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Theorem List for Metamath Proof Explorer - 25301-25400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremisperp2d 25301 One direction of isperp2 25300. (Contributed by Thierry Arnoux, 10-Nov-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴 ∈ ran 𝐿)    &   (𝜑𝐵 ∈ ran 𝐿)    &   (𝜑𝑋 ∈ (𝐴𝐵))    &   (𝜑𝑈𝐴)    &   (𝜑𝑉𝐵)    &   (𝜑𝐴(⟂G‘𝐺)𝐵)       (𝜑 → ⟨“𝑈𝑋𝑉”⟩ ∈ (∟G‘𝐺))
 
Theoremragperp 25302 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)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴 ∈ ran 𝐿)    &   (𝜑𝐵 ∈ ran 𝐿)    &   (𝜑𝑋 ∈ (𝐴𝐵))    &   (𝜑𝑈𝐴)    &   (𝜑𝑉𝐵)    &   (𝜑𝑈𝑋)    &   (𝜑𝑉𝑋)    &   (𝜑 → ⟨“𝑈𝑋𝑉”⟩ ∈ (∟G‘𝐺))       (𝜑𝐴(⟂G‘𝐺)𝐵)
 
Theoremfootex 25303* Lemma for foot 25304: existence part. (Contributed by Thierry Arnoux, 19-Oct-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴 ∈ ran 𝐿)    &   (𝜑𝐶𝑃)    &   (𝜑 → ¬ 𝐶𝐴)       (𝜑 → ∃𝑥𝐴 (𝐶𝐿𝑥)(⟂G‘𝐺)𝐴)
 
Theoremfoot 25304* 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)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴 ∈ ran 𝐿)    &   (𝜑𝐶𝑃)    &   (𝜑 → ¬ 𝐶𝐴)       (𝜑 → ∃!𝑥𝐴 (𝐶𝐿𝑥)(⟂G‘𝐺)𝐴)
 
Theoremfootne 25305 Uniqueness of the foot point. (Contributed by Thierry Arnoux, 28-Feb-2020.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴 ∈ ran 𝐿)    &   (𝜑𝑋𝐴)    &   (𝜑𝑌𝑃)    &   (𝜑 → (𝑋𝐿𝑌)(⟂G‘𝐺)𝐴)       (𝜑 → ¬ 𝑌𝐴)
 
Theoremfooteq 25306 Uniqueness of the foot point. (Contributed by Thierry Arnoux, 1-Mar-2020.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴 ∈ ran 𝐿)    &   (𝜑𝑋𝐴)    &   (𝜑𝑌𝐴)    &   (𝜑𝑍𝑃)    &   (𝜑 → (𝑋𝐿𝑍)(⟂G‘𝐺)𝐴)    &   (𝜑 → (𝑌𝐿𝑍)(⟂G‘𝐺)𝐴)       (𝜑𝑋 = 𝑌)
 
Theoremhlperpnel 25307 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)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴 ∈ ran 𝐿)    &   𝐾 = (hlG‘𝐺)    &   (𝜑𝑈𝐴)    &   (𝜑𝑉𝑃)    &   (𝜑𝑊𝑃)    &   (𝜑𝐴(⟂G‘𝐺)(𝑈𝐿𝑉))    &   (𝜑𝑉(𝐾𝑈)𝑊)       (𝜑 → ¬ 𝑊𝐴)
 
Theoremperprag 25308 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)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶 ∈ (𝐴𝐿𝐵))    &   (𝜑𝐷𝑃)    &   (𝜑 → (𝐴𝐿𝐵)(⟂G‘𝐺)(𝐶𝐿𝐷))       (𝜑 → ⟨“𝐴𝐶𝐷”⟩ ∈ (∟G‘𝐺))
 
TheoremperpdragALT 25309 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)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝐷)    &   (𝜑𝐵𝐷)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷(⟂G‘𝐺)(𝐵𝐿𝐶))       (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))
 
Theoremperpdrag 25310 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)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝐷)    &   (𝜑𝐵𝐷)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷(⟂G‘𝐺)(𝐵𝐿𝐶))       (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))
 
Theoremcolperp 25311 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)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑 → (𝐴𝐿𝐵)(⟂G‘𝐺)𝐷)    &   (𝜑 → (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))    &   (𝜑𝐴𝐶)       (𝜑 → (𝐴𝐿𝐶)(⟂G‘𝐺)𝐷)
 
Theoremcolperpexlem1 25312 Lemma for colperp 25311. 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)    &   (𝜑𝐺 ∈ TarskiG)    &   𝑆 = (pInvG‘𝐺)    &   𝑀 = (𝑆𝐴)    &   𝑁 = (𝑆𝐵)    &   𝐾 = (𝑆𝑄)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝑄𝑃)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))    &   (𝜑 → (𝐾‘(𝑀𝐶)) = (𝑁𝐶))       (𝜑 → ⟨“𝐵𝐴𝑄”⟩ ∈ (∟G‘𝐺))
 
Theoremcolperpexlem2 25313 Lemma for colperpex 25315. 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)    &   (𝜑𝐺 ∈ TarskiG)    &   𝑆 = (pInvG‘𝐺)    &   𝑀 = (𝑆𝐴)    &   𝑁 = (𝑆𝐵)    &   𝐾 = (𝑆𝑄)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝑄𝑃)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))    &   (𝜑 → (𝐾‘(𝑀𝐶)) = (𝑁𝐶))    &   (𝜑𝐵𝐶)       (𝜑𝐴𝑄)
 
Theoremcolperpexlem3 25314* Lemma for colperpex 25315. 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)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐴𝐵)    &   (𝜑 → ¬ 𝐶 ∈ (𝐴𝐿𝐵))       (𝜑 → ∃𝑝𝑃 ((𝐴𝐿𝑝)(⟂G‘𝐺)(𝐴𝐿𝐵) ∧ ∃𝑡𝑃 ((𝑡 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵) ∧ 𝑡 ∈ (𝐶𝐼𝑝))))
 
Theoremcolperpex 25315* In dimension 2 and above, on a line (𝐴𝐿𝐵) there is always a perpendicular 𝑃 from 𝐴 on a given plane (here given by 𝐶, in case 𝐶 does not lie on the line). 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)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐴𝐵)    &   (𝜑𝐺DimTarskiG≥2)       (𝜑 → ∃𝑝𝑃 ((𝐴𝐿𝑝)(⟂G‘𝐺)(𝐴𝐿𝐵) ∧ ∃𝑡𝑃 ((𝑡 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵) ∧ 𝑡 ∈ (𝐶𝐼𝑝))))
 
Theoremmideulem2 25316 Lemma for opphllem 25317, which is itself used for mideu 25320. (Contributed by Thierry Arnoux, 19-Feb-2020.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   𝑆 = (pInvG‘𝐺)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐴𝐵)    &   (𝜑𝑄𝑃)    &   (𝜑𝑂𝑃)    &   (𝜑𝑇𝑃)    &   (𝜑 → (𝐴𝐿𝐵)(⟂G‘𝐺)(𝑄𝐿𝐵))    &   (𝜑 → (𝐴𝐿𝐵)(⟂G‘𝐺)(𝐴𝐿𝑂))    &   (𝜑𝑇 ∈ (𝐴𝐿𝐵))    &   (𝜑𝑇 ∈ (𝑄𝐼𝑂))    &   (𝜑𝑅𝑃)    &   (𝜑𝑅 ∈ (𝐵𝐼𝑄))    &   (𝜑 → (𝐴 𝑂) = (𝐵 𝑅))    &   (𝜑𝑋𝑃)    &   (𝜑𝑋 ∈ (𝑇𝐼𝐵))    &   (𝜑𝑋 ∈ (𝑅𝐼𝑂))    &   (𝜑𝑍𝑃)    &   (𝜑𝑋 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑍))    &   (𝜑 → (𝑋 𝑍) = (𝑋 𝑅))    &   (𝜑𝑀𝑃)    &   (𝜑𝑅 = ((𝑆𝑀)‘𝑍))       (𝜑𝐵 = 𝑀)
 
Theoremopphllem 25317* Lemma 8.24 of [Schwabhauser] p. 66. This is used later for mideulem 25318 and later for opphl 25336. (Contributed by Thierry Arnoux, 21-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   𝑆 = (pInvG‘𝐺)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐴𝐵)    &   (𝜑𝑄𝑃)    &   (𝜑𝑂𝑃)    &   (𝜑𝑇𝑃)    &   (𝜑 → (𝐴𝐿𝐵)(⟂G‘𝐺)(𝑄𝐿𝐵))    &   (𝜑 → (𝐴𝐿𝐵)(⟂G‘𝐺)(𝐴𝐿𝑂))    &   (𝜑𝑇 ∈ (𝐴𝐿𝐵))    &   (𝜑𝑇 ∈ (𝑄𝐼𝑂))    &   (𝜑𝑅𝑃)    &   (𝜑𝑅 ∈ (𝐵𝐼𝑄))    &   (𝜑 → (𝐴 𝑂) = (𝐵 𝑅))       (𝜑 → ∃𝑥𝑃 (𝐵 = ((𝑆𝑥)‘𝐴) ∧ 𝑂 = ((𝑆𝑥)‘𝑅)))
 
Theoremmideulem 25318* Lemma for mideu 25320. We can assume mideulem.9 "without loss of generality" (Contributed by Thierry Arnoux, 25-Nov-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   𝑆 = (pInvG‘𝐺)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐴𝐵)    &   (𝜑𝑄𝑃)    &   (𝜑𝑂𝑃)    &   (𝜑𝑇𝑃)    &   (𝜑 → (𝐴𝐿𝐵)(⟂G‘𝐺)(𝑄𝐿𝐵))    &   (𝜑 → (𝐴𝐿𝐵)(⟂G‘𝐺)(𝐴𝐿𝑂))    &   (𝜑𝑇 ∈ (𝐴𝐿𝐵))    &   (𝜑𝑇 ∈ (𝑄𝐼𝑂))    &   (𝜑 → (𝐴 𝑂)(≤G‘𝐺)(𝐵 𝑄))       (𝜑 → ∃𝑥𝑃 𝐵 = ((𝑆𝑥)‘𝐴))
 
Theoremmidex 25319* Existence of the midpoint, part Theorem 8.22 of [Schwabhauser] p. 64. Note that this proof requires a construction in 2 dimensions or more, i.e. it does not prove the existence of a midpoint in dimension 1, for a geometry restricted to a line. (Contributed by Thierry Arnoux, 25-Nov-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   𝑆 = (pInvG‘𝐺)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐺DimTarskiG≥2)       (𝜑 → ∃𝑥𝑃 𝐵 = ((𝑆𝑥)‘𝐴))
 
Theoremmideu 25320* Existence and uniqueness of the midpoint, Theorem 8.22 of [Schwabhauser] p. 64. (Contributed by Thierry Arnoux, 25-Nov-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   𝑆 = (pInvG‘𝐺)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐺DimTarskiG≥2)       (𝜑 → ∃!𝑥𝑃 𝐵 = ((𝑆𝑥)‘𝐴))
 
15.2.14  Half-planes
 
Theoremislnopp 25321* The property for two points 𝐴 and 𝐵 to lie on the opposite sides of a set 𝐷 Definition 9.1 of [Schwabhauser] p. 67. (Contributed by Thierry Arnoux, 19-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)       (𝜑 → (𝐴𝑂𝐵 ↔ ((¬ 𝐴𝐷 ∧ ¬ 𝐵𝐷) ∧ ∃𝑡𝐷 𝑡 ∈ (𝐴𝐼𝐵))))
 
Theoremislnoppd 25322* Deduce that 𝐴 and 𝐵 lie on opposite sides of line 𝐿. (Contributed by Thierry Arnoux, 16-Aug-2020.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝐷)    &   (𝜑 → ¬ 𝐴𝐷)    &   (𝜑 → ¬ 𝐵𝐷)    &   (𝜑𝐶 ∈ (𝐴𝐼𝐵))       (𝜑𝐴𝑂𝐵)
 
Theoremoppne1 25323* Points lying on opposite sides of a line cannot be on the line. (Contributed by Thierry Arnoux, 3-Mar-2020.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐴𝑂𝐵)       (𝜑 → ¬ 𝐴𝐷)
 
Theoremoppne2 25324* Points lying on opposite sides of a line cannot be on the line. (Contributed by Thierry Arnoux, 3-Mar-2020.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐴𝑂𝐵)       (𝜑 → ¬ 𝐵𝐷)
 
Theoremoppne3 25325* Points lying on opposite sides of a line cannot be equal. (Contributed by Thierry Arnoux, 3-Aug-2020.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐴𝑂𝐵)       (𝜑𝐴𝐵)
 
Theoremoppcom 25326* Commutativity rule for "opposite" Theorem 9.2 of [Schwabhauser] p. 67. (Contributed by Thierry Arnoux, 19-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐴𝑂𝐵)       (𝜑𝐵𝑂𝐴)
 
Theoremopptgdim2 25327* If two points opposite to a line exist, dimension must be 2 or more. (Contributed by Thierry Arnoux, 3-Mar-2020.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐴𝑂𝐵)       (𝜑𝐺DimTarskiG≥2)
 
Theoremoppnid 25328* The "opposite to a line" relation is irreflexive. (Contributed by Thierry Arnoux, 4-Mar-2020.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)       (𝜑 → ¬ 𝐴𝑂𝐴)
 
Theoremopphllem1 25329* Lemma for opphl 25336. (Contributed by Thierry Arnoux, 20-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐺 ∈ TarskiG)    &   𝑆 = ((pInvG‘𝐺)‘𝑀)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝑅𝐷)    &   (𝜑𝐴𝑂𝐶)    &   (𝜑𝑀𝐷)    &   (𝜑𝐴 = (𝑆𝐶))    &   (𝜑𝐴𝑅)    &   (𝜑𝐵𝑅)    &   (𝜑𝐵 ∈ (𝑅𝐼𝐴))       (𝜑𝐵𝑂𝐶)
 
Theoremopphllem2 25330* Lemma for opphl 25336. Lemma 9.3 of [Schwabhauser] p. 68. (Contributed by Thierry Arnoux, 21-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐺 ∈ TarskiG)    &   𝑆 = ((pInvG‘𝐺)‘𝑀)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝑅𝐷)    &   (𝜑𝐴𝑂𝐶)    &   (𝜑𝑀𝐷)    &   (𝜑𝐴 = (𝑆𝐶))    &   (𝜑𝐴𝑅)    &   (𝜑𝐵𝑅)    &   (𝜑 → (𝐴 ∈ (𝑅𝐼𝐵) ∨ 𝐵 ∈ (𝑅𝐼𝐴)))       (𝜑𝐵𝑂𝐶)
 
Theoremopphllem3 25331* Lemma for opphl 25336: We assume opphllem3.l "without loss of generality". (Contributed by Thierry Arnoux, 21-Feb-2020.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐺 ∈ TarskiG)    &   𝐾 = (hlG‘𝐺)    &   𝑁 = ((pInvG‘𝐺)‘𝑀)    &   (𝜑𝐴𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝑅𝐷)    &   (𝜑𝑆𝐷)    &   (𝜑𝑀𝑃)    &   (𝜑𝐴𝑂𝐶)    &   (𝜑𝐷(⟂G‘𝐺)(𝐴𝐿𝑅))    &   (𝜑𝐷(⟂G‘𝐺)(𝐶𝐿𝑆))    &   (𝜑𝑅𝑆)    &   (𝜑 → (𝑆 𝐶)(≤G‘𝐺)(𝑅 𝐴))    &   (𝜑𝑈𝑃)    &   (𝜑 → (𝑁𝑅) = 𝑆)       (𝜑 → (𝑈(𝐾𝑅)𝐴 ↔ (𝑁𝑈)(𝐾𝑆)𝐶))
 
Theoremopphllem4 25332* Lemma for opphl 25336. (Contributed by Thierry Arnoux, 22-Feb-2020.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐺 ∈ TarskiG)    &   𝐾 = (hlG‘𝐺)    &   𝑁 = ((pInvG‘𝐺)‘𝑀)    &   (𝜑𝐴𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝑅𝐷)    &   (𝜑𝑆𝐷)    &   (𝜑𝑀𝑃)    &   (𝜑𝐴𝑂𝐶)    &   (𝜑𝐷(⟂G‘𝐺)(𝐴𝐿𝑅))    &   (𝜑𝐷(⟂G‘𝐺)(𝐶𝐿𝑆))    &   (𝜑𝑅𝑆)    &   (𝜑 → (𝑆 𝐶)(≤G‘𝐺)(𝑅 𝐴))    &   (𝜑𝑈𝑃)    &   (𝜑 → (𝑁𝑅) = 𝑆)    &   (𝜑𝑉𝑃)    &   (𝜑𝑈(𝐾𝑅)𝐴)    &   (𝜑𝑉(𝐾𝑆)𝐶)       (𝜑𝑈𝑂𝑉)
 
Theoremopphllem5 25333* Second part of Lemma 9.4 of [Schwabhauser] p. 68. (Contributed by Thierry Arnoux, 2-Mar-2020.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐺 ∈ TarskiG)    &   𝐾 = (hlG‘𝐺)    &   𝑁 = ((pInvG‘𝐺)‘𝑀)    &   (𝜑𝐴𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝑅𝐷)    &   (𝜑𝑆𝐷)    &   (𝜑𝑀𝑃)    &   (𝜑𝐴𝑂𝐶)    &   (𝜑𝐷(⟂G‘𝐺)(𝐴𝐿𝑅))    &   (𝜑𝐷(⟂G‘𝐺)(𝐶𝐿𝑆))    &   (𝜑𝑈𝑃)    &   (𝜑𝑉𝑃)    &   (𝜑𝑈(𝐾𝑅)𝐴)    &   (𝜑𝑉(𝐾𝑆)𝐶)       (𝜑𝑈𝑂𝑉)
 
Theoremopphllem6 25334* First part of Lemma 9.4 of [Schwabhauser] p. 68. (Contributed by Thierry Arnoux, 3-Mar-2020.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐺 ∈ TarskiG)    &   𝐾 = (hlG‘𝐺)    &   𝑁 = ((pInvG‘𝐺)‘𝑀)    &   (𝜑𝐴𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝑅𝐷)    &   (𝜑𝑆𝐷)    &   (𝜑𝑀𝑃)    &   (𝜑𝐴𝑂𝐶)    &   (𝜑𝐷(⟂G‘𝐺)(𝐴𝐿𝑅))    &   (𝜑𝐷(⟂G‘𝐺)(𝐶𝐿𝑆))    &   (𝜑𝑈𝑃)    &   (𝜑 → (𝑁𝑅) = 𝑆)       (𝜑 → (𝑈(𝐾𝑅)𝐴 ↔ (𝑁𝑈)(𝐾𝑆)𝐶))
 
Theoremoppperpex 25335* Restating colperpex 25315 using the "opposite side of a line" relation. (Contributed by Thierry Arnoux, 2-Aug-2020.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐺 ∈ TarskiG)    &   𝐾 = (hlG‘𝐺)    &   (𝜑𝐴𝐷)    &   (𝜑𝐶𝑃)    &   (𝜑 → ¬ 𝐶𝐷)    &   (𝜑𝐺DimTarskiG≥2)       (𝜑 → ∃𝑝𝑃 ((𝐴𝐿𝑝)(⟂G‘𝐺)𝐷𝐶𝑂𝑝))
 
Theoremopphl 25336* If two points 𝐴 and 𝐶 lie on the opposite side of a line 𝐷 then any point of the half line (𝑅 𝐴) also lies opposite to 𝐶. Theorem 9.5 of [Schwabhauser] p. 69. (Contributed by Thierry Arnoux, 3-Mar-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐺 ∈ TarskiG)    &   𝐾 = (hlG‘𝐺)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐴𝑂𝐶)    &   (𝜑𝑅𝐷)    &   (𝜑𝐴(𝐾𝑅)𝐵)       (𝜑𝐵𝑂𝐶)
 
Theoremoutpasch 25337* Axiom of Pasch, outer form. This was proven by Gupta from other axioms and is therefore presented as Theorem 9.6 in [Schwabhauser] p. 70. (Contributed by Thierry Arnoux, 16-Aug-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝑅𝑃)    &   (𝜑𝑄𝑃)    &   (𝜑𝐶 ∈ (𝐴𝐼𝑅))    &   (𝜑𝑄 ∈ (𝐵𝐼𝐶))       (𝜑 → ∃𝑥𝑃 (𝑥 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝑥)))
 
Theoremhlpasch 25338* An application of the axiom of Pasch for half-lines. (Contributed by Thierry Arnoux, 15-Sep-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐾 = (hlG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝑋𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐴𝐵)    &   (𝜑𝐶(𝐾𝐵)𝐷)    &   (𝜑𝐴 ∈ (𝑋𝐼𝐶))       (𝜑 → ∃𝑒𝑃 (𝐴(𝐾𝐵)𝑒𝑒 ∈ (𝑋𝐼𝐷)))
 
Syntaxchpg 25339 "Belong to the same open half-plane" relation for points in a geometry.
class hpG
 
Definitiondf-hpg 25340* Define the open half plane relation for a geometry 𝐺. Definition 9.7 of [Schwabhauser] p. 71. See hpgbr 25342 to find the same formulation. (Contributed by Thierry Arnoux, 4-Mar-2020.)
hpG = (𝑔 ∈ V ↦ (𝑑 ∈ ran (LineG‘𝑔) ↦ {⟨𝑎, 𝑏⟩ ∣ [(Base‘𝑔) / 𝑝][(Itv‘𝑔) / 𝑖]𝑐𝑝 (((𝑎 ∈ (𝑝𝑑) ∧ 𝑐 ∈ (𝑝𝑑)) ∧ ∃𝑡𝑑 𝑡 ∈ (𝑎𝑖𝑐)) ∧ ((𝑏 ∈ (𝑝𝑑) ∧ 𝑐 ∈ (𝑝𝑑)) ∧ ∃𝑡𝑑 𝑡 ∈ (𝑏𝑖𝑐)))}))
 
Theoremishpg 25341* Value of the half-plane relation for a given line 𝐷. (Contributed by Thierry Arnoux, 4-Mar-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐷 ∈ ran 𝐿)       (𝜑 → ((hpG‘𝐺)‘𝐷) = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐𝑃 (𝑎𝑂𝑐𝑏𝑂𝑐)})
 
Theoremhpgbr 25342* Half-planes : property for points 𝐴 and 𝐵 to belong to the same open half plane delimited by line 𝐷. Definition 9.7 of [Schwabhauser] p. 71. (Contributed by Thierry Arnoux, 4-Mar-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)       (𝜑 → (𝐴((hpG‘𝐺)‘𝐷)𝐵 ↔ ∃𝑐𝑃 (𝐴𝑂𝑐𝐵𝑂𝑐)))
 
Theoremhpgne1 25343* Points on the open half plane cannot lie on its border. (Contributed by Thierry Arnoux, 4-Mar-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐴((hpG‘𝐺)‘𝐷)𝐵)       (𝜑 → ¬ 𝐴𝐷)
 
Theoremhpgne2 25344* Points on the open half plane cannot lie on its border. (Contributed by Thierry Arnoux, 4-Mar-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐴((hpG‘𝐺)‘𝐷)𝐵)       (𝜑 → ¬ 𝐵𝐷)
 
Theoremlnopp2hpgb 25345* Theorem 9.8 of [Schwabhauser] p. 71. (Contributed by Thierry Arnoux, 4-Mar-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐴𝑂𝐶)       (𝜑 → (𝐵𝑂𝐶𝐴((hpG‘𝐺)‘𝐷)𝐵))
 
Theoremlnoppnhpg 25346* If two points lie on the opposite side of a line 𝐷, they are not on the same half-plane. Theorem 9.9 of [Schwabhauser] p. 72. (Contributed by Thierry Arnoux, 4-Mar-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐴𝑂𝐵)       (𝜑 → ¬ 𝐴((hpG‘𝐺)‘𝐷)𝐵)
 
Theoremhpgerlem 25347* Lemma for the proof that the half-plane relation is an equivalence relation. Lemma 9.10 of [Schwabhauser] p. 72. (Contributed by Thierry Arnoux, 4-Mar-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐴𝑃)    &   𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   (𝜑 → ¬ 𝐴𝐷)       (𝜑 → ∃𝑐𝑃 𝐴𝑂𝑐)
 
Theoremhpgid 25348* The half-plane relation is reflexive. Theorem 9.11 of [Schwabhauser] p. 72. (Contributed by Thierry Arnoux, 4-Mar-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐴𝑃)    &   𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   (𝜑 → ¬ 𝐴𝐷)       (𝜑𝐴((hpG‘𝐺)‘𝐷)𝐴)
 
Theoremhpgcom 25349* The half-plane relation commutes. Theorem 9.12 of [Schwabhauser] p. 72. (Contributed by Thierry Arnoux, 4-Mar-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐴𝑃)    &   𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   (𝜑𝐵𝑃)    &   (𝜑𝐴((hpG‘𝐺)‘𝐷)𝐵)       (𝜑𝐵((hpG‘𝐺)‘𝐷)𝐴)
 
Theoremhpgtr 25350* The half-plane relation is transitive. Theorem 9.13 of [Schwabhauser] p. 72. (Contributed by Thierry Arnoux, 4-Mar-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐴𝑃)    &   𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   (𝜑𝐵𝑃)    &   (𝜑𝐴((hpG‘𝐺)‘𝐷)𝐵)    &   (𝜑𝐶𝑃)    &   (𝜑𝐵((hpG‘𝐺)‘𝐷)𝐶)       (𝜑𝐴((hpG‘𝐺)‘𝐷)𝐶)
 
Theoremcolopp 25351* Opposite sides of a line for colinear points. Theorem 9.18 of [Schwabhauser] p. 73. (Contributed by Thierry Arnoux, 3-Aug-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐴𝑃)    &   𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝐷)    &   (𝜑 → (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))       (𝜑 → (𝐴𝑂𝐵 ↔ (𝐶 ∈ (𝐴𝐼𝐵) ∧ ¬ 𝐴𝐷 ∧ ¬ 𝐵𝐷)))
 
Theoremcolhp 25352* Half-plane relation for colinear points. Theorem 9.19 of [Schwabhauser] p. 73. (Contributed by Thierry Arnoux, 3-Aug-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐴𝑃)    &   𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝐷)    &   (𝜑 → (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))    &   𝐾 = (hlG‘𝐺)       (𝜑 → (𝐴((hpG‘𝐺)‘𝐷)𝐵 ↔ (𝐴(𝐾𝐶)𝐵 ∧ ¬ 𝐴𝐷)))
 
Theoremhphl 25353* If two points are on the same half-line with endpoint on a line, they are on the same half-plane defined by this line. (Contributed by Thierry Arnoux, 9-Aug-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐴𝑃)    &   𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   𝐾 = (hlG‘𝐺)    &   (𝜑𝐴𝐷)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑 → ¬ 𝐵𝐷)    &   (𝜑𝐵(𝐾𝐴)𝐶)       (𝜑𝐵((hpG‘𝐺)‘𝐷)𝐶)
 
15.2.15  Midpoints and Line Mirroring
 
Syntaxcmid 25354 Declare the constant for the midpoint operation.
class midG
 
Syntaxclmi 25355 Declare the constant for the line mirroring function.
class lInvG
 
Definitiondf-mid 25356* Define the midpoint operation. Definition 10.1 of [Schwabhauser] p. 88. See ismidb 25360, midbtwn 25361, and midcgr 25362. (Contributed by Thierry Arnoux, 9-Jun-2019.)
midG = (𝑔 ∈ V ↦ (𝑎 ∈ (Base‘𝑔), 𝑏 ∈ (Base‘𝑔) ↦ (𝑚 ∈ (Base‘𝑔)𝑏 = (((pInvG‘𝑔)‘𝑚)‘𝑎))))
 
Definitiondf-lmi 25357* Define the line mirroring function. Definition 10.3 of [Schwabhauser] p. 89. See islmib 25369. (Contributed by Thierry Arnoux, 1-Dec-2019.)
lInvG = (𝑔 ∈ V ↦ (𝑚 ∈ ran (LineG‘𝑔) ↦ (𝑎 ∈ (Base‘𝑔) ↦ (𝑏 ∈ (Base‘𝑔)((𝑎(midG‘𝑔)𝑏) ∈ 𝑚 ∧ (𝑚(⟂G‘𝑔)(𝑎(LineG‘𝑔)𝑏) ∨ 𝑎 = 𝑏))))))
 
Theoremmidf 25358 Midpoint as a function. (Contributed by Thierry Arnoux, 1-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐺DimTarskiG≥2)       (𝜑 → (midG‘𝐺):(𝑃 × 𝑃)⟶𝑃)
 
Theoremmidcl 25359 Closure of the midpoint. (Contributed by Thierry Arnoux, 1-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐺DimTarskiG≥2)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)       (𝜑 → (𝐴(midG‘𝐺)𝐵) ∈ 𝑃)
 
Theoremismidb 25360 Property of the midpoint. (Contributed by Thierry Arnoux, 1-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐺DimTarskiG≥2)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   𝑆 = (pInvG‘𝐺)    &   (𝜑𝑀𝑃)       (𝜑 → (𝐵 = ((𝑆𝑀)‘𝐴) ↔ (𝐴(midG‘𝐺)𝐵) = 𝑀))
 
Theoremmidbtwn 25361 Betweenness of midpoint. (Contributed by Thierry Arnoux, 7-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐺DimTarskiG≥2)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)       (𝜑 → (𝐴(midG‘𝐺)𝐵) ∈ (𝐴𝐼𝐵))
 
Theoremmidcgr 25362 Congruence of midpoint. (Contributed by Thierry Arnoux, 7-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐺DimTarskiG≥2)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑 → (𝐴(midG‘𝐺)𝐵) = 𝐶)       (𝜑 → (𝐶 𝐴) = (𝐶 𝐵))
 
Theoremmidid 25363 Midpoint of a null segment. (Contributed by Thierry Arnoux, 7-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐺DimTarskiG≥2)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)       (𝜑 → (𝐴(midG‘𝐺)𝐴) = 𝐴)
 
Theoremmidcom 25364 Commutativity rule for the midpoint. (Contributed by Thierry Arnoux, 2-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐺DimTarskiG≥2)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)       (𝜑 → (𝐴(midG‘𝐺)𝐵) = (𝐵(midG‘𝐺)𝐴))
 
Theoremmirmid 25365 Point inversion preserves midpoints. (Contributed by Thierry Arnoux, 12-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐺DimTarskiG≥2)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   𝑆 = ((pInvG‘𝐺)‘𝑀)    &   (𝜑𝑀𝑃)       (𝜑 → ((𝑆𝐴)(midG‘𝐺)(𝑆𝐵)) = (𝑆‘(𝐴(midG‘𝐺)𝐵)))
 
Theoremlmieu 25366* Uniqueness of the line mirror point. Theorem 10.2 of [Schwabhauser] p. 88. (Contributed by Thierry Arnoux, 1-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐺DimTarskiG≥2)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐴𝑃)       (𝜑 → ∃!𝑏𝑃 ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏)))
 
Theoremlmif 25367 Line mirror as a function. (Contributed by Thierry Arnoux, 11-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐺DimTarskiG≥2)    &   𝑀 = ((lInvG‘𝐺)‘𝐷)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐷 ∈ ran 𝐿)       (𝜑𝑀:𝑃𝑃)
 
Theoremlmicl 25368 Closure of the line mirror. (Contributed by Thierry Arnoux, 11-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐺DimTarskiG≥2)    &   𝑀 = ((lInvG‘𝐺)‘𝐷)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐴𝑃)       (𝜑 → (𝑀𝐴) ∈ 𝑃)
 
Theoremislmib 25369 Property of the line mirror. (Contributed by Thierry Arnoux, 11-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐺DimTarskiG≥2)    &   𝑀 = ((lInvG‘𝐺)‘𝐷)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)       (𝜑 → (𝐵 = (𝑀𝐴) ↔ ((𝐴(midG‘𝐺)𝐵) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))))
 
Theoremlmicom 25370 The line mirroring function is an involution. Theorem 10.4 of [Schwabhauser] p. 89. (Contributed by Thierry Arnoux, 11-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐺DimTarskiG≥2)    &   𝑀 = ((lInvG‘𝐺)‘𝐷)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑 → (𝑀𝐴) = 𝐵)       (𝜑 → (𝑀𝐵) = 𝐴)
 
Theoremlmilmi 25371 Line mirroring is an involution. Theorem 10.5 of [Schwabhauser] p. 89. (Contributed by Thierry Arnoux, 11-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐺DimTarskiG≥2)    &   𝑀 = ((lInvG‘𝐺)‘𝐷)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐴𝑃)       (𝜑 → (𝑀‘(𝑀𝐴)) = 𝐴)
 
Theoremlmireu 25372* Any point has a unique antecedent through line mirroring. Theorem 10.6 of [Schwabhauser] p. 89. (Contributed by Thierry Arnoux, 11-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐺DimTarskiG≥2)    &   𝑀 = ((lInvG‘𝐺)‘𝐷)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐴𝑃)       (𝜑 → ∃!𝑏𝑃 (𝑀𝑏) = 𝐴)
 
Theoremlmieq 25373 Equality deduction for line mirroring. Theorem 10.7 of [Schwabhauser] p. 89. (Contributed by Thierry Arnoux, 11-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐺DimTarskiG≥2)    &   𝑀 = ((lInvG‘𝐺)‘𝐷)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑 → (𝑀𝐴) = (𝑀𝐵))       (𝜑𝐴 = 𝐵)
 
Theoremlmiinv 25374 The invariants of the line mirroring lie on the mirror line. Theorem 10.8 of [Schwabhauser] p. 89. (Contributed by Thierry Arnoux, 11-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐺DimTarskiG≥2)    &   𝑀 = ((lInvG‘𝐺)‘𝐷)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐴𝑃)       (𝜑 → ((𝑀𝐴) = 𝐴𝐴𝐷))
 
Theoremlmicinv 25375 The mirroring line is an invariant. (Contributed by Thierry Arnoux, 8-Aug-2020.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐺DimTarskiG≥2)    &   𝑀 = ((lInvG‘𝐺)‘𝐷)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐴𝑃)    &   (𝜑𝐴𝐷)       (𝜑 → (𝑀𝐴) = 𝐴)
 
Theoremlmimid 25376 If we have a right angle, then the mirror point is the point inversion. (Contributed by Thierry Arnoux, 15-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐺DimTarskiG≥2)    &   𝑀 = ((lInvG‘𝐺)‘𝐷)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐴𝑃)    &   𝑆 = ((pInvG‘𝐺)‘𝐵)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))    &   (𝜑𝐴𝐷)    &   (𝜑𝐵𝐷)    &   (𝜑𝐶𝑃)    &   (𝜑𝐴𝐵)       (𝜑 → (𝑀𝐶) = (𝑆𝐶))
 
Theoremlmif1o 25377 The line mirroring function 𝑀 is a bijection. Theorem 10.9 of [Schwabhauser] p. 89. (Contributed by Thierry Arnoux, 11-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐺DimTarskiG≥2)    &   𝑀 = ((lInvG‘𝐺)‘𝐷)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐷 ∈ ran 𝐿)       (𝜑𝑀:𝑃1-1-onto𝑃)
 
Theoremlmiisolem 25378 Lemma for lmiiso 25379. (Contributed by Thierry Arnoux, 14-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐺DimTarskiG≥2)    &   𝑀 = ((lInvG‘𝐺)‘𝐷)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   𝑆 = ((pInvG‘𝐺)‘𝑍)    &   𝑍 = ((𝐴(midG‘𝐺)(𝑀𝐴))(midG‘𝐺)(𝐵(midG‘𝐺)(𝑀𝐵)))       (𝜑 → ((𝑀𝐴) (𝑀𝐵)) = (𝐴 𝐵))
 
Theoremlmiiso 25379 The line mirroring function is an isometry, i.e. it is conserves congruence. Because it is also a bijection, it is also a motion. Theorem 10.10 of [Schwabhauser] p. 89. (Contributed by Thierry Arnoux, 11-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐺DimTarskiG≥2)    &   𝑀 = ((lInvG‘𝐺)‘𝐷)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐷 ∈ ran 𝐿)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)       (𝜑 → ((𝑀𝐴) (𝑀𝐵)) = (𝐴 𝐵))
 
Theoremlmimot 25380 Line mirroring is a motion of the geometric space. Theorem 10.11 of [Schwabhauser] p. 90. (Contributed by Thierry Arnoux, 15-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐺DimTarskiG≥2)    &   𝑀 = ((lInvG‘𝐺)‘𝐷)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐷 ∈ ran 𝐿)       (𝜑𝑀 ∈ (𝐺Ismt𝐺))
 
Theoremhypcgrlem1 25381 Lemma for hypcgr 25383, case where triangles share a cathetus. (Contributed by Thierry Arnoux, 15-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐺DimTarskiG≥2)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))    &   (𝜑 → ⟨“𝐷𝐸𝐹”⟩ ∈ (∟G‘𝐺))    &   (𝜑 → (𝐴 𝐵) = (𝐷 𝐸))    &   (𝜑 → (𝐵 𝐶) = (𝐸 𝐹))    &   (𝜑𝐵 = 𝐸)    &   𝑆 = ((lInvG‘𝐺)‘((𝐴(midG‘𝐺)𝐷)(LineG‘𝐺)𝐵))    &   (𝜑𝐶 = 𝐹)       (𝜑 → (𝐴 𝐶) = (𝐷 𝐹))
 
Theoremhypcgrlem2 25382 Lemma for hypcgr 25383, case where triangles share one vertex 𝐵. (Contributed by Thierry Arnoux, 16-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐺DimTarskiG≥2)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))    &   (𝜑 → ⟨“𝐷𝐸𝐹”⟩ ∈ (∟G‘𝐺))    &   (𝜑 → (𝐴 𝐵) = (𝐷 𝐸))    &   (𝜑 → (𝐵 𝐶) = (𝐸 𝐹))    &   (𝜑𝐵 = 𝐸)    &   𝑆 = ((lInvG‘𝐺)‘((𝐶(midG‘𝐺)𝐹)(LineG‘𝐺)𝐵))       (𝜑 → (𝐴 𝐶) = (𝐷 𝐹))
 
Theoremhypcgr 25383 If the catheti of two right-angled triangles are congruent, so is their hypothenuse. Theorem 10.12 of [Schwabhauser] p. 91. (Contributed by Thierry Arnoux, 16-Dec-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐺DimTarskiG≥2)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))    &   (𝜑 → ⟨“𝐷𝐸𝐹”⟩ ∈ (∟G‘𝐺))    &   (𝜑 → (𝐴 𝐵) = (𝐷 𝐸))    &   (𝜑 → (𝐵 𝐶) = (𝐸 𝐹))       (𝜑 → (𝐴 𝐶) = (𝐷 𝐹))
 
Theoremlmiopp 25384* Line mirroring produces points on the opposite side of the mirroring line. Theorem 10.14 of [Schwabhauser] p. 92. (Contributed by Thierry Arnoux, 2-Aug-2020.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐺DimTarskiG≥2)    &   (𝜑𝐷 ∈ ran 𝐿)    &   𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   𝑀 = ((lInvG‘𝐺)‘𝐷)    &   (𝜑𝐴𝑃)    &   (𝜑 → ¬ 𝐴𝐷)       (𝜑𝐴𝑂(𝑀𝐴))
 
Theoremlnperpex 25385* Existence of a perpendicular to a line 𝐿 at a given point 𝐴. Theorem 10.15 of [Schwabhauser] p. 92. (Contributed by Thierry Arnoux, 2-Aug-2020.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐺DimTarskiG≥2)    &   (𝜑𝐷 ∈ ran 𝐿)    &   𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   (𝜑𝐴𝐷)    &   (𝜑𝑄𝑃)    &   (𝜑 → ¬ 𝑄𝐷)       (𝜑 → ∃𝑝𝑃 (𝐷(⟂G‘𝐺)(𝑝𝐿𝐴) ∧ 𝑝((hpG‘𝐺)‘𝐷)𝑄))
 
Theoremtrgcopy 25386* Triangle construction: a copy of a given triangle can always be constructed in such a way that one side is lying on a half-line, and the third vertex is on a given half-plane: existence part. First part of Theorem 10.16 of [Schwabhauser] p. 92. (Contributed by Thierry Arnoux, 4-Aug-2020.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝐾 = (hlG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑 → ¬ (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶))    &   (𝜑 → ¬ (𝐷 ∈ (𝐸𝐿𝐹) ∨ 𝐸 = 𝐹))    &   (𝜑 → (𝐴 𝐵) = (𝐷 𝐸))       (𝜑 → ∃𝑓𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑓”⟩ ∧ 𝑓((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹))
 
Theoremtrgcopyeulem 25387* Lemma for trgcopyeu 25388. (Contributed by Thierry Arnoux, 8-Aug-2020.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝐾 = (hlG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑 → ¬ (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶))    &   (𝜑 → ¬ (𝐷 ∈ (𝐸𝐿𝐹) ∨ 𝐸 = 𝐹))    &   (𝜑 → (𝐴 𝐵) = (𝐷 𝐸))    &   𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ (𝐷𝐿𝐸)) ∧ 𝑏 ∈ (𝑃 ∖ (𝐷𝐿𝐸))) ∧ ∃𝑡 ∈ (𝐷𝐿𝐸)𝑡 ∈ (𝑎𝐼𝑏))}    &   (𝜑𝑋𝑃)    &   (𝜑𝑌𝑃)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑋”⟩)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑌”⟩)    &   (𝜑𝑋((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)    &   (𝜑𝑌((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)       (𝜑𝑋 = 𝑌)
 
Theoremtrgcopyeu 25388* Triangle construction: a copy of a given triangle can always be constructed in such a way that one side is lying on a half-line, and the third vertex is on a given half-plane: uniqueness part. Second part of Theorem 10.16 of [Schwabhauser] p. 92. (Contributed by Thierry Arnoux, 8-Aug-2020.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝐾 = (hlG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑 → ¬ (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶))    &   (𝜑 → ¬ (𝐷 ∈ (𝐸𝐿𝐹) ∨ 𝐸 = 𝐹))    &   (𝜑 → (𝐴 𝐵) = (𝐷 𝐸))       (𝜑 → ∃!𝑓𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑓”⟩ ∧ 𝑓((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹))
 
15.2.16  Congruence of angles
 
Syntaxccgra 25389 Declare the constant for the congruence between angles relation.
class cgrA
 
Definitiondf-cgra 25390* Define the congruence relation bewteen angles. As for triangles we use "words of points". See iscgra 25391 for a more human readable version. (Contributed by Thierry Arnoux, 30-Jul-2020.)
cgrA = (𝑔 ∈ V ↦ {⟨𝑎, 𝑏⟩ ∣ [(Base‘𝑔) / 𝑝][(hlG‘𝑔) / 𝑘]((𝑎 ∈ (𝑝𝑚 (0..^3)) ∧ 𝑏 ∈ (𝑝𝑚 (0..^3))) ∧ ∃𝑥𝑝𝑦𝑝 (𝑎(cgrG‘𝑔)⟨“𝑥(𝑏‘1)𝑦”⟩ ∧ 𝑥(𝑘‘(𝑏‘1))(𝑏‘0) ∧ 𝑦(𝑘‘(𝑏‘1))(𝑏‘2)))})
 
Theoremiscgra 25391* Property for two angles ABC and DEF to be congruent. This is a modified version of the definition 11.3 of [Schwabhauser] p. 95. where the number of constructed points has been reduced to two. We chose this version rather than the textbook version because it is shorter and therefore simpler to work with. Theorem dfcgra2 25411 shows that those definitions are indeed equivalent. (Contributed by Thierry Arnoux, 31-Jul-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐾 = (hlG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)       (𝜑 → (⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩ ↔ ∃𝑥𝑃𝑦𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾𝐸)𝐷𝑦(𝐾𝐸)𝐹)))
 
Theoremiscgra1 25392* A special version of iscgra 25391 where one distance is known to be equal. In this case, angle congruence can be written with only one quantifier. (Contributed by Thierry Arnoux, 9-Aug-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐾 = (hlG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &    = (dist‘𝐺)    &   (𝜑𝐴𝐵)    &   (𝜑 → (𝐴 𝐵) = (𝐷 𝐸))       (𝜑 → (⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩ ↔ ∃𝑥𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝐹)))
 
Theoremiscgrad 25393 Sufficient conditions for angle congruence, deduction version. (Contributed by Thierry Arnoux, 1-Aug-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐾 = (hlG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑𝑋𝑃)    &   (𝜑𝑌𝑃)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑋𝐸𝑌”⟩)    &   (𝜑𝑋(𝐾𝐸)𝐷)    &   (𝜑𝑌(𝐾𝐸)𝐹)       (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)
 
Theoremcgrane1 25394 Angles imply inequality. (Contributed by Thierry Arnoux, 1-Aug-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐾 = (hlG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)       (𝜑𝐴𝐵)
 
Theoremcgrane2 25395 Angles imply inequality. (Contributed by Thierry Arnoux, 1-Aug-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐾 = (hlG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)       (𝜑𝐵𝐶)
 
Theoremcgrane3 25396 Angles imply inequality. (Contributed by Thierry Arnoux, 1-Aug-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐾 = (hlG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)       (𝜑𝐸𝐷)
 
Theoremcgrane4 25397 Angles imply inequality. (Contributed by Thierry Arnoux, 1-Aug-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐾 = (hlG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)       (𝜑𝐸𝐹)
 
Theoremcgrahl1 25398 Angle congruence is independent of the choice of points on the rays. Proposition 11.10 of [Schwabhauser] p. 95. (Contributed by Thierry Arnoux, 1-Aug-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐾 = (hlG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)    &   (𝜑𝑋𝑃)    &   (𝜑𝑋(𝐾𝐸)𝐷)       (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝑋𝐸𝐹”⟩)
 
Theoremcgrahl2 25399 Angle congruence is independent of the choice of points on the rays. Proposition 11.10 of [Schwabhauser] p. 95. (Contributed by Thierry Arnoux, 1-Aug-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐾 = (hlG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)    &   (𝜑𝑋𝑃)    &   (𝜑𝑋(𝐾𝐸)𝐹)       (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝑋”⟩)
 
Theoremcgracgr 25400 First direction of proposition 11.4 of [Schwabhauser] p. 95. Again, this is "half" of the proposition, i.e. only two additional points are used, while Schwabhauser has four. (Contributed by Thierry Arnoux, 31-Jul-2020.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐾 = (hlG‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)    &   (𝜑𝑋𝑃)    &    = (dist‘𝐺)    &   (𝜑𝑌𝑃)    &   (𝜑𝑋(𝐾𝐵)𝐴)    &   (𝜑𝑌(𝐾𝐵)𝐶)    &   (𝜑 → (𝐵 𝑋) = (𝐸 𝐷))    &   (𝜑 → (𝐵 𝑌) = (𝐸 𝐹))       (𝜑 → (𝑋 𝑌) = (𝐷 𝐹))
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