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Theorem List for Metamath Proof Explorer - 45701-45800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
20.41.20.3  The ` ZZ `-module ` ZZ X. ZZ `
 
Theoremzlmodzxzlmod 45701 The -module ℤ × ℤ is a (left) module with the ring of integers as base set. (Contributed by AV, 20-May-2019.) (Revised by AV, 10-Jun-2019.)
𝑍 = (ℤring freeLMod {0, 1})       (𝑍 ∈ LMod ∧ ℤring = (Scalar‘𝑍))
 
Theoremzlmodzxzel 45702 An element of the (base set of the) -module ℤ × ℤ. (Contributed by AV, 21-May-2019.) (Revised by AV, 10-Jun-2019.)
𝑍 = (ℤring freeLMod {0, 1})       ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → {⟨0, 𝐴⟩, ⟨1, 𝐵⟩} ∈ (Base‘𝑍))
 
Theoremzlmodzxz0 45703 The 0 of the -module ℤ × ℤ. (Contributed by AV, 20-May-2019.) (Revised by AV, 10-Jun-2019.)
𝑍 = (ℤring freeLMod {0, 1})    &    0 = {⟨0, 0⟩, ⟨1, 0⟩}        0 = (0g𝑍)
 
Theoremzlmodzxzscm 45704 The scalar multiplication of the -module ℤ × ℤ. (Contributed by AV, 20-May-2019.) (Revised by AV, 10-Jun-2019.)
𝑍 = (ℤring freeLMod {0, 1})    &    = ( ·𝑠𝑍)       ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (𝐴 {⟨0, 𝐵⟩, ⟨1, 𝐶⟩}) = {⟨0, (𝐴 · 𝐵)⟩, ⟨1, (𝐴 · 𝐶)⟩})
 
Theoremzlmodzxzadd 45705 The addition of the -module ℤ × ℤ. (Contributed by AV, 22-May-2019.) (Revised by AV, 10-Jun-2019.)
𝑍 = (ℤring freeLMod {0, 1})    &    + = (+g𝑍)       (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → ({⟨0, 𝐴⟩, ⟨1, 𝐶⟩} + {⟨0, 𝐵⟩, ⟨1, 𝐷⟩}) = {⟨0, (𝐴 + 𝐵)⟩, ⟨1, (𝐶 + 𝐷)⟩})
 
Theoremzlmodzxzsubm 45706 The subtraction of the -module ℤ × ℤ expressed as addition. (Contributed by AV, 24-May-2019.) (Revised by AV, 10-Jun-2019.)
𝑍 = (ℤring freeLMod {0, 1})    &    = (-g𝑍)       (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → ({⟨0, 𝐴⟩, ⟨1, 𝐶⟩} {⟨0, 𝐵⟩, ⟨1, 𝐷⟩}) = ({⟨0, 𝐴⟩, ⟨1, 𝐶⟩} (+g𝑍)(-1( ·𝑠𝑍){⟨0, 𝐵⟩, ⟨1, 𝐷⟩})))
 
Theoremzlmodzxzsub 45707 The subtraction of the -module ℤ × ℤ. (Contributed by AV, 22-May-2019.) (Revised by AV, 10-Jun-2019.)
𝑍 = (ℤring freeLMod {0, 1})    &    = (-g𝑍)       (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → ({⟨0, 𝐴⟩, ⟨1, 𝐶⟩} {⟨0, 𝐵⟩, ⟨1, 𝐷⟩}) = {⟨0, (𝐴𝐵)⟩, ⟨1, (𝐶𝐷)⟩})
 
20.41.20.4  Group sum operation (extension 2)
 
Theoremmgpsumunsn 45708* Extract a summand/factor from the group sum for the multiplicative group of a unital ring. (Contributed by AV, 29-Dec-2018.)
𝑀 = (mulGrp‘𝑅)    &    · = (.r𝑅)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝐼𝑁)    &   ((𝜑𝑘𝑁) → 𝐴 ∈ (Base‘𝑅))    &   (𝜑𝑋 ∈ (Base‘𝑅))    &   (𝑘 = 𝐼𝐴 = 𝑋)       (𝜑 → (𝑀 Σg (𝑘𝑁𝐴)) = ((𝑀 Σg (𝑘 ∈ (𝑁 ∖ {𝐼}) ↦ 𝐴)) · 𝑋))
 
Theoremmgpsumz 45709* If the group sum for the multiplicative group of a unital ring contains a summand/factor that is the zero of the ring, the group sum itself is zero. (Contributed by AV, 29-Dec-2018.)
𝑀 = (mulGrp‘𝑅)    &    · = (.r𝑅)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝐼𝑁)    &   ((𝜑𝑘𝑁) → 𝐴 ∈ (Base‘𝑅))    &    0 = (0g𝑅)    &   (𝑘 = 𝐼𝐴 = 0 )       (𝜑 → (𝑀 Σg (𝑘𝑁𝐴)) = 0 )
 
Theoremmgpsumn 45710* If the group sum for the multiplicative group of a unital ring contains a summand/factor that is the one of the ring, this summand/ factor can be removed from the group sum. (Contributed by AV, 29-Dec-2018.)
𝑀 = (mulGrp‘𝑅)    &    · = (.r𝑅)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝐼𝑁)    &   ((𝜑𝑘𝑁) → 𝐴 ∈ (Base‘𝑅))    &    1 = (1r𝑅)    &   (𝑘 = 𝐼𝐴 = 1 )       (𝜑 → (𝑀 Σg (𝑘𝑁𝐴)) = (𝑀 Σg (𝑘 ∈ (𝑁 ∖ {𝐼}) ↦ 𝐴)))
 
20.41.20.5  Symmetric groups (extension)
 
Theoremexple2lt6 45711 A nonnegative integer to the power of itself is less than 6 if it is less than or equal to 2. (Contributed by AV, 16-Mar-2019.)
((𝑁 ∈ ℕ0𝑁 ≤ 2) → (𝑁𝑁) < 6)
 
Theorempgrple2abl 45712 Every symmetric group on a set with at most 2 elements is abelian. (Contributed by AV, 16-Mar-2019.)
𝐺 = (SymGrp‘𝐴)       ((𝐴𝑉 ∧ (♯‘𝐴) ≤ 2) → 𝐺 ∈ Abel)
 
Theorempgrpgt2nabl 45713 Every symmetric group on a set with more than 2 elements is not abelian, see also the remark in [Rotman] p. 28. (Contributed by AV, 21-Mar-2019.)
𝐺 = (SymGrp‘𝐴)       ((𝐴𝑉 ∧ 2 < (♯‘𝐴)) → 𝐺 ∉ Abel)
 
20.41.20.6  Divisibility (extension)
 
Theoreminvginvrid 45714 Identity for a multiplication with additive and multiplicative inverses in a ring. (Contributed by AV, 18-May-2018.)
𝐵 = (Base‘𝑅)    &   𝑈 = (Unit‘𝑅)    &   𝑁 = (invg𝑅)    &   𝐼 = (invr𝑅)    &    · = (.r𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝐵𝑌𝑈) → ((𝑁𝑌) · ((𝐼‘(𝑁𝑌)) · 𝑋)) = 𝑋)
 
20.41.20.7  The support of functions (extension)
 
Theoremrmsupp0 45715* The support of a mapping of a multiplication of zero with a function into a ring is empty. (Contributed by AV, 10-Apr-2019.)
𝑅 = (Base‘𝑀)       (((𝑀 ∈ Ring ∧ 𝑉𝑋𝐶 = (0g𝑀)) ∧ 𝐴 ∈ (𝑅m 𝑉)) → ((𝑣𝑉 ↦ (𝐶(.r𝑀)(𝐴𝑣))) supp (0g𝑀)) = ∅)
 
Theoremdomnmsuppn0 45716* The support of a mapping of a multiplication of a nonzero constant with a function into a (ring theoretic) domain equals the support of the function. (Contributed by AV, 11-Apr-2019.)
𝑅 = (Base‘𝑀)       (((𝑀 ∈ Domn ∧ 𝑉𝑋) ∧ (𝐶𝑅𝐶 ≠ (0g𝑀)) ∧ 𝐴 ∈ (𝑅m 𝑉)) → ((𝑣𝑉 ↦ (𝐶(.r𝑀)(𝐴𝑣))) supp (0g𝑀)) = (𝐴 supp (0g𝑀)))
 
Theoremrmsuppss 45717* The support of a mapping of a multiplication of a constant with a function into a ring is a subset of the support of the function. (Contributed by AV, 11-Apr-2019.)
𝑅 = (Base‘𝑀)       (((𝑀 ∈ Ring ∧ 𝑉𝑋𝐶𝑅) ∧ 𝐴 ∈ (𝑅m 𝑉)) → ((𝑣𝑉 ↦ (𝐶(.r𝑀)(𝐴𝑣))) supp (0g𝑀)) ⊆ (𝐴 supp (0g𝑀)))
 
Theoremmndpsuppss 45718 The support of a mapping of a scalar multiplication with a function of scalars is a subset of the support of the function of scalars. (Contributed by AV, 5-Apr-2019.)
𝑅 = (Base‘𝑀)       (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → ((𝐴f (+g𝑀)𝐵) supp (0g𝑀)) ⊆ ((𝐴 supp (0g𝑀)) ∪ (𝐵 supp (0g𝑀))))
 
Theoremscmsuppss 45719* The support of a mapping of a scalar multiplication with a function of scalars is a subset of the support of the function of scalars. (Contributed by AV, 5-Apr-2019.)
𝑆 = (Scalar‘𝑀)    &   𝑅 = (Base‘𝑆)       ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝐴 ∈ (𝑅m 𝑉)) → ((𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣)) supp (0g𝑀)) ⊆ (𝐴 supp (0g𝑆)))
 
20.41.20.8  Finitely supported functions (extension)
 
Theoremrmsuppfi 45720* The support of a mapping of a multiplication of a constant with a function into a ring is finite if the support of the function is finite. (Contributed by AV, 11-Apr-2019.)
𝑅 = (Base‘𝑀)       (((𝑀 ∈ Ring ∧ 𝑉𝑋𝐶𝑅) ∧ 𝐴 ∈ (𝑅m 𝑉) ∧ (𝐴 supp (0g𝑀)) ∈ Fin) → ((𝑣𝑉 ↦ (𝐶(.r𝑀)(𝐴𝑣))) supp (0g𝑀)) ∈ Fin)
 
Theoremrmfsupp 45721* A mapping of a multiplication of a constant with a function into a ring is finitely supported if the function is finitely supported. (Contributed by AV, 9-Jun-2019.)
𝑅 = (Base‘𝑀)       (((𝑀 ∈ Ring ∧ 𝑉𝑋𝐶𝑅) ∧ 𝐴 ∈ (𝑅m 𝑉) ∧ 𝐴 finSupp (0g𝑀)) → (𝑣𝑉 ↦ (𝐶(.r𝑀)(𝐴𝑣))) finSupp (0g𝑀))
 
Theoremmndpsuppfi 45722 The support of a mapping of a scalar multiplication with a function of scalars is finite if the support of the function of scalars is finite. (Contributed by AV, 5-Apr-2019.)
𝑅 = (Base‘𝑀)       (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉)) ∧ ((𝐴 supp (0g𝑀)) ∈ Fin ∧ (𝐵 supp (0g𝑀)) ∈ Fin)) → ((𝐴f (+g𝑀)𝐵) supp (0g𝑀)) ∈ Fin)
 
Theoremmndpfsupp 45723 A mapping of a scalar multiplication with a function of scalars is finitely supported if the function of scalars is finitely supported. (Contributed by AV, 9-Jun-2019.)
𝑅 = (Base‘𝑀)       (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉)) ∧ (𝐴 finSupp (0g𝑀) ∧ 𝐵 finSupp (0g𝑀))) → (𝐴f (+g𝑀)𝐵) finSupp (0g𝑀))
 
Theoremscmsuppfi 45724* The support of a mapping of a scalar multiplication with a function of scalars is finite if the support of the function of scalars is finite. (Contributed by AV, 5-Apr-2019.)
𝑆 = (Scalar‘𝑀)    &   𝑅 = (Base‘𝑆)       (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐴 ∈ (𝑅m 𝑉) ∧ (𝐴 supp (0g𝑆)) ∈ Fin) → ((𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣)) supp (0g𝑀)) ∈ Fin)
 
Theoremscmfsupp 45725* A mapping of a scalar multiplication with a function of scalars is finitely supported if the function of scalars is finitely supported. (Contributed by AV, 9-Jun-2019.)
𝑆 = (Scalar‘𝑀)    &   𝑅 = (Base‘𝑆)       (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐴 ∈ (𝑅m 𝑉) ∧ 𝐴 finSupp (0g𝑆)) → (𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣)) finSupp (0g𝑀))
 
Theoremsuppmptcfin 45726* The support of a mapping with value 0 except of one is finite. (Contributed by AV, 27-Apr-2019.)
𝐵 = (Base‘𝑀)    &   𝑅 = (Scalar‘𝑀)    &    0 = (0g𝑅)    &    1 = (1r𝑅)    &   𝐹 = (𝑥𝑉 ↦ if(𝑥 = 𝑋, 1 , 0 ))       ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵𝑋𝑉) → (𝐹 supp 0 ) ∈ Fin)
 
Theoremmptcfsupp 45727* A mapping with value 0 except of one is finitely supported. (Contributed by AV, 9-Jun-2019.)
𝐵 = (Base‘𝑀)    &   𝑅 = (Scalar‘𝑀)    &    0 = (0g𝑅)    &    1 = (1r𝑅)    &   𝐹 = (𝑥𝑉 ↦ if(𝑥 = 𝑋, 1 , 0 ))       ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵𝑋𝑉) → 𝐹 finSupp 0 )
 
Theoremfsuppmptdmf 45728* A mapping with a finite domain is finitely supported. (Contributed by AV, 4-Sep-2019.)
𝑥𝜑    &   𝐹 = (𝑥𝐴𝑌)    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑥𝐴) → 𝑌𝑉)    &   (𝜑𝑍𝑊)       (𝜑𝐹 finSupp 𝑍)
 
20.41.20.9  Left modules (extension)
 
Theoremlmodvsmdi 45729 Multiple distributive law for scalar product (left-distributivity). (Contributed by AV, 5-Sep-2019.)
𝑉 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐾 = (Base‘𝐹)    &    = (.g𝑊)    &   𝐸 = (.g𝐹)       ((𝑊 ∈ LMod ∧ (𝑅𝐾𝑁 ∈ ℕ0𝑋𝑉)) → (𝑅 · (𝑁 𝑋)) = ((𝑁𝐸𝑅) · 𝑋))
 
Theoremgsumlsscl 45730* Closure of a group sum in a linear subspace: A (finitely supported) sum of scalar multiplications of vectors of a subset of a linear subspace is also contained in the linear subspace. (Contributed by AV, 20-Apr-2019.) (Revised by AV, 28-Jul-2019.)
𝑆 = (LSubSp‘𝑀)    &   𝑅 = (Scalar‘𝑀)    &   𝐵 = (Base‘𝑅)       ((𝑀 ∈ LMod ∧ 𝑍𝑆𝑉𝑍) → ((𝐹 ∈ (𝐵m 𝑉) ∧ 𝐹 finSupp (0g𝑅)) → (𝑀 Σg (𝑣𝑉 ↦ ((𝐹𝑣)( ·𝑠𝑀)𝑣))) ∈ 𝑍))
 
20.41.20.10  Associative algebras (extension)
 
Theoremassaascl0 45731 The scalar 0 embedded into an associative algebra corresponds to the 0 of the associative algebra. (Contributed by AV, 31-Jul-2019.)
𝐴 = (algSc‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &   (𝜑𝑊 ∈ AssAlg)       (𝜑 → (𝐴‘(0g𝐹)) = (0g𝑊))
 
Theoremassaascl1 45732 The scalar 1 embedded into an associative algebra corresponds to the 1 of the an associative algebra. (Contributed by AV, 31-Jul-2019.)
𝐴 = (algSc‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &   (𝜑𝑊 ∈ AssAlg)       (𝜑 → (𝐴‘(1r𝐹)) = (1r𝑊))
 
20.41.20.11  Univariate polynomials (extension)
 
Theoremply1vr1smo 45733 The variable in a polynomial expressed as scaled monomial. (Contributed by AV, 12-Aug-2019.)
𝑃 = (Poly1𝑅)    &    1 = (1r𝑅)    &    · = ( ·𝑠𝑃)    &   𝐺 = (mulGrp‘𝑃)    &    = (.g𝐺)    &   𝑋 = (var1𝑅)       (𝑅 ∈ Ring → ( 1 · (1 𝑋)) = 𝑋)
 
Theoremply1ass23l 45734 Associative identity with scalar and ring multiplication for the polynomial ring. (Contributed by AV, 14-Aug-2019.)
𝑃 = (Poly1𝑅)    &    × = (.r𝑃)    &   𝐵 = (Base‘𝑃)    &   𝐾 = (Base‘𝑅)    &    · = ( ·𝑠𝑃)       ((𝑅 ∈ Ring ∧ (𝐴𝐾𝑋𝐵𝑌𝐵)) → ((𝐴 · 𝑋) × 𝑌) = (𝐴 · (𝑋 × 𝑌)))
 
Theoremply1sclrmsm 45735 The ring multiplication of a polynomial with a scalar polynomial is equal to the scalar multiplication of the polynomial with the corresponding scalar. (Contributed by AV, 14-Aug-2019.)
𝐾 = (Base‘𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐸 = (Base‘𝑃)    &   𝑋 = (var1𝑅)    &    · = ( ·𝑠𝑃)    &    × = (.r𝑃)    &   𝑁 = (mulGrp‘𝑃)    &    = (.g𝑁)    &   𝐴 = (algSc‘𝑃)       ((𝑅 ∈ Ring ∧ 𝐹𝐾𝑍𝐸) → ((𝐴𝐹) × 𝑍) = (𝐹 · 𝑍))
 
Theoremcoe1id 45736* Coefficient vector of the unit polynomial. (Contributed by AV, 9-Aug-2019.)
𝑃 = (Poly1𝑅)    &   𝐼 = (1r𝑃)    &    0 = (0g𝑅)    &    1 = (1r𝑅)       (𝑅 ∈ Ring → (coe1𝐼) = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, 1 , 0 )))
 
Theoremcoe1sclmulval 45737 The value of the coefficient vector of a polynomial multiplied on the left by a scalar. (Contributed by AV, 14-Aug-2019.)
𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &   𝐾 = (Base‘𝑅)    &   𝐴 = (algSc‘𝑃)    &   𝑆 = ( ·𝑠𝑃)    &    = (.r𝑃)    &    · = (.r𝑅)       ((𝑅 ∈ Ring ∧ (𝑌𝐾𝑍𝐵) ∧ 𝑁 ∈ ℕ0) → ((coe1‘(𝑌𝑆𝑍))‘𝑁) = (𝑌 · ((coe1𝑍)‘𝑁)))
 
Theoremply1mulgsumlem1 45738* Lemma 1 for ply1mulgsum 45742. (Contributed by AV, 19-Oct-2019.)
𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &   𝐴 = (coe1𝐾)    &   𝐶 = (coe1𝐿)    &   𝑋 = (var1𝑅)    &    × = (.r𝑃)    &    · = ( ·𝑠𝑃)    &    = (.r𝑅)    &   𝑀 = (mulGrp‘𝑃)    &    = (.g𝑀)       ((𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵) → ∃𝑠 ∈ ℕ0𝑛 ∈ ℕ0 (𝑠 < 𝑛 → ((𝐴𝑛) = (0g𝑅) ∧ (𝐶𝑛) = (0g𝑅))))
 
Theoremply1mulgsumlem2 45739* Lemma 2 for ply1mulgsum 45742. (Contributed by AV, 19-Oct-2019.)
𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &   𝐴 = (coe1𝐾)    &   𝐶 = (coe1𝐿)    &   𝑋 = (var1𝑅)    &    × = (.r𝑃)    &    · = ( ·𝑠𝑃)    &    = (.r𝑅)    &   𝑀 = (mulGrp‘𝑃)    &    = (.g𝑀)       ((𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵) → ∃𝑠 ∈ ℕ0𝑛 ∈ ℕ0 (𝑠 < 𝑛 → (𝑅 Σg (𝑙 ∈ (0...𝑛) ↦ ((𝐴𝑙) (𝐶‘(𝑛𝑙))))) = (0g𝑅)))
 
Theoremply1mulgsumlem3 45740* Lemma 3 for ply1mulgsum 45742. (Contributed by AV, 20-Oct-2019.)
𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &   𝐴 = (coe1𝐾)    &   𝐶 = (coe1𝐿)    &   𝑋 = (var1𝑅)    &    × = (.r𝑃)    &    · = ( ·𝑠𝑃)    &    = (.r𝑅)    &   𝑀 = (mulGrp‘𝑃)    &    = (.g𝑀)       ((𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵) → (𝑘 ∈ ℕ0 ↦ (𝑅 Σg (𝑙 ∈ (0...𝑘) ↦ ((𝐴𝑙) (𝐶‘(𝑘𝑙)))))) finSupp (0g𝑅))
 
Theoremply1mulgsumlem4 45741* Lemma 4 for ply1mulgsum 45742. (Contributed by AV, 19-Oct-2019.)
𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &   𝐴 = (coe1𝐾)    &   𝐶 = (coe1𝐿)    &   𝑋 = (var1𝑅)    &    × = (.r𝑃)    &    · = ( ·𝑠𝑃)    &    = (.r𝑅)    &   𝑀 = (mulGrp‘𝑃)    &    = (.g𝑀)       ((𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵) → (𝑘 ∈ ℕ0 ↦ ((𝑅 Σg (𝑙 ∈ (0...𝑘) ↦ ((𝐴𝑙) (𝐶‘(𝑘𝑙))))) · (𝑘 𝑋))) finSupp (0g𝑃))
 
Theoremply1mulgsum 45742* The product of two polynomials expressed as group sum of scaled monomials. (Contributed by AV, 20-Oct-2019.)
𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &   𝐴 = (coe1𝐾)    &   𝐶 = (coe1𝐿)    &   𝑋 = (var1𝑅)    &    × = (.r𝑃)    &    · = ( ·𝑠𝑃)    &    = (.r𝑅)    &   𝑀 = (mulGrp‘𝑃)    &    = (.g𝑀)       ((𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵) → (𝐾 × 𝐿) = (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑅 Σg (𝑙 ∈ (0...𝑘) ↦ ((𝐴𝑙) (𝐶‘(𝑘𝑙))))) · (𝑘 𝑋)))))
 
Theoremevl1at0 45743 Polynomial evaluation for the 0 scalar. (Contributed by AV, 10-Aug-2019.)
𝑂 = (eval1𝑅)    &   𝑃 = (Poly1𝑅)    &    0 = (0g𝑅)    &   𝑍 = (0g𝑃)       (𝑅 ∈ CRing → ((𝑂𝑍)‘ 0 ) = 0 )
 
Theoremevl1at1 45744 Polynomial evaluation for the 1 scalar. (Contributed by AV, 10-Aug-2019.)
𝑂 = (eval1𝑅)    &   𝑃 = (Poly1𝑅)    &    1 = (1r𝑅)    &   𝐼 = (1r𝑃)       (𝑅 ∈ CRing → ((𝑂𝐼)‘ 1 ) = 1 )
 
20.41.20.12  Univariate polynomials (examples)
 
Theoremlinply1 45745 A term of the form 𝑥𝐶 is a (univariate) polynomial, also called "linear polynomial". (Part of ply1remlem 25336). (Contributed by AV, 3-Jul-2019.)
𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &   𝐾 = (Base‘𝑅)    &   𝑋 = (var1𝑅)    &    = (-g𝑃)    &   𝐴 = (algSc‘𝑃)    &   𝐺 = (𝑋 (𝐴𝐶))    &   (𝜑𝐶𝐾)    &   (𝜑𝑅 ∈ Ring)       (𝜑𝐺𝐵)
 
Theoremlineval 45746 A term of the form 𝑥𝐶 evaluated for 𝑥 = 𝑉 results in 𝑉𝐶 (part of ply1remlem 25336). (Contributed by AV, 3-Jul-2019.)
𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &   𝐾 = (Base‘𝑅)    &   𝑋 = (var1𝑅)    &    = (-g𝑃)    &   𝐴 = (algSc‘𝑃)    &   𝐺 = (𝑋 (𝐴𝐶))    &   (𝜑𝐶𝐾)    &   𝑂 = (eval1𝑅)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑉𝐾)       (𝜑 → ((𝑂𝐺)‘𝑉) = (𝑉(-g𝑅)𝐶))
 
Theoremlinevalexample 45747 The polynomial 𝑥 − 3 over evaluated for 𝑥 = 5 results in 2. (Contributed by AV, 3-Jul-2019.)
𝑃 = (Poly1‘ℤring)    &   𝐵 = (Base‘𝑃)    &   𝑋 = (var1‘ℤring)    &    = (-g𝑃)    &   𝐴 = (algSc‘𝑃)    &   𝐺 = (𝑋 (𝐴‘3))    &   𝑂 = (eval1‘ℤring)       ((𝑂‘(𝑋 (𝐴‘3)))‘5) = 2
 
20.41.21  Linear algebra (extension)
 
20.41.21.1  The subalgebras of diagonal and scalar matrices (extension)

In the following, alternative definitions for diagonal and scalar matrices are provided. These definitions define diagonal and scalar matrices as extensible structures, whereas Definitions df-dmat 21648 and df-scmat 21649 define diagonal and scalar matrices as sets.

 
Syntaxcdmatalt 45748 Alternative notation for the algebra of diagonal matrices.
class DMatALT
 
Syntaxcscmatalt 45749 Alternative notation for the algebra of scalar matrices.
class ScMatALT
 
Definitiondf-dmatalt 45750* Define the set of n x n diagonal (square) matrices over a set (usually a ring) r, see definition in [Roman] p. 4 or Definition 3.12 in [Hefferon] p. 240. (Contributed by AV, 8-Dec-2019.)
DMatALT = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ (𝑛 Mat 𝑟) / 𝑎(𝑎s {𝑚 ∈ (Base‘𝑎) ∣ ∀𝑖𝑛𝑗𝑛 (𝑖𝑗 → (𝑖𝑚𝑗) = (0g𝑟))}))
 
Definitiondf-scmatalt 45751* Define the algebra of n x n scalar matrices over a set (usually a ring) r, see definition in [Connell] p. 57: "A scalar matrix is a diagonal matrix for which all the diagonal terms are equal, i.e., a matrix of the form cIn";. (Contributed by AV, 8-Dec-2019.)
ScMatALT = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ (𝑛 Mat 𝑟) / 𝑎(𝑎s {𝑚 ∈ (Base‘𝑎) ∣ ∃𝑐 ∈ (Base‘𝑟)∀𝑖𝑛𝑗𝑛 (𝑖𝑚𝑗) = if(𝑖 = 𝑗, 𝑐, (0g𝑟))}))
 
TheoremdmatALTval 45752* The algebra of 𝑁 x 𝑁 diagonal matrices over a ring 𝑅. (Contributed by AV, 8-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &    0 = (0g𝑅)    &   𝐷 = (𝑁 DMatALT 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) → 𝐷 = (𝐴s {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑚𝑗) = 0 )}))
 
TheoremdmatALTbas 45753* The base set of the algebra of 𝑁 x 𝑁 diagonal matrices over a ring 𝑅, i.e. the set of all 𝑁 x 𝑁 diagonal matrices over the ring 𝑅. (Contributed by AV, 8-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &    0 = (0g𝑅)    &   𝐷 = (𝑁 DMatALT 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) → (Base‘𝐷) = {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑚𝑗) = 0 )})
 
TheoremdmatALTbasel 45754* An element of the base set of the algebra of 𝑁 x 𝑁 diagonal matrices over a ring 𝑅, i.e. an 𝑁 x 𝑁 diagonal matrix over the ring 𝑅. (Contributed by AV, 8-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &    0 = (0g𝑅)    &   𝐷 = (𝑁 DMatALT 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) → (𝑀 ∈ (Base‘𝐷) ↔ (𝑀𝐵 ∧ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑀𝑗) = 0 ))))
 
Theoremdmatbas 45755 The set of all 𝑁 x 𝑁 diagonal matrices over (the ring) 𝑅 is the base set of the algebra of 𝑁 x 𝑁 diagonal matrices over (the ring) 𝑅. (Contributed by AV, 8-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &    0 = (0g𝑅)    &   𝐷 = (𝑁 DMat 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝐷 = (Base‘(𝑁 DMatALT 𝑅)))
 
20.41.21.2  Linear combinations

According to Wikipedia ("Linear combination", 29-Mar-2019, https://en.wikipedia.org/wiki/Linear_combination) "In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results (e.g., a linear combination of x and y would be any expression of the form ax + by, where a and b are constants). The concept of linear combinations is central to linear algebra and related fields of mathematics." In linear algebra, these "terms" are "vectors" (elements from vector spaces or left modules), and the constants are elements of the underlying field resp. ring. This corresponds to the definition in [Lang] p. 129: "Let M be a module over a ring A and let S be a subset of M. By a linear combination of elements of S (with coefficients in A) one means a sum ∑x ∈S axx where {ax} is a set of elements of A, ...". In the definition in [Lang] p. 129, it is additionally claimed that "..., almost all of which [elements of A] are equal to 0.". This is not necessarily required in the following definition df-linc 45758, but it is essential if additions and scalar multiplications of linear combinations are considered. Therefore, we define the set of all linear combinations with finite support in df-lco 45759, so that we can show that such sets are submodules of the corresponding modules, see lincolss 45786.
Remark:According to Wikipedia ("Linear span", 28-Apr-2019, https://en.wikipedia.org/wiki/Linear_span 45786) "In linear algebra, the linear span (also called the linear hull or just span) of a set of vectors in a vector space [or module] is the intersection of all linear subspaces which each contain every vector in that set.", and "Alternately, the span of [a set] S may be defined as the set of all finite linear combinations of elements (vectors) of S". Whereas spans are defined according to the first approach in df-lsp 20243, the set of all linear combinations as defined by df-lco 45759 follows the alternative approach. That both definitions are equivalent is shown by lspeqlco 45791.

 
Syntaxclinc 45756 Extend class notation with the operation constructing a linear combination (of vectors from a left module).
class linC
 
Syntaxclinco 45757 Extend class notation with the operation constructing a set of linear combinations (of vectors from a left module) with finite support.
class LinCo
 
Definitiondf-linc 45758* Define the operation constructing a linear combination. Although this definition is taylored for linear combinations of vectors from left modules, it can be used for any structure having a Base, Scalar s and a scalar multiplication ·𝑠. (Contributed by AV, 29-Mar-2019.)
linC = (𝑚 ∈ V ↦ (𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑m 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑚) ↦ (𝑚 Σg (𝑥𝑣 ↦ ((𝑠𝑥)( ·𝑠𝑚)𝑥)))))
 
Definitiondf-lco 45759* Define the operation constructing the set of all linear combinations for a set of vectors. (Contributed by AV, 31-Mar-2019.) (Revised by AV, 28-Jul-2019.)
LinCo = (𝑚 ∈ V, 𝑣 ∈ 𝒫 (Base‘𝑚) ↦ {𝑐 ∈ (Base‘𝑚) ∣ ∃𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑m 𝑣)(𝑠 finSupp (0g‘(Scalar‘𝑚)) ∧ 𝑐 = (𝑠( linC ‘𝑚)𝑣))})
 
Theoremlincop 45760* A linear combination as operation. (Contributed by AV, 30-Mar-2019.)
(𝑀𝑋 → ( linC ‘𝑀) = (𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑀) ↦ (𝑀 Σg (𝑥𝑣 ↦ ((𝑠𝑥)( ·𝑠𝑀)𝑥)))))
 
Theoremlincval 45761* The value of a linear combination. (Contributed by AV, 30-Mar-2019.)
((𝑀𝑋𝑆 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉) ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (𝑆( linC ‘𝑀)𝑉) = (𝑀 Σg (𝑥𝑉 ↦ ((𝑆𝑥)( ·𝑠𝑀)𝑥))))
 
Theoremdflinc2 45762* Alternative definition of linear combinations using the function operation. (Contributed by AV, 1-Apr-2019.)
linC = (𝑚 ∈ V ↦ (𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑m 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑚) ↦ (𝑚 Σg (𝑠f ( ·𝑠𝑚)( I ↾ 𝑣)))))
 
Theoremlcoop 45763* A linear combination as operation. (Contributed by AV, 5-Apr-2019.) (Revised by AV, 28-Jul-2019.)
𝐵 = (Base‘𝑀)    &   𝑆 = (Scalar‘𝑀)    &   𝑅 = (Base‘𝑆)       ((𝑀𝑋𝑉 ∈ 𝒫 𝐵) → (𝑀 LinCo 𝑉) = {𝑐𝐵 ∣ ∃𝑠 ∈ (𝑅m 𝑉)(𝑠 finSupp (0g𝑆) ∧ 𝑐 = (𝑠( linC ‘𝑀)𝑉))})
 
Theoremlcoval 45764* The value of a linear combination. (Contributed by AV, 5-Apr-2019.) (Revised by AV, 28-Jul-2019.)
𝐵 = (Base‘𝑀)    &   𝑆 = (Scalar‘𝑀)    &   𝑅 = (Base‘𝑆)       ((𝑀𝑋𝑉 ∈ 𝒫 𝐵) → (𝐶 ∈ (𝑀 LinCo 𝑉) ↔ (𝐶𝐵 ∧ ∃𝑠 ∈ (𝑅m 𝑉)(𝑠 finSupp (0g𝑆) ∧ 𝐶 = (𝑠( linC ‘𝑀)𝑉)))))
 
Theoremlincfsuppcl 45765 A linear combination of vectors (with finite support) is a vector. (Contributed by AV, 25-Apr-2019.) (Revised by AV, 28-Jul-2019.)
𝐵 = (Base‘𝑀)    &   𝑅 = (Scalar‘𝑀)    &   𝑆 = (Base‘𝑅)    &    0 = (0g𝑅)       ((𝑀 ∈ LMod ∧ (𝑉𝑊𝑉𝐵) ∧ (𝐹 ∈ (𝑆m 𝑉) ∧ 𝐹 finSupp 0 )) → (𝐹( linC ‘𝑀)𝑉) ∈ 𝐵)
 
Theoremlinccl 45766 A linear combination of vectors is a vector. (Contributed by AV, 31-Mar-2019.)
𝐵 = (Base‘𝑀)    &   𝑅 = (Base‘(Scalar‘𝑀))       ((𝑀 ∈ LMod ∧ (𝑉 ∈ Fin ∧ 𝑉𝐵𝑆 ∈ (𝑅m 𝑉))) → (𝑆( linC ‘𝑀)𝑉) ∈ 𝐵)
 
Theoremlincval0 45767 The value of an empty linear combination. (Contributed by AV, 12-Apr-2019.)
(𝑀𝑋 → (∅( linC ‘𝑀)∅) = (0g𝑀))
 
Theoremlincvalsng 45768 The linear combination over a singleton. (Contributed by AV, 25-May-2019.)
𝐵 = (Base‘𝑀)    &   𝑆 = (Scalar‘𝑀)    &   𝑅 = (Base‘𝑆)    &    · = ( ·𝑠𝑀)       ((𝑀 ∈ LMod ∧ 𝑉𝐵𝑌𝑅) → ({⟨𝑉, 𝑌⟩} ( linC ‘𝑀){𝑉}) = (𝑌 · 𝑉))
 
Theoremlincvalsn 45769 The linear combination over a singleton. (Contributed by AV, 12-Apr-2019.) (Proof shortened by AV, 25-May-2019.)
𝐵 = (Base‘𝑀)    &   𝑆 = (Scalar‘𝑀)    &   𝑅 = (Base‘𝑆)    &    · = ( ·𝑠𝑀)    &   𝐹 = {⟨𝑉, 𝑌⟩}       ((𝑀 ∈ LMod ∧ 𝑉𝐵𝑌𝑅) → (𝐹( linC ‘𝑀){𝑉}) = (𝑌 · 𝑉))
 
Theoremlincvalpr 45770 The linear combination over an unordered pair. (Contributed by AV, 16-Apr-2019.)
𝐵 = (Base‘𝑀)    &   𝑆 = (Scalar‘𝑀)    &   𝑅 = (Base‘𝑆)    &    · = ( ·𝑠𝑀)    &    + = (+g𝑀)    &   𝐹 = {⟨𝑉, 𝑋⟩, ⟨𝑊, 𝑌⟩}       (((𝑀 ∈ LMod ∧ 𝑉𝑊) ∧ (𝑉𝐵𝑋𝑅) ∧ (𝑊𝐵𝑌𝑅)) → (𝐹( linC ‘𝑀){𝑉, 𝑊}) = ((𝑋 · 𝑉) + (𝑌 · 𝑊)))
 
Theoremlincval1 45771 The linear combination over a singleton mapping to 0. (Contributed by AV, 12-Apr-2019.)
𝐵 = (Base‘𝑀)    &   𝑆 = (Scalar‘𝑀)    &   𝑅 = (Base‘𝑆)    &   𝐹 = {⟨𝑉, (0g𝑆)⟩}       ((𝑀 ∈ LMod ∧ 𝑉𝐵) → (𝐹( linC ‘𝑀){𝑉}) = (0g𝑀))
 
Theoremlcosn0 45772 Properties of a linear combination over a singleton mapping to 0. (Contributed by AV, 12-Apr-2019.) (Revised by AV, 28-Jul-2019.)
𝐵 = (Base‘𝑀)    &   𝑆 = (Scalar‘𝑀)    &   𝑅 = (Base‘𝑆)    &   𝐹 = {⟨𝑉, (0g𝑆)⟩}       ((𝑀 ∈ LMod ∧ 𝑉𝐵) → (𝐹 ∈ (𝑅m {𝑉}) ∧ 𝐹 finSupp (0g𝑆) ∧ (𝐹( linC ‘𝑀){𝑉}) = (0g𝑀)))
 
Theoremlincvalsc0 45773* The linear combination where all scalars are 0. (Contributed by AV, 12-Apr-2019.)
𝐵 = (Base‘𝑀)    &   𝑆 = (Scalar‘𝑀)    &    0 = (0g𝑆)    &   𝑍 = (0g𝑀)    &   𝐹 = (𝑥𝑉0 )       ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐹( linC ‘𝑀)𝑉) = 𝑍)
 
Theoremlcoc0 45774* Properties of a linear combination where all scalars are 0. (Contributed by AV, 12-Apr-2019.) (Revised by AV, 28-Jul-2019.)
𝐵 = (Base‘𝑀)    &   𝑆 = (Scalar‘𝑀)    &    0 = (0g𝑆)    &   𝑍 = (0g𝑀)    &   𝐹 = (𝑥𝑉0 )    &   𝑅 = (Base‘𝑆)       ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐹 ∈ (𝑅m 𝑉) ∧ 𝐹 finSupp 0 ∧ (𝐹( linC ‘𝑀)𝑉) = 𝑍))
 
Theoremlinc0scn0 45775* If a set contains the zero element of a module, there is a linear combination being 0 where not all scalars are 0. (Contributed by AV, 13-Apr-2019.)
𝐵 = (Base‘𝑀)    &   𝑆 = (Scalar‘𝑀)    &    0 = (0g𝑆)    &    1 = (1r𝑆)    &   𝑍 = (0g𝑀)    &   𝐹 = (𝑥𝑉 ↦ if(𝑥 = 𝑍, 1 , 0 ))       ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐹( linC ‘𝑀)𝑉) = 𝑍)
 
Theoremlincdifsn 45776 A vector is a linear combination of a set containing this vector. (Contributed by AV, 21-Apr-2019.) (Revised by AV, 28-Jul-2019.)
𝐵 = (Base‘𝑀)    &   𝑅 = (Scalar‘𝑀)    &   𝑆 = (Base‘𝑅)    &    · = ( ·𝑠𝑀)    &    + = (+g𝑀)    &    0 = (0g𝑅)       (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵𝑋𝑉) ∧ (𝐹 ∈ (𝑆m 𝑉) ∧ 𝐹 finSupp 0 ) ∧ 𝐺 = (𝐹 ↾ (𝑉 ∖ {𝑋}))) → (𝐹( linC ‘𝑀)𝑉) = ((𝐺( linC ‘𝑀)(𝑉 ∖ {𝑋})) + ((𝐹𝑋) · 𝑋)))
 
Theoremlinc1 45777* A vector is a linear combination of a set containing this vector. (Contributed by AV, 18-Apr-2019.) (Proof shortened by AV, 28-Jul-2019.)
𝐵 = (Base‘𝑀)    &   𝑆 = (Scalar‘𝑀)    &    0 = (0g𝑆)    &    1 = (1r𝑆)    &   𝐹 = (𝑥𝑉 ↦ if(𝑥 = 𝑋, 1 , 0 ))       ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵𝑋𝑉) → (𝐹( linC ‘𝑀)𝑉) = 𝑋)
 
Theoremlincellss 45778 A linear combination of a subset of a linear subspace is also contained in the linear subspace. (Contributed by AV, 20-Apr-2019.) (Revised by AV, 28-Jul-2019.)
((𝑀 ∈ LMod ∧ 𝑆 ∈ (LSubSp‘𝑀) ∧ 𝑉𝑆) → ((𝐹 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉) ∧ 𝐹 finSupp (0g‘(Scalar‘𝑀))) → (𝐹( linC ‘𝑀)𝑉) ∈ 𝑆))
 
Theoremlco0 45779 The set of empty linear combinations over a monoid is the singleton with the identity element of the monoid. (Contributed by AV, 12-Apr-2019.)
(𝑀 ∈ Mnd → (𝑀 LinCo ∅) = {(0g𝑀)})
 
Theoremlcoel0 45780 The zero vector is always a linear combination. (Contributed by AV, 12-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.)
((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (0g𝑀) ∈ (𝑀 LinCo 𝑉))
 
Theoremlincsum 45781 The sum of two linear combinations is a linear combination, see also the proof in [Lang] p. 129. (Contributed by AV, 4-Apr-2019.) (Revised by AV, 28-Jul-2019.)
+ = (+g𝑀)    &   𝑋 = (𝐴( linC ‘𝑀)𝑉)    &   𝑌 = (𝐵( linC ‘𝑀)𝑉)    &   𝑆 = (Scalar‘𝑀)    &   𝑅 = (Base‘𝑆)    &    = (+g𝑆)       (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉)) ∧ (𝐴 finSupp (0g𝑆) ∧ 𝐵 finSupp (0g𝑆))) → (𝑋 + 𝑌) = ((𝐴f 𝐵)( linC ‘𝑀)𝑉))
 
Theoremlincscm 45782* A linear combinations multiplied with a scalar is a linear combination, see also the proof in [Lang] p. 129. (Contributed by AV, 9-Apr-2019.) (Revised by AV, 28-Jul-2019.)
= ( ·𝑠𝑀)    &    · = (.r‘(Scalar‘𝑀))    &   𝑋 = (𝐴( linC ‘𝑀)𝑉)    &   𝑅 = (Base‘(Scalar‘𝑀))    &   𝐹 = (𝑥𝑉 ↦ (𝑆 · (𝐴𝑥)))       (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝑆𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) → (𝑆 𝑋) = (𝐹( linC ‘𝑀)𝑉))
 
Theoremlincsumcl 45783 The sum of two linear combinations is a linear combination, see also the proof in [Lang] p. 129. (Contributed by AV, 4-Apr-2019.) (Proof shortened by AV, 28-Jul-2019.)
+ = (+g𝑀)       (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐶 ∈ (𝑀 LinCo 𝑉) ∧ 𝐷 ∈ (𝑀 LinCo 𝑉))) → (𝐶 + 𝐷) ∈ (𝑀 LinCo 𝑉))
 
Theoremlincscmcl 45784 The multiplication of a linear combination with a scalar is a linear combination, see also the proof in [Lang] p. 129. (Contributed by AV, 11-Apr-2019.) (Proof shortened by AV, 28-Jul-2019.)
· = ( ·𝑠𝑀)    &   𝑅 = (Base‘(Scalar‘𝑀))       (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶𝑅𝐷 ∈ (𝑀 LinCo 𝑉)) → (𝐶 · 𝐷) ∈ (𝑀 LinCo 𝑉))
 
Theoremlincsumscmcl 45785 The sum of a linear combination and a multiplication of a linear combination with a scalar is a linear combination. (Contributed by AV, 11-Apr-2019.)
· = ( ·𝑠𝑀)    &   𝑅 = (Base‘(Scalar‘𝑀))    &    + = (+g𝑀)       (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐶𝑅𝐷 ∈ (𝑀 LinCo 𝑉) ∧ 𝐵 ∈ (𝑀 LinCo 𝑉))) → ((𝐶 · 𝐷) + 𝐵) ∈ (𝑀 LinCo 𝑉))
 
Theoremlincolss 45786 According to the statement in [Lang] p. 129, the set (LSubSp‘𝑀) of all linear combinations of a set of vectors V is a submodule (generated by V) of the module M. The elements of V are called generators of (LSubSp‘𝑀). (Contributed by AV, 12-Apr-2019.)
((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (𝑀 LinCo 𝑉) ∈ (LSubSp‘𝑀))
 
Theoremellcoellss 45787* Every linear combination of a subset of a linear subspace is also contained in the linear subspace. (Contributed by AV, 20-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.)
((𝑀 ∈ LMod ∧ 𝑆 ∈ (LSubSp‘𝑀) ∧ 𝑉𝑆) → ∀𝑥 ∈ (𝑀 LinCo 𝑉)𝑥𝑆)
 
Theoremlcoss 45788 A set of vectors of a module is a subset of the set of all linear combinations of the set. (Contributed by AV, 18-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.)
((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → 𝑉 ⊆ (𝑀 LinCo 𝑉))
 
Theoremlspsslco 45789 Lemma for lspeqlco 45791. (Contributed by AV, 17-Apr-2019.)
𝐵 = (Base‘𝑀)       ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → ((LSpan‘𝑀)‘𝑉) ⊆ (𝑀 LinCo 𝑉))
 
Theoremlcosslsp 45790 Lemma for lspeqlco 45791. (Contributed by AV, 20-Apr-2019.)
𝐵 = (Base‘𝑀)       ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝑀 LinCo 𝑉) ⊆ ((LSpan‘𝑀)‘𝑉))
 
Theoremlspeqlco 45791 Equivalence of a span of a set of vectors of a left module defined as the intersection of all linear subspaces which each contain every vector in that set (see df-lsp 20243) and as the set of all linear combinations of the vectors of the set with finite support. (Contributed by AV, 20-Apr-2019.)
𝐵 = (Base‘𝑀)       ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝑀 LinCo 𝑉) = ((LSpan‘𝑀)‘𝑉))
 
20.41.21.3  Linear independence

According to the definition in [Lang] p. 129: "A subset S of a module M is said to be linearly independent (over [the ring] A) if whenever we have a linear combination ∑x ∈S axx which is equal to 0, then ax=0 for all x∈S." This definition does not care for the finiteness of the set S (because the definition of a linear combination in [Lang] p.129 does already assure that only a finite number of coefficients can be 0 in the sum). Our definition df-lininds 45794 does also neither claim that the subset must be finite, nor that almost all coefficients within the linear combination are 0. If this is required, it must be explicitly stated as precondition in the corresponding theorems.

Usually, the linear independence is defined for vector spaces, see Wikipedia ("Linear independence", 15-Apr-2019, https://en.wikipedia.org/wiki/Linear_independence 45794): "In the theory of vector spaces, a set of vectors is said to be linearly dependent if at least one of the vectors in the set can be defined as a linear combination of the others; if no vector in the set can be written in this way, then the vectors are said to be linearly independent." Furthermore, "In order to allow the number of linearly independent vectors in a vector space to be countably infinite, it is useful to define linear dependence as follows. More generally, let V be a vector space over a field K, and let {vi | i∈I} be a family of elements of V. The family is linearly dependent over K if there exists a finite family {aj | j∈J} of elements of K, all nonzero, such that ∑j∈J ajvj=0. A set X of elements of V is linearly independent if the corresponding family{x}x∈X is linearly independent".
Remark 1: There are already definitions of (linearly) independent families (df-lindf 21022) and (linearly) independent sets (df-linds 21023). These definitions are based on the principle "of vectors, no nonzero multiple of which can be expressed as a linear combination of other elements" or (see lbsind2 20352) "every element is not in the span of the remainder of the [set]". The equivalence of Definitions df-linds 21023 and df-lininds 45794 for (linear) independence for (left) modules is shown in lindslininds 45816.
Remark 2: Subsets of the base set of a (left) module are linearly dependent if they are not linearly independent (see df-lindeps 45796) or, according to Wikipedia, "if at least one of the vectors in the set can be defined as a linear combination of the others", see islindeps2 45835. The reversed implication is not valid for arbitrary modules (but for arbitrary vector spaces), because it requires a division by a coefficient. Therefore, the definition of Wikipedia is equivalent to our definition for (left) vector spaces (see isldepslvec2 45837) and not for (left) modules in general.

 
Syntaxclininds 45792 Extend class notation with the relation between a module and its linearly independent subsets.
class linIndS
 
Syntaxclindeps 45793 Extend class notation with the relation between a module and its linearly dependent subsets.
class linDepS
 
Definitiondf-lininds 45794* Define the relation between a module and its linearly independent subsets. (Contributed by AV, 12-Apr-2019.) (Revised by AV, 24-Apr-2019.) (Revised by AV, 30-Jul-2019.)
linIndS = {⟨𝑠, 𝑚⟩ ∣ (𝑠 ∈ 𝒫 (Base‘𝑚) ∧ ∀𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑m 𝑠)((𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)𝑠) = (0g𝑚)) → ∀𝑥𝑠 (𝑓𝑥) = (0g‘(Scalar‘𝑚))))}
 
Theoremrellininds 45795 The class defining the relation between a module and its linearly independent subsets is a relation. (Contributed by AV, 13-Apr-2019.)
Rel linIndS
 
Definitiondf-lindeps 45796* Define the relation between a module and its linearly dependent subsets. (Contributed by AV, 26-Apr-2019.)
linDepS = {⟨𝑠, 𝑚⟩ ∣ ¬ 𝑠 linIndS 𝑚}
 
Theoremlinindsv 45797 The classes of the module and its linearly independent subsets are sets. (Contributed by AV, 13-Apr-2019.)
(𝑆 linIndS 𝑀 → (𝑆 ∈ V ∧ 𝑀 ∈ V))
 
Theoremislininds 45798* The property of being a linearly independent subset. (Contributed by AV, 13-Apr-2019.) (Revised by AV, 30-Jul-2019.)
𝐵 = (Base‘𝑀)    &   𝑍 = (0g𝑀)    &   𝑅 = (Scalar‘𝑀)    &   𝐸 = (Base‘𝑅)    &    0 = (0g𝑅)       ((𝑆𝑉𝑀𝑊) → (𝑆 linIndS 𝑀 ↔ (𝑆 ∈ 𝒫 𝐵 ∧ ∀𝑓 ∈ (𝐸m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 ))))
 
Theoremlinindsi 45799* The implications of being a linearly independent subset. (Contributed by AV, 13-Apr-2019.) (Revised by AV, 30-Jul-2019.)
𝐵 = (Base‘𝑀)    &   𝑍 = (0g𝑀)    &   𝑅 = (Scalar‘𝑀)    &   𝐸 = (Base‘𝑅)    &    0 = (0g𝑅)       (𝑆 linIndS 𝑀 → (𝑆 ∈ 𝒫 𝐵 ∧ ∀𝑓 ∈ (𝐸m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 )))
 
Theoremlinindslinci 45800* The implications of being a linearly independent subset and a linear combination of this subset being 0. (Contributed by AV, 24-Apr-2019.) (Revised by AV, 30-Jul-2019.)
𝐵 = (Base‘𝑀)    &   𝑍 = (0g𝑀)    &   𝑅 = (Scalar‘𝑀)    &   𝐸 = (Base‘𝑅)    &    0 = (0g𝑅)       ((𝑆 linIndS 𝑀 ∧ (𝐹 ∈ (𝐸m 𝑆) ∧ 𝐹 finSupp 0 ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍)) → ∀𝑥𝑆 (𝐹𝑥) = 0 )
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