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	minplymindeg 33701	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 33702	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 33703	Lemma for minplyirred 33704. (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 33704	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 33705	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 33706	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	irredminply 33707	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 33708	Lemma for algextdeg 33716. (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 33709*	Lemma for algextdeg 33716. 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 33710*	Lemma for algextdeg 33716. 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 33711*	Lemma for algextdeg 33716. By lmhmqusker 33410, the surjective module homomorphism 𝐺 described in algextdeglem2 33709 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 33712*	Lemma for algextdeg 33716. 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 33713*	Lemma for algextdeg 33716. By r1pquslmic 33596, 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 33714*	Lemma for algextdeg 33716. 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 33715*	Lemma for algextdeg 33716. 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 33716	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)))
21.3.10.3  Quadratic Field Extensions
Theorem	rtelextdg2lem 33717	Lemma for rtelextdg2 33718: 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 33718	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))
21.3.10.4  Towers of quadratic extentions
Theorem	fldext2chn 33719*	In a non-empty tower 𝑇 of quadratic field extensions, the degree of the extension of the first member by the last is a power of two. (Contributed by Thierry Arnoux, 19-Jun-2025.)
⊢ < = {⟨𝑓, 𝑒⟩ ∣ (𝑒/FldExt𝑓 ∧ (𝑒[:]𝑓) = 2)}    &   ⊢ (𝜑 → 𝑇 ∈ ( < ChainField))    &   ⊢ (𝜑 → (𝑇‘0) = 𝑄)    &   ⊢ (𝜑 → (lastS‘𝑇) = 𝐹)    &   ⊢ (𝜑 → 0 < (♯‘𝑇))    ⇒   ⊢ (𝜑 → ∃𝑛 ∈ ℕ0 (𝐹[:]𝑄) = (2↑𝑛))
21.3.11  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 ( * imp2cube ), and of angle trisection ( * imp3ang )
Syntax	cconstr 33720	Extend class notation with the set of constructible points.
class Constr
Definition	df-constr 33721*	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 33722	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 33723	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 33724	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 33725	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 33726	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	constr0 33727	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 33728*	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 33729*	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 33730*	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 33731*	Lemma for constrss 33733. This lemma requires the additional condition that 0 is the constructible number; that condition is removed in constrss 33733. (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 33732*	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 33733*	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 33734*	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 33735*	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 33736*	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 33737*	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))
21.3.11.1  Impossible constructions
Theorem	2sqr3minply 33738	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)
21.3.12  Matrices
21.3.12.1  Submatrices
Syntax	csmat 33739	Syntax for a function generating submatrices.
class subMat1
Definition	df-smat 33740*	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 22604. (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 33741*	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 33742	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 33743	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 33744	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 33745	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 33746	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 33747	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 33748	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 33749*	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 33750	The submatrix of the identity matrix obtained by removing the ith row and the ith column is an identity matrix. Cf. 1marepvsma1 22610. (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 33751	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 33752	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 33753	Lemma for submateq 33755. (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 33754	Lemma for submateq 33755. (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 33755*	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 33756	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))
21.3.12.2  Matrix literals
Syntax	clmat 33757	Extend class notation with the literal matrix conversion function.
class litMat
Definition	df-lmat 33758*	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 33759*	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 33760*	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 33761*	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 33762*	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 33763*	Lemma for lmat22e11 33764 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 33764	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 33765	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 33766	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 33767	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 33768	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)))
21.3.12.3  Laplace expansion of determinants
Theorem	mdetpmtr1 33769*	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)))
Theorem	mdetpmtr2 33770*	The determinant of a matrix with permuted columns 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 ↦ (𝑖𝑀(𝑃‘𝑗)))    ⇒   ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) → (𝐷‘𝑀) = (((𝑍 ∘ 𝑆)‘𝑃) · (𝐷‘𝐸)))
Theorem	mdetpmtr12 33771*	The determinant of a matrix with permuted rows and columns is the determinant of the original matrix multiplied by the product of the signs of the permutations. (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)𝑀(𝑄‘𝑗)))    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑁 ∈ Fin)    &   ⊢ (𝜑 → 𝑀 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑃 ∈ 𝐺)    &   ⊢ (𝜑 → 𝑄 ∈ 𝐺)    ⇒   ⊢ (𝜑 → (𝐷‘𝑀) = ((𝑍‘((𝑆‘𝑃) · (𝑆‘𝑄))) · (𝐷‘𝐸)))
Theorem	mdetlap1 33772*	A Laplace expansion of the determinant of a matrix, using the adjunct (cofactor) matrix. (Contributed by Thierry Arnoux, 16-Aug-2020.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝐷 = (𝑁 maDet 𝑅)    &   ⊢ 𝐾 = (𝑁 maAdju 𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ∧ 𝐼 ∈ 𝑁) → (𝐷‘𝑀) = (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝐼𝑀𝑗) · (𝑗(𝐾‘𝑀)𝐼)))))
Theorem	madjusmdetlem1 33773*	Lemma for madjusmdet 33777. (Contributed by Thierry Arnoux, 22-Aug-2020.)
⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝐴 = ((1...𝑁) Mat 𝑅)    &   ⊢ 𝐷 = ((1...𝑁) maDet 𝑅)    &   ⊢ 𝐾 = ((1...𝑁) maAdju 𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝑍 = (ℤRHom‘𝑅)    &   ⊢ 𝐸 = ((1...(𝑁 − 1)) maDet 𝑅)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝐼 ∈ (1...𝑁))    &   ⊢ (𝜑 → 𝐽 ∈ (1...𝑁))    &   ⊢ (𝜑 → 𝑀 ∈ 𝐵)    &   ⊢ 𝐺 = (Base‘(SymGrp‘(1...𝑁)))    &   ⊢ 𝑆 = (pmSgn‘(1...𝑁))    &   ⊢ 𝑈 = (𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)    &   ⊢ 𝑊 = (𝑖 ∈ (1...𝑁), 𝑗 ∈ (1...𝑁) ↦ ((𝑃‘𝑖)𝑈(𝑄‘𝑗)))    &   ⊢ (𝜑 → 𝑃 ∈ 𝐺)    &   ⊢ (𝜑 → 𝑄 ∈ 𝐺)    &   ⊢ (𝜑 → (𝑃‘𝑁) = 𝐼)    &   ⊢ (𝜑 → (𝑄‘𝑁) = 𝐽)    &   ⊢ (𝜑 → (𝐼(subMat1‘𝑈)𝐽) = (𝑁(subMat1‘𝑊)𝑁))    ⇒   ⊢ (𝜑 → (𝐽(𝐾‘𝑀)𝐼) = ((𝑍‘((𝑆‘𝑃) · (𝑆‘𝑄))) · (𝐸‘(𝐼(subMat1‘𝑀)𝐽))))
Theorem	madjusmdetlem2 33774*	Lemma for madjusmdet 33777. (Contributed by Thierry Arnoux, 26-Aug-2020.)
⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝐴 = ((1...𝑁) Mat 𝑅)    &   ⊢ 𝐷 = ((1...𝑁) maDet 𝑅)    &   ⊢ 𝐾 = ((1...𝑁) maAdju 𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝑍 = (ℤRHom‘𝑅)    &   ⊢ 𝐸 = ((1...(𝑁 − 1)) maDet 𝑅)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝐼 ∈ (1...𝑁))    &   ⊢ (𝜑 → 𝐽 ∈ (1...𝑁))    &   ⊢ (𝜑 → 𝑀 ∈ 𝐵)    &   ⊢ 𝑃 = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝐼, if(𝑖 ≤ 𝐼, (𝑖 − 1), 𝑖)))    &   ⊢ 𝑆 = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝑁, if(𝑖 ≤ 𝑁, (𝑖 − 1), 𝑖)))    ⇒   ⊢ ((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) → if(𝑋 < 𝐼, 𝑋, (𝑋 + 1)) = ((𝑃 ∘ ◡𝑆)‘𝑋))
Theorem	madjusmdetlem3 33775*	Lemma for madjusmdet 33777. (Contributed by Thierry Arnoux, 27-Aug-2020.)
⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝐴 = ((1...𝑁) Mat 𝑅)    &   ⊢ 𝐷 = ((1...𝑁) maDet 𝑅)    &   ⊢ 𝐾 = ((1...𝑁) maAdju 𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝑍 = (ℤRHom‘𝑅)    &   ⊢ 𝐸 = ((1...(𝑁 − 1)) maDet 𝑅)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝐼 ∈ (1...𝑁))    &   ⊢ (𝜑 → 𝐽 ∈ (1...𝑁))    &   ⊢ (𝜑 → 𝑀 ∈ 𝐵)    &   ⊢ 𝑃 = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝐼, if(𝑖 ≤ 𝐼, (𝑖 − 1), 𝑖)))    &   ⊢ 𝑆 = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝑁, if(𝑖 ≤ 𝑁, (𝑖 − 1), 𝑖)))    &   ⊢ 𝑄 = (𝑗 ∈ (1...𝑁) ↦ if(𝑗 = 1, 𝐽, if(𝑗 ≤ 𝐽, (𝑗 − 1), 𝑗)))    &   ⊢ 𝑇 = (𝑗 ∈ (1...𝑁) ↦ if(𝑗 = 1, 𝑁, if(𝑗 ≤ 𝑁, (𝑗 − 1), 𝑗)))    &   ⊢ 𝑊 = (𝑖 ∈ (1...𝑁), 𝑗 ∈ (1...𝑁) ↦ (((𝑃 ∘ ◡𝑆)‘𝑖)𝑈((𝑄 ∘ ◡𝑇)‘𝑗)))    &   ⊢ (𝜑 → 𝑈 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐼(subMat1‘𝑈)𝐽) = (𝑁(subMat1‘𝑊)𝑁))
Theorem	madjusmdetlem4 33776*	Lemma for madjusmdet 33777. (Contributed by Thierry Arnoux, 22-Aug-2020.)
⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝐴 = ((1...𝑁) Mat 𝑅)    &   ⊢ 𝐷 = ((1...𝑁) maDet 𝑅)    &   ⊢ 𝐾 = ((1...𝑁) maAdju 𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝑍 = (ℤRHom‘𝑅)    &   ⊢ 𝐸 = ((1...(𝑁 − 1)) maDet 𝑅)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝐼 ∈ (1...𝑁))    &   ⊢ (𝜑 → 𝐽 ∈ (1...𝑁))    &   ⊢ (𝜑 → 𝑀 ∈ 𝐵)    &   ⊢ 𝑃 = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝐼, if(𝑖 ≤ 𝐼, (𝑖 − 1), 𝑖)))    &   ⊢ 𝑆 = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝑁, if(𝑖 ≤ 𝑁, (𝑖 − 1), 𝑖)))    &   ⊢ 𝑄 = (𝑗 ∈ (1...𝑁) ↦ if(𝑗 = 1, 𝐽, if(𝑗 ≤ 𝐽, (𝑗 − 1), 𝑗)))    &   ⊢ 𝑇 = (𝑗 ∈ (1...𝑁) ↦ if(𝑗 = 1, 𝑁, if(𝑗 ≤ 𝑁, (𝑗 − 1), 𝑗)))    ⇒   ⊢ (𝜑 → (𝐽(𝐾‘𝑀)𝐼) = ((𝑍‘(-1↑(𝐼 + 𝐽))) · (𝐸‘(𝐼(subMat1‘𝑀)𝐽))))
Theorem	madjusmdet 33777	Express the cofactor of the matrix, i.e. the entries of its adjunct matrix, using determinant of submatrices. (Contributed by Thierry Arnoux, 23-Aug-2020.)
⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝐴 = ((1...𝑁) Mat 𝑅)    &   ⊢ 𝐷 = ((1...𝑁) maDet 𝑅)    &   ⊢ 𝐾 = ((1...𝑁) maAdju 𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝑍 = (ℤRHom‘𝑅)    &   ⊢ 𝐸 = ((1...(𝑁 − 1)) maDet 𝑅)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝐼 ∈ (1...𝑁))    &   ⊢ (𝜑 → 𝐽 ∈ (1...𝑁))    &   ⊢ (𝜑 → 𝑀 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐽(𝐾‘𝑀)𝐼) = ((𝑍‘(-1↑(𝐼 + 𝐽))) · (𝐸‘(𝐼(subMat1‘𝑀)𝐽))))
Theorem	mdetlap 33778*	Laplace expansion of the determinant of a square matrix. (Contributed by Thierry Arnoux, 19-Aug-2020.)
⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝐴 = ((1...𝑁) Mat 𝑅)    &   ⊢ 𝐷 = ((1...𝑁) maDet 𝑅)    &   ⊢ 𝐾 = ((1...𝑁) maAdju 𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝑍 = (ℤRHom‘𝑅)    &   ⊢ 𝐸 = ((1...(𝑁 − 1)) maDet 𝑅)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝐼 ∈ (1...𝑁))    &   ⊢ (𝜑 → 𝐽 ∈ (1...𝑁))    &   ⊢ (𝜑 → 𝑀 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐷‘𝑀) = (𝑅 Σg (𝑗 ∈ (1...𝑁) ↦ ((𝑍‘(-1↑(𝐼 + 𝑗))) · ((𝐼𝑀𝑗) · (𝐸‘(𝐼(subMat1‘𝑀)𝑗)))))))
21.3.13  Topology
Theorem	ist0cld 33779*	The predicate "is a T0 space", using closed sets. (Contributed by Thierry Arnoux, 16-Aug-2020.)
⊢ (𝜑 → 𝐵 = ∪ 𝐽)    &   ⊢ (𝜑 → 𝐷 = (Clsd‘𝐽))    ⇒   ⊢ (𝜑 → (𝐽 ∈ Kol2 ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (∀𝑑 ∈ 𝐷 (𝑥 ∈ 𝑑 ↔ 𝑦 ∈ 𝑑) → 𝑥 = 𝑦))))
21.3.13.1  Open maps
Theorem	txomap 33780*	Given two open maps 𝐹 and 𝐺, 𝐻 mapping pairs of sets, is also an open map for the product topology. (Contributed by Thierry Arnoux, 29-Dec-2019.)
⊢ (𝜑 → 𝐹:𝑋⟶𝑍)    &   ⊢ (𝜑 → 𝐺:𝑌⟶𝑇)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))    &   ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑍))    &   ⊢ (𝜑 → 𝑀 ∈ (TopOn‘𝑇))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (𝐹 “ 𝑥) ∈ 𝐿)    &   ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐾) → (𝐺 “ 𝑦) ∈ 𝑀)    &   ⊢ (𝜑 → 𝐴 ∈ (𝐽 ×t 𝐾))    &   ⊢ 𝐻 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩)    ⇒   ⊢ (𝜑 → (𝐻 “ 𝐴) ∈ (𝐿 ×t 𝑀))
21.3.13.2  Topology of the unit circle
Theorem	qtopt1 33781*	If every equivalence class is closed, then the quotient space is T1 . (Contributed by Thierry Arnoux, 5-Jan-2020.)
⊢ 𝑋 = ∪ 𝐽    &   ⊢ (𝜑 → 𝐽 ∈ Fre)    &   ⊢ (𝜑 → 𝐹:𝑋–onto→𝑌)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → (◡𝐹 “ {𝑥}) ∈ (Clsd‘𝐽))    ⇒   ⊢ (𝜑 → (𝐽 qTop 𝐹) ∈ Fre)
Theorem	qtophaus 33782*	If an open map's graph in the product space (𝐽 ×t 𝐽) is closed, then its quotient topology is Hausdorff. (Contributed by Thierry Arnoux, 4-Jan-2020.)
⊢ 𝑋 = ∪ 𝐽    &   ⊢ ∼ = (◡𝐹 ∘ 𝐹)    &   ⊢ 𝐻 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩)    &   ⊢ (𝜑 → 𝐽 ∈ Haus)    &   ⊢ (𝜑 → 𝐹:𝑋–onto→𝑌)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (𝐹 “ 𝑥) ∈ (𝐽 qTop 𝐹))    &   ⊢ (𝜑 → ∼ ∈ (Clsd‘(𝐽 ×t 𝐽)))    ⇒   ⊢ (𝜑 → (𝐽 qTop 𝐹) ∈ Haus)
Theorem	circtopn 33783*	The topology of the unit circle is generated by open intervals of the polar coordinate. (Contributed by Thierry Arnoux, 4-Jan-2020.)
⊢ 𝐼 = (0[,](2 · π))    &   ⊢ 𝐽 = (topGen‘ran (,))    &   ⊢ 𝐹 = (𝑥 ∈ ℝ ↦ (exp‘(i · 𝑥)))    &   ⊢ 𝐶 = (◡abs “ {1})    ⇒   ⊢ (𝐽 qTop 𝐹) = (TopOpen‘(𝐹 “s ℝfld))
Theorem	circcn 33784*	The function gluing the real line into the unit circle is continuous. (Contributed by Thierry Arnoux, 5-Jan-2020.)
⊢ 𝐼 = (0[,](2 · π))    &   ⊢ 𝐽 = (topGen‘ran (,))    &   ⊢ 𝐹 = (𝑥 ∈ ℝ ↦ (exp‘(i · 𝑥)))    &   ⊢ 𝐶 = (◡abs “ {1})    ⇒   ⊢ 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹))
21.3.13.3  Refinements
Theorem	reff 33785*	For any cover refinement, there exists a function associating with each set in the refinement a set in the original cover containing it. This is sometimes used as a definition of refinement. Note that this definition uses the axiom of choice through ac6sg 10557. (Contributed by Thierry Arnoux, 12-Jan-2020.)
⊢ (𝐴 ∈ 𝑉 → (𝐴Ref𝐵 ↔ (∪ 𝐵 ⊆ ∪ 𝐴 ∧ ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣)))))
Theorem	locfinreflem 33786*	A locally finite refinement of an open cover induces a locally finite open cover with the original index set. This is fact 2 of http://at.yorku.ca/p/a/c/a/02.pdf, it is expressed by exposing a function 𝑓 from the original cover 𝑈, which is taken as the index set. The solution is constructed by building unions, so the same method can be used to prove a similar theorem about closed covers. (Contributed by Thierry Arnoux, 29-Jan-2020.)
⊢ 𝑋 = ∪ 𝐽    &   ⊢ (𝜑 → 𝑈 ⊆ 𝐽)    &   ⊢ (𝜑 → 𝑋 = ∪ 𝑈)    &   ⊢ (𝜑 → 𝑉 ⊆ 𝐽)    &   ⊢ (𝜑 → 𝑉Ref𝑈)    &   ⊢ (𝜑 → 𝑉 ∈ (LocFin‘𝐽))    ⇒   ⊢ (𝜑 → ∃𝑓((Fun 𝑓 ∧ dom 𝑓 ⊆ 𝑈 ∧ ran 𝑓 ⊆ 𝐽) ∧ (ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽))))
Theorem	locfinref 33787*	A locally finite refinement of an open cover induces a locally finite open cover with the original index set. This is fact 2 of http://at.yorku.ca/p/a/c/a/02.pdf, it is expressed by exposing a function 𝑓 from the original cover 𝑈, which is taken as the index set. (Contributed by Thierry Arnoux, 31-Jan-2020.)
⊢ 𝑋 = ∪ 𝐽    &   ⊢ (𝜑 → 𝑈 ⊆ 𝐽)    &   ⊢ (𝜑 → 𝑋 = ∪ 𝑈)    &   ⊢ (𝜑 → 𝑉 ⊆ 𝐽)    &   ⊢ (𝜑 → 𝑉Ref𝑈)    &   ⊢ (𝜑 → 𝑉 ∈ (LocFin‘𝐽))    ⇒   ⊢ (𝜑 → ∃𝑓(𝑓:𝑈⟶𝐽 ∧ ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽)))
21.3.13.4  Open cover refinement property
Syntax	ccref 33788	The "every open cover has an 𝐴 refinement" predicate.
class CovHasRef𝐴
Definition	df-cref 33789*	Define a statement "every open cover has an 𝐴 refinement" , where 𝐴 is a property for refinements like "finite", "countable", "point finite" or "locally finite". (Contributed by Thierry Arnoux, 7-Jan-2020.)
⊢ CovHasRef𝐴 = {𝑗 ∈ Top ∣ ∀𝑦 ∈ 𝒫 𝑗(∪ 𝑗 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑗 ∩ 𝐴)𝑧Ref𝑦)}
Theorem	iscref 33790*	The property that every open cover has an 𝐴 refinement for the topological space 𝐽. (Contributed by Thierry Arnoux, 7-Jan-2020.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ (𝐽 ∈ CovHasRef𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝑦)))
Theorem	crefeq 33791	Equality theorem for the "every open cover has an A refinement" predicate. (Contributed by Thierry Arnoux, 7-Jan-2020.)
⊢ (𝐴 = 𝐵 → CovHasRef𝐴 = CovHasRef𝐵)
Theorem	creftop 33792	A space where every open cover has an 𝐴 refinement is a topological space. (Contributed by Thierry Arnoux, 7-Jan-2020.)
⊢ (𝐽 ∈ CovHasRef𝐴 → 𝐽 ∈ Top)
Theorem	crefi 33793*	The property that every open cover has an 𝐴 refinement for the topological space 𝐽. (Contributed by Thierry Arnoux, 7-Jan-2020.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ CovHasRef𝐴 ∧ 𝐶 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝐶) → ∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝐶)
Theorem	crefdf 33794*	A formulation of crefi 33793 easier to use for definitions. (Contributed by Thierry Arnoux, 7-Jan-2020.)
⊢ 𝑋 = ∪ 𝐽    &   ⊢ 𝐵 = CovHasRef𝐴    &   ⊢ (𝑧 ∈ 𝐴 → 𝜑)    ⇒   ⊢ ((𝐽 ∈ 𝐵 ∧ 𝐶 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝐶) → ∃𝑧 ∈ 𝒫 𝐽(𝜑 ∧ 𝑧Ref𝐶))
Theorem	crefss 33795	The "every open cover has an 𝐴 refinement" predicate respects inclusion. (Contributed by Thierry Arnoux, 7-Jan-2020.)
⊢ (𝐴 ⊆ 𝐵 → CovHasRef𝐴 ⊆ CovHasRef𝐵)
Theorem	cmpcref 33796	Equivalent definition of compact space in terms of open cover refinements. Compact spaces are topologies with finite open cover refinements. (Contributed by Thierry Arnoux, 7-Jan-2020.)
⊢ Comp = CovHasRefFin
Theorem	cmpfiref 33797*	Every open cover of a Compact space has a finite refinement. (Contributed by Thierry Arnoux, 1-Feb-2020.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Comp ∧ 𝑈 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑈) → ∃𝑣 ∈ 𝒫 𝐽(𝑣 ∈ Fin ∧ 𝑣Ref𝑈))
21.3.13.5  Lindelöf spaces
Syntax	cldlf 33798	Extend class notation with the class of all Lindelöf spaces.
class Ldlf
Definition	df-ldlf 33799	Definition of a Lindelöf space. A Lindelöf space is a topological space in which every open cover has a countable subcover. Definition 1 of [BourbakiTop2] p. 195. (Contributed by Thierry Arnoux, 30-Jan-2020.)
⊢ Ldlf = CovHasRef{𝑥 ∣ 𝑥 ≼ ω}
Theorem	ldlfcntref 33800*	Every open cover of a Lindelöf space has a countable refinement. (Contributed by Thierry Arnoux, 1-Feb-2020.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Ldlf ∧ 𝑈 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑈) → ∃𝑣 ∈ 𝒫 𝐽(𝑣 ≼ ω ∧ 𝑣Ref𝑈))