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Theorem List for Metamath Proof Explorer - 20001-20100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdomnmuln0 20001 In a domain, a product of nonzero elements is nonzero. (Contributed by Mario Carneiro, 6-May-2015.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    0 = (0g𝑅)       ((𝑅 ∈ Domn ∧ (𝑋𝐵𝑋0 ) ∧ (𝑌𝐵𝑌0 )) → (𝑋 · 𝑌) ≠ 0 )
 
Theoremisdomn2 20002 A ring is a domain iff all nonzero elements are nonzero-divisors. (Contributed by Mario Carneiro, 28-Mar-2015.)
𝐵 = (Base‘𝑅)    &   𝐸 = (RLReg‘𝑅)    &    0 = (0g𝑅)       (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ (𝐵 ∖ { 0 }) ⊆ 𝐸))
 
Theoremdomnrrg 20003 In a domain, any nonzero element is a nonzero-divisor. (Contributed by Mario Carneiro, 28-Mar-2015.)
𝐵 = (Base‘𝑅)    &   𝐸 = (RLReg‘𝑅)    &    0 = (0g𝑅)       ((𝑅 ∈ Domn ∧ 𝑋𝐵𝑋0 ) → 𝑋𝐸)
 
Theoremopprdomn 20004 The opposite of a domain is also a domain. (Contributed by Mario Carneiro, 15-Jun-2015.)
𝑂 = (oppr𝑅)       (𝑅 ∈ Domn → 𝑂 ∈ Domn)
 
Theoremabvn0b 20005 Another characterization of domains, hinted at in abvtriv 19543: a nonzero ring is a domain iff it has an absolute value. (Contributed by Mario Carneiro, 6-May-2015.)
𝐴 = (AbsVal‘𝑅)       (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ 𝐴 ≠ ∅))
 
Theoremdrngdomn 20006 A division ring is a domain. (Contributed by Mario Carneiro, 29-Mar-2015.)
(𝑅 ∈ DivRing → 𝑅 ∈ Domn)
 
Theoremisidom 20007 An integral domain is a commutative domain. (Contributed by Mario Carneiro, 17-Jun-2015.)
(𝑅 ∈ IDomn ↔ (𝑅 ∈ CRing ∧ 𝑅 ∈ Domn))
 
Theoremfldidom 20008 A field is an integral domain. (Contributed by Mario Carneiro, 29-Mar-2015.)
(𝑅 ∈ Field → 𝑅 ∈ IDomn)
 
Theoremfidomndrnglem 20009* Lemma for fidomndrng 20010. (Contributed by Mario Carneiro, 15-Jun-2015.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &    1 = (1r𝑅)    &    = (∥r𝑅)    &    · = (.r𝑅)    &   (𝜑𝑅 ∈ Domn)    &   (𝜑𝐵 ∈ Fin)    &   (𝜑𝐴 ∈ (𝐵 ∖ { 0 }))    &   𝐹 = (𝑥𝐵 ↦ (𝑥 · 𝐴))       (𝜑𝐴 1 )
 
Theoremfidomndrng 20010 A finite domain is a division ring. (Contributed by Mario Carneiro, 15-Jun-2015.)
𝐵 = (Base‘𝑅)       (𝐵 ∈ Fin → (𝑅 ∈ Domn ↔ 𝑅 ∈ DivRing))
 
Theoremfiidomfld 20011 A finite integral domain is a field. (Contributed by Mario Carneiro, 15-Jun-2015.)
𝐵 = (Base‘𝑅)       (𝐵 ∈ Fin → (𝑅 ∈ IDomn ↔ 𝑅 ∈ Field))
 
10.8  Associative algebras
 
10.8.1  Definition and basic properties
 
Syntaxcasa 20012 Associative algebra.
class AssAlg
 
Syntaxcasp 20013 Algebraic span function.
class AlgSpan
 
Syntaxcascl 20014 Class of algebra scalar injection function.
class algSc
 
Definitiondf-assa 20015* Definition of an associative algebra. An associative algebra is a set equipped with a left-module structure on a (commutative) ring, coupled with a multiplicative internal operation on the vectors of the module that is associative and distributive for the additive structure of the left-module (so giving the vectors a ring structure) and that is also bilinear under the scalar product. (Contributed by Mario Carneiro, 29-Dec-2014.)
AssAlg = {𝑤 ∈ (LMod ∩ Ring) ∣ [(Scalar‘𝑤) / 𝑓](𝑓 ∈ CRing ∧ ∀𝑟 ∈ (Base‘𝑓)∀𝑥 ∈ (Base‘𝑤)∀𝑦 ∈ (Base‘𝑤)[( ·𝑠𝑤) / 𝑠][(.r𝑤) / 𝑡](((𝑟𝑠𝑥)𝑡𝑦) = (𝑟𝑠(𝑥𝑡𝑦)) ∧ (𝑥𝑡(𝑟𝑠𝑦)) = (𝑟𝑠(𝑥𝑡𝑦))))}
 
Definitiondf-asp 20016* Define the algebraic span of a set of vectors in an algebra. (Contributed by Mario Carneiro, 7-Jan-2015.)
AlgSpan = (𝑤 ∈ AssAlg ↦ (𝑠 ∈ 𝒫 (Base‘𝑤) ↦ {𝑡 ∈ ((SubRing‘𝑤) ∩ (LSubSp‘𝑤)) ∣ 𝑠𝑡}))
 
Definitiondf-ascl 20017* Every unital algebra contains a canonical homomorphic image of its ring of scalars as scalar multiples of the unit. This names the homomorphism. (Contributed by Mario Carneiro, 8-Mar-2015.)
algSc = (𝑤 ∈ V ↦ (𝑥 ∈ (Base‘(Scalar‘𝑤)) ↦ (𝑥( ·𝑠𝑤)(1r𝑤))))
 
Theoremisassa 20018* The properties of an associative algebra. (Contributed by Mario Carneiro, 29-Dec-2014.)
𝑉 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐵 = (Base‘𝐹)    &    · = ( ·𝑠𝑊)    &    × = (.r𝑊)       (𝑊 ∈ AssAlg ↔ ((𝑊 ∈ LMod ∧ 𝑊 ∈ Ring ∧ 𝐹 ∈ CRing) ∧ ∀𝑟𝐵𝑥𝑉𝑦𝑉 (((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)) ∧ (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦)))))
 
Theoremassalem 20019 The properties of an associative algebra. (Contributed by Mario Carneiro, 29-Dec-2014.)
𝑉 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐵 = (Base‘𝐹)    &    · = ( ·𝑠𝑊)    &    × = (.r𝑊)       ((𝑊 ∈ AssAlg ∧ (𝐴𝐵𝑋𝑉𝑌𝑉)) → (((𝐴 · 𝑋) × 𝑌) = (𝐴 · (𝑋 × 𝑌)) ∧ (𝑋 × (𝐴 · 𝑌)) = (𝐴 · (𝑋 × 𝑌))))
 
Theoremassaass 20020 Left-associative property of an associative algebra. (Contributed by Mario Carneiro, 29-Dec-2014.)
𝑉 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐵 = (Base‘𝐹)    &    · = ( ·𝑠𝑊)    &    × = (.r𝑊)       ((𝑊 ∈ AssAlg ∧ (𝐴𝐵𝑋𝑉𝑌𝑉)) → ((𝐴 · 𝑋) × 𝑌) = (𝐴 · (𝑋 × 𝑌)))
 
Theoremassaassr 20021 Right-associative property of an associative algebra. (Contributed by Mario Carneiro, 29-Dec-2014.)
𝑉 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐵 = (Base‘𝐹)    &    · = ( ·𝑠𝑊)    &    × = (.r𝑊)       ((𝑊 ∈ AssAlg ∧ (𝐴𝐵𝑋𝑉𝑌𝑉)) → (𝑋 × (𝐴 · 𝑌)) = (𝐴 · (𝑋 × 𝑌)))
 
Theoremassalmod 20022 An associative algebra is a left module. (Contributed by Mario Carneiro, 5-Dec-2014.)
(𝑊 ∈ AssAlg → 𝑊 ∈ LMod)
 
Theoremassaring 20023 An associative algebra is a ring. (Contributed by Mario Carneiro, 5-Dec-2014.)
(𝑊 ∈ AssAlg → 𝑊 ∈ Ring)
 
Theoremassasca 20024 An associative algebra's scalar field is a commutative ring. (Contributed by Mario Carneiro, 7-Jan-2015.)
𝐹 = (Scalar‘𝑊)       (𝑊 ∈ AssAlg → 𝐹 ∈ CRing)
 
Theoremassa2ass 20025 Left- and right-associative property of an associative algebra. Notice that the scalars are commuted! (Contributed by AV, 14-Aug-2019.)
𝑉 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐵 = (Base‘𝐹)    &    = (.r𝐹)    &    · = ( ·𝑠𝑊)    &    × = (.r𝑊)       ((𝑊 ∈ AssAlg ∧ (𝐴𝐵𝐶𝐵) ∧ (𝑋𝑉𝑌𝑉)) → ((𝐴 · 𝑋) × (𝐶 · 𝑌)) = ((𝐶 𝐴) · (𝑋 × 𝑌)))
 
Theoremisassad 20026* Sufficient condition for being an associative algebra. (Contributed by Mario Carneiro, 5-Dec-2014.)
(𝜑𝑉 = (Base‘𝑊))    &   (𝜑𝐹 = (Scalar‘𝑊))    &   (𝜑𝐵 = (Base‘𝐹))    &   (𝜑· = ( ·𝑠𝑊))    &   (𝜑× = (.r𝑊))    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑊 ∈ Ring)    &   (𝜑𝐹 ∈ CRing)    &   ((𝜑 ∧ (𝑟𝐵𝑥𝑉𝑦𝑉)) → ((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)))    &   ((𝜑 ∧ (𝑟𝐵𝑥𝑉𝑦𝑉)) → (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦)))       (𝜑𝑊 ∈ AssAlg)
 
Theoremissubassa3 20027 A subring that is also a subspace is a subalgebra. The key theorem is islss3 19662. (Contributed by Mario Carneiro, 7-Jan-2015.)
𝑆 = (𝑊s 𝐴)    &   𝐿 = (LSubSp‘𝑊)       ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴𝐿)) → 𝑆 ∈ AssAlg)
 
Theoremissubassa 20028 The subalgebras of an associative algebra are exactly the subrings (under the ring multiplication) that are simultaneously subspaces (under the scalar multiplication from the vector space). (Contributed by Mario Carneiro, 7-Jan-2015.)
𝑆 = (𝑊s 𝐴)    &   𝐿 = (LSubSp‘𝑊)    &   𝑉 = (Base‘𝑊)    &    1 = (1r𝑊)       ((𝑊 ∈ AssAlg ∧ 1𝐴𝐴𝑉) → (𝑆 ∈ AssAlg ↔ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴𝐿)))
 
Theoremsraassa 20029 The subring algebra over a commutative ring is an associative algebra. (Contributed by Mario Carneiro, 6-Oct-2015.)
𝐴 = ((subringAlg ‘𝑊)‘𝑆)       ((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝐴 ∈ AssAlg)
 
Theoremrlmassa 20030 The ring module over a commutative ring is an associative algebra. (Contributed by Mario Carneiro, 6-Oct-2015.)
(𝑅 ∈ CRing → (ringLMod‘𝑅) ∈ AssAlg)
 
Theoremassapropd 20031* If two structures have the same components (properties), one is an associative algebra iff the other one is. (Contributed by Mario Carneiro, 8-Feb-2015.)
(𝜑𝐵 = (Base‘𝐾))    &   (𝜑𝐵 = (Base‘𝐿))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(.r𝐾)𝑦) = (𝑥(.r𝐿)𝑦))    &   (𝜑𝐹 = (Scalar‘𝐾))    &   (𝜑𝐹 = (Scalar‘𝐿))    &   𝑃 = (Base‘𝐹)    &   ((𝜑 ∧ (𝑥𝑃𝑦𝐵)) → (𝑥( ·𝑠𝐾)𝑦) = (𝑥( ·𝑠𝐿)𝑦))       (𝜑 → (𝐾 ∈ AssAlg ↔ 𝐿 ∈ AssAlg))
 
Theoremaspval 20032* Value of the algebraic closure operation inside an associative algebra. (Contributed by Mario Carneiro, 7-Jan-2015.)
𝐴 = (AlgSpan‘𝑊)    &   𝑉 = (Base‘𝑊)    &   𝐿 = (LSubSp‘𝑊)       ((𝑊 ∈ AssAlg ∧ 𝑆𝑉) → (𝐴𝑆) = {𝑡 ∈ ((SubRing‘𝑊) ∩ 𝐿) ∣ 𝑆𝑡})
 
Theoremasplss 20033 The algebraic span of a set of vectors is a vector subspace. (Contributed by Mario Carneiro, 7-Jan-2015.)
𝐴 = (AlgSpan‘𝑊)    &   𝑉 = (Base‘𝑊)    &   𝐿 = (LSubSp‘𝑊)       ((𝑊 ∈ AssAlg ∧ 𝑆𝑉) → (𝐴𝑆) ∈ 𝐿)
 
Theoremaspid 20034 The algebraic span of a subalgebra is itself. (spanid 29052 analog.) (Contributed by Mario Carneiro, 7-Jan-2015.)
𝐴 = (AlgSpan‘𝑊)    &   𝑉 = (Base‘𝑊)    &   𝐿 = (LSubSp‘𝑊)       ((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊) ∧ 𝑆𝐿) → (𝐴𝑆) = 𝑆)
 
Theoremaspsubrg 20035 The algebraic span of a set of vectors is a subring of the algebra. (Contributed by Mario Carneiro, 7-Jan-2015.)
𝐴 = (AlgSpan‘𝑊)    &   𝑉 = (Base‘𝑊)       ((𝑊 ∈ AssAlg ∧ 𝑆𝑉) → (𝐴𝑆) ∈ (SubRing‘𝑊))
 
Theoremaspss 20036 Span preserves subset ordering. (spanss 29053 analog.) (Contributed by Mario Carneiro, 7-Jan-2015.)
𝐴 = (AlgSpan‘𝑊)    &   𝑉 = (Base‘𝑊)       ((𝑊 ∈ AssAlg ∧ 𝑆𝑉𝑇𝑆) → (𝐴𝑇) ⊆ (𝐴𝑆))
 
Theoremaspssid 20037 A set of vectors is a subset of its span. (spanss2 29050 analog.) (Contributed by Mario Carneiro, 7-Jan-2015.)
𝐴 = (AlgSpan‘𝑊)    &   𝑉 = (Base‘𝑊)       ((𝑊 ∈ AssAlg ∧ 𝑆𝑉) → 𝑆 ⊆ (𝐴𝑆))
 
Theoremasclfval 20038* Function value of the algebraic scalars function. (Contributed by Mario Carneiro, 8-Mar-2015.)
𝐴 = (algSc‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &    · = ( ·𝑠𝑊)    &    1 = (1r𝑊)       𝐴 = (𝑥𝐾 ↦ (𝑥 · 1 ))
 
Theoremasclval 20039 Value of a mapped algebra scalar. (Contributed by Mario Carneiro, 8-Mar-2015.)
𝐴 = (algSc‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &    · = ( ·𝑠𝑊)    &    1 = (1r𝑊)       (𝑋𝐾 → (𝐴𝑋) = (𝑋 · 1 ))
 
Theoremasclfn 20040 Unconditional functionality of the algebra scalars function. (Contributed by Mario Carneiro, 9-Mar-2015.)
𝐴 = (algSc‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       𝐴 Fn 𝐾
 
Theoremasclf 20041 The algebra scalars function is a function into the base set. (Contributed by Mario Carneiro, 4-Jul-2015.)
𝐴 = (algSc‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &   (𝜑𝑊 ∈ Ring)    &   (𝜑𝑊 ∈ LMod)    &   𝐾 = (Base‘𝐹)    &   𝐵 = (Base‘𝑊)       (𝜑𝐴:𝐾𝐵)
 
Theoremasclghm 20042 The algebra scalars function is a group homomorphism. (Contributed by Mario Carneiro, 4-Jul-2015.)
𝐴 = (algSc‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &   (𝜑𝑊 ∈ Ring)    &   (𝜑𝑊 ∈ LMod)       (𝜑𝐴 ∈ (𝐹 GrpHom 𝑊))
 
Theoremascl0 20043 The scalar 0 embedded into a left module corresponds to the 0 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‘(0g𝐹)) = (0g𝑊))
 
Theoremasclmul1 20044 Left multiplication by a lifted scalar is the same as the scalar operation. (Contributed by Mario Carneiro, 9-Mar-2015.)
𝐴 = (algSc‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &   𝑉 = (Base‘𝑊)    &    × = (.r𝑊)    &    · = ( ·𝑠𝑊)       ((𝑊 ∈ AssAlg ∧ 𝑅𝐾𝑋𝑉) → ((𝐴𝑅) × 𝑋) = (𝑅 · 𝑋))
 
Theoremasclmul2 20045 Right multiplication by a lifted scalar is the same as the scalar operation. (Contributed by Mario Carneiro, 9-Mar-2015.)
𝐴 = (algSc‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &   𝑉 = (Base‘𝑊)    &    × = (.r𝑊)    &    · = ( ·𝑠𝑊)       ((𝑊 ∈ AssAlg ∧ 𝑅𝐾𝑋𝑉) → (𝑋 × (𝐴𝑅)) = (𝑅 · 𝑋))
 
Theoremascldimul 20046 The algebra scalars function distributes over multiplication. (Contributed by Mario Carneiro, 8-Mar-2015.) (Proof shortened by SN, 5-Nov-2023.)
𝐴 = (algSc‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &    × = (.r𝑊)    &    · = (.r𝐹)       ((𝑊 ∈ AssAlg ∧ 𝑅𝐾𝑆𝐾) → (𝐴‘(𝑅 · 𝑆)) = ((𝐴𝑅) × (𝐴𝑆)))
 
TheoremascldimulOLD 20047 The algebra scalars function distributes over multiplication. (Contributed by Mario Carneiro, 8-Mar-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐴 = (algSc‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &    × = (.r𝑊)    &    · = (.r𝐹)       ((𝑊 ∈ AssAlg ∧ 𝑅𝐾𝑆𝐾) → (𝐴‘(𝑅 · 𝑆)) = ((𝐴𝑅) × (𝐴𝑆)))
 
Theoremasclinvg 20048 The group inverse (negation) of a lifted scalar is the lifted negation of the scalar. (Contributed by AV, 2-Sep-2019.)
𝐴 = (algSc‘𝑊)    &   𝑅 = (Scalar‘𝑊)    &   𝐵 = (Base‘𝑅)    &   𝐼 = (invg𝑅)    &   𝐽 = (invg𝑊)       ((𝑊 ∈ LMod ∧ 𝑊 ∈ Ring ∧ 𝐶𝐵) → (𝐽‘(𝐴𝐶)) = (𝐴‘(𝐼𝐶)))
 
Theoremasclrhm 20049 The scalar injection is a ring homomorphism. (Contributed by Mario Carneiro, 8-Mar-2015.)
𝐴 = (algSc‘𝑊)    &   𝐹 = (Scalar‘𝑊)       (𝑊 ∈ AssAlg → 𝐴 ∈ (𝐹 RingHom 𝑊))
 
Theoremrnascl 20050 The set of injected scalars is also interpretable as the span of the identity. (Contributed by Mario Carneiro, 9-Mar-2015.)
𝐴 = (algSc‘𝑊)    &    1 = (1r𝑊)    &   𝑁 = (LSpan‘𝑊)       (𝑊 ∈ AssAlg → ran 𝐴 = (𝑁‘{ 1 }))
 
Theoremissubassa2 20051 A subring of a unital algebra is a subspace and thus a subalgebra iff it contains all scalar multiples of the identity. (Contributed by Mario Carneiro, 9-Mar-2015.)
𝐴 = (algSc‘𝑊)    &   𝐿 = (LSubSp‘𝑊)       ((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) → (𝑆𝐿 ↔ ran 𝐴𝑆))
 
Theoremrnasclsubrg 20052 The scalar multiples of the unit vector form a subring of the vectors. (Contributed by SN, 5-Nov-2023.)
𝐶 = (algSc‘𝑊)    &   (𝜑𝑊 ∈ AssAlg)       (𝜑 → ran 𝐶 ∈ (SubRing‘𝑊))
 
Theoremrnasclmulcl 20053 (Vector) multiplication is closed for scalar multiples of the unit vector. (Contributed by SN, 5-Nov-2023.)
𝐶 = (algSc‘𝑊)    &    × = (.r𝑊)    &   (𝜑𝑊 ∈ AssAlg)       ((𝜑 ∧ (𝑋 ∈ ran 𝐶𝑌 ∈ ran 𝐶)) → (𝑋 × 𝑌) ∈ ran 𝐶)
 
Theoremrnasclassa 20054 The scalar multiples of the unit vector form a subalgebra of the vectors. (Contributed by SN, 16-Nov-2023.)
𝐴 = (algSc‘𝑊)    &   𝑈 = (𝑊s ran 𝐴)    &   (𝜑𝑊 ∈ AssAlg)       (𝜑𝑈 ∈ AssAlg)
 
Theoremressascl 20055 The injection of scalars is invariant between subalgebras and superalgebras. (Contributed by Mario Carneiro, 9-Mar-2015.)
𝐴 = (algSc‘𝑊)    &   𝑋 = (𝑊s 𝑆)       (𝑆 ∈ (SubRing‘𝑊) → 𝐴 = (algSc‘𝑋))
 
Theoremasclpropd 20056* If two structures have the same components (properties), one is an associative algebra iff the other one is. The last hypotheses on 1r can be discharged either by letting 𝑊 = V (if strong equality is known on ·𝑠) or assuming 𝐾 is a ring. (Contributed by Mario Carneiro, 5-Jul-2015.)
𝐹 = (Scalar‘𝐾)    &   𝐺 = (Scalar‘𝐿)    &   (𝜑𝑃 = (Base‘𝐹))    &   (𝜑𝑃 = (Base‘𝐺))    &   ((𝜑 ∧ (𝑥𝑃𝑦𝑊)) → (𝑥( ·𝑠𝐾)𝑦) = (𝑥( ·𝑠𝐿)𝑦))    &   (𝜑 → (1r𝐾) = (1r𝐿))    &   (𝜑 → (1r𝐾) ∈ 𝑊)       (𝜑 → (algSc‘𝐾) = (algSc‘𝐿))
 
Theoremaspval2 20057 The algebraic closure is the ring closure when the generating set is expanded to include all scalars. EDITORIAL : In light of this, is AlgSpan independently needed? (Contributed by Stefan O'Rear, 9-Mar-2015.)
𝐴 = (AlgSpan‘𝑊)    &   𝐶 = (algSc‘𝑊)    &   𝑅 = (mrCls‘(SubRing‘𝑊))    &   𝑉 = (Base‘𝑊)       ((𝑊 ∈ AssAlg ∧ 𝑆𝑉) → (𝐴𝑆) = (𝑅‘(ran 𝐶𝑆)))
 
Theoremassamulgscmlem1 20058 Lemma 1 for assamulgscm 20060 (induction base). (Contributed by AV, 26-Aug-2019.)
𝑉 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐵 = (Base‘𝐹)    &    · = ( ·𝑠𝑊)    &   𝐺 = (mulGrp‘𝐹)    &    = (.g𝐺)    &   𝐻 = (mulGrp‘𝑊)    &   𝐸 = (.g𝐻)       (((𝐴𝐵𝑋𝑉) ∧ 𝑊 ∈ AssAlg) → (0𝐸(𝐴 · 𝑋)) = ((0 𝐴) · (0𝐸𝑋)))
 
Theoremassamulgscmlem2 20059 Lemma for assamulgscm 20060 (induction step). (Contributed by AV, 26-Aug-2019.)
𝑉 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐵 = (Base‘𝐹)    &    · = ( ·𝑠𝑊)    &   𝐺 = (mulGrp‘𝐹)    &    = (.g𝐺)    &   𝐻 = (mulGrp‘𝑊)    &   𝐸 = (.g𝐻)       (𝑦 ∈ ℕ0 → (((𝐴𝐵𝑋𝑉) ∧ 𝑊 ∈ AssAlg) → ((𝑦𝐸(𝐴 · 𝑋)) = ((𝑦 𝐴) · (𝑦𝐸𝑋)) → ((𝑦 + 1)𝐸(𝐴 · 𝑋)) = (((𝑦 + 1) 𝐴) · ((𝑦 + 1)𝐸𝑋)))))
 
Theoremassamulgscm 20060 Exponentiation of a scalar multiplication in an associative algebra: (𝑎 · 𝑋)↑𝑁 = (𝑎𝑁) × (𝑋𝑁). (Contributed by AV, 26-Aug-2019.)
𝑉 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐵 = (Base‘𝐹)    &    · = ( ·𝑠𝑊)    &   𝐺 = (mulGrp‘𝐹)    &    = (.g𝐺)    &   𝐻 = (mulGrp‘𝑊)    &   𝐸 = (.g𝐻)       ((𝑊 ∈ AssAlg ∧ (𝑁 ∈ ℕ0𝐴𝐵𝑋𝑉)) → (𝑁𝐸(𝐴 · 𝑋)) = ((𝑁 𝐴) · (𝑁𝐸𝑋)))
 
10.9  Abstract multivariate polynomials
 
10.9.1  Definition and basic properties
 
Syntaxcmps 20061 Multivariate power series.
class mPwSer
 
Syntaxcmvr 20062 Multivariate power series variables.
class mVar
 
Syntaxcmpl 20063 Multivariate polynomials.
class mPoly
 
Syntaxcltb 20064 Ordering on terms of a multivariate polynomial.
class <bag
 
Syntaxcopws 20065 Ordered set of power series.
class ordPwSer
 
Definitiondf-psr 20066* Define the algebra of power series over the index set 𝑖 and with coefficients from the ring 𝑟. (Contributed by Mario Carneiro, 21-Mar-2015.)
mPwSer = (𝑖 ∈ V, 𝑟 ∈ V ↦ { ∈ (ℕ0m 𝑖) ∣ ( “ ℕ) ∈ Fin} / 𝑑((Base‘𝑟) ↑m 𝑑) / 𝑏({⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), ( ∘f (+g𝑟) ↾ (𝑏 × 𝑏))⟩, ⟨(.r‘ndx), (𝑓𝑏, 𝑔𝑏 ↦ (𝑘𝑑 ↦ (𝑟 Σg (𝑥 ∈ {𝑦𝑑𝑦r𝑘} ↦ ((𝑓𝑥)(.r𝑟)(𝑔‘(𝑘f𝑥)))))))⟩} ∪ {⟨(Scalar‘ndx), 𝑟⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑟), 𝑓𝑏 ↦ ((𝑑 × {𝑥}) ∘f (.r𝑟)𝑓))⟩, ⟨(TopSet‘ndx), (∏t‘(𝑑 × {(TopOpen‘𝑟)}))⟩}))
 
Definitiondf-mvr 20067* Define the generating elements of the power series algebra. (Contributed by Mario Carneiro, 7-Jan-2015.)
mVar = (𝑖 ∈ V, 𝑟 ∈ V ↦ (𝑥𝑖 ↦ (𝑓 ∈ { ∈ (ℕ0m 𝑖) ∣ ( “ ℕ) ∈ Fin} ↦ if(𝑓 = (𝑦𝑖 ↦ if(𝑦 = 𝑥, 1, 0)), (1r𝑟), (0g𝑟)))))
 
Definitiondf-mpl 20068* Define the subalgebra of the power series algebra generated by the variables; this is the polynomial algebra (the set of power series with finite degree). (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by AV, 25-Jun-2019.)
mPoly = (𝑖 ∈ V, 𝑟 ∈ V ↦ (𝑖 mPwSer 𝑟) / 𝑤(𝑤s {𝑓 ∈ (Base‘𝑤) ∣ 𝑓 finSupp (0g𝑟)}))
 
Definitiondf-ltbag 20069* Define a well-order on the set of all finite bags from the index set 𝑖 given a wellordering 𝑟 of 𝑖. (Contributed by Mario Carneiro, 8-Feb-2015.)
<bag = (𝑟 ∈ V, 𝑖 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ { ∈ (ℕ0m 𝑖) ∣ ( “ ℕ) ∈ Fin} ∧ ∃𝑧𝑖 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝑖 (𝑧𝑟𝑤 → (𝑥𝑤) = (𝑦𝑤))))})
 
Definitiondf-opsr 20070* Define a total order on the set of all power series in 𝑠 from the index set 𝑖 given a wellordering 𝑟 of 𝑖 and a totally ordered base ring 𝑠. (Contributed by Mario Carneiro, 8-Feb-2015.)
ordPwSer = (𝑖 ∈ V, 𝑠 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑖 × 𝑖) ↦ (𝑖 mPwSer 𝑠) / 𝑝(𝑝 sSet ⟨(le‘ndx), {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (Base‘𝑝) ∧ ([{ ∈ (ℕ0m 𝑖) ∣ ( “ ℕ) ∈ Fin} / 𝑑]𝑧𝑑 ((𝑥𝑧)(lt‘𝑠)(𝑦𝑧) ∧ ∀𝑤𝑑 (𝑤(𝑟 <bag 𝑖)𝑧 → (𝑥𝑤) = (𝑦𝑤))) ∨ 𝑥 = 𝑦))}⟩)))
 
Theoremreldmpsr 20071 The multivariate power series constructor is a proper binary operator. (Contributed by Mario Carneiro, 21-Mar-2015.)
Rel dom mPwSer
 
Theorempsrval 20072* Value of the multivariate power series structure. (Contributed by Mario Carneiro, 29-Dec-2014.)
𝑆 = (𝐼 mPwSer 𝑅)    &   𝐾 = (Base‘𝑅)    &    + = (+g𝑅)    &    · = (.r𝑅)    &   𝑂 = (TopOpen‘𝑅)    &   𝐷 = { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}    &   (𝜑𝐵 = (𝐾m 𝐷))    &    = ( ∘f + ↾ (𝐵 × 𝐵))    &    × = (𝑓𝐵, 𝑔𝐵 ↦ (𝑘𝐷 ↦ (𝑅 Σg (𝑥 ∈ {𝑦𝐷𝑦r𝑘} ↦ ((𝑓𝑥) · (𝑔‘(𝑘f𝑥)))))))    &    = (𝑥𝐾, 𝑓𝐵 ↦ ((𝐷 × {𝑥}) ∘f · 𝑓))    &   (𝜑𝐽 = (∏t‘(𝐷 × {𝑂})))    &   (𝜑𝐼𝑊)    &   (𝜑𝑅𝑋)       (𝜑𝑆 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑅⟩, ⟨( ·𝑠 ‘ndx), ⟩, ⟨(TopSet‘ndx), 𝐽⟩}))
 
Theorempsrvalstr 20073 The multivariate power series structure is a function. (Contributed by Mario Carneiro, 8-Feb-2015.)
({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑅⟩, ⟨( ·𝑠 ‘ndx), · ⟩, ⟨(TopSet‘ndx), 𝐽⟩}) Struct ⟨1, 9⟩
 
Theorempsrbag 20074* Elementhood in the set of finite bags. (Contributed by Mario Carneiro, 29-Dec-2014.)
𝐷 = {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}       (𝐼𝑉 → (𝐹𝐷 ↔ (𝐹:𝐼⟶ℕ0 ∧ (𝐹 “ ℕ) ∈ Fin)))
 
Theorempsrbagf 20075* A finite bag is a function. (Contributed by Mario Carneiro, 29-Dec-2014.)
𝐷 = {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}       ((𝐼𝑉𝐹𝐷) → 𝐹:𝐼⟶ℕ0)
 
Theoremsnifpsrbag 20076* A bag containing one element is a finite bag. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by AV, 8-Jul-2019.)
𝐷 = {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}       ((𝐼𝑉𝑁 ∈ ℕ0) → (𝑦𝐼 ↦ if(𝑦 = 𝑋, 𝑁, 0)) ∈ 𝐷)
 
Theoremfczpsrbag 20077* The constant function equal to zero is a finite bag. (Contributed by AV, 8-Jul-2019.)
𝐷 = {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}       (𝐼𝑉 → (𝑥𝐼 ↦ 0) ∈ 𝐷)
 
Theorempsrbaglesupp 20078* The support of a dominated bag is smaller than the dominating bag. (Contributed by Mario Carneiro, 29-Dec-2014.)
𝐷 = {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}       ((𝐼𝑉 ∧ (𝐹𝐷𝐺:𝐼⟶ℕ0𝐺r𝐹)) → (𝐺 “ ℕ) ⊆ (𝐹 “ ℕ))
 
Theorempsrbaglecl 20079* The set of finite bags is downward-closed. (Contributed by Mario Carneiro, 29-Dec-2014.)
𝐷 = {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}       ((𝐼𝑉 ∧ (𝐹𝐷𝐺:𝐼⟶ℕ0𝐺r𝐹)) → 𝐺𝐷)
 
Theorempsrbagaddcl 20080* The sum of two finite bags is a finite bag. (Contributed by Mario Carneiro, 9-Jan-2015.)
𝐷 = {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}       ((𝐼𝑉𝐹𝐷𝐺𝐷) → (𝐹f + 𝐺) ∈ 𝐷)
 
Theorempsrbagcon 20081* The analogue of the statement "0 ≤ 𝐺𝐹 implies 0 ≤ 𝐹𝐺𝐹 " for finite bags. (Contributed by Mario Carneiro, 29-Dec-2014.)
𝐷 = {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}       ((𝐼𝑉 ∧ (𝐹𝐷𝐺:𝐼⟶ℕ0𝐺r𝐹)) → ((𝐹f𝐺) ∈ 𝐷 ∧ (𝐹f𝐺) ∘r𝐹))
 
Theorempsrbaglefi 20082* There are finitely many bags dominated by a given bag. (Contributed by Mario Carneiro, 29-Dec-2014.) (Revised by Mario Carneiro, 25-Jan-2015.)
𝐷 = {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}       ((𝐼𝑉𝐹𝐷) → {𝑦𝐷𝑦r𝐹} ∈ Fin)
 
Theorempsrbagconcl 20083* The complement of a bag is a bag. (Contributed by Mario Carneiro, 29-Dec-2014.)
𝐷 = {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}    &   𝑆 = {𝑦𝐷𝑦r𝐹}       ((𝐼𝑉𝐹𝐷𝑋𝑆) → (𝐹f𝑋) ∈ 𝑆)
 
Theorempsrbagconf1o 20084* Bag complementation is a bijection on the set of bags dominated by a given bag 𝐹. (Contributed by Mario Carneiro, 29-Dec-2014.)
𝐷 = {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}    &   𝑆 = {𝑦𝐷𝑦r𝐹}       ((𝐼𝑉𝐹𝐷) → (𝑥𝑆 ↦ (𝐹f𝑥)):𝑆1-1-onto𝑆)
 
Theoremgsumbagdiaglem 20085* Lemma for gsumbagdiag 20086. (Contributed by Mario Carneiro, 5-Jan-2015.)
𝐷 = {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}    &   𝑆 = {𝑦𝐷𝑦r𝐹}    &   (𝜑𝐼𝑉)    &   (𝜑𝐹𝐷)       ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑋)})) → (𝑌𝑆𝑋 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑌)}))
 
Theoremgsumbagdiag 20086* Two-dimensional commutation of a group sum over a "triangular" region. fsum0diag 15122 analogue for finite bags. (Contributed by Mario Carneiro, 5-Jan-2015.)
𝐷 = {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}    &   𝑆 = {𝑦𝐷𝑦r𝐹}    &   (𝜑𝐼𝑉)    &   (𝜑𝐹𝐷)    &   𝐵 = (Base‘𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   ((𝜑 ∧ (𝑗𝑆𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)})) → 𝑋𝐵)       (𝜑 → (𝐺 Σg (𝑗𝑆, 𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑋)) = (𝐺 Σg (𝑘𝑆, 𝑗 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑘)} ↦ 𝑋)))
 
Theorempsrass1lem 20087* A group sum commutation used by psrass1 20115. (Contributed by Mario Carneiro, 5-Jan-2015.)
𝐷 = {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}    &   𝑆 = {𝑦𝐷𝑦r𝐹}    &   (𝜑𝐼𝑉)    &   (𝜑𝐹𝐷)    &   𝐵 = (Base‘𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   ((𝜑 ∧ (𝑗𝑆𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)})) → 𝑋𝐵)    &   (𝑘 = (𝑛f𝑗) → 𝑋 = 𝑌)       (𝜑 → (𝐺 Σg (𝑛𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥𝐷𝑥r𝑛} ↦ 𝑌)))) = (𝐺 Σg (𝑗𝑆 ↦ (𝐺 Σg (𝑘 ∈ {𝑥𝐷𝑥r ≤ (𝐹f𝑗)} ↦ 𝑋)))))
 
Theorempsrbas 20088* The base set of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by Mario Carneiro, 2-Oct-2015.) (Proof shortened by AV, 8-Jul-2019.)
𝑆 = (𝐼 mPwSer 𝑅)    &   𝐾 = (Base‘𝑅)    &   𝐷 = {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}    &   𝐵 = (Base‘𝑆)    &   (𝜑𝐼𝑉)       (𝜑𝐵 = (𝐾m 𝐷))
 
Theorempsrelbas 20089* An element of the set of power series is a function on the coefficients. (Contributed by Mario Carneiro, 28-Dec-2014.)
𝑆 = (𝐼 mPwSer 𝑅)    &   𝐾 = (Base‘𝑅)    &   𝐷 = {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}    &   𝐵 = (Base‘𝑆)    &   (𝜑𝑋𝐵)       (𝜑𝑋:𝐷𝐾)
 
Theorempsrelbasfun 20090 An element of the set of power series is a function. (Contributed by AV, 17-Jul-2019.)
𝑆 = (𝐼 mPwSer 𝑅)    &   𝐵 = (Base‘𝑆)       (𝑋𝐵 → Fun 𝑋)
 
Theorempsrplusg 20091 The addition operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by Mario Carneiro, 2-Oct-2015.)
𝑆 = (𝐼 mPwSer 𝑅)    &   𝐵 = (Base‘𝑆)    &    + = (+g𝑅)    &    = (+g𝑆)        = ( ∘f + ↾ (𝐵 × 𝐵))
 
Theorempsradd 20092 The addition operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.)
𝑆 = (𝐼 mPwSer 𝑅)    &   𝐵 = (Base‘𝑆)    &    + = (+g𝑅)    &    = (+g𝑆)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → (𝑋 𝑌) = (𝑋f + 𝑌))
 
Theorempsraddcl 20093 Closure of the power series addition operation. (Contributed by Mario Carneiro, 28-Dec-2014.)
𝑆 = (𝐼 mPwSer 𝑅)    &   𝐵 = (Base‘𝑆)    &    + = (+g𝑆)    &   (𝜑𝑅 ∈ Grp)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → (𝑋 + 𝑌) ∈ 𝐵)
 
Theorempsrmulr 20094* The multiplication operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by Mario Carneiro, 2-Oct-2015.) (Proof shortened by AV, 2-Mar-2024.)
𝑆 = (𝐼 mPwSer 𝑅)    &   𝐵 = (Base‘𝑆)    &    · = (.r𝑅)    &    = (.r𝑆)    &   𝐷 = { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}        = (𝑓𝐵, 𝑔𝐵 ↦ (𝑘𝐷 ↦ (𝑅 Σg (𝑥 ∈ {𝑦𝐷𝑦r𝑘} ↦ ((𝑓𝑥) · (𝑔‘(𝑘f𝑥)))))))
 
Theorempsrmulfval 20095* The multiplication operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.)
𝑆 = (𝐼 mPwSer 𝑅)    &   𝐵 = (Base‘𝑆)    &    · = (.r𝑅)    &    = (.r𝑆)    &   𝐷 = { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}    &   (𝜑𝐹𝐵)    &   (𝜑𝐺𝐵)       (𝜑 → (𝐹 𝐺) = (𝑘𝐷 ↦ (𝑅 Σg (𝑥 ∈ {𝑦𝐷𝑦r𝑘} ↦ ((𝐹𝑥) · (𝐺‘(𝑘f𝑥)))))))
 
Theorempsrmulval 20096* The multiplication operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.)
𝑆 = (𝐼 mPwSer 𝑅)    &   𝐵 = (Base‘𝑆)    &    · = (.r𝑅)    &    = (.r𝑆)    &   𝐷 = { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}    &   (𝜑𝐹𝐵)    &   (𝜑𝐺𝐵)    &   (𝜑𝑋𝐷)       (𝜑 → ((𝐹 𝐺)‘𝑋) = (𝑅 Σg (𝑘 ∈ {𝑦𝐷𝑦r𝑋} ↦ ((𝐹𝑘) · (𝐺‘(𝑋f𝑘))))))
 
Theorempsrmulcllem 20097* Closure of the power series multiplication operation. (Contributed by Mario Carneiro, 29-Dec-2014.)
𝑆 = (𝐼 mPwSer 𝑅)    &   𝐵 = (Base‘𝑆)    &    · = (.r𝑆)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   𝐷 = {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}       (𝜑 → (𝑋 · 𝑌) ∈ 𝐵)
 
Theorempsrmulcl 20098 Closure of the power series multiplication operation. (Contributed by Mario Carneiro, 29-Dec-2014.)
𝑆 = (𝐼 mPwSer 𝑅)    &   𝐵 = (Base‘𝑆)    &    · = (.r𝑆)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → (𝑋 · 𝑌) ∈ 𝐵)
 
Theorempsrsca 20099 The scalar field of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.)
𝑆 = (𝐼 mPwSer 𝑅)    &   (𝜑𝐼𝑉)    &   (𝜑𝑅𝑊)       (𝜑𝑅 = (Scalar‘𝑆))
 
Theorempsrvscafval 20100* The scalar multiplication operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by Mario Carneiro, 2-Oct-2015.)
𝑆 = (𝐼 mPwSer 𝑅)    &    = ( ·𝑠𝑆)    &   𝐾 = (Base‘𝑅)    &   𝐵 = (Base‘𝑆)    &    · = (.r𝑅)    &   𝐷 = { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}        = (𝑥𝐾, 𝑓𝐵 ↦ ((𝐷 × {𝑥}) ∘f · 𝑓))
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