P. 338 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 33701-33800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	algextdeglem5 33701*	Lemma for algextdeg 33705. The subspace 𝑍 of annihilators of 𝐴 is a principal ideal generated by the minimal polynomial. (Contributed by Thierry Arnoux, 2-Apr-2025.)
⊢ 𝐾 = (𝐸 ↾s 𝐹)    &   ⊢ 𝐿 = (𝐸 ↾s (𝐸 fldGen (𝐹 ∪ {𝐴})))    &   ⊢ 𝐷 = (deg1‘𝐸)    &   ⊢ 𝑀 = (𝐸 minPoly 𝐹)    &   ⊢ (𝜑 → 𝐸 ∈ Field)    &   ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))    &   ⊢ (𝜑 → 𝐴 ∈ (𝐸 IntgRing 𝐹))    &   ⊢ 𝑂 = (𝐸 evalSub1 𝐹)    &   ⊢ 𝑃 = (Poly1‘𝐾)    &   ⊢ 𝑈 = (Base‘𝑃)    &   ⊢ 𝐺 = (𝑝 ∈ 𝑈 ↦ ((𝑂‘𝑝)‘𝐴))    &   ⊢ 𝑁 = (𝑥 ∈ 𝑈 ↦ [𝑥](𝑃 ~QG 𝑍))    &   ⊢ 𝑍 = (◡𝐺 “ {(0g‘𝐿)})    &   ⊢ 𝑄 = (𝑃 /s (𝑃 ~QG 𝑍))    &   ⊢ 𝐽 = (𝑝 ∈ (Base‘𝑄) ↦ ∪ (𝐺 “ 𝑝))    ⇒   ⊢ (𝜑 → 𝑍 = ((RSpan‘𝑃)‘{(𝑀‘𝐴)}))
Theorem	algextdeglem6 33702*	Lemma for algextdeg 33705. By r1pquslmic 33566, the univariate polynomial remainder ring (𝐻 “s 𝑃) is isomorphic with the quotient ring 𝑄. (Contributed by Thierry Arnoux, 2-Apr-2025.)
⊢ 𝐾 = (𝐸 ↾s 𝐹)    &   ⊢ 𝐿 = (𝐸 ↾s (𝐸 fldGen (𝐹 ∪ {𝐴})))    &   ⊢ 𝐷 = (deg1‘𝐸)    &   ⊢ 𝑀 = (𝐸 minPoly 𝐹)    &   ⊢ (𝜑 → 𝐸 ∈ Field)    &   ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))    &   ⊢ (𝜑 → 𝐴 ∈ (𝐸 IntgRing 𝐹))    &   ⊢ 𝑂 = (𝐸 evalSub1 𝐹)    &   ⊢ 𝑃 = (Poly1‘𝐾)    &   ⊢ 𝑈 = (Base‘𝑃)    &   ⊢ 𝐺 = (𝑝 ∈ 𝑈 ↦ ((𝑂‘𝑝)‘𝐴))    &   ⊢ 𝑁 = (𝑥 ∈ 𝑈 ↦ [𝑥](𝑃 ~QG 𝑍))    &   ⊢ 𝑍 = (◡𝐺 “ {(0g‘𝐿)})    &   ⊢ 𝑄 = (𝑃 /s (𝑃 ~QG 𝑍))    &   ⊢ 𝐽 = (𝑝 ∈ (Base‘𝑄) ↦ ∪ (𝐺 “ 𝑝))    &   ⊢ 𝑅 = (rem1p‘𝐾)    &   ⊢ 𝐻 = (𝑝 ∈ 𝑈 ↦ (𝑝𝑅(𝑀‘𝐴)))    ⇒   ⊢ (𝜑 → (dim‘𝑄) = (dim‘(𝐻 “s 𝑃)))
Theorem	algextdeglem7 33703*	Lemma for algextdeg 33705. The polynomials 𝑋 of lower degree than the minimal polynomial are left unchanged when taking the remainder of the division by that minimal polynomial. (Contributed by Thierry Arnoux, 2-Apr-2025.)
⊢ 𝐾 = (𝐸 ↾s 𝐹)    &   ⊢ 𝐿 = (𝐸 ↾s (𝐸 fldGen (𝐹 ∪ {𝐴})))    &   ⊢ 𝐷 = (deg1‘𝐸)    &   ⊢ 𝑀 = (𝐸 minPoly 𝐹)    &   ⊢ (𝜑 → 𝐸 ∈ Field)    &   ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))    &   ⊢ (𝜑 → 𝐴 ∈ (𝐸 IntgRing 𝐹))    &   ⊢ 𝑂 = (𝐸 evalSub1 𝐹)    &   ⊢ 𝑃 = (Poly1‘𝐾)    &   ⊢ 𝑈 = (Base‘𝑃)    &   ⊢ 𝐺 = (𝑝 ∈ 𝑈 ↦ ((𝑂‘𝑝)‘𝐴))    &   ⊢ 𝑁 = (𝑥 ∈ 𝑈 ↦ [𝑥](𝑃 ~QG 𝑍))    &   ⊢ 𝑍 = (◡𝐺 “ {(0g‘𝐿)})    &   ⊢ 𝑄 = (𝑃 /s (𝑃 ~QG 𝑍))    &   ⊢ 𝐽 = (𝑝 ∈ (Base‘𝑄) ↦ ∪ (𝐺 “ 𝑝))    &   ⊢ 𝑅 = (rem1p‘𝐾)    &   ⊢ 𝐻 = (𝑝 ∈ 𝑈 ↦ (𝑝𝑅(𝑀‘𝐴)))    &   ⊢ 𝑇 = (◡(deg1‘𝐾) “ (-∞[,)(𝐷‘(𝑀‘𝐴))))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑈)    ⇒   ⊢ (𝜑 → (𝑋 ∈ 𝑇 ↔ (𝐻‘𝑋) = 𝑋))
Theorem	algextdeglem8 33704*	Lemma for algextdeg 33705. The dimension of the univariate polynomial remainder ring (𝐻 “s 𝑃) is the degree of the minimal polynomial. (Contributed by Thierry Arnoux, 2-Apr-2025.)
⊢ 𝐾 = (𝐸 ↾s 𝐹)    &   ⊢ 𝐿 = (𝐸 ↾s (𝐸 fldGen (𝐹 ∪ {𝐴})))    &   ⊢ 𝐷 = (deg1‘𝐸)    &   ⊢ 𝑀 = (𝐸 minPoly 𝐹)    &   ⊢ (𝜑 → 𝐸 ∈ Field)    &   ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))    &   ⊢ (𝜑 → 𝐴 ∈ (𝐸 IntgRing 𝐹))    &   ⊢ 𝑂 = (𝐸 evalSub1 𝐹)    &   ⊢ 𝑃 = (Poly1‘𝐾)    &   ⊢ 𝑈 = (Base‘𝑃)    &   ⊢ 𝐺 = (𝑝 ∈ 𝑈 ↦ ((𝑂‘𝑝)‘𝐴))    &   ⊢ 𝑁 = (𝑥 ∈ 𝑈 ↦ [𝑥](𝑃 ~QG 𝑍))    &   ⊢ 𝑍 = (◡𝐺 “ {(0g‘𝐿)})    &   ⊢ 𝑄 = (𝑃 /s (𝑃 ~QG 𝑍))    &   ⊢ 𝐽 = (𝑝 ∈ (Base‘𝑄) ↦ ∪ (𝐺 “ 𝑝))    &   ⊢ 𝑅 = (rem1p‘𝐾)    &   ⊢ 𝐻 = (𝑝 ∈ 𝑈 ↦ (𝑝𝑅(𝑀‘𝐴)))    &   ⊢ 𝑇 = (◡(deg1‘𝐾) “ (-∞[,)(𝐷‘(𝑀‘𝐴))))    ⇒   ⊢ (𝜑 → (dim‘(𝐻 “s 𝑃)) = (𝐷‘(𝑀‘𝐴)))
Theorem	algextdeg 33705	The degree of an algebraic field extension (noted [𝐿:𝐾]) is the degree of the minimal polynomial 𝑀(𝐴), whereas 𝐿 is the field generated by 𝐾 and the algebraic element 𝐴. (Contributed by Thierry Arnoux, 2-Apr-2025.)
⊢ 𝐾 = (𝐸 ↾s 𝐹)    &   ⊢ 𝐿 = (𝐸 ↾s (𝐸 fldGen (𝐹 ∪ {𝐴})))    &   ⊢ 𝐷 = (deg1‘𝐸)    &   ⊢ 𝑀 = (𝐸 minPoly 𝐹)    &   ⊢ (𝜑 → 𝐸 ∈ Field)    &   ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))    &   ⊢ (𝜑 → 𝐴 ∈ (𝐸 IntgRing 𝐹))    ⇒   ⊢ (𝜑 → (𝐿[:]𝐾) = (𝐷‘(𝑀‘𝐴)))
21.3.11.3  Quadratic Field Extensions
Theorem	rtelextdg2lem 33706	Lemma for rtelextdg2 33707: If an element 𝑋 is a solution of a quadratic equation, then the degree of its field extension is at most 2. (Contributed by Thierry Arnoux, 22-Jun-2025.)
⊢ 𝐾 = (𝐸 ↾s 𝐹)    &   ⊢ 𝐿 = (𝐸 ↾s (𝐸 fldGen (𝐹 ∪ {𝑋})))    &   ⊢ 0 = (0g‘𝐸)    &   ⊢ 𝑃 = (Poly1‘𝐾)    &   ⊢ 𝑉 = (Base‘𝐸)    &   ⊢ · = (.r‘𝐸)    &   ⊢ + = (+g‘𝐸)    &   ⊢ ↑ = (.g‘(mulGrp‘𝐸))    &   ⊢ (𝜑 → 𝐸 ∈ Field)    &   ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐹)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐹)    &   ⊢ (𝜑 → ((2 ↑ 𝑋) + ((𝐴 · 𝑋) + 𝐵)) = 0 )    &   ⊢ 𝑌 = (var1‘𝐾)    &   ⊢ ⊕ = (+g‘𝑃)    &   ⊢ ⊗ = (.r‘𝑃)    &   ⊢ ∧ = (.g‘(mulGrp‘𝑃))    &   ⊢ 𝑈 = (algSc‘𝑃)    &   ⊢ 𝐺 = ((2 ∧ 𝑌) ⊕ (((𝑈‘𝐴) ⊗ 𝑌) ⊕ (𝑈‘𝐵)))    ⇒   ⊢ (𝜑 → (𝐿[:]𝐾) ≤ 2)
Theorem	rtelextdg2 33707	If an element 𝑋 is a solution of a quadratic equation, then it is either in the base field, or the degree of its field extension is exactly 2. (Contributed by Thierry Arnoux, 22-Jun-2025.)
⊢ 𝐾 = (𝐸 ↾s 𝐹)    &   ⊢ 𝐿 = (𝐸 ↾s (𝐸 fldGen (𝐹 ∪ {𝑋})))    &   ⊢ 0 = (0g‘𝐸)    &   ⊢ 𝑃 = (Poly1‘𝐾)    &   ⊢ 𝑉 = (Base‘𝐸)    &   ⊢ · = (.r‘𝐸)    &   ⊢ + = (+g‘𝐸)    &   ⊢ ↑ = (.g‘(mulGrp‘𝐸))    &   ⊢ (𝜑 → 𝐸 ∈ Field)    &   ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐹)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐹)    &   ⊢ (𝜑 → ((2 ↑ 𝑋) + ((𝐴 · 𝑋) + 𝐵)) = 0 )    ⇒   ⊢ (𝜑 → (𝑋 ∈ 𝐹 ∨ (𝐿[:]𝐾) = 2))
21.3.11.4  Towers of quadratic extentions
Theorem	fldext2chn 33708*	In a non-empty chain 𝑇 of quadratic field extensions, the degree of the final extension is always a power of two. (Contributed by Thierry Arnoux, 19-Oct-2025.)
⊢ 𝐸 = (𝑊 ↾s 𝑒)    &   ⊢ 𝐹 = (𝑊 ↾s 𝑓)    &   ⊢ < = {⟨𝑓, 𝑒⟩ ∣ (𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 2)}    &   ⊢ (𝜑 → 𝑇 ∈ ( < Chain(SubDRing‘𝑊)))    &   ⊢ (𝜑 → 𝑊 ∈ Field)    &   ⊢ (𝜑 → (𝑊 ↾s (𝑇‘0)) = 𝑄)    &   ⊢ (𝜑 → (𝑊 ↾s (lastS‘𝑇)) = 𝐿)    &   ⊢ (𝜑 → 0 < (♯‘𝑇))    ⇒   ⊢ (𝜑 → (𝐿/FldExt𝑄 ∧ ∃𝑛 ∈ ℕ0 (𝐿[:]𝑄) = (2↑𝑛)))
21.3.12  Constructible Numbers This section defines the set of constructible points as complex numbers which can be drawn starting from two points (we take 0 and 1), and taking intersections of circles and lines. This construction is useful for proving the impossibility of doubling the cube (2sqr3nconstr 33761), and of angle trisection ( * cos9thpinconstr )
Syntax	cconstr 33709	Extend class notation with the set of constructible points.
class Constr
Definition	df-constr 33710*	Define the set of geometrically constructible points, by recursively adding the line-line, line-circle and circle-circle intersections constructions using points in a previous iteration. (Contributed by Saveliy Skresanov, 19-Jan-2025.)
⊢ Constr = ∪ (rec((𝑠 ∈ V ↦ {𝑥 ∈ ℂ ∣ (∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑡 ∈ ℝ ∃𝑟 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ 𝑥 = (𝑐 + (𝑟 · (𝑑 − 𝑐))) ∧ (ℑ‘((∗‘(𝑏 − 𝑎)) · (𝑑 − 𝑐))) ≠ 0) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 ∃𝑡 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ (abs‘(𝑥 − 𝑐)) = (abs‘(𝑒 − 𝑓))) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 (𝑎 ≠ 𝑑 ∧ (abs‘(𝑥 − 𝑎)) = (abs‘(𝑏 − 𝑐)) ∧ (abs‘(𝑥 − 𝑑)) = (abs‘(𝑒 − 𝑓))))}), {0, 1}) “ ω)
Theorem	constrrtll 33711	In the construction of constructible numbers, line-line intersections are solutions of linear equations, and can therefore be completely constructed. (Contributed by Thierry Arnoux, 6-Jul-2025.)
⊢ (𝜑 → 𝑆 ⊆ ℂ)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑇 ∈ ℝ)    &   ⊢ (𝜑 → 𝑅 ∈ ℝ)    &   ⊢ (𝜑 → 𝑋 = (𝐴 + (𝑇 · (𝐵 − 𝐴))))    &   ⊢ (𝜑 → 𝑋 = (𝐶 + (𝑅 · (𝐷 − 𝐶))))    &   ⊢ (𝜑 → (ℑ‘((∗‘(𝐵 − 𝐴)) · (𝐷 − 𝐶))) ≠ 0)    &   ⊢ 𝑁 = (𝐴 + (((((𝐴 − 𝐶) · ((∗‘𝐷) − (∗‘𝐶))) − (((∗‘𝐴) − (∗‘𝐶)) · (𝐷 − 𝐶))) / ((((∗‘𝐵) − (∗‘𝐴)) · (𝐷 − 𝐶)) − ((𝐵 − 𝐴) · ((∗‘𝐷) − (∗‘𝐶))))) · (𝐵 − 𝐴)))    ⇒   ⊢ (𝜑 → 𝑋 = 𝑁)
Theorem	constrrtlc1 33712	In the construction of constructible numbers, line-circle intersections are roots of a quadratic equation, non-degenerate case. (Contributed by Thierry Arnoux, 6-Jul-2025.)
⊢ (𝜑 → 𝑆 ⊆ ℂ)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑇 ∈ ℝ)    &   ⊢ (𝜑 → 𝑋 = (𝐴 + (𝑇 · (𝐵 − 𝐴))))    &   ⊢ (𝜑 → (abs‘(𝑋 − 𝐶)) = (abs‘(𝐸 − 𝐹)))    &   ⊢ 𝑄 = (((∗‘𝐵) − (∗‘𝐴)) / (𝐵 − 𝐴))    &   ⊢ 𝑀 = (((((∗‘𝐴) − (𝐴 · 𝑄)) − (∗‘𝐶)) − (𝐶 · 𝑄)) / 𝑄)    &   ⊢ 𝑁 = (-((𝐶 · (((∗‘𝐴) − (𝐴 · 𝑄)) − (∗‘𝐶))) + ((𝐸 − 𝐹) · ((∗‘𝐸) − (∗‘𝐹)))) / 𝑄)    &   ⊢ (𝜑 → 𝐴 ≠ 𝐵)    ⇒   ⊢ (𝜑 → (((𝑋↑2) + ((𝑀 · 𝑋) + 𝑁)) = 0 ∧ 𝑄 ≠ 0))
Theorem	constrrtlc2 33713	In the construction of constructible numbers, line-circle intersections are one of the original points, in a degenerate case. (Contributed by Thierry Arnoux, 6-Jul-2025.)
⊢ (𝜑 → 𝑆 ⊆ ℂ)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑇 ∈ ℝ)    &   ⊢ (𝜑 → 𝑋 = (𝐴 + (𝑇 · (𝐵 − 𝐴))))    &   ⊢ (𝜑 → (abs‘(𝑋 − 𝐶)) = (abs‘(𝐸 − 𝐹)))    &   ⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → 𝑋 = 𝐴)
Theorem	constrrtcclem 33714	In the construction of constructible numbers, circle-circle intersections are roots of a quadratic equation. Case of non-degenerate circles. (Contributed by Thierry Arnoux, 6-Jul-2025.)
⊢ (𝜑 → 𝑆 ⊆ ℂ)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑋 ∈ ℂ)    &   ⊢ (𝜑 → 𝐴 ≠ 𝐷)    &   ⊢ (𝜑 → (abs‘(𝑋 − 𝐴)) = (abs‘(𝐵 − 𝐶)))    &   ⊢ (𝜑 → (abs‘(𝑋 − 𝐷)) = (abs‘(𝐸 − 𝐹)))    &   ⊢ 𝑃 = ((𝐵 − 𝐶) · (∗‘(𝐵 − 𝐶)))    &   ⊢ 𝑄 = ((𝐸 − 𝐹) · (∗‘(𝐸 − 𝐹)))    &   ⊢ 𝑀 = (((𝑄 − ((∗‘𝐷) · (𝐷 + 𝐴))) − (𝑃 − ((∗‘𝐴) · (𝐷 + 𝐴)))) / ((∗‘𝐷) − (∗‘𝐴)))    &   ⊢ 𝑁 = -(((((∗‘𝐴) · (𝐷 · 𝐴)) − (𝑃 · 𝐷)) − (((∗‘𝐷) · (𝐷 · 𝐴)) − (𝑄 · 𝐴))) / ((∗‘𝐷) − (∗‘𝐴)))    &   ⊢ (𝜑 → 𝐵 ≠ 𝐶)    &   ⊢ (𝜑 → 𝐸 ≠ 𝐹)    ⇒   ⊢ (𝜑 → ((𝑋↑2) + ((𝑀 · 𝑋) + 𝑁)) = 0)
Theorem	constrrtcc 33715	In the construction of constructible numbers, circle-circle intersections are roots of a quadratic equation. (Contributed by Thierry Arnoux, 6-Jul-2025.)
⊢ (𝜑 → 𝑆 ⊆ ℂ)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑋 ∈ ℂ)    &   ⊢ (𝜑 → 𝐴 ≠ 𝐷)    &   ⊢ (𝜑 → (abs‘(𝑋 − 𝐴)) = (abs‘(𝐵 − 𝐶)))    &   ⊢ (𝜑 → (abs‘(𝑋 − 𝐷)) = (abs‘(𝐸 − 𝐹)))    &   ⊢ 𝑃 = ((𝐵 − 𝐶) · (∗‘(𝐵 − 𝐶)))    &   ⊢ 𝑄 = ((𝐸 − 𝐹) · (∗‘(𝐸 − 𝐹)))    &   ⊢ 𝑀 = (((𝑄 − ((∗‘𝐷) · (𝐷 + 𝐴))) − (𝑃 − ((∗‘𝐴) · (𝐷 + 𝐴)))) / ((∗‘𝐷) − (∗‘𝐴)))    &   ⊢ 𝑁 = -(((((∗‘𝐴) · (𝐷 · 𝐴)) − (𝑃 · 𝐷)) − (((∗‘𝐷) · (𝐷 · 𝐴)) − (𝑄 · 𝐴))) / ((∗‘𝐷) − (∗‘𝐴)))    ⇒   ⊢ (𝜑 → ((𝑋↑2) + ((𝑀 · 𝑋) + 𝑁)) = 0)
Theorem	isconstr 33716*	Property of being a constructible number. (Contributed by Thierry Arnoux, 19-Oct-2025.)
⊢ 𝐶 = rec((𝑠 ∈ V ↦ {𝑥 ∈ ℂ ∣ (∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑡 ∈ ℝ ∃𝑟 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ 𝑥 = (𝑐 + (𝑟 · (𝑑 − 𝑐))) ∧ (ℑ‘((∗‘(𝑏 − 𝑎)) · (𝑑 − 𝑐))) ≠ 0) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 ∃𝑡 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ (abs‘(𝑥 − 𝑐)) = (abs‘(𝑒 − 𝑓))) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 (𝑎 ≠ 𝑑 ∧ (abs‘(𝑥 − 𝑎)) = (abs‘(𝑏 − 𝑐)) ∧ (abs‘(𝑥 − 𝑑)) = (abs‘(𝑒 − 𝑓))))}), {0, 1})    ⇒   ⊢ (𝐴 ∈ Constr ↔ ∃𝑚 ∈ ω 𝐴 ∈ (𝐶‘𝑚))
Theorem	constr0 33717	The first step of the construction of constructible numbers is the pair {0, 1}. In this theorem and the following, we use (𝐶‘𝑁) for the 𝑁-th intermediate iteration of the constructible number. (Contributed by Thierry Arnoux, 25-Jun-2025.)
⊢ 𝐶 = rec((𝑠 ∈ V ↦ {𝑥 ∈ ℂ ∣ (∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑡 ∈ ℝ ∃𝑟 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ 𝑥 = (𝑐 + (𝑟 · (𝑑 − 𝑐))) ∧ (ℑ‘((∗‘(𝑏 − 𝑎)) · (𝑑 − 𝑐))) ≠ 0) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 ∃𝑡 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ (abs‘(𝑥 − 𝑐)) = (abs‘(𝑒 − 𝑓))) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 (𝑎 ≠ 𝑑 ∧ (abs‘(𝑥 − 𝑎)) = (abs‘(𝑏 − 𝑐)) ∧ (abs‘(𝑥 − 𝑑)) = (abs‘(𝑒 − 𝑓))))}), {0, 1})    ⇒   ⊢ (𝐶‘∅) = {0, 1}
Theorem	constrsuc 33718*	Membership in the successor step of the construction of constructible numbers. (Contributed by Thierry Arnoux, 25-Jun-2025.)
⊢ 𝐶 = rec((𝑠 ∈ V ↦ {𝑥 ∈ ℂ ∣ (∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑡 ∈ ℝ ∃𝑟 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ 𝑥 = (𝑐 + (𝑟 · (𝑑 − 𝑐))) ∧ (ℑ‘((∗‘(𝑏 − 𝑎)) · (𝑑 − 𝑐))) ≠ 0) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 ∃𝑡 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ (abs‘(𝑥 − 𝑐)) = (abs‘(𝑒 − 𝑓))) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 (𝑎 ≠ 𝑑 ∧ (abs‘(𝑥 − 𝑎)) = (abs‘(𝑏 − 𝑐)) ∧ (abs‘(𝑥 − 𝑑)) = (abs‘(𝑒 − 𝑓))))}), {0, 1})    &   ⊢ (𝜑 → 𝑁 ∈ On)    &   ⊢ 𝑆 = (𝐶‘𝑁)    ⇒   ⊢ (𝜑 → (𝑋 ∈ (𝐶‘suc 𝑁) ↔ (𝑋 ∈ ℂ ∧ (∃𝑎 ∈ 𝑆 ∃𝑏 ∈ 𝑆 ∃𝑐 ∈ 𝑆 ∃𝑑 ∈ 𝑆 ∃𝑡 ∈ ℝ ∃𝑟 ∈ ℝ (𝑋 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ 𝑋 = (𝑐 + (𝑟 · (𝑑 − 𝑐))) ∧ (ℑ‘((∗‘(𝑏 − 𝑎)) · (𝑑 − 𝑐))) ≠ 0) ∨ ∃𝑎 ∈ 𝑆 ∃𝑏 ∈ 𝑆 ∃𝑐 ∈ 𝑆 ∃𝑒 ∈ 𝑆 ∃𝑓 ∈ 𝑆 ∃𝑡 ∈ ℝ (𝑋 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ (abs‘(𝑋 − 𝑐)) = (abs‘(𝑒 − 𝑓))) ∨ ∃𝑎 ∈ 𝑆 ∃𝑏 ∈ 𝑆 ∃𝑐 ∈ 𝑆 ∃𝑑 ∈ 𝑆 ∃𝑒 ∈ 𝑆 ∃𝑓 ∈ 𝑆 (𝑎 ≠ 𝑑 ∧ (abs‘(𝑋 − 𝑎)) = (abs‘(𝑏 − 𝑐)) ∧ (abs‘(𝑋 − 𝑑)) = (abs‘(𝑒 − 𝑓)))))))
Theorem	constrlim 33719*	Limit step of the construction of constructible numbers. (Contributed by Thierry Arnoux, 25-Jun-2025.)
⊢ 𝐶 = rec((𝑠 ∈ V ↦ {𝑥 ∈ ℂ ∣ (∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑡 ∈ ℝ ∃𝑟 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ 𝑥 = (𝑐 + (𝑟 · (𝑑 − 𝑐))) ∧ (ℑ‘((∗‘(𝑏 − 𝑎)) · (𝑑 − 𝑐))) ≠ 0) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 ∃𝑡 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ (abs‘(𝑥 − 𝑐)) = (abs‘(𝑒 − 𝑓))) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 (𝑎 ≠ 𝑑 ∧ (abs‘(𝑥 − 𝑎)) = (abs‘(𝑏 − 𝑐)) ∧ (abs‘(𝑥 − 𝑑)) = (abs‘(𝑒 − 𝑓))))}), {0, 1})    &   ⊢ (𝜑 → 𝑁 ∈ 𝑉)    &   ⊢ (𝜑 → Lim 𝑁)    ⇒   ⊢ (𝜑 → (𝐶‘𝑁) = ∪ 𝑛 ∈ 𝑁 (𝐶‘𝑛))
Theorem	constrsscn 33720*	Closure of the constructible points in the complex numbers. (Contributed by Thierry Arnoux, 25-Jun-2025.)
⊢ 𝐶 = rec((𝑠 ∈ V ↦ {𝑥 ∈ ℂ ∣ (∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑡 ∈ ℝ ∃𝑟 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ 𝑥 = (𝑐 + (𝑟 · (𝑑 − 𝑐))) ∧ (ℑ‘((∗‘(𝑏 − 𝑎)) · (𝑑 − 𝑐))) ≠ 0) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 ∃𝑡 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ (abs‘(𝑥 − 𝑐)) = (abs‘(𝑒 − 𝑓))) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 (𝑎 ≠ 𝑑 ∧ (abs‘(𝑥 − 𝑎)) = (abs‘(𝑏 − 𝑐)) ∧ (abs‘(𝑥 − 𝑑)) = (abs‘(𝑒 − 𝑓))))}), {0, 1})    &   ⊢ (𝜑 → 𝑁 ∈ On)    ⇒   ⊢ (𝜑 → (𝐶‘𝑁) ⊆ ℂ)
Theorem	constrsslem 33721*	Lemma for constrss 33723. This lemma requires the additional condition that 0 is the constructible number; that condition is removed in constrss 33723. (Proposed by Saveliy Skresanov, 23-JUn-2025.) (Contributed by Thierry Arnoux, 25-Jun-2025.)
⊢ 𝐶 = rec((𝑠 ∈ V ↦ {𝑥 ∈ ℂ ∣ (∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑡 ∈ ℝ ∃𝑟 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ 𝑥 = (𝑐 + (𝑟 · (𝑑 − 𝑐))) ∧ (ℑ‘((∗‘(𝑏 − 𝑎)) · (𝑑 − 𝑐))) ≠ 0) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 ∃𝑡 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ (abs‘(𝑥 − 𝑐)) = (abs‘(𝑒 − 𝑓))) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 (𝑎 ≠ 𝑑 ∧ (abs‘(𝑥 − 𝑎)) = (abs‘(𝑏 − 𝑐)) ∧ (abs‘(𝑥 − 𝑑)) = (abs‘(𝑒 − 𝑓))))}), {0, 1})    &   ⊢ (𝜑 → 𝑁 ∈ On)    &   ⊢ (𝜑 → 0 ∈ (𝐶‘𝑁))    ⇒   ⊢ (𝜑 → (𝐶‘𝑁) ⊆ (𝐶‘suc 𝑁))
Theorem	constr01 33722*	0 and 1 are in all steps of the construction of constructible points. (Contributed by Thierry Arnoux, 25-Jun-2025.)
⊢ 𝐶 = rec((𝑠 ∈ V ↦ {𝑥 ∈ ℂ ∣ (∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑡 ∈ ℝ ∃𝑟 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ 𝑥 = (𝑐 + (𝑟 · (𝑑 − 𝑐))) ∧ (ℑ‘((∗‘(𝑏 − 𝑎)) · (𝑑 − 𝑐))) ≠ 0) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 ∃𝑡 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ (abs‘(𝑥 − 𝑐)) = (abs‘(𝑒 − 𝑓))) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 (𝑎 ≠ 𝑑 ∧ (abs‘(𝑥 − 𝑎)) = (abs‘(𝑏 − 𝑐)) ∧ (abs‘(𝑥 − 𝑑)) = (abs‘(𝑒 − 𝑓))))}), {0, 1})    &   ⊢ (𝜑 → 𝑁 ∈ On)    ⇒   ⊢ (𝜑 → {0, 1} ⊆ (𝐶‘𝑁))
Theorem	constrss 33723*	Constructed points are in the next generation constructed points. (Contributed by Thierry Arnoux, 25-Jun-2025.)
⊢ 𝐶 = rec((𝑠 ∈ V ↦ {𝑥 ∈ ℂ ∣ (∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑡 ∈ ℝ ∃𝑟 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ 𝑥 = (𝑐 + (𝑟 · (𝑑 − 𝑐))) ∧ (ℑ‘((∗‘(𝑏 − 𝑎)) · (𝑑 − 𝑐))) ≠ 0) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 ∃𝑡 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ (abs‘(𝑥 − 𝑐)) = (abs‘(𝑒 − 𝑓))) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 (𝑎 ≠ 𝑑 ∧ (abs‘(𝑥 − 𝑎)) = (abs‘(𝑏 − 𝑐)) ∧ (abs‘(𝑥 − 𝑑)) = (abs‘(𝑒 − 𝑓))))}), {0, 1})    &   ⊢ (𝜑 → 𝑁 ∈ On)    ⇒   ⊢ (𝜑 → (𝐶‘𝑁) ⊆ (𝐶‘suc 𝑁))
Theorem	constrmon 33724*	The construction of constructible numbers is monotonous, i.e. if the ordinal 𝑀 is less than the ordinal 𝑁, which is denoted by 𝑀 ∈ 𝑁, then the 𝑀-th step of the constructible numbers is included in the 𝑁-th step. (Contributed by Thierry Arnoux, 1-Jul-2025.)
⊢ 𝐶 = rec((𝑠 ∈ V ↦ {𝑥 ∈ ℂ ∣ (∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑡 ∈ ℝ ∃𝑟 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ 𝑥 = (𝑐 + (𝑟 · (𝑑 − 𝑐))) ∧ (ℑ‘((∗‘(𝑏 − 𝑎)) · (𝑑 − 𝑐))) ≠ 0) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 ∃𝑡 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ (abs‘(𝑥 − 𝑐)) = (abs‘(𝑒 − 𝑓))) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 (𝑎 ≠ 𝑑 ∧ (abs‘(𝑥 − 𝑎)) = (abs‘(𝑏 − 𝑐)) ∧ (abs‘(𝑥 − 𝑑)) = (abs‘(𝑒 − 𝑓))))}), {0, 1})    &   ⊢ (𝜑 → 𝑁 ∈ On)    &   ⊢ (𝜑 → 𝑀 ∈ 𝑁)    ⇒   ⊢ (𝜑 → (𝐶‘𝑀) ⊆ (𝐶‘𝑁))
Theorem	constrconj 33725*	If a point 𝑋 of the complex plane is constructible, so is its conjugate (∗‘𝑋). (Proposed by Saveliy Skresanov, 25-Jun-2025.) (Contributed by Thierry Arnoux, 1-Jul-2025.)
⊢ 𝐶 = rec((𝑠 ∈ V ↦ {𝑥 ∈ ℂ ∣ (∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑡 ∈ ℝ ∃𝑟 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ 𝑥 = (𝑐 + (𝑟 · (𝑑 − 𝑐))) ∧ (ℑ‘((∗‘(𝑏 − 𝑎)) · (𝑑 − 𝑐))) ≠ 0) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 ∃𝑡 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ (abs‘(𝑥 − 𝑐)) = (abs‘(𝑒 − 𝑓))) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 (𝑎 ≠ 𝑑 ∧ (abs‘(𝑥 − 𝑎)) = (abs‘(𝑏 − 𝑐)) ∧ (abs‘(𝑥 − 𝑑)) = (abs‘(𝑒 − 𝑓))))}), {0, 1})    &   ⊢ (𝜑 → 𝑁 ∈ On)    &   ⊢ (𝜑 → 𝑋 ∈ (𝐶‘𝑁))    ⇒   ⊢ (𝜑 → (∗‘𝑋) ∈ (𝐶‘𝑁))
Theorem	constrfin 33726*	Each step of the construction of constructible numbers is finite. (Contributed by Thierry Arnoux, 6-Jul-2025.)
⊢ 𝐶 = rec((𝑠 ∈ V ↦ {𝑥 ∈ ℂ ∣ (∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑡 ∈ ℝ ∃𝑟 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ 𝑥 = (𝑐 + (𝑟 · (𝑑 − 𝑐))) ∧ (ℑ‘((∗‘(𝑏 − 𝑎)) · (𝑑 − 𝑐))) ≠ 0) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 ∃𝑡 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ (abs‘(𝑥 − 𝑐)) = (abs‘(𝑒 − 𝑓))) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 (𝑎 ≠ 𝑑 ∧ (abs‘(𝑥 − 𝑎)) = (abs‘(𝑏 − 𝑐)) ∧ (abs‘(𝑥 − 𝑑)) = (abs‘(𝑒 − 𝑓))))}), {0, 1})    &   ⊢ (𝜑 → 𝑁 ∈ ω)    ⇒   ⊢ (𝜑 → (𝐶‘𝑁) ∈ Fin)
Theorem	constrelextdg2 33727*	If the 𝑁-th step (𝐶‘𝑁) of the construction of constuctible numbers is included in a subfield 𝐹 of the complex numbers, then any element 𝑋 of the next step (𝐶‘suc 𝑁) is either in 𝐹 or in a quadratic extension of 𝐹. (Contributed by Thierry Arnoux, 6-Jul-2025.)
⊢ 𝐶 = rec((𝑠 ∈ V ↦ {𝑥 ∈ ℂ ∣ (∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑡 ∈ ℝ ∃𝑟 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ 𝑥 = (𝑐 + (𝑟 · (𝑑 − 𝑐))) ∧ (ℑ‘((∗‘(𝑏 − 𝑎)) · (𝑑 − 𝑐))) ≠ 0) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 ∃𝑡 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ (abs‘(𝑥 − 𝑐)) = (abs‘(𝑒 − 𝑓))) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 (𝑎 ≠ 𝑑 ∧ (abs‘(𝑥 − 𝑎)) = (abs‘(𝑏 − 𝑐)) ∧ (abs‘(𝑥 − 𝑑)) = (abs‘(𝑒 − 𝑓))))}), {0, 1})    &   ⊢ 𝐾 = (ℂfld ↾s 𝐹)    &   ⊢ 𝐿 = (ℂfld ↾s (ℂfld fldGen (𝐹 ∪ {𝑋})))    &   ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘ℂfld))    &   ⊢ (𝜑 → 𝑁 ∈ On)    &   ⊢ (𝜑 → (𝐶‘𝑁) ⊆ 𝐹)    &   ⊢ (𝜑 → 𝑋 ∈ (𝐶‘suc 𝑁))    ⇒   ⊢ (𝜑 → (𝑋 ∈ 𝐹 ∨ (𝐿[:]𝐾) = 2))
Theorem	constrextdg2lem 33728*	Lemma for constrextdg2 33729 (Contributed by Thierry Arnoux, 19-Oct-2025.)
⊢ 𝐶 = rec((𝑠 ∈ V ↦ {𝑥 ∈ ℂ ∣ (∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑡 ∈ ℝ ∃𝑟 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ 𝑥 = (𝑐 + (𝑟 · (𝑑 − 𝑐))) ∧ (ℑ‘((∗‘(𝑏 − 𝑎)) · (𝑑 − 𝑐))) ≠ 0) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 ∃𝑡 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ (abs‘(𝑥 − 𝑐)) = (abs‘(𝑒 − 𝑓))) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 (𝑎 ≠ 𝑑 ∧ (abs‘(𝑥 − 𝑎)) = (abs‘(𝑏 − 𝑐)) ∧ (abs‘(𝑥 − 𝑑)) = (abs‘(𝑒 − 𝑓))))}), {0, 1})    &   ⊢ 𝐸 = (ℂfld ↾s 𝑒)    &   ⊢ 𝐹 = (ℂfld ↾s 𝑓)    &   ⊢ < = {⟨𝑓, 𝑒⟩ ∣ (𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 2)}    &   ⊢ (𝜑 → 𝑁 ∈ ω)    &   ⊢ (𝜑 → 𝑅 ∈ ( < Chain(SubDRing‘ℂfld)))    &   ⊢ (𝜑 → (𝑅‘0) = ℚ)    &   ⊢ (𝜑 → (𝐶‘𝑁) ⊆ (lastS‘𝑅))    ⇒   ⊢ (𝜑 → ∃𝑣 ∈ ( < Chain(SubDRing‘ℂfld))((𝑣‘0) = ℚ ∧ (𝐶‘suc 𝑁) ⊆ (lastS‘𝑣)))
Theorem	constrextdg2 33729*	Any step (𝐶‘𝑁) of the construction of constructible numbers is contained in the last field of a tower of quadratic field extensions starting with ℚ. (Contributed by Thierry Arnoux, 19-Oct-2025.)
⊢ 𝐶 = rec((𝑠 ∈ V ↦ {𝑥 ∈ ℂ ∣ (∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑡 ∈ ℝ ∃𝑟 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ 𝑥 = (𝑐 + (𝑟 · (𝑑 − 𝑐))) ∧ (ℑ‘((∗‘(𝑏 − 𝑎)) · (𝑑 − 𝑐))) ≠ 0) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 ∃𝑡 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ (abs‘(𝑥 − 𝑐)) = (abs‘(𝑒 − 𝑓))) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 (𝑎 ≠ 𝑑 ∧ (abs‘(𝑥 − 𝑎)) = (abs‘(𝑏 − 𝑐)) ∧ (abs‘(𝑥 − 𝑑)) = (abs‘(𝑒 − 𝑓))))}), {0, 1})    &   ⊢ 𝐸 = (ℂfld ↾s 𝑒)    &   ⊢ 𝐹 = (ℂfld ↾s 𝑓)    &   ⊢ < = {⟨𝑓, 𝑒⟩ ∣ (𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 2)}    &   ⊢ (𝜑 → 𝑁 ∈ ω)    ⇒   ⊢ (𝜑 → ∃𝑣 ∈ ( < Chain(SubDRing‘ℂfld))((𝑣‘0) = ℚ ∧ (𝐶‘𝑁) ⊆ (lastS‘𝑣)))
Theorem	constrext2chnlem 33730*	Lemma for constrext2chn 33739. (Contributed by Thierry Arnoux, 26-Oct-2025.)
⊢ 𝐶 = rec((𝑠 ∈ V ↦ {𝑥 ∈ ℂ ∣ (∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑡 ∈ ℝ ∃𝑟 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ 𝑥 = (𝑐 + (𝑟 · (𝑑 − 𝑐))) ∧ (ℑ‘((∗‘(𝑏 − 𝑎)) · (𝑑 − 𝑐))) ≠ 0) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 ∃𝑡 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ (abs‘(𝑥 − 𝑐)) = (abs‘(𝑒 − 𝑓))) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 (𝑎 ≠ 𝑑 ∧ (abs‘(𝑥 − 𝑎)) = (abs‘(𝑏 − 𝑐)) ∧ (abs‘(𝑥 − 𝑑)) = (abs‘(𝑒 − 𝑓))))}), {0, 1})    &   ⊢ 𝐸 = (ℂfld ↾s 𝑒)    &   ⊢ 𝐹 = (ℂfld ↾s 𝑓)    &   ⊢ < = {⟨𝑓, 𝑒⟩ ∣ (𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 2)}    &   ⊢ (𝜑 → 𝑁 ∈ ω)    &   ⊢ 𝑄 = (ℂfld ↾s ℚ)    &   ⊢ 𝐿 = (ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴})))    &   ⊢ (𝜑 → 𝐴 ∈ Constr)    ⇒   ⊢ (𝜑 → ∃𝑛 ∈ ℕ0 (𝐿[:]𝑄) = (2↑𝑛))
Theorem	constrfiss 33731*	For any finite set 𝐴 of constructible numbers, there is a 𝑛 -th step (𝐶‘𝑛) containing all numbers in 𝐴. (Contributed by Thierry Arnoux, 2-Nov-2025.)
⊢ 𝐶 = rec((𝑠 ∈ V ↦ {𝑥 ∈ ℂ ∣ (∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑡 ∈ ℝ ∃𝑟 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ 𝑥 = (𝑐 + (𝑟 · (𝑑 − 𝑐))) ∧ (ℑ‘((∗‘(𝑏 − 𝑎)) · (𝑑 − 𝑐))) ≠ 0) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 ∃𝑡 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ (abs‘(𝑥 − 𝑐)) = (abs‘(𝑒 − 𝑓))) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 (𝑎 ≠ 𝑑 ∧ (abs‘(𝑥 − 𝑎)) = (abs‘(𝑏 − 𝑐)) ∧ (abs‘(𝑥 − 𝑑)) = (abs‘(𝑒 − 𝑓))))}), {0, 1})    &   ⊢ (𝜑 → 𝐴 ⊆ Constr)    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    ⇒   ⊢ (𝜑 → ∃𝑛 ∈ ω 𝐴 ⊆ (𝐶‘𝑛))
Theorem	constrllcllem 33732*	Constructible numbers are closed under line-line intersections. (Contributed by Thierry Arnoux, 2-Nov-2025.)
⊢ 𝐶 = rec((𝑠 ∈ V ↦ {𝑥 ∈ ℂ ∣ (∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑡 ∈ ℝ ∃𝑟 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ 𝑥 = (𝑐 + (𝑟 · (𝑑 − 𝑐))) ∧ (ℑ‘((∗‘(𝑏 − 𝑎)) · (𝑑 − 𝑐))) ≠ 0) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 ∃𝑡 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ (abs‘(𝑥 − 𝑐)) = (abs‘(𝑒 − 𝑓))) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 (𝑎 ≠ 𝑑 ∧ (abs‘(𝑥 − 𝑎)) = (abs‘(𝑏 − 𝑐)) ∧ (abs‘(𝑥 − 𝑑)) = (abs‘(𝑒 − 𝑓))))}), {0, 1})    &   ⊢ (𝜑 → 𝐴 ∈ Constr)    &   ⊢ (𝜑 → 𝐵 ∈ Constr)    &   ⊢ (𝜑 → 𝐺 ∈ Constr)    &   ⊢ (𝜑 → 𝐷 ∈ Constr)    &   ⊢ (𝜑 → 𝑇 ∈ ℝ)    &   ⊢ (𝜑 → 𝑅 ∈ ℝ)    &   ⊢ (𝜑 → 𝑋 ∈ ℂ)    &   ⊢ (𝜑 → 𝑋 = (𝐴 + (𝑇 · (𝐵 − 𝐴))))    &   ⊢ (𝜑 → 𝑋 = (𝐺 + (𝑅 · (𝐷 − 𝐺))))    &   ⊢ (𝜑 → (ℑ‘((∗‘(𝐵 − 𝐴)) · (𝐷 − 𝐺))) ≠ 0)    ⇒   ⊢ (𝜑 → 𝑋 ∈ Constr)
Theorem	constrlccllem 33733*	Constructible numbers are closed under line-circle intersections. (Contributed by Thierry Arnoux, 2-Nov-2025.)
⊢ 𝐶 = rec((𝑠 ∈ V ↦ {𝑥 ∈ ℂ ∣ (∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑡 ∈ ℝ ∃𝑟 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ 𝑥 = (𝑐 + (𝑟 · (𝑑 − 𝑐))) ∧ (ℑ‘((∗‘(𝑏 − 𝑎)) · (𝑑 − 𝑐))) ≠ 0) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 ∃𝑡 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ (abs‘(𝑥 − 𝑐)) = (abs‘(𝑒 − 𝑓))) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 (𝑎 ≠ 𝑑 ∧ (abs‘(𝑥 − 𝑎)) = (abs‘(𝑏 − 𝑐)) ∧ (abs‘(𝑥 − 𝑑)) = (abs‘(𝑒 − 𝑓))))}), {0, 1})    &   ⊢ (𝜑 → 𝐴 ∈ Constr)    &   ⊢ (𝜑 → 𝐵 ∈ Constr)    &   ⊢ (𝜑 → 𝐺 ∈ Constr)    &   ⊢ (𝜑 → 𝐸 ∈ Constr)    &   ⊢ (𝜑 → 𝐹 ∈ Constr)    &   ⊢ (𝜑 → 𝑇 ∈ ℝ)    &   ⊢ (𝜑 → 𝑋 ∈ ℂ)    &   ⊢ (𝜑 → 𝑋 = (𝐴 + (𝑇 · (𝐵 − 𝐴))))    &   ⊢ (𝜑 → (abs‘(𝑋 − 𝐺)) = (abs‘(𝐸 − 𝐹)))    ⇒   ⊢ (𝜑 → 𝑋 ∈ Constr)
Theorem	constrcccllem 33734*	Constructible numbers are closed under circle-circle intersections. (Contributed by Thierry Arnoux, 2-Nov-2025.)
⊢ 𝐶 = rec((𝑠 ∈ V ↦ {𝑥 ∈ ℂ ∣ (∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑡 ∈ ℝ ∃𝑟 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ 𝑥 = (𝑐 + (𝑟 · (𝑑 − 𝑐))) ∧ (ℑ‘((∗‘(𝑏 − 𝑎)) · (𝑑 − 𝑐))) ≠ 0) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 ∃𝑡 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ (abs‘(𝑥 − 𝑐)) = (abs‘(𝑒 − 𝑓))) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 (𝑎 ≠ 𝑑 ∧ (abs‘(𝑥 − 𝑎)) = (abs‘(𝑏 − 𝑐)) ∧ (abs‘(𝑥 − 𝑑)) = (abs‘(𝑒 − 𝑓))))}), {0, 1})    &   ⊢ (𝜑 → 𝐴 ∈ Constr)    &   ⊢ (𝜑 → 𝐵 ∈ Constr)    &   ⊢ (𝜑 → 𝐺 ∈ Constr)    &   ⊢ (𝜑 → 𝐷 ∈ Constr)    &   ⊢ (𝜑 → 𝐸 ∈ Constr)    &   ⊢ (𝜑 → 𝐹 ∈ Constr)    &   ⊢ (𝜑 → 𝑋 ∈ ℂ)    &   ⊢ (𝜑 → 𝐴 ≠ 𝐷)    &   ⊢ (𝜑 → (abs‘(𝑋 − 𝐴)) = (abs‘(𝐵 − 𝐺)))    &   ⊢ (𝜑 → (abs‘(𝑋 − 𝐷)) = (abs‘(𝐸 − 𝐹)))    ⇒   ⊢ (𝜑 → 𝑋 ∈ Constr)
Theorem	constrcbvlem 33735*	Technical lemma for eliminating the hypothesis of constr0 33717 and co. (Contributed by Thierry Arnoux, 2-Nov-2025.)
⊢ rec((𝑧 ∈ V ↦ {𝑦 ∈ ℂ ∣ (∃𝑖 ∈ 𝑧 ∃𝑗 ∈ 𝑧 ∃𝑘 ∈ 𝑧 ∃𝑙 ∈ 𝑧 ∃𝑜 ∈ ℝ ∃𝑝 ∈ ℝ (𝑦 = (𝑖 + (𝑜 · (𝑗 − 𝑖))) ∧ 𝑦 = (𝑘 + (𝑝 · (𝑙 − 𝑘))) ∧ (ℑ‘((∗‘(𝑗 − 𝑖)) · (𝑙 − 𝑘))) ≠ 0) ∨ ∃𝑖 ∈ 𝑧 ∃𝑗 ∈ 𝑧 ∃𝑘 ∈ 𝑧 ∃𝑚 ∈ 𝑧 ∃𝑞 ∈ 𝑧 ∃𝑜 ∈ ℝ (𝑦 = (𝑖 + (𝑜 · (𝑗 − 𝑖))) ∧ (abs‘(𝑦 − 𝑘)) = (abs‘(𝑚 − 𝑞))) ∨ ∃𝑖 ∈ 𝑧 ∃𝑗 ∈ 𝑧 ∃𝑘 ∈ 𝑧 ∃𝑙 ∈ 𝑧 ∃𝑚 ∈ 𝑧 ∃𝑞 ∈ 𝑧 (𝑖 ≠ 𝑙 ∧ (abs‘(𝑦 − 𝑖)) = (abs‘(𝑗 − 𝑘)) ∧ (abs‘(𝑦 − 𝑙)) = (abs‘(𝑚 − 𝑞))))}), {0, 1}) = rec((𝑠 ∈ V ↦ {𝑥 ∈ ℂ ∣ (∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑡 ∈ ℝ ∃𝑟 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ 𝑥 = (𝑐 + (𝑟 · (𝑑 − 𝑐))) ∧ (ℑ‘((∗‘(𝑏 − 𝑎)) · (𝑑 − 𝑐))) ≠ 0) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 ∃𝑡 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ (abs‘(𝑥 − 𝑐)) = (abs‘(𝑒 − 𝑓))) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 (𝑎 ≠ 𝑑 ∧ (abs‘(𝑥 − 𝑎)) = (abs‘(𝑏 − 𝑐)) ∧ (abs‘(𝑥 − 𝑑)) = (abs‘(𝑒 − 𝑓))))}), {0, 1})
Theorem	constrllcl 33736	Constructible numbers are closed under line-line intersections. (Contributed by Thierry Arnoux, 2-Nov-2025.)
⊢ (𝜑 → 𝐴 ∈ Constr)    &   ⊢ (𝜑 → 𝐵 ∈ Constr)    &   ⊢ (𝜑 → 𝐺 ∈ Constr)    &   ⊢ (𝜑 → 𝐷 ∈ Constr)    &   ⊢ (𝜑 → 𝑇 ∈ ℝ)    &   ⊢ (𝜑 → 𝑅 ∈ ℝ)    &   ⊢ (𝜑 → 𝑋 ∈ ℂ)    &   ⊢ (𝜑 → 𝑋 = (𝐴 + (𝑇 · (𝐵 − 𝐴))))    &   ⊢ (𝜑 → 𝑋 = (𝐺 + (𝑅 · (𝐷 − 𝐺))))    &   ⊢ (𝜑 → (ℑ‘((∗‘(𝐵 − 𝐴)) · (𝐷 − 𝐺))) ≠ 0)    ⇒   ⊢ (𝜑 → 𝑋 ∈ Constr)
Theorem	constrlccl 33737	Constructible numbers are closed under line-circle intersections. (Contributed by Thierry Arnoux, 2-Nov-2025.)
⊢ (𝜑 → 𝐴 ∈ Constr)    &   ⊢ (𝜑 → 𝐵 ∈ Constr)    &   ⊢ (𝜑 → 𝐺 ∈ Constr)    &   ⊢ (𝜑 → 𝐸 ∈ Constr)    &   ⊢ (𝜑 → 𝐹 ∈ Constr)    &   ⊢ (𝜑 → 𝑇 ∈ ℝ)    &   ⊢ (𝜑 → 𝑋 ∈ ℂ)    &   ⊢ (𝜑 → 𝑋 = (𝐴 + (𝑇 · (𝐵 − 𝐴))))    &   ⊢ (𝜑 → (abs‘(𝑋 − 𝐺)) = (abs‘(𝐸 − 𝐹)))    ⇒   ⊢ (𝜑 → 𝑋 ∈ Constr)
Theorem	constrcccl 33738	Constructible numbers are closed under circle-circle intersections. (Contributed by Thierry Arnoux, 2-Nov-2025.)
⊢ (𝜑 → 𝐴 ∈ Constr)    &   ⊢ (𝜑 → 𝐵 ∈ Constr)    &   ⊢ (𝜑 → 𝐶 ∈ Constr)    &   ⊢ (𝜑 → 𝐷 ∈ Constr)    &   ⊢ (𝜑 → 𝐸 ∈ Constr)    &   ⊢ (𝜑 → 𝐹 ∈ Constr)    &   ⊢ (𝜑 → 𝑋 ∈ ℂ)    &   ⊢ (𝜑 → 𝐴 ≠ 𝐷)    &   ⊢ (𝜑 → (abs‘(𝑋 − 𝐴)) = (abs‘(𝐵 − 𝐶)))    &   ⊢ (𝜑 → (abs‘(𝑋 − 𝐷)) = (abs‘(𝐸 − 𝐹)))    ⇒   ⊢ (𝜑 → 𝑋 ∈ Constr)
Theorem	constrext2chn 33739*	If a constructible number generates some subfield 𝐿 of ℂ, then the degree of the extension of 𝐿 over ℚ is a power of two. (Contributed by Thierry Arnoux, 26-Oct-2025.)
⊢ 𝑄 = (ℂfld ↾s ℚ)    &   ⊢ 𝐿 = (ℂfld ↾s 𝑆)    &   ⊢ 𝑆 = (ℂfld fldGen (ℚ ∪ {𝐴}))    &   ⊢ (𝜑 → 𝐴 ∈ Constr)    ⇒   ⊢ (𝜑 → ∃𝑛 ∈ ℕ0 (𝐿[:]𝑄) = (2↑𝑛))
Theorem	constrcn 33740	Constructible numbers are complex numbers. (Contributed by Thierry Arnoux, 2-Nov-2025.)
⊢ (𝜑 → 𝑋 ∈ Constr)    ⇒   ⊢ (𝜑 → 𝑋 ∈ ℂ)
Theorem	nn0constr 33741	Nonnegative integers are constructible. (Contributed by Thierry Arnoux, 2-Nov-2025.)
⊢ (𝜑 → 𝑁 ∈ ℕ0)    ⇒   ⊢ (𝜑 → 𝑁 ∈ Constr)
Theorem	constraddcl 33742	Constructive numbers are closed under complex addition. (Contributed by Thierry Arnoux, 2-Nov-2025.)
⊢ (𝜑 → 𝑋 ∈ Constr)    &   ⊢ (𝜑 → 𝑌 ∈ Constr)    ⇒   ⊢ (𝜑 → (𝑋 + 𝑌) ∈ Constr)
Theorem	constrnegcl 33743	Constructible numbers are closed under additive inverse. (Contributed by Thierry Arnoux, 2-Nov-2025.)
⊢ (𝜑 → 𝑋 ∈ Constr)    ⇒   ⊢ (𝜑 → -𝑋 ∈ Constr)
Theorem	zconstr 33744	Integers are constructible. (Contributed by Thierry Arnoux, 3-Nov-2025.)
⊢ (𝜑 → 𝑋 ∈ ℤ)    ⇒   ⊢ (𝜑 → 𝑋 ∈ Constr)
Theorem	constrdircl 33745	Constructible numbers are closed under taking the point on the unit circle having the same argument. (Contributed by Thierry Arnoux, 2-Nov-2025.)
⊢ (𝜑 → 𝑋 ∈ Constr)    &   ⊢ (𝜑 → 𝑋 ≠ 0)    ⇒   ⊢ (𝜑 → (𝑋 / (abs‘𝑋)) ∈ Constr)
Theorem	iconstr 33746	The imaginary unit i is constructible. (Contributed by Thierry Arnoux, 2-Nov-2025.)
⊢ i ∈ Constr
Theorem	constrremulcl 33747	If two real numbers 𝑋 and 𝑌 are constructible, then, so is their product. (Contributed by Thierry Arnoux, 2-Nov-2025.)
⊢ (𝜑 → 𝑋 ∈ Constr)    &   ⊢ (𝜑 → 𝑌 ∈ Constr)    &   ⊢ (𝜑 → 𝑋 ∈ ℝ)    &   ⊢ (𝜑 → 𝑌 ∈ ℝ)    ⇒   ⊢ (𝜑 → (𝑋 · 𝑌) ∈ Constr)
Theorem	constrcjcl 33748	Constructible numbers are closed under complex conjugate. (Contributed by Thierry Arnoux, 5-Nov-2025.)
⊢ (𝜑 → 𝑋 ∈ Constr)    ⇒   ⊢ (𝜑 → (∗‘𝑋) ∈ Constr)
Theorem	constrrecl 33749	Constructible numbers are closed under taking the real part. (Contributed by Thierry Arnoux, 5-Nov-2025.)
⊢ (𝜑 → 𝑋 ∈ Constr)    ⇒   ⊢ (𝜑 → (ℜ‘𝑋) ∈ Constr)
Theorem	constrimcl 33750	Constructible numbers are closed under taking the imaginary part. (Contributed by Thierry Arnoux, 5-Nov-2025.)
⊢ (𝜑 → 𝑋 ∈ Constr)    ⇒   ⊢ (𝜑 → (ℑ‘𝑋) ∈ Constr)
Theorem	constrmulcl 33751	Constructible numbers are closed under complex multiplication. (Contributed by Thierry Arnoux, 5-Nov-2025.)
⊢ (𝜑 → 𝑋 ∈ Constr)    &   ⊢ (𝜑 → 𝑌 ∈ Constr)    ⇒   ⊢ (𝜑 → (𝑋 · 𝑌) ∈ Constr)
Theorem	constrreinvcl 33752	If a real number 𝑋 is constructible, then, so is its inverse. (Contributed by Thierry Arnoux, 5-Nov-2025.)
⊢ (𝜑 → 𝑋 ∈ Constr)    &   ⊢ (𝜑 → 𝑋 ≠ 0)    &   ⊢ (𝜑 → 𝑋 ∈ ℝ)    ⇒   ⊢ (𝜑 → (1 / 𝑋) ∈ Constr)
Theorem	constrinvcl 33753	Constructible numbers are closed under complex inverse. (Contributed by Thierry Arnoux, 5-Nov-2025.)
⊢ (𝜑 → 𝑋 ∈ Constr)    &   ⊢ (𝜑 → 𝑋 ≠ 0)    ⇒   ⊢ (𝜑 → (1 / 𝑋) ∈ Constr)
Theorem	constrcon 33754*	Contradiction of constructibility: If a complex number 𝐴 has minimal polynomial 𝐹 over ℚ of a degree that is not a power of 2, then 𝐴 is not constructible. (Contributed by Thierry Arnoux, 26-Oct-2025.)
⊢ 𝐷 = (deg1‘(ℂfld ↾s ℚ))    &   ⊢ 𝑀 = (ℂfld minPoly ℚ)    &   ⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐹 = (𝑀‘𝐴))    &   ⊢ (𝜑 → (𝐷‘𝐹) ∈ ℕ0)    &   ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (𝐷‘𝐹) ≠ (2↑𝑛))    ⇒   ⊢ (𝜑 → ¬ 𝐴 ∈ Constr)
Theorem	constrsdrg 33755	Constructible numbers form a subfield of the complex numbers. (Contributed by Thierry Arnoux, 5-Nov-2025.)
⊢ Constr ∈ (SubDRing‘ℂfld)
Theorem	constrfld 33756	The constructible numbers form a field. (Contributed by Thierry Arnoux, 5-Nov-2025.)
⊢ (ℂfld ↾s Constr) ∈ Field
Theorem	constrresqrtcl 33757	If a positive real number 𝑋 is constructible, then, so is its square root. (Contributed by Thierry Arnoux, 5-Nov-2025.)
⊢ (𝜑 → 𝑋 ∈ Constr)    &   ⊢ (𝜑 → 𝑋 ∈ ℝ)    &   ⊢ (𝜑 → 0 ≤ 𝑋)    ⇒   ⊢ (𝜑 → (√‘𝑋) ∈ Constr)
Theorem	constrabscl 33758	Constructible numbers are closed under absolute value (modulus). (Contributed by Thierry Arnoux, 6-Nov-2025.)
⊢ (𝜑 → 𝑋 ∈ Constr)    ⇒   ⊢ (𝜑 → (abs‘𝑋) ∈ Constr)
Theorem	constrsqrtcl 33759	Constructible numbers are closed under taking the square root. This is not generally the case for the cubic root operation, see 2sqr3nconstr 33761. (Proposed by Saveliy Skresanov, 3-Nov-2025.) (Contributed by Thierry Arnoux, 6-Nov-2025.)
⊢ (𝜑 → 𝑋 ∈ Constr)    ⇒   ⊢ (𝜑 → (√‘𝑋) ∈ Constr)
21.3.12.1  Impossible constructions
Theorem	2sqr3minply 33760	The polynomial ((𝑋↑3) − 2) is the minimal polynomial for (2↑𝑐(1 / 3)) over ℚ, and its degree is 3. (Contributed by Thierry Arnoux, 14-Jun-2025.)
⊢ 𝑄 = (ℂfld ↾s ℚ)    &   ⊢ − = (-g‘𝑃)    &   ⊢ ↑ = (.g‘(mulGrp‘𝑃))    &   ⊢ 𝑃 = (Poly1‘𝑄)    &   ⊢ 𝐾 = (algSc‘𝑃)    &   ⊢ 𝑋 = (var1‘𝑄)    &   ⊢ 𝐷 = (deg1‘𝑄)    &   ⊢ 𝐹 = ((3 ↑ 𝑋) − (𝐾‘2))    &   ⊢ 𝐴 = (2↑𝑐(1 / 3))    &   ⊢ 𝑀 = (ℂfld minPoly ℚ)    ⇒   ⊢ (𝐹 = (𝑀‘𝐴) ∧ (𝐷‘𝐹) = 3)
Theorem	2sqr3nconstr 33761	Doubling the cube is an impossible construction, i.e. the cube root of 2 is not constructible with straightedge and compass. Given a cube of edge of length one, a cube of double volume would have an edge of length (2↑𝑐(1 / 3)), however that number is not constructible. This is the first part of Metamath 100 proof #8. (Contributed by Thierry Arnoux and Saveliy Skresanov, 26-Oct-2025.)
⊢ (2↑𝑐(1 / 3)) ∉ Constr
Theorem	cos9thpiminplylem1 33762	The polynomial ((𝑋↑3) + ((-3 · (𝑋↑2)) + 1)) has no integer roots. (Contributed by Thierry Arnoux, 9-Nov-2025.)
⊢ (𝜑 → 𝑋 ∈ ℤ)    ⇒   ⊢ (𝜑 → ((𝑋↑3) + ((-3 · (𝑋↑2)) + 1)) ≠ 0)
Theorem	cos9thpiminplylem2 33763	The polynomial ((𝑋↑3) + ((-3 · 𝑋) + 1)) has no rational roots. (Contributed by Thierry Arnoux, 9-Nov-2025.)
⊢ (𝜑 → 𝑋 ∈ ℚ)    ⇒   ⊢ (𝜑 → ((𝑋↑3) + ((-3 · 𝑋) + 1)) ≠ 0)
Theorem	cos9thpiminplylem3 33764	Lemma for cos9thpiminply 33768. (Contributed by Thierry Arnoux, 14-Nov-2025.)
⊢ 𝑂 = (exp‘((i · (2 · π)) / 3))    ⇒   ⊢ ((𝑂↑2) + (𝑂 + 1)) = 0
Theorem	cos9thpiminplylem4 33765	Lemma for cos9thpiminply 33768. (Contributed by Thierry Arnoux, 14-Nov-2025.)
⊢ 𝑂 = (exp‘((i · (2 · π)) / 3))    &   ⊢ 𝑍 = (𝑂↑𝑐(1 / 3))    ⇒   ⊢ ((𝑍↑6) + (𝑍↑3)) = -1
Theorem	cos9thpiminplylem5 33766	The constructed complex number 𝐴 is a root of the polynomial ((𝑋↑3) + ((-3 · 𝑋) + 1)). (Contributed by Thierry Arnoux, 14-Nov-2025.)
⊢ 𝑂 = (exp‘((i · (2 · π)) / 3))    &   ⊢ 𝑍 = (𝑂↑𝑐(1 / 3))    &   ⊢ 𝐴 = (𝑍 + (1 / 𝑍))    ⇒   ⊢ ((𝐴↑3) + ((-3 · 𝐴) + 1)) = 0
Theorem	cos9thpiminplylem6 33767	Evaluation of the polynomial ((𝑋↑3) + ((-3 · 𝑋) + 1)). (Contributed by Thierry Arnoux, 14-Nov-2025.)
⊢ 𝑂 = (exp‘((i · (2 · π)) / 3))    &   ⊢ 𝑍 = (𝑂↑𝑐(1 / 3))    &   ⊢ 𝐴 = (𝑍 + (1 / 𝑍))    &   ⊢ 𝑄 = (ℂfld ↾s ℚ)    &   ⊢ + = (+g‘𝑃)    &   ⊢ · = (.r‘𝑃)    &   ⊢ ↑ = (.g‘(mulGrp‘𝑃))    &   ⊢ 𝑃 = (Poly1‘𝑄)    &   ⊢ 𝐾 = (algSc‘𝑃)    &   ⊢ 𝑋 = (var1‘𝑄)    &   ⊢ 𝐷 = (deg1‘𝑄)    &   ⊢ 𝐹 = ((3 ↑ 𝑋) + (((𝐾‘-3) · 𝑋) + (𝐾‘1)))    &   ⊢ (𝜑 → 𝑌 ∈ ℂ)    ⇒   ⊢ (𝜑 → (((ℂfld evalSub1 ℚ)‘𝐹)‘𝑌) = ((𝑌↑3) + ((-3 · 𝑌) + 1)))
Theorem	cos9thpiminply 33768	The polynomial ((𝑋↑3) + ((-3 · 𝑋) + 1)) is the minimal polynomial for 𝐴 over ℚ, and its degree is 3. (Contributed by Thierry Arnoux, 14-Nov-2025.)
⊢ 𝑂 = (exp‘((i · (2 · π)) / 3))    &   ⊢ 𝑍 = (𝑂↑𝑐(1 / 3))    &   ⊢ 𝐴 = (𝑍 + (1 / 𝑍))    &   ⊢ 𝑄 = (ℂfld ↾s ℚ)    &   ⊢ + = (+g‘𝑃)    &   ⊢ · = (.r‘𝑃)    &   ⊢ ↑ = (.g‘(mulGrp‘𝑃))    &   ⊢ 𝑃 = (Poly1‘𝑄)    &   ⊢ 𝐾 = (algSc‘𝑃)    &   ⊢ 𝑋 = (var1‘𝑄)    &   ⊢ 𝐷 = (deg1‘𝑄)    &   ⊢ 𝐹 = ((3 ↑ 𝑋) + (((𝐾‘-3) · 𝑋) + (𝐾‘1)))    &   ⊢ 𝑀 = (ℂfld minPoly ℚ)    ⇒   ⊢ (𝐹 = (𝑀‘𝐴) ∧ (𝐷‘𝐹) = 3)
Theorem	cos9thpinconstrlem1 33769	The complex number 𝑂, representing an angle of (2 · π) / 3, is constructible. (Contributed by Thierry Arnoux, 14-Nov-2025.)
⊢ 𝑂 = (exp‘((i · (2 · π)) / 3))    ⇒   ⊢ 𝑂 ∈ Constr
21.3.13  Matrices
21.3.13.1  Submatrices
Syntax	csmat 33770	Syntax for a function generating submatrices.
class subMat1
Definition	df-smat 33771*	Define a function generating submatrices of an integer-indexed matrix. The function maps an index in ((1...𝑀) × (1...𝑁)) into a new index in ((1...(𝑀 − 1)) × (1...(𝑁 − 1))). A submatrix is obtained by deleting a row and a column of the original matrix. Because this function re-indexes the matrix, the resulting submatrix still has the same index set for rows and columns, and its determinent is defined, unlike the current df-subma 22513. (Contributed by Thierry Arnoux, 18-Aug-2020.)
⊢ subMat1 = (𝑚 ∈ V ↦ (𝑘 ∈ ℕ, 𝑙 ∈ ℕ ↦ (𝑚 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝑘, 𝑖, (𝑖 + 1)), if(𝑗 < 𝑙, 𝑗, (𝑗 + 1))⟩))))
Theorem	smatfval 33772*	Value of the submatrix. (Contributed by Thierry Arnoux, 19-Aug-2020.)
⊢ ((𝐾 ∈ ℕ ∧ 𝐿 ∈ ℕ ∧ 𝑀 ∈ 𝑉) → (𝐾(subMat1‘𝑀)𝐿) = (𝑀 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)))
Theorem	smatrcl 33773	Closure of the rectangular submatrix. (Contributed by Thierry Arnoux, 19-Aug-2020.)
⊢ 𝑆 = (𝐾(subMat1‘𝐴)𝐿)    &   ⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐾 ∈ (1...𝑀))    &   ⊢ (𝜑 → 𝐿 ∈ (1...𝑁))    &   ⊢ (𝜑 → 𝐴 ∈ (𝐵 ↑m ((1...𝑀) × (1...𝑁))))    ⇒   ⊢ (𝜑 → 𝑆 ∈ (𝐵 ↑m ((1...(𝑀 − 1)) × (1...(𝑁 − 1)))))
Theorem	smatlem 33774	Lemma for the next theorems. (Contributed by Thierry Arnoux, 19-Aug-2020.)
⊢ 𝑆 = (𝐾(subMat1‘𝐴)𝐿)    &   ⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐾 ∈ (1...𝑀))    &   ⊢ (𝜑 → 𝐿 ∈ (1...𝑁))    &   ⊢ (𝜑 → 𝐴 ∈ (𝐵 ↑m ((1...𝑀) × (1...𝑁))))    &   ⊢ (𝜑 → 𝐼 ∈ ℕ)    &   ⊢ (𝜑 → 𝐽 ∈ ℕ)    &   ⊢ (𝜑 → if(𝐼 < 𝐾, 𝐼, (𝐼 + 1)) = 𝑋)    &   ⊢ (𝜑 → if(𝐽 < 𝐿, 𝐽, (𝐽 + 1)) = 𝑌)    ⇒   ⊢ (𝜑 → (𝐼𝑆𝐽) = (𝑋𝐴𝑌))
Theorem	smattl 33775	Entries of a submatrix, top left. (Contributed by Thierry Arnoux, 19-Aug-2020.)
⊢ 𝑆 = (𝐾(subMat1‘𝐴)𝐿)    &   ⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐾 ∈ (1...𝑀))    &   ⊢ (𝜑 → 𝐿 ∈ (1...𝑁))    &   ⊢ (𝜑 → 𝐴 ∈ (𝐵 ↑m ((1...𝑀) × (1...𝑁))))    &   ⊢ (𝜑 → 𝐼 ∈ (1..^𝐾))    &   ⊢ (𝜑 → 𝐽 ∈ (1..^𝐿))    ⇒   ⊢ (𝜑 → (𝐼𝑆𝐽) = (𝐼𝐴𝐽))
Theorem	smattr 33776	Entries of a submatrix, top right. (Contributed by Thierry Arnoux, 19-Aug-2020.)
⊢ 𝑆 = (𝐾(subMat1‘𝐴)𝐿)    &   ⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐾 ∈ (1...𝑀))    &   ⊢ (𝜑 → 𝐿 ∈ (1...𝑁))    &   ⊢ (𝜑 → 𝐴 ∈ (𝐵 ↑m ((1...𝑀) × (1...𝑁))))    &   ⊢ (𝜑 → 𝐼 ∈ (𝐾...𝑀))    &   ⊢ (𝜑 → 𝐽 ∈ (1..^𝐿))    ⇒   ⊢ (𝜑 → (𝐼𝑆𝐽) = ((𝐼 + 1)𝐴𝐽))
Theorem	smatbl 33777	Entries of a submatrix, bottom left. (Contributed by Thierry Arnoux, 19-Aug-2020.)
⊢ 𝑆 = (𝐾(subMat1‘𝐴)𝐿)    &   ⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐾 ∈ (1...𝑀))    &   ⊢ (𝜑 → 𝐿 ∈ (1...𝑁))    &   ⊢ (𝜑 → 𝐴 ∈ (𝐵 ↑m ((1...𝑀) × (1...𝑁))))    &   ⊢ (𝜑 → 𝐼 ∈ (1..^𝐾))    &   ⊢ (𝜑 → 𝐽 ∈ (𝐿...𝑁))    ⇒   ⊢ (𝜑 → (𝐼𝑆𝐽) = (𝐼𝐴(𝐽 + 1)))
Theorem	smatbr 33778	Entries of a submatrix, bottom right. (Contributed by Thierry Arnoux, 19-Aug-2020.)
⊢ 𝑆 = (𝐾(subMat1‘𝐴)𝐿)    &   ⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐾 ∈ (1...𝑀))    &   ⊢ (𝜑 → 𝐿 ∈ (1...𝑁))    &   ⊢ (𝜑 → 𝐴 ∈ (𝐵 ↑m ((1...𝑀) × (1...𝑁))))    &   ⊢ (𝜑 → 𝐼 ∈ (𝐾...𝑀))    &   ⊢ (𝜑 → 𝐽 ∈ (𝐿...𝑁))    ⇒   ⊢ (𝜑 → (𝐼𝑆𝐽) = ((𝐼 + 1)𝐴(𝐽 + 1)))
Theorem	smatcl 33779	Closure of the square submatrix: if 𝑀 is a square matrix of dimension 𝑁 with indices in (1...𝑁), then a submatrix of 𝑀 is of dimension (𝑁 − 1). (Contributed by Thierry Arnoux, 19-Aug-2020.)
⊢ 𝐴 = ((1...𝑁) Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝐶 = (Base‘((1...(𝑁 − 1)) Mat 𝑅))    &   ⊢ 𝑆 = (𝐾(subMat1‘𝑀)𝐿)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐾 ∈ (1...𝑁))    &   ⊢ (𝜑 → 𝐿 ∈ (1...𝑁))    &   ⊢ (𝜑 → 𝑀 ∈ 𝐵)    ⇒   ⊢ (𝜑 → 𝑆 ∈ 𝐶)
Theorem	matmpo 33780*	Write a square matrix as a mapping operation. (Contributed by Thierry Arnoux, 16-Aug-2020.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    ⇒   ⊢ (𝑀 ∈ 𝐵 → 𝑀 = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑖𝑀𝑗)))
Theorem	1smat1 33781	The submatrix of the identity matrix obtained by removing the ith row and the ith column is an identity matrix. Cf. 1marepvsma1 22519. (Contributed by Thierry Arnoux, 19-Aug-2020.)
⊢ 1 = (1r‘((1...𝑁) Mat 𝑅))    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐼 ∈ (1...𝑁))    ⇒   ⊢ (𝜑 → (𝐼(subMat1‘ 1 )𝐼) = (1r‘((1...(𝑁 − 1)) Mat 𝑅)))
Theorem	submat1n 33782	One case where the submatrix with integer indices, subMat1, and the general submatrix subMat, agree. (Contributed by Thierry Arnoux, 22-Aug-2020.)
⊢ 𝐴 = ((1...𝑁) Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    ⇒   ⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ 𝐵) → (𝑁(subMat1‘𝑀)𝑁) = (𝑁(((1...𝑁) subMat 𝑅)‘𝑀)𝑁))
Theorem	submatres 33783	Special case where the submatrix is a restriction of the initial matrix, and no renumbering occurs. (Contributed by Thierry Arnoux, 26-Aug-2020.)
⊢ 𝐴 = ((1...𝑁) Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    ⇒   ⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ 𝐵) → (𝑁(subMat1‘𝑀)𝑁) = (𝑀 ↾ ((1...(𝑁 − 1)) × (1...(𝑁 − 1)))))
Theorem	submateqlem1 33784	Lemma for submateq 33786. (Contributed by Thierry Arnoux, 25-Aug-2020.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐾 ∈ (1...𝑁))    &   ⊢ (𝜑 → 𝑀 ∈ (1...(𝑁 − 1)))    &   ⊢ (𝜑 → 𝐾 ≤ 𝑀)    ⇒   ⊢ (𝜑 → (𝑀 ∈ (𝐾...𝑁) ∧ (𝑀 + 1) ∈ ((1...𝑁) ∖ {𝐾})))
Theorem	submateqlem2 33785	Lemma for submateq 33786. (Contributed by Thierry Arnoux, 26-Aug-2020.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐾 ∈ (1...𝑁))    &   ⊢ (𝜑 → 𝑀 ∈ (1...(𝑁 − 1)))    &   ⊢ (𝜑 → 𝑀 < 𝐾)    ⇒   ⊢ (𝜑 → (𝑀 ∈ (1..^𝐾) ∧ 𝑀 ∈ ((1...𝑁) ∖ {𝐾})))
Theorem	submateq 33786*	Sufficient condition for two submatrices to be equal. (Contributed by Thierry Arnoux, 25-Aug-2020.)
⊢ 𝐴 = ((1...𝑁) Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐼 ∈ (1...𝑁))    &   ⊢ (𝜑 → 𝐽 ∈ (1...𝑁))    &   ⊢ (𝜑 → 𝐸 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑖 ∈ ((1...𝑁) ∖ {𝐼}) ∧ 𝑗 ∈ ((1...𝑁) ∖ {𝐽})) → (𝑖𝐸𝑗) = (𝑖𝐹𝑗))    ⇒   ⊢ (𝜑 → (𝐼(subMat1‘𝐸)𝐽) = (𝐼(subMat1‘𝐹)𝐽))
Theorem	submatminr1 33787	If we take a submatrix by removing the row 𝐼 and column 𝐽, then the result is the same on the matrix with row 𝐼 and column 𝐽 modified by the minMatR1 operator. (Contributed by Thierry Arnoux, 25-Aug-2020.)
⊢ 𝐴 = ((1...𝑁) Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐼 ∈ (1...𝑁))    &   ⊢ (𝜑 → 𝐽 ∈ (1...𝑁))    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝑀 ∈ 𝐵)    &   ⊢ 𝐸 = (𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)    ⇒   ⊢ (𝜑 → (𝐼(subMat1‘𝑀)𝐽) = (𝐼(subMat1‘𝐸)𝐽))
21.3.13.2  Matrix literals
Syntax	clmat 33788	Extend class notation with the literal matrix conversion function.
class litMat
Definition	df-lmat 33789*	Define a function converting words of words into matrices. (Contributed by Thierry Arnoux, 28-Aug-2020.)
⊢ litMat = (𝑚 ∈ V ↦ (𝑖 ∈ (1...(♯‘𝑚)), 𝑗 ∈ (1...(♯‘(𝑚‘0))) ↦ ((𝑚‘(𝑖 − 1))‘(𝑗 − 1))))
Theorem	lmatval 33790*	Value of the literal matrix conversion function. (Contributed by Thierry Arnoux, 28-Aug-2020.)
⊢ (𝑀 ∈ 𝑉 → (litMat‘𝑀) = (𝑖 ∈ (1...(♯‘𝑀)), 𝑗 ∈ (1...(♯‘(𝑀‘0))) ↦ ((𝑀‘(𝑖 − 1))‘(𝑗 − 1))))
Theorem	lmatfval 33791*	Entries of a literal matrix. (Contributed by Thierry Arnoux, 28-Aug-2020.)
⊢ 𝑀 = (litMat‘𝑊)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝑊 ∈ Word Word 𝑉)    &   ⊢ (𝜑 → (♯‘𝑊) = 𝑁)    &   ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (♯‘(𝑊‘𝑖)) = 𝑁)    &   ⊢ (𝜑 → 𝐼 ∈ (1...𝑁))    &   ⊢ (𝜑 → 𝐽 ∈ (1...𝑁))    ⇒   ⊢ (𝜑 → (𝐼𝑀𝐽) = ((𝑊‘(𝐼 − 1))‘(𝐽 − 1)))
Theorem	lmatfvlem 33792*	Useful lemma to extract literal matrix entries. Suggested by Mario Carneiro. (Contributed by Thierry Arnoux, 3-Sep-2020.)
⊢ 𝑀 = (litMat‘𝑊)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝑊 ∈ Word Word 𝑉)    &   ⊢ (𝜑 → (♯‘𝑊) = 𝑁)    &   ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (♯‘(𝑊‘𝑖)) = 𝑁)    &   ⊢ 𝐾 ∈ ℕ0    &   ⊢ 𝐿 ∈ ℕ0    &   ⊢ 𝐼 ≤ 𝑁    &   ⊢ 𝐽 ≤ 𝑁    &   ⊢ (𝐾 + 1) = 𝐼    &   ⊢ (𝐿 + 1) = 𝐽    &   ⊢ (𝑊‘𝐾) = 𝑋    &   ⊢ (𝜑 → (𝑋‘𝐿) = 𝑌)    ⇒   ⊢ (𝜑 → (𝐼𝑀𝐽) = 𝑌)
Theorem	lmatcl 33793*	Closure of the literal matrix. (Contributed by Thierry Arnoux, 12-Sep-2020.)
⊢ 𝑀 = (litMat‘𝑊)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝑊 ∈ Word Word 𝑉)    &   ⊢ (𝜑 → (♯‘𝑊) = 𝑁)    &   ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (♯‘(𝑊‘𝑖)) = 𝑁)    &   ⊢ 𝑉 = (Base‘𝑅)    &   ⊢ 𝑂 = ((1...𝑁) Mat 𝑅)    &   ⊢ 𝑃 = (Base‘𝑂)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑋)    ⇒   ⊢ (𝜑 → 𝑀 ∈ 𝑃)
Theorem	lmat22lem 33794*	Lemma for lmat22e11 33795 and co. (Contributed by Thierry Arnoux, 28-Aug-2020.)
⊢ 𝑀 = (litMat‘⟨“⟨“𝐴𝐵”⟩⟨“𝐶𝐷”⟩”⟩)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑉)    ⇒   ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^2)) → (♯‘(⟨“⟨“𝐴𝐵”⟩⟨“𝐶𝐷”⟩”⟩‘𝑖)) = 2)
Theorem	lmat22e11 33795	Entry of a 2x2 literal matrix. (Contributed by Thierry Arnoux, 28-Aug-2020.)
⊢ 𝑀 = (litMat‘⟨“⟨“𝐴𝐵”⟩⟨“𝐶𝐷”⟩”⟩)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (1𝑀1) = 𝐴)
Theorem	lmat22e12 33796	Entry of a 2x2 literal matrix. (Contributed by Thierry Arnoux, 12-Sep-2020.)
⊢ 𝑀 = (litMat‘⟨“⟨“𝐴𝐵”⟩⟨“𝐶𝐷”⟩”⟩)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (1𝑀2) = 𝐵)
Theorem	lmat22e21 33797	Entry of a 2x2 literal matrix. (Contributed by Thierry Arnoux, 12-Sep-2020.)
⊢ 𝑀 = (litMat‘⟨“⟨“𝐴𝐵”⟩⟨“𝐶𝐷”⟩”⟩)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (2𝑀1) = 𝐶)
Theorem	lmat22e22 33798	Entry of a 2x2 literal matrix. (Contributed by Thierry Arnoux, 12-Sep-2020.)
⊢ 𝑀 = (litMat‘⟨“⟨“𝐴𝐵”⟩⟨“𝐶𝐷”⟩”⟩)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (2𝑀2) = 𝐷)
Theorem	lmat22det 33799	The determinant of a literal 2x2 complex matrix. (Contributed by Thierry Arnoux, 1-Sep-2020.)
⊢ 𝑀 = (litMat‘⟨“⟨“𝐴𝐵”⟩⟨“𝐶𝐷”⟩”⟩)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑉)    &   ⊢ · = (.r‘𝑅)    &   ⊢ − = (-g‘𝑅)    &   ⊢ 𝑉 = (Base‘𝑅)    &   ⊢ 𝐽 = ((1...2) maDet 𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    ⇒   ⊢ (𝜑 → (𝐽‘𝑀) = ((𝐴 · 𝐷) − (𝐶 · 𝐵)))
21.3.13.3  Laplace expansion of determinants
Theorem	mdetpmtr1 33800*	The determinant of a matrix with permuted rows is the determinant of the original matrix multiplied by the sign of the permutation. (Contributed by Thierry Arnoux, 22-Aug-2020.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝐷 = (𝑁 maDet 𝑅)    &   ⊢ 𝐺 = (Base‘(SymGrp‘𝑁))    &   ⊢ 𝑆 = (pmSgn‘𝑁)    &   ⊢ 𝑍 = (ℤRHom‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝐸 = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑃‘𝑖)𝑀𝑗))    ⇒   ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) → (𝐷‘𝑀) = (((𝑍 ∘ 𝑆)‘𝑃) · (𝐷‘𝐸)))