwn 26372 Property of the image by the point inversion function. Definition 7.5 of [Schwabhauser] p. 49. (Contributed by Thierry Arnoux, 3-Jun-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝑆 = (pInvG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   𝑀 = (𝑆𝐴)    &   (𝜑𝐵𝑃)       (𝜑𝐴 ∈ ((𝑀𝐵)𝐼𝐵))
 
Theoremismir 26373 Property of the image by the point inversion function. Definition 7.5 of [Schwabhauser] p. 49. (Contributed by Thierry Arnoux, 3-Jun-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝑆 = (pInvG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   𝑀 = (𝑆𝐴)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑 → (𝐴 𝐶) = (𝐴 𝐵))    &   (𝜑𝐴 ∈ (𝐶𝐼𝐵))       (𝜑𝐶 = (𝑀𝐵))
 
Theoremmirf 26374 Point inversion as function. (Contributed by Thierry Arnoux, 30-May-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝑆 = (pInvG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   𝑀 = (𝑆𝐴)       (𝜑𝑀:𝑃𝑃)
 
Theoremmircl 26375 Closure of the point inversion function. (Contributed by Thierry Arnoux, 20-Oct-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝑆 = (pInvG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   𝑀 = (𝑆𝐴)    &   (𝜑𝑋𝑃)       (𝜑 → (𝑀𝑋) ∈ 𝑃)
 
Theoremmirmir 26376 The point inversion function is an involution. Theorem 7.7 of [Schwabhauser] p. 49. (Contributed by Thierry Arnoux, 3-Jun-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝑆 = (pInvG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   𝑀 = (𝑆𝐴)    &   (𝜑𝐵𝑃)       (𝜑 → (𝑀‘(𝑀𝐵)) = 𝐵)
 
Theoremmircom 26377 Variation on mirmir 26376. (Contributed by Thierry Arnoux, 10-Nov-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝑆 = (pInvG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   𝑀 = (𝑆𝐴)    &   (𝜑𝐵𝑃)    &   (𝜑 → (𝑀𝐵) = 𝐶)       (𝜑 → (𝑀𝐶) = 𝐵)
 
Theoremmirreu 26378* Any point has a unique antecedent through point inversion. Theorem 7.8 of [Schwabhauser] p. 50. (Contributed by Thierry Arnoux, 3-Jun-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝑆 = (pInvG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   𝑀 = (𝑆𝐴)    &   (𝜑𝐵𝑃)       (𝜑 → ∃!𝑎𝑃 (𝑀𝑎) = 𝐵)
 
Theoremmireq 26379 Equality deduction for point inversion. Theorem 7.9 of [Schwabhauser] p. 50. (Contributed by Thierry Arnoux, 30-May-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝑆 = (pInvG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   𝑀 = (𝑆𝐴)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑 → (𝑀𝐵) = (𝑀𝐶))       (𝜑𝐵 = 𝐶)
 
Theoremmirinv 26380 The only invariant point of a point inversion Theorem 7.3 of [Schwabhauser] p. 49, Theorem 7.10 of [Schwabhauser] p. 50. (Contributed by Thierry Arnoux, 30-Jul-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝑆 = (pInvG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   𝑀 = (𝑆𝐴)    &   (𝜑𝐵𝑃)       (𝜑 → ((𝑀𝐵) = 𝐵𝐴 = 𝐵))
 
Theoremmirne 26381 Mirror of non-center point cannot be the center point. (Contributed by Thierry Arnoux, 27-Sep-2020.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝑆 = (pInvG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   𝑀 = (𝑆𝐴)    &   (𝜑𝐵𝑃)    &   (𝜑𝐵𝐴)       (𝜑 → (𝑀𝐵) ≠ 𝐴)
 
Theoremmircinv 26382 The center point is invariant of a point inversion. (Contributed by Thierry Arnoux, 25-Aug-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝑆 = (pInvG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   𝑀 = (𝑆𝐴)       (𝜑 → (𝑀𝐴) = 𝐴)
 
Theoremmirf1o 26383 The point inversion function 𝑀 is a bijection. Theorem 7.11 of [Schwabhauser] p. 50. (Contributed by Thierry Arnoux, 6-Jun-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝑆 = (pInvG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   𝑀 = (𝑆𝐴)       (𝜑𝑀:𝑃1-1-onto𝑃)
 
Theoremmiriso 26384 The point inversion function is an isometry, i.e. it is conserves congruence. Because it is also a bijection, it is also a motion. Theorem 7.13 of [Schwabhauser] p. 50. (Contributed by Thierry Arnoux, 6-Jun-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝑆 = (pInvG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   𝑀 = (𝑆𝐴)    &   (𝜑𝑋𝑃)    &   (𝜑𝑌𝑃)       (𝜑 → ((𝑀𝑋) (𝑀𝑌)) = (𝑋 𝑌))
 
Theoremmirbtwni 26385 Point inversion preserves betweenness, first half of Theorem 7.15 of [Schwabhauser] p. 51. (Contributed by Thierry Arnoux, 9-Jun-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝑆 = (pInvG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   𝑀 = (𝑆𝐴)    &   (𝜑𝑋𝑃)    &   (𝜑𝑌𝑃)    &   (𝜑𝑍𝑃)    &   (𝜑𝑌 ∈ (𝑋𝐼𝑍))       (𝜑 → (𝑀𝑌) ∈ ((𝑀𝑋)𝐼(𝑀𝑍)))
 
Theoremmirbtwnb 26386 Point inversion preserves betweenness. Theorem 7.15 of [Schwabhauser] p. 51. (Contributed by Thierry Arnoux, 9-Jun-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝑆 = (pInvG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   𝑀 = (𝑆𝐴)    &   (𝜑𝑋𝑃)    &   (𝜑𝑌𝑃)    &   (𝜑𝑍𝑃)       (𝜑 → (𝑌 ∈ (𝑋𝐼𝑍) ↔ (𝑀𝑌) ∈ ((𝑀𝑋)𝐼(𝑀𝑍))))
 
Theoremmircgrs 26387 Point inversion preserves congruence. Theorem 7.16 of [Schwabhauser] p. 51. (Contributed by Thierry Arnoux, 30-Jul-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝑆 = (pInvG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   𝑀 = (𝑆𝐴)    &   (𝜑𝑋𝑃)    &   (𝜑𝑌𝑃)    &   (𝜑𝑍𝑃)    &   (𝜑𝑇𝑃)    &   (𝜑 → (𝑋 𝑌) = (𝑍 𝑇))       (𝜑 → ((𝑀𝑋) (𝑀𝑌)) = ((𝑀𝑍) (𝑀𝑇)))
 
Theoremmirmir2 26388 Point inversion of a point inversion through another point. (Contributed by Thierry Arnoux, 3-Nov-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝑆 = (pInvG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   𝑀 = (𝑆𝐴)    &   (𝜑𝑋𝑃)    &   (𝜑𝑌𝑃)       (𝜑 → (𝑀‘((𝑆𝑌)‘𝑋)) = ((𝑆‘(𝑀𝑌))‘(𝑀𝑋)))
 
Theoremmirmot 26389 Point investion is a motion of the geometric space. Theorem 7.14 of [Schwabhauser] p. 51. (Contributed by Thierry Arnoux, 15-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝑆 = (pInvG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   𝑀 = (𝑆𝐴)    &   (𝜑𝐴𝑃)       (𝜑𝑀 ∈ (𝐺Ismt𝐺))
 
Theoremmirln 26390 If two points are on the same line, so is the mirror point of one through the other. (Contributed by Thierry Arnoux, 21-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝑆 = (pInvG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   𝑀 = (𝑆𝐴)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐴𝐷)    &   (𝜑𝐵𝐷)       (𝜑 → (𝑀𝐵) ∈ 𝐷)
 
Theoremmirln2 26391 If a point and its mirror point are both on the same line, so is the center of the point inversion. (Contributed by Thierry Arnoux, 3-Mar-2020.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝑆 = (pInvG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   𝑀 = (𝑆𝐴)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝐷)    &   (𝜑 → (𝑀𝐵) ∈ 𝐷)       (𝜑𝐴𝐷)
 
Theoremmirconn 26392 Point inversion of connectedness. (Contributed by Thierry Arnoux, 2-Mar-2020.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝑆 = (pInvG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   𝑀 = (𝑆𝐴)    &   (𝜑𝐴𝑃)    &   (𝜑𝑋𝑃)    &   (𝜑𝑌𝑃)    &   (𝜑 → (𝑋 ∈ (𝐴𝐼𝑌) ∨ 𝑌 ∈ (𝐴𝐼𝑋)))       (𝜑𝐴 ∈ (𝑋𝐼(𝑀𝑌)))
 
Theoremmirhl 26393 If two points 𝑋 and 𝑌 are on the same half-line from 𝑍, the same applies to the mirror points. (Contributed by Thierry Arnoux, 21-Feb-2020.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝑆 = (pInvG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   𝑀 = (𝑆𝐴)    &   𝐾 = (hlG‘𝐺)    &   (𝜑𝐴𝑃)    &   (𝜑𝑋𝑃)    &   (𝜑𝑌𝑃)    &   (𝜑𝑍𝑃)    &   (𝜑𝑋(𝐾𝑍)𝑌)       (𝜑 → (𝑀𝑋)(𝐾‘(𝑀𝑍))(𝑀𝑌))
 
Theoremmirbtwnhl 26394 If the center of the point inversion 𝐴 is between two points 𝑋 and 𝑌, then the half lines are mirrored. (Contributed by Thierry Arnoux, 3-Mar-2020.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝑆 = (pInvG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   𝑀 = (𝑆𝐴)    &   𝐾 = (hlG‘𝐺)    &   (𝜑𝐴𝑃)    &   (𝜑𝑋𝑃)    &   (𝜑𝑌𝑃)    &   (𝜑𝑍𝑃)    &   (𝜑𝑋𝐴)    &   (𝜑𝑌𝐴)    &   (𝜑𝐴 ∈ (𝑋𝐼𝑌))       (𝜑 → (𝑍(𝐾𝐴)𝑋 ↔ (𝑀𝑍)(𝐾𝐴)𝑌))
 
Theoremmirhl2 26395 Deduce half-line relation from mirror point. (Contributed by Thierry Arnoux, 8-Aug-2020.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝑆 = (pInvG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   𝑀 = (𝑆𝐴)    &   𝐾 = (hlG‘𝐺)    &   (𝜑𝐴𝑃)    &   (𝜑𝑋𝑃)    &   (𝜑𝑌𝑃)    &   (𝜑𝑍𝑃)    &   (𝜑𝑋𝐴)    &   (𝜑𝑌𝐴)    &   (𝜑𝐴 ∈ (𝑋𝐼(𝑀𝑌)))       (𝜑𝑋(𝐾𝐴)𝑌)
 
Theoremmircgrextend 26396 Link congruence over a pair of mirror points. cf tgcgrextend 26199. (Contributed by Thierry Arnoux, 4-Oct-2020.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝑆 = (pInvG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &    = (cgrG‘𝐺)    &   𝑀 = (𝑆𝐵)    &   𝑁 = (𝑆𝑌)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝑋𝑃)    &   (𝜑𝑌𝑃)    &   (𝜑 → (𝐴 𝐵) = (𝑋 𝑌))       (𝜑 → (𝐴 (𝑀𝐴)) = (𝑋 (𝑁𝑋)))
 
Theoremmirtrcgr 26397 Point inversion of one point of a triangle around another point preserves triangle congruence. (Contributed by Thierry Arnoux, 4-Oct-2020.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝑆 = (pInvG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &    = (cgrG‘𝐺)    &   𝑀 = (𝑆𝐵)    &   𝑁 = (𝑆𝑌)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝑋𝑃)    &   (𝜑𝑌𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝑍𝑃)    &   (𝜑𝐴𝐵)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ⟨“𝑋𝑌𝑍”⟩)       (𝜑 → ⟨“(𝑀𝐴)𝐵𝐶”⟩ ⟨“(𝑁𝑋)𝑌𝑍”⟩)
 
Theoremmirauto 26398 Point inversion preserves point inversion. (Contributed by Thierry Arnoux, 30-Jul-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝑆 = (pInvG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   𝑀 = (𝑆𝑇)    &   𝑋 = (𝑀𝐴)    &   𝑌 = (𝑀𝐵)    &   𝑍 = (𝑀𝐶)    &   (𝜑𝑇𝑃)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑 → ((𝑆𝐴)‘𝐵) = 𝐶)       (𝜑 → ((𝑆𝑋)‘𝑌) = 𝑍)
 
Theoremmiduniq 26399 Uniqueness of the middle point, expressed with point inversion. Theorem 7.17 of [Schwabhauser] p. 51. (Contributed by Thierry Arnoux, 30-Jul-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝑆 = (pInvG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝑋𝑃)    &   (𝜑𝑌𝑃)    &   (𝜑 → ((𝑆𝐴)‘𝑋) = 𝑌)    &   (𝜑 → ((𝑆𝐵)‘𝑋) = 𝑌)       (𝜑𝐴 = 𝐵)
 
Theoremmiduniq1 26400 Uniqueness of the middle point, expressed with point inversion. Theorem 7.18 of [Schwabhauser] p. 52. (Contributed by Thierry Arnoux, 30-Jul-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝑆 = (pInvG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝑋𝑃)    &   (𝜑 → ((𝑆𝐴)‘𝑋) = ((𝑆𝐵)‘𝑋))       (𝜑𝐴 = 𝐵)
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