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Theorem List for Metamath Proof Explorer - 19601-19700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Definitiondf-coe1 19601* Define the coefficient function for a univariate polynomial. (Contributed by Stefan O'Rear, 21-Mar-2015.)
coe1 = (𝑓 ∈ V ↦ (𝑛 ∈ ℕ0 ↦ (𝑓‘(1𝑜 × {𝑛}))))
 
Definitiondf-toply1 19602* Define a function which maps a coefficient function for a univariate polynomial to the corresponding polynomial object. (Contributed by Mario Carneiro, 12-Jun-2015.)
toPoly1 = (𝑓 ∈ V ↦ (𝑛 ∈ (ℕ0𝑚 1𝑜) ↦ (𝑓‘(𝑛‘∅))))
 
Theorempsr1baslem 19603 The set of finite bags on 1𝑜 is just the set of all functions from 1𝑜 to 0. (Contributed by Mario Carneiro, 9-Feb-2015.)
(ℕ0𝑚 1𝑜) = {𝑓 ∈ (ℕ0𝑚 1𝑜) ∣ (𝑓 “ ℕ) ∈ Fin}
 
Theorempsr1val 19604 Value of the ring of univariate power series. (Contributed by Mario Carneiro, 8-Feb-2015.)
𝑆 = (PwSer1𝑅)       𝑆 = ((1𝑜 ordPwSer 𝑅)‘∅)
 
Theorempsr1crng 19605 The ring of univariate power series is a commutative ring. (Contributed by Mario Carneiro, 8-Feb-2015.)
𝑆 = (PwSer1𝑅)       (𝑅 ∈ CRing → 𝑆 ∈ CRing)
 
Theorempsr1assa 19606 The ring of univariate power series is an associative algebra. (Contributed by Mario Carneiro, 8-Feb-2015.)
𝑆 = (PwSer1𝑅)       (𝑅 ∈ CRing → 𝑆 ∈ AssAlg)
 
Theorempsr1tos 19607 The ordered power series structure is a totally ordered set. (Contributed by Mario Carneiro, 2-Jun-2015.)
𝑆 = (PwSer1𝑅)       (𝑅 ∈ Toset → 𝑆 ∈ Toset)
 
Theorempsr1bas2 19608 The base set of the ring of univariate power series. (Contributed by Mario Carneiro, 3-Jul-2015.)
𝑆 = (PwSer1𝑅)    &   𝐵 = (Base‘𝑆)    &   𝑂 = (1𝑜 mPwSer 𝑅)       𝐵 = (Base‘𝑂)
 
Theorempsr1bas 19609 The base set of the ring of univariate power series. (Contributed by Mario Carneiro, 8-Feb-2015.)
𝑆 = (PwSer1𝑅)    &   𝐵 = (Base‘𝑆)    &   𝐾 = (Base‘𝑅)       𝐵 = (𝐾𝑚 (ℕ0𝑚 1𝑜))
 
Theoremvr1val 19610 The value of the generator of the power series algebra (the 𝑋 in 𝑅[[𝑋]]). Since all univariate polynomial rings over a fixed base ring 𝑅 are isomorphic, we don't bother to pass this in as a parameter; internally we are actually using the empty set as this generator and 1𝑜 = {∅} is the index set (but for most purposes this choice should not be visible anyway). (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by Mario Carneiro, 12-Jun-2015.)
𝑋 = (var1𝑅)       𝑋 = ((1𝑜 mVar 𝑅)‘∅)
 
Theoremvr1cl2 19611 The variable 𝑋 is a member of the power series algebra 𝑅[[𝑋]]. (Contributed by Mario Carneiro, 8-Feb-2015.)
𝑋 = (var1𝑅)    &   𝑆 = (PwSer1𝑅)    &   𝐵 = (Base‘𝑆)       (𝑅 ∈ Ring → 𝑋𝐵)
 
Theoremply1val 19612 The value of the set of univariate polynomials. (Contributed by Mario Carneiro, 9-Feb-2015.)
𝑃 = (Poly1𝑅)    &   𝑆 = (PwSer1𝑅)       𝑃 = (𝑆s (Base‘(1𝑜 mPoly 𝑅)))
 
Theoremply1bas 19613 The value of the base set of univariate polynomials. (Contributed by Mario Carneiro, 9-Feb-2015.)
𝑃 = (Poly1𝑅)    &   𝑆 = (PwSer1𝑅)    &   𝑈 = (Base‘𝑃)       𝑈 = (Base‘(1𝑜 mPoly 𝑅))
 
Theoremply1lss 19614 Univariate polynomials form a linear subspace of the set of univariate power series. (Contributed by Mario Carneiro, 9-Feb-2015.)
𝑃 = (Poly1𝑅)    &   𝑆 = (PwSer1𝑅)    &   𝑈 = (Base‘𝑃)       (𝑅 ∈ Ring → 𝑈 ∈ (LSubSp‘𝑆))
 
Theoremply1subrg 19615 Univariate polynomials form a subring of the set of univariate power series. (Contributed by Mario Carneiro, 9-Feb-2015.)
𝑃 = (Poly1𝑅)    &   𝑆 = (PwSer1𝑅)    &   𝑈 = (Base‘𝑃)       (𝑅 ∈ Ring → 𝑈 ∈ (SubRing‘𝑆))
 
Theoremply1crng 19616 The ring of univariate polynomials is a commutative ring. (Contributed by Mario Carneiro, 9-Feb-2015.)
𝑃 = (Poly1𝑅)       (𝑅 ∈ CRing → 𝑃 ∈ CRing)
 
Theoremply1assa 19617 The ring of univariate polynomials is an associative algebra. (Contributed by Mario Carneiro, 9-Feb-2015.)
𝑃 = (Poly1𝑅)       (𝑅 ∈ CRing → 𝑃 ∈ AssAlg)
 
Theorempsr1bascl 19618 A univariate power series is a multivariate power series on one index. (Contributed by Stefan O'Rear, 25-Mar-2015.)
𝑃 = (PwSer1𝑅)    &   𝐵 = (Base‘𝑃)       (𝐹𝐵𝐹 ∈ (Base‘(1𝑜 mPwSer 𝑅)))
 
Theorempsr1basf 19619 Univariate power series base set elements are functions. (Contributed by Stefan O'Rear, 25-Mar-2015.)
𝑃 = (PwSer1𝑅)    &   𝐵 = (Base‘𝑃)    &   𝐾 = (Base‘𝑅)       (𝐹𝐵𝐹:(ℕ0𝑚 1𝑜)⟶𝐾)
 
Theoremply1basf 19620 Univariate polynomial base set elements are functions. (Contributed by Stefan O'Rear, 21-Mar-2015.)
𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &   𝐾 = (Base‘𝑅)       (𝐹𝐵𝐹:(ℕ0𝑚 1𝑜)⟶𝐾)
 
Theoremply1bascl 19621 A univariate polynomial is a univariate power series. (Contributed by Stefan O'Rear, 25-Mar-2015.)
𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)       (𝐹𝐵𝐹 ∈ (Base‘(PwSer1𝑅)))
 
Theoremply1bascl2 19622 A univariate polynomial is a multivariate polynomial on one index. (Contributed by Stefan O'Rear, 25-Mar-2015.)
𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)       (𝐹𝐵𝐹 ∈ (Base‘(1𝑜 mPoly 𝑅)))
 
Theoremcoe1fval 19623* Value of the univariate polynomial coefficient function. (Contributed by Stefan O'Rear, 21-Mar-2015.)
𝐴 = (coe1𝐹)       (𝐹𝑉𝐴 = (𝑛 ∈ ℕ0 ↦ (𝐹‘(1𝑜 × {𝑛}))))
 
Theoremcoe1fv 19624 Value of an evaluated coefficient in a polynomial coefficient vector. (Contributed by Stefan O'Rear, 21-Mar-2015.)
𝐴 = (coe1𝐹)       ((𝐹𝑉𝑁 ∈ ℕ0) → (𝐴𝑁) = (𝐹‘(1𝑜 × {𝑁})))
 
Theoremfvcoe1 19625 Value of a multivariate coefficient in terms of the coefficient vector. (Contributed by Stefan O'Rear, 21-Mar-2015.)
𝐴 = (coe1𝐹)       ((𝐹𝑉𝑋 ∈ (ℕ0𝑚 1𝑜)) → (𝐹𝑋) = (𝐴‘(𝑋‘∅)))
 
Theoremcoe1fval3 19626* Univariate power series coefficient vectors expressed as a function composition. (Contributed by Stefan O'Rear, 25-Mar-2015.)
𝐴 = (coe1𝐹)    &   𝐵 = (Base‘𝑃)    &   𝑃 = (PwSer1𝑅)    &   𝐺 = (𝑦 ∈ ℕ0 ↦ (1𝑜 × {𝑦}))       (𝐹𝐵𝐴 = (𝐹𝐺))
 
Theoremcoe1f2 19627 Functionality of univariate power series coefficient vectors. (Contributed by Stefan O'Rear, 25-Mar-2015.)
𝐴 = (coe1𝐹)    &   𝐵 = (Base‘𝑃)    &   𝑃 = (PwSer1𝑅)    &   𝐾 = (Base‘𝑅)       (𝐹𝐵𝐴:ℕ0𝐾)
 
Theoremcoe1fval2 19628* Univariate polynomial coefficient vectors expressed as a function composition. (Contributed by Stefan O'Rear, 21-Mar-2015.)
𝐴 = (coe1𝐹)    &   𝐵 = (Base‘𝑃)    &   𝑃 = (Poly1𝑅)    &   𝐺 = (𝑦 ∈ ℕ0 ↦ (1𝑜 × {𝑦}))       (𝐹𝐵𝐴 = (𝐹𝐺))
 
Theoremcoe1f 19629 Functionality of univariate polynomial coefficient vectors. (Contributed by Stefan O'Rear, 21-Mar-2015.)
𝐴 = (coe1𝐹)    &   𝐵 = (Base‘𝑃)    &   𝑃 = (Poly1𝑅)    &   𝐾 = (Base‘𝑅)       (𝐹𝐵𝐴:ℕ0𝐾)
 
Theoremcoe1fvalcl 19630 A coefficient of a univariate polynomial over a class/ring is an element of this class/ring. (Contributed by AV, 9-Oct-2019.)
𝐴 = (coe1𝐹)    &   𝐵 = (Base‘𝑃)    &   𝑃 = (Poly1𝑅)    &   𝐾 = (Base‘𝑅)       ((𝐹𝐵𝑁 ∈ ℕ0) → (𝐴𝑁) ∈ 𝐾)
 
Theoremcoe1sfi 19631 Finite support of univariate polynomial coefficient vectors. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by AV, 19-Jul-2019.)
𝐴 = (coe1𝐹)    &   𝐵 = (Base‘𝑃)    &   𝑃 = (Poly1𝑅)    &    0 = (0g𝑅)       (𝐹𝐵𝐴 finSupp 0 )
 
Theoremcoe1fsupp 19632* The coefficient vector of a univariate polynomial is a finitely supported mapping from the nonnegative integers to the elements of the coefficient class/ring for the polynomial. (Contributed by AV, 3-Oct-2019.)
𝐴 = (coe1𝐹)    &   𝐵 = (Base‘𝑃)    &   𝑃 = (Poly1𝑅)    &    0 = (0g𝑅)    &   𝐾 = (Base‘𝑅)       (𝐹𝐵𝐴 ∈ {𝑔 ∈ (𝐾𝑚0) ∣ 𝑔 finSupp 0 })
 
Theoremmptcoe1fsupp 19633* A mapping involving coefficients of polynomials is finitely supported. (Contributed by AV, 12-Oct-2019.)
𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &    0 = (0g𝑅)       ((𝑅 ∈ Ring ∧ 𝑀𝐵) → (𝑘 ∈ ℕ0 ↦ ((coe1𝑀)‘𝑘)) finSupp 0 )
 
Theoremcoe1ae0 19634* The coefficient vector of a univariate polynomial is 0 almost everywhere. (Contributed by AV, 19-Oct-2019.)
𝐴 = (coe1𝐹)    &   𝐵 = (Base‘𝑃)    &   𝑃 = (Poly1𝑅)    &    0 = (0g𝑅)       (𝐹𝐵 → ∃𝑠 ∈ ℕ0𝑛 ∈ ℕ0 (𝑠 < 𝑛 → (𝐴𝑛) = 0 ))
 
Theoremvr1cl 19635 The generator of a univariate polynomial algebra is contained in the base set. (Contributed by Stefan O'Rear, 19-Mar-2015.)
𝑋 = (var1𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)       (𝑅 ∈ Ring → 𝑋𝐵)
 
Theoremopsr0 19636 Zero in the ordered power series ring. (Contributed by Stefan O'Rear, 23-Mar-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
𝑆 = (𝐼 mPwSer 𝑅)    &   𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇)    &   (𝜑𝑇 ⊆ (𝐼 × 𝐼))       (𝜑 → (0g𝑆) = (0g𝑂))
 
Theoremopsr1 19637 One in the ordered power series ring. (Contributed by Stefan O'Rear, 23-Mar-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
𝑆 = (𝐼 mPwSer 𝑅)    &   𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇)    &   (𝜑𝑇 ⊆ (𝐼 × 𝐼))       (𝜑 → (1r𝑆) = (1r𝑂))
 
Theoremmplplusg 19638 Value of addition in a polynomial ring. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
𝑌 = (𝐼 mPoly 𝑅)    &   𝑆 = (𝐼 mPwSer 𝑅)    &    + = (+g𝑌)        + = (+g𝑆)
 
Theoremmplmulr 19639 Value of multiplication in a polynomial ring. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
𝑌 = (𝐼 mPoly 𝑅)    &   𝑆 = (𝐼 mPwSer 𝑅)    &    · = (.r𝑌)        · = (.r𝑆)
 
Theorempsr1plusg 19640 Value of addition in a univariate power series ring. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
𝑌 = (PwSer1𝑅)    &   𝑆 = (1𝑜 mPwSer 𝑅)    &    + = (+g𝑌)        + = (+g𝑆)
 
Theorempsr1vsca 19641 Value of scalar multiplication in a univariate power series ring. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
𝑌 = (PwSer1𝑅)    &   𝑆 = (1𝑜 mPwSer 𝑅)    &    · = ( ·𝑠𝑌)        · = ( ·𝑠𝑆)
 
Theorempsr1mulr 19642 Value of multiplication in a univariate power series ring. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
𝑌 = (PwSer1𝑅)    &   𝑆 = (1𝑜 mPwSer 𝑅)    &    · = (.r𝑌)        · = (.r𝑆)
 
Theoremply1plusg 19643 Value of addition in a univariate polynomial ring. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by Mario Carneiro, 4-Jul-2015.)
𝑌 = (Poly1𝑅)    &   𝑆 = (1𝑜 mPoly 𝑅)    &    + = (+g𝑌)        + = (+g𝑆)
 
Theoremply1vsca 19644 Value of scalar multiplication in a univariate polynomial ring. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by Mario Carneiro, 4-Jul-2015.)
𝑌 = (Poly1𝑅)    &   𝑆 = (1𝑜 mPoly 𝑅)    &    · = ( ·𝑠𝑌)        · = ( ·𝑠𝑆)
 
Theoremply1mulr 19645 Value of multiplication in a univariate polynomial ring. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by Mario Carneiro, 4-Jul-2015.)
𝑌 = (Poly1𝑅)    &   𝑆 = (1𝑜 mPoly 𝑅)    &    · = (.r𝑌)        · = (.r𝑆)
 
Theoremressply1bas2 19646 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 = (Poly1𝑅)    &   𝐻 = (𝑅s 𝑇)    &   𝑈 = (Poly1𝐻)    &   𝐵 = (Base‘𝑈)    &   (𝜑𝑇 ∈ (SubRing‘𝑅))    &   𝑊 = (PwSer1𝐻)    &   𝐶 = (Base‘𝑊)    &   𝐾 = (Base‘𝑆)       (𝜑𝐵 = (𝐶𝐾))
 
Theoremressply1bas 19647 A restricted polynomial algebra has the same base set. (Contributed by Mario Carneiro, 3-Jul-2015.)
𝑆 = (Poly1𝑅)    &   𝐻 = (𝑅s 𝑇)    &   𝑈 = (Poly1𝐻)    &   𝐵 = (Base‘𝑈)    &   (𝜑𝑇 ∈ (SubRing‘𝑅))    &   𝑃 = (𝑆s 𝐵)       (𝜑𝐵 = (Base‘𝑃))
 
Theoremressply1add 19648 A restricted polynomial algebra has the same addition operation. (Contributed by Mario Carneiro, 3-Jul-2015.)
𝑆 = (Poly1𝑅)    &   𝐻 = (𝑅s 𝑇)    &   𝑈 = (Poly1𝐻)    &   𝐵 = (Base‘𝑈)    &   (𝜑𝑇 ∈ (SubRing‘𝑅))    &   𝑃 = (𝑆s 𝐵)       ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋(+g𝑈)𝑌) = (𝑋(+g𝑃)𝑌))
 
Theoremressply1mul 19649 A restricted polynomial algebra has the same multiplication operation. (Contributed by Mario Carneiro, 3-Jul-2015.)
𝑆 = (Poly1𝑅)    &   𝐻 = (𝑅s 𝑇)    &   𝑈 = (Poly1𝐻)    &   𝐵 = (Base‘𝑈)    &   (𝜑𝑇 ∈ (SubRing‘𝑅))    &   𝑃 = (𝑆s 𝐵)       ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋(.r𝑈)𝑌) = (𝑋(.r𝑃)𝑌))
 
Theoremressply1vsca 19650 A restricted power series algebra has the same scalar multiplication operation. (Contributed by Mario Carneiro, 3-Jul-2015.)
𝑆 = (Poly1𝑅)    &   𝐻 = (𝑅s 𝑇)    &   𝑈 = (Poly1𝐻)    &   𝐵 = (Base‘𝑈)    &   (𝜑𝑇 ∈ (SubRing‘𝑅))    &   𝑃 = (𝑆s 𝐵)       ((𝜑 ∧ (𝑋𝑇𝑌𝐵)) → (𝑋( ·𝑠𝑈)𝑌) = (𝑋( ·𝑠𝑃)𝑌))
 
Theoremsubrgply1 19651 A subring of the base ring induces a subring of polynomials. (Contributed by Mario Carneiro, 3-Jul-2015.)
𝑆 = (Poly1𝑅)    &   𝐻 = (𝑅s 𝑇)    &   𝑈 = (Poly1𝐻)    &   𝐵 = (Base‘𝑈)       (𝑇 ∈ (SubRing‘𝑅) → 𝐵 ∈ (SubRing‘𝑆))
 
Theoremgsumply1subr 19652 Evaluate a group sum in a polynomial ring over a subring. (Contributed by AV, 22-Sep-2019.) (Proof shortened by AV, 31-Jan-2020.)
𝑆 = (Poly1𝑅)    &   𝐻 = (𝑅s 𝑇)    &   𝑈 = (Poly1𝐻)    &   𝐵 = (Base‘𝑈)    &   (𝜑𝑇 ∈ (SubRing‘𝑅))    &   (𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝐵)       (𝜑 → (𝑆 Σg 𝐹) = (𝑈 Σg 𝐹))
 
Theorempsrbaspropd 19653 Property deduction for power series base set. (Contributed by Stefan O'Rear, 27-Mar-2015.)
(𝜑 → (Base‘𝑅) = (Base‘𝑆))       (𝜑 → (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑆)))
 
Theorempsrplusgpropd 19654* Property deduction for power series addition. (Contributed by Stefan O'Rear, 27-Mar-2015.) (Revised by Mario Carneiro, 3-Oct-2015.)
(𝜑𝐵 = (Base‘𝑅))    &   (𝜑𝐵 = (Base‘𝑆))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝑅)𝑦) = (𝑥(+g𝑆)𝑦))       (𝜑 → (+g‘(𝐼 mPwSer 𝑅)) = (+g‘(𝐼 mPwSer 𝑆)))
 
Theoremmplbaspropd 19655* Property deduction for polynomial base set. (Contributed by Stefan O'Rear, 27-Mar-2015.) (Proof shortened by AV, 19-Jul-2019.)
(𝜑𝐵 = (Base‘𝑅))    &   (𝜑𝐵 = (Base‘𝑆))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝑅)𝑦) = (𝑥(+g𝑆)𝑦))       (𝜑 → (Base‘(𝐼 mPoly 𝑅)) = (Base‘(𝐼 mPoly 𝑆)))
 
Theorempsropprmul 19656 Reversing multiplication in a ring reverses multiplication in the power series ring. (Contributed by Stefan O'Rear, 27-Mar-2015.)
𝑌 = (𝐼 mPwSer 𝑅)    &   𝑆 = (oppr𝑅)    &   𝑍 = (𝐼 mPwSer 𝑆)    &    · = (.r𝑌)    &    = (.r𝑍)    &   𝐵 = (Base‘𝑌)       ((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) → (𝐹 𝐺) = (𝐺 · 𝐹))
 
Theoremply1opprmul 19657 Reversing multiplication in a ring reverses multiplication in the univariate polynomial ring. (Contributed by Stefan O'Rear, 27-Mar-2015.)
𝑌 = (Poly1𝑅)    &   𝑆 = (oppr𝑅)    &   𝑍 = (Poly1𝑆)    &    · = (.r𝑌)    &    = (.r𝑍)    &   𝐵 = (Base‘𝑌)       ((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) → (𝐹 𝐺) = (𝐺 · 𝐹))
 
Theorem00ply1bas 19658 Lemma for ply1basfvi 19659 and deg1fvi 23890. (Contributed by Stefan O'Rear, 28-Mar-2015.)
∅ = (Base‘(Poly1‘∅))
 
Theoremply1basfvi 19659 Protection compatibility of the univariate polynomial base set. (Contributed by Stefan O'Rear, 27-Mar-2015.)
(Base‘(Poly1𝑅)) = (Base‘(Poly1‘( I ‘𝑅)))
 
Theoremply1plusgfvi 19660 Protection compatibility of the univariate polynomial addition. (Contributed by Stefan O'Rear, 27-Mar-2015.)
(+g‘(Poly1𝑅)) = (+g‘(Poly1‘( I ‘𝑅)))
 
Theoremply1baspropd 19661* Property deduction for univariate polynomial base set. (Contributed by Stefan O'Rear, 27-Mar-2015.)
(𝜑𝐵 = (Base‘𝑅))    &   (𝜑𝐵 = (Base‘𝑆))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝑅)𝑦) = (𝑥(+g𝑆)𝑦))       (𝜑 → (Base‘(Poly1𝑅)) = (Base‘(Poly1𝑆)))
 
Theoremply1plusgpropd 19662* Property deduction for univariate polynomial addition. (Contributed by Stefan O'Rear, 27-Mar-2015.)
(𝜑𝐵 = (Base‘𝑅))    &   (𝜑𝐵 = (Base‘𝑆))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝑅)𝑦) = (𝑥(+g𝑆)𝑦))       (𝜑 → (+g‘(Poly1𝑅)) = (+g‘(Poly1𝑆)))
 
Theoremopsrring 19663 Ordered power series form a ring. (Contributed by Stefan O'Rear, 22-Mar-2015.)
𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇)    &   (𝜑𝐼𝑉)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑇 ⊆ (𝐼 × 𝐼))       (𝜑𝑂 ∈ Ring)
 
Theoremopsrlmod 19664 Ordered power series form a left module. (Contributed by Stefan O'Rear, 26-Mar-2015.)
𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇)    &   (𝜑𝐼𝑉)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑇 ⊆ (𝐼 × 𝐼))       (𝜑𝑂 ∈ LMod)
 
Theorempsr1ring 19665 Univariate power series form a ring. (Contributed by Stefan O'Rear, 22-Mar-2015.)
𝑆 = (PwSer1𝑅)       (𝑅 ∈ Ring → 𝑆 ∈ Ring)
 
Theoremply1ring 19666 Univariate polynomials form a ring. (Contributed by Stefan O'Rear, 22-Mar-2015.)
𝑃 = (Poly1𝑅)       (𝑅 ∈ Ring → 𝑃 ∈ Ring)
 
Theorempsr1lmod 19667 Univariate power series form a left module. (Contributed by Stefan O'Rear, 26-Mar-2015.)
𝑃 = (PwSer1𝑅)       (𝑅 ∈ Ring → 𝑃 ∈ LMod)
 
Theorempsr1sca 19668 Scalars of a univariate power series ring. (Contributed by Stefan O'Rear, 4-Jul-2015.)
𝑃 = (PwSer1𝑅)       (𝑅𝑉𝑅 = (Scalar‘𝑃))
 
Theorempsr1sca2 19669 Scalars of a univariate power series ring. (Contributed by Stefan O'Rear, 26-Mar-2015.) (Revised by Mario Carneiro, 4-Jul-2015.)
𝑃 = (PwSer1𝑅)       ( I ‘𝑅) = (Scalar‘𝑃)
 
Theoremply1lmod 19670 Univariate polynomials form a left module. (Contributed by Stefan O'Rear, 26-Mar-2015.)
𝑃 = (Poly1𝑅)       (𝑅 ∈ Ring → 𝑃 ∈ LMod)
 
Theoremply1sca 19671 Scalars of a univariate polynomial ring. (Contributed by Stefan O'Rear, 26-Mar-2015.)
𝑃 = (Poly1𝑅)       (𝑅𝑉𝑅 = (Scalar‘𝑃))
 
Theoremply1sca2 19672 Scalars of a univariate polynomial ring. (Contributed by Stefan O'Rear, 26-Mar-2015.)
𝑃 = (Poly1𝑅)       ( I ‘𝑅) = (Scalar‘𝑃)
 
Theoremply1mpl0 19673 The univariate polynomial ring has the same zero as the corresponding multivariate polynomial ring. (Contributed by Stefan O'Rear, 23-Mar-2015.) (Revised by Mario Carneiro, 3-Oct-2015.)
𝑀 = (1𝑜 mPoly 𝑅)    &   𝑃 = (Poly1𝑅)    &    0 = (0g𝑃)        0 = (0g𝑀)
 
Theoremply10s0 19674 Zero times a univariate polynomial is the zero polynomial (lmod0vs 18944 analog.) (Contributed by AV, 2-Dec-2019.)
𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &    = ( ·𝑠𝑃)    &    0 = (0g𝑅)       ((𝑅 ∈ Ring ∧ 𝑀𝐵) → ( 0 𝑀) = (0g𝑃))
 
Theoremply1mpl1 19675 The univariate polynomial ring has the same one as the corresponding multivariate polynomial ring. (Contributed by Stefan O'Rear, 23-Mar-2015.) (Revised by Mario Carneiro, 3-Oct-2015.)
𝑀 = (1𝑜 mPoly 𝑅)    &   𝑃 = (Poly1𝑅)    &    1 = (1r𝑃)        1 = (1r𝑀)
 
Theoremply1ascl 19676 The univariate polynomial ring inherits the multivariate ring's scalar function. (Contributed by Stefan O'Rear, 28-Mar-2015.) (Proof shortened by Fan Zheng, 26-Jun-2016.)
𝑃 = (Poly1𝑅)    &   𝐴 = (algSc‘𝑃)       𝐴 = (algSc‘(1𝑜 mPoly 𝑅))
 
Theoremsubrg1ascl 19677 The scalar injection function in a subring algebra is the same up to a restriction to the subring. (Contributed by Mario Carneiro, 4-Jul-2015.)
𝑃 = (Poly1𝑅)    &   𝐴 = (algSc‘𝑃)    &   𝐻 = (𝑅s 𝑇)    &   𝑈 = (Poly1𝐻)    &   (𝜑𝑇 ∈ (SubRing‘𝑅))    &   𝐶 = (algSc‘𝑈)       (𝜑𝐶 = (𝐴𝑇))
 
Theoremsubrg1asclcl 19678 The scalars in a polynomial algebra are in the subring algebra iff the scalar value is in the subring. (Contributed by Mario Carneiro, 4-Jul-2015.)
𝑃 = (Poly1𝑅)    &   𝐴 = (algSc‘𝑃)    &   𝐻 = (𝑅s 𝑇)    &   𝑈 = (Poly1𝐻)    &   (𝜑𝑇 ∈ (SubRing‘𝑅))    &   𝐵 = (Base‘𝑈)    &   𝐾 = (Base‘𝑅)    &   (𝜑𝑋𝐾)       (𝜑 → ((𝐴𝑋) ∈ 𝐵𝑋𝑇))
 
Theoremsubrgvr1 19679 The variables in a subring polynomial algebra are the same as the original ring. (Contributed by Mario Carneiro, 5-Jul-2015.)
𝑋 = (var1𝑅)    &   (𝜑𝑇 ∈ (SubRing‘𝑅))    &   𝐻 = (𝑅s 𝑇)       (𝜑𝑋 = (var1𝐻))
 
Theoremsubrgvr1cl 19680 The variables in a polynomial algebra are contained in every subring algebra. (Contributed by Mario Carneiro, 5-Jul-2015.)
𝑋 = (var1𝑅)    &   (𝜑𝑇 ∈ (SubRing‘𝑅))    &   𝐻 = (𝑅s 𝑇)    &   𝑈 = (Poly1𝐻)    &   𝐵 = (Base‘𝑈)       (𝜑𝑋𝐵)
 
Theoremcoe1z 19681 The coefficient vector of 0. (Contributed by Stefan O'Rear, 23-Mar-2015.)
𝑃 = (Poly1𝑅)    &    0 = (0g𝑃)    &   𝑌 = (0g𝑅)       (𝑅 ∈ Ring → (coe10 ) = (ℕ0 × {𝑌}))
 
Theoremcoe1add 19682 The coefficient vector of an addition. (Contributed by Stefan O'Rear, 24-Mar-2015.)
𝑌 = (Poly1𝑅)    &   𝐵 = (Base‘𝑌)    &    = (+g𝑌)    &    + = (+g𝑅)       ((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) → (coe1‘(𝐹 𝐺)) = ((coe1𝐹) ∘𝑓 + (coe1𝐺)))
 
Theoremcoe1addfv 19683 A particular coefficient of an addition. (Contributed by Stefan O'Rear, 23-Mar-2015.)
𝑌 = (Poly1𝑅)    &   𝐵 = (Base‘𝑌)    &    = (+g𝑌)    &    + = (+g𝑅)       (((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑋 ∈ ℕ0) → ((coe1‘(𝐹 𝐺))‘𝑋) = (((coe1𝐹)‘𝑋) + ((coe1𝐺)‘𝑋)))
 
Theoremcoe1subfv 19684 A particular coefficient of a subtraction. (Contributed by Stefan O'Rear, 23-Mar-2015.)
𝑌 = (Poly1𝑅)    &   𝐵 = (Base‘𝑌)    &    = (-g𝑌)    &   𝑁 = (-g𝑅)       (((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑋 ∈ ℕ0) → ((coe1‘(𝐹 𝐺))‘𝑋) = (((coe1𝐹)‘𝑋)𝑁((coe1𝐺)‘𝑋)))
 
Theoremcoe1mul2lem1 19685 An equivalence for coe1mul2 19687. (Contributed by Stefan O'Rear, 25-Mar-2015.)
((𝐴 ∈ ℕ0𝑋 ∈ (ℕ0𝑚 1𝑜)) → (𝑋𝑟 ≤ (1𝑜 × {𝐴}) ↔ (𝑋‘∅) ∈ (0...𝐴)))
 
Theoremcoe1mul2lem2 19686* An equivalence for coe1mul2 19687. (Contributed by Stefan O'Rear, 25-Mar-2015.)
𝐻 = {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟 ≤ (1𝑜 × {𝑘})}       (𝑘 ∈ ℕ0 → (𝑐𝐻 ↦ (𝑐‘∅)):𝐻1-1-onto→(0...𝑘))
 
Theoremcoe1mul2 19687* The coefficient vector of multiplication in the univariate power series ring. (Contributed by Stefan O'Rear, 25-Mar-2015.)
𝑆 = (PwSer1𝑅)    &    = (.r𝑆)    &    · = (.r𝑅)    &   𝐵 = (Base‘𝑆)       ((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) → (coe1‘(𝐹 𝐺)) = (𝑘 ∈ ℕ0 ↦ (𝑅 Σg (𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))))))
 
Theoremcoe1mul 19688* The coefficient vector of multiplication in the univariate polynomial ring. (Contributed by Stefan O'Rear, 25-Mar-2015.)
𝑌 = (Poly1𝑅)    &    = (.r𝑌)    &    · = (.r𝑅)    &   𝐵 = (Base‘𝑌)       ((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) → (coe1‘(𝐹 𝐺)) = (𝑘 ∈ ℕ0 ↦ (𝑅 Σg (𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))))))
 
Theoremply1moncl 19689 Closure of the expression for a univariate primitive monomial. (Contributed by AV, 14-Aug-2019.)
𝑃 = (Poly1𝑅)    &   𝑋 = (var1𝑅)    &   𝑁 = (mulGrp‘𝑃)    &    = (.g𝑁)    &   𝐵 = (Base‘𝑃)       ((𝑅 ∈ Ring ∧ 𝐷 ∈ ℕ0) → (𝐷 𝑋) ∈ 𝐵)
 
Theoremply1tmcl 19690 Closure of the expression for a univariate polynomial term. (Contributed by Stefan O'Rear, 27-Mar-2015.) (Proof shortened by AV, 25-Nov-2019.)
𝐾 = (Base‘𝑅)    &   𝑃 = (Poly1𝑅)    &   𝑋 = (var1𝑅)    &    · = ( ·𝑠𝑃)    &   𝑁 = (mulGrp‘𝑃)    &    = (.g𝑁)    &   𝐵 = (Base‘𝑃)       ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → (𝐶 · (𝐷 𝑋)) ∈ 𝐵)
 
Theoremcoe1tm 19691* Coefficient vector of a polynomial term. (Contributed by Stefan O'Rear, 27-Mar-2015.)
0 = (0g𝑅)    &   𝐾 = (Base‘𝑅)    &   𝑃 = (Poly1𝑅)    &   𝑋 = (var1𝑅)    &    · = ( ·𝑠𝑃)    &   𝑁 = (mulGrp‘𝑃)    &    = (.g𝑁)       ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → (coe1‘(𝐶 · (𝐷 𝑋))) = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 𝐷, 𝐶, 0 )))
 
Theoremcoe1tmfv1 19692 Nonzero coefficient of a polynomial term. (Contributed by Stefan O'Rear, 27-Mar-2015.)
0 = (0g𝑅)    &   𝐾 = (Base‘𝑅)    &   𝑃 = (Poly1𝑅)    &   𝑋 = (var1𝑅)    &    · = ( ·𝑠𝑃)    &   𝑁 = (mulGrp‘𝑃)    &    = (.g𝑁)       ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → ((coe1‘(𝐶 · (𝐷 𝑋)))‘𝐷) = 𝐶)
 
Theoremcoe1tmfv2 19693 Zero coefficient of a polynomial term. (Contributed by Stefan O'Rear, 27-Mar-2015.)
0 = (0g𝑅)    &   𝐾 = (Base‘𝑅)    &   𝑃 = (Poly1𝑅)    &   𝑋 = (var1𝑅)    &    · = ( ·𝑠𝑃)    &   𝑁 = (mulGrp‘𝑃)    &    = (.g𝑁)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐶𝐾)    &   (𝜑𝐷 ∈ ℕ0)    &   (𝜑𝐹 ∈ ℕ0)    &   (𝜑𝐷𝐹)       (𝜑 → ((coe1‘(𝐶 · (𝐷 𝑋)))‘𝐹) = 0 )
 
Theoremcoe1tmmul2 19694* Coefficient vector of a polynomial multiplied on the right by a term. (Contributed by Stefan O'Rear, 27-Mar-2015.)
0 = (0g𝑅)    &   𝐾 = (Base‘𝑅)    &   𝑃 = (Poly1𝑅)    &   𝑋 = (var1𝑅)    &    · = ( ·𝑠𝑃)    &   𝑁 = (mulGrp‘𝑃)    &    = (.g𝑁)    &   𝐵 = (Base‘𝑃)    &    = (.r𝑃)    &    × = (.r𝑅)    &   (𝜑𝐴𝐵)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐶𝐾)    &   (𝜑𝐷 ∈ ℕ0)       (𝜑 → (coe1‘(𝐴 (𝐶 · (𝐷 𝑋)))) = (𝑥 ∈ ℕ0 ↦ if(𝐷𝑥, (((coe1𝐴)‘(𝑥𝐷)) × 𝐶), 0 )))
 
Theoremcoe1tmmul 19695* Coefficient vector of a polynomial multiplied on the left by a term. (Contributed by Stefan O'Rear, 29-Mar-2015.)
0 = (0g𝑅)    &   𝐾 = (Base‘𝑅)    &   𝑃 = (Poly1𝑅)    &   𝑋 = (var1𝑅)    &    · = ( ·𝑠𝑃)    &   𝑁 = (mulGrp‘𝑃)    &    = (.g𝑁)    &   𝐵 = (Base‘𝑃)    &    = (.r𝑃)    &    × = (.r𝑅)    &   (𝜑𝐴𝐵)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐶𝐾)    &   (𝜑𝐷 ∈ ℕ0)       (𝜑 → (coe1‘((𝐶 · (𝐷 𝑋)) 𝐴)) = (𝑥 ∈ ℕ0 ↦ if(𝐷𝑥, (𝐶 × ((coe1𝐴)‘(𝑥𝐷))), 0 )))
 
Theoremcoe1tmmul2fv 19696 Function value of a right-multiplication by a term in the shifted domain. (Contributed by Stefan O'Rear, 27-Mar-2015.)
0 = (0g𝑅)    &   𝐾 = (Base‘𝑅)    &   𝑃 = (Poly1𝑅)    &   𝑋 = (var1𝑅)    &    · = ( ·𝑠𝑃)    &   𝑁 = (mulGrp‘𝑃)    &    = (.g𝑁)    &   𝐵 = (Base‘𝑃)    &    = (.r𝑃)    &    × = (.r𝑅)    &   (𝜑𝐴𝐵)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐶𝐾)    &   (𝜑𝐷 ∈ ℕ0)    &   (𝜑𝑌 ∈ ℕ0)       (𝜑 → ((coe1‘(𝐴 (𝐶 · (𝐷 𝑋))))‘(𝐷 + 𝑌)) = (((coe1𝐴)‘𝑌) × 𝐶))
 
Theoremcoe1pwmul 19697* Coefficient vector of a polynomial multiplied on the left by a variable power. (Contributed by Stefan O'Rear, 1-Apr-2015.)
0 = (0g𝑅)    &   𝑃 = (Poly1𝑅)    &   𝑋 = (var1𝑅)    &   𝑁 = (mulGrp‘𝑃)    &    = (.g𝑁)    &   𝐵 = (Base‘𝑃)    &    · = (.r𝑃)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐴𝐵)    &   (𝜑𝐷 ∈ ℕ0)       (𝜑 → (coe1‘((𝐷 𝑋) · 𝐴)) = (𝑥 ∈ ℕ0 ↦ if(𝐷𝑥, ((coe1𝐴)‘(𝑥𝐷)), 0 )))
 
Theoremcoe1pwmulfv 19698 Function value of a right-multiplication by a variable power in the shifted domain. (Contributed by Stefan O'Rear, 1-Apr-2015.)
0 = (0g𝑅)    &   𝑃 = (Poly1𝑅)    &   𝑋 = (var1𝑅)    &   𝑁 = (mulGrp‘𝑃)    &    = (.g𝑁)    &   𝐵 = (Base‘𝑃)    &    · = (.r𝑃)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐴𝐵)    &   (𝜑𝐷 ∈ ℕ0)    &   (𝜑𝑌 ∈ ℕ0)       (𝜑 → ((coe1‘((𝐷 𝑋) · 𝐴))‘(𝐷 + 𝑌)) = ((coe1𝐴)‘𝑌))
 
Theoremply1scltm 19699 A scalar is a term with zero exponent. (Contributed by Stefan O'Rear, 29-Mar-2015.)
𝐾 = (Base‘𝑅)    &   𝑃 = (Poly1𝑅)    &   𝑋 = (var1𝑅)    &    · = ( ·𝑠𝑃)    &   𝑁 = (mulGrp‘𝑃)    &    = (.g𝑁)    &   𝐴 = (algSc‘𝑃)       ((𝑅 ∈ Ring ∧ 𝐹𝐾) → (𝐴𝐹) = (𝐹 · (0 𝑋)))
 
Theoremcoe1sclmul 19700 Coefficient vector of a polynomial multiplied on the left by a scalar. (Contributed by Stefan O'Rear, 29-Mar-2015.)
𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &   𝐾 = (Base‘𝑅)    &   𝐴 = (algSc‘𝑃)    &    = (.r𝑃)    &    · = (.r𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝐾𝑌𝐵) → (coe1‘((𝐴𝑋) 𝑌)) = ((ℕ0 × {𝑋}) ∘𝑓 · (coe1𝑌)))
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