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	tglinerflx1 28701	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 28702	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 28703	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 28704	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 28705*	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 28706*	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 28707*	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 28708	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 28709*	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 28710*	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 28711	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 28712	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 28713	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 28714	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 28715	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 28716	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 28717*	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 28718*	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 28719*	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 28720	Declare the constant for the point inversion function.
class pInvG
Definition	df-mir 28721*	Define the point inversion ("mirror") function. Definition 7.5 of [Schwabhauser] p. 49. See mirval 28723 and ismir 28727. (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 28722*	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 28723*	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 28724*	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 28725	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 28726	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 28727	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 28728	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 28729	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 28730	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 28731	Variation on mirmir 28730. (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 28732*	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 28733	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 28734	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 28735	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 28736	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 28737	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 28738	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 28739	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 28740	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 28741	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 28742	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 28743	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 28744	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 28745	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 28746	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 28747	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 28748	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 28749	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 28750	Link congruence over a pair of mirror points. cf tgcgrextend 28553. (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 28751	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 28752	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 28753	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 28754	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 28755	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 28756	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 28757	Lemma of the symmetrical 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 28758	Lemma for krippen 28759. 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 28759	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 28760*	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 28806 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 28761	Declare the constant for the class of right angles.
class ∟G
Definition	df-rag 28762*	Define the class of right angles. Definition 8.1 of [Schwabhauser] p. 57. See israg 28765. (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 28763	Declare the constant for the perpendicular relation.
class ⟂G
Definition	df-perpg 28764*	Define the "perpendicular" relation. Definition 8.11 of [Schwabhauser] p. 59. See isperp 28780. (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 28765	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 28766	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 28767	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 28768	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 28769	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 28770	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 28771	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 28772	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 28773	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 28774	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 28775	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 28776	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 28777	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 28778	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 28779	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 28780*	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 28781	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 28782	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 28783*	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 28784	One direction of isperp2 28783. (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))
Theorem	ragperp 28785	Deduce that two lines are perpendicular from a right angle statement. One direction of theorem 8.13 of [Schwabhauser] p. 59. (Contributed by Thierry Arnoux, 20-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 → 𝑈 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑉 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑈 ≠ 𝑋)    &   ⊢ (𝜑 → 𝑉 ≠ 𝑋)    &   ⊢ (𝜑 → ⟨“𝑈𝑋𝑉”⟩ ∈ (∟G‘𝐺))    ⇒   ⊢ (𝜑 → 𝐴(⟂G‘𝐺)𝐵)
Theorem	footexALT 28786*	Alternative version of footex 28789 which minimization requires a notably long time. (Contributed by Thierry Arnoux, 19-Oct-2019.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ − = (dist‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑃)    &   ⊢ (𝜑 → ¬ 𝐶 ∈ 𝐴)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (𝐶𝐿𝑥)(⟂G‘𝐺)𝐴)
Theorem	footexlem1 28787	Lemma for footex 28789. (Contributed by Thierry Arnoux, 19-Oct-2019.) (Revised by Thierry Arnoux, 1-Jul-2023.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ − = (dist‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑃)    &   ⊢ (𝜑 → ¬ 𝐶 ∈ 𝐴)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐴 = (𝐸𝐿𝐹))    &   ⊢ (𝜑 → 𝐸 ≠ 𝐹)    &   ⊢ (𝜑 → 𝐸 ∈ (𝐹𝐼𝑌))    &   ⊢ (𝜑 → (𝐸 − 𝑌) = (𝐸 − 𝐶))    &   ⊢ (𝜑 → 𝐶 = (((pInvG‘𝐺)‘𝑅)‘𝑌))    &   ⊢ (𝜑 → 𝑌 ∈ (𝐸𝐼𝑍))    &   ⊢ (𝜑 → (𝑌 − 𝑍) = (𝑌 − 𝑅))    &   ⊢ (𝜑 → 𝑄 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑌 ∈ (𝑅𝐼𝑄))    &   ⊢ (𝜑 → (𝑌 − 𝑄) = (𝑌 − 𝐸))    &   ⊢ (𝜑 → 𝑌 ∈ ((((pInvG‘𝐺)‘𝑍)‘𝑄)𝐼𝐷))    &   ⊢ (𝜑 → (𝑌 − 𝐷) = (𝑌 − 𝐶))    &   ⊢ (𝜑 → 𝐷 = (((pInvG‘𝐺)‘𝑋)‘𝐶))    ⇒   ⊢ (𝜑 → 𝑋 ∈ 𝐴)
Theorem	footexlem2 28788	Lemma for footex 28789. (Contributed by Thierry Arnoux, 19-Oct-2019.) (Revised by Thierry Arnoux, 1-Jul-2023.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ − = (dist‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑃)    &   ⊢ (𝜑 → ¬ 𝐶 ∈ 𝐴)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐴 = (𝐸𝐿𝐹))    &   ⊢ (𝜑 → 𝐸 ≠ 𝐹)    &   ⊢ (𝜑 → 𝐸 ∈ (𝐹𝐼𝑌))    &   ⊢ (𝜑 → (𝐸 − 𝑌) = (𝐸 − 𝐶))    &   ⊢ (𝜑 → 𝐶 = (((pInvG‘𝐺)‘𝑅)‘𝑌))    &   ⊢ (𝜑 → 𝑌 ∈ (𝐸𝐼𝑍))    &   ⊢ (𝜑 → (𝑌 − 𝑍) = (𝑌 − 𝑅))    &   ⊢ (𝜑 → 𝑄 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑌 ∈ (𝑅𝐼𝑄))    &   ⊢ (𝜑 → (𝑌 − 𝑄) = (𝑌 − 𝐸))    &   ⊢ (𝜑 → 𝑌 ∈ ((((pInvG‘𝐺)‘𝑍)‘𝑄)𝐼𝐷))    &   ⊢ (𝜑 → (𝑌 − 𝐷) = (𝑌 − 𝐶))    &   ⊢ (𝜑 → 𝐷 = (((pInvG‘𝐺)‘𝑋)‘𝐶))    ⇒   ⊢ (𝜑 → (𝐶𝐿𝑋)(⟂G‘𝐺)𝐴)
Theorem	footex 28789*	From a point 𝐶 outside of a line 𝐴, there exists a point 𝑥 on 𝐴 such that (𝐶𝐿𝑥) is perpendicular to 𝐴. This point is unique, see foot 28790. (Contributed by Thierry Arnoux, 19-Oct-2019.) (Revised by Thierry Arnoux, 1-Jul-2023.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ − = (dist‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑃)    &   ⊢ (𝜑 → ¬ 𝐶 ∈ 𝐴)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (𝐶𝐿𝑥)(⟂G‘𝐺)𝐴)
Theorem	foot 28790*	From a point 𝐶 outside of a line 𝐴, there exists a unique point 𝑥 on 𝐴 such that (𝐶𝐿𝑥) is perpendicular to 𝐴. That point is called the foot from 𝐶 on 𝐴. Theorem 8.18 of [Schwabhauser] p. 60. (Contributed by Thierry Arnoux, 19-Oct-2019.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ − = (dist‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑃)    &   ⊢ (𝜑 → ¬ 𝐶 ∈ 𝐴)    ⇒   ⊢ (𝜑 → ∃!𝑥 ∈ 𝐴 (𝐶𝐿𝑥)(⟂G‘𝐺)𝐴)
Theorem	footne 28791	Uniqueness of the foot point. (Contributed by Thierry Arnoux, 28-Feb-2020.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ − = (dist‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑃)    &   ⊢ (𝜑 → (𝑋𝐿𝑌)(⟂G‘𝐺)𝐴)    ⇒   ⊢ (𝜑 → ¬ 𝑌 ∈ 𝐴)
Theorem	footeq 28792	Uniqueness of the foot point. (Contributed by Thierry Arnoux, 1-Mar-2020.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ − = (dist‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑃)    &   ⊢ (𝜑 → (𝑋𝐿𝑍)(⟂G‘𝐺)𝐴)    &   ⊢ (𝜑 → (𝑌𝐿𝑍)(⟂G‘𝐺)𝐴)    ⇒   ⊢ (𝜑 → 𝑋 = 𝑌)
Theorem	hlperpnel 28793	A point on a half-line which is perpendicular to a line cannot be on that line. (Contributed by Thierry Arnoux, 1-Mar-2020.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ − = (dist‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ ran 𝐿)    &   ⊢ 𝐾 = (hlG‘𝐺)    &   ⊢ (𝜑 → 𝑈 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑉 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑊 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐴(⟂G‘𝐺)(𝑈𝐿𝑉))    &   ⊢ (𝜑 → 𝑉(𝐾‘𝑈)𝑊)    ⇒   ⊢ (𝜑 → ¬ 𝑊 ∈ 𝐴)
Theorem	perprag 28794	Deduce a right angle from perpendicular lines. (Contributed by Thierry Arnoux, 10-Nov-2019.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ − = (dist‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐶 ∈ (𝐴𝐿𝐵))    &   ⊢ (𝜑 → 𝐷 ∈ 𝑃)    &   ⊢ (𝜑 → (𝐴𝐿𝐵)(⟂G‘𝐺)(𝐶𝐿𝐷))    ⇒   ⊢ (𝜑 → ⟨“𝐴𝐶𝐷”⟩ ∈ (∟G‘𝐺))
Theorem	perpdragALT 28795	Deduce a right angle from perpendicular lines. (Contributed by Thierry Arnoux, 12-Dec-2019.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ − = (dist‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐷)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐷)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐷(⟂G‘𝐺)(𝐵𝐿𝐶))    ⇒   ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))
Theorem	perpdrag 28796	Deduce a right angle from perpendicular lines. (Contributed by Thierry Arnoux, 12-Dec-2019.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ − = (dist‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐷)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐷)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐷(⟂G‘𝐺)(𝐵𝐿𝐶))    ⇒   ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))
Theorem	colperp 28797	Deduce a perpendicularity from perpendicularity and colinearity. (Contributed by Thierry Arnoux, 8-Dec-2019.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ − = (dist‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑃)    &   ⊢ (𝜑 → (𝐴𝐿𝐵)(⟂G‘𝐺)𝐷)    &   ⊢ (𝜑 → (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))    &   ⊢ (𝜑 → 𝐴 ≠ 𝐶)    ⇒   ⊢ (𝜑 → (𝐴𝐿𝐶)(⟂G‘𝐺)𝐷)
Theorem	colperpexlem1 28798	Lemma for colperp 28797. First part of lemma 8.20 of [Schwabhauser] p. 62. (Contributed by Thierry Arnoux, 27-Oct-2019.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ − = (dist‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ 𝑆 = (pInvG‘𝐺)    &   ⊢ 𝑀 = (𝑆‘𝐴)    &   ⊢ 𝑁 = (𝑆‘𝐵)    &   ⊢ 𝐾 = (𝑆‘𝑄)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑄 ∈ 𝑃)    &   ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))    &   ⊢ (𝜑 → (𝐾‘(𝑀‘𝐶)) = (𝑁‘𝐶))    ⇒   ⊢ (𝜑 → ⟨“𝐵𝐴𝑄”⟩ ∈ (∟G‘𝐺))
Theorem	colperpexlem2 28799	Lemma for colperpex 28801. Second part of lemma 8.20 of [Schwabhauser] p. 62. (Contributed by Thierry Arnoux, 10-Nov-2019.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ − = (dist‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ 𝑆 = (pInvG‘𝐺)    &   ⊢ 𝑀 = (𝑆‘𝐴)    &   ⊢ 𝑁 = (𝑆‘𝐵)    &   ⊢ 𝐾 = (𝑆‘𝑄)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑄 ∈ 𝑃)    &   ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))    &   ⊢ (𝜑 → (𝐾‘(𝑀‘𝐶)) = (𝑁‘𝐶))    &   ⊢ (𝜑 → 𝐵 ≠ 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 ≠ 𝑄)
Theorem	colperpexlem3 28800*	Lemma for colperpex 28801. Case 1 of theorem 8.21 of [Schwabhauser] p. 63. (Contributed by Thierry Arnoux, 20-Nov-2019.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ − = (dist‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐴 ≠ 𝐵)    &   ⊢ (𝜑 → ¬ 𝐶 ∈ (𝐴𝐿𝐵))    ⇒   ⊢ (𝜑 → ∃𝑝 ∈ 𝑃 ((𝐴𝐿𝑝)(⟂G‘𝐺)(𝐴𝐿𝐵) ∧ ∃𝑡 ∈ 𝑃 ((𝑡 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵) ∧ 𝑡 ∈ (𝐶𝐼𝑝))))