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Theorem List for Metamath Proof Explorer - 30601-30700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdrngdimgt0 30601 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))

Theoremlmhmlvec2 30602 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)

Theoremkerlmhm 30603 The kernel of a vector space homomorphism is a vector space itself. (Contributed by Thierry Arnoux, 7-May-2023.)
0 = (0g𝑈)    &   𝐾 = (𝑉s (𝐹 “ { 0 }))       ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → 𝐾 ∈ LVec)

Theoremimlmhm 30604 The image of a vector space homomorphism is a vector space itself. (Contributed by Thierry Arnoux, 7-May-2023.)
𝐼 = (𝑈s ran 𝐹)       ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → 𝐼 ∈ LVec)

Theoremlindsunlem 30605 Lemma for lindsun 30606. (Contributed by Thierry Arnoux, 9-May-2023.)
𝑁 = (LSpan‘𝑊)    &    0 = (0g𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑈 ∈ (LIndS‘𝑊))    &   (𝜑𝑉 ∈ (LIndS‘𝑊))    &   (𝜑 → ((𝑁𝑈) ∩ (𝑁𝑉)) = { 0 })    &   𝑂 = (0g‘(Scalar‘𝑊))    &   𝐹 = (Base‘(Scalar‘𝑊))    &   (𝜑𝐶𝑈)    &   (𝜑𝐾 ∈ (𝐹 ∖ {𝑂}))    &   (𝜑 → (𝐾( ·𝑠𝑊)𝐶) ∈ (𝑁‘((𝑈𝑉) ∖ {𝐶})))       (𝜑 → ⊥)

Theoremlindsun 30606 Condition for the union of two independent sets to be an independent set. (Contributed by Thierry Arnoux, 9-May-2023.)
𝑁 = (LSpan‘𝑊)    &    0 = (0g𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑈 ∈ (LIndS‘𝑊))    &   (𝜑𝑉 ∈ (LIndS‘𝑊))    &   (𝜑 → ((𝑁𝑈) ∩ (𝑁𝑉)) = { 0 })       (𝜑 → (𝑈𝑉) ∈ (LIndS‘𝑊))

Theoremlbsdiflsp0 30607 The linear spans of two disjunct independent sets only have a trivial intersection. This can be seen as the opposite direction of lindsun 30606. (Contributed by Thierry Arnoux, 17-May-2023.)
𝐽 = (LBasis‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &    0 = (0g𝑊)       ((𝑊 ∈ LVec ∧ 𝐵𝐽𝑉𝐵) → ((𝑁‘(𝐵𝑉)) ∩ (𝑁𝑉)) = { 0 })

Theoremdimkerim 30608 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 = (0g𝑈)    &   𝐾 = (𝑉s (𝐹 “ { 0 }))    &   𝐼 = (𝑈s ran 𝐹)       ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → (dim‘𝑉) = ((dim‘𝐾) +𝑒 (dim‘𝐼)))

Theoremqusdimsum 30609 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 = (𝑊s 𝑈)    &   𝑌 = (𝑊 /s (𝑊 ~QG 𝑈))       ((𝑊 ∈ LVec ∧ 𝑈 ∈ (LSubSp‘𝑊)) → (dim‘𝑊) = ((dim‘𝑋) +𝑒 (dim‘𝑌)))

Theoremfedgmullem1 30610* Lemma for fedgmul 30612 (Contributed by Thierry Arnoux, 20-Jul-2023.)
𝐴 = ((subringAlg ‘𝐸)‘𝑉)    &   𝐵 = ((subringAlg ‘𝐸)‘𝑈)    &   𝐶 = ((subringAlg ‘𝐹)‘𝑉)    &   𝐹 = (𝐸s 𝑈)    &   𝐾 = (𝐸s 𝑉)    &   (𝜑𝐸 ∈ DivRing)    &   (𝜑𝐹 ∈ DivRing)    &   (𝜑𝐾 ∈ DivRing)    &   (𝜑𝑈 ∈ (SubRing‘𝐸))    &   (𝜑𝑉 ∈ (SubRing‘𝐹))    &   𝐷 = (𝑗𝑌, 𝑖𝑋 ↦ (𝑖(.r𝐸)𝑗))    &   𝐻 = (𝑗𝑌, 𝑖𝑋 ↦ ((𝐺𝑗)‘𝑖))    &   (𝜑𝑋 ∈ (LBasis‘𝐶))    &   (𝜑𝑌 ∈ (LBasis‘𝐵))    &   (𝜑𝑍 ∈ (Base‘𝐴))    &   (𝜑𝐿:𝑌⟶(Base‘(Scalar‘𝐵)))    &   (𝜑𝐿 finSupp (0g‘(Scalar‘𝐵)))    &   (𝜑𝑍 = (𝐵 Σg (𝑗𝑌 ↦ ((𝐿𝑗)( ·𝑠𝐵)𝑗))))    &   (𝜑𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑𝑚 𝑋))    &   ((𝜑𝑗𝑌) → (𝐺𝑗) finSupp (0g‘(Scalar‘𝐶)))    &   ((𝜑𝑗𝑌) → (𝐿𝑗) = (𝐶 Σg (𝑖𝑋 ↦ (((𝐺𝑗)‘𝑖)( ·𝑠𝐶)𝑖))))       (𝜑 → (𝐻 finSupp (0g‘(Scalar‘𝐴)) ∧ 𝑍 = (𝐴 Σg (𝐻𝑓 ( ·𝑠𝐴)𝐷))))

Theoremfedgmullem2 30611* Lemma for fedgmul 30612 (Contributed by Thierry Arnoux, 20-Jul-2023.)
𝐴 = ((subringAlg ‘𝐸)‘𝑉)    &   𝐵 = ((subringAlg ‘𝐸)‘𝑈)    &   𝐶 = ((subringAlg ‘𝐹)‘𝑉)    &   𝐹 = (𝐸s 𝑈)    &   𝐾 = (𝐸s 𝑉)    &   (𝜑𝐸 ∈ DivRing)    &   (𝜑𝐹 ∈ DivRing)    &   (𝜑𝐾 ∈ DivRing)    &   (𝜑𝑈 ∈ (SubRing‘𝐸))    &   (𝜑𝑉 ∈ (SubRing‘𝐹))    &   𝐷 = (𝑗𝑌, 𝑖𝑋 ↦ (𝑖(.r𝐸)𝑗))    &   𝐻 = (𝑗𝑌, 𝑖𝑋 ↦ ((𝐺𝑗)‘𝑖))    &   (𝜑𝑋 ∈ (LBasis‘𝐶))    &   (𝜑𝑌 ∈ (LBasis‘𝐵))    &   (𝜑𝑊 ∈ (Base‘((Scalar‘𝐴) freeLMod (𝑌 × 𝑋))))    &   (𝜑 → (𝐴 Σg (𝑊𝑓 ( ·𝑠𝐴)𝐷)) = (0g𝐴))       (𝜑𝑊 = ((𝑌 × 𝑋) × {(0g‘(Scalar‘𝐴))}))

Theoremfedgmul 30612 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 = (𝐸s 𝑈)    &   𝐾 = (𝐸s 𝑉)    &   (𝜑𝐸 ∈ DivRing)    &   (𝜑𝐹 ∈ DivRing)    &   (𝜑𝐾 ∈ DivRing)    &   (𝜑𝑈 ∈ (SubRing‘𝐸))    &   (𝜑𝑉 ∈ (SubRing‘𝐹))       (𝜑 → (dim‘𝐴) = ((dim‘𝐵) ·e (dim‘𝐶)))

20.3.9  Field Extensions

Syntaxcfldext 30613 Syntax for the field extension relation.
class /FldExt

Syntaxcfinext 30614 Syntax for the finite field extension relation.
class /FinExt

Syntaxcalgext 30615 Syntax for the algebraic field extension relation.
class /AlgExt

Syntaxcextdg 30616 Syntax for the field extension degree operation.
class [:]

Definitiondf-fldext 30617* Definition of the field extension relation. (Contributed by Thierry Arnoux, 29-Jul-2023.)
/FldExt = {⟨𝑒, 𝑓⟩ ∣ ((𝑒 ∈ Field ∧ 𝑓 ∈ Field) ∧ (𝑓 = (𝑒s (Base‘𝑓)) ∧ (Base‘𝑓) ∈ (SubRing‘𝑒)))}

Definitiondf-extdg 30618* Definition of the field extension degree operation. (Contributed by Thierry Arnoux, 29-Jul-2023.)
[:] = (𝑒 ∈ V, 𝑓 ∈ (/FldExt “ {𝑒}) ↦ (dim‘((subringAlg ‘𝑒)‘(Base‘𝑓))))

Definitiondf-finext 30619* Definition of the finite field extension relation. (Contributed by Thierry Arnoux, 29-Jul-2023.)
/FinExt = {⟨𝑒, 𝑓⟩ ∣ (𝑒/FldExt𝑓 ∧ (𝑒[:]𝑓) ∈ ℕ0)}

Definitiondf-algext 30620* Definition of the algebraic extension relation. (Contributed by Thierry Arnoux, 29-Jul-2023.)
/AlgExt = {⟨𝑒, 𝑓⟩ ∣ (𝑒/FldExt𝑓 ∧ ∀𝑥 ∈ (Base‘𝑒)∃𝑝 ∈ (Poly1𝑓)(((eval1𝑓)‘𝑝)‘𝑥) = (0g𝑒))}

Theoremrelfldext 30621 The field extension is a relation. (Contributed by Thierry Arnoux, 29-Jul-2023.)
Rel /FldExt

Theorembrfldext 30622 The field extension relation explicited. (Contributed by Thierry Arnoux, 29-Jul-2023.)
((𝐸 ∈ Field ∧ 𝐹 ∈ Field) → (𝐸/FldExt𝐹 ↔ (𝐹 = (𝐸s (Base‘𝐹)) ∧ (Base‘𝐹) ∈ (SubRing‘𝐸))))

Theoremccfldextrr 30623 The field of the complex numbers is an extension of the field of the real numbers. (Contributed by Thierry Arnoux, 20-Jul-2023.)
fld/FldExtfld

Theoremfldextfld1 30624 A field extension is only defined if the extension is a field. (Contributed by Thierry Arnoux, 29-Jul-2023.)
(𝐸/FldExt𝐹𝐸 ∈ Field)

Theoremfldextfld2 30625 A field extension is only defined if the subfield is a field. (Contributed by Thierry Arnoux, 29-Jul-2023.)
(𝐸/FldExt𝐹𝐹 ∈ Field)

Theoremfldextsubrg 30626 Field extension implies a subring relation. (Contributed by Thierry Arnoux, 29-Jul-2023.)
𝑈 = (Base‘𝐹)       (𝐸/FldExt𝐹𝑈 ∈ (SubRing‘𝐸))

Theoremfldextress 30627 Field extension implies a structure restriction relation. (Contributed by Thierry Arnoux, 29-Jul-2023.)
(𝐸/FldExt𝐹𝐹 = (𝐸s (Base‘𝐹)))

Theorembrfinext 30628 The finite field extension relation explicited. (Contributed by Thierry Arnoux, 29-Jul-2023.)
(𝐸/FldExt𝐹 → (𝐸/FinExt𝐹 ↔ (𝐸[:]𝐹) ∈ ℕ0))

Theoremextdgval 30629 Value of the field extension degree operation. (Contributed by Thierry Arnoux, 29-Jul-2023.)
(𝐸/FldExt𝐹 → (𝐸[:]𝐹) = (dim‘((subringAlg ‘𝐸)‘(Base‘𝐹))))

Theoremfldextsralvec 30630 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)

Theoremextdgcl 30631 Closure of the field extension degree operation. (Contributed by Thierry Arnoux, 29-Jul-2023.)
(𝐸/FldExt𝐹 → (𝐸[:]𝐹) ∈ ℕ0*)

Theoremextdggt0 30632 Degrees of field extension are greater than zero. (Contributed by Thierry Arnoux, 30-Jul-2023.)
(𝐸/FldExt𝐹 → 0 < (𝐸[:]𝐹))

Theoremfldexttr 30633 Field extension is a transitive relation. (Contributed by Thierry Arnoux, 29-Jul-2023.)
((𝐸/FldExt𝐹𝐹/FldExt𝐾) → 𝐸/FldExt𝐾)

Theoremfldextid 30634 The field extension relation is reflexive. (Contributed by Thierry Arnoux, 30-Jul-2023.)
(𝐹 ∈ Field → 𝐹/FldExt𝐹)

Theoremextdgid 30635 A trivial field extension has degree one. (Contributed by Thierry Arnoux, 4-Aug-2023.)
(𝐸 ∈ Field → (𝐸[:]𝐸) = 1)

Theoremextdgmul 30636 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/FldExt𝐹𝐹/FldExt𝐾) → (𝐸[:]𝐾) = ((𝐸[:]𝐹) ·e (𝐹[:]𝐾)))

Theoremfinexttrb 30637 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/FldExt𝐹𝐹/FldExt𝐾) → (𝐸/FinExt𝐾 ↔ (𝐸/FinExt𝐹𝐹/FinExt𝐾)))

Theoremextdg1id 30638 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)

Theoremextdg1b 30639 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))

Theoremfldextchr 30640 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))

Theoremccfldsrarelvec 30641 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

Theoremccfldextdgrr 30642 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

20.3.10  Matrices

20.3.10.1  The symmetric group

Theoremsymgfcoeu 30643* Uniqueness property of permutations. (Contributed by Thierry Arnoux, 22-Aug-2020.)
𝐺 = (Base‘(SymGrp‘𝐷))       ((𝐷𝑉𝑃𝐺𝑄𝐺) → ∃!𝑝𝐺 𝑄 = (𝑃𝑝))

20.3.10.2  Permutation Signs

Theorempsgndmfi 30644 For a finite base set, the permutation sign is defined for all permutations. (Contributed by Thierry Arnoux, 22-Aug-2020.)
𝑆 = (pmSgn‘𝐷)    &   𝐺 = (Base‘(SymGrp‘𝐷))       (𝐷 ∈ Fin → 𝑆 Fn 𝐺)

Theorempsgnid 30645 Permutation sign of the identity. (Contributed by Thierry Arnoux, 21-Aug-2020.)
𝑆 = (pmSgn‘𝐷)       (𝐷 ∈ Fin → (𝑆‘( I ↾ 𝐷)) = 1)

Theorempmtrprfv2 30646 In a transposition of two given points, each maps to the other. (Contributed by Thierry Arnoux, 22-Aug-2020.)
𝑇 = (pmTrsp‘𝐷)       ((𝐷𝑉 ∧ (𝑋𝐷𝑌𝐷𝑋𝑌)) → ((𝑇‘{𝑋, 𝑌})‘𝑌) = 𝑋)

Theorempmtrto1cl 30647 Useful lemma for the following theorems. (Contributed by Thierry Arnoux, 21-Aug-2020.)
𝐷 = (1...𝑁)    &   𝑇 = (pmTrsp‘𝐷)       ((𝐾 ∈ ℕ ∧ (𝐾 + 1) ∈ 𝐷) → (𝑇‘{𝐾, (𝐾 + 1)}) ∈ ran 𝑇)

Theorempsgnfzto1stlem 30648* Lemma for psgnfzto1st 30653. Our permutation of rank (𝑛 + 1) can be written as a permutation of rank 𝑛 composed with a transposition. (Contributed by Thierry Arnoux, 21-Aug-2020.)
𝐷 = (1...𝑁)       ((𝐾 ∈ ℕ ∧ (𝐾 + 1) ∈ 𝐷) → (𝑖𝐷 ↦ if(𝑖 = 1, (𝐾 + 1), if(𝑖 ≤ (𝐾 + 1), (𝑖 − 1), 𝑖))) = (((pmTrsp‘𝐷)‘{𝐾, (𝐾 + 1)}) ∘ (𝑖𝐷 ↦ if(𝑖 = 1, 𝐾, if(𝑖𝐾, (𝑖 − 1), 𝑖)))))

Theoremfzto1stfv1 30649* Value of our permutation 𝑃 at 1. (Contributed by Thierry Arnoux, 23-Aug-2020.)
𝐷 = (1...𝑁)    &   𝑃 = (𝑖𝐷 ↦ if(𝑖 = 1, 𝐼, if(𝑖𝐼, (𝑖 − 1), 𝑖)))       (𝐼𝐷 → (𝑃‘1) = 𝐼)

Theoremfzto1st1 30650* Special case where the permutation defined in psgnfzto1st 30653 is the identity. (Contributed by Thierry Arnoux, 21-Aug-2020.)
𝐷 = (1...𝑁)    &   𝑃 = (𝑖𝐷 ↦ if(𝑖 = 1, 𝐼, if(𝑖𝐼, (𝑖 − 1), 𝑖)))       (𝐼 = 1 → 𝑃 = ( I ↾ 𝐷))

Theoremfzto1st 30651* The function moving one element to the first position (and shifting all elements before it) is a permutation. (Contributed by Thierry Arnoux, 21-Aug-2020.)
𝐷 = (1...𝑁)    &   𝑃 = (𝑖𝐷 ↦ if(𝑖 = 1, 𝐼, if(𝑖𝐼, (𝑖 − 1), 𝑖)))    &   𝐺 = (SymGrp‘𝐷)    &   𝐵 = (Base‘𝐺)       (𝐼𝐷𝑃𝐵)

Theoremfzto1stinvn 30652* Value of the inverse of our permutation 𝑃 at 𝐼 (Contributed by Thierry Arnoux, 23-Aug-2020.)
𝐷 = (1...𝑁)    &   𝑃 = (𝑖𝐷 ↦ if(𝑖 = 1, 𝐼, if(𝑖𝐼, (𝑖 − 1), 𝑖)))    &   𝐺 = (SymGrp‘𝐷)    &   𝐵 = (Base‘𝐺)       (𝐼𝐷 → (𝑃𝐼) = 1)

Theorempsgnfzto1st 30653* The permutation sign for moving one element to the first position. (Contributed by Thierry Arnoux, 21-Aug-2020.)
𝐷 = (1...𝑁)    &   𝑃 = (𝑖𝐷 ↦ if(𝑖 = 1, 𝐼, if(𝑖𝐼, (𝑖 − 1), 𝑖)))    &   𝐺 = (SymGrp‘𝐷)    &   𝐵 = (Base‘𝐺)    &   𝑆 = (pmSgn‘𝐷)       (𝐼𝐷 → (𝑆𝑃) = (-1↑(𝐼 + 1)))

20.3.10.3  Transpositions

Theorempmtridf1o 30654 Transpositions of 𝑋 and 𝑌 (understood to be the identity when 𝑋 = 𝑌), are bijections. (Contributed by Thierry Arnoux, 1-Jan-2022.)
(𝜑𝐴𝑉)    &   (𝜑𝑋𝐴)    &   (𝜑𝑌𝐴)    &   𝑇 = if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌}))       (𝜑𝑇:𝐴1-1-onto𝐴)

Theorempmtridfv1 30655 Value at X of the transposition of 𝑋 and 𝑌 (understood to be the identity when X = Y ). (Contributed by Thierry Arnoux, 3-Jan-2022.)
(𝜑𝐴𝑉)    &   (𝜑𝑋𝐴)    &   (𝜑𝑌𝐴)    &   𝑇 = if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌}))       (𝜑 → (𝑇𝑋) = 𝑌)

Theorempmtridfv2 30656 Value at Y of the transposition of 𝑋 and 𝑌 (understood to be the identity when X = Y ). (Contributed by Thierry Arnoux, 3-Jan-2022.)
(𝜑𝐴𝑉)    &   (𝜑𝑋𝐴)    &   (𝜑𝑌𝐴)    &   𝑇 = if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌}))       (𝜑 → (𝑇𝑌) = 𝑋)

20.3.10.4  Submatrices

Syntaxcsmat 30657 Syntax for a function generating submatrices.
class subMat1

Definitiondf-smat 30658* 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 20880. (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))⟩))))

Theoremsmatfval 30659* Value of the submatrix. (Contributed by Thierry Arnoux, 19-Aug-2020.)
((𝐾 ∈ ℕ ∧ 𝐿 ∈ ℕ ∧ 𝑀𝑉) → (𝐾(subMat1‘𝑀)𝐿) = (𝑀 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)))

Theoremsmatrcl 30660 Closure of the rectangular submatrix. (Contributed by Thierry Arnoux, 19-Aug-2020.)
𝑆 = (𝐾(subMat1‘𝐴)𝐿)    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐾 ∈ (1...𝑀))    &   (𝜑𝐿 ∈ (1...𝑁))    &   (𝜑𝐴 ∈ (𝐵𝑚 ((1...𝑀) × (1...𝑁))))       (𝜑𝑆 ∈ (𝐵𝑚 ((1...(𝑀 − 1)) × (1...(𝑁 − 1)))))

Theoremsmatlem 30661 Lemma for the next theorems. (Contributed by Thierry Arnoux, 19-Aug-2020.)
𝑆 = (𝐾(subMat1‘𝐴)𝐿)    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐾 ∈ (1...𝑀))    &   (𝜑𝐿 ∈ (1...𝑁))    &   (𝜑𝐴 ∈ (𝐵𝑚 ((1...𝑀) × (1...𝑁))))    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐽 ∈ ℕ)    &   (𝜑 → if(𝐼 < 𝐾, 𝐼, (𝐼 + 1)) = 𝑋)    &   (𝜑 → if(𝐽 < 𝐿, 𝐽, (𝐽 + 1)) = 𝑌)       (𝜑 → (𝐼𝑆𝐽) = (𝑋𝐴𝑌))

Theoremsmattl 30662 Entries of a submatrix, top left. (Contributed by Thierry Arnoux, 19-Aug-2020.)
𝑆 = (𝐾(subMat1‘𝐴)𝐿)    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐾 ∈ (1...𝑀))    &   (𝜑𝐿 ∈ (1...𝑁))    &   (𝜑𝐴 ∈ (𝐵𝑚 ((1...𝑀) × (1...𝑁))))    &   (𝜑𝐼 ∈ (1..^𝐾))    &   (𝜑𝐽 ∈ (1..^𝐿))       (𝜑 → (𝐼𝑆𝐽) = (𝐼𝐴𝐽))

Theoremsmattr 30663 Entries of a submatrix, top right. (Contributed by Thierry Arnoux, 19-Aug-2020.)
𝑆 = (𝐾(subMat1‘𝐴)𝐿)    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐾 ∈ (1...𝑀))    &   (𝜑𝐿 ∈ (1...𝑁))    &   (𝜑𝐴 ∈ (𝐵𝑚 ((1...𝑀) × (1...𝑁))))    &   (𝜑𝐼 ∈ (𝐾...𝑀))    &   (𝜑𝐽 ∈ (1..^𝐿))       (𝜑 → (𝐼𝑆𝐽) = ((𝐼 + 1)𝐴𝐽))

Theoremsmatbl 30664 Entries of a submatrix, bottom left. (Contributed by Thierry Arnoux, 19-Aug-2020.)
𝑆 = (𝐾(subMat1‘𝐴)𝐿)    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐾 ∈ (1...𝑀))    &   (𝜑𝐿 ∈ (1...𝑁))    &   (𝜑𝐴 ∈ (𝐵𝑚 ((1...𝑀) × (1...𝑁))))    &   (𝜑𝐼 ∈ (1..^𝐾))    &   (𝜑𝐽 ∈ (𝐿...𝑁))       (𝜑 → (𝐼𝑆𝐽) = (𝐼𝐴(𝐽 + 1)))

Theoremsmatbr 30665 Entries of a submatrix, bottom right. (Contributed by Thierry Arnoux, 19-Aug-2020.)
𝑆 = (𝐾(subMat1‘𝐴)𝐿)    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐾 ∈ (1...𝑀))    &   (𝜑𝐿 ∈ (1...𝑁))    &   (𝜑𝐴 ∈ (𝐵𝑚 ((1...𝑀) × (1...𝑁))))    &   (𝜑𝐼 ∈ (𝐾...𝑀))    &   (𝜑𝐽 ∈ (𝐿...𝑁))       (𝜑 → (𝐼𝑆𝐽) = ((𝐼 + 1)𝐴(𝐽 + 1)))

Theoremsmatcl 30666 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)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐾 ∈ (1...𝑁))    &   (𝜑𝐿 ∈ (1...𝑁))    &   (𝜑𝑀𝐵)       (𝜑𝑆𝐶)

Theoremmatmpo 30667* Write a square matrix as a mapping operation. (Contributed by Thierry Arnoux, 16-Aug-2020.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)       (𝑀𝐵𝑀 = (𝑖𝑁, 𝑗𝑁 ↦ (𝑖𝑀𝑗)))

Theorem1smat1 30668 The submatrix of the identity matrix obtained by removing the ith row and the ith column is an identity matrix. Cf. 1marepvsma1 20886. (Contributed by Thierry Arnoux, 19-Aug-2020.)
1 = (1r‘((1...𝑁) Mat 𝑅))    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐼 ∈ (1...𝑁))       (𝜑 → (𝐼(subMat1‘ 1 )𝐼) = (1r‘((1...(𝑁 − 1)) Mat 𝑅)))

Theoremsubmat1n 30669 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 ∧ 𝑀𝐵) → (𝑁(subMat1‘𝑀)𝑁) = (𝑁(((1...𝑁) subMat 𝑅)‘𝑀)𝑁))

Theoremsubmatres 30670 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 ∧ 𝑀𝐵) → (𝑁(subMat1‘𝑀)𝑁) = (𝑀 ↾ ((1...(𝑁 − 1)) × (1...(𝑁 − 1)))))

Theoremsubmateqlem1 30671 Lemma for submateq 30673. (Contributed by Thierry Arnoux, 25-Aug-2020.)
(𝜑𝑁 ∈ ℕ)    &   (𝜑𝐾 ∈ (1...𝑁))    &   (𝜑𝑀 ∈ (1...(𝑁 − 1)))    &   (𝜑𝐾𝑀)       (𝜑 → (𝑀 ∈ (𝐾...𝑁) ∧ (𝑀 + 1) ∈ ((1...𝑁) ∖ {𝐾})))

Theoremsubmateqlem2 30672 Lemma for submateq 30673. (Contributed by Thierry Arnoux, 26-Aug-2020.)
(𝜑𝑁 ∈ ℕ)    &   (𝜑𝐾 ∈ (1...𝑁))    &   (𝜑𝑀 ∈ (1...(𝑁 − 1)))    &   (𝜑𝑀 < 𝐾)       (𝜑 → (𝑀 ∈ (1..^𝐾) ∧ 𝑀 ∈ ((1...𝑁) ∖ {𝐾})))

Theoremsubmateq 30673* Sufficient condition for two submatrices to be equal. (Contributed by Thierry Arnoux, 25-Aug-2020.)
𝐴 = ((1...𝑁) Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐼 ∈ (1...𝑁))    &   (𝜑𝐽 ∈ (1...𝑁))    &   (𝜑𝐸𝐵)    &   (𝜑𝐹𝐵)    &   ((𝜑𝑖 ∈ ((1...𝑁) ∖ {𝐼}) ∧ 𝑗 ∈ ((1...𝑁) ∖ {𝐽})) → (𝑖𝐸𝑗) = (𝑖𝐹𝑗))       (𝜑 → (𝐼(subMat1‘𝐸)𝐽) = (𝐼(subMat1‘𝐹)𝐽))

Theoremsubmatminr1 30674 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)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐼 ∈ (1...𝑁))    &   (𝜑𝐽 ∈ (1...𝑁))    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑀𝐵)    &   𝐸 = (𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)       (𝜑 → (𝐼(subMat1‘𝑀)𝐽) = (𝐼(subMat1‘𝐸)𝐽))

20.3.10.5  Matrix literals

Syntaxclmat 30675 Extend class notation with the literal matrix conversion function.
class litMat

Definitiondf-lmat 30676* 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))))

Theoremlmatval 30677* Value of the literal matrix conversion function. (Contributed by Thierry Arnoux, 28-Aug-2020.)
(𝑀𝑉 → (litMat‘𝑀) = (𝑖 ∈ (1...(♯‘𝑀)), 𝑗 ∈ (1...(♯‘(𝑀‘0))) ↦ ((𝑀‘(𝑖 − 1))‘(𝑗 − 1))))

Theoremlmatfval 30678* Entries of a literal matrix. (Contributed by Thierry Arnoux, 28-Aug-2020.)
𝑀 = (litMat‘𝑊)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑊 ∈ Word Word 𝑉)    &   (𝜑 → (♯‘𝑊) = 𝑁)    &   ((𝜑𝑖 ∈ (0..^𝑁)) → (♯‘(𝑊𝑖)) = 𝑁)    &   (𝜑𝐼 ∈ (1...𝑁))    &   (𝜑𝐽 ∈ (1...𝑁))       (𝜑 → (𝐼𝑀𝐽) = ((𝑊‘(𝐼 − 1))‘(𝐽 − 1)))

Theoremlmatfvlem 30679* Useful lemma to extract literal matrix entries. Suggested by Mario Carneiro. (Contributed by Thierry Arnoux, 3-Sep-2020.)
𝑀 = (litMat‘𝑊)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑊 ∈ Word Word 𝑉)    &   (𝜑 → (♯‘𝑊) = 𝑁)    &   ((𝜑𝑖 ∈ (0..^𝑁)) → (♯‘(𝑊𝑖)) = 𝑁)    &   𝐾 ∈ ℕ0    &   𝐿 ∈ ℕ0    &   𝐼𝑁    &   𝐽𝑁    &   (𝐾 + 1) = 𝐼    &   (𝐿 + 1) = 𝐽    &   (𝑊𝐾) = 𝑋    &   (𝜑 → (𝑋𝐿) = 𝑌)       (𝜑 → (𝐼𝑀𝐽) = 𝑌)

Theoremlmatcl 30680* Closure of the literal matrix. (Contributed by Thierry Arnoux, 12-Sep-2020.)
𝑀 = (litMat‘𝑊)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑊 ∈ Word Word 𝑉)    &   (𝜑 → (♯‘𝑊) = 𝑁)    &   ((𝜑𝑖 ∈ (0..^𝑁)) → (♯‘(𝑊𝑖)) = 𝑁)    &   𝑉 = (Base‘𝑅)    &   𝑂 = ((1...𝑁) Mat 𝑅)    &   𝑃 = (Base‘𝑂)    &   (𝜑𝑅𝑋)       (𝜑𝑀𝑃)

Theoremlmat22lem 30681* Lemma for lmat22e11 30682 and co. (Contributed by Thierry Arnoux, 28-Aug-2020.)
𝑀 = (litMat‘⟨“⟨“𝐴𝐵”⟩⟨“𝐶𝐷”⟩”⟩)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑉)    &   (𝜑𝐶𝑉)    &   (𝜑𝐷𝑉)       ((𝜑𝑖 ∈ (0..^2)) → (♯‘(⟨“⟨“𝐴𝐵”⟩⟨“𝐶𝐷”⟩”⟩‘𝑖)) = 2)

Theoremlmat22e11 30682 Entry of a 2x2 literal matrix. (Contributed by Thierry Arnoux, 28-Aug-2020.)
𝑀 = (litMat‘⟨“⟨“𝐴𝐵”⟩⟨“𝐶𝐷”⟩”⟩)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑉)    &   (𝜑𝐶𝑉)    &   (𝜑𝐷𝑉)       (𝜑 → (1𝑀1) = 𝐴)

Theoremlmat22e12 30683 Entry of a 2x2 literal matrix. (Contributed by Thierry Arnoux, 12-Sep-2020.)
𝑀 = (litMat‘⟨“⟨“𝐴𝐵”⟩⟨“𝐶𝐷”⟩”⟩)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑉)    &   (𝜑𝐶𝑉)    &   (𝜑𝐷𝑉)       (𝜑 → (1𝑀2) = 𝐵)

Theoremlmat22e21 30684 Entry of a 2x2 literal matrix. (Contributed by Thierry Arnoux, 12-Sep-2020.)
𝑀 = (litMat‘⟨“⟨“𝐴𝐵”⟩⟨“𝐶𝐷”⟩”⟩)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑉)    &   (𝜑𝐶𝑉)    &   (𝜑𝐷𝑉)       (𝜑 → (2𝑀1) = 𝐶)

Theoremlmat22e22 30685 Entry of a 2x2 literal matrix. (Contributed by Thierry Arnoux, 12-Sep-2020.)
𝑀 = (litMat‘⟨“⟨“𝐴𝐵”⟩⟨“𝐶𝐷”⟩”⟩)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑉)    &   (𝜑𝐶𝑉)    &   (𝜑𝐷𝑉)       (𝜑 → (2𝑀2) = 𝐷)

Theoremlmat22det 30686 The determinant of a literal 2x2 complex matrix. (Contributed by Thierry Arnoux, 1-Sep-2020.)
𝑀 = (litMat‘⟨“⟨“𝐴𝐵”⟩⟨“𝐶𝐷”⟩”⟩)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑉)    &   (𝜑𝐶𝑉)    &   (𝜑𝐷𝑉)    &    · = (.r𝑅)    &    = (-g𝑅)    &   𝑉 = (Base‘𝑅)    &   𝐽 = ((1...2) maDet 𝑅)    &   (𝜑𝑅 ∈ Ring)       (𝜑 → (𝐽𝑀) = ((𝐴 · 𝐷) (𝐶 · 𝐵)))

20.3.10.6  Laplace expansion of determinants

Theoremmdetpmtr1 30687* 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)    &    · = (.r𝑅)    &   𝐸 = (𝑖𝑁, 𝑗𝑁 ↦ ((𝑃𝑖)𝑀𝑗))       (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀𝐵𝑃𝐺)) → (𝐷𝑀) = (((𝑍𝑆)‘𝑃) · (𝐷𝐸)))

Theoremmdetpmtr2 30688* 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)    &    · = (.r𝑅)    &   𝐸 = (𝑖𝑁, 𝑗𝑁 ↦ (𝑖𝑀(𝑃𝑗)))       (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀𝐵𝑃𝐺)) → (𝐷𝑀) = (((𝑍𝑆)‘𝑃) · (𝐷𝐸)))

Theoremmdetpmtr12 30689* 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)    &    · = (.r𝑅)    &   𝐸 = (𝑖𝑁, 𝑗𝑁 ↦ ((𝑃𝑖)𝑀(𝑄𝑗)))    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑀𝐵)    &   (𝜑𝑃𝐺)    &   (𝜑𝑄𝐺)       (𝜑 → (𝐷𝑀) = ((𝑍‘((𝑆𝑃) · (𝑆𝑄))) · (𝐷𝐸)))

Theoremmdetlap1 30690* 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)    &    · = (.r𝑅)       ((𝑅 ∈ CRing ∧ 𝑀𝐵𝐼𝑁) → (𝐷𝑀) = (𝑅 Σg (𝑗𝑁 ↦ ((𝐼𝑀𝑗) · (𝑗(𝐾𝑀)𝐼)))))

𝐵 = (Base‘𝐴)    &   𝐴 = ((1...𝑁) Mat 𝑅)    &   𝐷 = ((1...𝑁) maDet 𝑅)    &   𝐾 = ((1...𝑁) maAdju 𝑅)    &    · = (.r𝑅)    &   𝑍 = (ℤRHom‘𝑅)    &   𝐸 = ((1...(𝑁 − 1)) maDet 𝑅)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝐼 ∈ (1...𝑁))    &   (𝜑𝐽 ∈ (1...𝑁))    &   (𝜑𝑀𝐵)    &   𝐺 = (Base‘(SymGrp‘(1...𝑁)))    &   𝑆 = (pmSgn‘(1...𝑁))    &   𝑈 = (𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)    &   𝑊 = (𝑖 ∈ (1...𝑁), 𝑗 ∈ (1...𝑁) ↦ ((𝑃𝑖)𝑈(𝑄𝑗)))    &   (𝜑𝑃𝐺)    &   (𝜑𝑄𝐺)    &   (𝜑 → (𝑃𝑁) = 𝐼)    &   (𝜑 → (𝑄𝑁) = 𝐽)    &   (𝜑 → (𝐼(subMat1‘𝑈)𝐽) = (𝑁(subMat1‘𝑊)𝑁))       (𝜑 → (𝐽(𝐾𝑀)𝐼) = ((𝑍‘((𝑆𝑃) · (𝑆𝑄))) · (𝐸‘(𝐼(subMat1‘𝑀)𝐽))))

𝐵 = (Base‘𝐴)    &   𝐴 = ((1...𝑁) Mat 𝑅)    &   𝐷 = ((1...𝑁) maDet 𝑅)    &   𝐾 = ((1...𝑁) maAdju 𝑅)    &    · = (.r𝑅)    &   𝑍 = (ℤRHom‘𝑅)    &   𝐸 = ((1...(𝑁 − 1)) maDet 𝑅)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝐼 ∈ (1...𝑁))    &   (𝜑𝐽 ∈ (1...𝑁))    &   (𝜑𝑀𝐵)    &   𝑃 = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝐼, if(𝑖𝐼, (𝑖 − 1), 𝑖)))    &   𝑆 = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝑁, if(𝑖𝑁, (𝑖 − 1), 𝑖)))       ((𝜑𝑋 ∈ (1...(𝑁 − 1))) → if(𝑋 < 𝐼, 𝑋, (𝑋 + 1)) = ((𝑃𝑆)‘𝑋))

𝐵 = (Base‘𝐴)    &   𝐴 = ((1...𝑁) Mat 𝑅)    &   𝐷 = ((1...𝑁) maDet 𝑅)    &   𝐾 = ((1...𝑁) maAdju 𝑅)    &    · = (.r𝑅)    &   𝑍 = (ℤRHom‘𝑅)    &   𝐸 = ((1...(𝑁 − 1)) maDet 𝑅)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝐼 ∈ (1...𝑁))    &   (𝜑𝐽 ∈ (1...𝑁))    &   (𝜑𝑀𝐵)    &   𝑃 = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝐼, if(𝑖𝐼, (𝑖 − 1), 𝑖)))    &   𝑆 = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝑁, if(𝑖𝑁, (𝑖 − 1), 𝑖)))    &   𝑄 = (𝑗 ∈ (1...𝑁) ↦ if(𝑗 = 1, 𝐽, if(𝑗𝐽, (𝑗 − 1), 𝑗)))    &   𝑇 = (𝑗 ∈ (1...𝑁) ↦ if(𝑗 = 1, 𝑁, if(𝑗𝑁, (𝑗 − 1), 𝑗)))    &   𝑊 = (𝑖 ∈ (1...𝑁), 𝑗 ∈ (1...𝑁) ↦ (((𝑃𝑆)‘𝑖)𝑈((𝑄𝑇)‘𝑗)))    &   (𝜑𝑈𝐵)       (𝜑 → (𝐼(subMat1‘𝑈)𝐽) = (𝑁(subMat1‘𝑊)𝑁))

𝐵 = (Base‘𝐴)    &   𝐴 = ((1...𝑁) Mat 𝑅)    &   𝐷 = ((1...𝑁) maDet 𝑅)    &   𝐾 = ((1...𝑁) maAdju 𝑅)    &    · = (.r𝑅)    &   𝑍 = (ℤRHom‘𝑅)    &   𝐸 = ((1...(𝑁 − 1)) maDet 𝑅)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝐼 ∈ (1...𝑁))    &   (𝜑𝐽 ∈ (1...𝑁))    &   (𝜑𝑀𝐵)    &   𝑃 = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝐼, if(𝑖𝐼, (𝑖 − 1), 𝑖)))    &   𝑆 = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝑁, if(𝑖𝑁, (𝑖 − 1), 𝑖)))    &   𝑄 = (𝑗 ∈ (1...𝑁) ↦ if(𝑗 = 1, 𝐽, if(𝑗𝐽, (𝑗 − 1), 𝑗)))    &   𝑇 = (𝑗 ∈ (1...𝑁) ↦ if(𝑗 = 1, 𝑁, if(𝑗𝑁, (𝑗 − 1), 𝑗)))       (𝜑 → (𝐽(𝐾𝑀)𝐼) = ((𝑍‘(-1↑(𝐼 + 𝐽))) · (𝐸‘(𝐼(subMat1‘𝑀)𝐽))))

Theoremmadjusmdet 30695 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)    &    · = (.r𝑅)    &   𝑍 = (ℤRHom‘𝑅)    &   𝐸 = ((1...(𝑁 − 1)) maDet 𝑅)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝐼 ∈ (1...𝑁))    &   (𝜑𝐽 ∈ (1...𝑁))    &   (𝜑𝑀𝐵)       (𝜑 → (𝐽(𝐾𝑀)𝐼) = ((𝑍‘(-1↑(𝐼 + 𝐽))) · (𝐸‘(𝐼(subMat1‘𝑀)𝐽))))

Theoremmdetlap 30696* 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)    &    · = (.r𝑅)    &   𝑍 = (ℤRHom‘𝑅)    &   𝐸 = ((1...(𝑁 − 1)) maDet 𝑅)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝐼 ∈ (1...𝑁))    &   (𝜑𝐽 ∈ (1...𝑁))    &   (𝜑𝑀𝐵)       (𝜑 → (𝐷𝑀) = (𝑅 Σg (𝑗 ∈ (1...𝑁) ↦ ((𝑍‘(-1↑(𝐼 + 𝑗))) · ((𝐼𝑀𝑗) · (𝐸‘(𝐼(subMat1‘𝑀)𝑗)))))))

20.3.11  Topology

20.3.11.1  Open maps

Theoremfvproj 30697* Value of a function on pairs, given two projections 𝐹 and 𝐺. (Contributed by Thierry Arnoux, 30-Dec-2019.)
𝐻 = (𝑥𝐴, 𝑦𝐵 ↦ ⟨(𝐹𝑥), (𝐺𝑦)⟩)    &   (𝜑𝑋𝐴)    &   (𝜑𝑌𝐵)       (𝜑 → (𝐻‘⟨𝑋, 𝑌⟩) = ⟨(𝐹𝑋), (𝐺𝑌)⟩)

Theoremfimaproj 30698* Image of a cartesian product for a function on pairs, given two projections 𝐹 and 𝐺. (Contributed by Thierry Arnoux, 30-Dec-2019.)
𝐻 = (𝑥𝐴, 𝑦𝐵 ↦ ⟨(𝐹𝑥), (𝐺𝑦)⟩)    &   (𝜑𝐹 Fn 𝐴)    &   (𝜑𝐺 Fn 𝐵)    &   (𝜑𝑋𝐴)    &   (𝜑𝑌𝐵)       (𝜑 → (𝐻 “ (𝑋 × 𝑌)) = ((𝐹𝑋) × (𝐺𝑌)))

Theoremtxomap 30699* 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.)
(𝜑𝐹:𝑋𝑍)    &   (𝜑𝐺:𝑌𝑇)    &   (𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐾 ∈ (TopOn‘𝑌))    &   (𝜑𝐿 ∈ (TopOn‘𝑍))    &   (𝜑𝑀 ∈ (TopOn‘𝑇))    &   ((𝜑𝑥𝐽) → (𝐹𝑥) ∈ 𝐿)    &   ((𝜑𝑦𝐾) → (𝐺𝑦) ∈ 𝑀)    &   (𝜑𝐴 ∈ (𝐽 ×t 𝐾))    &   𝐻 = (𝑥𝑋, 𝑦𝑌 ↦ ⟨(𝐹𝑥), (𝐺𝑦)⟩)       (𝜑 → (𝐻𝐴) ∈ (𝐿 ×t 𝑀))

20.3.11.2  Topology of the unit circle

Theoremqtopt1 30700* If every equivalence class is closed, then the quotient space is T1 . (Contributed by Thierry Arnoux, 5-Jan-2020.)
𝑋 = 𝐽    &   (𝜑𝐽 ∈ Fre)    &   (𝜑𝐹:𝑋onto𝑌)    &   ((𝜑𝑥𝑌) → (𝐹 “ {𝑥}) ∈ (Clsd‘𝐽))       (𝜑 → (𝐽 qTop 𝐹) ∈ Fre)

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