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Theorem List for Metamath Proof Explorer - 26401-26500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremhlid 26401 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 26402 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 26403 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 26404 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 26405 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 26406 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 26407 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 26408* 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 26409 Lemma for hlcgreu 26410. (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 26410* The point constructed in hlcgrex 26408 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 26411 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 26412 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 26413 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 26414 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 26415 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 26416 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 26417 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 26418 Non-colinear points are different. (Contributed by Thierry Arnoux, 8-Aug-2019.) TODO (NM): maybe ncolne2 26418 could be simplified out and deleted, replaced by ncolcom 26353.
𝐵 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &   (𝜑 → ¬ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))       (𝜑𝑋𝑍)

Theoremtgisline 26419* 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 26420 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 26421 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 26422 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 26423 Transitivity law for lines, one half of tglineelsb2 26424. (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 26424 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 26425 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 26426 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 26427 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 26428 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 26429* 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 26430* 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 26431* 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 26432 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 26433* 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 26434* 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 26435 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 26436 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 26437 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 26438 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 26439 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 26440 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 26441* 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 26442* 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 26443* 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 26444 Declare the constant for the point inversion function.
class pInvG

Definitiondf-mir 26445* Define the point inversion ("mirror") function. Definition 7.5 of [Schwabhauser] p. 49. See mirval 26447 and ismir 26451. (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 26446* 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 26447* 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 26448* 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 26449 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 26450 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 26451 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 26452 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 26453 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 26454 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 26455 Variation on mirmir 26454. (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 26456* 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 26457 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 26458 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 26459 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 26460 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 26461 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 26462 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 26463 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 26464 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 26465 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 26466 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 26467 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 26468 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 26469 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 26470 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 26471 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 26472 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 26473 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 26474 Link congruence over a pair of mirror points. cf tgcgrextend 26277. (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 26475 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 26476 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 26477 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 26478 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)

Theoremmiduniq2 26479 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 26480 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 26481 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 26482 Lemma for krippen 26483. 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 26483 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 26484* 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 26530 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 26485 Declare the constant for the class of right angles.
class ∟G

Definitiondf-rag 26486* Define the class of right angles. Definition 8.1 of [Schwabhauser] p. 57. See israg 26489. (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 26487 Declare the constant for the perpendicular relation.
class ⟂G

Definitiondf-perpg 26488* Define the "perpendicular" relation. Definition 8.11 of [Schwabhauser] p. 59. See isperp 26504. (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 26489 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 26490 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 26491 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 26492 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 26493 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 26494 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 26495 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 26496 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 26497 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 26498 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 26499 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 26500 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))

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