P. 292 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 29101-29200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	mideu 29101*	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ 𝑆 = (pInvG‘𝐺)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐺DimTarskiG≥2)    ⇒   ⊢ (𝜑 → ∃!𝑥 ∈ 𝑃 𝐵 = ((𝑆‘𝑥)‘𝐴))
16.2.14  Half-planes
Theorem	islnopp 29102*	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 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    ⇒   ⊢ (𝜑 → (𝐴𝑂𝐵 ↔ ((¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐵 ∈ 𝐷) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐵))))
Theorem	islnoppd 29103*	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 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐷)    &   ⊢ (𝜑 → ¬ 𝐴 ∈ 𝐷)    &   ⊢ (𝜑 → ¬ 𝐵 ∈ 𝐷)    &   ⊢ (𝜑 → 𝐶 ∈ (𝐴𝐼𝐵))    ⇒   ⊢ (𝜑 → 𝐴𝑂𝐵)
Theorem	oppne1 29104*	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 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ (𝜑 → 𝐷 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐴𝑂𝐵)    ⇒   ⊢ (𝜑 → ¬ 𝐴 ∈ 𝐷)
Theorem	oppne2 29105*	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 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ (𝜑 → 𝐷 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐴𝑂𝐵)    ⇒   ⊢ (𝜑 → ¬ 𝐵 ∈ 𝐷)
Theorem	oppne3 29106*	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 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ (𝜑 → 𝐷 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐴𝑂𝐵)    ⇒   ⊢ (𝜑 → 𝐴 ≠ 𝐵)
Theorem	oppcom 29107*	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 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ (𝜑 → 𝐷 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐴𝑂𝐵)    ⇒   ⊢ (𝜑 → 𝐵𝑂𝐴)
Theorem	opptgdim2 29108*	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 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ (𝜑 → 𝐷 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐴𝑂𝐵)    ⇒   ⊢ (𝜑 → 𝐺DimTarskiG≥2)
Theorem	oppnid 29109*	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 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ (𝜑 → 𝐷 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    ⇒   ⊢ (𝜑 → ¬ 𝐴𝑂𝐴)
Theorem	opphllem1 29110*	Lemma for opphl 29117. (Contributed by Thierry Arnoux, 20-Dec-2019.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ − = (dist‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ (𝜑 → 𝐷 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ 𝑆 = ((pInvG‘𝐺)‘𝑀)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑅 ∈ 𝐷)    &   ⊢ (𝜑 → 𝐴𝑂𝐶)    &   ⊢ (𝜑 → 𝑀 ∈ 𝐷)    &   ⊢ (𝜑 → 𝐴 = (𝑆‘𝐶))    &   ⊢ (𝜑 → 𝐴 ≠ 𝑅)    &   ⊢ (𝜑 → 𝐵 ≠ 𝑅)    &   ⊢ (𝜑 → 𝐵 ∈ (𝑅𝐼𝐴))    ⇒   ⊢ (𝜑 → 𝐵𝑂𝐶)
Theorem	opphllem2 29111*	Lemma for opphl 29117. 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 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ (𝜑 → 𝐷 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ 𝑆 = ((pInvG‘𝐺)‘𝑀)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑅 ∈ 𝐷)    &   ⊢ (𝜑 → 𝐴𝑂𝐶)    &   ⊢ (𝜑 → 𝑀 ∈ 𝐷)    &   ⊢ (𝜑 → 𝐴 = (𝑆‘𝐶))    &   ⊢ (𝜑 → 𝐴 ≠ 𝑅)    &   ⊢ (𝜑 → 𝐵 ≠ 𝑅)    &   ⊢ (𝜑 → (𝐴 ∈ (𝑅𝐼𝐵) ∨ 𝐵 ∈ (𝑅𝐼𝐴)))    ⇒   ⊢ (𝜑 → 𝐵𝑂𝐶)
Theorem	opphllem3 29112*	Lemma for opphl 29117: 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 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ (𝜑 → 𝐷 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ 𝐾 = (hlG‘𝐺)    &   ⊢ 𝑁 = ((pInvG‘𝐺)‘𝑀)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑅 ∈ 𝐷)    &   ⊢ (𝜑 → 𝑆 ∈ 𝐷)    &   ⊢ (𝜑 → 𝑀 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐴𝑂𝐶)    &   ⊢ (𝜑 → 𝐷(⟂G‘𝐺)(𝐴𝐿𝑅))    &   ⊢ (𝜑 → 𝐷(⟂G‘𝐺)(𝐶𝐿𝑆))    &   ⊢ (𝜑 → 𝑅 ≠ 𝑆)    &   ⊢ (𝜑 → (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴))    &   ⊢ (𝜑 → 𝑈 ∈ 𝑃)    &   ⊢ (𝜑 → (𝑁‘𝑅) = 𝑆)    ⇒   ⊢ (𝜑 → (𝑈(𝐾‘𝑅)𝐴 ↔ (𝑁‘𝑈)(𝐾‘𝑆)𝐶))
Theorem	opphllem4 29113*	Lemma for opphl 29117. (Contributed by Thierry Arnoux, 22-Feb-2020.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ − = (dist‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ (𝜑 → 𝐷 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ 𝐾 = (hlG‘𝐺)    &   ⊢ 𝑁 = ((pInvG‘𝐺)‘𝑀)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑅 ∈ 𝐷)    &   ⊢ (𝜑 → 𝑆 ∈ 𝐷)    &   ⊢ (𝜑 → 𝑀 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐴𝑂𝐶)    &   ⊢ (𝜑 → 𝐷(⟂G‘𝐺)(𝐴𝐿𝑅))    &   ⊢ (𝜑 → 𝐷(⟂G‘𝐺)(𝐶𝐿𝑆))    &   ⊢ (𝜑 → 𝑅 ≠ 𝑆)    &   ⊢ (𝜑 → (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴))    &   ⊢ (𝜑 → 𝑈 ∈ 𝑃)    &   ⊢ (𝜑 → (𝑁‘𝑅) = 𝑆)    &   ⊢ (𝜑 → 𝑉 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑈(𝐾‘𝑅)𝐴)    &   ⊢ (𝜑 → 𝑉(𝐾‘𝑆)𝐶)    ⇒   ⊢ (𝜑 → 𝑈𝑂𝑉)
Theorem	opphllem5 29114*	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 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ (𝜑 → 𝐷 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ 𝐾 = (hlG‘𝐺)    &   ⊢ 𝑁 = ((pInvG‘𝐺)‘𝑀)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑅 ∈ 𝐷)    &   ⊢ (𝜑 → 𝑆 ∈ 𝐷)    &   ⊢ (𝜑 → 𝑀 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐴𝑂𝐶)    &   ⊢ (𝜑 → 𝐷(⟂G‘𝐺)(𝐴𝐿𝑅))    &   ⊢ (𝜑 → 𝐷(⟂G‘𝐺)(𝐶𝐿𝑆))    &   ⊢ (𝜑 → 𝑈 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑉 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑈(𝐾‘𝑅)𝐴)    &   ⊢ (𝜑 → 𝑉(𝐾‘𝑆)𝐶)    ⇒   ⊢ (𝜑 → 𝑈𝑂𝑉)
Theorem	opphllem6 29115*	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 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ (𝜑 → 𝐷 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ 𝐾 = (hlG‘𝐺)    &   ⊢ 𝑁 = ((pInvG‘𝐺)‘𝑀)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑅 ∈ 𝐷)    &   ⊢ (𝜑 → 𝑆 ∈ 𝐷)    &   ⊢ (𝜑 → 𝑀 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐴𝑂𝐶)    &   ⊢ (𝜑 → 𝐷(⟂G‘𝐺)(𝐴𝐿𝑅))    &   ⊢ (𝜑 → 𝐷(⟂G‘𝐺)(𝐶𝐿𝑆))    &   ⊢ (𝜑 → 𝑈 ∈ 𝑃)    &   ⊢ (𝜑 → (𝑁‘𝑅) = 𝑆)    ⇒   ⊢ (𝜑 → (𝑈(𝐾‘𝑅)𝐴 ↔ (𝑁‘𝑈)(𝐾‘𝑆)𝐶))
Theorem	oppperpex 29116*	Restating colperpex 29096 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 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ (𝜑 → 𝐷 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ 𝐾 = (hlG‘𝐺)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐷)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑃)    &   ⊢ (𝜑 → ¬ 𝐶 ∈ 𝐷)    &   ⊢ (𝜑 → 𝐺DimTarskiG≥2)    ⇒   ⊢ (𝜑 → ∃𝑝 ∈ 𝑃 ((𝐴𝐿𝑝)(⟂G‘𝐺)𝐷 ∧ 𝐶𝑂𝑝))
Theorem	opphl 29117*	If two points 𝐴 and 𝐶 lie on opposite sides of a line 𝐷, then any point of the half line (𝑅𝐴) 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 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ (𝜑 → 𝐷 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ 𝐾 = (hlG‘𝐺)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐴𝑂𝐶)    &   ⊢ (𝜑 → 𝑅 ∈ 𝐷)    &   ⊢ (𝜑 → 𝐴(𝐾‘𝑅)𝐵)    ⇒   ⊢ (𝜑 → 𝐵𝑂𝐶)
Theorem	lnoppinn0 29118*	The segment between two points 𝑋 and 𝑌 on opposite sides of a line 𝐷 intersects 𝐷. (Contributed by Thierry Arnoux, 23-Aug-2026.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   ⊢ (𝜑 → 𝐺 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐷 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑋𝑂𝑌)    ⇒   ⊢ (𝜑 → (𝐷 ∩ (𝑋𝐼𝑌)) ≠ ∅)
Theorem	oppmir 29119*	The mirror point with regard to a point 𝑋 on a line 𝐴 lies on the other side of 𝐴. (Contributed by Thierry Arnoux, 5-Jul-2026.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ 𝑆 = (pInvG‘𝐺)    &   ⊢ 𝑀 = (𝑆‘𝑋)    &   ⊢ 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ 𝐴) ∧ 𝑏 ∈ (𝑃 ∖ 𝐴)) ∧ ∃𝑡 ∈ 𝐴 𝑡 ∈ (𝑎𝐼𝑏))}    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑌 ∈ (𝑃 ∖ 𝐴))    &   ⊢ 𝐿 = (LineG‘𝐺)    ⇒   ⊢ (𝜑 → 𝑌𝑂(𝑀‘𝑌))
Theorem	outpasch 29120*	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑄 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐶 ∈ (𝐴𝐼𝑅))    &   ⊢ (𝜑 → 𝑄 ∈ (𝐵𝐼𝐶))    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 (𝑥 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝑥)))
Theorem	hlpasch 29121*	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐴 ≠ 𝐵)    &   ⊢ (𝜑 → 𝐶(𝐾‘𝐵)𝐷)    &   ⊢ (𝜑 → 𝐴 ∈ (𝑋𝐼𝐶))    ⇒   ⊢ (𝜑 → ∃𝑒 ∈ 𝑃 (𝐴(𝐾‘𝐵)𝑒 ∧ 𝑒 ∈ (𝑋𝐼𝐷)))
Syntax	chpg 29122	"Belong to the same open half-plane" relation for points in a geometry.
class hpG
Definition	df-hpg 29123*	Define the open half plane relation for a geometry 𝐺. Definition 9.7 of [Schwabhauser] p. 71. See hpgbr 29125 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]∃𝑐 ∈ 𝑝 (((𝑎 ∈ (𝑝 ∖ 𝑑) ∧ 𝑐 ∈ (𝑝 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑎𝑖𝑐)) ∧ ((𝑏 ∈ (𝑝 ∖ 𝑑) ∧ 𝑐 ∈ (𝑝 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑏𝑖𝑐)))}))
Theorem	ishpg 29124*	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 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐷 ∈ ran 𝐿)    ⇒   ⊢ (𝜑 → ((hpG‘𝐺)‘𝐷) = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ 𝑃 (𝑎𝑂𝑐 ∧ 𝑏𝑂𝑐)})
Theorem	hpgbr 29125*	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 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐷 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    ⇒   ⊢ (𝜑 → (𝐴((hpG‘𝐺)‘𝐷)𝐵 ↔ ∃𝑐 ∈ 𝑃 (𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐)))
Theorem	hpgne1 29126*	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 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐷 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐴((hpG‘𝐺)‘𝐷)𝐵)    ⇒   ⊢ (𝜑 → ¬ 𝐴 ∈ 𝐷)
Theorem	hpgne2 29127*	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 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐷 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐴((hpG‘𝐺)‘𝐷)𝐵)    ⇒   ⊢ (𝜑 → ¬ 𝐵 ∈ 𝐷)
Theorem	lnopp2hpgb 29128*	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 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐷 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐴𝑂𝐶)    ⇒   ⊢ (𝜑 → (𝐵𝑂𝐶 ↔ 𝐴((hpG‘𝐺)‘𝐷)𝐵))
Theorem	lnoppnhpg 29129*	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 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐷 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐴𝑂𝐵)    ⇒   ⊢ (𝜑 → ¬ 𝐴((hpG‘𝐺)‘𝐷)𝐵)
Theorem	hpgerlem 29130*	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐷 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   ⊢ (𝜑 → ¬ 𝐴 ∈ 𝐷)    ⇒   ⊢ (𝜑 → ∃𝑐 ∈ 𝑃 𝐴𝑂𝑐)
Theorem	hpgid 29131*	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐷 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   ⊢ (𝜑 → ¬ 𝐴 ∈ 𝐷)    ⇒   ⊢ (𝜑 → 𝐴((hpG‘𝐺)‘𝐷)𝐴)
Theorem	hpgcom 29132*	The half-plane relation is symmetric. Theorem 9.12 of [Schwabhauser] p. 72. (Contributed by Thierry Arnoux, 4-Mar-2020.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐷 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐴((hpG‘𝐺)‘𝐷)𝐵)    ⇒   ⊢ (𝜑 → 𝐵((hpG‘𝐺)‘𝐷)𝐴)
Theorem	hpgtr 29133*	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐷 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐴((hpG‘𝐺)‘𝐷)𝐵)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵((hpG‘𝐺)‘𝐷)𝐶)    ⇒   ⊢ (𝜑 → 𝐴((hpG‘𝐺)‘𝐷)𝐶)
Theorem	colopp 29134*	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐷 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐷)    &   ⊢ (𝜑 → (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))    ⇒   ⊢ (𝜑 → (𝐴𝑂𝐵 ↔ (𝐶 ∈ (𝐴𝐼𝐵) ∧ ¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐵 ∈ 𝐷)))
Theorem	colhp 29135*	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐷 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐷)    &   ⊢ (𝜑 → (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))    &   ⊢ 𝐾 = (hlG‘𝐺)    ⇒   ⊢ (𝜑 → (𝐴((hpG‘𝐺)‘𝐷)𝐵 ↔ (𝐴(𝐾‘𝐶)𝐵 ∧ ¬ 𝐴 ∈ 𝐷)))
Theorem	hphl 29136*	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐷 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   ⊢ 𝐾 = (hlG‘𝐺)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐷)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑃)    &   ⊢ (𝜑 → ¬ 𝐵 ∈ 𝐷)    &   ⊢ (𝜑 → 𝐵(𝐾‘𝐴)𝐶)    ⇒   ⊢ (𝜑 → 𝐵((hpG‘𝐺)‘𝐷)𝐶)
Theorem	hlopp 29137*	If two points 𝑋 and 𝑌 lie on opposite sides of a line 𝐴, then given a point 𝑍 on 𝐴, any point 𝑊 on the line (𝑋𝐿𝑍) opposite to 𝑌 lies on the half line (𝑍𝑋) (Contributed by Thierry Arnoux, 20-Jul-2026.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ 𝐴) ∧ 𝑏 ∈ (𝑃 ∖ 𝐴)) ∧ ∃𝑡 ∈ 𝐴 𝑡 ∈ (𝑎𝐼𝑏))}    &   ⊢ 𝐾 = (hlG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑋𝑂𝑌)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑊𝑂𝑌)    &   ⊢ (𝜑 → 𝑊 ∈ (𝑋𝐿𝑍))    ⇒   ⊢ (𝜑 → 𝑊(𝐾‘𝑍)𝑋)
16.2.15  Planes
Syntax	cplng 29138	Declare the constant for the class of planes.
class hlG
Definition	df-plng 29139*	Define the function building a plane from a line and a point not on that line. Definition 9.20 of [Schwabhauser] p. 74. (Contributed by Thierry Arnoux, 17-Jun-2026.)
⊢ hlG = (𝑔 ∈ V ↦ (𝑎 ∈ ran (LineG‘𝑔), 𝑟 ∈ ((Base‘𝑔) ∖ 𝑎) ↦ {𝑥 ∈ (Base‘𝑔) ∣ (𝑥 ∈ 𝑎 ∨ 𝑥((hpG‘𝑔)‘𝑎)𝑟 ∨ ∃𝑡 ∈ 𝑎 𝑡 ∈ (𝑥(Itv‘𝑔)𝑟))}))
Theorem	tgplnfn 29140	The plane generating function as a function. (Contributed by Thierry Arnoux, 17-Jun-2026.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ 𝐸 = (hlG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑉)    ⇒   ⊢ (𝜑 → 𝐸 Fn ((ran 𝐿 × 𝑃) ∖ ◡ E ))
Theorem	tgelrnpln 29141	The property of being a plane, generated by a line and a point. (Contributed by Thierry Arnoux, 17-Jun-2026.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ 𝐸 = (hlG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐴 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝑅 ∈ (𝑃 ∖ 𝐴))    ⇒   ⊢ (𝜑 → (𝐴𝐸𝑅) ∈ ran 𝐸)
Theorem	plngval 29142*	The plane defined by a line 𝐴 and a point 𝑅 outside of 𝐴. This is defined as the union of 3 parts: the line itself, the open half-plane containing 𝑅, and the points opposite to 𝑅 (see islnopp 29102). (Contributed by Thierry Arnoux, 17-Jun-2026.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ 𝐸 = (hlG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝑅 ∈ (𝑃 ∖ 𝐴))    ⇒   ⊢ (𝜑 → (𝐴𝐸𝑅) = {𝑥 ∈ 𝑃 ∣ (𝑥 ∈ 𝐴 ∨ 𝑥((hpG‘𝐺)‘𝐴)𝑅 ∨ ∃𝑡 ∈ 𝐴 𝑡 ∈ (𝑥𝐼𝑅))})
Theorem	isplng 29143*	The property of being a plane. (Contributed by Thierry Arnoux, 17-Jun-2026.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ 𝐸 = (hlG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐻 ∈ ran 𝐸)    ⇒   ⊢ (𝜑 → ∃𝑎 ∈ ran 𝐿∃𝑟 ∈ (𝑃 ∖ 𝑎)𝐻 = (𝑎𝐸𝑟))
Theorem	plngrnssp 29144	Planes are sets of points. (Contributed by Thierry Arnoux, 17-Jun-2026.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ 𝐸 = (hlG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐻 ∈ ran 𝐸)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐻)    ⇒   ⊢ (𝜑 → 𝑋 ∈ 𝑃)
Theorem	elplng 29145*	Elementhood in the plane defined by a line 𝐴 and a point 𝑅. Definition 9.20 of [Schwabhauser] p. 74. (Contributed by Thierry Arnoux, 17-Jun-2026.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ 𝐸 = (hlG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝑅 ∈ (𝑃 ∖ 𝐴))    &   ⊢ 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ 𝐴) ∧ 𝑏 ∈ (𝑃 ∖ 𝐴)) ∧ ∃𝑡 ∈ 𝐴 𝑡 ∈ (𝑎𝐼𝑏))}    &   ⊢ (𝜑 → 𝑋 ∈ 𝑃)    ⇒   ⊢ (𝜑 → (𝑋 ∈ (𝐴𝐸𝑅) ↔ (𝑋 ∈ 𝐴 ∨ 𝑋((hpG‘𝐺)‘𝐴)𝑅 ∨ 𝑋𝑂𝑅)))
Theorem	plngssp 29146	Planes are sets of points. (Contributed by Thierry Arnoux, 17-Jun-2026.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ 𝐸 = (hlG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝑅 ∈ (𝑃 ∖ 𝐴))    &   ⊢ (𝜑 → 𝑋 ∈ (𝐴𝐸𝑅))    ⇒   ⊢ (𝜑 → 𝑋 ∈ 𝑃)
Theorem	elplngid 29147	The point 𝑅 is itself an element of a plane defined by a line 𝐴 and the point 𝑅. (Contributed by Thierry Arnoux, 17-Jun-2026.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ 𝐸 = (hlG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝑅 ∈ (𝑃 ∖ 𝐴))    ⇒   ⊢ (𝜑 → 𝑅 ∈ (𝐴𝐸𝑅))
Theorem	elplnglnid 29148	The line 𝐴 itself is a subset of a plane defined by the line 𝐴 and a point 𝑅. (Contributed by Thierry Arnoux, 17-Jun-2026.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ 𝐸 = (hlG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝑅 ∈ (𝑃 ∖ 𝐴))    ⇒   ⊢ (𝜑 → 𝐴 ⊆ (𝐴𝐸𝑅))
Theorem	lnincplng 29149	If two lines 𝐴 and 𝐵 intersect, then 𝐵 is in a plane defined by 𝐴 and any point of 𝐵. Lemma 9.22 of [Schwabhauser] p. 74. (Contributed by Thierry Arnoux, 17-Jun-2026.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ 𝐸 = (hlG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝐵 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑋 ≠ 𝑌)    &   ⊢ (𝜑 → (𝐴 ∩ 𝐵) = {𝑌})    ⇒   ⊢ (𝜑 → 𝐵 ⊆ (𝐴𝐸𝑋))
Theorem	plngcplem 29150*	Lemma for plngcp 29151. (Contributed by Thierry Arnoux, 17-Jun-2026.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ 𝐸 = (hlG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝑅 ∈ (𝑃 ∖ 𝐴))    &   ⊢ (𝜑 → 𝑆 ∈ ((𝐴𝐸𝑅) ∖ 𝐴))    &   ⊢ 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ 𝐴) ∧ 𝑏 ∈ (𝑃 ∖ 𝐴)) ∧ ∃𝑡 ∈ 𝐴 𝑡 ∈ (𝑎𝐼𝑏))}    ⇒   ⊢ (𝜑 → (𝐴𝐸𝑅) = (𝐴𝐸𝑆))
Theorem	plngcp 29151	The plane defined by a line 𝐴 and a point 𝑅 can also be defined using a different point 𝑅 on the same plane: changes the point used to define the plane. Theorem 9.21 of [Schwabhauser] p. 74. (Contributed by Thierry Arnoux, 17-Jun-2026.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ 𝐸 = (hlG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝑅 ∈ (𝑃 ∖ 𝐴))    &   ⊢ (𝜑 → 𝑆 ∈ ((𝐴𝐸𝑅) ∖ 𝐴))    ⇒   ⊢ (𝜑 → (𝐴𝐸𝑅) = (𝐴𝐸𝑆))
Theorem	plngrotlem1 29152*	Lemma for plngrot 29155. (Contributed by Thierry Arnoux, 17-Jun-2026.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ 𝐸 = (hlG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝑋 ∈ (𝑃 ∖ (𝑍𝐿𝑌)))    &   ⊢ (𝜑 → 𝑌 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑍 ∈ (𝑃 ∖ (𝑋𝐿𝑌)))    &   ⊢ (𝜑 → 𝑋 ≠ 𝑌)    &   ⊢ 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ (𝑋𝐿𝑌)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑋𝐿𝑌))) ∧ ∃𝑡 ∈ (𝑋𝐿𝑌)𝑡 ∈ (𝑎𝐼𝑏))}    &   ⊢ (𝜑 → 𝑊 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑌 ∈ (𝑍𝐼𝑊))    &   ⊢ (𝜑 → 𝑌 ≠ 𝑊)    &   ⊢ (𝜑 → 𝑆 ∈ ((𝑋𝐿𝑌)𝐸𝑍))    &   ⊢ (𝜑 → (𝑆 ∈ (𝑋𝐿𝑌) ∨ 𝑆((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑍))    ⇒   ⊢ (𝜑 → 𝑆 ∈ ((𝑍𝐿𝑌)𝐸𝑋))
Theorem	plngrotlem2 29153*	Lemma for plngrot 29155. (Contributed by Thierry Arnoux, 17-Jun-2026.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ 𝐸 = (hlG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝑋 ∈ (𝑃 ∖ (𝑍𝐿𝑌)))    &   ⊢ (𝜑 → 𝑌 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑍 ∈ (𝑃 ∖ (𝑋𝐿𝑌)))    &   ⊢ (𝜑 → 𝑋 ≠ 𝑌)    &   ⊢ 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ (𝑋𝐿𝑌)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑋𝐿𝑌))) ∧ ∃𝑡 ∈ (𝑋𝐿𝑌)𝑡 ∈ (𝑎𝐼𝑏))}    &   ⊢ (𝜑 → 𝑊 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑌 ∈ (𝑍𝐼𝑊))    &   ⊢ (𝜑 → 𝑌 ≠ 𝑊)    ⇒   ⊢ (𝜑 → ((𝑋𝐿𝑌)𝐸𝑍) ⊆ ((𝑍𝐿𝑌)𝐸𝑋))
Theorem	plngrotlem3 29154*	Lemma for plngrot 29155. (Contributed by Thierry Arnoux, 17-Jun-2026.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ 𝐸 = (hlG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝑋 ∈ (𝑃 ∖ (𝑍𝐿𝑌)))    &   ⊢ (𝜑 → 𝑌 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑍 ∈ (𝑃 ∖ (𝑋𝐿𝑌)))    &   ⊢ (𝜑 → 𝑋 ≠ 𝑌)    &   ⊢ 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ (𝑋𝐿𝑌)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑋𝐿𝑌))) ∧ ∃𝑡 ∈ (𝑋𝐿𝑌)𝑡 ∈ (𝑎𝐼𝑏))}    ⇒   ⊢ (𝜑 → ((𝑋𝐿𝑌)𝐸𝑍) ⊆ ((𝑍𝐿𝑌)𝐸𝑋))
Theorem	plngrot 29155	The plane defined by a line (𝑋𝐿𝑌) and a point 𝑍 is also defined by the line (𝑍𝐿𝑌) and the point 𝑋. See first part of Theorem 9.24 of [Schwabhauser] p. 74. (Contributed by Thierry Arnoux, 17-Jun-2026.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ 𝐸 = (hlG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝑋 ∈ (𝑃 ∖ (𝑍𝐿𝑌)))    &   ⊢ (𝜑 → 𝑌 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑍 ∈ (𝑃 ∖ (𝑋𝐿𝑌)))    &   ⊢ (𝜑 → 𝑋 ≠ 𝑌)    ⇒   ⊢ (𝜑 → ((𝑋𝐿𝑌)𝐸𝑍) = ((𝑍𝐿𝑌)𝐸𝑋))
Theorem	lnssplnglem 29156*	Lemma for lnssplng 29157. (Contributed by Thierry Arnoux, 17-Jun-2026.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ 𝐸 = (hlG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝑋 ∈ (𝐴𝐸𝑅))    &   ⊢ (𝜑 → 𝑌 ∈ (𝐴𝐸𝑅))    &   ⊢ (𝜑 → 𝑋 ≠ 𝑌)    &   ⊢ (𝜑 → 𝐴 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝑅 ∈ (𝑃 ∖ 𝐴))    &   ⊢ (𝜑 → 𝐴 ≠ (𝑋𝐿𝑌))    &   ⊢ (𝜑 → ¬ 𝑌 ∈ 𝐴)    ⇒   ⊢ (𝜑 → ((𝑋𝐿𝑌) ⊆ (𝐴𝐸𝑅) ∧ ∃𝑠 ∈ (𝑃 ∖ (𝑋𝐿𝑌))(𝐴𝐸𝑅) = ((𝑋𝐿𝑌)𝐸𝑠)))
Theorem	lnssplng 29157*	A line defined by two points 𝑋 and 𝑌, both on a plane 𝐻, is entirely contained in 𝐻. Theorem 9.25 of [Schwabhauser] p. 75. (Contributed by Thierry Arnoux, 17-Jun-2026.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ 𝐸 = (hlG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐻 ∈ ran 𝐸)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐻)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐻)    &   ⊢ (𝜑 → 𝑋 ≠ 𝑌)    ⇒   ⊢ (𝜑 → ((𝑋𝐿𝑌) ⊆ 𝐻 ∧ ∃𝑠 ∈ (𝑃 ∖ (𝑋𝐿𝑌))𝐻 = ((𝑋𝐿𝑌)𝐸𝑠)))
Theorem	lnssplng1 29158	A line defined by two points 𝑋 and 𝑌, both on a plane 𝐻, is entirely contained in 𝐻. First part of Theorem 9.25 of [Schwabhauser] p. 75. (Contributed by Thierry Arnoux, 17-Jun-2026.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ 𝐸 = (hlG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐻 ∈ ran 𝐸)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐻)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐻)    &   ⊢ (𝜑 → 𝑋 ≠ 𝑌)    ⇒   ⊢ (𝜑 → (𝑋𝐿𝑌) ⊆ 𝐻)
Theorem	plngmiropp 29159*	Given a line 𝐴 and a point 𝑋 not on 𝐴, then a point 𝑌 on the plane defined by 𝐴 and 𝑋 is either opposite to 𝑋, or opposite to the mirror point of 𝑋 by any point 𝑍 of 𝐴. (Contributed by Thierry Arnoux, 5-Jul-2026.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ 𝐸 = (hlG‘𝐺)    &   ⊢ 𝑆 = (pInvG‘𝐺)    &   ⊢ 𝑀 = (𝑆‘𝑍)    &   ⊢ 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ 𝐴) ∧ 𝑏 ∈ (𝑃 ∖ 𝐴)) ∧ ∃𝑡 ∈ 𝐴 𝑡 ∈ (𝑎𝐼𝑏))}    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝑋 ∈ (𝑃 ∖ 𝐴))    &   ⊢ (𝜑 → 𝑌 ∈ ((𝐴𝐸𝑋) ∖ 𝐴))    &   ⊢ (𝜑 → 𝑍 ∈ 𝐴)    ⇒   ⊢ (𝜑 → (𝑋𝑂𝑌 ∨ (𝑀‘𝑋)𝑂𝑌))
Theorem	mirplncl 29160	The mirror of a point with regard to another point is in the same plane as the two points. (Contributed by Thierry Arnoux, 5-Jul-2026.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐸 = (hlG‘𝐺)    &   ⊢ 𝑆 = (pInvG‘𝐺)    &   ⊢ 𝑀 = (𝑆‘𝑋)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐻 ∈ ran 𝐸)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐻)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐻)    ⇒   ⊢ (𝜑 → (𝑀‘𝑌) ∈ 𝐻)
Theorem	hpgssplng 29161	Any point 𝑋 on a half plane defined by a line 𝐴 and another point 𝑌 is on the plane defined by 𝐴 and 𝑌. (Contributed by Thierry Arnoux, 5-Jul-2026.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ 𝐸 = (hlG‘𝐺)    &   ⊢ (𝜑 → 𝐴 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑌 ∈ (𝑃 ∖ 𝐴))    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝑋((hpG‘𝐺)‘𝐴)𝑌)    ⇒   ⊢ (𝜑 → 𝑋 ∈ (𝐴𝐸𝑌))
Theorem	plng3p 29162	If 𝐻 is a plane containing a line 𝐴 and a point 𝑅 not on 𝐴, then 𝐻 is the plane defined by 𝐴 and 𝑅. Theorem 9.26 of [Schwabhauser] p. 76. See tglinethru 28991 for the 2-point line equivalent. (Contributed by Thierry Arnoux, 17-Jun-2026.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ 𝐸 = (hlG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐻 ∈ ran 𝐸)    &   ⊢ (𝜑 → 𝐴 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝑅 ∈ (𝐻 ∖ 𝐴))    &   ⊢ (𝜑 → 𝐴 ⊆ 𝐻)    ⇒   ⊢ (𝜑 → 𝐻 = (𝐴𝐸𝑅))
Theorem	nhpmirhp 29163	If a point 𝑍 is on the plane defined by a line 𝐴 and a point 𝑌, but not on the same half-plane as 𝑌, then its mirror point (𝑀‘𝑍) by a point 𝑋 on 𝐴 is on the same half-plane as 𝑌. (Contributed by Thierry Arnoux, 5-Jul-2026.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ 𝑆 = (pInvG‘𝐺)    &   ⊢ 𝐸 = (hlG‘𝐺)    &   ⊢ 𝑀 = (𝑆‘𝑋)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑌 ∈ (𝑃 ∖ 𝐴))    &   ⊢ (𝜑 → 𝑍 ∈ ((𝐴𝐸𝑌) ∖ 𝐴))    &   ⊢ (𝜑 → ¬ 𝑌((hpG‘𝐺)‘𝐴)𝑍)    ⇒   ⊢ (𝜑 → 𝑌((hpG‘𝐺)‘𝐴)(𝑀‘𝑍))
16.2.16  Midpoints and Line Mirroring
Syntax	cmid 29164	Declare the constant for the midpoint operation.
class midG
Syntax	clmi 29165	Declare the constant for the line mirroring function.
class lInvG
Definition	df-mid 29166*	Define the midpoint operation. Definition 10.1 of [Schwabhauser] p. 88. See ismidb 29170, midbtwn 29171, and midcgr 29172. (Contributed by Thierry Arnoux, 9-Jun-2019.)
⊢ midG = (𝑔 ∈ V ↦ (𝑎 ∈ (Base‘𝑔), 𝑏 ∈ (Base‘𝑔) ↦ (℩𝑚 ∈ (Base‘𝑔)𝑏 = (((pInvG‘𝑔)‘𝑚)‘𝑎))))
Definition	df-lmi 29167*	Define the line mirroring function. Definition 10.3 of [Schwabhauser] p. 89. See islmib 29179. (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))))))
Theorem	midf 29168	Midpoint as a function. (Contributed by Thierry Arnoux, 1-Dec-2019.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ − = (dist‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐺DimTarskiG≥2)    ⇒   ⊢ (𝜑 → (midG‘𝐺):(𝑃 × 𝑃)⟶𝑃)
Theorem	midcl 29169	Closure of the midpoint. (Contributed by Thierry Arnoux, 1-Dec-2019.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ − = (dist‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐺DimTarskiG≥2)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    ⇒   ⊢ (𝜑 → (𝐴(midG‘𝐺)𝐵) ∈ 𝑃)
Theorem	ismidb 29170	Property of the midpoint. (Contributed by Thierry Arnoux, 1-Dec-2019.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ − = (dist‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐺DimTarskiG≥2)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ 𝑆 = (pInvG‘𝐺)    &   ⊢ (𝜑 → 𝑀 ∈ 𝑃)    ⇒   ⊢ (𝜑 → (𝐵 = ((𝑆‘𝑀)‘𝐴) ↔ (𝐴(midG‘𝐺)𝐵) = 𝑀))
Theorem	midbtwn 29171	Betweenness of midpoint. (Contributed by Thierry Arnoux, 7-Dec-2019.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ − = (dist‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐺DimTarskiG≥2)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    ⇒   ⊢ (𝜑 → (𝐴(midG‘𝐺)𝐵) ∈ (𝐴𝐼𝐵))
Theorem	midcgr 29172	Congruence of midpoint. (Contributed by Thierry Arnoux, 7-Dec-2019.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ − = (dist‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐺DimTarskiG≥2)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → (𝐴(midG‘𝐺)𝐵) = 𝐶)    ⇒   ⊢ (𝜑 → (𝐶 − 𝐴) = (𝐶 − 𝐵))
Theorem	midid 29173	Midpoint of a null segment. (Contributed by Thierry Arnoux, 7-Dec-2019.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ − = (dist‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐺DimTarskiG≥2)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    ⇒   ⊢ (𝜑 → (𝐴(midG‘𝐺)𝐴) = 𝐴)
Theorem	midcom 29174	Commutativity rule for the midpoint. (Contributed by Thierry Arnoux, 2-Dec-2019.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ − = (dist‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐺DimTarskiG≥2)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    ⇒   ⊢ (𝜑 → (𝐴(midG‘𝐺)𝐵) = (𝐵(midG‘𝐺)𝐴))
Theorem	mirmid 29175	Point inversion preserves midpoints. (Contributed by Thierry Arnoux, 12-Dec-2019.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ − = (dist‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐺DimTarskiG≥2)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ 𝑆 = ((pInvG‘𝐺)‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ 𝑃)    ⇒   ⊢ (𝜑 → ((𝑆‘𝐴)(midG‘𝐺)(𝑆‘𝐵)) = (𝑆‘(𝐴(midG‘𝐺)𝐵)))
Theorem	lmieu 29176*	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐺DimTarskiG≥2)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ (𝜑 → 𝐷 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    ⇒   ⊢ (𝜑 → ∃!𝑏 ∈ 𝑃 ((𝐴(midG‘𝐺)𝑏) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝑏) ∨ 𝐴 = 𝑏)))
Theorem	lmif 29177	Line mirror as a function. (Contributed by Thierry Arnoux, 11-Dec-2019.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ − = (dist‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐺DimTarskiG≥2)    &   ⊢ 𝑀 = ((lInvG‘𝐺)‘𝐷)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ (𝜑 → 𝐷 ∈ ran 𝐿)    ⇒   ⊢ (𝜑 → 𝑀:𝑃⟶𝑃)
Theorem	lmicl 29178	Closure of the line mirror. (Contributed by Thierry Arnoux, 11-Dec-2019.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ − = (dist‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐺DimTarskiG≥2)    &   ⊢ 𝑀 = ((lInvG‘𝐺)‘𝐷)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ (𝜑 → 𝐷 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    ⇒   ⊢ (𝜑 → (𝑀‘𝐴) ∈ 𝑃)
Theorem	islmib 29179	Property of the line mirror. (Contributed by Thierry Arnoux, 11-Dec-2019.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ − = (dist‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐺DimTarskiG≥2)    &   ⊢ 𝑀 = ((lInvG‘𝐺)‘𝐷)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ (𝜑 → 𝐷 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    ⇒   ⊢ (𝜑 → (𝐵 = (𝑀‘𝐴) ↔ ((𝐴(midG‘𝐺)𝐵) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))))
Theorem	lmicom 29180	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐺DimTarskiG≥2)    &   ⊢ 𝑀 = ((lInvG‘𝐺)‘𝐷)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ (𝜑 → 𝐷 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → (𝑀‘𝐴) = 𝐵)    ⇒   ⊢ (𝜑 → (𝑀‘𝐵) = 𝐴)
Theorem	lmilmi 29181	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐺DimTarskiG≥2)    &   ⊢ 𝑀 = ((lInvG‘𝐺)‘𝐷)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ (𝜑 → 𝐷 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    ⇒   ⊢ (𝜑 → (𝑀‘(𝑀‘𝐴)) = 𝐴)
Theorem	lmireu 29182*	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐺DimTarskiG≥2)    &   ⊢ 𝑀 = ((lInvG‘𝐺)‘𝐷)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ (𝜑 → 𝐷 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    ⇒   ⊢ (𝜑 → ∃!𝑏 ∈ 𝑃 (𝑀‘𝑏) = 𝐴)
Theorem	lmieq 29183	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐺DimTarskiG≥2)    &   ⊢ 𝑀 = ((lInvG‘𝐺)‘𝐷)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ (𝜑 → 𝐷 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → (𝑀‘𝐴) = (𝑀‘𝐵))    ⇒   ⊢ (𝜑 → 𝐴 = 𝐵)
Theorem	lmiinv 29184	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐺DimTarskiG≥2)    &   ⊢ 𝑀 = ((lInvG‘𝐺)‘𝐷)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ (𝜑 → 𝐷 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    ⇒   ⊢ (𝜑 → ((𝑀‘𝐴) = 𝐴 ↔ 𝐴 ∈ 𝐷))
Theorem	lmicinv 29185	The mirroring line is an invariant. (Contributed by Thierry Arnoux, 8-Aug-2020.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ − = (dist‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐺DimTarskiG≥2)    &   ⊢ 𝑀 = ((lInvG‘𝐺)‘𝐷)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ (𝜑 → 𝐷 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐷)    ⇒   ⊢ (𝜑 → (𝑀‘𝐴) = 𝐴)
Theorem	lmimid 29186	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐺DimTarskiG≥2)    &   ⊢ 𝑀 = ((lInvG‘𝐺)‘𝐷)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ (𝜑 → 𝐷 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ 𝑆 = ((pInvG‘𝐺)‘𝐵)    &   ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))    &   ⊢ (𝜑 → 𝐴 ∈ 𝐷)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐷)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐴 ≠ 𝐵)    ⇒   ⊢ (𝜑 → (𝑀‘𝐶) = (𝑆‘𝐶))
Theorem	lmif1o 29187	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐺DimTarskiG≥2)    &   ⊢ 𝑀 = ((lInvG‘𝐺)‘𝐷)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ (𝜑 → 𝐷 ∈ ran 𝐿)    ⇒   ⊢ (𝜑 → 𝑀:𝑃–1-1-onto→𝑃)
Theorem	lmiisolem 29188	Lemma for lmiiso 29189. (Contributed by Thierry Arnoux, 14-Dec-2019.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ − = (dist‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐺DimTarskiG≥2)    &   ⊢ 𝑀 = ((lInvG‘𝐺)‘𝐷)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ (𝜑 → 𝐷 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ 𝑆 = ((pInvG‘𝐺)‘𝑍)    &   ⊢ 𝑍 = ((𝐴(midG‘𝐺)(𝑀‘𝐴))(midG‘𝐺)(𝐵(midG‘𝐺)(𝑀‘𝐵)))    ⇒   ⊢ (𝜑 → ((𝑀‘𝐴) − (𝑀‘𝐵)) = (𝐴 − 𝐵))
Theorem	lmiiso 29189	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐺DimTarskiG≥2)    &   ⊢ 𝑀 = ((lInvG‘𝐺)‘𝐷)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ (𝜑 → 𝐷 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    ⇒   ⊢ (𝜑 → ((𝑀‘𝐴) − (𝑀‘𝐵)) = (𝐴 − 𝐵))
Theorem	lmimot 29190	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐺DimTarskiG≥2)    &   ⊢ 𝑀 = ((lInvG‘𝐺)‘𝐷)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ (𝜑 → 𝐷 ∈ ran 𝐿)    ⇒   ⊢ (𝜑 → 𝑀 ∈ (𝐺Ismt𝐺))
Theorem	symquadmid 29191*	In a symmetrical quadrilateral, the midpoints of the diagonals coincide. Corollary of Lemma 7.21 of [Schwabhauser] p. 52. (Contributed by Thierry Arnoux, 20-Jul-2026.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ − = (dist‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ 𝑀 = (midG‘𝐺)    &   ⊢ 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ (𝑋𝐿𝑍)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑋𝐿𝑍))) ∧ ∃𝑡 ∈ (𝑋𝐿𝑍)𝑡 ∈ (𝑎𝐼𝑏))}    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑊 ∈ 𝑃)    &   ⊢ (𝜑 → ¬ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))    &   ⊢ (𝜑 → 𝑌 ≠ 𝑊)    &   ⊢ (𝜑 → (𝑋 − 𝑌) = (𝑍 − 𝑊))    &   ⊢ (𝜑 → (𝑌 − 𝑍) = (𝑊 − 𝑋))    &   ⊢ (𝜑 → 𝑌𝑂𝑊)    ⇒   ⊢ (𝜑 → (𝑋𝑀𝑍) = (𝑌𝑀𝑊))
Theorem	hypcgrlem1 29192	Lemma for hypcgr 29194, case where triangles share a cathetus. (Contributed by Thierry Arnoux, 15-Dec-2019.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ − = (dist‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐺DimTarskiG≥2)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑃)    &   ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))    &   ⊢ (𝜑 → ⟨“𝐷𝐸𝐹”⟩ ∈ (∟G‘𝐺))    &   ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐷 − 𝐸))    &   ⊢ (𝜑 → (𝐵 − 𝐶) = (𝐸 − 𝐹))    &   ⊢ (𝜑 → 𝐵 = 𝐸)    &   ⊢ 𝑆 = ((lInvG‘𝐺)‘((𝐴(midG‘𝐺)𝐷)(LineG‘𝐺)𝐵))    &   ⊢ (𝜑 → 𝐶 = 𝐹)    ⇒   ⊢ (𝜑 → (𝐴 − 𝐶) = (𝐷 − 𝐹))
Theorem	hypcgrlem2 29193	Lemma for hypcgr 29194, case where triangles share one vertex 𝐵. (Contributed by Thierry Arnoux, 16-Dec-2019.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ − = (dist‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐺DimTarskiG≥2)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑃)    &   ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))    &   ⊢ (𝜑 → ⟨“𝐷𝐸𝐹”⟩ ∈ (∟G‘𝐺))    &   ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐷 − 𝐸))    &   ⊢ (𝜑 → (𝐵 − 𝐶) = (𝐸 − 𝐹))    &   ⊢ (𝜑 → 𝐵 = 𝐸)    &   ⊢ 𝑆 = ((lInvG‘𝐺)‘((𝐶(midG‘𝐺)𝐹)(LineG‘𝐺)𝐵))    ⇒   ⊢ (𝜑 → (𝐴 − 𝐶) = (𝐷 − 𝐹))
Theorem	hypcgr 29194	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐺DimTarskiG≥2)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑃)    &   ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))    &   ⊢ (𝜑 → ⟨“𝐷𝐸𝐹”⟩ ∈ (∟G‘𝐺))    &   ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐷 − 𝐸))    &   ⊢ (𝜑 → (𝐵 − 𝐶) = (𝐸 − 𝐹))    ⇒   ⊢ (𝜑 → (𝐴 − 𝐶) = (𝐷 − 𝐹))
Theorem	lmiopp 29195*	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐺DimTarskiG≥2)    &   ⊢ (𝜑 → 𝐷 ∈ ran 𝐿)    &   ⊢ 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   ⊢ 𝑀 = ((lInvG‘𝐺)‘𝐷)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → ¬ 𝐴 ∈ 𝐷)    ⇒   ⊢ (𝜑 → 𝐴𝑂(𝑀‘𝐴))
Theorem	lnperpex 29196*	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐺DimTarskiG≥2)    &   ⊢ (𝜑 → 𝐷 ∈ ran 𝐿)    &   ⊢ 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   ⊢ (𝜑 → 𝐴 ∈ 𝐷)    &   ⊢ (𝜑 → 𝑄 ∈ 𝑃)    &   ⊢ (𝜑 → ¬ 𝑄 ∈ 𝐷)    ⇒   ⊢ (𝜑 → ∃𝑝 ∈ 𝑃 (𝐷(⟂G‘𝐺)(𝑝𝐿𝐴) ∧ 𝑝((hpG‘𝐺)‘𝐷)𝑄))
Theorem	lnperpexs 29197*	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)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐺DimTarskiG≥2)    &   ⊢ (𝜑 → 𝐷 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐷)    &   ⊢ (𝜑 → 𝑄 ∈ 𝑃)    &   ⊢ (𝜑 → ¬ 𝑄 ∈ 𝐷)    ⇒   ⊢ (𝜑 → ∃𝑝 ∈ 𝑃 (𝐷(⟂G‘𝐺)(𝑝𝐿𝐴) ∧ 𝑝((hpG‘𝐺)‘𝐷)𝑄))
Theorem	trgcopy 29198*	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑃)    &   ⊢ (𝜑 → ¬ (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶))    &   ⊢ (𝜑 → ¬ (𝐷 ∈ (𝐸𝐿𝐹) ∨ 𝐸 = 𝐹))    &   ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐷 − 𝐸))    ⇒   ⊢ (𝜑 → ∃𝑓 ∈ 𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑓”⟩ ∧ 𝑓((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹))
Theorem	trgcopyeulem 29199*	Lemma for trgcopyeu 29200. (Contributed by Thierry Arnoux, 8-Aug-2020.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ − = (dist‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ 𝐾 = (hlG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑃)    &   ⊢ (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))}    &   ⊢ (𝜑 → 𝑋 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑃)    &   ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑋”⟩)    &   ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑌”⟩)    &   ⊢ (𝜑 → 𝑋((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)    &   ⊢ (𝜑 → 𝑌((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)    ⇒   ⊢ (𝜑 → 𝑋 = 𝑌)
Theorem	trgcopyeu 29200*	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)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑃)    &   ⊢ (𝜑 → ¬ (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶))    &   ⊢ (𝜑 → ¬ (𝐷 ∈ (𝐸𝐿𝐹) ∨ 𝐸 = 𝐹))    &   ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐷 − 𝐸))    ⇒   ⊢ (𝜑 → ∃!𝑓 ∈ 𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑓”⟩ ∧ 𝑓((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹))