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Theorem List for Metamath Proof Explorer - 21501-21600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcoe1fval3 21501* Univariate power series coefficient vectors expressed as a function composition. (Contributed by Stefan O'Rear, 25-Mar-2015.)
𝐴 = (coe1β€˜πΉ)    &   π΅ = (Baseβ€˜π‘ƒ)    &   π‘ƒ = (PwSer1β€˜π‘…)    &   πΊ = (𝑦 ∈ β„•0 ↦ (1o Γ— {𝑦}))    β‡’   (𝐹 ∈ 𝐡 β†’ 𝐴 = (𝐹 ∘ 𝐺))
 
Theoremcoe1f2 21502 Functionality of univariate power series coefficient vectors. (Contributed by Stefan O'Rear, 25-Mar-2015.)
𝐴 = (coe1β€˜πΉ)    &   π΅ = (Baseβ€˜π‘ƒ)    &   π‘ƒ = (PwSer1β€˜π‘…)    &   πΎ = (Baseβ€˜π‘…)    β‡’   (𝐹 ∈ 𝐡 β†’ 𝐴:β„•0⟢𝐾)
 
Theoremcoe1fval2 21503* Univariate polynomial coefficient vectors expressed as a function composition. (Contributed by Stefan O'Rear, 21-Mar-2015.)
𝐴 = (coe1β€˜πΉ)    &   π΅ = (Baseβ€˜π‘ƒ)    &   π‘ƒ = (Poly1β€˜π‘…)    &   πΊ = (𝑦 ∈ β„•0 ↦ (1o Γ— {𝑦}))    β‡’   (𝐹 ∈ 𝐡 β†’ 𝐴 = (𝐹 ∘ 𝐺))
 
Theoremcoe1f 21504 Functionality of univariate polynomial coefficient vectors. (Contributed by Stefan O'Rear, 21-Mar-2015.)
𝐴 = (coe1β€˜πΉ)    &   π΅ = (Baseβ€˜π‘ƒ)    &   π‘ƒ = (Poly1β€˜π‘…)    &   πΎ = (Baseβ€˜π‘…)    β‡’   (𝐹 ∈ 𝐡 β†’ 𝐴:β„•0⟢𝐾)
 
Theoremcoe1fvalcl 21505 A coefficient of a univariate polynomial over a class/ring is an element of this class/ring. (Contributed by AV, 9-Oct-2019.)
𝐴 = (coe1β€˜πΉ)    &   π΅ = (Baseβ€˜π‘ƒ)    &   π‘ƒ = (Poly1β€˜π‘…)    &   πΎ = (Baseβ€˜π‘…)    β‡’   ((𝐹 ∈ 𝐡 ∧ 𝑁 ∈ β„•0) β†’ (π΄β€˜π‘) ∈ 𝐾)
 
Theoremcoe1sfi 21506 Finite support of univariate polynomial coefficient vectors. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by AV, 19-Jul-2019.)
𝐴 = (coe1β€˜πΉ)    &   π΅ = (Baseβ€˜π‘ƒ)    &   π‘ƒ = (Poly1β€˜π‘…)    &    0 = (0gβ€˜π‘…)    β‡’   (𝐹 ∈ 𝐡 β†’ 𝐴 finSupp 0 )
 
Theoremcoe1fsupp 21507* 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 = (coe1β€˜πΉ)    &   π΅ = (Baseβ€˜π‘ƒ)    &   π‘ƒ = (Poly1β€˜π‘…)    &    0 = (0gβ€˜π‘…)    &   πΎ = (Baseβ€˜π‘…)    β‡’   (𝐹 ∈ 𝐡 β†’ 𝐴 ∈ {𝑔 ∈ (𝐾 ↑m β„•0) ∣ 𝑔 finSupp 0 })
 
Theoremmptcoe1fsupp 21508* A mapping involving coefficients of polynomials is finitely supported. (Contributed by AV, 12-Oct-2019.)
𝑃 = (Poly1β€˜π‘…)    &   π΅ = (Baseβ€˜π‘ƒ)    &    0 = (0gβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐡) β†’ (π‘˜ ∈ β„•0 ↦ ((coe1β€˜π‘€)β€˜π‘˜)) finSupp 0 )
 
Theoremcoe1ae0 21509* The coefficient vector of a univariate polynomial is 0 almost everywhere. (Contributed by AV, 19-Oct-2019.)
𝐴 = (coe1β€˜πΉ)    &   π΅ = (Baseβ€˜π‘ƒ)    &   π‘ƒ = (Poly1β€˜π‘…)    &    0 = (0gβ€˜π‘…)    β‡’   (𝐹 ∈ 𝐡 β†’ βˆƒπ‘  ∈ β„•0 βˆ€π‘› ∈ β„•0 (𝑠 < 𝑛 β†’ (π΄β€˜π‘›) = 0 ))
 
Theoremvr1cl 21510 The generator of a univariate polynomial algebra is contained in the base set. (Contributed by Stefan O'Rear, 19-Mar-2015.)
𝑋 = (var1β€˜π‘…)    &   π‘ƒ = (Poly1β€˜π‘…)    &   π΅ = (Baseβ€˜π‘ƒ)    β‡’   (𝑅 ∈ Ring β†’ 𝑋 ∈ 𝐡)
 
Theoremopsr0 21511 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)    &   π‘‚ = ((𝐼 ordPwSer 𝑅)β€˜π‘‡)    &   (πœ‘ β†’ 𝑇 βŠ† (𝐼 Γ— 𝐼))    β‡’   (πœ‘ β†’ (0gβ€˜π‘†) = (0gβ€˜π‘‚))
 
Theoremopsr1 21512 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)    &   π‘‚ = ((𝐼 ordPwSer 𝑅)β€˜π‘‡)    &   (πœ‘ β†’ 𝑇 βŠ† (𝐼 Γ— 𝐼))    β‡’   (πœ‘ β†’ (1rβ€˜π‘†) = (1rβ€˜π‘‚))
 
Theoremmplplusg 21513 Value of addition in a polynomial ring. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
π‘Œ = (𝐼 mPoly 𝑅)    &   π‘† = (𝐼 mPwSer 𝑅)    &    + = (+gβ€˜π‘Œ)    β‡’    + = (+gβ€˜π‘†)
 
Theoremmplmulr 21514 Value of multiplication in a polynomial ring. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
π‘Œ = (𝐼 mPoly 𝑅)    &   π‘† = (𝐼 mPwSer 𝑅)    &    Β· = (.rβ€˜π‘Œ)    β‡’    Β· = (.rβ€˜π‘†)
 
Theorempsr1plusg 21515 Value of addition in a univariate power series ring. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
π‘Œ = (PwSer1β€˜π‘…)    &   π‘† = (1o mPwSer 𝑅)    &    + = (+gβ€˜π‘Œ)    β‡’    + = (+gβ€˜π‘†)
 
Theorempsr1vsca 21516 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.)
π‘Œ = (PwSer1β€˜π‘…)    &   π‘† = (1o mPwSer 𝑅)    &    Β· = ( ·𝑠 β€˜π‘Œ)    β‡’    Β· = ( ·𝑠 β€˜π‘†)
 
Theorempsr1mulr 21517 Value of multiplication in a univariate power series ring. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
π‘Œ = (PwSer1β€˜π‘…)    &   π‘† = (1o mPwSer 𝑅)    &    Β· = (.rβ€˜π‘Œ)    β‡’    Β· = (.rβ€˜π‘†)
 
Theoremply1plusg 21518 Value of addition in a univariate polynomial ring. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by Mario Carneiro, 4-Jul-2015.)
π‘Œ = (Poly1β€˜π‘…)    &   π‘† = (1o mPoly 𝑅)    &    + = (+gβ€˜π‘Œ)    β‡’    + = (+gβ€˜π‘†)
 
Theoremply1vsca 21519 Value of scalar multiplication in a univariate polynomial ring. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by Mario Carneiro, 4-Jul-2015.)
π‘Œ = (Poly1β€˜π‘…)    &   π‘† = (1o mPoly 𝑅)    &    Β· = ( ·𝑠 β€˜π‘Œ)    β‡’    Β· = ( ·𝑠 β€˜π‘†)
 
Theoremply1mulr 21520 Value of multiplication in a univariate polynomial ring. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by Mario Carneiro, 4-Jul-2015.)
π‘Œ = (Poly1β€˜π‘…)    &   π‘† = (1o mPoly 𝑅)    &    Β· = (.rβ€˜π‘Œ)    β‡’    Β· = (.rβ€˜π‘†)
 
Theoremressply1bas2 21521 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 = (Poly1β€˜π‘…)    &   π» = (𝑅 β†Ύs 𝑇)    &   π‘ˆ = (Poly1β€˜π»)    &   π΅ = (Baseβ€˜π‘ˆ)    &   (πœ‘ β†’ 𝑇 ∈ (SubRingβ€˜π‘…))    &   π‘Š = (PwSer1β€˜π»)    &   πΆ = (Baseβ€˜π‘Š)    &   πΎ = (Baseβ€˜π‘†)    β‡’   (πœ‘ β†’ 𝐡 = (𝐢 ∩ 𝐾))
 
Theoremressply1bas 21522 A restricted polynomial algebra has the same base set. (Contributed by Mario Carneiro, 3-Jul-2015.)
𝑆 = (Poly1β€˜π‘…)    &   π» = (𝑅 β†Ύs 𝑇)    &   π‘ˆ = (Poly1β€˜π»)    &   π΅ = (Baseβ€˜π‘ˆ)    &   (πœ‘ β†’ 𝑇 ∈ (SubRingβ€˜π‘…))    &   π‘ƒ = (𝑆 β†Ύs 𝐡)    β‡’   (πœ‘ β†’ 𝐡 = (Baseβ€˜π‘ƒ))
 
Theoremressply1add 21523 A restricted polynomial algebra has the same addition operation. (Contributed by Mario Carneiro, 3-Jul-2015.)
𝑆 = (Poly1β€˜π‘…)    &   π» = (𝑅 β†Ύs 𝑇)    &   π‘ˆ = (Poly1β€˜π»)    &   π΅ = (Baseβ€˜π‘ˆ)    &   (πœ‘ β†’ 𝑇 ∈ (SubRingβ€˜π‘…))    &   π‘ƒ = (𝑆 β†Ύs 𝐡)    β‡’   ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡)) β†’ (𝑋(+gβ€˜π‘ˆ)π‘Œ) = (𝑋(+gβ€˜π‘ƒ)π‘Œ))
 
Theoremressply1mul 21524 A restricted polynomial algebra has the same multiplication operation. (Contributed by Mario Carneiro, 3-Jul-2015.)
𝑆 = (Poly1β€˜π‘…)    &   π» = (𝑅 β†Ύs 𝑇)    &   π‘ˆ = (Poly1β€˜π»)    &   π΅ = (Baseβ€˜π‘ˆ)    &   (πœ‘ β†’ 𝑇 ∈ (SubRingβ€˜π‘…))    &   π‘ƒ = (𝑆 β†Ύs 𝐡)    β‡’   ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡)) β†’ (𝑋(.rβ€˜π‘ˆ)π‘Œ) = (𝑋(.rβ€˜π‘ƒ)π‘Œ))
 
Theoremressply1vsca 21525 A restricted power series algebra has the same scalar multiplication operation. (Contributed by Mario Carneiro, 3-Jul-2015.)
𝑆 = (Poly1β€˜π‘…)    &   π» = (𝑅 β†Ύs 𝑇)    &   π‘ˆ = (Poly1β€˜π»)    &   π΅ = (Baseβ€˜π‘ˆ)    &   (πœ‘ β†’ 𝑇 ∈ (SubRingβ€˜π‘…))    &   π‘ƒ = (𝑆 β†Ύs 𝐡)    β‡’   ((πœ‘ ∧ (𝑋 ∈ 𝑇 ∧ π‘Œ ∈ 𝐡)) β†’ (𝑋( ·𝑠 β€˜π‘ˆ)π‘Œ) = (𝑋( ·𝑠 β€˜π‘ƒ)π‘Œ))
 
Theoremsubrgply1 21526 A subring of the base ring induces a subring of polynomials. (Contributed by Mario Carneiro, 3-Jul-2015.)
𝑆 = (Poly1β€˜π‘…)    &   π» = (𝑅 β†Ύs 𝑇)    &   π‘ˆ = (Poly1β€˜π»)    &   π΅ = (Baseβ€˜π‘ˆ)    β‡’   (𝑇 ∈ (SubRingβ€˜π‘…) β†’ 𝐡 ∈ (SubRingβ€˜π‘†))
 
Theoremgsumply1subr 21527 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 = (Poly1β€˜π‘…)    &   π» = (𝑅 β†Ύs 𝑇)    &   π‘ˆ = (Poly1β€˜π»)    &   π΅ = (Baseβ€˜π‘ˆ)    &   (πœ‘ β†’ 𝑇 ∈ (SubRingβ€˜π‘…))    &   (πœ‘ β†’ 𝐴 ∈ 𝑉)    &   (πœ‘ β†’ 𝐹:𝐴⟢𝐡)    β‡’   (πœ‘ β†’ (𝑆 Ξ£g 𝐹) = (π‘ˆ Ξ£g 𝐹))
 
Theorempsrbaspropd 21528 Property deduction for power series base set. (Contributed by Stefan O'Rear, 27-Mar-2015.)
(πœ‘ β†’ (Baseβ€˜π‘…) = (Baseβ€˜π‘†))    β‡’   (πœ‘ β†’ (Baseβ€˜(𝐼 mPwSer 𝑅)) = (Baseβ€˜(𝐼 mPwSer 𝑆)))
 
Theorempsrplusgpropd 21529* Property deduction for power series addition. (Contributed by Stefan O'Rear, 27-Mar-2015.) (Revised by Mario Carneiro, 3-Oct-2015.)
(πœ‘ β†’ 𝐡 = (Baseβ€˜π‘…))    &   (πœ‘ β†’ 𝐡 = (Baseβ€˜π‘†))    &   ((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ (π‘₯(+gβ€˜π‘…)𝑦) = (π‘₯(+gβ€˜π‘†)𝑦))    β‡’   (πœ‘ β†’ (+gβ€˜(𝐼 mPwSer 𝑅)) = (+gβ€˜(𝐼 mPwSer 𝑆)))
 
Theoremmplbaspropd 21530* Property deduction for polynomial base set. (Contributed by Stefan O'Rear, 27-Mar-2015.) (Proof shortened by AV, 19-Jul-2019.)
(πœ‘ β†’ 𝐡 = (Baseβ€˜π‘…))    &   (πœ‘ β†’ 𝐡 = (Baseβ€˜π‘†))    &   ((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ (π‘₯(+gβ€˜π‘…)𝑦) = (π‘₯(+gβ€˜π‘†)𝑦))    β‡’   (πœ‘ β†’ (Baseβ€˜(𝐼 mPoly 𝑅)) = (Baseβ€˜(𝐼 mPoly 𝑆)))
 
Theorempsropprmul 21531 Reversing multiplication in a ring reverses multiplication in the power series ring. (Contributed by Stefan O'Rear, 27-Mar-2015.)
π‘Œ = (𝐼 mPwSer 𝑅)    &   π‘† = (opprβ€˜π‘…)    &   π‘ = (𝐼 mPwSer 𝑆)    &    Β· = (.rβ€˜π‘Œ)    &    βˆ™ = (.rβ€˜π‘)    &   π΅ = (Baseβ€˜π‘Œ)    β‡’   ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐡 ∧ 𝐺 ∈ 𝐡) β†’ (𝐹 βˆ™ 𝐺) = (𝐺 Β· 𝐹))
 
Theoremply1opprmul 21532 Reversing multiplication in a ring reverses multiplication in the univariate polynomial ring. (Contributed by Stefan O'Rear, 27-Mar-2015.)
π‘Œ = (Poly1β€˜π‘…)    &   π‘† = (opprβ€˜π‘…)    &   π‘ = (Poly1β€˜π‘†)    &    Β· = (.rβ€˜π‘Œ)    &    βˆ™ = (.rβ€˜π‘)    &   π΅ = (Baseβ€˜π‘Œ)    β‡’   ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐡 ∧ 𝐺 ∈ 𝐡) β†’ (𝐹 βˆ™ 𝐺) = (𝐺 Β· 𝐹))
 
Theorem00ply1bas 21533 Lemma for ply1basfvi 21534 and deg1fvi 25372. (Contributed by Stefan O'Rear, 28-Mar-2015.)
βˆ… = (Baseβ€˜(Poly1β€˜βˆ…))
 
Theoremply1basfvi 21534 Protection compatibility of the univariate polynomial base set. (Contributed by Stefan O'Rear, 27-Mar-2015.)
(Baseβ€˜(Poly1β€˜π‘…)) = (Baseβ€˜(Poly1β€˜( I β€˜π‘…)))
 
Theoremply1plusgfvi 21535 Protection compatibility of the univariate polynomial addition. (Contributed by Stefan O'Rear, 27-Mar-2015.)
(+gβ€˜(Poly1β€˜π‘…)) = (+gβ€˜(Poly1β€˜( I β€˜π‘…)))
 
Theoremply1baspropd 21536* Property deduction for univariate polynomial base set. (Contributed by Stefan O'Rear, 27-Mar-2015.)
(πœ‘ β†’ 𝐡 = (Baseβ€˜π‘…))    &   (πœ‘ β†’ 𝐡 = (Baseβ€˜π‘†))    &   ((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ (π‘₯(+gβ€˜π‘…)𝑦) = (π‘₯(+gβ€˜π‘†)𝑦))    β‡’   (πœ‘ β†’ (Baseβ€˜(Poly1β€˜π‘…)) = (Baseβ€˜(Poly1β€˜π‘†)))
 
Theoremply1plusgpropd 21537* Property deduction for univariate polynomial addition. (Contributed by Stefan O'Rear, 27-Mar-2015.)
(πœ‘ β†’ 𝐡 = (Baseβ€˜π‘…))    &   (πœ‘ β†’ 𝐡 = (Baseβ€˜π‘†))    &   ((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ (π‘₯(+gβ€˜π‘…)𝑦) = (π‘₯(+gβ€˜π‘†)𝑦))    β‡’   (πœ‘ β†’ (+gβ€˜(Poly1β€˜π‘…)) = (+gβ€˜(Poly1β€˜π‘†)))
 
Theoremopsrring 21538 Ordered power series form a ring. (Contributed by Stefan O'Rear, 22-Mar-2015.)
𝑂 = ((𝐼 ordPwSer 𝑅)β€˜π‘‡)    &   (πœ‘ β†’ 𝐼 ∈ 𝑉)    &   (πœ‘ β†’ 𝑅 ∈ Ring)    &   (πœ‘ β†’ 𝑇 βŠ† (𝐼 Γ— 𝐼))    β‡’   (πœ‘ β†’ 𝑂 ∈ Ring)
 
Theoremopsrlmod 21539 Ordered power series form a left module. (Contributed by Stefan O'Rear, 26-Mar-2015.)
𝑂 = ((𝐼 ordPwSer 𝑅)β€˜π‘‡)    &   (πœ‘ β†’ 𝐼 ∈ 𝑉)    &   (πœ‘ β†’ 𝑅 ∈ Ring)    &   (πœ‘ β†’ 𝑇 βŠ† (𝐼 Γ— 𝐼))    β‡’   (πœ‘ β†’ 𝑂 ∈ LMod)
 
Theorempsr1ring 21540 Univariate power series form a ring. (Contributed by Stefan O'Rear, 22-Mar-2015.)
𝑆 = (PwSer1β€˜π‘…)    β‡’   (𝑅 ∈ Ring β†’ 𝑆 ∈ Ring)
 
Theoremply1ring 21541 Univariate polynomials form a ring. (Contributed by Stefan O'Rear, 22-Mar-2015.)
𝑃 = (Poly1β€˜π‘…)    β‡’   (𝑅 ∈ Ring β†’ 𝑃 ∈ Ring)
 
Theorempsr1lmod 21542 Univariate power series form a left module. (Contributed by Stefan O'Rear, 26-Mar-2015.)
𝑃 = (PwSer1β€˜π‘…)    β‡’   (𝑅 ∈ Ring β†’ 𝑃 ∈ LMod)
 
Theorempsr1sca 21543 Scalars of a univariate power series ring. (Contributed by Stefan O'Rear, 4-Jul-2015.)
𝑃 = (PwSer1β€˜π‘…)    β‡’   (𝑅 ∈ 𝑉 β†’ 𝑅 = (Scalarβ€˜π‘ƒ))
 
Theorempsr1sca2 21544 Scalars of a univariate power series ring. (Contributed by Stefan O'Rear, 26-Mar-2015.) (Revised by Mario Carneiro, 4-Jul-2015.)
𝑃 = (PwSer1β€˜π‘…)    β‡’   ( I β€˜π‘…) = (Scalarβ€˜π‘ƒ)
 
Theoremply1lmod 21545 Univariate polynomials form a left module. (Contributed by Stefan O'Rear, 26-Mar-2015.)
𝑃 = (Poly1β€˜π‘…)    β‡’   (𝑅 ∈ Ring β†’ 𝑃 ∈ LMod)
 
Theoremply1sca 21546 Scalars of a univariate polynomial ring. (Contributed by Stefan O'Rear, 26-Mar-2015.)
𝑃 = (Poly1β€˜π‘…)    β‡’   (𝑅 ∈ 𝑉 β†’ 𝑅 = (Scalarβ€˜π‘ƒ))
 
Theoremply1sca2 21547 Scalars of a univariate polynomial ring. (Contributed by Stefan O'Rear, 26-Mar-2015.)
𝑃 = (Poly1β€˜π‘…)    β‡’   ( I β€˜π‘…) = (Scalarβ€˜π‘ƒ)
 
Theoremply1mpl0 21548 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)    &   π‘ƒ = (Poly1β€˜π‘…)    &    0 = (0gβ€˜π‘ƒ)    β‡’    0 = (0gβ€˜π‘€)
 
Theoremply10s0 21549 Zero times a univariate polynomial is the zero polynomial (lmod0vs 20278 analog.) (Contributed by AV, 2-Dec-2019.)
𝑃 = (Poly1β€˜π‘…)    &   π΅ = (Baseβ€˜π‘ƒ)    &    βˆ— = ( ·𝑠 β€˜π‘ƒ)    &    0 = (0gβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐡) β†’ ( 0 βˆ— 𝑀) = (0gβ€˜π‘ƒ))
 
Theoremply1mpl1 21550 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)    &   π‘ƒ = (Poly1β€˜π‘…)    &    1 = (1rβ€˜π‘ƒ)    β‡’    1 = (1rβ€˜π‘€)
 
Theoremply1ascl 21551 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 = (Poly1β€˜π‘…)    &   π΄ = (algScβ€˜π‘ƒ)    β‡’   π΄ = (algScβ€˜(1o mPoly 𝑅))
 
Theoremsubrg1ascl 21552 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 = (Poly1β€˜π‘…)    &   π΄ = (algScβ€˜π‘ƒ)    &   π» = (𝑅 β†Ύs 𝑇)    &   π‘ˆ = (Poly1β€˜π»)    &   (πœ‘ β†’ 𝑇 ∈ (SubRingβ€˜π‘…))    &   πΆ = (algScβ€˜π‘ˆ)    β‡’   (πœ‘ β†’ 𝐢 = (𝐴 β†Ύ 𝑇))
 
Theoremsubrg1asclcl 21553 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 = (Poly1β€˜π‘…)    &   π΄ = (algScβ€˜π‘ƒ)    &   π» = (𝑅 β†Ύs 𝑇)    &   π‘ˆ = (Poly1β€˜π»)    &   (πœ‘ β†’ 𝑇 ∈ (SubRingβ€˜π‘…))    &   π΅ = (Baseβ€˜π‘ˆ)    &   πΎ = (Baseβ€˜π‘…)    &   (πœ‘ β†’ 𝑋 ∈ 𝐾)    β‡’   (πœ‘ β†’ ((π΄β€˜π‘‹) ∈ 𝐡 ↔ 𝑋 ∈ 𝑇))
 
Theoremsubrgvr1 21554 The variables in a subring polynomial algebra are the same as the original ring. (Contributed by Mario Carneiro, 5-Jul-2015.)
𝑋 = (var1β€˜π‘…)    &   (πœ‘ β†’ 𝑇 ∈ (SubRingβ€˜π‘…))    &   π» = (𝑅 β†Ύs 𝑇)    β‡’   (πœ‘ β†’ 𝑋 = (var1β€˜π»))
 
Theoremsubrgvr1cl 21555 The variables in a polynomial algebra are contained in every subring algebra. (Contributed by Mario Carneiro, 5-Jul-2015.)
𝑋 = (var1β€˜π‘…)    &   (πœ‘ β†’ 𝑇 ∈ (SubRingβ€˜π‘…))    &   π» = (𝑅 β†Ύs 𝑇)    &   π‘ˆ = (Poly1β€˜π»)    &   π΅ = (Baseβ€˜π‘ˆ)    β‡’   (πœ‘ β†’ 𝑋 ∈ 𝐡)
 
Theoremcoe1z 21556 The coefficient vector of 0. (Contributed by Stefan O'Rear, 23-Mar-2015.)
𝑃 = (Poly1β€˜π‘…)    &    0 = (0gβ€˜π‘ƒ)    &   π‘Œ = (0gβ€˜π‘…)    β‡’   (𝑅 ∈ Ring β†’ (coe1β€˜ 0 ) = (β„•0 Γ— {π‘Œ}))
 
Theoremcoe1add 21557 The coefficient vector of an addition. (Contributed by Stefan O'Rear, 24-Mar-2015.)
π‘Œ = (Poly1β€˜π‘…)    &   π΅ = (Baseβ€˜π‘Œ)    &    ✚ = (+gβ€˜π‘Œ)    &    + = (+gβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐡 ∧ 𝐺 ∈ 𝐡) β†’ (coe1β€˜(𝐹 ✚ 𝐺)) = ((coe1β€˜πΉ) ∘f + (coe1β€˜πΊ)))
 
Theoremcoe1addfv 21558 A particular coefficient of an addition. (Contributed by Stefan O'Rear, 23-Mar-2015.)
π‘Œ = (Poly1β€˜π‘…)    &   π΅ = (Baseβ€˜π‘Œ)    &    ✚ = (+gβ€˜π‘Œ)    &    + = (+gβ€˜π‘…)    β‡’   (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐡 ∧ 𝐺 ∈ 𝐡) ∧ 𝑋 ∈ β„•0) β†’ ((coe1β€˜(𝐹 ✚ 𝐺))β€˜π‘‹) = (((coe1β€˜πΉ)β€˜π‘‹) + ((coe1β€˜πΊ)β€˜π‘‹)))
 
Theoremcoe1subfv 21559 A particular coefficient of a subtraction. (Contributed by Stefan O'Rear, 23-Mar-2015.)
π‘Œ = (Poly1β€˜π‘…)    &   π΅ = (Baseβ€˜π‘Œ)    &    βˆ’ = (-gβ€˜π‘Œ)    &   π‘ = (-gβ€˜π‘…)    β‡’   (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐡 ∧ 𝐺 ∈ 𝐡) ∧ 𝑋 ∈ β„•0) β†’ ((coe1β€˜(𝐹 βˆ’ 𝐺))β€˜π‘‹) = (((coe1β€˜πΉ)β€˜π‘‹)𝑁((coe1β€˜πΊ)β€˜π‘‹)))
 
Theoremcoe1mul2lem1 21560 An equivalence for coe1mul2 21562. (Contributed by Stefan O'Rear, 25-Mar-2015.)
((𝐴 ∈ β„•0 ∧ 𝑋 ∈ (β„•0 ↑m 1o)) β†’ (𝑋 ∘r ≀ (1o Γ— {𝐴}) ↔ (π‘‹β€˜βˆ…) ∈ (0...𝐴)))
 
Theoremcoe1mul2lem2 21561* An equivalence for coe1mul2 21562. (Contributed by Stefan O'Rear, 25-Mar-2015.)
𝐻 = {𝑑 ∈ (β„•0 ↑m 1o) ∣ 𝑑 ∘r ≀ (1o Γ— {π‘˜})}    β‡’   (π‘˜ ∈ β„•0 β†’ (𝑐 ∈ 𝐻 ↦ (π‘β€˜βˆ…)):𝐻–1-1-ontoβ†’(0...π‘˜))
 
Theoremcoe1mul2 21562* The coefficient vector of multiplication in the univariate power series ring. (Contributed by Stefan O'Rear, 25-Mar-2015.)
𝑆 = (PwSer1β€˜π‘…)    &    βˆ™ = (.rβ€˜π‘†)    &    Β· = (.rβ€˜π‘…)    &   π΅ = (Baseβ€˜π‘†)    β‡’   ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐡 ∧ 𝐺 ∈ 𝐡) β†’ (coe1β€˜(𝐹 βˆ™ 𝐺)) = (π‘˜ ∈ β„•0 ↦ (𝑅 Ξ£g (π‘₯ ∈ (0...π‘˜) ↦ (((coe1β€˜πΉ)β€˜π‘₯) Β· ((coe1β€˜πΊ)β€˜(π‘˜ βˆ’ π‘₯)))))))
 
Theoremcoe1mul 21563* The coefficient vector of multiplication in the univariate polynomial ring. (Contributed by Stefan O'Rear, 25-Mar-2015.)
π‘Œ = (Poly1β€˜π‘…)    &    βˆ™ = (.rβ€˜π‘Œ)    &    Β· = (.rβ€˜π‘…)    &   π΅ = (Baseβ€˜π‘Œ)    β‡’   ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐡 ∧ 𝐺 ∈ 𝐡) β†’ (coe1β€˜(𝐹 βˆ™ 𝐺)) = (π‘˜ ∈ β„•0 ↦ (𝑅 Ξ£g (π‘₯ ∈ (0...π‘˜) ↦ (((coe1β€˜πΉ)β€˜π‘₯) Β· ((coe1β€˜πΊ)β€˜(π‘˜ βˆ’ π‘₯)))))))
 
Theoremply1moncl 21564 Closure of the expression for a univariate primitive monomial. (Contributed by AV, 14-Aug-2019.)
𝑃 = (Poly1β€˜π‘…)    &   π‘‹ = (var1β€˜π‘…)    &   π‘ = (mulGrpβ€˜π‘ƒ)    &    ↑ = (.gβ€˜π‘)    &   π΅ = (Baseβ€˜π‘ƒ)    β‡’   ((𝑅 ∈ Ring ∧ 𝐷 ∈ β„•0) β†’ (𝐷 ↑ 𝑋) ∈ 𝐡)
 
Theoremply1tmcl 21565 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 = (Baseβ€˜π‘…)    &   π‘ƒ = (Poly1β€˜π‘…)    &   π‘‹ = (var1β€˜π‘…)    &    Β· = ( ·𝑠 β€˜π‘ƒ)    &   π‘ = (mulGrpβ€˜π‘ƒ)    &    ↑ = (.gβ€˜π‘)    &   π΅ = (Baseβ€˜π‘ƒ)    β‡’   ((𝑅 ∈ Ring ∧ 𝐢 ∈ 𝐾 ∧ 𝐷 ∈ β„•0) β†’ (𝐢 Β· (𝐷 ↑ 𝑋)) ∈ 𝐡)
 
Theoremcoe1tm 21566* Coefficient vector of a polynomial term. (Contributed by Stefan O'Rear, 27-Mar-2015.)
0 = (0gβ€˜π‘…)    &   πΎ = (Baseβ€˜π‘…)    &   π‘ƒ = (Poly1β€˜π‘…)    &   π‘‹ = (var1β€˜π‘…)    &    Β· = ( ·𝑠 β€˜π‘ƒ)    &   π‘ = (mulGrpβ€˜π‘ƒ)    &    ↑ = (.gβ€˜π‘)    β‡’   ((𝑅 ∈ Ring ∧ 𝐢 ∈ 𝐾 ∧ 𝐷 ∈ β„•0) β†’ (coe1β€˜(𝐢 Β· (𝐷 ↑ 𝑋))) = (π‘₯ ∈ β„•0 ↦ if(π‘₯ = 𝐷, 𝐢, 0 )))
 
Theoremcoe1tmfv1 21567 Nonzero coefficient of a polynomial term. (Contributed by Stefan O'Rear, 27-Mar-2015.)
0 = (0gβ€˜π‘…)    &   πΎ = (Baseβ€˜π‘…)    &   π‘ƒ = (Poly1β€˜π‘…)    &   π‘‹ = (var1β€˜π‘…)    &    Β· = ( ·𝑠 β€˜π‘ƒ)    &   π‘ = (mulGrpβ€˜π‘ƒ)    &    ↑ = (.gβ€˜π‘)    β‡’   ((𝑅 ∈ Ring ∧ 𝐢 ∈ 𝐾 ∧ 𝐷 ∈ β„•0) β†’ ((coe1β€˜(𝐢 Β· (𝐷 ↑ 𝑋)))β€˜π·) = 𝐢)
 
Theoremcoe1tmfv2 21568 Zero coefficient of a polynomial term. (Contributed by Stefan O'Rear, 27-Mar-2015.)
0 = (0gβ€˜π‘…)    &   πΎ = (Baseβ€˜π‘…)    &   π‘ƒ = (Poly1β€˜π‘…)    &   π‘‹ = (var1β€˜π‘…)    &    Β· = ( ·𝑠 β€˜π‘ƒ)    &   π‘ = (mulGrpβ€˜π‘ƒ)    &    ↑ = (.gβ€˜π‘)    &   (πœ‘ β†’ 𝑅 ∈ Ring)    &   (πœ‘ β†’ 𝐢 ∈ 𝐾)    &   (πœ‘ β†’ 𝐷 ∈ β„•0)    &   (πœ‘ β†’ 𝐹 ∈ β„•0)    &   (πœ‘ β†’ 𝐷 β‰  𝐹)    β‡’   (πœ‘ β†’ ((coe1β€˜(𝐢 Β· (𝐷 ↑ 𝑋)))β€˜πΉ) = 0 )
 
Theoremcoe1tmmul2 21569* Coefficient vector of a polynomial multiplied on the right by a term. (Contributed by Stefan O'Rear, 27-Mar-2015.)
0 = (0gβ€˜π‘…)    &   πΎ = (Baseβ€˜π‘…)    &   π‘ƒ = (Poly1β€˜π‘…)    &   π‘‹ = (var1β€˜π‘…)    &    Β· = ( ·𝑠 β€˜π‘ƒ)    &   π‘ = (mulGrpβ€˜π‘ƒ)    &    ↑ = (.gβ€˜π‘)    &   π΅ = (Baseβ€˜π‘ƒ)    &    βˆ™ = (.rβ€˜π‘ƒ)    &    Γ— = (.rβ€˜π‘…)    &   (πœ‘ β†’ 𝐴 ∈ 𝐡)    &   (πœ‘ β†’ 𝑅 ∈ Ring)    &   (πœ‘ β†’ 𝐢 ∈ 𝐾)    &   (πœ‘ β†’ 𝐷 ∈ β„•0)    β‡’   (πœ‘ β†’ (coe1β€˜(𝐴 βˆ™ (𝐢 Β· (𝐷 ↑ 𝑋)))) = (π‘₯ ∈ β„•0 ↦ if(𝐷 ≀ π‘₯, (((coe1β€˜π΄)β€˜(π‘₯ βˆ’ 𝐷)) Γ— 𝐢), 0 )))
 
Theoremcoe1tmmul 21570* Coefficient vector of a polynomial multiplied on the left by a term. (Contributed by Stefan O'Rear, 29-Mar-2015.)
0 = (0gβ€˜π‘…)    &   πΎ = (Baseβ€˜π‘…)    &   π‘ƒ = (Poly1β€˜π‘…)    &   π‘‹ = (var1β€˜π‘…)    &    Β· = ( ·𝑠 β€˜π‘ƒ)    &   π‘ = (mulGrpβ€˜π‘ƒ)    &    ↑ = (.gβ€˜π‘)    &   π΅ = (Baseβ€˜π‘ƒ)    &    βˆ™ = (.rβ€˜π‘ƒ)    &    Γ— = (.rβ€˜π‘…)    &   (πœ‘ β†’ 𝐴 ∈ 𝐡)    &   (πœ‘ β†’ 𝑅 ∈ Ring)    &   (πœ‘ β†’ 𝐢 ∈ 𝐾)    &   (πœ‘ β†’ 𝐷 ∈ β„•0)    β‡’   (πœ‘ β†’ (coe1β€˜((𝐢 Β· (𝐷 ↑ 𝑋)) βˆ™ 𝐴)) = (π‘₯ ∈ β„•0 ↦ if(𝐷 ≀ π‘₯, (𝐢 Γ— ((coe1β€˜π΄)β€˜(π‘₯ βˆ’ 𝐷))), 0 )))
 
Theoremcoe1tmmul2fv 21571 Function value of a right-multiplication by a term in the shifted domain. (Contributed by Stefan O'Rear, 27-Mar-2015.)
0 = (0gβ€˜π‘…)    &   πΎ = (Baseβ€˜π‘…)    &   π‘ƒ = (Poly1β€˜π‘…)    &   π‘‹ = (var1β€˜π‘…)    &    Β· = ( ·𝑠 β€˜π‘ƒ)    &   π‘ = (mulGrpβ€˜π‘ƒ)    &    ↑ = (.gβ€˜π‘)    &   π΅ = (Baseβ€˜π‘ƒ)    &    βˆ™ = (.rβ€˜π‘ƒ)    &    Γ— = (.rβ€˜π‘…)    &   (πœ‘ β†’ 𝐴 ∈ 𝐡)    &   (πœ‘ β†’ 𝑅 ∈ Ring)    &   (πœ‘ β†’ 𝐢 ∈ 𝐾)    &   (πœ‘ β†’ 𝐷 ∈ β„•0)    &   (πœ‘ β†’ π‘Œ ∈ β„•0)    β‡’   (πœ‘ β†’ ((coe1β€˜(𝐴 βˆ™ (𝐢 Β· (𝐷 ↑ 𝑋))))β€˜(𝐷 + π‘Œ)) = (((coe1β€˜π΄)β€˜π‘Œ) Γ— 𝐢))
 
Theoremcoe1pwmul 21572* Coefficient vector of a polynomial multiplied on the left by a variable power. (Contributed by Stefan O'Rear, 1-Apr-2015.)
0 = (0gβ€˜π‘…)    &   π‘ƒ = (Poly1β€˜π‘…)    &   π‘‹ = (var1β€˜π‘…)    &   π‘ = (mulGrpβ€˜π‘ƒ)    &    ↑ = (.gβ€˜π‘)    &   π΅ = (Baseβ€˜π‘ƒ)    &    Β· = (.rβ€˜π‘ƒ)    &   (πœ‘ β†’ 𝑅 ∈ Ring)    &   (πœ‘ β†’ 𝐴 ∈ 𝐡)    &   (πœ‘ β†’ 𝐷 ∈ β„•0)    β‡’   (πœ‘ β†’ (coe1β€˜((𝐷 ↑ 𝑋) Β· 𝐴)) = (π‘₯ ∈ β„•0 ↦ if(𝐷 ≀ π‘₯, ((coe1β€˜π΄)β€˜(π‘₯ βˆ’ 𝐷)), 0 )))
 
Theoremcoe1pwmulfv 21573 Function value of a right-multiplication by a variable power in the shifted domain. (Contributed by Stefan O'Rear, 1-Apr-2015.)
0 = (0gβ€˜π‘…)    &   π‘ƒ = (Poly1β€˜π‘…)    &   π‘‹ = (var1β€˜π‘…)    &   π‘ = (mulGrpβ€˜π‘ƒ)    &    ↑ = (.gβ€˜π‘)    &   π΅ = (Baseβ€˜π‘ƒ)    &    Β· = (.rβ€˜π‘ƒ)    &   (πœ‘ β†’ 𝑅 ∈ Ring)    &   (πœ‘ β†’ 𝐴 ∈ 𝐡)    &   (πœ‘ β†’ 𝐷 ∈ β„•0)    &   (πœ‘ β†’ π‘Œ ∈ β„•0)    β‡’   (πœ‘ β†’ ((coe1β€˜((𝐷 ↑ 𝑋) Β· 𝐴))β€˜(𝐷 + π‘Œ)) = ((coe1β€˜π΄)β€˜π‘Œ))
 
Theoremply1scltm 21574 A scalar is a term with zero exponent. (Contributed by Stefan O'Rear, 29-Mar-2015.)
𝐾 = (Baseβ€˜π‘…)    &   π‘ƒ = (Poly1β€˜π‘…)    &   π‘‹ = (var1β€˜π‘…)    &    Β· = ( ·𝑠 β€˜π‘ƒ)    &   π‘ = (mulGrpβ€˜π‘ƒ)    &    ↑ = (.gβ€˜π‘)    &   π΄ = (algScβ€˜π‘ƒ)    β‡’   ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐾) β†’ (π΄β€˜πΉ) = (𝐹 Β· (0 ↑ 𝑋)))
 
Theoremcoe1sclmul 21575 Coefficient vector of a polynomial multiplied on the left by a scalar. (Contributed by Stefan O'Rear, 29-Mar-2015.)
𝑃 = (Poly1β€˜π‘…)    &   π΅ = (Baseβ€˜π‘ƒ)    &   πΎ = (Baseβ€˜π‘…)    &   π΄ = (algScβ€˜π‘ƒ)    &    βˆ™ = (.rβ€˜π‘ƒ)    &    Β· = (.rβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ π‘Œ ∈ 𝐡) β†’ (coe1β€˜((π΄β€˜π‘‹) βˆ™ π‘Œ)) = ((β„•0 Γ— {𝑋}) ∘f Β· (coe1β€˜π‘Œ)))
 
Theoremcoe1sclmulfv 21576 A single coefficient of a polynomial multiplied on the left by a scalar. (Contributed by Stefan O'Rear, 1-Apr-2015.)
𝑃 = (Poly1β€˜π‘…)    &   π΅ = (Baseβ€˜π‘ƒ)    &   πΎ = (Baseβ€˜π‘…)    &   π΄ = (algScβ€˜π‘ƒ)    &    βˆ™ = (.rβ€˜π‘ƒ)    &    Β· = (.rβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐾 ∧ π‘Œ ∈ 𝐡) ∧ 0 ∈ β„•0) β†’ ((coe1β€˜((π΄β€˜π‘‹) βˆ™ π‘Œ))β€˜ 0 ) = (𝑋 Β· ((coe1β€˜π‘Œ)β€˜ 0 )))
 
Theoremcoe1sclmul2 21577 Coefficient vector of a polynomial multiplied on the right by a scalar. (Contributed by Stefan O'Rear, 29-Mar-2015.)
𝑃 = (Poly1β€˜π‘…)    &   π΅ = (Baseβ€˜π‘ƒ)    &   πΎ = (Baseβ€˜π‘…)    &   π΄ = (algScβ€˜π‘ƒ)    &    βˆ™ = (.rβ€˜π‘ƒ)    &    Β· = (.rβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ π‘Œ ∈ 𝐡) β†’ (coe1β€˜(π‘Œ βˆ™ (π΄β€˜π‘‹))) = ((coe1β€˜π‘Œ) ∘f Β· (β„•0 Γ— {𝑋})))
 
Theoremply1sclf 21578 A scalar polynomial is a polynomial. (Contributed by Stefan O'Rear, 28-Mar-2015.)
𝑃 = (Poly1β€˜π‘…)    &   π΄ = (algScβ€˜π‘ƒ)    &   πΎ = (Baseβ€˜π‘…)    &   π΅ = (Baseβ€˜π‘ƒ)    β‡’   (𝑅 ∈ Ring β†’ 𝐴:𝐾⟢𝐡)
 
Theoremply1sclcl 21579 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 = (Poly1β€˜π‘…)    &   π΄ = (algScβ€˜π‘ƒ)    &   πΎ = (Baseβ€˜π‘…)    &   π΅ = (Baseβ€˜π‘ƒ)    β‡’   ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐾) β†’ (π΄β€˜π‘†) ∈ 𝐡)
 
Theoremcoe1scl 21580* Coefficient vector of a scalar. (Contributed by Stefan O'Rear, 28-Mar-2015.)
𝑃 = (Poly1β€˜π‘…)    &   π΄ = (algScβ€˜π‘ƒ)    &   πΎ = (Baseβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾) β†’ (coe1β€˜(π΄β€˜π‘‹)) = (π‘₯ ∈ β„•0 ↦ if(π‘₯ = 0, 𝑋, 0 )))
 
Theoremply1sclid 21581 Recover the base scalar from a scalar polynomial. (Contributed by Stefan O'Rear, 28-Mar-2015.)
𝑃 = (Poly1β€˜π‘…)    &   π΄ = (algScβ€˜π‘ƒ)    &   πΎ = (Baseβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾) β†’ 𝑋 = ((coe1β€˜(π΄β€˜π‘‹))β€˜0))
 
Theoremply1sclf1 21582 The polynomial scalar function is injective. (Contributed by Stefan O'Rear, 28-Mar-2015.)
𝑃 = (Poly1β€˜π‘…)    &   π΄ = (algScβ€˜π‘ƒ)    &   πΎ = (Baseβ€˜π‘…)    &   π΅ = (Baseβ€˜π‘ƒ)    β‡’   (𝑅 ∈ Ring β†’ 𝐴:𝐾–1-1→𝐡)
 
Theoremply1scl0 21583 The zero scalar is zero. (Contributed by Stefan O'Rear, 29-Mar-2015.)
𝑃 = (Poly1β€˜π‘…)    &   π΄ = (algScβ€˜π‘ƒ)    &    0 = (0gβ€˜π‘…)    &   π‘Œ = (0gβ€˜π‘ƒ)    β‡’   (𝑅 ∈ Ring β†’ (π΄β€˜ 0 ) = π‘Œ)
 
Theoremply1scln0 21584 Nonzero scalars create nonzero polynomials. (Contributed by Stefan O'Rear, 29-Mar-2015.)
𝑃 = (Poly1β€˜π‘…)    &   π΄ = (algScβ€˜π‘ƒ)    &    0 = (0gβ€˜π‘…)    &   π‘Œ = (0gβ€˜π‘ƒ)    &   πΎ = (Baseβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑋 β‰  0 ) β†’ (π΄β€˜π‘‹) β‰  π‘Œ)
 
Theoremply1scl1 21585 The one scalar is the unit polynomial. (Contributed by Stefan O'Rear, 1-Apr-2015.)
𝑃 = (Poly1β€˜π‘…)    &   π΄ = (algScβ€˜π‘ƒ)    &    1 = (1rβ€˜π‘…)    &   π‘ = (1rβ€˜π‘ƒ)    β‡’   (𝑅 ∈ Ring β†’ (π΄β€˜ 1 ) = 𝑁)
 
Theoremply1idvr1 21586 The identity of a polynomial ring expressed as power of the polynomial variable. (Contributed by AV, 14-Aug-2019.)
𝑃 = (Poly1β€˜π‘…)    &   π‘‹ = (var1β€˜π‘…)    &   π‘ = (mulGrpβ€˜π‘ƒ)    &    ↑ = (.gβ€˜π‘)    β‡’   (𝑅 ∈ Ring β†’ (0 ↑ 𝑋) = (1rβ€˜π‘ƒ))
 
Theoremcply1mul 21587* The product of two constant polynomials is a constant polynomial. (Contributed by AV, 18-Nov-2019.)
𝑃 = (Poly1β€˜π‘…)    &   π΅ = (Baseβ€˜π‘ƒ)    &    0 = (0gβ€˜π‘…)    &    Γ— = (.rβ€˜π‘ƒ)    β‡’   ((𝑅 ∈ Ring ∧ (𝐹 ∈ 𝐡 ∧ 𝐺 ∈ 𝐡)) β†’ (βˆ€π‘ ∈ β„• (((coe1β€˜πΉ)β€˜π‘) = 0 ∧ ((coe1β€˜πΊ)β€˜π‘) = 0 ) β†’ βˆ€π‘ ∈ β„• ((coe1β€˜(𝐹 Γ— 𝐺))β€˜π‘) = 0 ))
 
Theoremply1coefsupp 21588* The decomposition of a univariate polynomial is finitely supported. Formerly part of proof for ply1coe 21589. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by AV, 8-Aug-2019.)
𝑃 = (Poly1β€˜π‘…)    &   π‘‹ = (var1β€˜π‘…)    &   π΅ = (Baseβ€˜π‘ƒ)    &    Β· = ( ·𝑠 β€˜π‘ƒ)    &   π‘€ = (mulGrpβ€˜π‘ƒ)    &    ↑ = (.gβ€˜π‘€)    &   π΄ = (coe1β€˜πΎ)    β‡’   ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) β†’ (π‘˜ ∈ β„•0 ↦ ((π΄β€˜π‘˜) Β· (π‘˜ ↑ 𝑋))) finSupp (0gβ€˜π‘ƒ))
 
Theoremply1coe 21589* Decompose a univariate polynomial as a sum of powers. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by AV, 7-Oct-2019.)
𝑃 = (Poly1β€˜π‘…)    &   π‘‹ = (var1β€˜π‘…)    &   π΅ = (Baseβ€˜π‘ƒ)    &    Β· = ( ·𝑠 β€˜π‘ƒ)    &   π‘€ = (mulGrpβ€˜π‘ƒ)    &    ↑ = (.gβ€˜π‘€)    &   π΄ = (coe1β€˜πΎ)    β‡’   ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) β†’ 𝐾 = (𝑃 Ξ£g (π‘˜ ∈ β„•0 ↦ ((π΄β€˜π‘˜) Β· (π‘˜ ↑ 𝑋)))))
 
Theoremeqcoe1ply1eq 21590* Two polynomials over the same ring are equal if they have identical coefficients. (Contributed by AV, 7-Oct-2019.)
𝑃 = (Poly1β€˜π‘…)    &   π΅ = (Baseβ€˜π‘ƒ)    &   π΄ = (coe1β€˜πΎ)    &   πΆ = (coe1β€˜πΏ)    β‡’   ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡 ∧ 𝐿 ∈ 𝐡) β†’ (βˆ€π‘˜ ∈ β„•0 (π΄β€˜π‘˜) = (πΆβ€˜π‘˜) β†’ 𝐾 = 𝐿))
 
Theoremply1coe1eq 21591* Two polynomials over the same ring are equal iff they have identical coefficients. (Contributed by AV, 13-Oct-2019.)
𝑃 = (Poly1β€˜π‘…)    &   π΅ = (Baseβ€˜π‘ƒ)    &   π΄ = (coe1β€˜πΎ)    &   πΆ = (coe1β€˜πΏ)    β‡’   ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡 ∧ 𝐿 ∈ 𝐡) β†’ (βˆ€π‘˜ ∈ β„•0 (π΄β€˜π‘˜) = (πΆβ€˜π‘˜) ↔ 𝐾 = 𝐿))
 
Theoremcply1coe0 21592* All but the first coefficient of a constant polynomial ( i.e. a "lifted scalar") are zero. (Contributed by AV, 16-Nov-2019.)
𝐾 = (Baseβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    &   π‘ƒ = (Poly1β€˜π‘…)    &   π΅ = (Baseβ€˜π‘ƒ)    &   π΄ = (algScβ€˜π‘ƒ)    β‡’   ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐾) β†’ βˆ€π‘› ∈ β„• ((coe1β€˜(π΄β€˜π‘†))β€˜π‘›) = 0 )
 
Theoremcply1coe0bi 21593* A polynomial is constant (i.e. a "lifted scalar") iff all but the first coefficient are zero. (Contributed by AV, 16-Nov-2019.)
𝐾 = (Baseβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    &   π‘ƒ = (Poly1β€˜π‘…)    &   π΅ = (Baseβ€˜π‘ƒ)    &   π΄ = (algScβ€˜π‘ƒ)    β‡’   ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐡) β†’ (βˆƒπ‘  ∈ 𝐾 𝑀 = (π΄β€˜π‘ ) ↔ βˆ€π‘› ∈ β„• ((coe1β€˜π‘€)β€˜π‘›) = 0 ))
 
Theoremcoe1fzgsumdlem 21594* Lemma for coe1fzgsumd 21595 (induction step). (Contributed by AV, 8-Oct-2019.)
𝑃 = (Poly1β€˜π‘…)    &   π΅ = (Baseβ€˜π‘ƒ)    &   (πœ‘ β†’ 𝑅 ∈ Ring)    &   (πœ‘ β†’ 𝐾 ∈ β„•0)    β‡’   ((π‘š ∈ Fin ∧ Β¬ π‘Ž ∈ π‘š ∧ πœ‘) β†’ ((βˆ€π‘₯ ∈ π‘š 𝑀 ∈ 𝐡 β†’ ((coe1β€˜(𝑃 Ξ£g (π‘₯ ∈ π‘š ↦ 𝑀)))β€˜πΎ) = (𝑅 Ξ£g (π‘₯ ∈ π‘š ↦ ((coe1β€˜π‘€)β€˜πΎ)))) β†’ (βˆ€π‘₯ ∈ (π‘š βˆͺ {π‘Ž})𝑀 ∈ 𝐡 β†’ ((coe1β€˜(𝑃 Ξ£g (π‘₯ ∈ (π‘š βˆͺ {π‘Ž}) ↦ 𝑀)))β€˜πΎ) = (𝑅 Ξ£g (π‘₯ ∈ (π‘š βˆͺ {π‘Ž}) ↦ ((coe1β€˜π‘€)β€˜πΎ))))))
 
Theoremcoe1fzgsumd 21595* Value of an evaluated coefficient in a finite group sum of polynomials. (Contributed by AV, 8-Oct-2019.)
𝑃 = (Poly1β€˜π‘…)    &   π΅ = (Baseβ€˜π‘ƒ)    &   (πœ‘ β†’ 𝑅 ∈ Ring)    &   (πœ‘ β†’ 𝐾 ∈ β„•0)    &   (πœ‘ β†’ βˆ€π‘₯ ∈ 𝑁 𝑀 ∈ 𝐡)    &   (πœ‘ β†’ 𝑁 ∈ Fin)    β‡’   (πœ‘ β†’ ((coe1β€˜(𝑃 Ξ£g (π‘₯ ∈ 𝑁 ↦ 𝑀)))β€˜πΎ) = (𝑅 Ξ£g (π‘₯ ∈ 𝑁 ↦ ((coe1β€˜π‘€)β€˜πΎ))))
 
Theoremgsumsmonply1 21596* A finite group sum of scaled monomials is a univariate polynomial. (Contributed by AV, 8-Oct-2019.)
𝑃 = (Poly1β€˜π‘…)    &   π΅ = (Baseβ€˜π‘ƒ)    &   π‘‹ = (var1β€˜π‘…)    &    ↑ = (.gβ€˜(mulGrpβ€˜π‘ƒ))    &   (πœ‘ β†’ 𝑅 ∈ Ring)    &   πΎ = (Baseβ€˜π‘…)    &    βˆ— = ( ·𝑠 β€˜π‘ƒ)    &    0 = (0gβ€˜π‘…)    &   (πœ‘ β†’ βˆ€π‘˜ ∈ β„•0 𝐴 ∈ 𝐾)    &   (πœ‘ β†’ (π‘˜ ∈ β„•0 ↦ 𝐴) finSupp 0 )    β‡’   (πœ‘ β†’ (𝑃 Ξ£g (π‘˜ ∈ β„•0 ↦ (𝐴 βˆ— (π‘˜ ↑ 𝑋)))) ∈ 𝐡)
 
Theoremgsummoncoe1 21597* A coefficient of the polynomial represented as a sum of scaled monomials is the coefficient of the corresponding scaled monomial. (Contributed by AV, 13-Oct-2019.)
𝑃 = (Poly1β€˜π‘…)    &   π΅ = (Baseβ€˜π‘ƒ)    &   π‘‹ = (var1β€˜π‘…)    &    ↑ = (.gβ€˜(mulGrpβ€˜π‘ƒ))    &   (πœ‘ β†’ 𝑅 ∈ Ring)    &   πΎ = (Baseβ€˜π‘…)    &    βˆ— = ( ·𝑠 β€˜π‘ƒ)    &    0 = (0gβ€˜π‘…)    &   (πœ‘ β†’ βˆ€π‘˜ ∈ β„•0 𝐴 ∈ 𝐾)    &   (πœ‘ β†’ (π‘˜ ∈ β„•0 ↦ 𝐴) finSupp 0 )    &   (πœ‘ β†’ 𝐿 ∈ β„•0)    β‡’   (πœ‘ β†’ ((coe1β€˜(𝑃 Ξ£g (π‘˜ ∈ β„•0 ↦ (𝐴 βˆ— (π‘˜ ↑ 𝑋)))))β€˜πΏ) = ⦋𝐿 / π‘˜β¦Œπ΄)
 
Theoremgsumply1eq 21598* 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 = (Poly1β€˜π‘…)    &   π‘‹ = (var1β€˜π‘…)    &    ↑ = (.gβ€˜(mulGrpβ€˜π‘ƒ))    &   (πœ‘ β†’ 𝑅 ∈ Ring)    &   πΎ = (Baseβ€˜π‘…)    &    βˆ— = ( ·𝑠 β€˜π‘ƒ)    &    0 = (0gβ€˜π‘…)    &   (πœ‘ β†’ βˆ€π‘˜ ∈ β„•0 𝐴 ∈ 𝐾)    &   (πœ‘ β†’ (π‘˜ ∈ β„•0 ↦ 𝐴) finSupp 0 )    &   (πœ‘ β†’ βˆ€π‘˜ ∈ β„•0 𝐡 ∈ 𝐾)    &   (πœ‘ β†’ (π‘˜ ∈ β„•0 ↦ 𝐡) finSupp 0 )    &   (πœ‘ β†’ 𝑂 = (𝑃 Ξ£g (π‘˜ ∈ β„•0 ↦ (𝐴 βˆ— (π‘˜ ↑ 𝑋)))))    &   (πœ‘ β†’ 𝑄 = (𝑃 Ξ£g (π‘˜ ∈ β„•0 ↦ (𝐡 βˆ— (π‘˜ ↑ 𝑋)))))    β‡’   (πœ‘ β†’ (𝑂 = 𝑄 ↔ βˆ€π‘˜ ∈ β„•0 𝐴 = 𝐡))
 
Theoremlply1binom 21599* The binomial theorem for linear polynomials (monic polynomials of degree 1) over commutative rings: (𝑋 + 𝐴)↑𝑁 is the sum from π‘˜ = 0 to 𝑁 of (𝑁Cπ‘˜) Β· ((𝐴↑(𝑁 βˆ’ π‘˜)) Β· (π‘‹β†‘π‘˜)). (Contributed by AV, 25-Aug-2019.)
𝑃 = (Poly1β€˜π‘…)    &   π‘‹ = (var1β€˜π‘…)    &    + = (+gβ€˜π‘ƒ)    &    Γ— = (.rβ€˜π‘ƒ)    &    Β· = (.gβ€˜π‘ƒ)    &   πΊ = (mulGrpβ€˜π‘ƒ)    &    ↑ = (.gβ€˜πΊ)    &   π΅ = (Baseβ€˜π‘ƒ)    β‡’   ((𝑅 ∈ CRing ∧ 𝑁 ∈ β„•0 ∧ 𝐴 ∈ 𝐡) β†’ (𝑁 ↑ (𝑋 + 𝐴)) = (𝑃 Ξ£g (π‘˜ ∈ (0...𝑁) ↦ ((𝑁Cπ‘˜) Β· (((𝑁 βˆ’ π‘˜) ↑ 𝐴) Γ— (π‘˜ ↑ 𝑋))))))
 
Theoremlply1binomsc 21600* The binomial theorem for linear polynomials (monic polynomials of degree 1) over commutative rings, expressed by an element of this ring: (𝑋 + 𝐴)↑𝑁 is the sum from π‘˜ = 0 to 𝑁 of (𝑁Cπ‘˜) Β· ((𝐴↑(𝑁 βˆ’ π‘˜)) Β· (π‘‹β†‘π‘˜)). (Contributed by AV, 25-Aug-2019.)
𝑃 = (Poly1β€˜π‘…)    &   π‘‹ = (var1β€˜π‘…)    &    + = (+gβ€˜π‘ƒ)    &    Γ— = (.rβ€˜π‘ƒ)    &    Β· = (.gβ€˜π‘ƒ)    &   πΊ = (mulGrpβ€˜π‘ƒ)    &    ↑ = (.gβ€˜πΊ)    &   πΎ = (Baseβ€˜π‘…)    &   π‘† = (algScβ€˜π‘ƒ)    &   π» = (mulGrpβ€˜π‘…)    &   πΈ = (.gβ€˜π»)    β‡’   ((𝑅 ∈ CRing ∧ 𝑁 ∈ β„•0 ∧ 𝐴 ∈ 𝐾) β†’ (𝑁 ↑ (𝑋 + (π‘†β€˜π΄))) = (𝑃 Ξ£g (π‘˜ ∈ (0...𝑁) ↦ ((𝑁Cπ‘˜) Β· ((π‘†β€˜((𝑁 βˆ’ π‘˜)𝐸𝐴)) Γ— (π‘˜ ↑ 𝑋))))))
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144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 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 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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 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