P. 324 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 32301-32400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	dimvalfi 32301	The dimension of a vector space 𝐹 is the cardinality of one of its bases. This version of dimval 32300 does not depend on the axiom of choice, but it is limited to the case where the base 𝑆 is finite. (Contributed by Thierry Arnoux, 24-May-2023.)
⊢ 𝐽 = (LBasis‘𝐹)    ⇒   ⊢ ((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽 ∧ 𝑆 ∈ Fin) → (dim‘𝐹) = (♯‘𝑆))
Theorem	dimcl 32302	Closure of the vector space dimension. (Contributed by Thierry Arnoux, 18-May-2023.)
⊢ (𝑉 ∈ LVec → (dim‘𝑉) ∈ ℕ0*)
Theorem	lvecdim0i 32303	A vector space of dimension zero is reduced to its identity element. (Contributed by Thierry Arnoux, 31-Jul-2023.)
⊢ 0 = (0g‘𝑉)    ⇒   ⊢ ((𝑉 ∈ LVec ∧ (dim‘𝑉) = 0) → (Base‘𝑉) = { 0 })
Theorem	lvecdim0 32304	A vector space of dimension zero is reduced to its identity element, biconditional version. (Contributed by Thierry Arnoux, 31-Jul-2023.)
⊢ 0 = (0g‘𝑉)    ⇒   ⊢ (𝑉 ∈ LVec → ((dim‘𝑉) = 0 ↔ (Base‘𝑉) = { 0 }))
Theorem	lssdimle 32305	The dimension of a linear subspace is less than or equal to the dimension of the parent vector space. This is corollary 5.4 of [Lang] p. 141. (Contributed by Thierry Arnoux, 20-May-2023.)
⊢ 𝑋 = (𝑊 ↾s 𝑈)    ⇒   ⊢ ((𝑊 ∈ LVec ∧ 𝑈 ∈ (LSubSp‘𝑊)) → (dim‘𝑋) ≤ (dim‘𝑊))
Theorem	dimpropd 32306*	If two structures have the same components (properties), they have the same dimension. (Contributed by Thierry Arnoux, 18-May-2023.)
⊢ (𝜑 → 𝐵 = (Base‘𝐾))    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐿))    &   ⊢ (𝜑 → 𝐵 ⊆ 𝑊)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝐾)𝑦) ∈ 𝑊)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝐾)𝑦) = (𝑥( ·𝑠 ‘𝐿)𝑦))    &   ⊢ 𝐹 = (Scalar‘𝐾)    &   ⊢ 𝐺 = (Scalar‘𝐿)    &   ⊢ (𝜑 → 𝑃 = (Base‘𝐹))    &   ⊢ (𝜑 → 𝑃 = (Base‘𝐺))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃)) → (𝑥(+g‘𝐹)𝑦) = (𝑥(+g‘𝐺)𝑦))    &   ⊢ (𝜑 → 𝐾 ∈ LVec)    &   ⊢ (𝜑 → 𝐿 ∈ LVec)    ⇒   ⊢ (𝜑 → (dim‘𝐾) = (dim‘𝐿))
Theorem	rgmoddim 32307	The left vector space induced by a ring over itself has dimension 1. (Contributed by Thierry Arnoux, 5-Aug-2023.)
⊢ 𝑉 = (ringLMod‘𝐹)    ⇒   ⊢ (𝐹 ∈ Field → (dim‘𝑉) = 1)
Theorem	frlmdim 32308	Dimension of a free left module. (Contributed by Thierry Arnoux, 20-May-2023.)
⊢ 𝐹 = (𝑅 freeLMod 𝐼)    ⇒   ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑉) → (dim‘𝐹) = (♯‘𝐼))
Theorem	tnglvec 32309	Augmenting a structure with a norm conserves left vector spaces. (Contributed by Thierry Arnoux, 20-May-2023.)
⊢ 𝑇 = (𝐺 toNrmGrp 𝑁)    ⇒   ⊢ (𝑁 ∈ 𝑉 → (𝐺 ∈ LVec ↔ 𝑇 ∈ LVec))
Theorem	tngdim 32310	Dimension of a left vector space augmented with a norm. (Contributed by Thierry Arnoux, 20-May-2023.)
⊢ 𝑇 = (𝐺 toNrmGrp 𝑁)    ⇒   ⊢ ((𝐺 ∈ LVec ∧ 𝑁 ∈ 𝑉) → (dim‘𝐺) = (dim‘𝑇))
Theorem	rrxdim 32311	Dimension of the generalized Euclidean space. (Contributed by Thierry Arnoux, 20-May-2023.)
⊢ 𝐻 = (ℝ^‘𝐼)    ⇒   ⊢ (𝐼 ∈ 𝑉 → (dim‘𝐻) = (♯‘𝐼))
Theorem	matdim 32312	Dimension of the space of square matrices. (Contributed by Thierry Arnoux, 20-May-2023.)
⊢ 𝐴 = (𝐼 Mat 𝑅)    &   ⊢ 𝑁 = (♯‘𝐼)    ⇒   ⊢ ((𝐼 ∈ Fin ∧ 𝑅 ∈ DivRing) → (dim‘𝐴) = (𝑁 · 𝑁))
Theorem	lbslsat 32313	A nonzero vector 𝑋 is a basis of a line spanned by the singleton 𝑋. Spans of nonzero singletons are sometimes called "atoms", see df-lsatoms 37438 and for example lsatlspsn 37455. (Contributed by Thierry Arnoux, 20-May-2023.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ 𝑌 = (𝑊 ↾s (𝑁‘{𝑋}))    ⇒   ⊢ ((𝑊 ∈ LVec ∧ 𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 ) → {𝑋} ∈ (LBasis‘𝑌))
Theorem	lsatdim 32314	A line, spanned by a nonzero singleton, has dimension 1. (Contributed by Thierry Arnoux, 20-May-2023.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ 𝑌 = (𝑊 ↾s (𝑁‘{𝑋}))    ⇒   ⊢ ((𝑊 ∈ LVec ∧ 𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 ) → (dim‘𝑌) = 1)
Theorem	drngdimgt0 32315	The dimension of a vector space that is also a division ring is greater than zero. (Contributed by Thierry Arnoux, 29-Jul-2023.)
⊢ ((𝐹 ∈ LVec ∧ 𝐹 ∈ DivRing) → 0 < (dim‘𝐹))
Theorem	lmhmlvec2 32316	A homomorphism of left vector spaces has a left vector space as codomain. (Contributed by Thierry Arnoux, 7-May-2023.)
⊢ ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → 𝑈 ∈ LVec)
Theorem	kerlmhm 32317	The kernel of a vector space homomorphism is a vector space itself. (Contributed by Thierry Arnoux, 7-May-2023.)
⊢ 0 = (0g‘𝑈)    &   ⊢ 𝐾 = (𝑉 ↾s (◡𝐹 “ { 0 }))    ⇒   ⊢ ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → 𝐾 ∈ LVec)
Theorem	imlmhm 32318	The image of a vector space homomorphism is a vector space itself. (Contributed by Thierry Arnoux, 7-May-2023.)
⊢ 𝐼 = (𝑈 ↾s ran 𝐹)    ⇒   ⊢ ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → 𝐼 ∈ LVec)
Theorem	lindsunlem 32319	Lemma for lindsun 32320. (Contributed by Thierry Arnoux, 9-May-2023.)
⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝑈 ∈ (LIndS‘𝑊))    &   ⊢ (𝜑 → 𝑉 ∈ (LIndS‘𝑊))    &   ⊢ (𝜑 → ((𝑁‘𝑈) ∩ (𝑁‘𝑉)) = { 0 })    &   ⊢ 𝑂 = (0g‘(Scalar‘𝑊))    &   ⊢ 𝐹 = (Base‘(Scalar‘𝑊))    &   ⊢ (𝜑 → 𝐶 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐾 ∈ (𝐹 ∖ {𝑂}))    &   ⊢ (𝜑 → (𝐾( ·𝑠 ‘𝑊)𝐶) ∈ (𝑁‘((𝑈 ∪ 𝑉) ∖ {𝐶})))    ⇒   ⊢ (𝜑 → ⊥)
Theorem	lindsun 32320	Condition for the union of two independent sets to be an independent set. (Contributed by Thierry Arnoux, 9-May-2023.)
⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝑈 ∈ (LIndS‘𝑊))    &   ⊢ (𝜑 → 𝑉 ∈ (LIndS‘𝑊))    &   ⊢ (𝜑 → ((𝑁‘𝑈) ∩ (𝑁‘𝑉)) = { 0 })    ⇒   ⊢ (𝜑 → (𝑈 ∪ 𝑉) ∈ (LIndS‘𝑊))
Theorem	lbsdiflsp0 32321	The linear spans of two disjunct independent sets only have a trivial intersection. This can be seen as the opposite direction of lindsun 32320. (Contributed by Thierry Arnoux, 17-May-2023.)
⊢ 𝐽 = (LBasis‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽 ∧ 𝑉 ⊆ 𝐵) → ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉)) = { 0 })
Theorem	dimkerim 32322	Given a linear map 𝐹 between vector spaces 𝑉 and 𝑈, the dimension of the vector space 𝑉 is the sum of the dimension of 𝐹 's kernel and the dimension of 𝐹's image. Second part of theorem 5.3 in [Lang] p. 141 This can also be described as the Rank-nullity theorem, (dim‘𝐼) being the rank of 𝐹 (the dimension of its image), and (dim‘𝐾) its nullity (the dimension of its kernel). (Contributed by Thierry Arnoux, 17-May-2023.)
⊢ 0 = (0g‘𝑈)    &   ⊢ 𝐾 = (𝑉 ↾s (◡𝐹 “ { 0 }))    &   ⊢ 𝐼 = (𝑈 ↾s ran 𝐹)    ⇒   ⊢ ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → (dim‘𝑉) = ((dim‘𝐾) +𝑒 (dim‘𝐼)))
Theorem	qusdimsum 32323	Let 𝑊 be a vector space, and let 𝑋 be a subspace. Then the dimension of 𝑊 is the sum of the dimension of 𝑋 and the dimension of the quotient space of 𝑋. First part of theorem 5.3 in [Lang] p. 141. (Contributed by Thierry Arnoux, 20-May-2023.)
⊢ 𝑋 = (𝑊 ↾s 𝑈)    &   ⊢ 𝑌 = (𝑊 /s (𝑊 ~QG 𝑈))    ⇒   ⊢ ((𝑊 ∈ LVec ∧ 𝑈 ∈ (LSubSp‘𝑊)) → (dim‘𝑊) = ((dim‘𝑋) +𝑒 (dim‘𝑌)))
Theorem	fedgmullem1 32324*	Lemma for fedgmul 32326. (Contributed by Thierry Arnoux, 20-Jul-2023.)
⊢ 𝐴 = ((subringAlg ‘𝐸)‘𝑉)    &   ⊢ 𝐵 = ((subringAlg ‘𝐸)‘𝑈)    &   ⊢ 𝐶 = ((subringAlg ‘𝐹)‘𝑉)    &   ⊢ 𝐹 = (𝐸 ↾s 𝑈)    &   ⊢ 𝐾 = (𝐸 ↾s 𝑉)    &   ⊢ (𝜑 → 𝐸 ∈ DivRing)    &   ⊢ (𝜑 → 𝐹 ∈ DivRing)    &   ⊢ (𝜑 → 𝐾 ∈ DivRing)    &   ⊢ (𝜑 → 𝑈 ∈ (SubRing‘𝐸))    &   ⊢ (𝜑 → 𝑉 ∈ (SubRing‘𝐹))    &   ⊢ 𝐷 = (𝑗 ∈ 𝑌, 𝑖 ∈ 𝑋 ↦ (𝑖(.r‘𝐸)𝑗))    &   ⊢ 𝐻 = (𝑗 ∈ 𝑌, 𝑖 ∈ 𝑋 ↦ ((𝐺‘𝑗)‘𝑖))    &   ⊢ (𝜑 → 𝑋 ∈ (LBasis‘𝐶))    &   ⊢ (𝜑 → 𝑌 ∈ (LBasis‘𝐵))    &   ⊢ (𝜑 → 𝑍 ∈ (Base‘𝐴))    &   ⊢ (𝜑 → 𝐿:𝑌⟶(Base‘(Scalar‘𝐵)))    &   ⊢ (𝜑 → 𝐿 finSupp (0g‘(Scalar‘𝐵)))    &   ⊢ (𝜑 → 𝑍 = (𝐵 Σg (𝑗 ∈ 𝑌 ↦ ((𝐿‘𝑗)( ·𝑠 ‘𝐵)𝑗))))    &   ⊢ (𝜑 → 𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑m 𝑋))    &   ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝐺‘𝑗) finSupp (0g‘(Scalar‘𝐶)))    &   ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝐿‘𝑗) = (𝐶 Σg (𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)( ·𝑠 ‘𝐶)𝑖))))    ⇒   ⊢ (𝜑 → (𝐻 finSupp (0g‘(Scalar‘𝐴)) ∧ 𝑍 = (𝐴 Σg (𝐻 ∘f ( ·𝑠 ‘𝐴)𝐷))))
Theorem	fedgmullem2 32325*	Lemma for fedgmul 32326. (Contributed by Thierry Arnoux, 20-Jul-2023.)
⊢ 𝐴 = ((subringAlg ‘𝐸)‘𝑉)    &   ⊢ 𝐵 = ((subringAlg ‘𝐸)‘𝑈)    &   ⊢ 𝐶 = ((subringAlg ‘𝐹)‘𝑉)    &   ⊢ 𝐹 = (𝐸 ↾s 𝑈)    &   ⊢ 𝐾 = (𝐸 ↾s 𝑉)    &   ⊢ (𝜑 → 𝐸 ∈ DivRing)    &   ⊢ (𝜑 → 𝐹 ∈ DivRing)    &   ⊢ (𝜑 → 𝐾 ∈ DivRing)    &   ⊢ (𝜑 → 𝑈 ∈ (SubRing‘𝐸))    &   ⊢ (𝜑 → 𝑉 ∈ (SubRing‘𝐹))    &   ⊢ 𝐷 = (𝑗 ∈ 𝑌, 𝑖 ∈ 𝑋 ↦ (𝑖(.r‘𝐸)𝑗))    &   ⊢ 𝐻 = (𝑗 ∈ 𝑌, 𝑖 ∈ 𝑋 ↦ ((𝐺‘𝑗)‘𝑖))    &   ⊢ (𝜑 → 𝑋 ∈ (LBasis‘𝐶))    &   ⊢ (𝜑 → 𝑌 ∈ (LBasis‘𝐵))    &   ⊢ (𝜑 → 𝑊 ∈ (Base‘((Scalar‘𝐴) freeLMod (𝑌 × 𝑋))))    &   ⊢ (𝜑 → (𝐴 Σg (𝑊 ∘f ( ·𝑠 ‘𝐴)𝐷)) = (0g‘𝐴))    ⇒   ⊢ (𝜑 → 𝑊 = ((𝑌 × 𝑋) × {(0g‘(Scalar‘𝐴))}))
Theorem	fedgmul 32326	The multiplicativity formula for degrees of field extensions. Given 𝐸 a field extension of 𝐹, itself a field extension of 𝐾, we have [𝐸:𝐾] = [𝐸:𝐹][𝐹:𝐾]. Proposition 1.2 of [Lang], p. 224. Here (dim‘𝐴) is the degree of the extension 𝐸 of 𝐾, (dim‘𝐵) is the degree of the extension 𝐸 of 𝐹, and (dim‘𝐶) is the degree of the extension 𝐹 of 𝐾. This proof is valid for infinite dimensions, and is actually valid for division ring extensions, not just field extensions. (Contributed by Thierry Arnoux, 25-Jul-2023.)
⊢ 𝐴 = ((subringAlg ‘𝐸)‘𝑉)    &   ⊢ 𝐵 = ((subringAlg ‘𝐸)‘𝑈)    &   ⊢ 𝐶 = ((subringAlg ‘𝐹)‘𝑉)    &   ⊢ 𝐹 = (𝐸 ↾s 𝑈)    &   ⊢ 𝐾 = (𝐸 ↾s 𝑉)    &   ⊢ (𝜑 → 𝐸 ∈ DivRing)    &   ⊢ (𝜑 → 𝐹 ∈ DivRing)    &   ⊢ (𝜑 → 𝐾 ∈ DivRing)    &   ⊢ (𝜑 → 𝑈 ∈ (SubRing‘𝐸))    &   ⊢ (𝜑 → 𝑉 ∈ (SubRing‘𝐹))    ⇒   ⊢ (𝜑 → (dim‘𝐴) = ((dim‘𝐵) ·e (dim‘𝐶)))
21.3.10  Field Extensions
Syntax	cfldext 32327	Syntax for the field extension relation.
class /FldExt
Syntax	cfinext 32328	Syntax for the finite field extension relation.
class /FinExt
Syntax	calgext 32329	Syntax for the algebraic field extension relation.
class /AlgExt
Syntax	cextdg 32330	Syntax for the field extension degree operation.
class [:]
Definition	df-fldext 32331*	Definition of the field extension relation. (Contributed by Thierry Arnoux, 29-Jul-2023.)
⊢ /FldExt = {⟨𝑒, 𝑓⟩ ∣ ((𝑒 ∈ Field ∧ 𝑓 ∈ Field) ∧ (𝑓 = (𝑒 ↾s (Base‘𝑓)) ∧ (Base‘𝑓) ∈ (SubRing‘𝑒)))}
Definition	df-extdg 32332*	Definition of the field extension degree operation. (Contributed by Thierry Arnoux, 29-Jul-2023.)
⊢ [:] = (𝑒 ∈ V, 𝑓 ∈ (/FldExt “ {𝑒}) ↦ (dim‘((subringAlg ‘𝑒)‘(Base‘𝑓))))
Definition	df-finext 32333*	Definition of the finite field extension relation. (Contributed by Thierry Arnoux, 29-Jul-2023.)
⊢ /FinExt = {⟨𝑒, 𝑓⟩ ∣ (𝑒/FldExt𝑓 ∧ (𝑒[:]𝑓) ∈ ℕ0)}
Definition	df-algext 32334*	Definition of the algebraic extension relation. (Contributed by Thierry Arnoux, 29-Jul-2023.)
⊢ /AlgExt = {⟨𝑒, 𝑓⟩ ∣ (𝑒/FldExt𝑓 ∧ ∀𝑥 ∈ (Base‘𝑒)∃𝑝 ∈ (Poly1‘𝑓)(((eval1‘𝑓)‘𝑝)‘𝑥) = (0g‘𝑒))}
Theorem	relfldext 32335	The field extension is a relation. (Contributed by Thierry Arnoux, 29-Jul-2023.)
⊢ Rel /FldExt
Theorem	brfldext 32336	The field extension relation explicited. (Contributed by Thierry Arnoux, 29-Jul-2023.)
⊢ ((𝐸 ∈ Field ∧ 𝐹 ∈ Field) → (𝐸/FldExt𝐹 ↔ (𝐹 = (𝐸 ↾s (Base‘𝐹)) ∧ (Base‘𝐹) ∈ (SubRing‘𝐸))))
Theorem	ccfldextrr 32337	The field of the complex numbers is an extension of the field of the real numbers. (Contributed by Thierry Arnoux, 20-Jul-2023.)
⊢ ℂfld/FldExtℝfld
Theorem	fldextfld1 32338	A field extension is only defined if the extension is a field. (Contributed by Thierry Arnoux, 29-Jul-2023.)
⊢ (𝐸/FldExt𝐹 → 𝐸 ∈ Field)
Theorem	fldextfld2 32339	A field extension is only defined if the subfield is a field. (Contributed by Thierry Arnoux, 29-Jul-2023.)
⊢ (𝐸/FldExt𝐹 → 𝐹 ∈ Field)
Theorem	fldextsubrg 32340	Field extension implies a subring relation. (Contributed by Thierry Arnoux, 29-Jul-2023.)
⊢ 𝑈 = (Base‘𝐹)    ⇒   ⊢ (𝐸/FldExt𝐹 → 𝑈 ∈ (SubRing‘𝐸))
Theorem	fldextress 32341	Field extension implies a structure restriction relation. (Contributed by Thierry Arnoux, 29-Jul-2023.)
⊢ (𝐸/FldExt𝐹 → 𝐹 = (𝐸 ↾s (Base‘𝐹)))
Theorem	brfinext 32342	The finite field extension relation explicited. (Contributed by Thierry Arnoux, 29-Jul-2023.)
⊢ (𝐸/FldExt𝐹 → (𝐸/FinExt𝐹 ↔ (𝐸[:]𝐹) ∈ ℕ0))
Theorem	extdgval 32343	Value of the field extension degree operation. (Contributed by Thierry Arnoux, 29-Jul-2023.)
⊢ (𝐸/FldExt𝐹 → (𝐸[:]𝐹) = (dim‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
Theorem	fldextsralvec 32344	The subring algebra associated with a field extension is a vector space. (Contributed by Thierry Arnoux, 3-Aug-2023.)
⊢ (𝐸/FldExt𝐹 → ((subringAlg ‘𝐸)‘(Base‘𝐹)) ∈ LVec)
Theorem	extdgcl 32345	Closure of the field extension degree operation. (Contributed by Thierry Arnoux, 29-Jul-2023.)
⊢ (𝐸/FldExt𝐹 → (𝐸[:]𝐹) ∈ ℕ0*)
Theorem	extdggt0 32346	Degrees of field extension are greater than zero. (Contributed by Thierry Arnoux, 30-Jul-2023.)
⊢ (𝐸/FldExt𝐹 → 0 < (𝐸[:]𝐹))
Theorem	fldexttr 32347	Field extension is a transitive relation. (Contributed by Thierry Arnoux, 29-Jul-2023.)
⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → 𝐸/FldExt𝐾)
Theorem	fldextid 32348	The field extension relation is reflexive. (Contributed by Thierry Arnoux, 30-Jul-2023.)
⊢ (𝐹 ∈ Field → 𝐹/FldExt𝐹)
Theorem	extdgid 32349	A trivial field extension has degree one. (Contributed by Thierry Arnoux, 4-Aug-2023.)
⊢ (𝐸 ∈ Field → (𝐸[:]𝐸) = 1)
Theorem	extdgmul 32350	The multiplicativity formula for degrees of field extensions. Given 𝐸 a field extension of 𝐹, itself a field extension of 𝐾, the degree of the extension 𝐸/FldExt𝐾 is the product of the degrees of the extensions 𝐸/FldExt𝐹 and 𝐹/FldExt𝐾. Proposition 1.2 of [Lang], p. 224. (Contributed by Thierry Arnoux, 30-Jul-2023.)
⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (𝐸[:]𝐾) = ((𝐸[:]𝐹) ·e (𝐹[:]𝐾)))
Theorem	finexttrb 32351	The extension 𝐸 of 𝐾 is finite if and only if 𝐸 is finite over 𝐹 and 𝐹 is finite over 𝐾. Corollary 1.3 of [Lang] , p. 225. (Contributed by Thierry Arnoux, 30-Jul-2023.)
⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (𝐸/FinExt𝐾 ↔ (𝐸/FinExt𝐹 ∧ 𝐹/FinExt𝐾)))
Theorem	extdg1id 32352	If the degree of the extension 𝐸/FldExt𝐹 is 1, then 𝐸 and 𝐹 are identical. (Contributed by Thierry Arnoux, 6-Aug-2023.)
⊢ ((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) → 𝐸 = 𝐹)
Theorem	extdg1b 32353	The degree of the extension 𝐸/FldExt𝐹 is 1 iff 𝐸 and 𝐹 are the same structure. (Contributed by Thierry Arnoux, 6-Aug-2023.)
⊢ (𝐸/FldExt𝐹 → ((𝐸[:]𝐹) = 1 ↔ 𝐸 = 𝐹))
Theorem	fldextchr 32354	The characteristic of a subfield is the same as the characteristic of the larger field. (Contributed by Thierry Arnoux, 20-Aug-2023.)
⊢ (𝐸/FldExt𝐹 → (chr‘𝐹) = (chr‘𝐸))
Theorem	ccfldsrarelvec 32355	The subring algebra of the complex numbers over the real numbers is a left vector space. (Contributed by Thierry Arnoux, 20-Aug-2023.)
⊢ ((subringAlg ‘ℂfld)‘ℝ) ∈ LVec
Theorem	ccfldextdgrr 32356	The degree of the field extension of the complex numbers over the real numbers is 2. (Suggested by GL, 4-Aug-2023.) (Contributed by Thierry Arnoux, 20-Aug-2023.)
⊢ (ℂfld[:]ℝfld) = 2
21.3.10.1  Algebraic numbers
Syntax	cirng 32357	Integral subring of a ring.
class IntgRing
Definition	df-irng 32358*	Define the subring of elements of a ring 𝑟 integral over a subset 𝑠. (Contributed by Mario Carneiro, 2-Dec-2014.) (Revised by Thierry Arnoux, 28-Jan-2025.)
⊢ IntgRing = (𝑟 ∈ V, 𝑠 ∈ V ↦ ∪ 𝑓 ∈ (Monic1p‘(𝑟 ↾s 𝑠))(◡((𝑟 evalSub1 𝑠)‘𝑓) “ {(0g‘𝑟)}))
Theorem	irngval 32359*	The elements of a field 𝑅 integral over a subset 𝑆. In the case of a subfield, those are the algebraic numbers over the field 𝑆 within the field 𝑅. That is, the numbers 𝑋 which are roots of monic polynomials 𝑃(𝑋) with coefficients in 𝑆. (Contributed by Thierry Arnoux, 28-Jan-2025.)
⊢ 𝑂 = (𝑅 evalSub1 𝑆)    &   ⊢ 𝑈 = (𝑅 ↾s 𝑆)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝑆 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → (𝑅 IntgRing 𝑆) = ∪ 𝑓 ∈ (Monic1p‘𝑈)(◡(𝑂‘𝑓) “ { 0 }))
Theorem	elirng 32360*	Property for an element 𝑋 of a field 𝑅 to be integral over a subring 𝑆. (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 → (𝑋 ∈ (𝑅 IntgRing 𝑆) ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑓 ∈ (Monic1p‘𝑈)((𝑂‘𝑓)‘𝑋) = 0 )))
Theorem	irngss 32361	All elements of a subring 𝑆 are integral over 𝑆. This is only true in the case of a nonzero ring, since there are no integral elements in a zero ring (see 0ringirng 32363). (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 ∈ NzRing)    ⇒   ⊢ (𝜑 → 𝑆 ⊆ (𝑅 IntgRing 𝑆))
Theorem	irngssv 32362	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 32363	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 32364	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 32365*	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 }))
21.3.10.2  Minimal polynomials
Syntax	cminply 32366	Extend class notation with the minimal polynomial builder function.
class minPoly
Definition	df-minply 32367*	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	evls1maprhm 32368*	The function 𝐹 mapping polynomials 𝑝 to their subring evaluation at a given point 𝑋 is a ring homomorphism. (Contributed by Thierry Arnoux, 8-Feb-2025.)
⊢ 𝑂 = (𝑅 evalSub1 𝑆)    &   ⊢ 𝑃 = (Poly1‘(𝑅 ↾s 𝑆))    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑈 = (Base‘𝑃)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑆 ∈ (SubRing‘𝑅))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ 𝐹 = (𝑝 ∈ 𝑈 ↦ ((𝑂‘𝑝)‘𝑋))    ⇒   ⊢ (𝜑 → 𝐹 ∈ (𝑃 RingHom 𝑅))
Theorem	ply1annidllem 32369*	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 32370*	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	ply1annig1p 32371*	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 32372*	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 32371, 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	ply1annprmidl 32373*	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))
21.3.11  Matrices
21.3.11.1  Submatrices
Syntax	csmat 32374	Syntax for a function generating submatrices.
class subMat1
Definition	df-smat 32375*	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 21926. (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 32376*	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 32377	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 32378	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 32379	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 32380	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 32381	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 32382	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 32383	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 32384*	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 32385	The submatrix of the identity matrix obtained by removing the ith row and the ith column is an identity matrix. Cf. 1marepvsma1 21932. (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 32386	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 32387	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 32388	Lemma for submateq 32390. (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 32389	Lemma for submateq 32390. (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 32390*	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 32391	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))
21.3.11.2  Matrix literals
Syntax	clmat 32392	Extend class notation with the literal matrix conversion function.
class litMat
Definition	df-lmat 32393*	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 32394*	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 32395*	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 32396*	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 32397*	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 32398*	Lemma for lmat22e11 32399 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 32399	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 32400	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)