HomeHome Metamath Proof Explorer
Theorem List (p. 256 of 429)
< Previous  Next >
Bad symbols? Try the
GIF version.

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

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-27903)
  Hilbert Space Explorer  Hilbert Space Explorer
(27904-29428)
  Users' Mathboxes  Users' Mathboxes
(29429-42879)
 

Theorem List for Metamath Proof Explorer - 25501-25600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremncolcom 25501 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 25502 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 25503 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 25504 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 25505 If there are 3 non-colinear points, dimension must be 2 or more. tglowdim2l 25590 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 25506 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 25507* 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 25508 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 25509 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 25510 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 25511 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 25512 Lemma for tgbtwnconn1 25515. (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 25513 Lemma for tgbtwnconn1 25515. (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 25514 Lemma for tgbtwnconn1 25515. (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 25515 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 25516 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 25517 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 25518 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 25519 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 25520 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 25521 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 25522 Less-than relation for geometric congruences.
class ≤G
 
Definitiondf-leg 25523* Define the less-than relationship between geometric distance congruence classes. See legval 25524. (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 25524* 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 25525* 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 25526* 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 25527 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 25528 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 25529 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 25530 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 25531 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 25532 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 25533 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 25534 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 25535 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 25536* 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 25537 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 25538 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 25539 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 25540 Function producing the relation "belong to the same half-line".
class hlG
 
Definitiondf-hlg 25541* 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 25542 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 25543 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 25544 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 25545 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 25546 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 25547 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 25548 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 25549 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 25550 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 25551 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 25552 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 25553 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 25554 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 25555 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 25556* 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 25557 Lemma for hlcgreu 25558. (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 25558* The point constructed in hlcgrex 25556 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 25559 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 25560 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 25561 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 25562 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 25563 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 25564 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 25565 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 25566 Non-colinear points are different. (Contributed by Thierry Arnoux, 8-Aug-2019.) TODO (NM): maybe ncolne2 25566 could be simplified out and deleted, replaced by ncolcom 25501.
𝐵 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &   (𝜑 → ¬ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))       (𝜑𝑋𝑍)
 
Theoremtgisline 25567* 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 25568 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 25569 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 25570 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 25571 Transitivity law for lines, one half of tglineelsb2 25572. (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 25572 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 25573 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 25574 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 25575 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 25576 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 25577* 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 25578* 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 25579* 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 25580 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 25581* 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 25582* 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 25583 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 25584 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 25585 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 25586 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 25587 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 25588 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 25589* 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 25590* Reformulation of the lower dimension axiom for dimension 2. 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 25591* 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 25592 Declare the constant for the point inversion function.
class pInvG
 
Definitiondf-mir 25593* Define the point inversion ("mirror") function. Definition 7.5 of [Schwabhauser] p. 49. See mirval 25595 and ismir 25599. (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 25594* 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 25595* 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 25596* 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 25597 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 25598 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 25599 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 25600 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)
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42879
  Copyright terms: Public domain < Previous  Next >