HomeHome Metamath Proof Explorer
Theorem List (p. 253 of 425)
< 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
(1-26947)
  Hilbert Space Explorer  Hilbert Space Explorer
(26948-28472)
  Users' Mathboxes  Users' Mathboxes
(28473-42426)
 

Theorem List for Metamath Proof Explorer - 25201-25300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremhlcgrex 25201* 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 25202 Lemma for hlcgreu 25203. (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 25203* The point constructed in hlcgrex 25201 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 25204 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 25205 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 25206 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 25207 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 25208 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 25209 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 25210 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 25211 Non-colinear points are different. (Contributed by Thierry Arnoux, 8-Aug-2019.) TODO (NM): maybe ncolne2 25211 could be simplified out and deleted, replaced by ncolcom 25146.
𝐵 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &   (𝜑 → ¬ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))       (𝜑𝑋𝑍)
 
Theoremtgisline 25212* 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 25213 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 25214 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 25215 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 25216 Transitivity law for lines, one half of tglineelsb2 25217. (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 25217 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 25218 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 25219 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 25220 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 25221 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 25222* 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 25223* 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 25224* 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 25225 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 25226* 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 25227* 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 25228 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 25229 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 25230 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 25231 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 25232 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 25233 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 25234* 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 25235* 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 25236* 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 25237 Declare the constant for the point inversion function.
class pInvG
 
Definitiondf-mir 25238* Define the point inversion ("mirror") function. Definition 7.5 of [Schwabhauser] p. 49. See mirval 25240 and ismir 25244. (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 25239* 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 25240* 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 25241* 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 25242 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 25243 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 25244 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 25245 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 25246 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 25247 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 25248 Variation on mirmir 25247. (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 25249* 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 25250 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 25251 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 25252 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 25253 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 25254 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 25255 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 25256 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 25257 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 25258 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 25259 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 25260 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 25261 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 25262 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 25263 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 25264 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 25265 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 25266 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 25267 Link congruence over a pair of mirror points. cf tgcgrextend 25069. (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 25268 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 25269 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 25270 Unicity 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 25271 Unicity 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)
 
Theoremmiduniq2 25272 If two point inversions commute, they are identical. Theorem 7.19 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)‘((𝑆𝐵)‘𝑋)) = ((𝑆𝐵)‘((𝑆𝐴)‘𝑋)))       (𝜑𝐴 = 𝐵)
 
Theoremcolmid 25273 Colinearity and equidistance implies midpoint. Theorem 7.20 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)    &   𝑀 = (𝑆𝑋)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝑋𝑃)    &   (𝜑 → (𝑋 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))    &   (𝜑 → (𝑋 𝐴) = (𝑋 𝐵))       (𝜑 → (𝐵 = (𝑀𝐴) ∨ 𝐴 = 𝐵))
 
Theoremsymquadlem 25274 Lemma of the symetrial quadrilateral. The diagonals of quadrilaterals with congruent opposing sides intersect at their middle point. In Euclidean geometry, such quadrilaterals are called parallelograms, as opposing sides are parallel. However, this is not necessarily true in the case of absolute geometry. Lemma 7.21 of [Schwabhauser] p. 52. (Contributed by Thierry Arnoux, 6-Aug-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝑆 = (pInvG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   𝑀 = (𝑆𝑋)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝑋𝑃)    &   (𝜑 → ¬ (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶))    &   (𝜑𝐵𝐷)    &   (𝜑 → (𝐴 𝐵) = (𝐶 𝐷))    &   (𝜑 → (𝐵 𝐶) = (𝐷 𝐴))    &   (𝜑 → (𝑋 ∈ (𝐴𝐿𝐶) ∨ 𝐴 = 𝐶))    &   (𝜑 → (𝑋 ∈ (𝐵𝐿𝐷) ∨ 𝐵 = 𝐷))       (𝜑𝐴 = (𝑀𝐶))
 
Theoremkrippenlem 25275 Lemma for krippen 25276. We can assume krippen.7 "without loss of generality" (Contributed by Thierry Arnoux, 12-Aug-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝑆 = (pInvG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   𝑀 = (𝑆𝑋)    &   𝑁 = (𝑆𝑌)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑𝑋𝑃)    &   (𝜑𝑌𝑃)    &   (𝜑𝐶 ∈ (𝐴𝐼𝐸))    &   (𝜑𝐶 ∈ (𝐵𝐼𝐹))    &   (𝜑 → (𝐶 𝐴) = (𝐶 𝐵))    &   (𝜑 → (𝐶 𝐸) = (𝐶 𝐹))    &   (𝜑𝐵 = (𝑀𝐴))    &   (𝜑𝐹 = (𝑁𝐸))    &    = (≤G‘𝐺)    &   (𝜑 → (𝐶 𝐴) (𝐶 𝐸))       (𝜑𝐶 ∈ (𝑋𝐼𝑌))
 
Theoremkrippen 25276 Krippenlemma (German for crib's lemma) Lemma 7.22 of [Schwabhauser] p. 53. proven by Gupta 1965 as Theorem 3.45. (Contributed by Thierry Arnoux, 12-Aug-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝑆 = (pInvG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   𝑀 = (𝑆𝑋)    &   𝑁 = (𝑆𝑌)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑𝑋𝑃)    &   (𝜑𝑌𝑃)    &   (𝜑𝐶 ∈ (𝐴𝐼𝐸))    &   (𝜑𝐶 ∈ (𝐵𝐼𝐹))    &   (𝜑 → (𝐶 𝐴) = (𝐶 𝐵))    &   (𝜑 → (𝐶 𝐸) = (𝐶 𝐹))    &   (𝜑𝐵 = (𝑀𝐴))    &   (𝜑𝐹 = (𝑁𝐸))       (𝜑𝐶 ∈ (𝑋𝐼𝑌))
 
Theoremmidexlem 25277* Lemma for the existence of a middle point. Lemma 7.25 of [Schwabhauser] p. 55. This proof of the existence of a midpoint requires the existence of a third point 𝐶 equidistant to 𝐴 and 𝐵 This condition will be removed later. Because the operation notation (𝐴(midG‘𝐺)𝐵) for a midpoint implies its uniqueness, it cannot be used until uniqueness is proven, and until then, an equivalent mirror point notation 𝐵 = (𝑀𝐴) has to be used. See mideu 25320 for the existence and uniqueness of the midpoint. (Contributed by Thierry Arnoux, 25-Aug-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝑆 = (pInvG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   𝑀 = (𝑆𝑥)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑 → (𝐶 𝐴) = (𝐶 𝐵))       (𝜑 → ∃𝑥𝑃 𝐵 = (𝑀𝐴))
 
15.2.13  Right angles
 
Syntaxcrag 25278 Declare the constant for the class of right angles.
class ∟G
 
Definitiondf-rag 25279* Define the class of right angles. Definition 8.1 of [Schwabhauser] p. 57. See israg 25282. (Contributed by Thierry Arnoux, 25-Aug-2019.)
∟G = (𝑔 ∈ V ↦ {𝑤 ∈ Word (Base‘𝑔) ∣ ((#‘𝑤) = 3 ∧ ((𝑤‘0)(dist‘𝑔)(𝑤‘2)) = ((𝑤‘0)(dist‘𝑔)(((pInvG‘𝑔)‘(𝑤‘1))‘(𝑤‘2))))})
 
Syntaxcperpg 25280 Declare the constant for the perpendicular relation.
class ⟂G
 
Definitiondf-perpg 25281* Define the "perpendicular" relation. Definition 8.11 of [Schwabhauser] p. 59. See isperp 25297. (Contributed by Thierry Arnoux, 8-Sep-2019.)
⟂G = (𝑔 ∈ V ↦ {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ran (LineG‘𝑔) ∧ 𝑏 ∈ ran (LineG‘𝑔)) ∧ ∃𝑥 ∈ (𝑎𝑏)∀𝑢𝑎𝑣𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝑔))})
 
Theoremisrag 25282 Property for 3 points A, B, C to form a right angle. Definition 8.1 of [Schwabhauser] p. 57. (Contributed by Thierry Arnoux, 25-Aug-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝑆 = (pInvG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)       (𝜑 → (⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺) ↔ (𝐴 𝐶) = (𝐴 ((𝑆𝐵)‘𝐶))))
 
Theoremragcom 25283 Commutative rule for right angles. Theorem 8.2 of [Schwabhauser] p. 57. (Contributed by Thierry Arnoux, 25-Aug-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝑆 = (pInvG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))       (𝜑 → ⟨“𝐶𝐵𝐴”⟩ ∈ (∟G‘𝐺))
 
Theoremragcol 25284 The right angle property is independent of the choice of point on one side. Theorem 8.3 of [Schwabhauser] p. 58. (Contributed by Thierry Arnoux, 25-Aug-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝑆 = (pInvG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))    &   (𝜑𝐴𝐵)    &   (𝜑 → (𝐴 ∈ (𝐵𝐿𝐷) ∨ 𝐵 = 𝐷))       (𝜑 → ⟨“𝐷𝐵𝐶”⟩ ∈ (∟G‘𝐺))
 
Theoremragmir 25285 Right angle property is preserved by point inversion. Theorem 8.4 of [Schwabhauser] p. 58. (Contributed by Thierry Arnoux, 25-Aug-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝑆 = (pInvG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))       (𝜑 → ⟨“𝐴𝐵((𝑆𝐵)‘𝐶)”⟩ ∈ (∟G‘𝐺))
 
Theoremmirrag 25286 Right angle is conserved by point inversion. (Contributed by Thierry Arnoux, 3-Nov-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝑆 = (pInvG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))    &   𝑀 = (𝑆𝐷)    &   (𝜑𝐷𝑃)       (𝜑 → ⟨“(𝑀𝐴)(𝑀𝐵)(𝑀𝐶)”⟩ ∈ (∟G‘𝐺))
 
Theoremragtrivb 25287 Trivial right angle. Theorem 8.5 of [Schwabhauser] p. 58. (Contributed by Thierry Arnoux, 25-Aug-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝑆 = (pInvG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)       (𝜑 → ⟨“𝐴𝐵𝐵”⟩ ∈ (∟G‘𝐺))
 
Theoremragflat2 25288 Deduce equality from two right angles. Theorem 8.6 of [Schwabhauser] p. 58. (Contributed by Thierry Arnoux, 3-Sep-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝑆 = (pInvG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))    &   (𝜑 → ⟨“𝐷𝐵𝐶”⟩ ∈ (∟G‘𝐺))    &   (𝜑𝐶 ∈ (𝐴𝐼𝐷))       (𝜑𝐵 = 𝐶)
 
Theoremragflat 25289 Deduce equality from two right angles. Theorem 8.7 of [Schwabhauser] p. 58. (Contributed by Thierry Arnoux, 3-Sep-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝑆 = (pInvG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))    &   (𝜑 → ⟨“𝐴𝐶𝐵”⟩ ∈ (∟G‘𝐺))       (𝜑𝐵 = 𝐶)
 
Theoremragtriva 25290 Trivial right angle. Theorem 8.8 of [Schwabhauser] p. 58. (Contributed by Thierry Arnoux, 3-Sep-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝑆 = (pInvG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑 → ⟨“𝐴𝐵𝐴”⟩ ∈ (∟G‘𝐺))       (𝜑𝐴 = 𝐵)
 
Theoremragflat3 25291 Right angle and colinearity. Theorem 8.9 of [Schwabhauser] p. 58. (Contributed by Thierry Arnoux, 4-Sep-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝑆 = (pInvG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))    &   (𝜑 → (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))       (𝜑 → (𝐴 = 𝐵𝐶 = 𝐵))
 
Theoremragcgr 25292 Right angle and colinearity. Theorem 8.10 of [Schwabhauser] p. 58. (Contributed by Thierry Arnoux, 4-Sep-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝑆 = (pInvG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &    = (cgrG‘𝐺)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ⟨“𝐷𝐸𝐹”⟩)       (𝜑 → ⟨“𝐷𝐸𝐹”⟩ ∈ (∟G‘𝐺))
 
Theoremmotrag 25293 Right angles are preserved by motions. (Contributed by Thierry Arnoux, 16-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝑆 = (pInvG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐹 ∈ (𝐺Ismt𝐺))    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))       (𝜑 → ⟨“(𝐹𝐴)(𝐹𝐵)(𝐹𝐶)”⟩ ∈ (∟G‘𝐺))
 
Theoremragncol 25294 Right angle implies non-colinearity. A consequence of theorem 8.9 of [Schwabhauser] p. 58. (Contributed by Thierry Arnoux, 1-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝑆 = (pInvG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))    &   (𝜑𝐴𝐵)    &   (𝜑𝐶𝐵)       (𝜑 → ¬ (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))
 
Theoremperpln1 25295 Derive a line from perpendicularity. (Contributed by Thierry Arnoux, 27-Nov-2019.)
𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴(⟂G‘𝐺)𝐵)       (𝜑𝐴 ∈ ran 𝐿)
 
Theoremperpln2 25296 Derive a line from perpendicularity. (Contributed by Thierry Arnoux, 27-Nov-2019.)
𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴(⟂G‘𝐺)𝐵)       (𝜑𝐵 ∈ ran 𝐿)
 
Theoremisperp 25297* Property for 2 lines A, B to be perpendicular. Item (ii) of definition 8.11 of [Schwabhauser] p. 59. (Contributed by Thierry Arnoux, 16-Oct-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴 ∈ ran 𝐿)    &   (𝜑𝐵 ∈ ran 𝐿)       (𝜑 → (𝐴(⟂G‘𝐺)𝐵 ↔ ∃𝑥 ∈ (𝐴𝐵)∀𝑢𝐴𝑣𝐵 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)))
 
Theoremperpcom 25298 The "perpendicular" relation commutes. Theorem 8.12 of [Schwabhauser] p. 59. (Contributed by Thierry Arnoux, 16-Oct-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴 ∈ ran 𝐿)    &   (𝜑𝐵 ∈ ran 𝐿)    &   (𝜑𝐴(⟂G‘𝐺)𝐵)       (𝜑𝐵(⟂G‘𝐺)𝐴)
 
Theoremperpneq 25299 Two perpendicular lines are different. Theorem 8.14 of [Schwabhauser] p. 59. (Contributed by Thierry Arnoux, 18-Oct-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴 ∈ ran 𝐿)    &   (𝜑𝐵 ∈ ran 𝐿)    &   (𝜑𝐴(⟂G‘𝐺)𝐵)       (𝜑𝐴𝐵)
 
Theoremisperp2 25300* Property for 2 lines A, B, intersecting at a point X to be perpendicular. Item (i) of definition 8.13 of [Schwabhauser] p. 59. (Contributed by Thierry Arnoux, 16-Oct-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴 ∈ ran 𝐿)    &   (𝜑𝐵 ∈ ran 𝐿)    &   (𝜑𝑋 ∈ (𝐴𝐵))       (𝜑 → (𝐴(⟂G‘𝐺)𝐵 ↔ ∀𝑢𝐴𝑣𝐵 ⟨“𝑢𝑋𝑣”⟩ ∈ (∟G‘𝐺)))
    < 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 42401-42426
  Copyright terms: Public domain < Previous  Next >