P. 288 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 28701-28800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	hlcgreulem 28701	Lemma for hlcgreu 28702. (Contributed by Thierry Arnoux, 9-Aug-2020.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ 𝐾 = (hlG‘𝐺)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑃)    &   ⊢ − = (dist‘𝐺)    &   ⊢ (𝜑 → 𝐷 ≠ 𝐴)    &   ⊢ (𝜑 → 𝐵 ≠ 𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑋(𝐾‘𝐴)𝐷)    &   ⊢ (𝜑 → 𝑌(𝐾‘𝐴)𝐷)    &   ⊢ (𝜑 → (𝐴 − 𝑋) = (𝐵 − 𝐶))    &   ⊢ (𝜑 → (𝐴 − 𝑌) = (𝐵 − 𝐶))    ⇒   ⊢ (𝜑 → 𝑋 = 𝑌)
Theorem	hlcgreu 28702*	The point constructed in hlcgrex 28700 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)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑃)    &   ⊢ − = (dist‘𝐺)    &   ⊢ (𝜑 → 𝐷 ≠ 𝐴)    &   ⊢ (𝜑 → 𝐵 ≠ 𝐶)    ⇒   ⊢ (𝜑 → ∃!𝑥 ∈ 𝑃 (𝑥(𝐾‘𝐴)𝐷 ∧ (𝐴 − 𝑥) = (𝐵 − 𝐶)))
16.2.11  Lines
Theorem	btwnlng1 28703	Betweenness implies colinearity. (Contributed by Thierry Arnoux, 28-Mar-2019.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑋 ≠ 𝑌)    &   ⊢ (𝜑 → 𝑍 ∈ (𝑋𝐼𝑌))    ⇒   ⊢ (𝜑 → 𝑍 ∈ (𝑋𝐿𝑌))
Theorem	btwnlng2 28704	Betweenness implies colinearity. (Contributed by Thierry Arnoux, 28-Mar-2019.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑋 ≠ 𝑌)    &   ⊢ (𝜑 → 𝑋 ∈ (𝑍𝐼𝑌))    ⇒   ⊢ (𝜑 → 𝑍 ∈ (𝑋𝐿𝑌))
Theorem	btwnlng3 28705	Betweenness implies colinearity. (Contributed by Thierry Arnoux, 28-Mar-2019.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑋 ≠ 𝑌)    &   ⊢ (𝜑 → 𝑌 ∈ (𝑋𝐼𝑍))    ⇒   ⊢ (𝜑 → 𝑍 ∈ (𝑋𝐿𝑌))
Theorem	lncom 28706	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑋 ≠ 𝑌)    &   ⊢ (𝜑 → 𝑍 ∈ (𝑌𝐿𝑋))    ⇒   ⊢ (𝜑 → 𝑍 ∈ (𝑋𝐿𝑌))
Theorem	lnrot1 28707	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑋 ≠ 𝑌)    &   ⊢ (𝜑 → 𝑌 ∈ (𝑍𝐿𝑋))    &   ⊢ (𝜑 → 𝑍 ≠ 𝑋)    ⇒   ⊢ (𝜑 → 𝑍 ∈ (𝑋𝐿𝑌))
Theorem	lnrot2 28708	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑋 ≠ 𝑌)    &   ⊢ (𝜑 → 𝑋 ∈ (𝑌𝐿𝑍))    &   ⊢ (𝜑 → 𝑌 ≠ 𝑍)    ⇒   ⊢ (𝜑 → 𝑍 ∈ (𝑋𝐿𝑌))
Theorem	ncolne1 28709	Non-colinear points are different. (Contributed by Thierry Arnoux, 8-Aug-2019.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ (𝜑 → ¬ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))    ⇒   ⊢ (𝜑 → 𝑋 ≠ 𝑌)
Theorem	ncolne2 28710	Non-colinear points are different. (Contributed by Thierry Arnoux, 8-Aug-2019.) TODO (NM): maybe ncolne2 28710 could be simplified out and deleted, replaced by ncolcom 28645.
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ (𝜑 → ¬ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))    ⇒   ⊢ (𝜑 → 𝑋 ≠ 𝑍)
Theorem	tgisline 28711*	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ ran 𝐿)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 (𝐴 = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦))
Theorem	tglnne 28712	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → (𝑋𝐿𝑌) ∈ ran 𝐿)    ⇒   ⊢ (𝜑 → 𝑋 ≠ 𝑌)
Theorem	tglndim0 28713	There are no lines in dimension 0. (Contributed by Thierry Arnoux, 18-Oct-2019.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → (♯‘𝐵) = 1)    ⇒   ⊢ (𝜑 → ¬ 𝐴 ∈ ran 𝐿)
Theorem	tgelrnln 28714	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑋 ≠ 𝑌)    ⇒   ⊢ (𝜑 → (𝑋𝐿𝑌) ∈ ran 𝐿)
Theorem	tglineeltr 28715	Transitivity law for lines, one half of tglineelsb2 28716. (Contributed by Thierry Arnoux, 25-May-2019.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝑃 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑄 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑃 ≠ 𝑄)    &   ⊢ (𝜑 → 𝑆 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑆 ≠ 𝑃)    &   ⊢ (𝜑 → 𝑆 ∈ (𝑃𝐿𝑄))    &   ⊢ (𝜑 → 𝑅 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑅 ∈ (𝑃𝐿𝑆))    ⇒   ⊢ (𝜑 → 𝑅 ∈ (𝑃𝐿𝑄))
Theorem	tglineelsb2 28716	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝑃 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑄 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑃 ≠ 𝑄)    &   ⊢ (𝜑 → 𝑆 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑆 ≠ 𝑃)    &   ⊢ (𝜑 → 𝑆 ∈ (𝑃𝐿𝑄))    ⇒   ⊢ (𝜑 → (𝑃𝐿𝑄) = (𝑃𝐿𝑆))
Theorem	tglinerflx1 28717	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝑃 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑄 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑃 ≠ 𝑄)    ⇒   ⊢ (𝜑 → 𝑃 ∈ (𝑃𝐿𝑄))
Theorem	tglinerflx2 28718	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝑃 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑄 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑃 ≠ 𝑄)    ⇒   ⊢ (𝜑 → 𝑄 ∈ (𝑃𝐿𝑄))
Theorem	tglinecom 28719	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝑃 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑄 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑃 ≠ 𝑄)    ⇒   ⊢ (𝜑 → (𝑃𝐿𝑄) = (𝑄𝐿𝑃))
Theorem	tglinethru 28720	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝑃 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑄 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑃 ≠ 𝑄)    &   ⊢ (𝜑 → 𝑃 ≠ 𝑄)    &   ⊢ (𝜑 → 𝐴 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝑃 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑄 ∈ 𝐴)    ⇒   ⊢ (𝜑 → 𝐴 = (𝑃𝐿𝑄))
Theorem	tghilberti1 28721*	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝑃 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑄 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑃 ≠ 𝑄)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ ran 𝐿(𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥))
Theorem	tghilberti2 28722*	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝑃 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑄 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑃 ≠ 𝑄)    ⇒   ⊢ (𝜑 → ∃*𝑥 ∈ ran 𝐿(𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥))
Theorem	tglinethrueu 28723*	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝑃 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑄 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑃 ≠ 𝑄)    ⇒   ⊢ (𝜑 → ∃!𝑥 ∈ ran 𝐿(𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥))
Theorem	tglnne0 28724	A line 𝐴 has at least one point. (Contributed by Thierry Arnoux, 4-Mar-2020.)
⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ ran 𝐿)    ⇒   ⊢ (𝜑 → 𝐴 ≠ ∅)
Theorem	tglnpt2 28725*	Find a second point on a line. (Contributed by Thierry Arnoux, 18-Oct-2019.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    ⇒   ⊢ (𝜑 → ∃𝑦 ∈ 𝐴 𝑋 ≠ 𝑦)
Theorem	tglineintmo 28726*	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝐵 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝐴 ≠ 𝐵)    ⇒   ⊢ (𝜑 → ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
Theorem	tglineineq 28727	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝐵 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝐴 ≠ 𝐵)    &   ⊢ (𝜑 → 𝑋 ∈ (𝐴 ∩ 𝐵))    &   ⊢ (𝜑 → 𝑌 ∈ (𝐴 ∩ 𝐵))    ⇒   ⊢ (𝜑 → 𝑋 = 𝑌)
Theorem	tglineneq 28728	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑃)    &   ⊢ (𝜑 → ¬ (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶))    ⇒   ⊢ (𝜑 → (𝐴𝐿𝐵) ≠ (𝐶𝐿𝐷))
Theorem	tglineinteq 28729	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑃)    &   ⊢ (𝜑 → ¬ (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶))    &   ⊢ (𝜑 → 𝑋 ∈ (𝐴𝐿𝐵))    &   ⊢ (𝜑 → 𝑌 ∈ (𝐴𝐿𝐵))    &   ⊢ (𝜑 → 𝑋 ∈ (𝐶𝐿𝐷))    &   ⊢ (𝜑 → 𝑌 ∈ (𝐶𝐿𝐷))    ⇒   ⊢ (𝜑 → 𝑋 = 𝑌)
Theorem	ncolncol 28730	Deduce non-colinearity from non-colinearity and colinearity. (Contributed by Thierry Arnoux, 27-Aug-2019.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑃)    &   ⊢ (𝜑 → ¬ (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶))    &   ⊢ (𝜑 → 𝐷 ∈ (𝐴𝐿𝐵))    &   ⊢ (𝜑 → 𝐷 ≠ 𝐵)    ⇒   ⊢ (𝜑 → ¬ (𝐷 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶))
Theorem	coltr 28731	A transitivity law for colinearity. (Contributed by Thierry Arnoux, 27-Nov-2019.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐴 ∈ (𝐵𝐿𝐶))    &   ⊢ (𝜑 → (𝐵 ∈ (𝐶𝐿𝐷) ∨ 𝐶 = 𝐷))    ⇒   ⊢ (𝜑 → (𝐴 ∈ (𝐶𝐿𝐷) ∨ 𝐶 = 𝐷))
Theorem	coltr3 28732	A transitivity law for colinearity. (Contributed by Thierry Arnoux, 27-Nov-2019.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐴 ∈ (𝐵𝐿𝐶))    &   ⊢ (𝜑 → 𝐷 ∈ (𝐴𝐼𝐶))    ⇒   ⊢ (𝜑 → 𝐷 ∈ (𝐵𝐿𝐶))
Theorem	colline 28733*	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑃)    &   ⊢ (𝜑 → 2 ≤ (♯‘𝑃))    ⇒   ⊢ (𝜑 → ((𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍) ↔ ∃𝑎 ∈ ran 𝐿(𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎 ∧ 𝑍 ∈ 𝑎)))
Theorem	tglowdim2l 28734*	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐺DimTarskiG≥2)    ⇒   ⊢ (𝜑 → ∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 ∃𝑐 ∈ 𝑃 ¬ (𝑐 ∈ (𝑎𝐿𝑏) ∨ 𝑎 = 𝑏))
Theorem	tglowdim2ln 28735*	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐺DimTarskiG≥2)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐴 ≠ 𝐵)    ⇒   ⊢ (𝜑 → ∃𝑐 ∈ 𝑃 ¬ 𝑐 ∈ (𝐴𝐿𝐵))
16.2.12  Point inversions
Syntax	cmir 28736	Declare the constant for the point inversion function.
class pInvG
Definition	df-mir 28737*	Define the point inversion ("mirror") function. Definition 7.5 of [Schwabhauser] p. 49. See mirval 28739 and ismir 28743. (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))))))
Theorem	mirreu3 28738*	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑀 ∈ 𝑃)    ⇒   ⊢ (𝜑 → ∃!𝑏 ∈ 𝑃 ((𝑀 − 𝑏) = (𝑀 − 𝐴) ∧ 𝑀 ∈ (𝑏𝐼𝐴)))
Theorem	mirval 28739*	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    ⇒   ⊢ (𝜑 → (𝑆‘𝐴) = (𝑦 ∈ 𝑃 ↦ (℩𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦)))))
Theorem	mirfv 28740*	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ 𝑀 = (𝑆‘𝐴)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    ⇒   ⊢ (𝜑 → (𝑀‘𝐵) = (℩𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))))
Theorem	mircgr 28741	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ 𝑀 = (𝑆‘𝐴)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    ⇒   ⊢ (𝜑 → (𝐴 − (𝑀‘𝐵)) = (𝐴 − 𝐵))
Theorem	mirbtwn 28742	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ 𝑀 = (𝑆‘𝐴)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ((𝑀‘𝐵)𝐼𝐵))
Theorem	ismir 28743	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ 𝑀 = (𝑆‘𝐴)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑃)    &   ⊢ (𝜑 → (𝐴 − 𝐶) = (𝐴 − 𝐵))    &   ⊢ (𝜑 → 𝐴 ∈ (𝐶𝐼𝐵))    ⇒   ⊢ (𝜑 → 𝐶 = (𝑀‘𝐵))
Theorem	mirf 28744	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ 𝑀 = (𝑆‘𝐴)    ⇒   ⊢ (𝜑 → 𝑀:𝑃⟶𝑃)
Theorem	mircl 28745	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ 𝑀 = (𝑆‘𝐴)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑃)    ⇒   ⊢ (𝜑 → (𝑀‘𝑋) ∈ 𝑃)
Theorem	mirmir 28746	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ 𝑀 = (𝑆‘𝐴)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    ⇒   ⊢ (𝜑 → (𝑀‘(𝑀‘𝐵)) = 𝐵)
Theorem	mircom 28747	Variation on mirmir 28746. (Contributed by Thierry Arnoux, 10-Nov-2019.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ − = (dist‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ 𝑆 = (pInvG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ 𝑀 = (𝑆‘𝐴)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → (𝑀‘𝐵) = 𝐶)    ⇒   ⊢ (𝜑 → (𝑀‘𝐶) = 𝐵)
Theorem	mirreu 28748*	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ 𝑀 = (𝑆‘𝐴)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    ⇒   ⊢ (𝜑 → ∃!𝑎 ∈ 𝑃 (𝑀‘𝑎) = 𝐵)
Theorem	mireq 28749	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ 𝑀 = (𝑆‘𝐴)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑃)    &   ⊢ (𝜑 → (𝑀‘𝐵) = (𝑀‘𝐶))    ⇒   ⊢ (𝜑 → 𝐵 = 𝐶)
Theorem	mirinv 28750	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ 𝑀 = (𝑆‘𝐴)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    ⇒   ⊢ (𝜑 → ((𝑀‘𝐵) = 𝐵 ↔ 𝐴 = 𝐵))
Theorem	mirne 28751	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ 𝑀 = (𝑆‘𝐴)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ≠ 𝐴)    ⇒   ⊢ (𝜑 → (𝑀‘𝐵) ≠ 𝐴)
Theorem	mircinv 28752	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ 𝑀 = (𝑆‘𝐴)    ⇒   ⊢ (𝜑 → (𝑀‘𝐴) = 𝐴)
Theorem	mirf1o 28753	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ 𝑀 = (𝑆‘𝐴)    ⇒   ⊢ (𝜑 → 𝑀:𝑃–1-1-onto→𝑃)
Theorem	miriso 28754	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ 𝑀 = (𝑆‘𝐴)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑃)    ⇒   ⊢ (𝜑 → ((𝑀‘𝑋) − (𝑀‘𝑌)) = (𝑋 − 𝑌))
Theorem	mirbtwni 28755	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ 𝑀 = (𝑆‘𝐴)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑌 ∈ (𝑋𝐼𝑍))    ⇒   ⊢ (𝜑 → (𝑀‘𝑌) ∈ ((𝑀‘𝑋)𝐼(𝑀‘𝑍)))
Theorem	mirbtwnb 28756	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ 𝑀 = (𝑆‘𝐴)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑃)    ⇒   ⊢ (𝜑 → (𝑌 ∈ (𝑋𝐼𝑍) ↔ (𝑀‘𝑌) ∈ ((𝑀‘𝑋)𝐼(𝑀‘𝑍))))
Theorem	mircgrs 28757	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ 𝑀 = (𝑆‘𝐴)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑇 ∈ 𝑃)    &   ⊢ (𝜑 → (𝑋 − 𝑌) = (𝑍 − 𝑇))    ⇒   ⊢ (𝜑 → ((𝑀‘𝑋) − (𝑀‘𝑌)) = ((𝑀‘𝑍) − (𝑀‘𝑇)))
Theorem	mirmir2 28758	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ 𝑀 = (𝑆‘𝐴)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑃)    ⇒   ⊢ (𝜑 → (𝑀‘((𝑆‘𝑌)‘𝑋)) = ((𝑆‘(𝑀‘𝑌))‘(𝑀‘𝑋)))
Theorem	mirmot 28759	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ 𝑀 = (𝑆‘𝐴)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    ⇒   ⊢ (𝜑 → 𝑀 ∈ (𝐺Ismt𝐺))
Theorem	mirln 28760	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ 𝑀 = (𝑆‘𝐴)    &   ⊢ (𝜑 → 𝐷 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐷)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐷)    ⇒   ⊢ (𝜑 → (𝑀‘𝐵) ∈ 𝐷)
Theorem	mirln2 28761	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ 𝑀 = (𝑆‘𝐴)    &   ⊢ (𝜑 → 𝐷 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐷)    &   ⊢ (𝜑 → (𝑀‘𝐵) ∈ 𝐷)    ⇒   ⊢ (𝜑 → 𝐴 ∈ 𝐷)
Theorem	mirconn 28762	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ 𝑀 = (𝑆‘𝐴)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑃)    &   ⊢ (𝜑 → (𝑋 ∈ (𝐴𝐼𝑌) ∨ 𝑌 ∈ (𝐴𝐼𝑋)))    ⇒   ⊢ (𝜑 → 𝐴 ∈ (𝑋𝐼(𝑀‘𝑌)))
Theorem	mirhl 28763	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ 𝑀 = (𝑆‘𝐴)    &   ⊢ 𝐾 = (hlG‘𝐺)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑋(𝐾‘𝑍)𝑌)    ⇒   ⊢ (𝜑 → (𝑀‘𝑋)(𝐾‘(𝑀‘𝑍))(𝑀‘𝑌))
Theorem	mirbtwnhl 28764	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ 𝑀 = (𝑆‘𝐴)    &   ⊢ 𝐾 = (hlG‘𝐺)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑋 ≠ 𝐴)    &   ⊢ (𝜑 → 𝑌 ≠ 𝐴)    &   ⊢ (𝜑 → 𝐴 ∈ (𝑋𝐼𝑌))    ⇒   ⊢ (𝜑 → (𝑍(𝐾‘𝐴)𝑋 ↔ (𝑀‘𝑍)(𝐾‘𝐴)𝑌))
Theorem	mirhl2 28765	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ 𝑀 = (𝑆‘𝐴)    &   ⊢ 𝐾 = (hlG‘𝐺)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑋 ≠ 𝐴)    &   ⊢ (𝜑 → 𝑌 ≠ 𝐴)    &   ⊢ (𝜑 → 𝐴 ∈ (𝑋𝐼(𝑀‘𝑌)))    ⇒   ⊢ (𝜑 → 𝑋(𝐾‘𝐴)𝑌)
Theorem	mircgrextend 28766	Link congruence over a pair of mirror points. cf tgcgrextend 28569. (Contributed by Thierry Arnoux, 4-Oct-2020.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ − = (dist‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ 𝑆 = (pInvG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ ∼ = (cgrG‘𝐺)    &   ⊢ 𝑀 = (𝑆‘𝐵)    &   ⊢ 𝑁 = (𝑆‘𝑌)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑃)    &   ⊢ (𝜑 → (𝐴 − 𝐵) = (𝑋 − 𝑌))    ⇒   ⊢ (𝜑 → (𝐴 − (𝑀‘𝐴)) = (𝑋 − (𝑁‘𝑋)))
Theorem	mirtrcgr 28767	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ ∼ = (cgrG‘𝐺)    &   ⊢ 𝑀 = (𝑆‘𝐵)    &   ⊢ 𝑁 = (𝑆‘𝑌)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐴 ≠ 𝐵)    &   ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∼ ⟨“𝑋𝑌𝑍”⟩)    ⇒   ⊢ (𝜑 → ⟨“(𝑀‘𝐴)𝐵𝐶”⟩ ∼ ⟨“(𝑁‘𝑋)𝑌𝑍”⟩)
Theorem	mirauto 28768	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ 𝑀 = (𝑆‘𝑇)    &   ⊢ 𝑋 = (𝑀‘𝐴)    &   ⊢ 𝑌 = (𝑀‘𝐵)    &   ⊢ 𝑍 = (𝑀‘𝐶)    &   ⊢ (𝜑 → 𝑇 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑃)    &   ⊢ (𝜑 → ((𝑆‘𝐴)‘𝐵) = 𝐶)    ⇒   ⊢ (𝜑 → ((𝑆‘𝑋)‘𝑌) = 𝑍)
Theorem	miduniq 28769	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑃)    &   ⊢ (𝜑 → ((𝑆‘𝐴)‘𝑋) = 𝑌)    &   ⊢ (𝜑 → ((𝑆‘𝐵)‘𝑋) = 𝑌)    ⇒   ⊢ (𝜑 → 𝐴 = 𝐵)
Theorem	miduniq1 28770	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑃)    &   ⊢ (𝜑 → ((𝑆‘𝐴)‘𝑋) = ((𝑆‘𝐵)‘𝑋))    ⇒   ⊢ (𝜑 → 𝐴 = 𝐵)
Theorem	miduniq2 28771	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑃)    &   ⊢ (𝜑 → ((𝑆‘𝐴)‘((𝑆‘𝐵)‘𝑋)) = ((𝑆‘𝐵)‘((𝑆‘𝐴)‘𝑋)))    ⇒   ⊢ (𝜑 → 𝐴 = 𝐵)
Theorem	colmid 28772	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ 𝑀 = (𝑆‘𝑋)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑃)    &   ⊢ (𝜑 → (𝑋 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))    &   ⊢ (𝜑 → (𝑋 − 𝐴) = (𝑋 − 𝐵))    ⇒   ⊢ (𝜑 → (𝐵 = (𝑀‘𝐴) ∨ 𝐴 = 𝐵))
Theorem	symquadlem 28773	Lemma of the symmetrial 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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ 𝑀 = (𝑆‘𝑋)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑃)    &   ⊢ (𝜑 → ¬ (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶))    &   ⊢ (𝜑 → 𝐵 ≠ 𝐷)    &   ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐶 − 𝐷))    &   ⊢ (𝜑 → (𝐵 − 𝐶) = (𝐷 − 𝐴))    &   ⊢ (𝜑 → (𝑋 ∈ (𝐴𝐿𝐶) ∨ 𝐴 = 𝐶))    &   ⊢ (𝜑 → (𝑋 ∈ (𝐵𝐿𝐷) ∨ 𝐵 = 𝐷))    ⇒   ⊢ (𝜑 → 𝐴 = (𝑀‘𝐶))
Theorem	krippenlem 28774	Lemma for krippen 28775. 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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ 𝑀 = (𝑆‘𝑋)    &   ⊢ 𝑁 = (𝑆‘𝑌)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐶 ∈ (𝐴𝐼𝐸))    &   ⊢ (𝜑 → 𝐶 ∈ (𝐵𝐼𝐹))    &   ⊢ (𝜑 → (𝐶 − 𝐴) = (𝐶 − 𝐵))    &   ⊢ (𝜑 → (𝐶 − 𝐸) = (𝐶 − 𝐹))    &   ⊢ (𝜑 → 𝐵 = (𝑀‘𝐴))    &   ⊢ (𝜑 → 𝐹 = (𝑁‘𝐸))    &   ⊢ ≤ = (≤G‘𝐺)    &   ⊢ (𝜑 → (𝐶 − 𝐴) ≤ (𝐶 − 𝐸))    ⇒   ⊢ (𝜑 → 𝐶 ∈ (𝑋𝐼𝑌))
Theorem	krippen 28775	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ 𝑀 = (𝑆‘𝑋)    &   ⊢ 𝑁 = (𝑆‘𝑌)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐶 ∈ (𝐴𝐼𝐸))    &   ⊢ (𝜑 → 𝐶 ∈ (𝐵𝐼𝐹))    &   ⊢ (𝜑 → (𝐶 − 𝐴) = (𝐶 − 𝐵))    &   ⊢ (𝜑 → (𝐶 − 𝐸) = (𝐶 − 𝐹))    &   ⊢ (𝜑 → 𝐵 = (𝑀‘𝐴))    &   ⊢ (𝜑 → 𝐹 = (𝑁‘𝐸))    ⇒   ⊢ (𝜑 → 𝐶 ∈ (𝑋𝐼𝑌))
Theorem	midexlem 28776*	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 = (𝑀‘𝐴) has to be used. See mideu 28822 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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ 𝑀 = (𝑆‘𝑥)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑃)    &   ⊢ (𝜑 → (𝐶 − 𝐴) = (𝐶 − 𝐵))    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 𝐵 = (𝑀‘𝐴))
16.2.13  Right angles
Syntax	crag 28777	Declare the constant for the class of right angles.
class ∟G
Definition	df-rag 28778*	Define the class of right angles. Definition 8.1 of [Schwabhauser] p. 57. See israg 28781. (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))))})
Syntax	cperpg 28779	Declare the constant for the perpendicular relation.
class ⟂G
Definition	df-perpg 28780*	Define the "perpendicular" relation. Definition 8.11 of [Schwabhauser] p. 59. See isperp 28796. (Contributed by Thierry Arnoux, 8-Sep-2019.)
⊢ ⟂G = (𝑔 ∈ V ↦ {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ran (LineG‘𝑔) ∧ 𝑏 ∈ ran (LineG‘𝑔)) ∧ ∃𝑥 ∈ (𝑎 ∩ 𝑏)∀𝑢 ∈ 𝑎 ∀𝑣 ∈ 𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝑔))})
Theorem	israg 28781	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑃)    ⇒   ⊢ (𝜑 → (⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺) ↔ (𝐴 − 𝐶) = (𝐴 − ((𝑆‘𝐵)‘𝐶))))
Theorem	ragcom 28782	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑃)    &   ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))    ⇒   ⊢ (𝜑 → ⟨“𝐶𝐵𝐴”⟩ ∈ (∟G‘𝐺))
Theorem	ragcol 28783	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑃)    &   ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))    &   ⊢ (𝜑 → 𝐴 ≠ 𝐵)    &   ⊢ (𝜑 → (𝐴 ∈ (𝐵𝐿𝐷) ∨ 𝐵 = 𝐷))    ⇒   ⊢ (𝜑 → ⟨“𝐷𝐵𝐶”⟩ ∈ (∟G‘𝐺))
Theorem	ragmir 28784	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑃)    &   ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))    ⇒   ⊢ (𝜑 → ⟨“𝐴𝐵((𝑆‘𝐵)‘𝐶)”⟩ ∈ (∟G‘𝐺))
Theorem	mirrag 28785	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑃)    &   ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))    &   ⊢ 𝑀 = (𝑆‘𝐷)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑃)    ⇒   ⊢ (𝜑 → ⟨“(𝑀‘𝐴)(𝑀‘𝐵)(𝑀‘𝐶)”⟩ ∈ (∟G‘𝐺))
Theorem	ragtrivb 28786	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑃)    ⇒   ⊢ (𝜑 → ⟨“𝐴𝐵𝐵”⟩ ∈ (∟G‘𝐺))
Theorem	ragflat2 28787	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑃)    &   ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))    &   ⊢ (𝜑 → ⟨“𝐷𝐵𝐶”⟩ ∈ (∟G‘𝐺))    &   ⊢ (𝜑 → 𝐶 ∈ (𝐴𝐼𝐷))    ⇒   ⊢ (𝜑 → 𝐵 = 𝐶)
Theorem	ragflat 28788	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑃)    &   ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))    &   ⊢ (𝜑 → ⟨“𝐴𝐶𝐵”⟩ ∈ (∟G‘𝐺))    ⇒   ⊢ (𝜑 → 𝐵 = 𝐶)
Theorem	ragtriva 28789	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑃)    &   ⊢ (𝜑 → ⟨“𝐴𝐵𝐴”⟩ ∈ (∟G‘𝐺))    ⇒   ⊢ (𝜑 → 𝐴 = 𝐵)
Theorem	ragflat3 28790	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑃)    &   ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))    &   ⊢ (𝜑 → (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))    ⇒   ⊢ (𝜑 → (𝐴 = 𝐵 ∨ 𝐶 = 𝐵))
Theorem	ragcgr 28791	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑃)    &   ⊢ ∼ = (cgrG‘𝐺)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑃)    &   ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))    &   ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∼ ⟨“𝐷𝐸𝐹”⟩)    ⇒   ⊢ (𝜑 → ⟨“𝐷𝐸𝐹”⟩ ∈ (∟G‘𝐺))
Theorem	motrag 28792	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐺Ismt𝐺))    &   ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))    ⇒   ⊢ (𝜑 → ⟨“(𝐹‘𝐴)(𝐹‘𝐵)(𝐹‘𝐶)”⟩ ∈ (∟G‘𝐺))
Theorem	ragncol 28793	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑃)    &   ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))    &   ⊢ (𝜑 → 𝐴 ≠ 𝐵)    &   ⊢ (𝜑 → 𝐶 ≠ 𝐵)    ⇒   ⊢ (𝜑 → ¬ (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))
Theorem	perpln1 28794	Derive a line from perpendicularity. (Contributed by Thierry Arnoux, 27-Nov-2019.)
⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴(⟂G‘𝐺)𝐵)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ran 𝐿)
Theorem	perpln2 28795	Derive a line from perpendicularity. (Contributed by Thierry Arnoux, 27-Nov-2019.)
⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴(⟂G‘𝐺)𝐵)    ⇒   ⊢ (𝜑 → 𝐵 ∈ ran 𝐿)
Theorem	isperp 28796*	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝐵 ∈ ran 𝐿)    ⇒   ⊢ (𝜑 → (𝐴(⟂G‘𝐺)𝐵 ↔ ∃𝑥 ∈ (𝐴 ∩ 𝐵)∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐵 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)))
Theorem	perpcom 28797	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝐵 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝐴(⟂G‘𝐺)𝐵)    ⇒   ⊢ (𝜑 → 𝐵(⟂G‘𝐺)𝐴)
Theorem	perpneq 28798	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝐵 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝐴(⟂G‘𝐺)𝐵)    ⇒   ⊢ (𝜑 → 𝐴 ≠ 𝐵)
Theorem	isperp2 28799*	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝐵 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝑋 ∈ (𝐴 ∩ 𝐵))    ⇒   ⊢ (𝜑 → (𝐴(⟂G‘𝐺)𝐵 ↔ ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐵 ⟨“𝑢𝑋𝑣”⟩ ∈ (∟G‘𝐺)))
Theorem	isperp2d 28800	One direction of isperp2 28799. (Contributed by Thierry Arnoux, 10-Nov-2019.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ − = (dist‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝐵 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝑋 ∈ (𝐴 ∩ 𝐵))    &   ⊢ (𝜑 → 𝑈 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑉 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐴(⟂G‘𝐺)𝐵)    ⇒   ⊢ (𝜑 → ⟨“𝑈𝑋𝑉”⟩ ∈ (∟G‘𝐺))