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Theorem List for Metamath Proof Explorer - 20301-20400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcoe1fsupp 20301* 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 20302* 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 20303* 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 20304 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 20305 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 20306 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 20307 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 20308 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 20309 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 20310 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 20311 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 20312 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 20313 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 20314 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 20315 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 20316 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 20317 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 20318 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 20319 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 20320 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 20321 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 20322 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 20323* 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 20324* 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 20325 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 20326 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 20327 Lemma for ply1basfvi 20328 and deg1fvi 24596. (Contributed by Stefan O'Rear, 28-Mar-2015.)
∅ = (Base‘(Poly1‘∅))

Theoremply1basfvi 20328 Protection compatibility of the univariate polynomial base set. (Contributed by Stefan O'Rear, 27-Mar-2015.)
(Base‘(Poly1𝑅)) = (Base‘(Poly1‘( I ‘𝑅)))

Theoremply1plusgfvi 20329 Protection compatibility of the univariate polynomial addition. (Contributed by Stefan O'Rear, 27-Mar-2015.)
(+g‘(Poly1𝑅)) = (+g‘(Poly1‘( I ‘𝑅)))

Theoremply1baspropd 20330* 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 20331* 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 20332 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 20333 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 20334 Univariate power series form a ring. (Contributed by Stefan O'Rear, 22-Mar-2015.)
𝑆 = (PwSer1𝑅)       (𝑅 ∈ Ring → 𝑆 ∈ Ring)

Theoremply1ring 20335 Univariate polynomials form a ring. (Contributed by Stefan O'Rear, 22-Mar-2015.)
𝑃 = (Poly1𝑅)       (𝑅 ∈ Ring → 𝑃 ∈ Ring)

Theorempsr1lmod 20336 Univariate power series form a left module. (Contributed by Stefan O'Rear, 26-Mar-2015.)
𝑃 = (PwSer1𝑅)       (𝑅 ∈ Ring → 𝑃 ∈ LMod)

Theorempsr1sca 20337 Scalars of a univariate power series ring. (Contributed by Stefan O'Rear, 4-Jul-2015.)
𝑃 = (PwSer1𝑅)       (𝑅𝑉𝑅 = (Scalar‘𝑃))

Theorempsr1sca2 20338 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 20339 Univariate polynomials form a left module. (Contributed by Stefan O'Rear, 26-Mar-2015.)
𝑃 = (Poly1𝑅)       (𝑅 ∈ Ring → 𝑃 ∈ LMod)

Theoremply1sca 20340 Scalars of a univariate polynomial ring. (Contributed by Stefan O'Rear, 26-Mar-2015.)
𝑃 = (Poly1𝑅)       (𝑅𝑉𝑅 = (Scalar‘𝑃))

Theoremply1sca2 20341 Scalars of a univariate polynomial ring. (Contributed by Stefan O'Rear, 26-Mar-2015.)
𝑃 = (Poly1𝑅)       ( I ‘𝑅) = (Scalar‘𝑃)

Theoremply1mpl0 20342 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 20343 Zero times a univariate polynomial is the zero polynomial (lmod0vs 19589 analog.) (Contributed by AV, 2-Dec-2019.)
𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &    = ( ·𝑠𝑃)    &    0 = (0g𝑅)       ((𝑅 ∈ Ring ∧ 𝑀𝐵) → ( 0 𝑀) = (0g𝑃))

Theoremply1mpl1 20344 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 20345 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 20346 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 20347 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 20348 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 20349 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 20350 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 20351 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 20352 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 20353 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 20354 An equivalence for coe1mul2 20356. (Contributed by Stefan O'Rear, 25-Mar-2015.)
((𝐴 ∈ ℕ0𝑋 ∈ (ℕ0m 1o)) → (𝑋r ≤ (1o × {𝐴}) ↔ (𝑋‘∅) ∈ (0...𝐴)))

Theoremcoe1mul2lem2 20355* An equivalence for coe1mul2 20356. (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 20356* 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 20357* 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 20358 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 20359 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 20360* 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 20361 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 20362 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 20363* 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 20364* 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 20365 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 20366* 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 20367 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 20368 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 20369 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 20370 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 20371 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 20372 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 20373 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 20374* 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 20375 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 20376 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 20377 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 20378 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 20379 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 20380 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 20381* 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 20382* The decomposition of a univariate polynomial is finitely supported. Formerly part of proof for ply1coe 20383. (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 20383* 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 20384* 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 20385* 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 20386* 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 20387* 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 20388* Lemma for coe1fzgsumd 20389 (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 20389* 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 20390* 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 20391* 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 = (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 20392* 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))

Theoremlply1binom 20393* The binomial theorem for linear polynomials (monic polynomials of degree 1) over commutative rings: (𝑋 + 𝐴)↑𝑁 is the sum from 𝑘 = 0 to 𝑁 of (𝑁C𝑘) · ((𝐴↑(𝑁𝑘)) · (𝑋𝑘)). (Contributed by AV, 25-Aug-2019.)
𝑃 = (Poly1𝑅)    &   𝑋 = (var1𝑅)    &    + = (+g𝑃)    &    × = (.r𝑃)    &    · = (.g𝑃)    &   𝐺 = (mulGrp‘𝑃)    &    = (.g𝐺)    &   𝐵 = (Base‘𝑃)       ((𝑅 ∈ CRing ∧ 𝑁 ∈ ℕ0𝐴𝐵) → (𝑁 (𝑋 + 𝐴)) = (𝑃 Σg (𝑘 ∈ (0...𝑁) ↦ ((𝑁C𝑘) · (((𝑁𝑘) 𝐴) × (𝑘 𝑋))))))

Theoremlply1binomsc 20394* 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 𝑘 = 0 to 𝑁 of (𝑁C𝑘) · ((𝐴↑(𝑁𝑘)) · (𝑋𝑘)). (Contributed by AV, 25-Aug-2019.)
𝑃 = (Poly1𝑅)    &   𝑋 = (var1𝑅)    &    + = (+g𝑃)    &    × = (.r𝑃)    &    · = (.g𝑃)    &   𝐺 = (mulGrp‘𝑃)    &    = (.g𝐺)    &   𝐾 = (Base‘𝑅)    &   𝑆 = (algSc‘𝑃)    &   𝐻 = (mulGrp‘𝑅)    &   𝐸 = (.g𝐻)       ((𝑅 ∈ CRing ∧ 𝑁 ∈ ℕ0𝐴𝐾) → (𝑁 (𝑋 + (𝑆𝐴))) = (𝑃 Σg (𝑘 ∈ (0...𝑁) ↦ ((𝑁C𝑘) · ((𝑆‘((𝑁𝑘)𝐸𝐴)) × (𝑘 𝑋))))))

10.9.5  Univariate polynomial evaluation

Syntaxces1 20395 Evaluation of a univariate polynomial in a subring.
class evalSub1

Syntaxce1 20396 Evaluation of a univariate polynomial.
class eval1

Definitiondf-evls1 20397* Define the evaluation map for the univariate polynomial algebra. The function (𝑆 evalSub1 𝑅):𝑉⟶(𝑆m 𝑆) makes sense when 𝑆 is a ring and 𝑅 is a subring of 𝑆, and where 𝑉 is the set of polynomials in (Poly1𝑅). This function maps an element of the formal polynomial algebra (with coefficients in 𝑅) to a function from assignments to the variable from 𝑆 into an element of 𝑆 formed by evaluating the polynomial with the given assignment. (Contributed by Mario Carneiro, 12-Jun-2015.)
evalSub1 = (𝑠 ∈ V, 𝑟 ∈ 𝒫 (Base‘𝑠) ↦ (Base‘𝑠) / 𝑏((𝑥 ∈ (𝑏m (𝑏m 1o)) ↦ (𝑥 ∘ (𝑦𝑏 ↦ (1o × {𝑦})))) ∘ ((1o evalSub 𝑠)‘𝑟)))

Definitiondf-evl1 20398* Define the evaluation map for the univariate polynomial algebra. The function (eval1𝑅):𝑉⟶(𝑅m 𝑅) makes sense when 𝑅 is a ring, and 𝑉 is the set of polynomials in (Poly1𝑅). This function maps an element of the formal polynomial algebra (with coefficients in 𝑅) to a function from assignments to the variable from 𝑅 into an element of 𝑅 formed by evaluating the polynomial with the given assignment. (Contributed by Mario Carneiro, 12-Jun-2015.)
eval1 = (𝑟 ∈ V ↦ (Base‘𝑟) / 𝑏((𝑥 ∈ (𝑏m (𝑏m 1o)) ↦ (𝑥 ∘ (𝑦𝑏 ↦ (1o × {𝑦})))) ∘ (1o eval 𝑟)))

Theoremreldmevls1 20399 Well-behaved binary operation property of evalSub1. (Contributed by AV, 7-Sep-2019.)
Rel dom evalSub1

Theoremply1frcl 20400 Reverse closure for the set of univariate polynomial functions. (Contributed by AV, 9-Sep-2019.)
𝑄 = ran (𝑆 evalSub1 𝑅)       (𝑋𝑄 → (𝑆 ∈ V ∧ 𝑅 ∈ 𝒫 (Base‘𝑆)))

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