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	irngssv 33701	An integral element is an element of the base set. (Contributed by Thierry Arnoux, 28-Jan-2025.)
⊢ 𝑂 = (𝑅 evalSub1 𝑆)    &   ⊢ 𝑈 = (𝑅 ↾s 𝑆)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑆 ∈ (SubRing‘𝑅))    ⇒   ⊢ (𝜑 → (𝑅 IntgRing 𝑆) ⊆ 𝐵)
Theorem	0ringirng 33702	A zero ring 𝑅 has no integral elements. (Contributed by Thierry Arnoux, 5-Feb-2025.)
⊢ 𝑂 = (𝑅 evalSub1 𝑆)    &   ⊢ 𝑈 = (𝑅 ↾s 𝑆)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑆 ∈ (SubRing‘𝑅))    &   ⊢ (𝜑 → ¬ 𝑅 ∈ NzRing)    ⇒   ⊢ (𝜑 → (𝑅 IntgRing 𝑆) = ∅)
Theorem	irngnzply1lem 33703	In the case of a field 𝐸, a root 𝑋 of some nonzero polynomial 𝑃 with coefficients in a subfield 𝐹 is integral over 𝐹. (Contributed by Thierry Arnoux, 5-Feb-2025.)
⊢ 𝑂 = (𝐸 evalSub1 𝐹)    &   ⊢ 𝑍 = (0g‘(Poly1‘𝐸))    &   ⊢ 0 = (0g‘𝐸)    &   ⊢ (𝜑 → 𝐸 ∈ Field)    &   ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))    &   ⊢ 𝐵 = (Base‘𝐸)    &   ⊢ (𝜑 → 𝑃 ∈ dom 𝑂)    &   ⊢ (𝜑 → 𝑃 ≠ 𝑍)    &   ⊢ (𝜑 → ((𝑂‘𝑃)‘𝑋) = 0 )    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → 𝑋 ∈ (𝐸 IntgRing 𝐹))
Theorem	irngnzply1 33704*	In the case of a field 𝐸, the roots of nonzero polynomials 𝑝 with coefficients in a subfield 𝐹 are exactly the integral elements over 𝐹. Roots of nonzero polynomials are called algebraic numbers, so this shows that in the case of a field, elements integral over 𝐹 are exactly the algebraic numbers. In this formula, dom 𝑂 represents the polynomials, and 𝑍 the zero polynomial. (Contributed by Thierry Arnoux, 5-Feb-2025.)
⊢ 𝑂 = (𝐸 evalSub1 𝐹)    &   ⊢ 𝑍 = (0g‘(Poly1‘𝐸))    &   ⊢ 0 = (0g‘𝐸)    &   ⊢ (𝜑 → 𝐸 ∈ Field)    &   ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))    ⇒   ⊢ (𝜑 → (𝐸 IntgRing 𝐹) = ∪ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})(◡(𝑂‘𝑝) “ { 0 }))
Theorem	extdgfialglem1 33705*	Lemma for extdgfialg 33707. (Contributed by Thierry Arnoux, 10-Jan-2026.)
⊢ 𝐵 = (Base‘𝐸)    &   ⊢ 𝐷 = (dim‘((subringAlg ‘𝐸)‘𝐹))    &   ⊢ (𝜑 → 𝐸 ∈ Field)    &   ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))    &   ⊢ (𝜑 → 𝐷 ∈ ℕ0)    &   ⊢ 𝑍 = (0g‘𝐸)    &   ⊢ · = (.r‘𝐸)    &   ⊢ 𝐺 = (𝑛 ∈ (0...𝐷) ↦ (𝑛(.g‘(mulGrp‘((subringAlg ‘𝐸)‘𝐹)))𝑋))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ∃𝑎 ∈ (𝐹 ↑m (0...𝐷))(𝑎 finSupp 𝑍 ∧ ((𝐸 Σg (𝑎 ∘f · 𝐺)) = 𝑍 ∧ 𝑎 ≠ ((0...𝐷) × {𝑍}))))
Theorem	extdgfialglem2 33706*	Lemma for extdgfialg 33707. (Contributed by Thierry Arnoux, 10-Jan-2026.)
⊢ 𝐵 = (Base‘𝐸)    &   ⊢ 𝐷 = (dim‘((subringAlg ‘𝐸)‘𝐹))    &   ⊢ (𝜑 → 𝐸 ∈ Field)    &   ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))    &   ⊢ (𝜑 → 𝐷 ∈ ℕ0)    &   ⊢ 𝑍 = (0g‘𝐸)    &   ⊢ · = (.r‘𝐸)    &   ⊢ 𝐺 = (𝑛 ∈ (0...𝐷) ↦ (𝑛(.g‘(mulGrp‘((subringAlg ‘𝐸)‘𝐹)))𝑋))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐴:(0...𝐷)⟶𝐹)    &   ⊢ (𝜑 → 𝐴 finSupp 𝑍)    &   ⊢ (𝜑 → (𝐸 Σg (𝐴 ∘f · 𝐺)) = 𝑍)    &   ⊢ (𝜑 → 𝐴 ≠ ((0...𝐷) × {𝑍}))    ⇒   ⊢ (𝜑 → 𝑋 ∈ (𝐸 IntgRing 𝐹))
Theorem	extdgfialg 33707	A finite field extension 𝐸 / 𝐹 is algebraic. Part of the proof of Proposition 1.1 of [Lang], p. 224. (Contributed by Thierry Arnoux, 10-Jan-2026.)
⊢ 𝐵 = (Base‘𝐸)    &   ⊢ 𝐷 = (dim‘((subringAlg ‘𝐸)‘𝐹))    &   ⊢ (𝜑 → 𝐸 ∈ Field)    &   ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))    &   ⊢ (𝜑 → 𝐷 ∈ ℕ0)    ⇒   ⊢ (𝜑 → (𝐸 IntgRing 𝐹) = 𝐵)
21.3.11.2  Algebraic extensions
Syntax	calgext 33708	Syntax for the algebraic field extension relation.
class /AlgExt
Definition	df-algext 33709*	Definition of the algebraic extension relation. (Contributed by Thierry Arnoux, 29-Jul-2023.)
⊢ /AlgExt = {⟨𝑒, 𝑓⟩ ∣ (𝑒/FldExt𝑓 ∧ (𝑒 IntgRing (Base‘𝑓)) = (Base‘𝑒))}
Theorem	bralgext 33710	Express the fact that a field extension 𝐸 / 𝐹 is algebraic. (Contributed by Thierry Arnoux, 10-Jan-2026.)
⊢ 𝐵 = (Base‘𝐸)    &   ⊢ 𝐶 = (Base‘𝐹)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝐸/AlgExt𝐹 ↔ (𝐸/FldExt𝐹 ∧ (𝐸 IntgRing 𝐶) = 𝐵)))
Theorem	finextalg 33711	A finite field extension is algebraic. Proposition 1.1 of [Lang], p. 224. (Contributed by Thierry Arnoux, 10-Jan-2026.)
⊢ (𝜑 → 𝐸/FinExt𝐹)    ⇒   ⊢ (𝜑 → 𝐸/AlgExt𝐹)
21.3.11.3  Minimal polynomials
Syntax	cminply 33712	Extend class notation with the minimal polynomial builder function.
class minPoly
Definition	df-minply 33713*	Define the minimal polynomial builder function. (Contributed by Thierry Arnoux, 19-Jan-2025.)
⊢ minPoly = (𝑒 ∈ V, 𝑓 ∈ V ↦ (𝑥 ∈ (Base‘𝑒) ↦ ((idlGen1p‘(𝑒 ↾s 𝑓))‘{𝑝 ∈ dom (𝑒 evalSub1 𝑓) ∣ (((𝑒 evalSub1 𝑓)‘𝑝)‘𝑥) = (0g‘𝑒)})))
Theorem	ply1annidllem 33714*	Write the set 𝑄 of polynomials annihilating an element 𝐴 as the kernel of the ring homomorphism 𝐹 mapping polynomials 𝑝 to their subring evaluation at a given point 𝐴. (Contributed by Thierry Arnoux, 9-Feb-2025.)
⊢ 𝑂 = (𝑅 evalSub1 𝑆)    &   ⊢ 𝑃 = (Poly1‘(𝑅 ↾s 𝑆))    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑆 ∈ (SubRing‘𝑅))    &   ⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝑄 = {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = 0 }    &   ⊢ 𝐹 = (𝑝 ∈ (Base‘𝑃) ↦ ((𝑂‘𝑝)‘𝐴))    ⇒   ⊢ (𝜑 → 𝑄 = (◡𝐹 “ { 0 }))
Theorem	ply1annidl 33715*	The set 𝑄 of polynomials annihilating an element 𝐴 forms an ideal. (Contributed by Thierry Arnoux, 9-Feb-2025.)
⊢ 𝑂 = (𝑅 evalSub1 𝑆)    &   ⊢ 𝑃 = (Poly1‘(𝑅 ↾s 𝑆))    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑆 ∈ (SubRing‘𝑅))    &   ⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝑄 = {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = 0 }    ⇒   ⊢ (𝜑 → 𝑄 ∈ (LIdeal‘𝑃))
Theorem	ply1annnr 33716*	The set 𝑄 of polynomials annihilating an element 𝐴 is not the whole polynomial ring. (Contributed by Thierry Arnoux, 22-Mar-2025.)
⊢ 𝑂 = (𝑅 evalSub1 𝑆)    &   ⊢ 𝑃 = (Poly1‘(𝑅 ↾s 𝑆))    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑆 ∈ (SubRing‘𝑅))    &   ⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝑄 = {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = 0 }    &   ⊢ 𝑈 = (Base‘𝑃)    &   ⊢ (𝜑 → 𝑅 ∈ NzRing)    ⇒   ⊢ (𝜑 → 𝑄 ≠ 𝑈)
Theorem	ply1annig1p 33717*	The ideal 𝑄 of polynomials annihilating an element 𝐴 is generated by the ideal's canonical generator. (Contributed by Thierry Arnoux, 9-Feb-2025.)
⊢ 𝑂 = (𝐸 evalSub1 𝐹)    &   ⊢ 𝑃 = (Poly1‘(𝐸 ↾s 𝐹))    &   ⊢ 𝐵 = (Base‘𝐸)    &   ⊢ (𝜑 → 𝐸 ∈ Field)    &   ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))    &   ⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ 0 = (0g‘𝐸)    &   ⊢ 𝑄 = {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = 0 }    &   ⊢ 𝐾 = (RSpan‘𝑃)    &   ⊢ 𝐺 = (idlGen1p‘(𝐸 ↾s 𝐹))    ⇒   ⊢ (𝜑 → 𝑄 = (𝐾‘{(𝐺‘𝑄)}))
Theorem	minplyval 33718*	Expand the value of the minimal polynomial (𝑀‘𝐴) for a given element 𝐴. It is defined as the unique monic polynomial of minimal degree which annihilates 𝐴. By ply1annig1p 33717, that polynomial generates the ideal of the annihilators of 𝐴. (Contributed by Thierry Arnoux, 9-Feb-2025.)
⊢ 𝑂 = (𝐸 evalSub1 𝐹)    &   ⊢ 𝑃 = (Poly1‘(𝐸 ↾s 𝐹))    &   ⊢ 𝐵 = (Base‘𝐸)    &   ⊢ (𝜑 → 𝐸 ∈ Field)    &   ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))    &   ⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ 0 = (0g‘𝐸)    &   ⊢ 𝑄 = {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = 0 }    &   ⊢ 𝐾 = (RSpan‘𝑃)    &   ⊢ 𝐺 = (idlGen1p‘(𝐸 ↾s 𝐹))    &   ⊢ 𝑀 = (𝐸 minPoly 𝐹)    ⇒   ⊢ (𝜑 → (𝑀‘𝐴) = (𝐺‘𝑄))
Theorem	minplycl 33719*	The minimal polynomial is a polynomial. (Contributed by Thierry Arnoux, 22-Mar-2025.)
⊢ 𝑂 = (𝐸 evalSub1 𝐹)    &   ⊢ 𝑃 = (Poly1‘(𝐸 ↾s 𝐹))    &   ⊢ 𝐵 = (Base‘𝐸)    &   ⊢ (𝜑 → 𝐸 ∈ Field)    &   ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))    &   ⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ 0 = (0g‘𝐸)    &   ⊢ 𝑄 = {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = 0 }    &   ⊢ 𝐾 = (RSpan‘𝑃)    &   ⊢ 𝐺 = (idlGen1p‘(𝐸 ↾s 𝐹))    &   ⊢ 𝑀 = (𝐸 minPoly 𝐹)    ⇒   ⊢ (𝜑 → (𝑀‘𝐴) ∈ (Base‘𝑃))
Theorem	ply1annprmidl 33720*	The set 𝑄 of polynomials annihilating an element 𝐴 is a prime ideal. (Contributed by Thierry Arnoux, 9-Feb-2025.)
⊢ 𝑂 = (𝐸 evalSub1 𝐹)    &   ⊢ 𝑃 = (Poly1‘(𝐸 ↾s 𝐹))    &   ⊢ 𝐵 = (Base‘𝐸)    &   ⊢ (𝜑 → 𝐸 ∈ Field)    &   ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))    &   ⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ 0 = (0g‘𝐸)    &   ⊢ 𝑄 = {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = 0 }    ⇒   ⊢ (𝜑 → 𝑄 ∈ (PrmIdeal‘𝑃))
Theorem	minplymindeg 33721	The minimal polynomial of 𝐴 is minimal among the nonzero annihilators of 𝐴 with regard to degree. (Contributed by Thierry Arnoux, 22-Jun-2025.)
⊢ 𝑂 = (𝐸 evalSub1 𝐹)    &   ⊢ 𝑃 = (Poly1‘(𝐸 ↾s 𝐹))    &   ⊢ 𝐵 = (Base‘𝐸)    &   ⊢ (𝜑 → 𝐸 ∈ Field)    &   ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))    &   ⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ 0 = (0g‘𝐸)    &   ⊢ 𝑀 = (𝐸 minPoly 𝐹)    &   ⊢ 𝐷 = (deg1‘(𝐸 ↾s 𝐹))    &   ⊢ 𝑍 = (0g‘𝑃)    &   ⊢ 𝑈 = (Base‘𝑃)    &   ⊢ (𝜑 → ((𝑂‘𝐻)‘𝐴) = 0 )    &   ⊢ (𝜑 → 𝐻 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐻 ≠ 𝑍)    ⇒   ⊢ (𝜑 → (𝐷‘(𝑀‘𝐴)) ≤ (𝐷‘𝐻))
Theorem	minplyann 33722	The minimal polynomial for 𝐴 annihilates 𝐴 (Contributed by Thierry Arnoux, 25-Apr-2025.)
⊢ 𝑂 = (𝐸 evalSub1 𝐹)    &   ⊢ 𝑃 = (Poly1‘(𝐸 ↾s 𝐹))    &   ⊢ 𝐵 = (Base‘𝐸)    &   ⊢ (𝜑 → 𝐸 ∈ Field)    &   ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))    &   ⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ 0 = (0g‘𝐸)    &   ⊢ 𝑀 = (𝐸 minPoly 𝐹)    ⇒   ⊢ (𝜑 → ((𝑂‘(𝑀‘𝐴))‘𝐴) = 0 )
Theorem	minplyirredlem 33723	Lemma for minplyirred 33724. (Contributed by Thierry Arnoux, 22-Mar-2025.)
⊢ 𝑂 = (𝐸 evalSub1 𝐹)    &   ⊢ 𝑃 = (Poly1‘(𝐸 ↾s 𝐹))    &   ⊢ 𝐵 = (Base‘𝐸)    &   ⊢ (𝜑 → 𝐸 ∈ Field)    &   ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))    &   ⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ 𝑀 = (𝐸 minPoly 𝐹)    &   ⊢ 𝑍 = (0g‘𝑃)    &   ⊢ (𝜑 → (𝑀‘𝐴) ≠ 𝑍)    &   ⊢ (𝜑 → 𝐺 ∈ (Base‘𝑃))    &   ⊢ (𝜑 → 𝐻 ∈ (Base‘𝑃))    &   ⊢ (𝜑 → (𝐺(.r‘𝑃)𝐻) = (𝑀‘𝐴))    &   ⊢ (𝜑 → ((𝑂‘𝐺)‘𝐴) = (0g‘𝐸))    &   ⊢ (𝜑 → 𝐺 ≠ 𝑍)    &   ⊢ (𝜑 → 𝐻 ≠ 𝑍)    ⇒   ⊢ (𝜑 → 𝐻 ∈ (Unit‘𝑃))
Theorem	minplyirred 33724	A nonzero minimal polynomial is irreducible. (Contributed by Thierry Arnoux, 22-Mar-2025.)
⊢ 𝑂 = (𝐸 evalSub1 𝐹)    &   ⊢ 𝑃 = (Poly1‘(𝐸 ↾s 𝐹))    &   ⊢ 𝐵 = (Base‘𝐸)    &   ⊢ (𝜑 → 𝐸 ∈ Field)    &   ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))    &   ⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ 𝑀 = (𝐸 minPoly 𝐹)    &   ⊢ 𝑍 = (0g‘𝑃)    &   ⊢ (𝜑 → (𝑀‘𝐴) ≠ 𝑍)    ⇒   ⊢ (𝜑 → (𝑀‘𝐴) ∈ (Irred‘𝑃))
Theorem	irngnminplynz 33725	Integral elements have nonzero minimal polynomials. (Contributed by Thierry Arnoux, 22-Mar-2025.)
⊢ 𝑍 = (0g‘(Poly1‘𝐸))    &   ⊢ (𝜑 → 𝐸 ∈ Field)    &   ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))    &   ⊢ 𝑀 = (𝐸 minPoly 𝐹)    &   ⊢ (𝜑 → 𝐴 ∈ (𝐸 IntgRing 𝐹))    ⇒   ⊢ (𝜑 → (𝑀‘𝐴) ≠ 𝑍)
Theorem	minplym1p 33726	A minimal polynomial is monic. (Contributed by Thierry Arnoux, 2-Apr-2025.)
⊢ 𝑍 = (0g‘(Poly1‘𝐸))    &   ⊢ (𝜑 → 𝐸 ∈ Field)    &   ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))    &   ⊢ 𝑀 = (𝐸 minPoly 𝐹)    &   ⊢ (𝜑 → 𝐴 ∈ (𝐸 IntgRing 𝐹))    &   ⊢ 𝑈 = (Monic1p‘(𝐸 ↾s 𝐹))    ⇒   ⊢ (𝜑 → (𝑀‘𝐴) ∈ 𝑈)
Theorem	minplynzm1p 33727	If a minimal polynomial is nonzero, then it is monic. (Contributed by Thierry Arnoux, 26-Oct-2025.)
⊢ 𝐵 = (Base‘𝐸)    &   ⊢ 𝑍 = (0g‘(Poly1‘𝐸))    &   ⊢ (𝜑 → 𝐸 ∈ Field)    &   ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))    &   ⊢ 𝑀 = (𝐸 minPoly 𝐹)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ (𝜑 → (𝑀‘𝐴) ≠ 𝑍)    &   ⊢ 𝑈 = (Monic1p‘(𝐸 ↾s 𝐹))    ⇒   ⊢ (𝜑 → (𝑀‘𝐴) ∈ 𝑈)
Theorem	minplyelirng 33728	If the minimial polynomial 𝐹 of an element 𝑋 of a field 𝑅 has nonnegative degree, then 𝑋 is integral. (Contributed by Thierry Arnoux, 26-Oct-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑀 = (𝑅 minPoly 𝑆)    &   ⊢ 𝐷 = (deg1‘(𝑅 ↾s 𝑆))    &   ⊢ (𝜑 → 𝑅 ∈ Field)    &   ⊢ (𝜑 → 𝑆 ∈ (SubDRing‘𝑅))    &   ⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐷‘(𝑀‘𝐴)) ∈ ℕ0)    ⇒   ⊢ (𝜑 → 𝐴 ∈ (𝑅 IntgRing 𝑆))
Theorem	irredminply 33729	An irreducible, monic, annihilating polynomial is the minimal polynomial. (Contributed by Thierry Arnoux, 27-Apr-2025.)
⊢ 𝑂 = (𝐸 evalSub1 𝐹)    &   ⊢ 𝑃 = (Poly1‘(𝐸 ↾s 𝐹))    &   ⊢ 𝐵 = (Base‘𝐸)    &   ⊢ (𝜑 → 𝐸 ∈ Field)    &   ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))    &   ⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ 0 = (0g‘𝐸)    &   ⊢ 𝑀 = (𝐸 minPoly 𝐹)    &   ⊢ 𝑍 = (0g‘𝑃)    &   ⊢ (𝜑 → ((𝑂‘𝐺)‘𝐴) = 0 )    &   ⊢ (𝜑 → 𝐺 ∈ (Irred‘𝑃))    &   ⊢ (𝜑 → 𝐺 ∈ (Monic1p‘(𝐸 ↾s 𝐹)))    ⇒   ⊢ (𝜑 → 𝐺 = (𝑀‘𝐴))
Theorem	algextdeglem1 33730	Lemma for algextdeg 33738. (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 ↾s 𝐹) = 𝐾)
Theorem	algextdeglem2 33731*	Lemma for algextdeg 33738. Both the ring of polynomials 𝑃 and the field 𝐿 generated by 𝐾 and the algebraic element 𝐴 can be considered as modules over the elements of 𝐹. Then, the evaluation map 𝐺, mapping polynomials to their evaluation at 𝐴, is a module homomorphism between those modules. (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 → 𝐺 ∈ (𝑃 LMHom ((subringAlg ‘𝐿)‘𝐹)))
Theorem	algextdeglem3 33732*	Lemma for algextdeg 33738. The quotient 𝑃 / 𝑍 of the vector space 𝑃 of polynomials by the subspace 𝑍 of polynomials annihilating 𝐴 is itself a vector space. (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 → 𝑄 ∈ LVec)
Theorem	algextdeglem4 33733*	Lemma for algextdeg 33738. By lmhmqusker 33382, the surjective module homomorphism 𝐺 described in algextdeglem2 33731 induces an isomorphism with the quotient space. Therefore, the dimension of that quotient space 𝑃 / 𝑍 is the degree of the algebraic field extension. (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 → (dim‘𝑄) = (𝐿[:]𝐾))
Theorem	algextdeglem5 33734*	Lemma for algextdeg 33738. 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 33735*	Lemma for algextdeg 33738. By r1pquslmic 33571, 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 33736*	Lemma for algextdeg 33738. 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 33737*	Lemma for algextdeg 33738. 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 33738	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. Part of Proposition 1.4 of [Lang], p. 225. (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)))
21.3.11.4  Quadratic Field Extensions
Theorem	rtelextdg2lem 33739	Lemma for rtelextdg2 33740: 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 33740	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))
21.3.11.5  Towers of quadratic extentions
Theorem	fldext2chn 33741*	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 33794), and of angle trisection (cos9thpinconstr 33804)
Syntax	cconstr 33742	Extend class notation with the set of constructible points.
class Constr
Definition	df-constr 33743*	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. Definition 7.4. in [Stewart] p. 92 (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 33744	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 33745	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 33746	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 33747	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 33748	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 33749*	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 33750	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 33751*	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 33752*	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 33753*	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 33754*	Lemma for constrss 33756. This lemma requires the additional condition that 0 is a constructible number; that condition is removed in constrss 33756. (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 33755*	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 33756*	Constructed points are in the next generation constructed points. Lemma 7.3 of [Stewart] p. 91 (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 33757*	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 33758*	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 33759*	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 33760*	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 33761*	Lemma for constrextdg2 33762 (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 33762*	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. See Theorem 7.11 of [Stewart] p. 97. (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 33763*	Lemma for constrext2chn 33772. (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 33764*	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 33765*	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 33766*	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 33767*	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 33768*	Technical lemma for eliminating the hypothesis of constr0 33750 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 33769	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 33770	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 33771	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 33772*	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. Theorem 7.12 of [Stewart] p. 98. (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 33773	Constructible numbers are complex numbers. (Contributed by Thierry Arnoux, 2-Nov-2025.)
⊢ (𝜑 → 𝑋 ∈ Constr)    ⇒   ⊢ (𝜑 → 𝑋 ∈ ℂ)
Theorem	nn0constr 33774	Nonnegative integers are constructible. (Contributed by Thierry Arnoux, 2-Nov-2025.)
⊢ (𝜑 → 𝑁 ∈ ℕ0)    ⇒   ⊢ (𝜑 → 𝑁 ∈ Constr)
Theorem	constraddcl 33775	Constructive numbers are closed under complex addition. Item (1) of Theorem 7.10 of [Stewart] p. 96 (Contributed by Thierry Arnoux, 2-Nov-2025.)
⊢ (𝜑 → 𝑋 ∈ Constr)    &   ⊢ (𝜑 → 𝑌 ∈ Constr)    ⇒   ⊢ (𝜑 → (𝑋 + 𝑌) ∈ Constr)
Theorem	constrnegcl 33776	Constructible numbers are closed under additive inverse. Item (2) of Theorem 7.10 of [Stewart] p. 96 (Contributed by Thierry Arnoux, 2-Nov-2025.)
⊢ (𝜑 → 𝑋 ∈ Constr)    ⇒   ⊢ (𝜑 → -𝑋 ∈ Constr)
Theorem	zconstr 33777	Integers are constructible. (Contributed by Thierry Arnoux, 3-Nov-2025.)
⊢ (𝜑 → 𝑋 ∈ ℤ)    ⇒   ⊢ (𝜑 → 𝑋 ∈ Constr)
Theorem	constrdircl 33778	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 33779	The imaginary unit i is constructible. (Contributed by Thierry Arnoux, 2-Nov-2025.)
⊢ i ∈ Constr
Theorem	constrremulcl 33780	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 33781	Constructible numbers are closed under complex conjugate. (Contributed by Thierry Arnoux, 5-Nov-2025.)
⊢ (𝜑 → 𝑋 ∈ Constr)    ⇒   ⊢ (𝜑 → (∗‘𝑋) ∈ Constr)
Theorem	constrrecl 33782	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 33783	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 33784	Constructible numbers are closed under complex multiplication. Item (3) of Theorem 7.10 of [Stewart] p. 96 (Contributed by Thierry Arnoux, 5-Nov-2025.)
⊢ (𝜑 → 𝑋 ∈ Constr)    &   ⊢ (𝜑 → 𝑌 ∈ Constr)    ⇒   ⊢ (𝜑 → (𝑋 · 𝑌) ∈ Constr)
Theorem	constrreinvcl 33785	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 33786	Constructible numbers are closed under complex inverse. Item (4) of Theorem 7.10 of [Stewart] p. 96 (Contributed by Thierry Arnoux, 5-Nov-2025.)
⊢ (𝜑 → 𝑋 ∈ Constr)    &   ⊢ (𝜑 → 𝑋 ≠ 0)    ⇒   ⊢ (𝜑 → (1 / 𝑋) ∈ Constr)
Theorem	constrcon 33787*	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 33788	Constructible numbers form a subfield of the complex numbers. (Contributed by Thierry Arnoux, 5-Nov-2025.)
⊢ Constr ∈ (SubDRing‘ℂfld)
Theorem	constrfld 33789	The constructible numbers form a field. (Contributed by Thierry Arnoux, 5-Nov-2025.)
⊢ (ℂfld ↾s Constr) ∈ Field
Theorem	constrresqrtcl 33790	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 33791	Constructible numbers are closed under absolute value (modulus). (Contributed by Thierry Arnoux, 6-Nov-2025.)
⊢ (𝜑 → 𝑋 ∈ Constr)    ⇒   ⊢ (𝜑 → (abs‘𝑋) ∈ Constr)
Theorem	constrsqrtcl 33792	Constructible numbers are closed under taking the square root. This is not generally the case for the cubic root operation, see 2sqr3nconstr 33794. Item (5) of Theorem 7.10 of [Stewart] p. 96 (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 33793	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 33794	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. Theorem 7.13 of [Stewart] p. 99. (Contributed by Thierry Arnoux and Saveliy Skresanov, 26-Oct-2025.)
⊢ (2↑𝑐(1 / 3)) ∉ Constr
Theorem	cos9thpiminplylem1 33795	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 33796	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 33797	Lemma for cos9thpiminply 33801. (Contributed by Thierry Arnoux, 14-Nov-2025.)
⊢ 𝑂 = (exp‘((i · (2 · π)) / 3))    ⇒   ⊢ ((𝑂↑2) + (𝑂 + 1)) = 0
Theorem	cos9thpiminplylem4 33798	Lemma for cos9thpiminply 33801. (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 33799	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 33800	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)))