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Theorem List for Metamath Proof Explorer - 19401-19500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Syntaxcmpl 19401 Multivariate polynomials.
class mPoly

Syntaxcltb 19402 Ordering on terms of a multivariate polynomial.
class <bag

Syntaxcopws 19403 Ordered set of power series.
class ordPwSer

Definitiondf-psr 19404* 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 19405* 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 19406* 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 19407* 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 19408* 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 19409 The multivariate power series constructor is a proper binary operator. (Contributed by Mario Carneiro, 21-Mar-2015.)
Rel dom mPwSer

Theorempsrval 19410* 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 19411 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 19412* 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 19413* 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 19414* 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 19415* 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 19416* 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 19417* 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 19418* 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 19419* 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 19420* 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 19421* 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 19422* 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 19423* Lemma for gsumbagdiag 19424. (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 19424* Two-dimensional commutation of a group sum over a "triangular" region. fsum0diag 14553 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 19425* A group sum commutation used by psrass1 19453. (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 19426* 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 19427* 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 19428 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 19429 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 19430 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 19431 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 19432* 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 19433* 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 19434* 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 19435* 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 19436 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 19437 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 19438* 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 19439* 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 19440* 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 19441 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 19442* 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 19443* 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 19444* 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 19445* 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 19446 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 19447* 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 19448* 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 19449 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 19450* 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 19451* 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 19452* 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 19453* 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 19454* 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 19455* 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 19456* Associative identity for the ring of power series. Part of psrass23 19458 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 19457* 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 19458* 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 19459 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 19460* 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 19461 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 19462 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 19463 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 19464 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 19465 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 19466 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 19467 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))

Theoremmvrfval 19468* Value of the generating elements of the power series structure. (Contributed by Mario Carneiro, 7-Jan-2015.)
𝑉 = (𝐼 mVar 𝑅)    &   𝐷 = { ∈ (ℕ0𝑚 𝐼) ∣ ( “ ℕ) ∈ Fin}    &    0 = (0g𝑅)    &    1 = (1r𝑅)    &   (𝜑𝐼𝑊)    &   (𝜑𝑅𝑌)       (𝜑𝑉 = (𝑥𝐼 ↦ (𝑓𝐷 ↦ if(𝑓 = (𝑦𝐼 ↦ if(𝑦 = 𝑥, 1, 0)), 1 , 0 ))))

Theoremmvrval 19469* Value of the generating elements of the power series structure. (Contributed by Mario Carneiro, 7-Jan-2015.)
𝑉 = (𝐼 mVar 𝑅)    &   𝐷 = { ∈ (ℕ0𝑚 𝐼) ∣ ( “ ℕ) ∈ Fin}    &    0 = (0g𝑅)    &    1 = (1r𝑅)    &   (𝜑𝐼𝑊)    &   (𝜑𝑅𝑌)    &   (𝜑𝑋𝐼)       (𝜑 → (𝑉𝑋) = (𝑓𝐷 ↦ if(𝑓 = (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 )))

Theoremmvrval2 19470* Value of the generating elements of the power series structure. (Contributed by Mario Carneiro, 7-Jan-2015.)
𝑉 = (𝐼 mVar 𝑅)    &   𝐷 = { ∈ (ℕ0𝑚 𝐼) ∣ ( “ ℕ) ∈ Fin}    &    0 = (0g𝑅)    &    1 = (1r𝑅)    &   (𝜑𝐼𝑊)    &   (𝜑𝑅𝑌)    &   (𝜑𝑋𝐼)    &   (𝜑𝐹𝐷)       (𝜑 → ((𝑉𝑋)‘𝐹) = if(𝐹 = (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 ))

Theoremmvrid 19471* The 𝑋𝑖-th coefficient of the term 𝑋𝑖 is 1. (Contributed by Mario Carneiro, 7-Jan-2015.)
𝑉 = (𝐼 mVar 𝑅)    &   𝐷 = { ∈ (ℕ0𝑚 𝐼) ∣ ( “ ℕ) ∈ Fin}    &    0 = (0g𝑅)    &    1 = (1r𝑅)    &   (𝜑𝐼𝑊)    &   (𝜑𝑅𝑌)    &   (𝜑𝑋𝐼)       (𝜑 → ((𝑉𝑋)‘(𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) = 1 )

Theoremmvrf 19472 The power series variable function is a function from the index set to elements of the power series structure representing 𝑋𝑖 for each 𝑖. (Contributed by Mario Carneiro, 29-Dec-2014.)
𝑆 = (𝐼 mPwSer 𝑅)    &   𝑉 = (𝐼 mVar 𝑅)    &   𝐵 = (Base‘𝑆)    &   (𝜑𝐼𝑊)    &   (𝜑𝑅 ∈ Ring)       (𝜑𝑉:𝐼𝐵)

Theoremmvrf1 19473 The power series variable function is injective if the base ring is nonzero. (Contributed by Mario Carneiro, 29-Dec-2014.)
𝑆 = (𝐼 mPwSer 𝑅)    &   𝑉 = (𝐼 mVar 𝑅)    &   𝐵 = (Base‘𝑆)    &   (𝜑𝐼𝑊)    &   (𝜑𝑅 ∈ Ring)    &    0 = (0g𝑅)    &    1 = (1r𝑅)    &   (𝜑10 )       (𝜑𝑉:𝐼1-1𝐵)

Theoremmvrcl2 19474 A power series variable is an element of the base set. (Contributed by Mario Carneiro, 29-Dec-2014.)
𝑆 = (𝐼 mPwSer 𝑅)    &   𝑉 = (𝐼 mVar 𝑅)    &   𝐵 = (Base‘𝑆)    &   (𝜑𝐼𝑊)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑋𝐼)       (𝜑 → (𝑉𝑋) ∈ 𝐵)

Theoremreldmmpl 19475 The multivariate polynomial constructor is a proper binary operator. (Contributed by Mario Carneiro, 21-Mar-2015.)
Rel dom mPoly

Theoremmplval 19476* Value of the set of multivariate polynomials. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 25-Jun-2019.)
𝑃 = (𝐼 mPoly 𝑅)    &   𝑆 = (𝐼 mPwSer 𝑅)    &   𝐵 = (Base‘𝑆)    &    0 = (0g𝑅)    &   𝑈 = {𝑓𝐵𝑓 finSupp 0 }       𝑃 = (𝑆s 𝑈)

Theoremmplbas 19477* Base set of the set of multivariate polynomials. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 25-Jun-2019.)
𝑃 = (𝐼 mPoly 𝑅)    &   𝑆 = (𝐼 mPwSer 𝑅)    &   𝐵 = (Base‘𝑆)    &    0 = (0g𝑅)    &   𝑈 = (Base‘𝑃)       𝑈 = {𝑓𝐵𝑓 finSupp 0 }

Theoremmplelbas 19478 Property of being a polynomial. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 25-Jun-2019.)
𝑃 = (𝐼 mPoly 𝑅)    &   𝑆 = (𝐼 mPwSer 𝑅)    &   𝐵 = (Base‘𝑆)    &    0 = (0g𝑅)    &   𝑈 = (Base‘𝑃)       (𝑋𝑈 ↔ (𝑋𝐵𝑋 finSupp 0 ))

Theoremmplval2 19479 Self-referential expression for the set of multivariate polynomials. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
𝑃 = (𝐼 mPoly 𝑅)    &   𝑆 = (𝐼 mPwSer 𝑅)    &   𝑈 = (Base‘𝑃)       𝑃 = (𝑆s 𝑈)

Theoremmplbasss 19480 The set of polynomials is a subset of the set of power series. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
𝑃 = (𝐼 mPoly 𝑅)    &   𝑆 = (𝐼 mPwSer 𝑅)    &   𝑈 = (Base‘𝑃)    &   𝐵 = (Base‘𝑆)       𝑈𝐵

Theoremmplelf 19481* A polynomial is defined as a function on the coefficients. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
𝑃 = (𝐼 mPoly 𝑅)    &   𝐾 = (Base‘𝑅)    &   𝐵 = (Base‘𝑃)    &   𝐷 = {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}    &   (𝜑𝑋𝐵)       (𝜑𝑋:𝐷𝐾)

Theoremmplsubglem 19482* If 𝐴 is an ideal of sets (a nonempty collection closed under subset and binary union) of the set 𝐷 of finite bags (the primary applications being 𝐴 = Fin and 𝐴 = 𝒫 𝐵 for some 𝐵), then the set of all power series whose coefficient functions are supported on an element of 𝐴 is a subgroup of the set of all power series. (Contributed by Mario Carneiro, 12-Jan-2015.) (Revised by AV, 16-Jul-2019.)
𝑆 = (𝐼 mPwSer 𝑅)    &   𝐵 = (Base‘𝑆)    &    0 = (0g𝑅)    &   𝐷 = {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}    &   (𝜑𝐼𝑊)    &   (𝜑 → ∅ ∈ 𝐴)    &   ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → (𝑥𝑦) ∈ 𝐴)    &   ((𝜑 ∧ (𝑥𝐴𝑦𝑥)) → 𝑦𝐴)    &   (𝜑𝑈 = {𝑔𝐵 ∣ (𝑔 supp 0 ) ∈ 𝐴})    &   (𝜑𝑅 ∈ Grp)       (𝜑𝑈 ∈ (SubGrp‘𝑆))

Theoremmpllsslem 19483* If 𝐴 is an ideal of subsets (a nonempty collection closed under subset and binary union) of the set 𝐷 of finite bags (the primary applications being 𝐴 = Fin and 𝐴 = 𝒫 𝐵 for some 𝐵), then the set of all power series whose coefficient functions are supported on an element of 𝐴 is a linear subspace of the set of all power series. (Contributed by Mario Carneiro, 12-Jan-2015.) (Revised by AV, 16-Jul-2019.)
𝑆 = (𝐼 mPwSer 𝑅)    &   𝐵 = (Base‘𝑆)    &    0 = (0g𝑅)    &   𝐷 = {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}    &   (𝜑𝐼𝑊)    &   (𝜑 → ∅ ∈ 𝐴)    &   ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → (𝑥𝑦) ∈ 𝐴)    &   ((𝜑 ∧ (𝑥𝐴𝑦𝑥)) → 𝑦𝐴)    &   (𝜑𝑈 = {𝑔𝐵 ∣ (𝑔 supp 0 ) ∈ 𝐴})    &   (𝜑𝑅 ∈ Ring)       (𝜑𝑈 ∈ (LSubSp‘𝑆))

Theoremmplsubglem2 19484* Lemma for mplsubg 19485 and mpllss 19486. (Contributed by AV, 16-Jul-2019.)
𝑆 = (𝐼 mPwSer 𝑅)    &   𝑃 = (𝐼 mPoly 𝑅)    &   𝑈 = (Base‘𝑃)    &   (𝜑𝐼𝑊)       (𝜑𝑈 = {𝑔 ∈ (Base‘𝑆) ∣ (𝑔 supp (0g𝑅)) ∈ Fin})

Theoremmplsubg 19485 The set of polynomials is closed under addition, i.e. it is a subgroup of the set of power series. (Contributed by Mario Carneiro, 8-Jan-2015.) (Proof shortened by AV, 16-Jul-2019.)
𝑆 = (𝐼 mPwSer 𝑅)    &   𝑃 = (𝐼 mPoly 𝑅)    &   𝑈 = (Base‘𝑃)    &   (𝜑𝐼𝑊)    &   (𝜑𝑅 ∈ Grp)       (𝜑𝑈 ∈ (SubGrp‘𝑆))

Theoremmpllss 19486 The set of polynomials is closed under scalar multiplication, i.e. it is a linear subspace of the set of power series. (Contributed by Mario Carneiro, 7-Jan-2015.) (Proof shortened by AV, 16-Jul-2019.)
𝑆 = (𝐼 mPwSer 𝑅)    &   𝑃 = (𝐼 mPoly 𝑅)    &   𝑈 = (Base‘𝑃)    &   (𝜑𝐼𝑊)    &   (𝜑𝑅 ∈ Ring)       (𝜑𝑈 ∈ (LSubSp‘𝑆))

Theoremmplsubrglem 19487* Lemma for mplsubrg 19488. (Contributed by Mario Carneiro, 9-Jan-2015.) (Revised by AV, 18-Jul-2019.)
𝑆 = (𝐼 mPwSer 𝑅)    &   𝑃 = (𝐼 mPoly 𝑅)    &   𝑈 = (Base‘𝑃)    &   (𝜑𝐼𝑊)    &   (𝜑𝑅 ∈ Ring)    &   𝐷 = {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}    &    0 = (0g𝑅)    &   𝐴 = ( ∘𝑓 + “ ((𝑋 supp 0 ) × (𝑌 supp 0 )))    &    · = (.r𝑅)    &   (𝜑𝑋𝑈)    &   (𝜑𝑌𝑈)       (𝜑 → (𝑋(.r𝑆)𝑌) ∈ 𝑈)

Theoremmplsubrg 19488 The set of polynomials is closed under multiplication, i.e. it is a subring of the set of power series. (Contributed by Mario Carneiro, 9-Jan-2015.)
𝑆 = (𝐼 mPwSer 𝑅)    &   𝑃 = (𝐼 mPoly 𝑅)    &   𝑈 = (Base‘𝑃)    &   (𝜑𝐼𝑊)    &   (𝜑𝑅 ∈ Ring)       (𝜑𝑈 ∈ (SubRing‘𝑆))

Theoremmpl0 19489* The zero polynomial. (Contributed by Mario Carneiro, 9-Jan-2015.)
𝑃 = (𝐼 mPoly 𝑅)    &   𝐷 = {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}    &   𝑂 = (0g𝑅)    &    0 = (0g𝑃)    &   (𝜑𝐼𝑊)    &   (𝜑𝑅 ∈ Grp)       (𝜑0 = (𝐷 × {𝑂}))

Theoremmpladd 19490 The addition operation on multivariate polynomials. (Contributed by Mario Carneiro, 9-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
𝑃 = (𝐼 mPoly 𝑅)    &   𝐵 = (Base‘𝑃)    &    + = (+g𝑅)    &    = (+g𝑃)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → (𝑋 𝑌) = (𝑋𝑓 + 𝑌))

Theoremmplmul 19491* The multiplication operation on multivariate polynomials. (Contributed by Mario Carneiro, 9-Jan-2015.)
𝑃 = (𝐼 mPoly 𝑅)    &   𝐵 = (Base‘𝑃)    &    · = (.r𝑅)    &    = (.r𝑃)    &   𝐷 = { ∈ (ℕ0𝑚 𝐼) ∣ ( “ ℕ) ∈ Fin}    &   (𝜑𝐹𝐵)    &   (𝜑𝐺𝐵)       (𝜑 → (𝐹 𝐺) = (𝑘𝐷 ↦ (𝑅 Σg (𝑥 ∈ {𝑦𝐷𝑦𝑟𝑘} ↦ ((𝐹𝑥) · (𝐺‘(𝑘𝑓𝑥)))))))

Theoremmpl1 19492* The identity element of the ring of polynomials. (Contributed by Mario Carneiro, 10-Jan-2015.)
𝑃 = (𝐼 mPoly 𝑅)    &   𝐷 = {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}    &    0 = (0g𝑅)    &    1 = (1r𝑅)    &   𝑈 = (1r𝑃)    &   (𝜑𝐼𝑊)    &   (𝜑𝑅 ∈ Ring)       (𝜑𝑈 = (𝑥𝐷 ↦ if(𝑥 = (𝐼 × {0}), 1 , 0 )))

Theoremmplsca 19493 The scalar field of a multivariate polynomial structure. (Contributed by Mario Carneiro, 9-Jan-2015.)
𝑃 = (𝐼 mPoly 𝑅)    &   (𝜑𝐼𝑉)    &   (𝜑𝑅𝑊)       (𝜑𝑅 = (Scalar‘𝑃))

Theoremmplvsca2 19494 The scalar multiplication operation on multivariate polynomials. (Contributed by Mario Carneiro, 9-Jan-2015.)
𝑃 = (𝐼 mPoly 𝑅)    &   𝑆 = (𝐼 mPwSer 𝑅)    &    · = ( ·𝑠𝑃)        · = ( ·𝑠𝑆)

Theoremmplvsca 19495* The scalar multiplication operation on multivariate polynomials. (Contributed by Mario Carneiro, 9-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
𝑃 = (𝐼 mPoly 𝑅)    &    = ( ·𝑠𝑃)    &   𝐾 = (Base‘𝑅)    &   𝐵 = (Base‘𝑃)    &    · = (.r𝑅)    &   𝐷 = { ∈ (ℕ0𝑚 𝐼) ∣ ( “ ℕ) ∈ Fin}    &   (𝜑𝑋𝐾)    &   (𝜑𝐹𝐵)       (𝜑 → (𝑋 𝐹) = ((𝐷 × {𝑋}) ∘𝑓 · 𝐹))

Theoremmplvscaval 19496* The scalar multiplication operation on multivariate polynomials. (Contributed by Mario Carneiro, 9-Jan-2015.)
𝑃 = (𝐼 mPoly 𝑅)    &    = ( ·𝑠𝑃)    &   𝐾 = (Base‘𝑅)    &   𝐵 = (Base‘𝑃)    &    · = (.r𝑅)    &   𝐷 = { ∈ (ℕ0𝑚 𝐼) ∣ ( “ ℕ) ∈ Fin}    &   (𝜑𝑋𝐾)    &   (𝜑𝐹𝐵)    &   (𝜑𝑌𝐷)       (𝜑 → ((𝑋 𝐹)‘𝑌) = (𝑋 · (𝐹𝑌)))

Theoremmvrcl 19497 A power series variable is a polynomial. (Contributed by Mario Carneiro, 9-Jan-2015.)
𝑃 = (𝐼 mPoly 𝑅)    &   𝑉 = (𝐼 mVar 𝑅)    &   𝐵 = (Base‘𝑃)    &   (𝜑𝐼𝑊)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑋𝐼)       (𝜑 → (𝑉𝑋) ∈ 𝐵)

Theoremmplgrp 19498 The polynomial ring is a group. (Contributed by Mario Carneiro, 9-Jan-2015.)
𝑃 = (𝐼 mPoly 𝑅)       ((𝐼𝑉𝑅 ∈ Grp) → 𝑃 ∈ Grp)

Theoremmpllmod 19499 The polynomial ring is a left module. (Contributed by Mario Carneiro, 9-Jan-2015.)
𝑃 = (𝐼 mPoly 𝑅)       ((𝐼𝑉𝑅 ∈ Ring) → 𝑃 ∈ LMod)

Theoremmplring 19500 The polynomial ring is a ring. (Contributed by Mario Carneiro, 9-Jan-2015.)
𝑃 = (𝐼 mPoly 𝑅)       ((𝐼𝑉𝑅 ∈ Ring) → 𝑃 ∈ Ring)

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