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Theorem List for Metamath Proof Explorer - 19301-19400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcoe1fv 19301 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 19302 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 19303* 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 19304 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 19305* 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 19306 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 19307 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 19308 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 19309* 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 19310* 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 19311* 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 19312 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 19313 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 19314 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 19315 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 19316 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 19317 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 19318 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 19319 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 19320 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 19321 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 19322 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 19323 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 19324 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 19325 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 19326 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 19327 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 19328 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 19329 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 19330 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 19331* 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 19332* 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 19333 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 19334 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 19335 Lemma for ply1basfvi 19336 and deg1fvi 23524. (Contributed by Stefan O'Rear, 28-Mar-2015.)
∅ = (Base‘(Poly1‘∅))
 
Theoremply1basfvi 19336 Protection compatibility of the univariate polynomial base set. (Contributed by Stefan O'Rear, 27-Mar-2015.)
(Base‘(Poly1𝑅)) = (Base‘(Poly1‘( I ‘𝑅)))
 
Theoremply1plusgfvi 19337 Protection compatibility of the univariate polynomial addition. (Contributed by Stefan O'Rear, 27-Mar-2015.)
(+g‘(Poly1𝑅)) = (+g‘(Poly1‘( I ‘𝑅)))
 
Theoremply1baspropd 19338* 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 19339* 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 19340 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 19341 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 19342 Univariate power series form a ring. (Contributed by Stefan O'Rear, 22-Mar-2015.)
𝑆 = (PwSer1𝑅)       (𝑅 ∈ Ring → 𝑆 ∈ Ring)
 
Theoremply1ring 19343 Univariate polynomials form a ring. (Contributed by Stefan O'Rear, 22-Mar-2015.)
𝑃 = (Poly1𝑅)       (𝑅 ∈ Ring → 𝑃 ∈ Ring)
 
Theorempsr1lmod 19344 Univariate power series form a left module. (Contributed by Stefan O'Rear, 26-Mar-2015.)
𝑃 = (PwSer1𝑅)       (𝑅 ∈ Ring → 𝑃 ∈ LMod)
 
Theorempsr1sca 19345 Scalars of a univariate power series ring. (Contributed by Stefan O'Rear, 4-Jul-2015.)
𝑃 = (PwSer1𝑅)       (𝑅𝑉𝑅 = (Scalar‘𝑃))
 
Theorempsr1sca2 19346 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 19347 Univariate polynomials form a left module. (Contributed by Stefan O'Rear, 26-Mar-2015.)
𝑃 = (Poly1𝑅)       (𝑅 ∈ Ring → 𝑃 ∈ LMod)
 
Theoremply1sca 19348 Scalars of a univariate polynomial ring. (Contributed by Stefan O'Rear, 26-Mar-2015.)
𝑃 = (Poly1𝑅)       (𝑅𝑉𝑅 = (Scalar‘𝑃))
 
Theoremply1sca2 19349 Scalars of a univariate polynomial ring. (Contributed by Stefan O'Rear, 26-Mar-2015.)
𝑃 = (Poly1𝑅)       ( I ‘𝑅) = (Scalar‘𝑃)
 
Theoremply1mpl0 19350 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 19351 Zero times a univariate polynomial is the zero polynomial (lmod0vs 18626 analog.) (Contributed by AV, 2-Dec-2019.)
𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &    = ( ·𝑠𝑃)    &    0 = (0g𝑅)       ((𝑅 ∈ Ring ∧ 𝑀𝐵) → ( 0 𝑀) = (0g𝑃))
 
Theoremply1mpl1 19352 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 19353 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 19354 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 19355 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 19356 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 19357 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 19358 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 19359 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 19360 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 19361 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 19362 An equivalence for coe1mul2 19364. (Contributed by Stefan O'Rear, 25-Mar-2015.)
((𝐴 ∈ ℕ0𝑋 ∈ (ℕ0𝑚 1𝑜)) → (𝑋𝑟 ≤ (1𝑜 × {𝐴}) ↔ (𝑋‘∅) ∈ (0...𝐴)))
 
Theoremcoe1mul2lem2 19363* An equivalence for coe1mul2 19364. (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 19364* 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 19365* 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 19366 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 19367 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 19368* 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 19369 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 19370 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 19371* 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 19372* 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 19373 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 19374* 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 19375 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 19376 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 19377 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)))
 
Theoremcoe1sclmulfv 19378 A single coefficient of a polynomial multiplied on the left by a scalar. (Contributed by Stefan O'Rear, 1-Apr-2015.)
𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &   𝐾 = (Base‘𝑅)    &   𝐴 = (algSc‘𝑃)    &    = (.r𝑃)    &    · = (.r𝑅)       ((𝑅 ∈ Ring ∧ (𝑋𝐾𝑌𝐵) ∧ 0 ∈ ℕ0) → ((coe1‘((𝐴𝑋) 𝑌))‘ 0 ) = (𝑋 · ((coe1𝑌)‘ 0 )))
 
Theoremcoe1sclmul2 19379 Coefficient vector of a polynomial multiplied on the right 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‘(𝑌 (𝐴𝑋))) = ((coe1𝑌) ∘𝑓 · (ℕ0 × {𝑋})))
 
Theoremply1sclf 19380 A scalar polynomial is a polynomial. (Contributed by Stefan O'Rear, 28-Mar-2015.)
𝑃 = (Poly1𝑅)    &   𝐴 = (algSc‘𝑃)    &   𝐾 = (Base‘𝑅)    &   𝐵 = (Base‘𝑃)       (𝑅 ∈ Ring → 𝐴:𝐾𝐵)
 
Theoremply1sclcl 19381 The value of the algebra scalars function for (univariate) polynomials applied to a scalar results in a constant polynomial. (Contributed by AV, 27-Nov-2019.)
𝑃 = (Poly1𝑅)    &   𝐴 = (algSc‘𝑃)    &   𝐾 = (Base‘𝑅)    &   𝐵 = (Base‘𝑃)       ((𝑅 ∈ Ring ∧ 𝑆𝐾) → (𝐴𝑆) ∈ 𝐵)
 
Theoremcoe1scl 19382* Coefficient vector of a scalar. (Contributed by Stefan O'Rear, 28-Mar-2015.)
𝑃 = (Poly1𝑅)    &   𝐴 = (algSc‘𝑃)    &   𝐾 = (Base‘𝑅)    &    0 = (0g𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝐾) → (coe1‘(𝐴𝑋)) = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, 𝑋, 0 )))
 
Theoremply1sclid 19383 Recover the base scalar from a scalar polynomial. (Contributed by Stefan O'Rear, 28-Mar-2015.)
𝑃 = (Poly1𝑅)    &   𝐴 = (algSc‘𝑃)    &   𝐾 = (Base‘𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝐾) → 𝑋 = ((coe1‘(𝐴𝑋))‘0))
 
Theoremply1sclf1 19384 The polynomial scalar function is injective. (Contributed by Stefan O'Rear, 28-Mar-2015.)
𝑃 = (Poly1𝑅)    &   𝐴 = (algSc‘𝑃)    &   𝐾 = (Base‘𝑅)    &   𝐵 = (Base‘𝑃)       (𝑅 ∈ Ring → 𝐴:𝐾1-1𝐵)
 
Theoremply1scl0 19385 The zero scalar is zero. (Contributed by Stefan O'Rear, 29-Mar-2015.)
𝑃 = (Poly1𝑅)    &   𝐴 = (algSc‘𝑃)    &    0 = (0g𝑅)    &   𝑌 = (0g𝑃)       (𝑅 ∈ Ring → (𝐴0 ) = 𝑌)
 
Theoremply1scln0 19386 Nonzero scalars create nonzero polynomials. (Contributed by Stefan O'Rear, 29-Mar-2015.)
𝑃 = (Poly1𝑅)    &   𝐴 = (algSc‘𝑃)    &    0 = (0g𝑅)    &   𝑌 = (0g𝑃)    &   𝐾 = (Base‘𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝐾𝑋0 ) → (𝐴𝑋) ≠ 𝑌)
 
Theoremply1scl1 19387 The one scalar is the unit polynomial. (Contributed by Stefan O'Rear, 1-Apr-2015.)
𝑃 = (Poly1𝑅)    &   𝐴 = (algSc‘𝑃)    &    1 = (1r𝑅)    &   𝑁 = (1r𝑃)       (𝑅 ∈ Ring → (𝐴1 ) = 𝑁)
 
Theoremply1idvr1 19388 The identity of a polynomial ring expressed as power of the polynomial variable. (Contributed by AV, 14-Aug-2019.)
𝑃 = (Poly1𝑅)    &   𝑋 = (var1𝑅)    &   𝑁 = (mulGrp‘𝑃)    &    = (.g𝑁)       (𝑅 ∈ Ring → (0 𝑋) = (1r𝑃))
 
Theoremcply1mul 19389* The product of two constant polynomials is a constant polynomial. (Contributed by AV, 18-Nov-2019.)
𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &    0 = (0g𝑅)    &    × = (.r𝑃)       ((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) → (∀𝑐 ∈ ℕ (((coe1𝐹)‘𝑐) = 0 ∧ ((coe1𝐺)‘𝑐) = 0 ) → ∀𝑐 ∈ ℕ ((coe1‘(𝐹 × 𝐺))‘𝑐) = 0 ))
 
Theoremply1coefsupp 19390* The decomposition of a univariate polynomial is finitely supported. Formerly part of proof for ply1coe 19391. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by AV, 8-Aug-2019.)
𝑃 = (Poly1𝑅)    &   𝑋 = (var1𝑅)    &   𝐵 = (Base‘𝑃)    &    · = ( ·𝑠𝑃)    &   𝑀 = (mulGrp‘𝑃)    &    = (.g𝑀)    &   𝐴 = (coe1𝐾)       ((𝑅 ∈ Ring ∧ 𝐾𝐵) → (𝑘 ∈ ℕ0 ↦ ((𝐴𝑘) · (𝑘 𝑋))) finSupp (0g𝑃))
 
Theoremply1coe 19391* Decompose a univariate polynomial as a sum of powers. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by AV, 7-Oct-2019.)
𝑃 = (Poly1𝑅)    &   𝑋 = (var1𝑅)    &   𝐵 = (Base‘𝑃)    &    · = ( ·𝑠𝑃)    &   𝑀 = (mulGrp‘𝑃)    &    = (.g𝑀)    &   𝐴 = (coe1𝐾)       ((𝑅 ∈ Ring ∧ 𝐾𝐵) → 𝐾 = (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝐴𝑘) · (𝑘 𝑋)))))
 
Theoremeqcoe1ply1eq 19392* Two polynomials over the same ring are equal if they have identical coefficients. (Contributed by AV, 7-Oct-2019.)
𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &   𝐴 = (coe1𝐾)    &   𝐶 = (coe1𝐿)       ((𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵) → (∀𝑘 ∈ ℕ0 (𝐴𝑘) = (𝐶𝑘) → 𝐾 = 𝐿))
 
Theoremply1coe1eq 19393* Two polynomials over the same ring are equal iff they have identical coefficients. (Contributed by AV, 13-Oct-2019.)
𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &   𝐴 = (coe1𝐾)    &   𝐶 = (coe1𝐿)       ((𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵) → (∀𝑘 ∈ ℕ0 (𝐴𝑘) = (𝐶𝑘) ↔ 𝐾 = 𝐿))
 
Theoremcply1coe0 19394* All but the first coefficient of a constant polynomial ( i.e. a "lifted scalar") are zero. (Contributed by AV, 16-Nov-2019.)
𝐾 = (Base‘𝑅)    &    0 = (0g𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &   𝐴 = (algSc‘𝑃)       ((𝑅 ∈ Ring ∧ 𝑆𝐾) → ∀𝑛 ∈ ℕ ((coe1‘(𝐴𝑆))‘𝑛) = 0 )
 
Theoremcply1coe0bi 19395* A polynomial is constant (i.e. a "lifted scalar") iff all but the first coefficient are zero. (Contributed by AV, 16-Nov-2019.)
𝐾 = (Base‘𝑅)    &    0 = (0g𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &   𝐴 = (algSc‘𝑃)       ((𝑅 ∈ Ring ∧ 𝑀𝐵) → (∃𝑠𝐾 𝑀 = (𝐴𝑠) ↔ ∀𝑛 ∈ ℕ ((coe1𝑀)‘𝑛) = 0 ))
 
Theoremcoe1fzgsumdlem 19396* Lemma for coe1fzgsumd 19397 (induction step). (Contributed by AV, 8-Oct-2019.)
𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐾 ∈ ℕ0)       ((𝑚 ∈ Fin ∧ ¬ 𝑎𝑚𝜑) → ((∀𝑥𝑚 𝑀𝐵 → ((coe1‘(𝑃 Σg (𝑥𝑚𝑀)))‘𝐾) = (𝑅 Σg (𝑥𝑚 ↦ ((coe1𝑀)‘𝐾)))) → (∀𝑥 ∈ (𝑚 ∪ {𝑎})𝑀𝐵 → ((coe1‘(𝑃 Σg (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ 𝑀)))‘𝐾) = (𝑅 Σg (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ ((coe1𝑀)‘𝐾))))))
 
Theoremcoe1fzgsumd 19397* Value of an evaluated coefficient in a finite group sum of polynomials. (Contributed by AV, 8-Oct-2019.)
𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐾 ∈ ℕ0)    &   (𝜑 → ∀𝑥𝑁 𝑀𝐵)    &   (𝜑𝑁 ∈ Fin)       (𝜑 → ((coe1‘(𝑃 Σg (𝑥𝑁𝑀)))‘𝐾) = (𝑅 Σg (𝑥𝑁 ↦ ((coe1𝑀)‘𝐾))))
 
Theoremgsumsmonply1 19398* A finite group sum of scaled monomials is a univariate polynomial. (Contributed by AV, 8-Oct-2019.)
𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &   𝑋 = (var1𝑅)    &    = (.g‘(mulGrp‘𝑃))    &   (𝜑𝑅 ∈ Ring)    &   𝐾 = (Base‘𝑅)    &    = ( ·𝑠𝑃)    &    0 = (0g𝑅)    &   (𝜑 → ∀𝑘 ∈ ℕ0 𝐴𝐾)    &   (𝜑 → (𝑘 ∈ ℕ0𝐴) finSupp 0 )       (𝜑 → (𝑃 Σg (𝑘 ∈ ℕ0 ↦ (𝐴 (𝑘 𝑋)))) ∈ 𝐵)
 
Theoremgsummoncoe1 19399* A coefficient of the polynomial represented as sum of scaled monomials is the coefficient of the corresponding scaled monomial. (Contributed by AV, 13-Oct-2019.)
𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &   𝑋 = (var1𝑅)    &    = (.g‘(mulGrp‘𝑃))    &   (𝜑𝑅 ∈ Ring)    &   𝐾 = (Base‘𝑅)    &    = ( ·𝑠𝑃)    &    0 = (0g𝑅)    &   (𝜑 → ∀𝑘 ∈ ℕ0 𝐴𝐾)    &   (𝜑 → (𝑘 ∈ ℕ0𝐴) finSupp 0 )    &   (𝜑𝐿 ∈ ℕ0)       (𝜑 → ((coe1‘(𝑃 Σg (𝑘 ∈ ℕ0 ↦ (𝐴 (𝑘 𝑋)))))‘𝐿) = 𝐿 / 𝑘𝐴)
 
Theoremgsumply1eq 19400* Two univariate polynomials given as (finitely supported) sum of scaled monomials are equal iff the corresponding coefficients are equal. (Contributed by AV, 21-Nov-2019.)
𝑃 = (Poly1𝑅)    &   𝑋 = (var1𝑅)    &    = (.g‘(mulGrp‘𝑃))    &   (𝜑𝑅 ∈ Ring)    &   𝐾 = (Base‘𝑅)    &    = ( ·𝑠𝑃)    &    0 = (0g𝑅)    &   (𝜑 → ∀𝑘 ∈ ℕ0 𝐴𝐾)    &   (𝜑 → (𝑘 ∈ ℕ0𝐴) finSupp 0 )    &   (𝜑 → ∀𝑘 ∈ ℕ0 𝐵𝐾)    &   (𝜑 → (𝑘 ∈ ℕ0𝐵) finSupp 0 )    &   (𝜑𝑂 = (𝑃 Σg (𝑘 ∈ ℕ0 ↦ (𝐴 (𝑘 𝑋)))))    &   (𝜑𝑄 = (𝑃 Σg (𝑘 ∈ ℕ0 ↦ (𝐵 (𝑘 𝑋)))))       (𝜑 → (𝑂 = 𝑄 ↔ ∀𝑘 ∈ ℕ0 𝐴 = 𝐵))
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