 Home Metamath Proof ExplorerTheorem List (p. 440 of 445) < Previous  Next > Bad symbols? Try the GIF version. Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

 Color key: Metamath Proof Explorer (1-28382) Hilbert Space Explorer (28383-29905) Users' Mathboxes (29906-44433)

Theorem List for Metamath Proof Explorer - 43901-44000   *Has distinct variable group(s)
TypeLabelDescription
Statement

TheoremaltgsumbcALT 43901* Alternate proof of altgsumbc 43900, using Pascal's rule (bcpascm1 43899) instead of the binomial theorem (binom 15022). (Contributed by AV, 8-Sep-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑁 ∈ ℕ → Σ𝑘 ∈ (0...𝑁)((-1↑𝑘) · (𝑁C𝑘)) = 0)

20.38.19.3  The ` ZZ `-module ` ZZ X. ZZ `

Theoremzlmodzxzlmod 43902 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 43903 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 43904 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 43905 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 43906 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 43907 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 43908 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)⟩})

20.38.19.4  Ordered group sum operation (extension)

Theoremmgpsumunsn 43909* 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 43910* 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 43911* 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)))

Theoremgsumsplit2f 43912* Split a group sum into two parts. (Contributed by AV, 4-Sep-2019.)
𝑘𝜑    &   𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &    + = (+g𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐴𝑉)    &   ((𝜑𝑘𝐴) → 𝑋𝐵)    &   (𝜑 → (𝑘𝐴𝑋) finSupp 0 )    &   (𝜑 → (𝐶𝐷) = ∅)    &   (𝜑𝐴 = (𝐶𝐷))       (𝜑 → (𝐺 Σg (𝑘𝐴𝑋)) = ((𝐺 Σg (𝑘𝐶𝑋)) + (𝐺 Σg (𝑘𝐷𝑋))))

Theoremgsumdifsndf 43913* Extract a summand from a finitely supported group sum. (Contributed by AV, 4-Sep-2019.)
𝑘𝑌    &   𝑘𝜑    &   𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐴𝑊)    &   (𝜑 → (𝑘𝐴𝑋) finSupp (0g𝐺))    &   ((𝜑𝑘𝐴) → 𝑋𝐵)    &   (𝜑𝑀𝐴)    &   (𝜑𝑌𝐵)    &   ((𝜑𝑘 = 𝑀) → 𝑋 = 𝑌)       (𝜑 → (𝐺 Σg (𝑘𝐴𝑋)) = ((𝐺 Σg (𝑘 ∈ (𝐴 ∖ {𝑀}) ↦ 𝑋)) + 𝑌))

20.38.19.5  Symmetric groups (extension)

Theoremexple2lt6 43914 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 43915 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 43916 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)

20.38.19.6  Divisibility (extension)

Theoreminvginvrid 43917 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)

20.38.19.7  The support of functions (extension)

Theoremrmsupp0 43918* 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 43919* 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 43920* 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 43921 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 43922* 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)))

20.38.19.8  Finitely supported functions (extension)

Theoremrmsuppfi 43923* 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 43924* 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 43925 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 43926 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 43927* 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 43928* 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 43929* 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 43930* 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 43931* 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)

20.38.19.9  Left modules (extension)

Theoremlmodvsmdi 43932 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 43933* 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))

20.38.19.10  Associative algebras (extension)

Theoremascl1 43934 The scalar 1 embedded into a left module corresponds to the 1 of the left module if the left module is also a ring. (Contributed by AV, 31-Jul-2019.)
𝐴 = (algSc‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑊 ∈ Ring)       (𝜑 → (𝐴‘(1r𝐹)) = (1r𝑊))

Theoremassaascl0 43935 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 43936 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))

20.38.19.11  Univariate polynomials (extension)

Theoremply1vr1smo 43937 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 43938 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 43939 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 43940* 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 43941 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 43942* Lemma 1 for ply1mulgsum 43946. (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 43943* Lemma 2 for ply1mulgsum 43946. (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 43944* Lemma 3 for ply1mulgsum 43946. (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 43945* Lemma 4 for ply1mulgsum 43946. (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 43946* 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 43947 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 43948 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 )

20.38.19.12  Univariate polynomials (examples)

Theoremlinply1 43949 A term of the form 𝑥𝐶 is a (univariate) polynomial, also called "linear polynomial". (Part of ply1remlem 24443). (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 43950 A term of the form 𝑥𝐶 evaluated for 𝑥 = 𝑉 results in 𝑉𝐶 (part of ply1remlem 24443). (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))

Theoremzringsubgval 43951 Subtraction in the ring of integers. (Contributed by AV, 3-Aug-2019.)
= (-g‘ℤring)       ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → (𝑋𝑌) = (𝑋 𝑌))

Theoremlinevalexample 43952 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

20.38.20  Linear algebra (extension)

20.38.20.1  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 the definitions df-dmat 20787 and df-scmat 20788 define diagonal and scalar matrices as sets.

Syntaxcdmatalt 43953 Alternative notation for the algebra of diagonal matrices.
class DMatALT

Syntaxcscmatalt 43954 Alternative notation for the algebra of scalar matrices.
class ScMatALT

Definitiondf-dmatalt 43955* 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 43956* 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 43957* 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 43958* 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 43959* 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 43960 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)))

20.38.20.2  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 43963, 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 43964, so that we can show that such sets are submodules of the corresponding modules, see lincolss 43991.
Remark:According to Wikipedia ("Linear span", 28-Apr-2019, https://en.wikipedia.org/wiki/Linear_span 43991) "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 19438, the set of all linear combinations as defined by df-lco 43964 follows the alternative approach. That both definitions are equivalent is shown by lspeqlco 43996.

Syntaxclinc 43961 Extend class notation with the operation constructing a linear combination (of vectors from a left module).
class linC

Syntaxclinco 43962 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 43963* 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 43964* 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 43965* 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 43966* 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 43967* 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 43968* 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 43969* 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 43970 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 43971 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 43972 The value of an empty linear combination. (Contributed by AV, 12-Apr-2019.)
(𝑀𝑋 → (∅( linC ‘𝑀)∅) = (0g𝑀))

Theoremlincvalsng 43973 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 43974 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 43975 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 43976 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 43977 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 43978* 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 43979* 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 43980* 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 43981 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 43982* 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 43983 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 43984 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 43985 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 43986 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 43987* 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 43988 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 43989 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 43990 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 43991 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 43992* 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 43993 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 43994 Lemma for lspeqlco 43996. (Contributed by AV, 17-Apr-2019.)
𝐵 = (Base‘𝑀)       ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → ((LSpan‘𝑀)‘𝑉) ⊆ (𝑀 LinCo 𝑉))

Theoremlcosslsp 43995 Lemma for lspeqlco 43996. (Contributed by AV, 20-Apr-2019.)
𝐵 = (Base‘𝑀)       ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝑀 LinCo 𝑉) ⊆ ((LSpan‘𝑀)‘𝑉))

Theoremlspeqlco 43996 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 19438) 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))

20.38.20.3  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 43999 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 43999): "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 20636) and (linearly) independent sets (df-linds 20637). 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 19547) "every element is not in the span of the remainder of the [set]". The equivalence of the definitions df-linds 20637 and df-lininds 43999 for (linear) independence for (left) modules is shown in lindslininds 44021.
Remark 2: Subsets of the base set of a (left) module are linearly dependent if they are not linearly independent (see df-lindeps 44001) 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 44040. 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 44042) and not for (left) modules in general.

Syntaxclininds 43997 Extend class notation with the relation between a module and its linearly independent subsets.
class linIndS

Syntaxclindeps 43998 Extend class notation with the relation between a module and its linearly dependent subsets.
class linDepS

Definitiondf-lininds 43999* 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 44000 The class defining the relation between a module and its linearly independent subsets is a relation. (Contributed by AV, 13-Apr-2019.)
Rel linIndS

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44433
 Copyright terms: Public domain < Previous  Next >