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Theorem List for Metamath Proof Explorer - 19301-19400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdrnglpir 19301 Division rings are principal ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
(𝑅 ∈ DivRing → 𝑅 ∈ LPIR)

Theoremrspsn 19302* Membership in principal ideals is closely related to divisibility. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Mario Carneiro, 6-May-2015.)
𝐵 = (Base‘𝑅)    &   𝐾 = (RSpan‘𝑅)    &    = (∥r𝑅)       ((𝑅 ∈ Ring ∧ 𝐺𝐵) → (𝐾‘{𝐺}) = {𝑥𝐺 𝑥})

Theoremlidldvgen 19303* An element generates an ideal iff it is contained in the ideal and all elements are right-divided by it. (Contributed by Stefan O'Rear, 3-Jan-2015.)
𝐵 = (Base‘𝑅)    &   𝑈 = (LIdeal‘𝑅)    &   𝐾 = (RSpan‘𝑅)    &    = (∥r𝑅)       ((𝑅 ∈ Ring ∧ 𝐼𝑈𝐺𝐵) → (𝐼 = (𝐾‘{𝐺}) ↔ (𝐺𝐼 ∧ ∀𝑥𝐼 𝐺 𝑥)))

Theoremlpigen 19304* An ideal is principal iff it contains an element which right-divides all elements. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Wolf Lammen, 6-Sep-2020.)
𝑈 = (LIdeal‘𝑅)    &   𝑃 = (LPIdeal‘𝑅)    &    = (∥r𝑅)       ((𝑅 ∈ Ring ∧ 𝐼𝑈) → (𝐼𝑃 ↔ ∃𝑥𝐼𝑦𝐼 𝑥 𝑦))

10.8.4  Nonzero rings and zero rings

Syntaxcnzr 19305 The class of nonzero rings.
class NzRing

Definitiondf-nzr 19306 A nonzero or nontrivial ring is a ring with at least two values, or equivalently where 1 and 0 are different. (Contributed by Stefan O'Rear, 24-Feb-2015.)
NzRing = {𝑟 ∈ Ring ∣ (1r𝑟) ≠ (0g𝑟)}

Theoremisnzr 19307 Property of a nonzero ring. (Contributed by Stefan O'Rear, 24-Feb-2015.)
1 = (1r𝑅)    &    0 = (0g𝑅)       (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ 10 ))

Theoremnzrnz 19308 One and zero are different in a nonzero ring. (Contributed by Stefan O'Rear, 24-Feb-2015.)
1 = (1r𝑅)    &    0 = (0g𝑅)       (𝑅 ∈ NzRing → 10 )

Theoremnzrring 19309 A nonzero ring is a ring. (Contributed by Stefan O'Rear, 24-Feb-2015.)
(𝑅 ∈ NzRing → 𝑅 ∈ Ring)

Theoremdrngnzr 19310 All division rings are nonzero. (Contributed by Stefan O'Rear, 24-Feb-2015.)
(𝑅 ∈ DivRing → 𝑅 ∈ NzRing)

Theoremisnzr2 19311 Equivalent characterization of nonzero rings: they have at least two elements. (Contributed by Stefan O'Rear, 24-Feb-2015.)
𝐵 = (Base‘𝑅)       (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ 2𝑜𝐵))

Theoremisnzr2hash 19312 Equivalent characterization of nonzero rings: they have at least two elements. Analogous to isnzr2 19311. (Contributed by AV, 14-Apr-2019.)
𝐵 = (Base‘𝑅)       (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ 1 < (#‘𝐵)))

Theoremopprnzr 19313 The opposite of a nonzero ring is nonzero. (Contributed by Mario Carneiro, 17-Jun-2015.)
𝑂 = (oppr𝑅)       (𝑅 ∈ NzRing → 𝑂 ∈ NzRing)

Theoremringelnzr 19314 A ring is nonzero if it has a nonzero element. (Contributed by Stefan O'Rear, 6-Feb-2015.) (Revised by Mario Carneiro, 13-Jun-2015.)
0 = (0g𝑅)    &   𝐵 = (Base‘𝑅)       ((𝑅 ∈ Ring ∧ 𝑋 ∈ (𝐵 ∖ { 0 })) → 𝑅 ∈ NzRing)

Theoremnzrunit 19315 A unit is nonzero in any nonzero ring. (Contributed by Mario Carneiro, 6-Oct-2015.)
𝑈 = (Unit‘𝑅)    &    0 = (0g𝑅)       ((𝑅 ∈ NzRing ∧ 𝐴𝑈) → 𝐴0 )

Theoremsubrgnzr 19316 A subring of a nonzero ring is nonzero. (Contributed by Mario Carneiro, 15-Jun-2015.)
𝑆 = (𝑅s 𝐴)       ((𝑅 ∈ NzRing ∧ 𝐴 ∈ (SubRing‘𝑅)) → 𝑆 ∈ NzRing)

Theorem0ringnnzr 19317 A ring is a zero ring iff it is not a nonzero ring. (Contributed by AV, 14-Apr-2019.)
(𝑅 ∈ Ring → ((#‘(Base‘𝑅)) = 1 ↔ ¬ 𝑅 ∈ NzRing))

Theorem0ring 19318 If a ring has only one element, it is the zero ring. According to Wikipedia ("Zero ring", 14-Apr-2019, https://en.wikipedia.org/wiki/Zero_ring): "The zero ring, denoted {0} or simply 0, consists of the one-element set {0} with the operations + and * defined so that 0 + 0 = 0 and 0 * 0 = 0.". (Contributed by AV, 14-Apr-2019.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)       ((𝑅 ∈ Ring ∧ (#‘𝐵) = 1) → 𝐵 = { 0 })

Theorem0ring01eq 19319 In a ring with only one element, i.e. a zero ring, the zero and the identity element are the same. (Contributed by AV, 14-Apr-2019.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &    1 = (1r𝑅)       ((𝑅 ∈ Ring ∧ (#‘𝐵) = 1) → 0 = 1 )

Theorem01eq0ring 19320 If the zero and the identity element of a ring are the same, the ring is the zero ring. (Contributed by AV, 16-Apr-2019.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &    1 = (1r𝑅)       ((𝑅 ∈ Ring ∧ 0 = 1 ) → 𝐵 = { 0 })

Theorem0ring01eqbi 19321 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 19322 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 19323 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 19324 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 19325 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 19326 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 19327 Set of left-regular elements in a ring.
class RLReg

Syntaxcdomn 19328 Class of (ring theoretic) domains.
class Domn

Syntaxcidom 19329 Class of integral domains.
class IDomn

Syntaxcpid 19330 Class of principal ideal domains.
class PID

Definitiondf-rlreg 19331* 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 19332* 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 19333 An integral domain is a commutative domain. (Contributed by Mario Carneiro, 17-Jun-2015.)
IDomn = (CRing ∩ Domn)

Definitiondf-pid 19334 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 19335* 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 19336* 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 19337 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 19338 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 19339 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 19340 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 19341 Units are regular elements. (Contributed by Stefan O'Rear, 22-Mar-2015.)
𝐸 = (RLReg‘𝑅)    &   𝑈 = (Unit‘𝑅)       (𝑅 ∈ Ring → 𝑈𝐸)

Theoremisdomn 19342* 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 19343 A domain is a nonzero ring. (Contributed by Mario Carneiro, 28-Mar-2015.)
(𝑅 ∈ Domn → 𝑅 ∈ NzRing)

Theoremdomnring 19344 A domain is a ring. (Contributed by Mario Carneiro, 28-Mar-2015.)
(𝑅 ∈ Domn → 𝑅 ∈ Ring)

Theoremdomneq0 19345 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 19346 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 19347 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 19348 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 19349 The opposite of a domain is also a domain. (Contributed by Mario Carneiro, 15-Jun-2015.)
𝑂 = (oppr𝑅)       (𝑅 ∈ Domn → 𝑂 ∈ Domn)

Theoremabvn0b 19350 Another characterization of domains, hinted at in abvtriv 18889: 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 19351 A division ring is a domain. (Contributed by Mario Carneiro, 29-Mar-2015.)
(𝑅 ∈ DivRing → 𝑅 ∈ Domn)

Theoremisidom 19352 An integral domain is a commutative domain. (Contributed by Mario Carneiro, 17-Jun-2015.)
(𝑅 ∈ IDomn ↔ (𝑅 ∈ CRing ∧ 𝑅 ∈ Domn))

Theoremfldidom 19353 A field is an integral domain. (Contributed by Mario Carneiro, 29-Mar-2015.)
(𝑅 ∈ Field → 𝑅 ∈ IDomn)

Theoremfidomndrnglem 19354* Lemma for fidomndrng 19355. (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 19355 A finite domain is a division ring. (Contributed by Mario Carneiro, 15-Jun-2015.)
𝐵 = (Base‘𝑅)       (𝐵 ∈ Fin → (𝑅 ∈ Domn ↔ 𝑅 ∈ DivRing))

Theoremfiidomfld 19356 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 19357 Associative algebra.
class AssAlg

Syntaxcasp 19358 Algebraic span function.
class AlgSpan

Syntaxcascl 19359 Class of algebra scalar injection function.
class algSc

Definitiondf-assa 19360* 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 19361* 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 19362* 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 19363* 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 19364 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 19365 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 19366 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 19367 An associative algebra is a left module. (Contributed by Mario Carneiro, 5-Dec-2014.)
(𝑊 ∈ AssAlg → 𝑊 ∈ LMod)

Theoremassaring 19368 An associative algebra is a ring. (Contributed by Mario Carneiro, 5-Dec-2014.)
(𝑊 ∈ AssAlg → 𝑊 ∈ Ring)

Theoremassasca 19369 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 19370 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 19371* 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 19372 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 19373 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 19374 The ring module over a commutative ring is an associative algebra. (Contributed by Mario Carneiro, 6-Oct-2015.)
(𝑅 ∈ CRing → (ringLMod‘𝑅) ∈ AssAlg)

Theoremassapropd 19375* 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 19376* 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 19377 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 19378 The algebraic span of a subalgebra is itself. (spanid 28334 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 19379 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 19380 Span preserves subset ordering. (spanss 28335 analog.) (Contributed by Mario Carneiro, 7-Jan-2015.)
𝐴 = (AlgSpan‘𝑊)    &   𝑉 = (Base‘𝑊)       ((𝑊 ∈ AssAlg ∧ 𝑆𝑉𝑇𝑆) → (𝐴𝑇) ⊆ (𝐴𝑆))

Theoremaspssid 19381 A set of vectors is a subset of its span. (spanss2 28332 analog.) (Contributed by Mario Carneiro, 7-Jan-2015.)
𝐴 = (AlgSpan‘𝑊)    &   𝑉 = (Base‘𝑊)       ((𝑊 ∈ AssAlg ∧ 𝑆𝑉) → 𝑆 ⊆ (𝐴𝑆))

Theoremasclfval 19382* 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 19383 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 19384 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 19385 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 19386 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 19387 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 19388 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 19389 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 19390 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 19391 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 19392 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 19393 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 19394* 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 19395 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 19396 Lemma 1 for assamulgscm 19398 (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 19397 Lemma for assamulgscm 19398 (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 19398 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 19399 Multivariate power series.
class mPwSer

Syntaxcmvr 19400 Multivariate power series variables.
class mVar

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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 42801-42879
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