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

Theoremassaring 19301 An associative algebra is a ring. (Contributed by Mario Carneiro, 5-Dec-2014.)
(𝑊 ∈ AssAlg → 𝑊 ∈ Ring)

Theoremassasca 19302 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 19303 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 19304* 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 19305 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 19306 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 19307 The ring module over a commutative ring is an associative algebra. (Contributed by Mario Carneiro, 6-Oct-2015.)
(𝑅 ∈ CRing → (ringLMod‘𝑅) ∈ AssAlg)

Theoremassapropd 19308* 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 19309* 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 19310 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 19311 The algebraic span of a subalgebra is itself. (spanid 28176 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 19312 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 19313 Span preserves subset ordering. (spanss 28177 analog.) (Contributed by Mario Carneiro, 7-Jan-2015.)
𝐴 = (AlgSpan‘𝑊)    &   𝑉 = (Base‘𝑊)       ((𝑊 ∈ AssAlg ∧ 𝑆𝑉𝑇𝑆) → (𝐴𝑇) ⊆ (𝐴𝑆))

Theoremaspssid 19314 A set of vectors is a subset of its span. (spanss2 28174 analog.) (Contributed by Mario Carneiro, 7-Jan-2015.)
𝐴 = (AlgSpan‘𝑊)    &   𝑉 = (Base‘𝑊)       ((𝑊 ∈ AssAlg ∧ 𝑆𝑉) → 𝑆 ⊆ (𝐴𝑆))

Theoremasclfval 19315* 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 19316 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 19317 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 19318 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 19319 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 19320 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 19321 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 19322 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 19323 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 19324 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 19325 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 19326 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 19327* 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 19328 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 19329 Lemma 1 for assamulgscm 19331 (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 19330 Lemma for assamulgscm 19331 (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 19331 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 19332 Multivariate power series.
class mPwSer

Syntaxcmvr 19333 Multivariate power series variables.
class mVar

Syntaxcmpl 19334 Multivariate polynomials.
class mPoly

Syntaxcltb 19335 Ordering on terms of a multivariate polynomial.
class <bag

Syntaxcopws 19336 Ordered set of power series.
class ordPwSer

Definitiondf-psr 19337* 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 19338* 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 19339* 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 19340* 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 19341* 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 19342 The multivariate power series constructor is a proper binary operator. (Contributed by Mario Carneiro, 21-Mar-2015.)
Rel dom mPwSer

Theorempsrval 19343* 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 19344 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 19345* 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 19346* 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 19347* 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 19348* 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 19349* 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 19350* 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 19351* 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 19352* 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 19353* 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 19354* 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 19355* 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 19356* Lemma for gsumbagdiag 19357. (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 19357* Two-dimensional commutation of a group sum over a "triangular" region. fsum0diag 14490 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 19358* A group sum commutation used by psrass1 19386. (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 19359* 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 19360* 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 19361 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 19362 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 19363 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 19364 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 19365* The 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)    &   𝐵 = (Base‘𝑆)    &    · = (.r𝑅)    &    = (.r𝑆)    &   𝐷 = { ∈ (ℕ0𝑚 𝐼) ∣ ( “ ℕ) ∈ Fin}        = (𝑓𝐵, 𝑔𝐵 ↦ (𝑘𝐷 ↦ (𝑅 Σg (𝑥 ∈ {𝑦𝐷𝑦𝑟𝑘} ↦ ((𝑓𝑥) · (𝑔‘(𝑘𝑓𝑥)))))))

Theorempsrmulfval 19366* 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 19367* 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 19368* 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 19369 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 19370 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 19371* 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))

Theorempsrvsca 19372* The scalar multiplication operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.)
𝑆 = (𝐼 mPwSer 𝑅)    &    = ( ·𝑠𝑆)    &   𝐾 = (Base‘𝑅)    &   𝐵 = (Base‘𝑆)    &    · = (.r𝑅)    &   𝐷 = { ∈ (ℕ0𝑚 𝐼) ∣ ( “ ℕ) ∈ Fin}    &   (𝜑𝑋𝐾)    &   (𝜑𝐹𝐵)       (𝜑 → (𝑋 𝐹) = ((𝐷 × {𝑋}) ∘𝑓 · 𝐹))

Theorempsrvscaval 19373* The scalar multiplication operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.)
𝑆 = (𝐼 mPwSer 𝑅)    &    = ( ·𝑠𝑆)    &   𝐾 = (Base‘𝑅)    &   𝐵 = (Base‘𝑆)    &    · = (.r𝑅)    &   𝐷 = { ∈ (ℕ0𝑚 𝐼) ∣ ( “ ℕ) ∈ Fin}    &   (𝜑𝑋𝐾)    &   (𝜑𝐹𝐵)    &   (𝜑𝑌𝐷)       (𝜑 → ((𝑋 𝐹)‘𝑌) = (𝑋 · (𝐹𝑌)))

Theorempsrvscacl 19374 Closure of the power series scalar multiplication operation. (Contributed by Mario Carneiro, 29-Dec-2014.)
𝑆 = (𝐼 mPwSer 𝑅)    &    · = ( ·𝑠𝑆)    &   𝐾 = (Base‘𝑅)    &   𝐵 = (Base‘𝑆)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑋𝐾)    &   (𝜑𝐹𝐵)       (𝜑 → (𝑋 · 𝐹) ∈ 𝐵)

Theorempsr0cl 19375* The zero element of the ring of power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
𝑆 = (𝐼 mPwSer 𝑅)    &   (𝜑𝐼𝑉)    &   (𝜑𝑅 ∈ Grp)    &   𝐷 = {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}    &    0 = (0g𝑅)    &   𝐵 = (Base‘𝑆)       (𝜑 → (𝐷 × { 0 }) ∈ 𝐵)

Theorempsr0lid 19376* The zero element of the ring of power series is a left identity. (Contributed by Mario Carneiro, 29-Dec-2014.)
𝑆 = (𝐼 mPwSer 𝑅)    &   (𝜑𝐼𝑉)    &   (𝜑𝑅 ∈ Grp)    &   𝐷 = {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}    &    0 = (0g𝑅)    &   𝐵 = (Base‘𝑆)    &    + = (+g𝑆)    &   (𝜑𝑋𝐵)       (𝜑 → ((𝐷 × { 0 }) + 𝑋) = 𝑋)

Theorempsrnegcl 19377* The negative function in the ring of power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
𝑆 = (𝐼 mPwSer 𝑅)    &   (𝜑𝐼𝑉)    &   (𝜑𝑅 ∈ Grp)    &   𝐷 = {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}    &   𝑁 = (invg𝑅)    &   𝐵 = (Base‘𝑆)    &   (𝜑𝑋𝐵)       (𝜑 → (𝑁𝑋) ∈ 𝐵)

Theorempsrlinv 19378* The negative function in the ring of power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
𝑆 = (𝐼 mPwSer 𝑅)    &   (𝜑𝐼𝑉)    &   (𝜑𝑅 ∈ Grp)    &   𝐷 = {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}    &   𝑁 = (invg𝑅)    &   𝐵 = (Base‘𝑆)    &   (𝜑𝑋𝐵)    &    0 = (0g𝑅)    &    + = (+g𝑆)       (𝜑 → ((𝑁𝑋) + 𝑋) = (𝐷 × { 0 }))

Theorempsrgrp 19379 The ring of power series is a group. (Contributed by Mario Carneiro, 29-Dec-2014.)
𝑆 = (𝐼 mPwSer 𝑅)    &   (𝜑𝐼𝑉)    &   (𝜑𝑅 ∈ Grp)       (𝜑𝑆 ∈ Grp)

Theorempsr0 19380* The zero element of the ring of power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
𝑆 = (𝐼 mPwSer 𝑅)    &   (𝜑𝐼𝑉)    &   (𝜑𝑅 ∈ Grp)    &   𝐷 = {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}    &   𝑂 = (0g𝑅)    &    0 = (0g𝑆)       (𝜑0 = (𝐷 × {𝑂}))

Theorempsrneg 19381* The negative function of the ring of power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
𝑆 = (𝐼 mPwSer 𝑅)    &   (𝜑𝐼𝑉)    &   (𝜑𝑅 ∈ Grp)    &   𝐷 = {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}    &   𝑁 = (invg𝑅)    &   𝐵 = (Base‘𝑆)    &   𝑀 = (invg𝑆)    &   (𝜑𝑋𝐵)       (𝜑 → (𝑀𝑋) = (𝑁𝑋))

Theorempsrlmod 19382 The ring of power series is a left module. (Contributed by Mario Carneiro, 29-Dec-2014.)
𝑆 = (𝐼 mPwSer 𝑅)    &   (𝜑𝐼𝑉)    &   (𝜑𝑅 ∈ Ring)       (𝜑𝑆 ∈ LMod)

Theorempsr1cl 19383* The identity element of the ring of power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
𝑆 = (𝐼 mPwSer 𝑅)    &   (𝜑𝐼𝑉)    &   (𝜑𝑅 ∈ Ring)    &   𝐷 = {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}    &    0 = (0g𝑅)    &    1 = (1r𝑅)    &   𝑈 = (𝑥𝐷 ↦ if(𝑥 = (𝐼 × {0}), 1 , 0 ))    &   𝐵 = (Base‘𝑆)       (𝜑𝑈𝐵)

Theorempsrlidm 19384* The identity element of the ring of power series is a left identity. (Contributed by Mario Carneiro, 29-Dec-2014.) (Proof shortened by AV, 8-Jul-2019.)
𝑆 = (𝐼 mPwSer 𝑅)    &   (𝜑𝐼𝑉)    &   (𝜑𝑅 ∈ Ring)    &   𝐷 = {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}    &    0 = (0g𝑅)    &    1 = (1r𝑅)    &   𝑈 = (𝑥𝐷 ↦ if(𝑥 = (𝐼 × {0}), 1 , 0 ))    &   𝐵 = (Base‘𝑆)    &    · = (.r𝑆)    &   (𝜑𝑋𝐵)       (𝜑 → (𝑈 · 𝑋) = 𝑋)

Theorempsrridm 19385* The identity element of the ring of power series is a right identity. (Contributed by Mario Carneiro, 29-Dec-2014.) (Proof shortened by AV, 8-Jul-2019.)
𝑆 = (𝐼 mPwSer 𝑅)    &   (𝜑𝐼𝑉)    &   (𝜑𝑅 ∈ Ring)    &   𝐷 = {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}    &    0 = (0g𝑅)    &    1 = (1r𝑅)    &   𝑈 = (𝑥𝐷 ↦ if(𝑥 = (𝐼 × {0}), 1 , 0 ))    &   𝐵 = (Base‘𝑆)    &    · = (.r𝑆)    &   (𝜑𝑋𝐵)       (𝜑 → (𝑋 · 𝑈) = 𝑋)

Theorempsrass1 19386* Associative identity for the ring of power series. (Contributed by Mario Carneiro, 5-Jan-2015.)
𝑆 = (𝐼 mPwSer 𝑅)    &   (𝜑𝐼𝑉)    &   (𝜑𝑅 ∈ Ring)    &   𝐷 = {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}    &    × = (.r𝑆)    &   𝐵 = (Base‘𝑆)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)       (𝜑 → ((𝑋 × 𝑌) × 𝑍) = (𝑋 × (𝑌 × 𝑍)))

Theorempsrdi 19387* Distributive law for the ring of power series (left-distributivity). (Contributed by Mario Carneiro, 7-Jan-2015.)
𝑆 = (𝐼 mPwSer 𝑅)    &   (𝜑𝐼𝑉)    &   (𝜑𝑅 ∈ Ring)    &   𝐷 = {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}    &    × = (.r𝑆)    &   𝐵 = (Base‘𝑆)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &    + = (+g𝑆)       (𝜑 → (𝑋 × (𝑌 + 𝑍)) = ((𝑋 × 𝑌) + (𝑋 × 𝑍)))

Theorempsrdir 19388* Distributive law for the ring of power series (right-distributivity). (Contributed by Mario Carneiro, 7-Jan-2015.)
𝑆 = (𝐼 mPwSer 𝑅)    &   (𝜑𝐼𝑉)    &   (𝜑𝑅 ∈ Ring)    &   𝐷 = {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}    &    × = (.r𝑆)    &   𝐵 = (Base‘𝑆)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &    + = (+g𝑆)       (𝜑 → ((𝑋 + 𝑌) × 𝑍) = ((𝑋 × 𝑍) + (𝑌 × 𝑍)))

Theorempsrass23l 19389* Associative identity for the ring of power series. Part of psrass23 19391 which does not require the scalar ring to be commutative. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by AV, 14-Aug-2019.)
𝑆 = (𝐼 mPwSer 𝑅)    &   (𝜑𝐼𝑉)    &   (𝜑𝑅 ∈ Ring)    &   𝐷 = {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}    &    × = (.r𝑆)    &   𝐵 = (Base‘𝑆)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   𝐾 = (Base‘𝑅)    &    · = ( ·𝑠𝑆)    &   (𝜑𝐴𝐾)       (𝜑 → ((𝐴 · 𝑋) × 𝑌) = (𝐴 · (𝑋 × 𝑌)))

Theorempsrcom 19390* Commutative law for the ring of power series. (Contributed by Mario Carneiro, 7-Jan-2015.)
𝑆 = (𝐼 mPwSer 𝑅)    &   (𝜑𝐼𝑉)    &   (𝜑𝑅 ∈ Ring)    &   𝐷 = {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}    &    × = (.r𝑆)    &   𝐵 = (Base‘𝑆)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑅 ∈ CRing)       (𝜑 → (𝑋 × 𝑌) = (𝑌 × 𝑋))

Theorempsrass23 19391* Associative identities for the ring of power series. (Contributed by Mario Carneiro, 7-Jan-2015.) (Proof shortened by AV, 25-Nov-2019.)
𝑆 = (𝐼 mPwSer 𝑅)    &   (𝜑𝐼𝑉)    &   (𝜑𝑅 ∈ Ring)    &   𝐷 = {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}    &    × = (.r𝑆)    &   𝐵 = (Base‘𝑆)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑅 ∈ CRing)    &   𝐾 = (Base‘𝑅)    &    · = ( ·𝑠𝑆)    &   (𝜑𝐴𝐾)       (𝜑 → (((𝐴 · 𝑋) × 𝑌) = (𝐴 · (𝑋 × 𝑌)) ∧ (𝑋 × (𝐴 · 𝑌)) = (𝐴 · (𝑋 × 𝑌))))

Theorempsrring 19392 The ring of power series is a ring. (Contributed by Mario Carneiro, 29-Dec-2014.)
𝑆 = (𝐼 mPwSer 𝑅)    &   (𝜑𝐼𝑉)    &   (𝜑𝑅 ∈ Ring)       (𝜑𝑆 ∈ Ring)

Theorempsr1 19393* The identity element of the ring of power series. (Contributed by Mario Carneiro, 8-Jan-2015.)
𝑆 = (𝐼 mPwSer 𝑅)    &   (𝜑𝐼𝑉)    &   (𝜑𝑅 ∈ Ring)    &   𝐷 = {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}    &    0 = (0g𝑅)    &    1 = (1r𝑅)    &   𝑈 = (1r𝑆)       (𝜑𝑈 = (𝑥𝐷 ↦ if(𝑥 = (𝐼 × {0}), 1 , 0 )))

Theorempsrcrng 19394 The ring of power series is commutative ring. (Contributed by Mario Carneiro, 10-Jan-2015.)
𝑆 = (𝐼 mPwSer 𝑅)    &   (𝜑𝐼𝑉)    &   (𝜑𝑅 ∈ CRing)       (𝜑𝑆 ∈ CRing)

Theorempsrassa 19395 The ring of power series is an associative algebra. (Contributed by Mario Carneiro, 29-Dec-2014.)
𝑆 = (𝐼 mPwSer 𝑅)    &   (𝜑𝐼𝑉)    &   (𝜑𝑅 ∈ CRing)       (𝜑𝑆 ∈ AssAlg)

Theoremresspsrbas 19396 A restricted power series algebra has the same base set. (Contributed by Mario Carneiro, 3-Jul-2015.)
𝑆 = (𝐼 mPwSer 𝑅)    &   𝐻 = (𝑅s 𝑇)    &   𝑈 = (𝐼 mPwSer 𝐻)    &   𝐵 = (Base‘𝑈)    &   𝑃 = (𝑆s 𝐵)    &   (𝜑𝑇 ∈ (SubRing‘𝑅))       (𝜑𝐵 = (Base‘𝑃))

Theoremresspsradd 19397 A restricted power series algebra has the same addition operation. (Contributed by Mario Carneiro, 3-Jul-2015.)
𝑆 = (𝐼 mPwSer 𝑅)    &   𝐻 = (𝑅s 𝑇)    &   𝑈 = (𝐼 mPwSer 𝐻)    &   𝐵 = (Base‘𝑈)    &   𝑃 = (𝑆s 𝐵)    &   (𝜑𝑇 ∈ (SubRing‘𝑅))       ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋(+g𝑈)𝑌) = (𝑋(+g𝑃)𝑌))

Theoremresspsrmul 19398 A restricted power series algebra has the same multiplication operation. (Contributed by Mario Carneiro, 3-Jul-2015.)
𝑆 = (𝐼 mPwSer 𝑅)    &   𝐻 = (𝑅s 𝑇)    &   𝑈 = (𝐼 mPwSer 𝐻)    &   𝐵 = (Base‘𝑈)    &   𝑃 = (𝑆s 𝐵)    &   (𝜑𝑇 ∈ (SubRing‘𝑅))       ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋(.r𝑈)𝑌) = (𝑋(.r𝑃)𝑌))

Theoremresspsrvsca 19399 A restricted power series algebra has the same scalar multiplication operation. (Contributed by Mario Carneiro, 3-Jul-2015.)
𝑆 = (𝐼 mPwSer 𝑅)    &   𝐻 = (𝑅s 𝑇)    &   𝑈 = (𝐼 mPwSer 𝐻)    &   𝐵 = (Base‘𝑈)    &   𝑃 = (𝑆s 𝐵)    &   (𝜑𝑇 ∈ (SubRing‘𝑅))       ((𝜑 ∧ (𝑋𝑇𝑌𝐵)) → (𝑋( ·𝑠𝑈)𝑌) = (𝑋( ·𝑠𝑃)𝑌))

Theoremsubrgpsr 19400 A subring of the base ring induces a subring of power series. (Contributed by Mario Carneiro, 3-Jul-2015.)
𝑆 = (𝐼 mPwSer 𝑅)    &   𝐻 = (𝑅s 𝑇)    &   𝑈 = (𝐼 mPwSer 𝐻)    &   𝐵 = (Base‘𝑈)       ((𝐼𝑉𝑇 ∈ (SubRing‘𝑅)) → 𝐵 ∈ (SubRing‘𝑆))

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