P. 295 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 29401-29500   *Has distinct variable group(s)

Type	Label	Description
Statement
Syntax	cleag 29401	Extend class relation with the "angle less than" relation.
class ≤∠
Definition	df-inag 29402*	Definition of the geometrical "in angle" relation. (Contributed by Thierry Arnoux, 15-Aug-2020.)
⊢ inA = (𝑔 ∈ V ↦ {⟨𝑝, 𝑡⟩ ∣ ((𝑝 ∈ (Base‘𝑔) ∧ 𝑡 ∈ ((Base‘𝑔) ↑m (0..^3))) ∧ (((𝑡‘0) ≠ (𝑡‘1) ∧ (𝑡‘2) ≠ (𝑡‘1) ∧ 𝑝 ≠ (𝑡‘1)) ∧ ∃𝑥 ∈ (Base‘𝑔)(𝑥 ∈ ((𝑡‘0)(Itv‘𝑔)(𝑡‘2)) ∧ (𝑥 = (𝑡‘1) ∨ 𝑥((hlG‘𝑔)‘(𝑡‘1))𝑝))))})
Theorem	isinag 29403*	Property for point 𝑋 to lie in the angle ⟨“𝐴𝐵𝐶”⟩. Definition 11.23 of [Schwabhauser] p. 101. (Contributed by Thierry Arnoux, 15-Aug-2020.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ 𝐾 = (hlG‘𝐺)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝑋(inA‘𝐺)⟨“𝐴𝐵𝐶”⟩ ↔ ((𝐴 ≠ 𝐵 ∧ 𝐶 ≠ 𝐵 ∧ 𝑋 ≠ 𝐵) ∧ ∃𝑥 ∈ 𝑃 (𝑥 ∈ (𝐴𝐼𝐶) ∧ (𝑥 = 𝐵 ∨ 𝑥(𝐾‘𝐵)𝑋)))))
Theorem	isinagd 29404	Sufficient conditions for in-angle relation, deduction version. (Contributed by Thierry Arnoux, 20-Oct-2020.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ 𝐾 = (hlG‘𝐺)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐴 ≠ 𝐵)    &   ⊢ (𝜑 → 𝐶 ≠ 𝐵)    &   ⊢ (𝜑 → 𝑋 ≠ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ (𝐴𝐼𝐶))    &   ⊢ (𝜑 → (𝑌 = 𝐵 ∨ 𝑌(𝐾‘𝐵)𝑋))    ⇒   ⊢ (𝜑 → 𝑋(inA‘𝐺)⟨“𝐴𝐵𝐶”⟩)
Theorem	inagflat 29405	Any point lies in a flat angle. (Contributed by Thierry Arnoux, 13-Feb-2023.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ 𝐾 = (hlG‘𝐺)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐴 ≠ 𝐵)    &   ⊢ (𝜑 → 𝐶 ≠ 𝐵)    &   ⊢ (𝜑 → 𝑋 ≠ 𝐵)    &   ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶))    ⇒   ⊢ (𝜑 → 𝑋(inA‘𝐺)⟨“𝐴𝐵𝐶”⟩)
Theorem	inagswap 29406	Swap the order of the half lines delimiting the angle. Theorem 11.24 of [Schwabhauser] p. 101. (Contributed by Thierry Arnoux, 15-Aug-2020.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ 𝐾 = (hlG‘𝐺)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝑋(inA‘𝐺)⟨“𝐴𝐵𝐶”⟩)    ⇒   ⊢ (𝜑 → 𝑋(inA‘𝐺)⟨“𝐶𝐵𝐴”⟩)
Theorem	inagne1 29407	Deduce inequality from the in-angle relation. (Contributed by Thierry Arnoux, 29-Oct-2021.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ 𝐾 = (hlG‘𝐺)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝑋(inA‘𝐺)⟨“𝐴𝐵𝐶”⟩)    ⇒   ⊢ (𝜑 → 𝐴 ≠ 𝐵)
Theorem	inagne2 29408	Deduce inequality from the in-angle relation. (Contributed by Thierry Arnoux, 29-Oct-2021.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ 𝐾 = (hlG‘𝐺)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝑋(inA‘𝐺)⟨“𝐴𝐵𝐶”⟩)    ⇒   ⊢ (𝜑 → 𝐶 ≠ 𝐵)
Theorem	inagne3 29409	Deduce inequality from the in-angle relation. (Contributed by Thierry Arnoux, 29-Oct-2021.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ 𝐾 = (hlG‘𝐺)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝑋(inA‘𝐺)⟨“𝐴𝐵𝐶”⟩)    ⇒   ⊢ (𝜑 → 𝑋 ≠ 𝐵)
Theorem	inaghl 29410	The "point lie in angle" relation is independent of the points chosen on the half lines starting from 𝐵. Theorem 11.25 of [Schwabhauser] p. 101. (Contributed by Thierry Arnoux, 27-Sep-2020.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ 𝐾 = (hlG‘𝐺)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝑋(inA‘𝐺)⟨“𝐴𝐵𝐶”⟩)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐷(𝐾‘𝐵)𝐴)    &   ⊢ (𝜑 → 𝐹(𝐾‘𝐵)𝐶)    &   ⊢ (𝜑 → 𝑌(𝐾‘𝐵)𝑋)    ⇒   ⊢ (𝜑 → 𝑌(inA‘𝐺)⟨“𝐷𝐵𝐹”⟩)
Definition	df-leag 29411*	Definition of the geometrical "angle less than" relation. Definition 11.27 of [Schwabhauser] p. 102. (Contributed by Thierry Arnoux, 7-Oct-2020.)
⊢ ≤∠ = (𝑔 ∈ V ↦ {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ((Base‘𝑔) ↑m (0..^3)) ∧ 𝑏 ∈ ((Base‘𝑔) ↑m (0..^3))) ∧ ∃𝑥 ∈ (Base‘𝑔)(𝑥(inA‘𝑔)⟨“(𝑏‘0)(𝑏‘1)(𝑏‘2)”⟩ ∧ ⟨“(𝑎‘0)(𝑎‘1)(𝑎‘2)”⟩(cgrA‘𝑔)⟨“(𝑏‘0)(𝑏‘1)𝑥”⟩))})
Theorem	isleag 29412*	Geometrical "less than" property for angles. Definition 11.27 of [Schwabhauser] p. 102. (Contributed by Thierry Arnoux, 7-Oct-2020.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑃)    ⇒   ⊢ (𝜑 → (⟨“𝐴𝐵𝐶”⟩(≤∠‘𝐺)⟨“𝐷𝐸𝐹”⟩ ↔ ∃𝑥 ∈ 𝑃 (𝑥(inA‘𝐺)⟨“𝐷𝐸𝐹”⟩ ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝑥”⟩)))
Theorem	isleagd 29413	Sufficient condition for "less than" angle relation, deduction version (Contributed by Thierry Arnoux, 12-Oct-2020.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑃)    &   ⊢ ≤ = (≤∠‘𝐺)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑋(inA‘𝐺)⟨“𝐷𝐸𝐹”⟩)    &   ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝑋”⟩)    ⇒   ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ≤ ⟨“𝐷𝐸𝐹”⟩)
Theorem	leagne1 29414	Deduce inequality from the less-than angle relation. (Contributed by Thierry Arnoux, 25-Feb-2023.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑃)    &   ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩(≤∠‘𝐺)⟨“𝐷𝐸𝐹”⟩)    ⇒   ⊢ (𝜑 → 𝐴 ≠ 𝐵)
Theorem	leagne2 29415	Deduce inequality from the less-than angle relation. (Contributed by Thierry Arnoux, 25-Feb-2023.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑃)    &   ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩(≤∠‘𝐺)⟨“𝐷𝐸𝐹”⟩)    ⇒   ⊢ (𝜑 → 𝐶 ≠ 𝐵)
Theorem	leagne3 29416	Deduce inequality from the less-than angle relation. (Contributed by Thierry Arnoux, 25-Feb-2023.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑃)    &   ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩(≤∠‘𝐺)⟨“𝐷𝐸𝐹”⟩)    ⇒   ⊢ (𝜑 → 𝐷 ≠ 𝐸)
Theorem	leagne4 29417	Deduce inequality from the less-than angle relation. (Contributed by Thierry Arnoux, 25-Feb-2023.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑃)    &   ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩(≤∠‘𝐺)⟨“𝐷𝐸𝐹”⟩)    ⇒   ⊢ (𝜑 → 𝐹 ≠ 𝐸)
Theorem	cgrg3col4 29418*	Lemma 11.28 of [Schwabhauser] p. 102. Extend a congruence of three points with a fourth colinear point. (Contributed by Thierry Arnoux, 8-Oct-2020.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑃)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑃)    &   ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩)    &   ⊢ (𝜑 → (𝑋 ∈ (𝐴𝐿𝐶) ∨ 𝐴 = 𝐶))    ⇒   ⊢ (𝜑 → ∃𝑦 ∈ 𝑃 ⟨“𝐴𝐵𝐶𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹𝑦”⟩)
16.2.19  Angle Addition
Syntax	cangmgm 29419	Extend class relation with the "angle addition" magma.
class AngMgm
Definition	df-angmgm 29420*	Definition of the angle addition magma. See following theorems for better readable properties. Because the textbook's definition of angle congruence does not consider orientation (cf. cgraswap 29373) , our notion of angle does not include reflex angles (angles larger than a straight angle). Therefore, like with ℕ0, at this point, we can only build a magma , which is e.g. not isomorphic with the rotation group SO(2) . (Contributed by Thierry Arnoux, 20-Jul-2026.)
⊢ AngMgm = (𝑔 ∈ V ↦ ⦋(Base‘𝑔) / 𝑝⦌⦋{𝑑 ∈ (𝑝 ↑m (0..^3)) ∣ ((𝑑‘0) ≠ (𝑑‘1) ∧ (𝑑‘1) ≠ (𝑑‘2))} / 𝑎⦌({⟨(Base‘ndx), 𝑎⟩, ⟨(+g‘ndx), (𝑒 ∈ 𝑎, 𝑓 ∈ 𝑎 ↦ if((𝑒‘0) ∈ ((𝑒‘1)(LineG‘𝑔)(𝑒‘2)), ⟨“(𝑓‘0)(𝑓‘1)(℩𝑧 ∈ 𝑝 (⟨“(𝑓‘2)(𝑓‘1)𝑧”⟩(cgrA‘𝑔)𝑒 ∧ ((𝑓‘1)(dist‘𝑔)𝑧) = ((𝑒‘1)(dist‘𝑔)(𝑒‘0))))”⟩, ⟨“(𝑒‘0)(𝑒‘1)(℩𝑧 ∈ 𝑝 (⟨“(𝑒‘2)(𝑒‘1)𝑧”⟩(cgrA‘𝑔)𝑓 ∧ ((𝑒‘1)(dist‘𝑔)𝑧) = ((𝑓‘1)(dist‘𝑔)(𝑓‘0)) ∧ (((𝑒‘1)(LineG‘𝑔)(𝑒‘2)) ∩ (𝑧(Itv‘𝑔)(𝑒‘0))) ≠ ∅))”⟩))⟩, ⟨(le‘ndx), (≤∠‘𝑔)⟩} /s (cgrA‘𝑔)))
Theorem	elcgrabasi 29421*	Helper theorem for the membership in the base set of the angle addition monoid. (Contributed by Thierry Arnoux, 23-Aug-2026.)
⊢ 𝑃 ∈ V    &   ⊢ 𝐴 = {𝑑 ∈ (𝑃 ↑m (0..^3)) ∣ ((𝑑‘0) ≠ (𝑑‘1) ∧ (𝑑‘1) ≠ (𝑑‘2))}    &   ⊢ (𝜑 → 𝐸 ∈ 𝐴)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 ∃𝑧 ∈ 𝑃 (𝐸 = ⟨“𝑥𝑦𝑧”⟩ ∧ (𝑥 ≠ 𝑦 ∧ 𝑦 ≠ 𝑧)))
Theorem	elcgrabasrd 29422*	Helper theorem for the membership in the base set of the angle addition monoid. (Contributed by Thierry Arnoux, 23-Aug-2026.)
⊢ 𝐴 = {𝑑 ∈ (𝑃 ↑m (0..^3)) ∣ ((𝑑‘0) ≠ (𝑑‘1) ∧ (𝑑‘1) ≠ (𝑑‘2))}    &   ⊢ (𝜑 → 𝑃 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑋 ≠ 𝑌)    &   ⊢ (𝜑 → 𝑌 ≠ 𝑍)    ⇒   ⊢ (𝜑 → ⟨“𝑋𝑌𝑍”⟩ ∈ 𝐴)
Theorem	cgraer 29423*	The angle congruence relation is an equivalence relation. (Contributed by Thierry Arnoux, 31-Aug-2026.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐴 = {𝑑 ∈ (𝑃 ↑m (0..^3)) ∣ ((𝑑‘0) ≠ (𝑑‘1) ∧ (𝑑‘1) ≠ (𝑑‘2))}    &   ⊢ ∼ = (cgrA‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    ⇒   ⊢ (𝜑 → ( ∼ ∩ (𝐴 × 𝐴)) Er 𝐴)
Theorem	cgrabasimass 29424*	The angle congruence relation is hereditary. (Contributed by Thierry Arnoux, 31-Aug-2026.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐴 = {𝑑 ∈ (𝑃 ↑m (0..^3)) ∣ ((𝑑‘0) ≠ (𝑑‘1) ∧ (𝑑‘1) ≠ (𝑑‘2))}    &   ⊢ ∼ = (cgrA‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    ⇒   ⊢ (𝜑 → ( ∼ “ 𝐴) ⊆ 𝐴)
Theorem	angmgmaddeu1 29425*	There exists a unique point 𝑠 satisfying the conditions of angle addition. General case. (Contributed by Thierry Arnoux, 23-Aug-2026.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐴 = {𝑑 ∈ (𝑃 ↑m (0..^3)) ∣ ((𝑑‘0) ≠ (𝑑‘1) ∧ (𝑑‘1) ≠ (𝑑‘2))}    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ − = (dist‘𝐺)    &   ⊢ ∼ = (cgrA‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑉 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑊 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑈 ≠ 𝑉)    &   ⊢ (𝜑 → 𝑉 ≠ 𝑊)    &   ⊢ (𝜑 → 𝑋 ≠ 𝑌)    &   ⊢ (𝜑 → 𝑌 ≠ 𝑍)    &   ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑌𝐿𝑍))    &   ⊢ (𝜑 → ¬ 𝑈 ∈ (𝑉𝐿𝑊))    ⇒   ⊢ (𝜑 → ∃!𝑠 ∈ 𝑃 (⟨“𝑍𝑌𝑠”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩ ∧ (𝑌 − 𝑠) = (𝑉 − 𝑈) ∧ ((𝑌𝐿𝑍) ∩ (𝑠𝐼𝑋)) ≠ ∅))
Theorem	angmgmaddeu2 29426*	Existence of a unique point for building angle addition. Case where the second angle is a zero angle. (Contributed by Thierry Arnoux, 23-Aug-2026.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐴 = {𝑑 ∈ (𝑃 ↑m (0..^3)) ∣ ((𝑑‘0) ≠ (𝑑‘1) ∧ (𝑑‘1) ≠ (𝑑‘2))}    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ − = (dist‘𝐺)    &   ⊢ ∼ = (cgrA‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑉 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑊 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑈 ≠ 𝑉)    &   ⊢ (𝜑 → 𝑉 ≠ 𝑊)    &   ⊢ (𝜑 → 𝑋 ≠ 𝑌)    &   ⊢ (𝜑 → 𝑌 ≠ 𝑍)    &   ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑌𝐿𝑍))    &   ⊢ (𝜑 → 𝑈((hlG‘𝐺)‘𝑉)𝑊)    ⇒   ⊢ (𝜑 → ∃!𝑠 ∈ 𝑃 (⟨“𝑍𝑌𝑠”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩ ∧ (𝑌 − 𝑠) = (𝑉 − 𝑈) ∧ ((𝑌𝐿𝑍) ∩ (𝑠𝐼𝑋)) ≠ ∅))
Theorem	angmgmaddeu3 29427*	Existence of a unique point for building angle addition. Case where the second angle is a flat angle. (Contributed by Thierry Arnoux, 23-Aug-2026.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐴 = {𝑑 ∈ (𝑃 ↑m (0..^3)) ∣ ((𝑑‘0) ≠ (𝑑‘1) ∧ (𝑑‘1) ≠ (𝑑‘2))}    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ − = (dist‘𝐺)    &   ⊢ ∼ = (cgrA‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑉 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑊 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑈 ≠ 𝑉)    &   ⊢ (𝜑 → 𝑉 ≠ 𝑊)    &   ⊢ (𝜑 → 𝑋 ≠ 𝑌)    &   ⊢ (𝜑 → 𝑌 ≠ 𝑍)    &   ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑌𝐿𝑍))    &   ⊢ (𝜑 → 𝑉 ∈ (𝑈𝐼𝑊))    ⇒   ⊢ (𝜑 → ∃!𝑠 ∈ 𝑃 (⟨“𝑍𝑌𝑠”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩ ∧ (𝑌 − 𝑠) = (𝑉 − 𝑈) ∧ ((𝑌𝐿𝑍) ∩ (𝑠𝐼𝑋)) ≠ ∅))
Theorem	angmgmaddeu4 29428*	There exists a unique point 𝑠 satisfying the conditions of angle addition. Case where both angles are zero angles. (Contributed by Thierry Arnoux, 23-Aug-2026.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐴 = {𝑑 ∈ (𝑃 ↑m (0..^3)) ∣ ((𝑑‘0) ≠ (𝑑‘1) ∧ (𝑑‘1) ≠ (𝑑‘2))}    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ − = (dist‘𝐺)    &   ⊢ ∼ = (cgrA‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑉 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑊 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑈 ≠ 𝑉)    &   ⊢ (𝜑 → 𝑉 ≠ 𝑊)    &   ⊢ (𝜑 → 𝑋 ≠ 𝑌)    &   ⊢ (𝜑 → 𝑌 ≠ 𝑍)    &   ⊢ (𝜑 → 𝑋((hlG‘𝐺)‘𝑌)𝑍)    &   ⊢ (𝜑 → 𝑈((hlG‘𝐺)‘𝑉)𝑊)    ⇒   ⊢ (𝜑 → ∃!𝑠 ∈ 𝑃 (⟨“𝑍𝑌𝑠”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩ ∧ (𝑌 − 𝑠) = (𝑉 − 𝑈)))
Theorem	angmgmaddeu5 29429*	There exists a unique point 𝑠 satisfying the conditions of angle addition. Case where one angle is a zero angle, and the other angle is a straight angle. (Contributed by Thierry Arnoux, 23-Aug-2026.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐴 = {𝑑 ∈ (𝑃 ↑m (0..^3)) ∣ ((𝑑‘0) ≠ (𝑑‘1) ∧ (𝑑‘1) ≠ (𝑑‘2))}    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ − = (dist‘𝐺)    &   ⊢ ∼ = (cgrA‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑉 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑊 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑈 ≠ 𝑉)    &   ⊢ (𝜑 → 𝑉 ≠ 𝑊)    &   ⊢ (𝜑 → 𝑋 ≠ 𝑌)    &   ⊢ (𝜑 → 𝑌 ≠ 𝑍)    &   ⊢ (𝜑 → 𝑋((hlG‘𝐺)‘𝑌)𝑍)    &   ⊢ (𝜑 → 𝑉 ∈ (𝑈𝐼𝑊))    ⇒   ⊢ (𝜑 → ∃!𝑠 ∈ 𝑃 (⟨“𝑍𝑌𝑠”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩ ∧ (𝑌 − 𝑠) = (𝑉 − 𝑈)))
Theorem	angmgmaddeu6 29430*	There exists a unique point 𝑠 satisfying the conditions of angle addition. Case where the first angle is a zero angle, and the second angle is a straight angle. (Contributed by Thierry Arnoux, 23-Aug-2026.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐴 = {𝑑 ∈ (𝑃 ↑m (0..^3)) ∣ ((𝑑‘0) ≠ (𝑑‘1) ∧ (𝑑‘1) ≠ (𝑑‘2))}    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ − = (dist‘𝐺)    &   ⊢ ∼ = (cgrA‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑉 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑊 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑈 ≠ 𝑉)    &   ⊢ (𝜑 → 𝑉 ≠ 𝑊)    &   ⊢ (𝜑 → 𝑋 ≠ 𝑌)    &   ⊢ (𝜑 → 𝑌 ≠ 𝑍)    &   ⊢ (𝜑 → 𝑌 ∈ (𝑋𝐼𝑍))    &   ⊢ (𝜑 → 𝑈((hlG‘𝐺)‘𝑉)𝑊)    ⇒   ⊢ (𝜑 → ∃!𝑠 ∈ 𝑃 (⟨“𝑍𝑌𝑠”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩ ∧ (𝑌 − 𝑠) = (𝑉 − 𝑈)))
Theorem	angmgmaddeu7 29431*	There exists a unique point 𝑠 satisfying the conditions of angle addition. Case where both angles are straight angles. (Contributed by Thierry Arnoux, 23-Aug-2026.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐴 = {𝑑 ∈ (𝑃 ↑m (0..^3)) ∣ ((𝑑‘0) ≠ (𝑑‘1) ∧ (𝑑‘1) ≠ (𝑑‘2))}    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ − = (dist‘𝐺)    &   ⊢ ∼ = (cgrA‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑉 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑊 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑈 ≠ 𝑉)    &   ⊢ (𝜑 → 𝑉 ≠ 𝑊)    &   ⊢ (𝜑 → 𝑋 ≠ 𝑌)    &   ⊢ (𝜑 → 𝑌 ≠ 𝑍)    &   ⊢ (𝜑 → 𝑌 ∈ (𝑋𝐼𝑍))    &   ⊢ (𝜑 → 𝑉 ∈ (𝑈𝐼𝑊))    ⇒   ⊢ (𝜑 → ∃!𝑠 ∈ 𝑃 (⟨“𝑍𝑌𝑠”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩ ∧ (𝑌 − 𝑠) = (𝑉 − 𝑈)))
Theorem	angmgmaddov1lem 29432*	Lemma for angmgmaddov1 29434. (Contributed by Thierry Arnoux, 23-Aug-2026.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐴 = {𝑑 ∈ (𝑃 ↑m (0..^3)) ∣ ((𝑑‘0) ≠ (𝑑‘1) ∧ (𝑑‘1) ≠ (𝑑‘2))}    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ − = (dist‘𝐺)    &   ⊢ ∼ = (cgrA‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑉 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑊 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑈 ≠ 𝑉)    &   ⊢ (𝜑 → 𝑉 ≠ 𝑊)    &   ⊢ (𝜑 → 𝑋 ≠ 𝑌)    &   ⊢ (𝜑 → 𝑌 ≠ 𝑍)    &   ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑌𝐿𝑍))    ⇒   ⊢ (𝜑 → ∃!𝑠 ∈ 𝑃 (⟨“𝑍𝑌𝑠”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩ ∧ (𝑌 − 𝑠) = (𝑉 − 𝑈) ∧ ((𝑌𝐿𝑍) ∩ (𝑠𝐼𝑋)) ≠ ∅))
Theorem	angmgmaddov2lem 29433*	Lemma for angmgmaddov2 29435. (Contributed by Thierry Arnoux, 23-Aug-2026.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐴 = {𝑑 ∈ (𝑃 ↑m (0..^3)) ∣ ((𝑑‘0) ≠ (𝑑‘1) ∧ (𝑑‘1) ≠ (𝑑‘2))}    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ − = (dist‘𝐺)    &   ⊢ ∼ = (cgrA‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑉 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑊 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑈 ≠ 𝑉)    &   ⊢ (𝜑 → 𝑉 ≠ 𝑊)    &   ⊢ (𝜑 → 𝑋 ≠ 𝑌)    &   ⊢ (𝜑 → 𝑌 ≠ 𝑍)    &   ⊢ (𝜑 → 𝑋 ∈ (𝑌𝐿𝑍))    ⇒   ⊢ (𝜑 → ∃!𝑠 ∈ 𝑃 (⟨“𝑊𝑉𝑠”⟩ ∼ ⟨“𝑋𝑌𝑍”⟩ ∧ (𝑉 − 𝑠) = (𝑌 − 𝑋)))
Theorem	angmgmaddov1 29434*	Value of the addition operation in the angle addition magma, in case the first angle is neither zero nor flat. (Contributed by Thierry Arnoux, 23-Aug-2026.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐴 = {𝑑 ∈ (𝑃 ↑m (0..^3)) ∣ ((𝑑‘0) ≠ (𝑑‘1) ∧ (𝑑‘1) ≠ (𝑑‘2))}    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ − = (dist‘𝐺)    &   ⊢ ∼ = (cgrA‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑉 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑊 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑈 ≠ 𝑉)    &   ⊢ (𝜑 → 𝑉 ≠ 𝑊)    &   ⊢ (𝜑 → 𝑋 ≠ 𝑌)    &   ⊢ (𝜑 → 𝑌 ≠ 𝑍)    &   ⊢ + = (𝑒 ∈ 𝐴, 𝑓 ∈ 𝐴 ↦ if((𝑒‘0) ∈ ((𝑒‘1)𝐿(𝑒‘2)), ⟨“(𝑓‘0)(𝑓‘1)(℩𝑠 ∈ 𝑃 (⟨“(𝑓‘2)(𝑓‘1)𝑠”⟩ ∼ 𝑒 ∧ ((𝑓‘1) − 𝑠) = ((𝑒‘1) − (𝑒‘0))))”⟩, ⟨“(𝑒‘0)(𝑒‘1)(℩𝑠 ∈ 𝑃 (⟨“(𝑒‘2)(𝑒‘1)𝑠”⟩ ∼ 𝑓 ∧ ((𝑒‘1) − 𝑠) = ((𝑓‘1) − (𝑓‘0)) ∧ (((𝑒‘1)𝐿(𝑒‘2)) ∩ (𝑠𝐼(𝑒‘0))) ≠ ∅))”⟩))    &   ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑌𝐿𝑍))    &   ⊢ (𝜑 → 𝑆 ∈ 𝑃)    &   ⊢ (𝜑 → ⟨“𝑍𝑌𝑆”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩)    &   ⊢ (𝜑 → (𝑌 − 𝑆) = (𝑉 − 𝑈))    &   ⊢ (𝜑 → ((𝑌𝐿𝑍) ∩ (𝑆𝐼𝑋)) ≠ ∅)    ⇒   ⊢ (𝜑 → (⟨“𝑋𝑌𝑍”⟩ + ⟨“𝑈𝑉𝑊”⟩) = ⟨“𝑋𝑌𝑆”⟩)
Theorem	angmgmaddov2 29435*	Value of the addition operation in the angle addition magma, in case the first angle is zero or flat. (Contributed by Thierry Arnoux, 23-Aug-2026.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐴 = {𝑑 ∈ (𝑃 ↑m (0..^3)) ∣ ((𝑑‘0) ≠ (𝑑‘1) ∧ (𝑑‘1) ≠ (𝑑‘2))}    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ − = (dist‘𝐺)    &   ⊢ ∼ = (cgrA‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑉 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑊 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑈 ≠ 𝑉)    &   ⊢ (𝜑 → 𝑉 ≠ 𝑊)    &   ⊢ (𝜑 → 𝑋 ≠ 𝑌)    &   ⊢ (𝜑 → 𝑌 ≠ 𝑍)    &   ⊢ + = (𝑒 ∈ 𝐴, 𝑓 ∈ 𝐴 ↦ if((𝑒‘0) ∈ ((𝑒‘1)𝐿(𝑒‘2)), ⟨“(𝑓‘0)(𝑓‘1)(℩𝑠 ∈ 𝑃 (⟨“(𝑓‘2)(𝑓‘1)𝑠”⟩ ∼ 𝑒 ∧ ((𝑓‘1) − 𝑠) = ((𝑒‘1) − (𝑒‘0))))”⟩, ⟨“(𝑒‘0)(𝑒‘1)(℩𝑠 ∈ 𝑃 (⟨“(𝑒‘2)(𝑒‘1)𝑠”⟩ ∼ 𝑓 ∧ ((𝑒‘1) − 𝑠) = ((𝑓‘1) − (𝑓‘0)) ∧ (((𝑒‘1)𝐿(𝑒‘2)) ∩ (𝑠𝐼(𝑒‘0))) ≠ ∅))”⟩))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑌𝐿𝑍))    &   ⊢ (𝜑 → 𝑆 ∈ 𝑃)    &   ⊢ (𝜑 → ⟨“𝑊𝑉𝑆”⟩ ∼ ⟨“𝑋𝑌𝑍”⟩)    &   ⊢ (𝜑 → (𝑉 − 𝑆) = (𝑌 − 𝑋))    ⇒   ⊢ (𝜑 → (⟨“𝑋𝑌𝑍”⟩ + ⟨“𝑈𝑉𝑊”⟩) = ⟨“𝑈𝑉𝑆”⟩)
Theorem	angmgmaddcpbl 29436*	Addition of angles is compatible with angle congruence. (Contributed by Thierry Arnoux, 23-Aug-2026.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐴 = {𝑑 ∈ (𝑃 ↑m (0..^3)) ∣ ((𝑑‘0) ≠ (𝑑‘1) ∧ (𝑑‘1) ≠ (𝑑‘2))}    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ − = (dist‘𝐺)    &   ⊢ ∼ = (cgrA‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ + = (𝑒 ∈ 𝐴, 𝑓 ∈ 𝐴 ↦ if((𝑒‘0) ∈ ((𝑒‘1)𝐿(𝑒‘2)), ⟨“(𝑓‘0)(𝑓‘1)(℩𝑠 ∈ 𝑃 (⟨“(𝑓‘2)(𝑓‘1)𝑠”⟩ ∼ 𝑒 ∧ ((𝑓‘1) − 𝑠) = ((𝑒‘1) − (𝑒‘0))))”⟩, ⟨“(𝑒‘0)(𝑒‘1)(℩𝑠 ∈ 𝑃 (⟨“(𝑒‘2)(𝑒‘1)𝑠”⟩ ∼ 𝑓 ∧ ((𝑒‘1) − 𝑠) = ((𝑓‘1) − (𝑓‘0)) ∧ (((𝑒‘1)𝐿(𝑒‘2)) ∩ (𝑠𝐼(𝑒‘0))) ≠ ∅))”⟩))    &   ⊢ (𝜑 → 𝐶 ∈ 𝐴)    &   ⊢ (𝜑 → 𝐷 ∈ 𝐴)    &   ⊢ (𝜑 → 𝐸 ∈ 𝐴)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐴)    &   ⊢ (𝜑 → 𝐸 ∼ 𝐶)    &   ⊢ (𝜑 → 𝐹 ∼ 𝐷)    ⇒   ⊢ (𝜑 → (𝐸 + 𝐹) ∼ (𝐶 + 𝐷))
Theorem	angmgmaddcl 29437*	Closure of the addition of angles. (Contributed by Thierry Arnoux, 31-Aug-2026.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐴 = {𝑑 ∈ (𝑃 ↑m (0..^3)) ∣ ((𝑑‘0) ≠ (𝑑‘1) ∧ (𝑑‘1) ≠ (𝑑‘2))}    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ − = (dist‘𝐺)    &   ⊢ ∼ = (cgrA‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ + = (𝑒 ∈ 𝐴, 𝑓 ∈ 𝐴 ↦ if((𝑒‘0) ∈ ((𝑒‘1)𝐿(𝑒‘2)), ⟨“(𝑓‘0)(𝑓‘1)(℩𝑠 ∈ 𝑃 (⟨“(𝑓‘2)(𝑓‘1)𝑠”⟩ ∼ 𝑒 ∧ ((𝑓‘1) − 𝑠) = ((𝑒‘1) − (𝑒‘0))))”⟩, ⟨“(𝑒‘0)(𝑒‘1)(℩𝑠 ∈ 𝑃 (⟨“(𝑒‘2)(𝑒‘1)𝑠”⟩ ∼ 𝑓 ∧ ((𝑒‘1) − 𝑠) = ((𝑓‘1) − (𝑓‘0)) ∧ (((𝑒‘1)𝐿(𝑒‘2)) ∩ (𝑠𝐼(𝑒‘0))) ≠ ∅))”⟩))    &   ⊢ (𝜑 → 𝐸 ∈ 𝐴)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐴)    ⇒   ⊢ (𝜑 → (𝐸 + 𝐹) ∈ 𝐴)
Theorem	angmgmaddlid 29438*	The left identity element for the addition of angles. (Contributed by Thierry Arnoux, 31-Aug-2026.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐴 = {𝑑 ∈ (𝑃 ↑m (0..^3)) ∣ ((𝑑‘0) ≠ (𝑑‘1) ∧ (𝑑‘1) ≠ (𝑑‘2))}    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ − = (dist‘𝐺)    &   ⊢ ∼ = (cgrA‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ + = (𝑒 ∈ 𝐴, 𝑓 ∈ 𝐴 ↦ if((𝑒‘0) ∈ ((𝑒‘1)𝐿(𝑒‘2)), ⟨“(𝑓‘0)(𝑓‘1)(℩𝑠 ∈ 𝑃 (⟨“(𝑓‘2)(𝑓‘1)𝑠”⟩ ∼ 𝑒 ∧ ((𝑓‘1) − 𝑠) = ((𝑒‘1) − (𝑒‘0))))”⟩, ⟨“(𝑒‘0)(𝑒‘1)(℩𝑠 ∈ 𝑃 (⟨“(𝑒‘2)(𝑒‘1)𝑠”⟩ ∼ 𝑓 ∧ ((𝑒‘1) − 𝑠) = ((𝑓‘1) − (𝑓‘0)) ∧ (((𝑒‘1)𝐿(𝑒‘2)) ∩ (𝑠𝐼(𝑒‘0))) ≠ ∅))”⟩))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑌 ∈ (𝑃 ∖ {𝑋}))    &   ⊢ (𝜑 → 𝐸 ∈ 𝐴)    ⇒   ⊢ (𝜑 → (⟨“𝑋𝑌𝑋”⟩ + 𝐸) ∼ 𝐸)
Theorem	angmgmaddrid 29439*	The right identity element for the addition of angles. (Contributed by Thierry Arnoux, 31-Aug-2026.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐴 = {𝑑 ∈ (𝑃 ↑m (0..^3)) ∣ ((𝑑‘0) ≠ (𝑑‘1) ∧ (𝑑‘1) ≠ (𝑑‘2))}    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ − = (dist‘𝐺)    &   ⊢ ∼ = (cgrA‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ + = (𝑒 ∈ 𝐴, 𝑓 ∈ 𝐴 ↦ if((𝑒‘0) ∈ ((𝑒‘1)𝐿(𝑒‘2)), ⟨“(𝑓‘0)(𝑓‘1)(℩𝑠 ∈ 𝑃 (⟨“(𝑓‘2)(𝑓‘1)𝑠”⟩ ∼ 𝑒 ∧ ((𝑓‘1) − 𝑠) = ((𝑒‘1) − (𝑒‘0))))”⟩, ⟨“(𝑒‘0)(𝑒‘1)(℩𝑠 ∈ 𝑃 (⟨“(𝑒‘2)(𝑒‘1)𝑠”⟩ ∼ 𝑓 ∧ ((𝑒‘1) − 𝑠) = ((𝑓‘1) − (𝑓‘0)) ∧ (((𝑒‘1)𝐿(𝑒‘2)) ∩ (𝑠𝐼(𝑒‘0))) ≠ ∅))”⟩))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑌 ∈ (𝑃 ∖ {𝑋}))    &   ⊢ (𝜑 → 𝐸 ∈ 𝐴)    ⇒   ⊢ (𝜑 → (𝐸 + ⟨“𝑋𝑌𝑋”⟩) ∼ 𝐸)
Theorem	angmgmval 29440*	Explicit the value of the angle addition magma for a given geometry 𝐺. (Contributed by Thierry Arnoux, 31-Aug-2026.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐴 = {𝑑 ∈ (𝑃 ↑m (0..^3)) ∣ ((𝑑‘0) ≠ (𝑑‘1) ∧ (𝑑‘1) ≠ (𝑑‘2))}    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ − = (dist‘𝐺)    &   ⊢ ∼ = (cgrA‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ + = (𝑒 ∈ 𝐴, 𝑓 ∈ 𝐴 ↦ if((𝑒‘0) ∈ ((𝑒‘1)𝐿(𝑒‘2)), ⟨“(𝑓‘0)(𝑓‘1)(℩𝑠 ∈ 𝑃 (⟨“(𝑓‘2)(𝑓‘1)𝑠”⟩ ∼ 𝑒 ∧ ((𝑓‘1) − 𝑠) = ((𝑒‘1) − (𝑒‘0))))”⟩, ⟨“(𝑒‘0)(𝑒‘1)(℩𝑠 ∈ 𝑃 (⟨“(𝑒‘2)(𝑒‘1)𝑠”⟩ ∼ 𝑓 ∧ ((𝑒‘1) − 𝑠) = ((𝑓‘1) − (𝑓‘0)) ∧ (((𝑒‘1)𝐿(𝑒‘2)) ∩ (𝑠𝐼(𝑒‘0))) ≠ ∅))”⟩))    &   ⊢ 𝐽 = (AngMgm‘𝐺)    &   ⊢ ≤ = (≤∠‘𝐺)    ⇒   ⊢ (𝐺 ∈ 𝑉 → 𝐽 = ({⟨(Base‘ndx), 𝐴⟩, ⟨(+g‘ndx), + ⟩, ⟨(le‘ndx), ≤ ⟩} /s ∼ ))
Theorem	angmgmlem 29441*	Lemma for angmgm 29443. (Contributed by Thierry Arnoux, 31-Aug-2026.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐴 = {𝑑 ∈ (𝑃 ↑m (0..^3)) ∣ ((𝑑‘0) ≠ (𝑑‘1) ∧ (𝑑‘1) ≠ (𝑑‘2))}    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ − = (dist‘𝐺)    &   ⊢ ∼ = (cgrA‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ + = (𝑒 ∈ 𝐴, 𝑓 ∈ 𝐴 ↦ if((𝑒‘0) ∈ ((𝑒‘1)𝐿(𝑒‘2)), ⟨“(𝑓‘0)(𝑓‘1)(℩𝑠 ∈ 𝑃 (⟨“(𝑓‘2)(𝑓‘1)𝑠”⟩ ∼ 𝑒 ∧ ((𝑓‘1) − 𝑠) = ((𝑒‘1) − (𝑒‘0))))”⟩, ⟨“(𝑒‘0)(𝑒‘1)(℩𝑠 ∈ 𝑃 (⟨“(𝑒‘2)(𝑒‘1)𝑠”⟩ ∼ 𝑓 ∧ ((𝑒‘1) − 𝑠) = ((𝑓‘1) − (𝑓‘0)) ∧ (((𝑒‘1)𝐿(𝑒‘2)) ∩ (𝑠𝐼(𝑒‘0))) ≠ ∅))”⟩))    &   ⊢ 𝐽 = (AngMgm‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑌 ∈ (𝑃 ∖ {𝑋}))    ⇒   ⊢ (𝜑 → (𝐽 ∈ Mgm ∧ [⟨“𝑋𝑌𝑋”⟩] ∼ = (0g‘𝐽)))
Theorem	angmgm0g 29442*	The identity element of the angle addition magma. (Contributed by Thierry Arnoux, 31-Aug-2026.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐴 = {𝑑 ∈ (𝑃 ↑m (0..^3)) ∣ ((𝑑‘0) ≠ (𝑑‘1) ∧ (𝑑‘1) ≠ (𝑑‘2))}    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ − = (dist‘𝐺)    &   ⊢ ∼ = (cgrA‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ + = (𝑒 ∈ 𝐴, 𝑓 ∈ 𝐴 ↦ if((𝑒‘0) ∈ ((𝑒‘1)𝐿(𝑒‘2)), ⟨“(𝑓‘0)(𝑓‘1)(℩𝑠 ∈ 𝑃 (⟨“(𝑓‘2)(𝑓‘1)𝑠”⟩ ∼ 𝑒 ∧ ((𝑓‘1) − 𝑠) = ((𝑒‘1) − (𝑒‘0))))”⟩, ⟨“(𝑒‘0)(𝑒‘1)(℩𝑠 ∈ 𝑃 (⟨“(𝑒‘2)(𝑒‘1)𝑠”⟩ ∼ 𝑓 ∧ ((𝑒‘1) − 𝑠) = ((𝑓‘1) − (𝑓‘0)) ∧ (((𝑒‘1)𝐿(𝑒‘2)) ∩ (𝑠𝐼(𝑒‘0))) ≠ ∅))”⟩))    &   ⊢ 𝐽 = (AngMgm‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑌 ∈ (𝑃 ∖ {𝑋}))    ⇒   ⊢ (𝜑 → [⟨“𝑋𝑌𝑋”⟩] ∼ = (0g‘𝐽))
Theorem	angmgm 29443*	The angle addition magma is a magma. (Contributed by Thierry Arnoux, 31-Aug-2026.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐴 = {𝑑 ∈ (𝑃 ↑m (0..^3)) ∣ ((𝑑‘0) ≠ (𝑑‘1) ∧ (𝑑‘1) ≠ (𝑑‘2))}    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ − = (dist‘𝐺)    &   ⊢ ∼ = (cgrA‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ + = (𝑒 ∈ 𝐴, 𝑓 ∈ 𝐴 ↦ if((𝑒‘0) ∈ ((𝑒‘1)𝐿(𝑒‘2)), ⟨“(𝑓‘0)(𝑓‘1)(℩𝑠 ∈ 𝑃 (⟨“(𝑓‘2)(𝑓‘1)𝑠”⟩ ∼ 𝑒 ∧ ((𝑓‘1) − 𝑠) = ((𝑒‘1) − (𝑒‘0))))”⟩, ⟨“(𝑒‘0)(𝑒‘1)(℩𝑠 ∈ 𝑃 (⟨“(𝑒‘2)(𝑒‘1)𝑠”⟩ ∼ 𝑓 ∧ ((𝑒‘1) − 𝑠) = ((𝑓‘1) − (𝑓‘0)) ∧ (((𝑒‘1)𝐿(𝑒‘2)) ∩ (𝑠𝐼(𝑒‘0))) ≠ ∅))”⟩))    &   ⊢ 𝐽 = (AngMgm‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 2 ≤ (♯‘𝑃))    ⇒   ⊢ (𝜑 → 𝐽 ∈ Mgm)
Theorem	angmgmbas 29444*	The base set of the angle addition magma. (Contributed by Thierry Arnoux, 31-Aug-2026.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐴 = {𝑑 ∈ (𝑃 ↑m (0..^3)) ∣ ((𝑑‘0) ≠ (𝑑‘1) ∧ (𝑑‘1) ≠ (𝑑‘2))}    &   ⊢ ∼ = (cgrA‘𝐺)    &   ⊢ 𝐽 = (AngMgm‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    ⇒   ⊢ (𝜑 → (𝐴 / ∼ ) = (Base‘𝐽))
16.2.20  Congruence Theorems
Theorem	tgsas1 29445	First congruence theorem: SAS (Side-Angle-Side): If two pairs of sides of two triangles are equal in length, and the included angles are equal in measurement, then third sides are equal in length. Theorem 11.49 of [Schwabhauser] p. 107. (Contributed by Thierry Arnoux, 1-Aug-2020.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ − = (dist‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑃)    &   ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐷 − 𝐸))    &   ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)    &   ⊢ (𝜑 → (𝐵 − 𝐶) = (𝐸 − 𝐹))    ⇒   ⊢ (𝜑 → (𝐶 − 𝐴) = (𝐹 − 𝐷))
Theorem	tgsas 29446	First congruence theorem: SAS (Side-Angle-Side): If two pairs of sides of two triangles are equal in length, and the included angles are equal in measurement, then the triangles are congruent. Theorem 11.49 of [Schwabhauser] p. 107. (Contributed by Thierry Arnoux, 1-Aug-2020.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ − = (dist‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑃)    &   ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐷 − 𝐸))    &   ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)    &   ⊢ (𝜑 → (𝐵 − 𝐶) = (𝐸 − 𝐹))    ⇒   ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩)
Theorem	tgsas2 29447	First congruence theorem: SAS. Theorem 11.49 of [Schwabhauser] p. 107. (Contributed by Thierry Arnoux, 1-Aug-2020.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ − = (dist‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑃)    &   ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐷 − 𝐸))    &   ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)    &   ⊢ (𝜑 → (𝐵 − 𝐶) = (𝐸 − 𝐹))    &   ⊢ (𝜑 → 𝐴 ≠ 𝐶)    ⇒   ⊢ (𝜑 → ⟨“𝐶𝐴𝐵”⟩(cgrA‘𝐺)⟨“𝐹𝐷𝐸”⟩)
Theorem	tgsas3 29448	First congruence theorem: SAS. Theorem 11.49 of [Schwabhauser] p. 107. (Contributed by Thierry Arnoux, 1-Aug-2020.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ − = (dist‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑃)    &   ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐷 − 𝐸))    &   ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)    &   ⊢ (𝜑 → (𝐵 − 𝐶) = (𝐸 − 𝐹))    &   ⊢ (𝜑 → 𝐴 ≠ 𝐶)    ⇒   ⊢ (𝜑 → ⟨“𝐵𝐶𝐴”⟩(cgrA‘𝐺)⟨“𝐸𝐹𝐷”⟩)
Theorem	tgasa1 29449	Second congruence theorem: ASA. (Angle-Side-Angle): If two pairs of angles of two triangles are equal in measurement, and the included sides are equal in length, then the triangles are congruent. Theorem 11.50 of [Schwabhauser] p. 108. (Contributed by Thierry Arnoux, 15-Aug-2020.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ − = (dist‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑃)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ (𝜑 → ¬ (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))    &   ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐷 − 𝐸))    &   ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)    &   ⊢ (𝜑 → ⟨“𝐶𝐴𝐵”⟩(cgrA‘𝐺)⟨“𝐹𝐷𝐸”⟩)    ⇒   ⊢ (𝜑 → (𝐵 − 𝐶) = (𝐸 − 𝐹))
Theorem	tgasa 29450	Second congruence theorem: ASA. (Angle-Side-Angle): If two pairs of angles of two triangles are equal in measurement, and the included sides are equal in length, then the triangles are congruent. Theorem 11.50 of [Schwabhauser] p. 108. (Contributed by Thierry Arnoux, 15-Aug-2020.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ − = (dist‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑃)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ (𝜑 → ¬ (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))    &   ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐷 − 𝐸))    &   ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)    &   ⊢ (𝜑 → ⟨“𝐶𝐴𝐵”⟩(cgrA‘𝐺)⟨“𝐹𝐷𝐸”⟩)    ⇒   ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩)
Theorem	tgsss1 29451	Third congruence theorem: SSS (Side-Side-Side): If the three pairs of sides of two triangles are equal in length, then the triangles are congruent. Theorem 11.51 of [Schwabhauser] p. 109. (Contributed by Thierry Arnoux, 1-Aug-2020.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ − = (dist‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑃)    &   ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐷 − 𝐸))    &   ⊢ (𝜑 → (𝐵 − 𝐶) = (𝐸 − 𝐹))    &   ⊢ (𝜑 → (𝐶 − 𝐴) = (𝐹 − 𝐷))    &   ⊢ (𝜑 → 𝐴 ≠ 𝐵)    &   ⊢ (𝜑 → 𝐵 ≠ 𝐶)    &   ⊢ (𝜑 → 𝐶 ≠ 𝐴)    ⇒   ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)
Theorem	tgsss2 29452	Third congruence theorem: SSS. Theorem 11.51 of [Schwabhauser] p. 109. (Contributed by Thierry Arnoux, 1-Aug-2020.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ − = (dist‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑃)    &   ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐷 − 𝐸))    &   ⊢ (𝜑 → (𝐵 − 𝐶) = (𝐸 − 𝐹))    &   ⊢ (𝜑 → (𝐶 − 𝐴) = (𝐹 − 𝐷))    &   ⊢ (𝜑 → 𝐴 ≠ 𝐵)    &   ⊢ (𝜑 → 𝐵 ≠ 𝐶)    &   ⊢ (𝜑 → 𝐶 ≠ 𝐴)    ⇒   ⊢ (𝜑 → ⟨“𝐶𝐴𝐵”⟩(cgrA‘𝐺)⟨“𝐹𝐷𝐸”⟩)
Theorem	tgsss3 29453	Third congruence theorem: SSS. Theorem 11.51 of [Schwabhauser] p. 109. (Contributed by Thierry Arnoux, 1-Aug-2020.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ − = (dist‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑃)    &   ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐷 − 𝐸))    &   ⊢ (𝜑 → (𝐵 − 𝐶) = (𝐸 − 𝐹))    &   ⊢ (𝜑 → (𝐶 − 𝐴) = (𝐹 − 𝐷))    &   ⊢ (𝜑 → 𝐴 ≠ 𝐵)    &   ⊢ (𝜑 → 𝐵 ≠ 𝐶)    &   ⊢ (𝜑 → 𝐶 ≠ 𝐴)    ⇒   ⊢ (𝜑 → ⟨“𝐵𝐶𝐴”⟩(cgrA‘𝐺)⟨“𝐸𝐹𝐷”⟩)
Theorem	dfcgrg2 29454	Congruence for two triangles can also be defined as congruence of sides and angles (6 parts). This is often the actual textbook definition of triangle congruence, see for example https://en.wikipedia.org/wiki/Congruence_(geometry)#Congruence_of_triangles. With this definition, the "SSS" congruence theorem has an additional part, namely, that triangle congruence implies congruence of the sides (which means equality of the lengths). Because our development of elementary geometry strives to closely follow Schwabhaeuser's, our original definition of shape congruence, df-cgrg 29020, already covers that part: see trgcgr 29025. This theorem is also named "CPCTC", which stands for "Corresponding Parts of Congruent Triangles are Congruent", see https://en.wikipedia.org/wiki/Congruence_(geometry)#CPCTC 29025. (Contributed by Thierry Arnoux, 18-Jan-2023.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ − = (dist‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐴 ≠ 𝐵)    &   ⊢ (𝜑 → 𝐵 ≠ 𝐶)    &   ⊢ (𝜑 → 𝐶 ≠ 𝐴)    ⇒   ⊢ (𝜑 → (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩ ↔ (((𝐴 − 𝐵) = (𝐷 − 𝐸) ∧ (𝐵 − 𝐶) = (𝐸 − 𝐹) ∧ (𝐶 − 𝐴) = (𝐹 − 𝐷)) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩ ∧ ⟨“𝐶𝐴𝐵”⟩(cgrA‘𝐺)⟨“𝐹𝐷𝐸”⟩ ∧ ⟨“𝐵𝐶𝐴”⟩(cgrA‘𝐺)⟨“𝐸𝐹𝐷”⟩))))
Theorem	isoas 29455	Congruence theorem for isocele triangles: if two angles of a triangle are congruent, then the corresponding sides also are. (Contributed by Thierry Arnoux, 5-Oct-2020.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ − = (dist‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑃)    &   ⊢ (𝜑 → ¬ (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))    &   ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐴𝐶𝐵”⟩)    ⇒   ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐴 − 𝐶))
16.2.21  Equilateral triangles
Syntax	ceqlg 29456	Declare the class of equilateral triangles.
class eqltrG
Definition	df-eqlg 29457*	Define the class of equilateral triangles. (Contributed by Thierry Arnoux, 27-Nov-2019.)
⊢ eqltrG = (𝑔 ∈ V ↦ {𝑥 ∈ ((Base‘𝑔) ↑m (0..^3)) ∣ 𝑥(cgrG‘𝑔)⟨“(𝑥‘1)(𝑥‘2)(𝑥‘0)”⟩})
Theorem	iseqlg 29458	Property of a triangle being equilateral. (Contributed by Thierry Arnoux, 5-Oct-2020.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ − = (dist‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑃)    ⇒   ⊢ (𝜑 → (⟨“𝐴𝐵𝐶”⟩ ∈ (eqltrG‘𝐺) ↔ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐵𝐶𝐴”⟩))
Theorem	iseqlgd 29459	Condition for a triangle to be equilateral. (Contributed by Thierry Arnoux, 5-Oct-2020.)
⊢ 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) = (𝐵 − 𝐶))    &   ⊢ (𝜑 → (𝐵 − 𝐶) = (𝐶 − 𝐴))    &   ⊢ (𝜑 → (𝐶 − 𝐴) = (𝐴 − 𝐵))    ⇒   ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (eqltrG‘𝐺))
16.2.22  Parallel lines
Syntax	cprlng 29460	Extend class notation for the parallel lines relation.
class parlnG
Definition	df-prlng 29461*	Define the parallel relation for lines. Definition 12.2 of [Schwabhauser] p. 121. Note that the textbook first defines a "strict" parallelism where equal lines are not considered parallel in the strict sense: here we jump directly to the more common definition which allows equality. (Contributed by Thierry Arnoux, 17-Jun-2026.)
⊢ parlnG = (𝑔 ∈ V ↦ {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ran (LineG‘𝑔) ∧ 𝑏 ∈ ran (LineG‘𝑔)) ∧ (𝑎 = 𝑏 ∨ (∃ℎ ∈ ran (hlG‘𝑔)(𝑎 ⊆ ℎ ∧ 𝑏 ⊆ ℎ) ∧ (𝑎 ∩ 𝑏) = ∅)))})
Theorem	brprlng 29462*	Property of two lines 𝐴 and 𝐵 to be parallel. (Contributed by Thierry Arnoux, 18-Jun-2026.)
⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ 𝐸 = (hlG‘𝐺)    &   ⊢ ∥ = (parlnG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝐴 ∥ 𝐵 ↔ ((𝐴 ∈ ran 𝐿 ∧ 𝐵 ∈ ran 𝐿) ∧ (𝐴 = 𝐵 ∨ (∃ℎ ∈ ran 𝐸(𝐴 ⊆ ℎ ∧ 𝐵 ⊆ ℎ) ∧ (𝐴 ∩ 𝐵) = ∅)))))
Theorem	prlngd 29463	Deduce parallelism between two lines 𝐴 and 𝐵. (Contributed by Thierry Arnoux, 18-Jun-2026.)
⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ 𝐸 = (hlG‘𝐺)    &   ⊢ ∥ = (parlnG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐴 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝐵 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝐻 ∈ ran 𝐸)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝐻)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐻)    &   ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅)    ⇒   ⊢ (𝜑 → 𝐴 ∥ 𝐵)
Theorem	prlngref 29464	Parallelism is reflexive. Theorem 12.4 of [Schwabhauser] p. 122. (Contributed by Thierry Arnoux, 18-Jun-2026.)
⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ 𝐸 = (hlG‘𝐺)    &   ⊢ ∥ = (parlnG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐴 ∈ ran 𝐿)    ⇒   ⊢ (𝜑 → 𝐴 ∥ 𝐴)
Theorem	prlngsym 29465	Parallelism is symmetric. Theorem 12.5 of [Schwabhauser] p. 122. (Contributed by Thierry Arnoux, 18-Jun-2026.)
⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ 𝐸 = (hlG‘𝐺)    &   ⊢ ∥ = (parlnG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐴 ∥ 𝐵)    ⇒   ⊢ (𝜑 → 𝐵 ∥ 𝐴)
Theorem	prlngrcl1 29466	Reverse closure for parallelism. (Contributed by Thierry Arnoux, 5-Jul-2026.)
⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ ∥ = (parlnG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐴 ∥ 𝐵)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ran 𝐿)
Theorem	prlngrcl2 29467	Reverse closure for parallelism. (Contributed by Thierry Arnoux, 5-Jul-2026.)
⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ ∥ = (parlnG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐴 ∥ 𝐵)    ⇒   ⊢ (𝜑 → 𝐵 ∈ ran 𝐿)
Theorem	prlngin0 29468	Two parallel lines do not intersect. (Contributed by Thierry Arnoux, 5-Jul-2026.)
⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ ∥ = (parlnG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐴 ∥ 𝐵)    &   ⊢ (𝜑 → 𝐴 ≠ 𝐵)    ⇒   ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅)
Theorem	prlngpln 29469*	Two parallel lines are on a common plane. (Contributed by Thierry Arnoux, 5-Jul-2026.)
⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ 𝐸 = (hlG‘𝐺)    &   ⊢ ∥ = (parlnG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐴 ∥ 𝐵)    &   ⊢ (𝜑 → 𝐴 ≠ 𝐵)    ⇒   ⊢ (𝜑 → ∃ℎ ∈ ran 𝐸(𝐴 ⊆ ℎ ∧ 𝐵 ⊆ ℎ))
Theorem	prlnghpg 29470	If two lines 𝐴 and 𝐵 are parallel, then any two points 𝑋 and 𝑌 of 𝐵 lie on the same half-plane limited by 𝐴. Theorem 12.6 of [Schwabhauser] p. 122. . (Contributed by Thierry Arnoux, 5-Jul-2026.)
⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ 𝐸 = (hlG‘𝐺)    &   ⊢ ∥ = (parlnG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∥ 𝐵)    &   ⊢ (𝜑 → 𝐴 ≠ 𝐵)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → 𝑋((hpG‘𝐺)‘𝐴)𝑌)
Theorem	dfprlng2 29471	Alternate definition of (strict) parallelism. Theorem 12.7 of [Schwabhauser] p. 122. (Contributed by Thierry Arnoux, 13-Jul-2026.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ ∥ = (parlnG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑌 ∈ (𝑃 ∖ {𝑋}))    &   ⊢ (𝜑 → 𝑍 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑊 ∈ (𝑃 ∖ {𝑍}))    &   ⊢ (𝜑 → (𝑋𝐿𝑌) ≠ (𝑍𝐿𝑊))    ⇒   ⊢ (𝜑 → ((𝑋𝐿𝑌) ∥ (𝑍𝐿𝑊) ↔ (𝑍((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑊 ∧ ((𝑋𝐿𝑌) ∩ (𝑍𝐿𝑊)) = ∅)))
Theorem	dfprlng3 29472	Alternate definition of (strict) parallelism. Theorem 12.7 of [Schwabhauser] p. 122. (Contributed by Thierry Arnoux, 13-Jul-2026.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ ∥ = (parlnG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑌 ∈ (𝑃 ∖ {𝑋}))    &   ⊢ (𝜑 → 𝐴 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝐴 ≠ (𝑋𝐿𝑌))    ⇒   ⊢ (𝜑 → (𝐴 ∥ (𝑋𝐿𝑌) ↔ (𝑋((hpG‘𝐺)‘𝐴)𝑌 ∧ (𝐴 ∩ (𝑋𝐿𝑌)) = ∅)))
Theorem	prlngpln3 29473	Two parallel lines are on a common plane. (Contributed by Thierry Arnoux, 5-Jul-2026.)
⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ 𝐸 = (hlG‘𝐺)    &   ⊢ ∥ = (parlnG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∥ 𝐵)    &   ⊢ (𝜑 → 𝐴 ≠ 𝐵)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → 𝐵 ⊆ (𝐴𝐸𝑋))
Theorem	perpprlng 29474	If two lines 𝐴 and 𝐵 have a common perpendicular 𝐶 and lie in the same plane 𝐻, then they are parallel. Theorem 12.9 of [Schwabhauser] p. 122. (Contributed by Thierry Arnoux, 5-Jul-2026.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ 𝐸 = (hlG‘𝐺)    &   ⊢ ∥ = (parlnG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐻 ∈ ran 𝐸)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝐻)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐻)    &   ⊢ (𝜑 → 𝐶 ⊆ 𝐻)    &   ⊢ (𝜑 → 𝐴(⟂G‘𝐺)𝐶)    &   ⊢ (𝜑 → 𝐵(⟂G‘𝐺)𝐶)    ⇒   ⊢ (𝜑 → 𝐴 ∥ 𝐵)
Theorem	prlngex 29475*	There exists at least one parallel line 𝑏 to a given line 𝐴 through a given point 𝑋. Theorem 12.10 of [Schwabhauser] p. 122. (Contributed by Thierry Arnoux, 5-Jul-2026.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ ∥ = (parlnG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑃)    ⇒   ⊢ (𝜑 → ∃𝑏 ∈ ran 𝐿(𝐴 ∥ 𝑏 ∧ 𝑋 ∈ 𝑏))
Theorem	prlngmolem1 29476*	Lemma for prlngmo 29478: Contradiction: Assuming two different parallels 𝐵 and 𝐶 having a common point 𝑋 exist to a line 𝐴, the geometry cannot be Euclidean (Contributed by Thierry Arnoux, 5-Jul-2026.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ ∥ = (parlnG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝑋 ∈ (𝑃 ∖ 𝐴))    &   ⊢ 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ 𝐵) ∧ 𝑏 ∈ (𝑃 ∖ 𝐵)) ∧ ∃𝑦 ∈ 𝐵 𝑦 ∈ (𝑎𝐼𝑏))}    &   ⊢ 𝑄 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ 𝐴) ∧ 𝑏 ∈ (𝑃 ∖ 𝐴)) ∧ ∃𝑤 ∈ 𝐴 𝑤 ∈ (𝑎𝐼𝑏))}    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ (𝜑 → 𝐵 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝐶 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝐴 ∥ 𝐵)    &   ⊢ (𝜑 → 𝐴 ∥ 𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐶)    &   ⊢ (𝜑 → 𝑇 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑊 ∈ (𝐶 ∖ 𝐵))    &   ⊢ (𝜑 → 𝐵 ≠ 𝐶)    &   ⊢ (𝜑 → 𝑊𝑂𝑇)    ⇒   ⊢ (𝜑 → ¬ 𝐺 ∈ TarskiGE)
Theorem	prlngmolem2 29477*	Lemma for prlngmo 29478. (Contributed by Thierry Arnoux, 5-Jul-2026.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ ∥ = (parlnG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝑋 ∈ (𝑃 ∖ 𝐴))    &   ⊢ (𝜑 → 𝐺 ∈ TarskiGE)    &   ⊢ 𝑂 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝑃 ∖ 𝑏) ∧ 𝑦 ∈ (𝑃 ∖ 𝑏)) ∧ ∃𝑟 ∈ 𝑏 𝑟 ∈ (𝑥(Itv‘𝐺)𝑦))}    &   ⊢ 𝑄 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝑃 ∖ 𝐴) ∧ 𝑦 ∈ (𝑃 ∖ 𝐴)) ∧ ∃𝑠 ∈ 𝐴 𝑠 ∈ (𝑥(Itv‘𝐺)𝑦))}    ⇒   ⊢ (𝜑 → ∃*𝑏 ∈ ran 𝐿(𝐴 ∥ 𝑏 ∧ 𝑋 ∈ 𝑏))
Theorem	prlngmo 29478*	Playfair's axiom. Given a line 𝐴 and a point 𝑋 not on 𝐴, at most one line parallel to 𝐴 can be drawn through 𝑋. Theorem 12.11 of [Schwabhauser] p. 123. Note that this is the first instance of a theorem where the geometry is required to be Euclidean, as expressed by 𝐺 ∈ TarskiGE. Theorem A10 of [Schwabhauser] p. 24 is used, in the form of axtgeucl 28980, in the proof of prlngmolem1 29476. See prlngex 29475 for the corresponding existence theorem. (Contributed by Thierry Arnoux, 5-Jul-2026.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ ∥ = (parlnG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝑋 ∈ (𝑃 ∖ 𝐴))    &   ⊢ (𝜑 → 𝐺 ∈ TarskiGE)    ⇒   ⊢ (𝜑 → ∃*𝑏 ∈ ran 𝐿(𝐴 ∥ 𝑏 ∧ 𝑋 ∈ 𝑏))
Theorem	prlngeu 29479*	Given a line 𝐴 and a point 𝑋 not on 𝐴, a unique line parallel to 𝐴 can be drawn through 𝑋. Theorem 12.13 of [Schwabhauser] p. 124. (Contributed by Thierry Arnoux, 5-Jul-2026.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ ∥ = (parlnG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝑋 ∈ (𝑃 ∖ 𝐴))    &   ⊢ (𝜑 → 𝐺 ∈ TarskiGE)    ⇒   ⊢ (𝜑 → ∃!𝑏 ∈ ran 𝐿(𝐴 ∥ 𝑏 ∧ 𝑋 ∈ 𝑏))
Theorem	prlngmo2 29480*	Playfair's axiom, without the restriction that the point 𝑋 is outside of the line 𝐴. Theorem 12.11 of [Schwabhauser] p. 123. (Contributed by Thierry Arnoux, 13-Jul-2026.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ ∥ = (parlnG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐴 ∈ ran 𝐿)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiGE)    ⇒   ⊢ (𝜑 → ∃*𝑏 ∈ ran 𝐿(𝐴 ∥ 𝑏 ∧ 𝑋 ∈ 𝑏))
Theorem	prlngeq 29481	Playfair's axiom, written as an equality: if two different lines are parallel to a given line at a given point, they are equal. (Contributed by Thierry Arnoux, 20-Jul-2026.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ ∥ = (parlnG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiGE)    &   ⊢ (𝜑 → 𝐴 ∥ 𝐵)    &   ⊢ (𝜑 → 𝐴 ∥ 𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐶)    ⇒   ⊢ (𝜑 → 𝐵 = 𝐶)
Theorem	prlngpln4 29482	Building a parallel line conserves planes, i.e. given a line 𝐴 and a point 𝑋 not on 𝐴, the (unique) parallel 𝐵 to 𝐴 through 𝑋 lies completely within the plane defined by 𝐴 and 𝑋. Theorem 12.14 of [Schwabhauser] p. 124. (Contributed by Thierry Arnoux, 13-Jul-2026.)
⊢ 𝐸 = (hlG‘𝐺)    &   ⊢ ∥ = (parlnG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐻 ∈ ran 𝐸)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝐻)    &   ⊢ (𝜑 → 𝐴 ∥ 𝐵)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐻)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → 𝐵 ⊆ 𝐻)
Theorem	prlngplngtr 29483	Transitivity of parallelism, for lines in the same plane 𝐻. This is case 1 of Theorem 12.15 of [Schwabhauser] p. 124. (Contributed by Thierry Arnoux, 13-Jul-2026.)
⊢ 𝐸 = (hlG‘𝐺)    &   ⊢ ∥ = (parlnG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐻 ∈ ran 𝐸)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝐻)    &   ⊢ (𝜑 → 𝐴 ∥ 𝐵)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiGE)    &   ⊢ (𝜑 → 𝐶 ⊆ 𝐻)    &   ⊢ (𝜑 → 𝐵 ∥ 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 ∥ 𝐶)
Theorem	prlnginn0 29484	A line 𝐶 intersecting another line 𝐴 also intersects any line 𝐵 parallel to 𝐴. Theorem 12.16 of [Schwabhauser] p. 125. (Contributed by Thierry Arnoux, 13-Jul-2026.)
⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ 𝐸 = (hlG‘𝐺)    &   ⊢ ∥ = (parlnG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiGE)    &   ⊢ (𝜑 → 𝐻 ∈ ran 𝐸)    &   ⊢ (𝜑 → 𝐶 ∈ ran 𝐿)    &   ⊢ (𝜑 → (𝐴 ∩ 𝐶) ≠ ∅)    &   ⊢ (𝜑 → 𝐴 ≠ 𝐶)    &   ⊢ (𝜑 → 𝐴 ∥ 𝐵)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝐻)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐻)    &   ⊢ (𝜑 → 𝐶 ⊆ 𝐻)    ⇒   ⊢ (𝜑 → (𝐵 ∩ 𝐶) ≠ ∅)
Theorem	prlngmid2 29485	If the midpoints of two segments (𝑋𝐼𝑍) and (𝑌𝐼𝑊) coincide, the points 𝑋, 𝑌, 𝑍 and 𝑊 form a parallelogram, i.e. the lines (𝑋𝐿𝑌) and (𝑍𝐿𝑊) are parallel. Theorem 12.17 of [Schwabhauser] p. 125. (Contributed by Thierry Arnoux, 13-Jul-2026.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ 𝐸 = (hlG‘𝐺)    &   ⊢ ∥ = (parlnG‘𝐺)    &   ⊢ 𝑀 = (midG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiGE)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑍 ∈ (𝑃 ∖ (𝑋𝐿𝑌)))    &   ⊢ (𝜑 → 𝑊 ∈ 𝑃)    &   ⊢ (𝜑 → (𝑋𝑀𝑍) = (𝑌𝑀𝑊))    &   ⊢ (𝜑 → 𝑋 ≠ 𝑌)    ⇒   ⊢ (𝜑 → (𝑋𝐿𝑌) ∥ (𝑍𝐿𝑊))
Theorem	symquadprlng 29486	Symmetrical quadrilaterals are parallelograms. Theorem 12.18 of [Schwabhauser] p. 126. (Contributed by Thierry Arnoux, 20-Jul-2026.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ − = (dist‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ ∥ = (parlnG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiGE)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑊 ∈ 𝑃)    &   ⊢ (𝜑 → (𝑋 − 𝑌) = (𝑍 − 𝑊))    &   ⊢ (𝜑 → (𝑌 − 𝑍) = (𝑊 − 𝑋))    &   ⊢ (𝜑 → ¬ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))    &   ⊢ (𝜑 → 𝑌 ≠ 𝑊)    &   ⊢ (𝜑 → 𝑇 ∈ (𝑋𝐿𝑍))    &   ⊢ (𝜑 → 𝑇 ∈ (𝑌𝐿𝑊))    ⇒   ⊢ (𝜑 → ((𝑋𝐿𝑌) ∥ (𝑍𝐿𝑊) ∧ (𝑌𝐿𝑍) ∥ (𝑊𝐿𝑋)))
Theorem	prlngsymquadlem 29487	Lemma for prlngsymquad 29488. (Contributed by Thierry Arnoux, 20-Jul-2026.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ − = (dist‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ ∥ = (parlnG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiGE)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑊 ∈ 𝑃)    &   ⊢ (𝜑 → ¬ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))    &   ⊢ (𝜑 → (𝑋𝐿𝑌) ∥ (𝑍𝐿𝑊))    &   ⊢ (𝜑 → (𝑌𝐿𝑍) ∥ (𝑊𝐿𝑋))    &   ⊢ 𝑇 = (((pInvG‘𝐺)‘(𝑋(midG‘𝐺)𝑍))‘𝑌)    ⇒   ⊢ (𝜑 → 𝑇 = 𝑊)
Theorem	prlngsymquad 29488	All parallelograms are symmetric quadrilaterals. First part of Theorem 12.19 of [Schwabhauser] p. 126. (Contributed by Thierry Arnoux, 20-Jul-2026.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ − = (dist‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ ∥ = (parlnG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiGE)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑊 ∈ 𝑃)    &   ⊢ (𝜑 → ¬ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))    &   ⊢ (𝜑 → (𝑋𝐿𝑌) ∥ (𝑍𝐿𝑊))    &   ⊢ (𝜑 → (𝑌𝐿𝑍) ∥ (𝑊𝐿𝑋))    ⇒   ⊢ (𝜑 → ((𝑋 − 𝑌) = (𝑍 − 𝑊) ∧ (𝑌 − 𝑍) = (𝑊 − 𝑋)))
Theorem	prlngsymquadopp 29489*	In parallelograms, opposing vertices are on opposite sides of the diagonal. Second part of Theorem 12.19 of [Schwabhauser] p. 126. (Contributed by Thierry Arnoux, 20-Jul-2026.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ − = (dist‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ ∥ = (parlnG‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiGE)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑊 ∈ 𝑃)    &   ⊢ (𝜑 → ¬ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))    &   ⊢ (𝜑 → (𝑋𝐿𝑌) ∥ (𝑍𝐿𝑊))    &   ⊢ (𝜑 → (𝑌𝐿𝑍) ∥ (𝑊𝐿𝑋))    &   ⊢ 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ (𝑋𝐿𝑍)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑋𝐿𝑍))) ∧ ∃𝑡 ∈ (𝑋𝐿𝑍)𝑡 ∈ (𝑎𝐼𝑏))}    &   ⊢ 𝐼 = (Itv‘𝐺)    ⇒   ⊢ (𝜑 → 𝑊𝑂𝑌)
Theorem	quadcgrprlng 29490*	Nontrivial quadrilaterals with congruent and parallel opposite sides are parallelograms. Theorem 12.20 of [Schwabhauser] p. 126. (Contributed by Thierry Arnoux, 20-Jul-2026.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ − = (dist‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ ∥ = (parlnG‘𝐺)    &   ⊢ 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ (𝑋𝐿𝑍)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑋𝐿𝑍))) ∧ ∃𝑡 ∈ (𝑋𝐿𝑍)𝑡 ∈ (𝑎𝐼𝑏))}    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiGE)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑊 ∈ 𝑃)    &   ⊢ (𝜑 → ¬ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))    &   ⊢ (𝜑 → (𝑋𝐿𝑌) ∥ (𝑍𝐿𝑊))    &   ⊢ (𝜑 → (𝑋 − 𝑌) = (𝑍 − 𝑊))    &   ⊢ (𝜑 → 𝑌𝑂𝑊)    ⇒   ⊢ (𝜑 → ((𝑌𝐿𝑍) ∥ (𝑊𝐿𝑋) ∧ (𝑌 − 𝑍) = (𝑊 − 𝑋)))
Theorem	tgaltai 29491*	Parallelism implies alternate angles congruence: if a line (𝑋𝐿𝑍) crosses two parallel lines (𝑋𝐿𝑌) and (𝑍𝐿𝑊), then the alternate angles are congruent. First direction of Theorem 12.21 of [Schwabhauser] p. 126. This is also Proposition 1.29 of Euclid's Elements . (Contributed by Thierry Arnoux, 20-Jul-2026.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ ∥ = (parlnG‘𝐺)    &   ⊢ 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ (𝑋𝐿𝑍)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑋𝐿𝑍))) ∧ ∃𝑡 ∈ (𝑋𝐿𝑍)𝑡 ∈ (𝑎𝐼𝑏))}    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiGE)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑊 ∈ 𝑃)    &   ⊢ (𝜑 → (𝑋𝐿𝑌) ∥ (𝑍𝐿𝑊))    &   ⊢ (𝜑 → 𝑌𝑂𝑊)    &   ⊢ (𝜑 → 𝑋 ≠ 𝑍)    ⇒   ⊢ (𝜑 → ⟨“𝑌𝑋𝑍”⟩(cgrA‘𝐺)⟨“𝑊𝑍𝑋”⟩)
16.3  Properties of geometries
16.3.1  Isomorphisms between geometries
Theorem	f1otrgds 29492*	Convenient lemma for f1otrg 29494. (Contributed by Thierry Arnoux, 19-Mar-2019.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐷 = (dist‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ 𝐵 = (Base‘𝐻)    &   ⊢ 𝐸 = (dist‘𝐻)    &   ⊢ 𝐽 = (Itv‘𝐻)    &   ⊢ (𝜑 → 𝐹:𝐵⟶1-1onto𝑃)    &   ⊢ ((𝜑 ∧ (𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐵)) → (𝑒𝐸𝑓) = ((𝐹‘𝑒)𝐷(𝐹‘𝑓)))    &   ⊢ ((𝜑 ∧ (𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → (𝑔 ∈ (𝑒𝐽𝑓) ↔ (𝐹‘𝑔) ∈ ((𝐹‘𝑒)𝐼(𝐹‘𝑓))))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑋𝐸𝑌) = ((𝐹‘𝑋)𝐷(𝐹‘𝑌)))
Theorem	f1otrgitv 29493*	Convenient lemma for f1otrg 29494. (Contributed by Thierry Arnoux, 19-Mar-2019.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐷 = (dist‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ 𝐵 = (Base‘𝐻)    &   ⊢ 𝐸 = (dist‘𝐻)    &   ⊢ 𝐽 = (Itv‘𝐻)    &   ⊢ (𝜑 → 𝐹:𝐵⟶1-1onto𝑃)    &   ⊢ ((𝜑 ∧ (𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐵)) → (𝑒𝐸𝑓) = ((𝐹‘𝑒)𝐷(𝐹‘𝑓)))    &   ⊢ ((𝜑 ∧ (𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → (𝑔 ∈ (𝑒𝐽𝑓) ↔ (𝐹‘𝑔) ∈ ((𝐹‘𝑒)𝐼(𝐹‘𝑓))))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐽𝑌) ↔ (𝐹‘𝑍) ∈ ((𝐹‘𝑋)𝐼(𝐹‘𝑌))))
Theorem	f1otrg 29494*	A bijection between bases which conserves distances and intervals conserves also geometries. (Contributed by Thierry Arnoux, 23-Mar-2019.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐷 = (dist‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ 𝐵 = (Base‘𝐻)    &   ⊢ 𝐸 = (dist‘𝐻)    &   ⊢ 𝐽 = (Itv‘𝐻)    &   ⊢ (𝜑 → 𝐹:𝐵⟶1-1onto𝑃)    &   ⊢ ((𝜑 ∧ (𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐵)) → (𝑒𝐸𝑓) = ((𝐹‘𝑒)𝐷(𝐹‘𝑓)))    &   ⊢ ((𝜑 ∧ (𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → (𝑔 ∈ (𝑒𝐽𝑓) ↔ (𝐹‘𝑔) ∈ ((𝐹‘𝑒)𝐼(𝐹‘𝑓))))    &   ⊢ (𝜑 → 𝐻 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ (𝜑 → (LineG‘𝐻) = (𝑥 ∈ 𝐵, 𝑦 ∈ (𝐵 ∖ {𝑥}) ↦ {𝑧 ∈ 𝐵 ∣ (𝑧 ∈ (𝑥𝐽𝑦) ∨ 𝑥 ∈ (𝑧𝐽𝑦) ∨ 𝑦 ∈ (𝑥𝐽𝑧))}))    ⇒   ⊢ (𝜑 → 𝐻 ∈ TarskiG)
Theorem	f1otrge 29495*	A bijection between bases which conserves distances and intervals conserves also the property of being a Euclidean geometry. (Contributed by Thierry Arnoux, 23-Mar-2019.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐷 = (dist‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ 𝐵 = (Base‘𝐻)    &   ⊢ 𝐸 = (dist‘𝐻)    &   ⊢ 𝐽 = (Itv‘𝐻)    &   ⊢ (𝜑 → 𝐹:𝐵⟶1-1onto𝑃)    &   ⊢ ((𝜑 ∧ (𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐵)) → (𝑒𝐸𝑓) = ((𝐹‘𝑒)𝐷(𝐹‘𝑓)))    &   ⊢ ((𝜑 ∧ (𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → (𝑔 ∈ (𝑒𝐽𝑓) ↔ (𝐹‘𝑔) ∈ ((𝐹‘𝑒)𝐼(𝐹‘𝑓))))    &   ⊢ (𝜑 → 𝐻 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiGE)    ⇒   ⊢ (𝜑 → 𝐻 ∈ TarskiGE)
16.4  Geometry in Hilbert spaces
Syntax	cttg 29496	Function to convert an algebraic structure to a Tarski geometry.
class toTG
Definition	df-ttg 29497*	Define a function converting a subcomplex Hilbert space to a Tarski Geometry. It does so by equipping the structure with a betweenness operation. Note that because the scalar product is applied over the interval (0[,]1), only spaces whose scalar field is a superset of that interval can be considered. (Contributed by Thierry Arnoux, 24-Mar-2019.)
⊢ toTG = (𝑤 ∈ V ↦ ⦋(𝑥 ∈ (Base‘𝑤), 𝑦 ∈ (Base‘𝑤) ↦ {𝑧 ∈ (Base‘𝑤) ∣ ∃𝑘 ∈ (0[,]1)(𝑧(-g‘𝑤)𝑥) = (𝑘( ·𝑠 ‘𝑤)(𝑦(-g‘𝑤)𝑥))}) / 𝑖⦌((𝑤 sSet ⟨(Itv‘ndx), 𝑖⟩) sSet ⟨(LineG‘ndx), (𝑥 ∈ (Base‘𝑤), 𝑦 ∈ (Base‘𝑤) ↦ {𝑧 ∈ (Base‘𝑤) ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})⟩))
Theorem	ttgval 29498*	Define a function to augment a subcomplex Hilbert space with betweenness and a line definition. (Contributed by Thierry Arnoux, 25-Mar-2019.) (Proof shortened by AV, 9-Nov-2024.)
⊢ 𝐺 = (toTG‘𝐻)    &   ⊢ 𝐵 = (Base‘𝐻)    &   ⊢ − = (-g‘𝐻)    &   ⊢ · = ( ·𝑠 ‘𝐻)    &   ⊢ 𝐼 = (Itv‘𝐺)    ⇒   ⊢ (𝐻 ∈ 𝑉 → (𝐺 = ((𝐻 sSet ⟨(Itv‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})⟩) sSet ⟨(LineG‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))})⟩) ∧ 𝐼 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})))
Theorem	ttglem 29499	Lemma for ttgbas 29500, ttgvsca 29503 etc. (Contributed by Thierry Arnoux, 15-Apr-2019.) (Revised by AV, 29-Oct-2024.)
⊢ 𝐺 = (toTG‘𝐻)    &   ⊢ 𝐸 = Slot (𝐸‘ndx)    &   ⊢ (𝐸‘ndx) ≠ (LineG‘ndx)    &   ⊢ (𝐸‘ndx) ≠ (Itv‘ndx)    ⇒   ⊢ (𝐸‘𝐻) = (𝐸‘𝐺)
Theorem	ttgbas 29500	The base set of a subcomplex Hilbert space augmented with betweenness. (Contributed by Thierry Arnoux, 25-Mar-2019.) (Revised by AV, 29-Oct-2024.)
⊢ 𝐺 = (toTG‘𝐻)    &   ⊢ 𝐵 = (Base‘𝐻)    ⇒   ⊢ 𝐵 = (Base‘𝐺)