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Theorem List for Metamath Proof Explorer - 19601-19700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsnifpsrbag 19601* 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 19602* 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 19603* 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 19604* 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 19605* 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 19606* 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 19607* 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 19608* 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 19609* 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 19610* Lemma for gsumbagdiag 19611. (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 19611* Two-dimensional commutation of a group sum over a "triangular" region. fsum0diag 14738 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 19612* A group sum commutation used by psrass1 19640. (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 19613* 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 19614* 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 19615 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 19616 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 19617 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 19618 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 19619* 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 19620* 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 19621* 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 19622* 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 19623 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 19624 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 19625* 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 19626* 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 19627* 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 19628 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 19629* 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 19630* 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 19631* 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 19632* 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 19633 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 19634* 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 19635* 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 19636 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 19637* 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 19638* 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 19639* 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 19640* 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 19641* 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 19642* 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 19643* Associative identity for the ring of power series. Part of psrass23 19645 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 19644* 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 19645* 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 19646 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 19647* 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 19648 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 19649 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 19650 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 19651 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 19652 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 19653 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 19654 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 19655* 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 19656* 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 19657* 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 19658* 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 19659 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 19660 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 19661 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 19662 The multivariate polynomial constructor is a proper binary operator. (Contributed by Mario Carneiro, 21-Mar-2015.)
Rel dom mPoly
 
Theoremmplval 19663* 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 19664* 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 19665 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 19666 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 19667 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 19668* 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 19669* 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 19670* 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 19671* Lemma for mplsubg 19672 and mpllss 19673. (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 19672 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 19673 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 19674* Lemma for mplsubrg 19675. (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 19675 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 19676* 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 19677 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 19678* 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 19679* 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 19680 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 19681 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 19682* 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 19683* 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 19684 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 19685 The polynomial ring is a group. (Contributed by Mario Carneiro, 9-Jan-2015.)
𝑃 = (𝐼 mPoly 𝑅)       ((𝐼𝑉𝑅 ∈ Grp) → 𝑃 ∈ Grp)
 
Theoremmpllmod 19686 The polynomial ring is a left module. (Contributed by Mario Carneiro, 9-Jan-2015.)
𝑃 = (𝐼 mPoly 𝑅)       ((𝐼𝑉𝑅 ∈ Ring) → 𝑃 ∈ LMod)
 
Theoremmplring 19687 The polynomial ring is a ring. (Contributed by Mario Carneiro, 9-Jan-2015.)
𝑃 = (𝐼 mPoly 𝑅)       ((𝐼𝑉𝑅 ∈ Ring) → 𝑃 ∈ Ring)
 
Theoremmplcrng 19688 The polynomial ring is a commutative ring. (Contributed by Mario Carneiro, 9-Jan-2015.)
𝑃 = (𝐼 mPoly 𝑅)       ((𝐼𝑉𝑅 ∈ CRing) → 𝑃 ∈ CRing)
 
Theoremmplassa 19689 The polynomial ring is an associative algebra. (Contributed by Mario Carneiro, 9-Jan-2015.)
𝑃 = (𝐼 mPoly 𝑅)       ((𝐼𝑉𝑅 ∈ CRing) → 𝑃 ∈ AssAlg)
 
Theoremressmplbas2 19690 The base set of a restricted polynomial algebra consists of power series in the subring which are also polynomials (in the parent ring). (Contributed by Mario Carneiro, 3-Jul-2015.)
𝑆 = (𝐼 mPoly 𝑅)    &   𝐻 = (𝑅s 𝑇)    &   𝑈 = (𝐼 mPoly 𝐻)    &   𝐵 = (Base‘𝑈)    &   (𝜑𝐼𝑉)    &   (𝜑𝑇 ∈ (SubRing‘𝑅))    &   𝑊 = (𝐼 mPwSer 𝐻)    &   𝐶 = (Base‘𝑊)    &   𝐾 = (Base‘𝑆)       (𝜑𝐵 = (𝐶𝐾))
 
Theoremressmplbas 19691 A restricted polynomial algebra has the same base set. (Contributed by Mario Carneiro, 3-Jul-2015.)
𝑆 = (𝐼 mPoly 𝑅)    &   𝐻 = (𝑅s 𝑇)    &   𝑈 = (𝐼 mPoly 𝐻)    &   𝐵 = (Base‘𝑈)    &   (𝜑𝐼𝑉)    &   (𝜑𝑇 ∈ (SubRing‘𝑅))    &   𝑃 = (𝑆s 𝐵)       (𝜑𝐵 = (Base‘𝑃))
 
Theoremressmpladd 19692 A restricted polynomial algebra has the same addition operation. (Contributed by Mario Carneiro, 3-Jul-2015.)
𝑆 = (𝐼 mPoly 𝑅)    &   𝐻 = (𝑅s 𝑇)    &   𝑈 = (𝐼 mPoly 𝐻)    &   𝐵 = (Base‘𝑈)    &   (𝜑𝐼𝑉)    &   (𝜑𝑇 ∈ (SubRing‘𝑅))    &   𝑃 = (𝑆s 𝐵)       ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋(+g𝑈)𝑌) = (𝑋(+g𝑃)𝑌))
 
Theoremressmplmul 19693 A restricted polynomial algebra has the same multiplication operation. (Contributed by Mario Carneiro, 3-Jul-2015.)
𝑆 = (𝐼 mPoly 𝑅)    &   𝐻 = (𝑅s 𝑇)    &   𝑈 = (𝐼 mPoly 𝐻)    &   𝐵 = (Base‘𝑈)    &   (𝜑𝐼𝑉)    &   (𝜑𝑇 ∈ (SubRing‘𝑅))    &   𝑃 = (𝑆s 𝐵)       ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋(.r𝑈)𝑌) = (𝑋(.r𝑃)𝑌))
 
Theoremressmplvsca 19694 A restricted power series algebra has the same scalar multiplication operation. (Contributed by Mario Carneiro, 3-Jul-2015.)
𝑆 = (𝐼 mPoly 𝑅)    &   𝐻 = (𝑅s 𝑇)    &   𝑈 = (𝐼 mPoly 𝐻)    &   𝐵 = (Base‘𝑈)    &   (𝜑𝐼𝑉)    &   (𝜑𝑇 ∈ (SubRing‘𝑅))    &   𝑃 = (𝑆s 𝐵)       ((𝜑 ∧ (𝑋𝑇𝑌𝐵)) → (𝑋( ·𝑠𝑈)𝑌) = (𝑋( ·𝑠𝑃)𝑌))
 
Theoremsubrgmpl 19695 A subring of the base ring induces a subring of polynomials. (Contributed by Mario Carneiro, 3-Jul-2015.)
𝑆 = (𝐼 mPoly 𝑅)    &   𝐻 = (𝑅s 𝑇)    &   𝑈 = (𝐼 mPoly 𝐻)    &   𝐵 = (Base‘𝑈)       ((𝐼𝑉𝑇 ∈ (SubRing‘𝑅)) → 𝐵 ∈ (SubRing‘𝑆))
 
Theoremsubrgmvr 19696 The variables in a subring polynomial algebra are the same as the original ring. (Contributed by Mario Carneiro, 4-Jul-2015.)
𝑉 = (𝐼 mVar 𝑅)    &   (𝜑𝐼𝑊)    &   (𝜑𝑇 ∈ (SubRing‘𝑅))    &   𝐻 = (𝑅s 𝑇)       (𝜑𝑉 = (𝐼 mVar 𝐻))
 
Theoremsubrgmvrf 19697 The variables in a polynomial algebra are contained in every subring algebra. (Contributed by Mario Carneiro, 4-Jul-2015.)
𝑉 = (𝐼 mVar 𝑅)    &   (𝜑𝐼𝑊)    &   (𝜑𝑇 ∈ (SubRing‘𝑅))    &   𝐻 = (𝑅s 𝑇)    &   𝑈 = (𝐼 mPoly 𝐻)    &   𝐵 = (Base‘𝑈)       (𝜑𝑉:𝐼𝐵)
 
Theoremmplmon 19698* A monomial is a polynomial. (Contributed by Mario Carneiro, 9-Jan-2015.)
𝑃 = (𝐼 mPoly 𝑅)    &   𝐵 = (Base‘𝑃)    &    0 = (0g𝑅)    &    1 = (1r𝑅)    &   𝐷 = {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}    &   (𝜑𝐼𝑊)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑋𝐷)       (𝜑 → (𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) ∈ 𝐵)
 
Theoremmplmonmul 19699* The product of two monomials adds the exponent vectors together. For example, the product of (𝑥↑2)(𝑦↑2) with (𝑦↑1)(𝑧↑3) is (𝑥↑2)(𝑦↑3)(𝑧↑3), where the exponent vectors ⟨2, 2, 0⟩ and ⟨0, 1, 3⟩ are added to give ⟨2, 3, 3⟩. (Contributed by Mario Carneiro, 9-Jan-2015.)
𝑃 = (𝐼 mPoly 𝑅)    &   𝐵 = (Base‘𝑃)    &    0 = (0g𝑅)    &    1 = (1r𝑅)    &   𝐷 = {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}    &   (𝜑𝐼𝑊)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑋𝐷)    &    · = (.r𝑃)    &   (𝜑𝑌𝐷)       (𝜑 → ((𝑦𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) · (𝑦𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))) = (𝑦𝐷 ↦ if(𝑦 = (𝑋𝑓 + 𝑌), 1 , 0 )))
 
Theoremmplcoe1 19700* Decompose a polynomial into a finite sum of monomials. (Contributed by Mario Carneiro, 9-Jan-2015.)
𝑃 = (𝐼 mPoly 𝑅)    &   𝐷 = {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}    &    0 = (0g𝑅)    &    1 = (1r𝑅)    &   (𝜑𝐼𝑊)    &   𝐵 = (Base‘𝑃)    &    · = ( ·𝑠𝑃)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑋𝐵)       (𝜑𝑋 = (𝑃 Σg (𝑘𝐷 ↦ ((𝑋𝑘) · (𝑦𝐷 ↦ if(𝑦 = 𝑘, 1 , 0 ))))))
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206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 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 42801-42900 430 42901-43000 431 43001-43100 432 43101-43106
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