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Theorem List for Metamath Proof Explorer - 19001-19100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem0ring01eqbi 19001 In a unital ring the zero equals the unity iff the ring is the zero ring. (Contributed by FL, 14-Feb-2010.) (Revised by AV, 23-Jan-2020.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &    1 = (1r𝑅)       (𝑅 ∈ Ring → (𝐵 ≈ 1𝑜1 = 0 ))
 
Theoremrng1nnzr 19002 The (smallest) structure representing a zero ring is not a nonzero ring. (Contributed by AV, 29-Apr-2019.)
𝑀 = {⟨(Base‘ndx), {𝑍}⟩, ⟨(+g‘ndx), {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩, ⟨(.r‘ndx), {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩}       (𝑍𝑉𝑀 ∉ NzRing)
 
Theoremring1zr 19003 The only (unital) ring with a base set consisting of one element is the zero ring (at least if its operations are internal binary operations). Note: The assumption 𝑅 ∈ Ring could be weakened if a definition of a non-unital ring ("Rng") was available (it would be sufficient that the multiplication is closed). (Contributed by FL, 13-Feb-2010.) (Revised by AV, 25-Jan-2020.) (Proof shortened by AV, 7-Feb-2020.)
𝐵 = (Base‘𝑅)    &    + = (+g𝑅)    &    = (.r𝑅)       (((𝑅 ∈ Ring ∧ + Fn (𝐵 × 𝐵) ∧ Fn (𝐵 × 𝐵)) ∧ 𝑍𝐵) → (𝐵 = {𝑍} ↔ ( + = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩} ∧ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩})))
 
Theoremrngen1zr 19004 The only (unital) ring with one element is the zero ring (at least if its operations are internal binary operations). Note: The assumption 𝑅 ∈ Ring could be weakened if a definition of a non-unital ring ("Rng") was available (it would be sufficient that the multiplication is closed). (Contributed by FL, 14-Feb-2010.) (Revised by AV, 25-Jan-2020.)
𝐵 = (Base‘𝑅)    &    + = (+g𝑅)    &    = (.r𝑅)       (((𝑅 ∈ Ring ∧ + Fn (𝐵 × 𝐵) ∧ Fn (𝐵 × 𝐵)) ∧ 𝑍𝐵) → (𝐵 ≈ 1𝑜 ↔ ( + = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩} ∧ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩})))
 
Theoremringen1zr 19005 The only unital ring with one element is the zero ring (at least if its operations are internal binary operations). Note: The assumption 𝑅 ∈ Ring could be weakened if a definition of a non-unital ring ("Rng") was available (it would be sufficient that the multiplication is closed). (Contributed by FL, 15-Feb-2010.) (Revised by AV, 25-Jan-2020.)
𝐵 = (Base‘𝑅)    &    + = (+g𝑅)    &    = (.r𝑅)    &   𝑍 = (0g𝑅)       ((𝑅 ∈ Ring ∧ + Fn (𝐵 × 𝐵) ∧ Fn (𝐵 × 𝐵)) → (𝐵 ≈ 1𝑜 ↔ ( + = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩} ∧ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩})))
 
Theoremrng1nfld 19006 The zero ring is not a field. (Contributed by AV, 29-Apr-2019.)
𝑀 = {⟨(Base‘ndx), {𝑍}⟩, ⟨(+g‘ndx), {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩, ⟨(.r‘ndx), {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩}       (𝑍𝑉𝑀 ∉ Field)
 
10.8.5  Left regular elements. More kinds of rings
 
Syntaxcrlreg 19007 Set of left-regular elements in a ring.
class RLReg
 
Syntaxcdomn 19008 Class of (ring theoretic) domains.
class Domn
 
Syntaxcidom 19009 Class of integral domains.
class IDomn
 
Syntaxcpid 19010 Class of principal ideal domains.
class PID
 
Definitiondf-rlreg 19011* Define the set of left-regular elements in a ring as those elements which are not left zero divisors, meaning that multiplying a nonzero element on the left by a left-regular element gives a nonzero product. (Contributed by Stefan O'Rear, 22-Mar-2015.)
RLReg = (𝑟 ∈ V ↦ {𝑥 ∈ (Base‘𝑟) ∣ ∀𝑦 ∈ (Base‘𝑟)((𝑥(.r𝑟)𝑦) = (0g𝑟) → 𝑦 = (0g𝑟))})
 
Definitiondf-domn 19012* A domain is a nonzero ring in which there are no nontrivial zero divisors. (Contributed by Mario Carneiro, 28-Mar-2015.)
Domn = {𝑟 ∈ NzRing ∣ [(Base‘𝑟) / 𝑏][(0g𝑟) / 𝑧]𝑥𝑏𝑦𝑏 ((𝑥(.r𝑟)𝑦) = 𝑧 → (𝑥 = 𝑧𝑦 = 𝑧))}
 
Definitiondf-idom 19013 An integral domain is a commutative domain. (Contributed by Mario Carneiro, 17-Jun-2015.)
IDomn = (CRing ∩ Domn)
 
Definitiondf-pid 19014 A principal ideal domain is an integral domain satisfying the left principal ideal property. (Contributed by Stefan O'Rear, 29-Mar-2015.)
PID = (IDomn ∩ LPIR)
 
Theoremrrgval 19015* Value of the set or left-regular elements in a ring. (Contributed by Stefan O'Rear, 22-Mar-2015.)
𝐸 = (RLReg‘𝑅)    &   𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    0 = (0g𝑅)       𝐸 = {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 · 𝑦) = 0𝑦 = 0 )}
 
Theoremisrrg 19016* Membership in the set of left-regular elements. (Contributed by Stefan O'Rear, 22-Mar-2015.)
𝐸 = (RLReg‘𝑅)    &   𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    0 = (0g𝑅)       (𝑋𝐸 ↔ (𝑋𝐵 ∧ ∀𝑦𝐵 ((𝑋 · 𝑦) = 0𝑦 = 0 )))
 
Theoremrrgeq0i 19017 Property of a left-regular element. (Contributed by Stefan O'Rear, 22-Mar-2015.)
𝐸 = (RLReg‘𝑅)    &   𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    0 = (0g𝑅)       ((𝑋𝐸𝑌𝐵) → ((𝑋 · 𝑌) = 0𝑌 = 0 ))
 
Theoremrrgeq0 19018 Left-multiplication by a left regular element does not change zeroness. (Contributed by Stefan O'Rear, 28-Mar-2015.)
𝐸 = (RLReg‘𝑅)    &   𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    0 = (0g𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝐸𝑌𝐵) → ((𝑋 · 𝑌) = 0𝑌 = 0 ))
 
Theoremrrgsupp 19019 Left multiplication by a left regular element does not change the support set of a vector. (Contributed by Stefan O'Rear, 28-Mar-2015.) (Revised by AV, 20-Jul-2019.)
𝐸 = (RLReg‘𝑅)    &   𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    0 = (0g𝑅)    &   (𝜑𝐼𝑉)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑋𝐸)    &   (𝜑𝑌:𝐼𝐵)       (𝜑 → (((𝐼 × {𝑋}) ∘𝑓 · 𝑌) supp 0 ) = (𝑌 supp 0 ))
 
Theoremrrgss 19020 Left-regular elements are a subset of the base set. (Contributed by Stefan O'Rear, 22-Mar-2015.)
𝐸 = (RLReg‘𝑅)    &   𝐵 = (Base‘𝑅)       𝐸𝐵
 
Theoremunitrrg 19021 Units are regular elements. (Contributed by Stefan O'Rear, 22-Mar-2015.)
𝐸 = (RLReg‘𝑅)    &   𝑈 = (Unit‘𝑅)       (𝑅 ∈ Ring → 𝑈𝐸)
 
Theoremisdomn 19022* Expand definition of a domain. (Contributed by Mario Carneiro, 28-Mar-2015.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    0 = (0g𝑅)       (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ ∀𝑥𝐵𝑦𝐵 ((𝑥 · 𝑦) = 0 → (𝑥 = 0𝑦 = 0 ))))
 
Theoremdomnnzr 19023 A domain is a nonzero ring. (Contributed by Mario Carneiro, 28-Mar-2015.)
(𝑅 ∈ Domn → 𝑅 ∈ NzRing)
 
Theoremdomnring 19024 A domain is a ring. (Contributed by Mario Carneiro, 28-Mar-2015.)
(𝑅 ∈ Domn → 𝑅 ∈ Ring)
 
Theoremdomneq0 19025 In a domain, a product is zero iff it has a zero factor. (Contributed by Mario Carneiro, 28-Mar-2015.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    0 = (0g𝑅)       ((𝑅 ∈ Domn ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 · 𝑌) = 0 ↔ (𝑋 = 0𝑌 = 0 )))
 
Theoremdomnmuln0 19026 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 19027 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 19028 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 19029 The opposite of a domain is also a domain. (Contributed by Mario Carneiro, 15-Jun-2015.)
𝑂 = (oppr𝑅)       (𝑅 ∈ Domn → 𝑂 ∈ Domn)
 
Theoremabvn0b 19030 Another characterization of domains, hinted at in abvtriv 18574: 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 19031 A division ring is a domain. (Contributed by Mario Carneiro, 29-Mar-2015.)
(𝑅 ∈ DivRing → 𝑅 ∈ Domn)
 
Theoremisidom 19032 An integral domain is a commutative domain. (Contributed by Mario Carneiro, 17-Jun-2015.)
(𝑅 ∈ IDomn ↔ (𝑅 ∈ CRing ∧ 𝑅 ∈ Domn))
 
Theoremfldidom 19033 A field is an integral domain. (Contributed by Mario Carneiro, 29-Mar-2015.)
(𝑅 ∈ Field → 𝑅 ∈ IDomn)
 
Theoremfidomndrnglem 19034* Lemma for fidomndrng 19035. (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 19035 A finite domain is a division ring. (Contributed by Mario Carneiro, 15-Jun-2015.)
𝐵 = (Base‘𝑅)       (𝐵 ∈ Fin → (𝑅 ∈ Domn ↔ 𝑅 ∈ DivRing))
 
Theoremfiidomfld 19036 A finite integral domain is a field. (Contributed by Mario Carneiro, 15-Jun-2015.)
𝐵 = (Base‘𝑅)       (𝐵 ∈ Fin → (𝑅 ∈ IDomn ↔ 𝑅 ∈ Field))
 
10.9  Associative algebras
 
10.9.1  Definition and basic properties
 
Syntaxcasa 19037 Associative algebra.
class AssAlg
 
Syntaxcasp 19038 Algebraic span function.
class AlgSpan
 
Syntaxcascl 19039 Class of algebra scalar injection function.
class algSc
 
Definitiondf-assa 19040* 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 19041* 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 19042* 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 19043* 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 19044 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 19045 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 19046 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 19047 An associative algebra is a left module. (Contributed by Mario Carneiro, 5-Dec-2014.)
(𝑊 ∈ AssAlg → 𝑊 ∈ LMod)
 
Theoremassaring 19048 An associative algebra is a ring. (Contributed by Mario Carneiro, 5-Dec-2014.)
(𝑊 ∈ AssAlg → 𝑊 ∈ Ring)
 
Theoremassasca 19049 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 19050 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 19051* 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)
 
Theoremissubassa 19052 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 19053 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 19054 The ring module over a commutative ring is an associative algebra. (Contributed by Mario Carneiro, 6-Oct-2015.)
(𝑅 ∈ CRing → (ringLMod‘𝑅) ∈ AssAlg)
 
Theoremassapropd 19055* 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 19056* 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 19057 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 19058 The algebraic span of a subalgebra is itself. (spanid 27372 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 19059 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 19060 Span preserves subset ordering. (spanss 27373 analog.) (Contributed by Mario Carneiro, 7-Jan-2015.)
𝐴 = (AlgSpan‘𝑊)    &   𝑉 = (Base‘𝑊)       ((𝑊 ∈ AssAlg ∧ 𝑆𝑉𝑇𝑆) → (𝐴𝑇) ⊆ (𝐴𝑆))
 
Theoremaspssid 19061 A set of vectors is a subset of its span. (spanss2 27370 analog.) (Contributed by Mario Carneiro, 7-Jan-2015.)
𝐴 = (AlgSpan‘𝑊)    &   𝑉 = (Base‘𝑊)       ((𝑊 ∈ AssAlg ∧ 𝑆𝑉) → 𝑆 ⊆ (𝐴𝑆))
 
Theoremasclfval 19062* 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 19063 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 19064 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 19065 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 19066 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))
 
Theoremasclmul1 19067 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 19068 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))
 
Theoremasclinvg 19069 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 19070 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 19071 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 }))
 
Theoremressascl 19072 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))
 
Theoremissubassa2 19073 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))
 
Theoremasclpropd 19074* 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 19075 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 19076 Lemma 1 for assamulgscm 19078 (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 19077 Lemma for assamulgscm 19078 (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 19078 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)))
 
10.10  Abstract multivariate polynomials
 
10.10.1  Definition and basic properties
 
Syntaxcmps 19079 Multivariate power series.
class mPwSer
 
Syntaxcmvr 19080 Multivariate power series variables.
class mVar
 
Syntaxcmpl 19081 Multivariate polynomials.
class mPoly
 
Syntaxcltb 19082 Ordering on terms of a multivariate polynomial.
class <bag
 
Syntaxcopws 19083 Ordered set of power series.
class ordPwSer
 
Definitiondf-psr 19084* 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 19085* 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 19086* 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 19087* 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 19088* 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 19089 The multivariate power series constructor is a proper binary operator. (Contributed by Mario Carneiro, 21-Mar-2015.)
Rel dom mPwSer
 
Theorempsrval 19090* 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 19091 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 19092* 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 19093* 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 19094* 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 19095* 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 19096* 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 19097* 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 19098* 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 19099* 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 19100* 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)
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