P. 287 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 28601-28700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	cgratr 28601	Angle congruence is transitive. Theorem 11.8 of [Schwabhauser] p. 97. (Contributed by Thierry Arnoux, 5-Mar-2020.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TarskiG)    &   ⊢ 𝐾 = (hlG‘𝐺)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑃)    &   ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)    &   ⊢ (𝜑 → 𝐻 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐽 ∈ 𝑃)    &   ⊢ (𝜑 → ⟨“𝐷𝐸𝐹”⟩(cgrA‘𝐺)⟨“𝐻𝑈𝐽”⟩)    ⇒   ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐻𝑈𝐽”⟩)
Theorem	flatcgra 28602	Flat angles are congruent. (Contributed by Thierry Arnoux, 13-Feb-2023.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐼 = (Itv‘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 → 𝐵 ∈ (𝐴𝐼𝐶))    &   ⊢ (𝜑 → 𝐸 ∈ (𝐷𝐼𝐹))    &   ⊢ (𝜑 → 𝐴 ≠ 𝐵)    &   ⊢ (𝜑 → 𝐶 ≠ 𝐵)    &   ⊢ (𝜑 → 𝐷 ≠ 𝐸)    &   ⊢ (𝜑 → 𝐹 ≠ 𝐸)    ⇒   ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)
Theorem	cgraswaplr 28603	Swap both side of angle congruence. (Contributed by Thierry Arnoux, 5-Oct-2020.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐼 = (Itv‘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 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)    ⇒   ⊢ (𝜑 → ⟨“𝐶𝐵𝐴”⟩(cgrA‘𝐺)⟨“𝐹𝐸𝐷”⟩)
Theorem	cgrabtwn 28604	Angle congruence preserves flat angles. Part of Theorem 11.21 of [Schwabhauser] p. 97. (Contributed by Thierry Arnoux, 9-Aug-2020.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐼 = (Itv‘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 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)    &   ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶))    ⇒   ⊢ (𝜑 → 𝐸 ∈ (𝐷𝐼𝐹))
Theorem	cgrahl 28605	Angle congruence preserves null angles. Part of Theorem 11.21 of [Schwabhauser] p. 97. (Contributed by Thierry Arnoux, 9-Aug-2020.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐼 = (Itv‘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 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)    &   ⊢ 𝐾 = (hlG‘𝐺)    &   ⊢ (𝜑 → 𝐴(𝐾‘𝐵)𝐶)    ⇒   ⊢ (𝜑 → 𝐷(𝐾‘𝐸)𝐹)
Theorem	cgracol 28606	Angle congruence preserves colinearity. (Contributed by Thierry Arnoux, 9-Aug-2020.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐼 = (Itv‘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 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ (𝜑 → (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))    ⇒   ⊢ (𝜑 → (𝐹 ∈ (𝐷𝐿𝐸) ∨ 𝐷 = 𝐸))
Theorem	cgrancol 28607	Angle congruence preserves non-colinearity. (Contributed by Thierry Arnoux, 9-Aug-2020.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐼 = (Itv‘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 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)    &   ⊢ 𝐿 = (LineG‘𝐺)    &   ⊢ (𝜑 → ¬ (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))    ⇒   ⊢ (𝜑 → ¬ (𝐹 ∈ (𝐷𝐿𝐸) ∨ 𝐷 = 𝐸))
Theorem	dfcgra2 28608*	This is the full statement of definition 11.2 of [Schwabhauser] p. 95. This proof serves to confirm that the definition we have chosen, df-cgra 28586 is indeed equivalent to the textbook's definition. (Contributed by Thierry Arnoux, 2-Aug-2020.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐼 = (Itv‘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 → (⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩ ↔ ((𝐴 ≠ 𝐵 ∧ 𝐶 ≠ 𝐵) ∧ (𝐷 ≠ 𝐸 ∧ 𝐹 ≠ 𝐸) ∧ ∃𝑎 ∈ 𝑃 ∃𝑐 ∈ 𝑃 ∃𝑑 ∈ 𝑃 ∃𝑓 ∈ 𝑃 (((𝐴 ∈ (𝐵𝐼𝑎) ∧ (𝐴 − 𝑎) = (𝐸 − 𝐷)) ∧ (𝐶 ∈ (𝐵𝐼𝑐) ∧ (𝐶 − 𝑐) = (𝐸 − 𝐹))) ∧ ((𝐷 ∈ (𝐸𝐼𝑑) ∧ (𝐷 − 𝑑) = (𝐵 − 𝐴)) ∧ (𝐹 ∈ (𝐸𝐼𝑓) ∧ (𝐹 − 𝑓) = (𝐵 − 𝐶))) ∧ (𝑎 − 𝑐) = (𝑑 − 𝑓)))))
Theorem	sacgr 28609	Supplementary angles of congruent angles are themselves congruent. Theorem 11.13 of [Schwabhauser] p. 98. (Contributed by Thierry Arnoux, 30-Sep-2020.) (Proof shortened by Igor Ieskov, 16-Feb-2023.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐼 = (Itv‘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 → 𝑋 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑃)    &   ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)    &   ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝑋))    &   ⊢ (𝜑 → 𝐸 ∈ (𝐷𝐼𝑌))    &   ⊢ (𝜑 → 𝐵 ≠ 𝑋)    &   ⊢ (𝜑 → 𝐸 ≠ 𝑌)    ⇒   ⊢ (𝜑 → ⟨“𝑋𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝑌𝐸𝐹”⟩)
Theorem	oacgr 28610	Vertical angle theorem. Vertical, or opposite angles are the facing pair of angles formed when two lines intersect. Eudemus of Rhodes attributed the proof to Thales of Miletus. The proposition showed that since both of a pair of vertical angles are supplementary to both of the adjacent angles, the vertical angles are equal in measure. We follow the same path. Theorem 11.14 of [Schwabhauser] p. 98. (Contributed by Thierry Arnoux, 27-Sep-2020.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐼 = (Itv‘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 → 𝐵 ∈ (𝐴𝐼𝐷))    &   ⊢ (𝜑 → 𝐵 ∈ (𝐶𝐼𝐹))    &   ⊢ (𝜑 → 𝐵 ≠ 𝐴)    &   ⊢ (𝜑 → 𝐵 ≠ 𝐶)    &   ⊢ (𝜑 → 𝐵 ≠ 𝐷)    &   ⊢ (𝜑 → 𝐵 ≠ 𝐹)    ⇒   ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐵𝐹”⟩)
Theorem	acopy 28611*	Angle construction. Theorem 11.15 of [Schwabhauser] p. 98. This is Hilbert's axiom III.4 for geometry. (Contributed by Thierry Arnoux, 9-Aug-2020.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ − = (dist‘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 → ¬ (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶))    &   ⊢ (𝜑 → ¬ (𝐷 ∈ (𝐸𝐿𝐹) ∨ 𝐸 = 𝐹))    ⇒   ⊢ (𝜑 → ∃𝑓 ∈ 𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝑓”⟩ ∧ 𝑓((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹))
Theorem	acopyeu 28612	Angle construction. Theorem 11.15 of [Schwabhauser] p. 98. This is Hilbert's axiom III.4 for geometry. Akin to a uniqueness theorem, this states that if two points 𝑋 and 𝑌 both fulfill the conditions, then they are on the same half-line. (Contributed by Thierry Arnoux, 9-Aug-2020.)
⊢ 𝑃 = (Base‘𝐺)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ − = (dist‘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 → ¬ (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶))    &   ⊢ (𝜑 → ¬ (𝐷 ∈ (𝐸𝐿𝐹) ∨ 𝐸 = 𝐹))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑃)    &   ⊢ 𝐾 = (hlG‘𝐺)    &   ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝑋”⟩)    &   ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝑌”⟩)    &   ⊢ (𝜑 → 𝑋((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)    &   ⊢ (𝜑 → 𝑌((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)    ⇒   ⊢ (𝜑 → 𝑋(𝐾‘𝐸)𝑌)
16.2.17  Angle Comparisons
Syntax	cinag 28613	Extend class relation with the geometrical "point in angle" relation.
class inA
Syntax	cleag 28614	Extend class relation with the "angle less than" relation.
class ≤∠
Definition	df-inag 28615*	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 28616*	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 28617	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 28618	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 28619	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 28620	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 28621	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 28622	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 28623	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 28624*	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 28625*	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 28626	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 28627	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 28628	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 28629	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 28630	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 28631*	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”⟩)
16.2.18  Congruence Theorems
Theorem	tgsas1 28632	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 28633	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 28634	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 28635	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 28636	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 28637	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 28638	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 28639	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 28640	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 28641	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 28289, already covers that part: see trgcgr 28294. 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 28294. (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 28642	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))
16.2.19  Equilateral triangles
Syntax	ceqlg 28643	Declare the class of equilateral triangles.
class eqltrG
Definition	df-eqlg 28644*	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 28645	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 28646	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))
16.3  Properties of geometries
16.3.1  Isomorphisms between geometries
Theorem	f1otrgds 28647*	Convenient lemma for f1otrg 28649. (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–1-1-onto→𝑃)    &   ⊢ ((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 28648*	Convenient lemma for f1otrg 28649. (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–1-1-onto→𝑃)    &   ⊢ ((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 28649*	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–1-1-onto→𝑃)    &   ⊢ ((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 28650*	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–1-1-onto→𝑃)    &   ⊢ ((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 28651	Function to convert an algebraic structure to a Tarski geometry.
class toTG
Definition	df-ttg 28652*	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 28653*	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	ttgvalOLD 28654*	Obsolete proof of ttgval 28653 as of 9-Nov-2024. Define a function to augment a subcomplex Hilbert space with betweenness and a line definition. (Contributed by Thierry Arnoux, 25-Mar-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 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 28655	Lemma for ttgbas 28657, ttgvsca 28662 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	ttglemOLD 28656	Obsolete version of ttglem 28655 as of 29-Oct-2024. Lemma for ttgbas 28657 and ttgvsca 28662. (Contributed by Thierry Arnoux, 15-Apr-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐺 = (toTG‘𝐻)    &   ⊢ 𝐸 = Slot 𝑁    &   ⊢ 𝑁 ∈ ℕ    &   ⊢ 𝑁 < ;16    ⇒   ⊢ (𝐸‘𝐻) = (𝐸‘𝐺)
Theorem	ttgbas 28657	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)
Theorem	ttgbasOLD 28658	Obsolete proof of ttgbas 28657 as of 29-Oct-2024. The base set of a subcomplex Hilbert space augmented with betweenness. (Contributed by Thierry Arnoux, 25-Mar-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐺 = (toTG‘𝐻)    &   ⊢ 𝐵 = (Base‘𝐻)    ⇒   ⊢ 𝐵 = (Base‘𝐺)
Theorem	ttgplusg 28659	The addition operation of a subcomplex Hilbert space augmented with betweenness. (Contributed by Thierry Arnoux, 25-Mar-2019.) (Revised by AV, 29-Oct-2024.)
⊢ 𝐺 = (toTG‘𝐻)    &   ⊢ + = (+g‘𝐻)    ⇒   ⊢ + = (+g‘𝐺)
Theorem	ttgplusgOLD 28660	Obsolete proof of ttgplusg 28659 as of 29-Oct-2024. The addition operation of a subcomplex Hilbert space augmented with betweenness. (Contributed by Thierry Arnoux, 25-Mar-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐺 = (toTG‘𝐻)    &   ⊢ + = (+g‘𝐻)    ⇒   ⊢ + = (+g‘𝐺)
Theorem	ttgsub 28661	The subtraction operation of a subcomplex Hilbert space augmented with betweenness. (Contributed by Thierry Arnoux, 25-Mar-2019.)
⊢ 𝐺 = (toTG‘𝐻)    &   ⊢ − = (-g‘𝐻)    ⇒   ⊢ − = (-g‘𝐺)
Theorem	ttgvsca 28662	The scalar product of a subcomplex Hilbert space augmented with betweenness. (Contributed by Thierry Arnoux, 25-Mar-2019.) (Revised by AV, 29-Oct-2024.)
⊢ 𝐺 = (toTG‘𝐻)    &   ⊢ · = ( ·𝑠 ‘𝐻)    ⇒   ⊢ · = ( ·𝑠 ‘𝐺)
Theorem	ttgvscaOLD 28663	Obsolete proof of ttgvsca 28662 as of 29-Oct-2024. The scalar product of a subcomplex Hilbert space augmented with betweenness. (Contributed by Thierry Arnoux, 25-Mar-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐺 = (toTG‘𝐻)    &   ⊢ · = ( ·𝑠 ‘𝐻)    ⇒   ⊢ · = ( ·𝑠 ‘𝐺)
Theorem	ttgds 28664	The metric of a subcomplex Hilbert space augmented with betweenness. (Contributed by Thierry Arnoux, 25-Mar-2019.) (Revised by AV, 29-Oct-2024.)
⊢ 𝐺 = (toTG‘𝐻)    &   ⊢ 𝐷 = (dist‘𝐻)    ⇒   ⊢ 𝐷 = (dist‘𝐺)
Theorem	ttgdsOLD 28665	Obsolete proof of ttgds 28664 as of 29-Oct-2024. The metric of a subcomplex Hilbert space augmented with betweenness. (Contributed by Thierry Arnoux, 25-Mar-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐺 = (toTG‘𝐻)    &   ⊢ 𝐷 = (dist‘𝐻)    ⇒   ⊢ 𝐷 = (dist‘𝐺)
Theorem	ttgitvval 28666*	Betweenness for a subcomplex Hilbert space augmented with betweenness. (Contributed by Thierry Arnoux, 25-Mar-2019.)
⊢ 𝐺 = (toTG‘𝐻)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ 𝑃 = (Base‘𝐻)    &   ⊢ − = (-g‘𝐻)    &   ⊢ · = ( ·𝑠 ‘𝐻)    ⇒   ⊢ ((𝐻 ∈ 𝑉 ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) → (𝑋𝐼𝑌) = {𝑧 ∈ 𝑃 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑋) = (𝑘 · (𝑌 − 𝑋))})
Theorem	ttgelitv 28667*	Betweenness for a subcomplex Hilbert space augmented with betweenness. (Contributed by Thierry Arnoux, 25-Mar-2019.)
⊢ 𝐺 = (toTG‘𝐻)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ 𝑃 = (Base‘𝐻)    &   ⊢ − = (-g‘𝐻)    &   ⊢ · = ( ·𝑠 ‘𝐻)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐻 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑃)    ⇒   ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐼𝑌) ↔ ∃𝑘 ∈ (0[,]1)(𝑍 − 𝑋) = (𝑘 · (𝑌 − 𝑋))))
Theorem	ttgbtwnid 28668	Any subcomplex module equipped with the betweenness operation fulfills the identity of betweenness (Axiom A6). (Contributed by Thierry Arnoux, 26-Mar-2019.)
⊢ 𝐺 = (toTG‘𝐻)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ 𝑃 = (Base‘𝐻)    &   ⊢ − = (-g‘𝐻)    &   ⊢ · = ( ·𝑠 ‘𝐻)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑃)    &   ⊢ 𝑅 = (Base‘(Scalar‘𝐻))    &   ⊢ (𝜑 → (0[,]1) ⊆ 𝑅)    &   ⊢ (𝜑 → 𝐻 ∈ ℂMod)    &   ⊢ (𝜑 → 𝑌 ∈ (𝑋𝐼𝑋))    ⇒   ⊢ (𝜑 → 𝑋 = 𝑌)
Theorem	ttgcontlem1 28669	Lemma for % ttgcont . (Contributed by Thierry Arnoux, 24-May-2019.)
⊢ 𝐺 = (toTG‘𝐻)    &   ⊢ 𝐼 = (Itv‘𝐺)    &   ⊢ 𝑃 = (Base‘𝐻)    &   ⊢ − = (-g‘𝐻)    &   ⊢ · = ( ·𝑠 ‘𝐻)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑃)    &   ⊢ 𝑅 = (Base‘(Scalar‘𝐻))    &   ⊢ (𝜑 → (0[,]1) ⊆ 𝑅)    &   ⊢ + = (+g‘𝐻)    &   ⊢ (𝜑 → 𝐻 ∈ ℂVec)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑁 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑀 ≠ 0)    &   ⊢ (𝜑 → 𝐾 ≠ 0)    &   ⊢ (𝜑 → 𝐾 ≠ 1)    &   ⊢ (𝜑 → 𝐿 ≠ 𝑀)    &   ⊢ (𝜑 → 𝐿 ≤ (𝑀 / 𝐾))    &   ⊢ (𝜑 → 𝐿 ∈ (0[,]1))    &   ⊢ (𝜑 → 𝐾 ∈ (0[,]1))    &   ⊢ (𝜑 → 𝑀 ∈ (0[,]𝐿))    &   ⊢ (𝜑 → (𝑋 − 𝐴) = (𝐾 · (𝑌 − 𝐴)))    &   ⊢ (𝜑 → (𝑋 − 𝐴) = (𝑀 · (𝑁 − 𝐴)))    &   ⊢ (𝜑 → 𝐵 = (𝐴 + (𝐿 · (𝑁 − 𝐴))))    ⇒   ⊢ (𝜑 → 𝐵 ∈ (𝑋𝐼𝑌))
Theorem	xmstrkgc 28670	Any metric space fulfills Tarski's geometry axioms of congruence. (Contributed by Thierry Arnoux, 13-Mar-2019.)
⊢ (𝐺 ∈ ∞MetSp → 𝐺 ∈ TarskiGC)
16.4.1  Geometry in the complex plane
Theorem	cchhllem 28671*	Lemma for chlbas and chlvsca . (Contributed by Thierry Arnoux, 15-Apr-2019.) (Revised by AV, 29-Oct-2024.)
⊢ 𝐶 = (((subringAlg ‘ℂfld)‘ℝ) sSet ⟨(·𝑖‘ndx), (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · (∗‘𝑦)))⟩)    &   ⊢ 𝐸 = Slot (𝐸‘ndx)    &   ⊢ (Scalar‘ndx) ≠ (𝐸‘ndx)    &   ⊢ ( ·𝑠 ‘ndx) ≠ (𝐸‘ndx)    &   ⊢ (·𝑖‘ndx) ≠ (𝐸‘ndx)    ⇒   ⊢ (𝐸‘ℂfld) = (𝐸‘𝐶)
Theorem	cchhllemOLD 28672*	Obsolete version of cchhllem 28671 as of 29-Oct-2024. Lemma for chlbas and chlvsca . (Contributed by Thierry Arnoux, 15-Apr-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐶 = (((subringAlg ‘ℂfld)‘ℝ) sSet ⟨(·𝑖‘ndx), (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · (∗‘𝑦)))⟩)    &   ⊢ 𝐸 = Slot 𝑁    &   ⊢ 𝑁 ∈ ℕ    &   ⊢ (𝑁 < 5 ∨ 8 < 𝑁)    ⇒   ⊢ (𝐸‘ℂfld) = (𝐸‘𝐶)
16.4.2  Geometry in Euclidean spaces
16.4.2.1  Definition of the Euclidean space
Syntax	cee 28673	Declare the syntax for the Euclidean space generator.
class 𝔼
Syntax	cbtwn 28674	Declare the syntax for the Euclidean betweenness predicate.
class Btwn
Syntax	ccgr 28675	Declare the syntax for the Euclidean congruence predicate.
class Cgr
Definition	df-ee 28676	Define the Euclidean space generator. For details, see elee 28679. (Contributed by Scott Fenton, 3-Jun-2013.)
⊢ 𝔼 = (𝑛 ∈ ℕ ↦ (ℝ ↑m (1...𝑛)))
Definition	df-btwn 28677*	Define the Euclidean betweenness predicate. For details, see brbtwn 28684. (Contributed by Scott Fenton, 3-Jun-2013.)
⊢ Btwn = ◡{⟨⟨𝑥, 𝑧⟩, 𝑦⟩ ∣ ∃𝑛 ∈ ℕ ((𝑥 ∈ (𝔼‘𝑛) ∧ 𝑧 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛)) ∧ ∃𝑡 ∈ (0[,]1)∀𝑖 ∈ (1...𝑛)(𝑦‘𝑖) = (((1 − 𝑡) · (𝑥‘𝑖)) + (𝑡 · (𝑧‘𝑖))))}
Definition	df-cgr 28678*	Define the Euclidean congruence predicate. For details, see brcgr 28685. (Contributed by Scott Fenton, 3-Jun-2013.)
⊢ Cgr = {⟨𝑥, 𝑦⟩ ∣ ∃𝑛 ∈ ℕ ((𝑥 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑦 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛))) ∧ Σ𝑖 ∈ (1...𝑛)((((1st ‘𝑥)‘𝑖) − ((2nd ‘𝑥)‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑛)((((1st ‘𝑦)‘𝑖) − ((2nd ‘𝑦)‘𝑖))↑2))}
Theorem	elee 28679	Membership in a Euclidean space. We define Euclidean space here using Cartesian coordinates over 𝑁 space. We later abstract away from this using Tarski's geometry axioms, so this exact definition is unimportant. (Contributed by Scott Fenton, 3-Jun-2013.)
⊢ (𝑁 ∈ ℕ → (𝐴 ∈ (𝔼‘𝑁) ↔ 𝐴:(1...𝑁)⟶ℝ))
Theorem	mptelee 28680*	A condition for a mapping to be an element of a Euclidean space. (Contributed by Scott Fenton, 7-Jun-2013.)
⊢ (𝑁 ∈ ℕ → ((𝑘 ∈ (1...𝑁) ↦ (𝐴𝐹𝐵)) ∈ (𝔼‘𝑁) ↔ ∀𝑘 ∈ (1...𝑁)(𝐴𝐹𝐵) ∈ ℝ))
Theorem	eleenn 28681	If 𝐴 is in (𝔼‘𝑁), then 𝑁 is a natural. (Contributed by Scott Fenton, 1-Jul-2013.)
⊢ (𝐴 ∈ (𝔼‘𝑁) → 𝑁 ∈ ℕ)
Theorem	eleei 28682	The forward direction of elee 28679. (Contributed by Scott Fenton, 1-Jul-2013.)
⊢ (𝐴 ∈ (𝔼‘𝑁) → 𝐴:(1...𝑁)⟶ℝ)
Theorem	eedimeq 28683	A point belongs to at most one Euclidean space. (Contributed by Scott Fenton, 1-Jul-2013.)
⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑀)) → 𝑁 = 𝑀)
Theorem	brbtwn 28684*	The binary relation form of the betweenness predicate. The statement 𝐴 Btwn ⟨𝐵, 𝐶⟩ should be informally read as "𝐴 lies on a line segment between 𝐵 and 𝐶. This exact definition is abstracted away by Tarski's geometry axioms later on. (Contributed by Scott Fenton, 3-Jun-2013.)
⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) → (𝐴 Btwn ⟨𝐵, 𝐶⟩ ↔ ∃𝑡 ∈ (0[,]1)∀𝑖 ∈ (1...𝑁)(𝐴‘𝑖) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝐶‘𝑖)))))
Theorem	brcgr 28685*	The binary relation form of the congruence predicate. The statement ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝐷⟩ should be read informally as "the 𝑁 dimensional point 𝐴 is as far from 𝐵 as 𝐶 is from 𝐷, or "the line segment 𝐴𝐵 is congruent to the line segment 𝐶𝐷. This particular definition is encapsulated by Tarski's axioms later on. (Contributed by Scott Fenton, 3-Jun-2013.)
⊢ (((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → (⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝐷⟩ ↔ Σ𝑖 ∈ (1...𝑁)(((𝐴‘𝑖) − (𝐵‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑁)(((𝐶‘𝑖) − (𝐷‘𝑖))↑2)))
Theorem	fveere 28686	The function value of a point is a real. (Contributed by Scott Fenton, 10-Jun-2013.)
⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐼 ∈ (1...𝑁)) → (𝐴‘𝐼) ∈ ℝ)
Theorem	fveecn 28687	The function value of a point is a complex. (Contributed by Scott Fenton, 10-Jun-2013.)
⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐼 ∈ (1...𝑁)) → (𝐴‘𝐼) ∈ ℂ)
Theorem	eqeefv 28688*	Two points are equal iff they agree in all dimensions. (Contributed by Scott Fenton, 10-Jun-2013.)
⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → (𝐴 = 𝐵 ↔ ∀𝑖 ∈ (1...𝑁)(𝐴‘𝑖) = (𝐵‘𝑖)))
Theorem	eqeelen 28689*	Two points are equal iff the square of the distance between them is zero. (Contributed by Scott Fenton, 10-Jun-2013.) (Revised by Mario Carneiro, 22-May-2014.)
⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → (𝐴 = 𝐵 ↔ Σ𝑖 ∈ (1...𝑁)(((𝐴‘𝑖) − (𝐵‘𝑖))↑2) = 0))
Theorem	brbtwn2 28690*	Alternate characterization of betweenness, with no existential quantifiers. (Contributed by Scott Fenton, 24-Jun-2013.)
⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) → (𝐴 Btwn ⟨𝐵, 𝐶⟩ ↔ (∀𝑖 ∈ (1...𝑁)(((𝐵‘𝑖) − (𝐴‘𝑖)) · ((𝐶‘𝑖) − (𝐴‘𝑖))) ≤ 0 ∧ ∀𝑖 ∈ (1...𝑁)∀𝑗 ∈ (1...𝑁)(((𝐵‘𝑖) − (𝐴‘𝑖)) · ((𝐶‘𝑗) − (𝐴‘𝑗))) = (((𝐵‘𝑗) − (𝐴‘𝑗)) · ((𝐶‘𝑖) − (𝐴‘𝑖))))))
Theorem	colinearalglem1 28691	Lemma for colinearalg 28695. Expand out a multiplication. (Contributed by Scott Fenton, 24-Jun-2013.)
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ ∧ 𝐹 ∈ ℂ)) → (((𝐵 − 𝐴) · (𝐹 − 𝐷)) = ((𝐸 − 𝐷) · (𝐶 − 𝐴)) ↔ ((𝐵 · 𝐹) − ((𝐴 · 𝐹) + (𝐵 · 𝐷))) = ((𝐶 · 𝐸) − ((𝐴 · 𝐸) + (𝐶 · 𝐷)))))
Theorem	colinearalglem2 28692*	Lemma for colinearalg 28695. Translate between two forms of the colinearity condition. (Contributed by Scott Fenton, 24-Jun-2013.)
⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) → (∀𝑖 ∈ (1...𝑁)∀𝑗 ∈ (1...𝑁)(((𝐵‘𝑖) − (𝐴‘𝑖)) · ((𝐶‘𝑗) − (𝐴‘𝑗))) = (((𝐵‘𝑗) − (𝐴‘𝑗)) · ((𝐶‘𝑖) − (𝐴‘𝑖))) ↔ ∀𝑖 ∈ (1...𝑁)∀𝑗 ∈ (1...𝑁)(((𝐶‘𝑖) − (𝐵‘𝑖)) · ((𝐴‘𝑗) − (𝐵‘𝑗))) = (((𝐶‘𝑗) − (𝐵‘𝑗)) · ((𝐴‘𝑖) − (𝐵‘𝑖)))))
Theorem	colinearalglem3 28693*	Lemma for colinearalg 28695. Translate between two forms of the colinearity condition. (Contributed by Scott Fenton, 24-Jun-2013.)
⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) → (∀𝑖 ∈ (1...𝑁)∀𝑗 ∈ (1...𝑁)(((𝐵‘𝑖) − (𝐴‘𝑖)) · ((𝐶‘𝑗) − (𝐴‘𝑗))) = (((𝐵‘𝑗) − (𝐴‘𝑗)) · ((𝐶‘𝑖) − (𝐴‘𝑖))) ↔ ∀𝑖 ∈ (1...𝑁)∀𝑗 ∈ (1...𝑁)(((𝐴‘𝑖) − (𝐶‘𝑖)) · ((𝐵‘𝑗) − (𝐶‘𝑗))) = (((𝐴‘𝑗) − (𝐶‘𝑗)) · ((𝐵‘𝑖) − (𝐶‘𝑖)))))
Theorem	colinearalglem4 28694*	Lemma for colinearalg 28695. Prove a disjunction that will be needed in the final proof. (Contributed by Scott Fenton, 27-Jun-2013.)
⊢ (((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ 𝐾 ∈ ℝ) → (∀𝑖 ∈ (1...𝑁)((((𝐾 · ((𝐶‘𝑖) − (𝐴‘𝑖))) + (𝐴‘𝑖)) − (𝐴‘𝑖)) · ((𝐶‘𝑖) − (𝐴‘𝑖))) ≤ 0 ∨ ∀𝑖 ∈ (1...𝑁)(((𝐶‘𝑖) − ((𝐾 · ((𝐶‘𝑖) − (𝐴‘𝑖))) + (𝐴‘𝑖))) · ((𝐴‘𝑖) − ((𝐾 · ((𝐶‘𝑖) − (𝐴‘𝑖))) + (𝐴‘𝑖)))) ≤ 0 ∨ ∀𝑖 ∈ (1...𝑁)(((𝐴‘𝑖) − (𝐶‘𝑖)) · (((𝐾 · ((𝐶‘𝑖) − (𝐴‘𝑖))) + (𝐴‘𝑖)) − (𝐶‘𝑖))) ≤ 0))
Theorem	colinearalg 28695*	An algebraic characterization of colinearity. Note the similarity to brbtwn2 28690. (Contributed by Scott Fenton, 24-Jun-2013.)
⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) → ((𝐴 Btwn ⟨𝐵, 𝐶⟩ ∨ 𝐵 Btwn ⟨𝐶, 𝐴⟩ ∨ 𝐶 Btwn ⟨𝐴, 𝐵⟩) ↔ ∀𝑖 ∈ (1...𝑁)∀𝑗 ∈ (1...𝑁)(((𝐵‘𝑖) − (𝐴‘𝑖)) · ((𝐶‘𝑗) − (𝐴‘𝑗))) = (((𝐵‘𝑗) − (𝐴‘𝑗)) · ((𝐶‘𝑖) − (𝐴‘𝑖)))))
Theorem	eleesub 28696*	Membership of a subtraction mapping in a Euclidean space. (Contributed by Scott Fenton, 17-Jul-2013.)
⊢ 𝐶 = (𝑖 ∈ (1...𝑁) ↦ ((𝐴‘𝑖) − (𝐵‘𝑖)))    ⇒   ⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → 𝐶 ∈ (𝔼‘𝑁))
Theorem	eleesubd 28697*	Membership of a subtraction mapping in a Euclidean space. Deduction form of eleesub 28696. (Contributed by Scott Fenton, 17-Jul-2013.)
⊢ (𝜑 → 𝐶 = (𝑖 ∈ (1...𝑁) ↦ ((𝐴‘𝑖) − (𝐵‘𝑖))))    ⇒   ⊢ ((𝜑 ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → 𝐶 ∈ (𝔼‘𝑁))
16.4.2.2  Tarski's axioms for geometry for the Euclidean space
Theorem	axdimuniq 28698	The unique dimension axiom. If a point is in 𝑁 dimensional space and in 𝑀 dimensional space, then 𝑁 = 𝑀. This axiom is not traditionally presented with Tarski's axioms, but we require it here as we are considering spaces in arbitrary dimensions. (Contributed by Scott Fenton, 24-Sep-2013.)
⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁)) ∧ (𝑀 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑀))) → 𝑁 = 𝑀)
Theorem	axcgrrflx 28699	𝐴 is as far from 𝐵 as 𝐵 is from 𝐴. Axiom A1 of [Schwabhauser] p. 10. (Contributed by Scott Fenton, 3-Jun-2013.)
⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → ⟨𝐴, 𝐵⟩Cgr⟨𝐵, 𝐴⟩)
Theorem	axcgrtr 28700	Congruence is transitive. Axiom A2 of [Schwabhauser] p. 10. (Contributed by Scott Fenton, 3-Jun-2013.)
⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) → ((⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐸, 𝐹⟩) → ⟨𝐶, 𝐷⟩Cgr⟨𝐸, 𝐹⟩))