P. 222 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 22101-22200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	coe1f2 22101	Functionality of univariate power series coefficient vectors. (Contributed by Stefan O'Rear, 25-Mar-2015.)
⊢ 𝐴 = (coe1‘𝐹)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝑃 = (PwSer1‘𝑅)    &   ⊢ 𝐾 = (Base‘𝑅)    ⇒   ⊢ (𝐹 ∈ 𝐵 → 𝐴:ℕ0⟶𝐾)
Theorem	coe1fval2 22102*	Univariate polynomial coefficient vectors expressed as a function composition. (Contributed by Stefan O'Rear, 21-Mar-2015.)
⊢ 𝐴 = (coe1‘𝐹)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐺 = (𝑦 ∈ ℕ0 ↦ (1o × {𝑦}))    ⇒   ⊢ (𝐹 ∈ 𝐵 → 𝐴 = (𝐹 ∘ 𝐺))
Theorem	coe1f 22103	Functionality of univariate polynomial coefficient vectors. (Contributed by Stefan O'Rear, 21-Mar-2015.)
⊢ 𝐴 = (coe1‘𝐹)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐾 = (Base‘𝑅)    ⇒   ⊢ (𝐹 ∈ 𝐵 → 𝐴:ℕ0⟶𝐾)
Theorem	coe1fvalcl 22104	A coefficient of a univariate polynomial over a class/ring is an element of this class/ring. (Contributed by AV, 9-Oct-2019.)
⊢ 𝐴 = (coe1‘𝐹)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐾 = (Base‘𝑅)    ⇒   ⊢ ((𝐹 ∈ 𝐵 ∧ 𝑁 ∈ ℕ0) → (𝐴‘𝑁) ∈ 𝐾)
Theorem	coe1sfi 22105	Finite support of univariate polynomial coefficient vectors. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by AV, 19-Jul-2019.)
⊢ 𝐴 = (coe1‘𝐹)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (𝐹 ∈ 𝐵 → 𝐴 finSupp 0 )
Theorem	coe1fsupp 22106*	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 = (coe1‘𝐹)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐾 = (Base‘𝑅)    ⇒   ⊢ (𝐹 ∈ 𝐵 → 𝐴 ∈ {𝑔 ∈ (𝐾 ↑m ℕ0) ∣ 𝑔 finSupp 0 })
Theorem	mptcoe1fsupp 22107*	A mapping involving coefficients of polynomials is finitely supported. (Contributed by AV, 12-Oct-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑘 ∈ ℕ0 ↦ ((coe1‘𝑀)‘𝑘)) finSupp 0 )
Theorem	coe1ae0 22108*	The coefficient vector of a univariate polynomial is 0 almost everywhere. (Contributed by AV, 19-Oct-2019.)
⊢ 𝐴 = (coe1‘𝐹)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (𝐹 ∈ 𝐵 → ∃𝑠 ∈ ℕ0 ∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → (𝐴‘𝑛) = 0 ))
Theorem	vr1cl 22109	The generator of a univariate polynomial algebra is contained in the base set. (Contributed by Stefan O'Rear, 19-Mar-2015.)
⊢ 𝑋 = (var1‘𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    ⇒   ⊢ (𝑅 ∈ Ring → 𝑋 ∈ 𝐵)
Theorem	opsr0 22110	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)    &   ⊢ (𝜑 → 𝑇 ⊆ (𝐼 × 𝐼))    ⇒   ⊢ (𝜑 → (0g‘𝑆) = (0g‘𝑂))
Theorem	opsr1 22111	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)    &   ⊢ (𝜑 → 𝑇 ⊆ (𝐼 × 𝐼))    ⇒   ⊢ (𝜑 → (1r‘𝑆) = (1r‘𝑂))
Theorem	psr1plusg 22112	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 = (PwSer1‘𝑅)    &   ⊢ 𝑆 = (1o mPwSer 𝑅)    &   ⊢ + = (+g‘𝑌)    ⇒   ⊢ + = (+g‘𝑆)
Theorem	psr1vsca 22113	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 = (PwSer1‘𝑅)    &   ⊢ 𝑆 = (1o mPwSer 𝑅)    &   ⊢ · = ( ·𝑠 ‘𝑌)    ⇒   ⊢ · = ( ·𝑠 ‘𝑆)
Theorem	psr1mulr 22114	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 = (PwSer1‘𝑅)    &   ⊢ 𝑆 = (1o mPwSer 𝑅)    &   ⊢ · = (.r‘𝑌)    ⇒   ⊢ · = (.r‘𝑆)
Theorem	ply1plusg 22115	Value of addition in a univariate polynomial ring. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by Mario Carneiro, 4-Jul-2015.)
⊢ 𝑌 = (Poly1‘𝑅)    &   ⊢ 𝑆 = (1o mPoly 𝑅)    &   ⊢ + = (+g‘𝑌)    ⇒   ⊢ + = (+g‘𝑆)
Theorem	ply1vsca 22116	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 = (Poly1‘𝑅)    &   ⊢ 𝑆 = (1o mPoly 𝑅)    &   ⊢ · = ( ·𝑠 ‘𝑌)    ⇒   ⊢ · = ( ·𝑠 ‘𝑆)
Theorem	ply1mulr 22117	Value of multiplication in a univariate polynomial ring. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by Mario Carneiro, 4-Jul-2015.)
⊢ 𝑌 = (Poly1‘𝑅)    &   ⊢ 𝑆 = (1o mPoly 𝑅)    &   ⊢ · = (.r‘𝑌)    ⇒   ⊢ · = (.r‘𝑆)
Theorem	ply1ass23l 22118	Associative identity with scalar and ring multiplication for the polynomial ring. (Contributed by AV, 14-Aug-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ × = (.r‘𝑃)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ · = ( ·𝑠 ‘𝑃)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ (𝐴 ∈ 𝐾 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ((𝐴 · 𝑋) × 𝑌) = (𝐴 · (𝑋 × 𝑌)))
Theorem	ressply1bas2 22119	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 = (Poly1‘𝑅)    &   ⊢ 𝐻 = (𝑅 ↾s 𝑇)    &   ⊢ 𝑈 = (Poly1‘𝐻)    &   ⊢ 𝐵 = (Base‘𝑈)    &   ⊢ (𝜑 → 𝑇 ∈ (SubRing‘𝑅))    &   ⊢ 𝑊 = (PwSer1‘𝐻)    &   ⊢ 𝐶 = (Base‘𝑊)    &   ⊢ 𝐾 = (Base‘𝑆)    ⇒   ⊢ (𝜑 → 𝐵 = (𝐶 ∩ 𝐾))
Theorem	ressply1bas 22120	A restricted polynomial algebra has the same base set. (Contributed by Mario Carneiro, 3-Jul-2015.)
⊢ 𝑆 = (Poly1‘𝑅)    &   ⊢ 𝐻 = (𝑅 ↾s 𝑇)    &   ⊢ 𝑈 = (Poly1‘𝐻)    &   ⊢ 𝐵 = (Base‘𝑈)    &   ⊢ (𝜑 → 𝑇 ∈ (SubRing‘𝑅))    &   ⊢ 𝑃 = (𝑆 ↾s 𝐵)    ⇒   ⊢ (𝜑 → 𝐵 = (Base‘𝑃))
Theorem	ressply1add 22121	A restricted polynomial algebra has the same addition operation. (Contributed by Mario Carneiro, 3-Jul-2015.)
⊢ 𝑆 = (Poly1‘𝑅)    &   ⊢ 𝐻 = (𝑅 ↾s 𝑇)    &   ⊢ 𝑈 = (Poly1‘𝐻)    &   ⊢ 𝐵 = (Base‘𝑈)    &   ⊢ (𝜑 → 𝑇 ∈ (SubRing‘𝑅))    &   ⊢ 𝑃 = (𝑆 ↾s 𝐵)    ⇒   ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋(+g‘𝑈)𝑌) = (𝑋(+g‘𝑃)𝑌))
Theorem	ressply1mul 22122	A restricted polynomial algebra has the same multiplication operation. (Contributed by Mario Carneiro, 3-Jul-2015.)
⊢ 𝑆 = (Poly1‘𝑅)    &   ⊢ 𝐻 = (𝑅 ↾s 𝑇)    &   ⊢ 𝑈 = (Poly1‘𝐻)    &   ⊢ 𝐵 = (Base‘𝑈)    &   ⊢ (𝜑 → 𝑇 ∈ (SubRing‘𝑅))    &   ⊢ 𝑃 = (𝑆 ↾s 𝐵)    ⇒   ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋(.r‘𝑈)𝑌) = (𝑋(.r‘𝑃)𝑌))
Theorem	ressply1vsca 22123	A restricted power series algebra has the same scalar multiplication operation. (Contributed by Mario Carneiro, 3-Jul-2015.)
⊢ 𝑆 = (Poly1‘𝑅)    &   ⊢ 𝐻 = (𝑅 ↾s 𝑇)    &   ⊢ 𝑈 = (Poly1‘𝐻)    &   ⊢ 𝐵 = (Base‘𝑈)    &   ⊢ (𝜑 → 𝑇 ∈ (SubRing‘𝑅))    &   ⊢ 𝑃 = (𝑆 ↾s 𝐵)    ⇒   ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝐵)) → (𝑋( ·𝑠 ‘𝑈)𝑌) = (𝑋( ·𝑠 ‘𝑃)𝑌))
Theorem	subrgply1 22124	A subring of the base ring induces a subring of polynomials. (Contributed by Mario Carneiro, 3-Jul-2015.)
⊢ 𝑆 = (Poly1‘𝑅)    &   ⊢ 𝐻 = (𝑅 ↾s 𝑇)    &   ⊢ 𝑈 = (Poly1‘𝐻)    &   ⊢ 𝐵 = (Base‘𝑈)    ⇒   ⊢ (𝑇 ∈ (SubRing‘𝑅) → 𝐵 ∈ (SubRing‘𝑆))
Theorem	gsumply1subr 22125	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 = (Poly1‘𝑅)    &   ⊢ 𝐻 = (𝑅 ↾s 𝑇)    &   ⊢ 𝑈 = (Poly1‘𝐻)    &   ⊢ 𝐵 = (Base‘𝑈)    &   ⊢ (𝜑 → 𝑇 ∈ (SubRing‘𝑅))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    ⇒   ⊢ (𝜑 → (𝑆 Σg 𝐹) = (𝑈 Σg 𝐹))
Theorem	psrbaspropd 22126	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)))
Theorem	psrplusgpropd 22127*	Property deduction for power series addition. (Contributed by Stefan O'Rear, 27-Mar-2015.) (Revised by Mario Carneiro, 3-Oct-2015.)
⊢ (𝜑 → 𝐵 = (Base‘𝑅))    &   ⊢ (𝜑 → 𝐵 = (Base‘𝑆))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝑅)𝑦) = (𝑥(+g‘𝑆)𝑦))    ⇒   ⊢ (𝜑 → (+g‘(𝐼 mPwSer 𝑅)) = (+g‘(𝐼 mPwSer 𝑆)))
Theorem	mplbaspropd 22128*	Property deduction for polynomial base set. (Contributed by Stefan O'Rear, 27-Mar-2015.) (Proof shortened by AV, 19-Jul-2019.)
⊢ (𝜑 → 𝐵 = (Base‘𝑅))    &   ⊢ (𝜑 → 𝐵 = (Base‘𝑆))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝑅)𝑦) = (𝑥(+g‘𝑆)𝑦))    ⇒   ⊢ (𝜑 → (Base‘(𝐼 mPoly 𝑅)) = (Base‘(𝐼 mPoly 𝑆)))
Theorem	psropprmul 22129	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 = (oppr‘𝑅)    &   ⊢ 𝑍 = (𝐼 mPwSer 𝑆)    &   ⊢ · = (.r‘𝑌)    &   ⊢ ∙ = (.r‘𝑍)    &   ⊢ 𝐵 = (Base‘𝑌)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝐹 ∙ 𝐺) = (𝐺 · 𝐹))
Theorem	ply1opprmul 22130	Reversing multiplication in a ring reverses multiplication in the univariate polynomial ring. (Contributed by Stefan O'Rear, 27-Mar-2015.)
⊢ 𝑌 = (Poly1‘𝑅)    &   ⊢ 𝑆 = (oppr‘𝑅)    &   ⊢ 𝑍 = (Poly1‘𝑆)    &   ⊢ · = (.r‘𝑌)    &   ⊢ ∙ = (.r‘𝑍)    &   ⊢ 𝐵 = (Base‘𝑌)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝐹 ∙ 𝐺) = (𝐺 · 𝐹))
Theorem	00ply1bas 22131	Lemma for ply1basfvi 22132 and deg1fvi 25997. (Contributed by Stefan O'Rear, 28-Mar-2015.)
⊢ ∅ = (Base‘(Poly1‘∅))
Theorem	ply1basfvi 22132	Protection compatibility of the univariate polynomial base set. (Contributed by Stefan O'Rear, 27-Mar-2015.)
⊢ (Base‘(Poly1‘𝑅)) = (Base‘(Poly1‘( I ‘𝑅)))
Theorem	ply1plusgfvi 22133	Protection compatibility of the univariate polynomial addition. (Contributed by Stefan O'Rear, 27-Mar-2015.)
⊢ (+g‘(Poly1‘𝑅)) = (+g‘(Poly1‘( I ‘𝑅)))
Theorem	ply1baspropd 22134*	Property deduction for univariate polynomial base set. (Contributed by Stefan O'Rear, 27-Mar-2015.)
⊢ (𝜑 → 𝐵 = (Base‘𝑅))    &   ⊢ (𝜑 → 𝐵 = (Base‘𝑆))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝑅)𝑦) = (𝑥(+g‘𝑆)𝑦))    ⇒   ⊢ (𝜑 → (Base‘(Poly1‘𝑅)) = (Base‘(Poly1‘𝑆)))
Theorem	ply1plusgpropd 22135*	Property deduction for univariate polynomial addition. (Contributed by Stefan O'Rear, 27-Mar-2015.)
⊢ (𝜑 → 𝐵 = (Base‘𝑅))    &   ⊢ (𝜑 → 𝐵 = (Base‘𝑆))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝑅)𝑦) = (𝑥(+g‘𝑆)𝑦))    ⇒   ⊢ (𝜑 → (+g‘(Poly1‘𝑅)) = (+g‘(Poly1‘𝑆)))
Theorem	opsrring 22136	Ordered power series form a ring. (Contributed by Stefan O'Rear, 22-Mar-2015.)
⊢ 𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝑇 ⊆ (𝐼 × 𝐼))    ⇒   ⊢ (𝜑 → 𝑂 ∈ Ring)
Theorem	opsrlmod 22137	Ordered power series form a left module. (Contributed by Stefan O'Rear, 26-Mar-2015.)
⊢ 𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝑇 ⊆ (𝐼 × 𝐼))    ⇒   ⊢ (𝜑 → 𝑂 ∈ LMod)
Theorem	psr1ring 22138	Univariate power series form a ring. (Contributed by Stefan O'Rear, 22-Mar-2015.)
⊢ 𝑆 = (PwSer1‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → 𝑆 ∈ Ring)
Theorem	ply1ring 22139	Univariate polynomials form a ring. (Contributed by Stefan O'Rear, 22-Mar-2015.)
⊢ 𝑃 = (Poly1‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
Theorem	psr1lmod 22140	Univariate power series form a left module. (Contributed by Stefan O'Rear, 26-Mar-2015.)
⊢ 𝑃 = (PwSer1‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → 𝑃 ∈ LMod)
Theorem	psr1sca 22141	Scalars of a univariate power series ring. (Contributed by Stefan O'Rear, 4-Jul-2015.)
⊢ 𝑃 = (PwSer1‘𝑅)    ⇒   ⊢ (𝑅 ∈ 𝑉 → 𝑅 = (Scalar‘𝑃))
Theorem	psr1sca2 22142	Scalars of a univariate power series ring. (Contributed by Stefan O'Rear, 26-Mar-2015.) (Revised by Mario Carneiro, 4-Jul-2015.)
⊢ 𝑃 = (PwSer1‘𝑅)    ⇒   ⊢ ( I ‘𝑅) = (Scalar‘𝑃)
Theorem	ply1lmod 22143	Univariate polynomials form a left module. (Contributed by Stefan O'Rear, 26-Mar-2015.)
⊢ 𝑃 = (Poly1‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → 𝑃 ∈ LMod)
Theorem	ply1sca 22144	Scalars of a univariate polynomial ring. (Contributed by Stefan O'Rear, 26-Mar-2015.)
⊢ 𝑃 = (Poly1‘𝑅)    ⇒   ⊢ (𝑅 ∈ 𝑉 → 𝑅 = (Scalar‘𝑃))
Theorem	ply1sca2 22145	Scalars of a univariate polynomial ring. (Contributed by Stefan O'Rear, 26-Mar-2015.)
⊢ 𝑃 = (Poly1‘𝑅)    ⇒   ⊢ ( I ‘𝑅) = (Scalar‘𝑃)
Theorem	ply1ascl0 22146	The zero scalar as a polynomial. (Contributed by Thierry Arnoux, 20-Jan-2025.)
⊢ 𝑊 = (Poly1‘𝑅)    &   ⊢ 𝐴 = (algSc‘𝑊)    &   ⊢ 𝑂 = (0g‘𝑅)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    ⇒   ⊢ (𝜑 → (𝐴‘𝑂) = 0 )
Theorem	ply1ascl1 22147	The multiplicative identity scalar as a univariate polynomial. (Contributed by Thierry Arnoux, 20-Jan-2025.)
⊢ 𝑊 = (Poly1‘𝑅)    &   ⊢ 𝐴 = (algSc‘𝑊)    &   ⊢ 𝐼 = (1r‘𝑅)    &   ⊢ 1 = (1r‘𝑊)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    ⇒   ⊢ (𝜑 → (𝐴‘𝐼) = 1 )
Theorem	ply1mpl0 22148	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 = (Poly1‘𝑅)    &   ⊢ 0 = (0g‘𝑃)    ⇒   ⊢ 0 = (0g‘𝑀)
Theorem	ply10s0 22149	Zero times a univariate polynomial is the zero polynomial (lmod0vs 20808 analog.) (Contributed by AV, 2-Dec-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ ∗ = ( ·𝑠 ‘𝑃)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ( 0 ∗ 𝑀) = (0g‘𝑃))
Theorem	ply1mpl1 22150	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 = (Poly1‘𝑅)    &   ⊢ 1 = (1r‘𝑃)    ⇒   ⊢ 1 = (1r‘𝑀)
Theorem	ply1ascl 22151	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 = (Poly1‘𝑅)    &   ⊢ 𝐴 = (algSc‘𝑃)    ⇒   ⊢ 𝐴 = (algSc‘(1o mPoly 𝑅))
Theorem	subrg1ascl 22152	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 = (Poly1‘𝑅)    &   ⊢ 𝐴 = (algSc‘𝑃)    &   ⊢ 𝐻 = (𝑅 ↾s 𝑇)    &   ⊢ 𝑈 = (Poly1‘𝐻)    &   ⊢ (𝜑 → 𝑇 ∈ (SubRing‘𝑅))    &   ⊢ 𝐶 = (algSc‘𝑈)    ⇒   ⊢ (𝜑 → 𝐶 = (𝐴 ↾ 𝑇))
Theorem	subrg1asclcl 22153	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 = (Poly1‘𝑅)    &   ⊢ 𝐴 = (algSc‘𝑃)    &   ⊢ 𝐻 = (𝑅 ↾s 𝑇)    &   ⊢ 𝑈 = (Poly1‘𝐻)    &   ⊢ (𝜑 → 𝑇 ∈ (SubRing‘𝑅))    &   ⊢ 𝐵 = (Base‘𝑈)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐾)    ⇒   ⊢ (𝜑 → ((𝐴‘𝑋) ∈ 𝐵 ↔ 𝑋 ∈ 𝑇))
Theorem	subrgvr1 22154	The variables in a subring polynomial algebra are the same as the original ring. (Contributed by Mario Carneiro, 5-Jul-2015.)
⊢ 𝑋 = (var1‘𝑅)    &   ⊢ (𝜑 → 𝑇 ∈ (SubRing‘𝑅))    &   ⊢ 𝐻 = (𝑅 ↾s 𝑇)    ⇒   ⊢ (𝜑 → 𝑋 = (var1‘𝐻))
Theorem	subrgvr1cl 22155	The variables in a polynomial algebra are contained in every subring algebra. (Contributed by Mario Carneiro, 5-Jul-2015.)
⊢ 𝑋 = (var1‘𝑅)    &   ⊢ (𝜑 → 𝑇 ∈ (SubRing‘𝑅))    &   ⊢ 𝐻 = (𝑅 ↾s 𝑇)    &   ⊢ 𝑈 = (Poly1‘𝐻)    &   ⊢ 𝐵 = (Base‘𝑈)    ⇒   ⊢ (𝜑 → 𝑋 ∈ 𝐵)
Theorem	coe1z 22156	The coefficient vector of 0. (Contributed by Stefan O'Rear, 23-Mar-2015.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 0 = (0g‘𝑃)    &   ⊢ 𝑌 = (0g‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → (coe1‘ 0 ) = (ℕ0 × {𝑌}))
Theorem	coe1add 22157	The coefficient vector of an addition. (Contributed by Stefan O'Rear, 24-Mar-2015.)
⊢ 𝑌 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑌)    &   ⊢ ✚ = (+g‘𝑌)    &   ⊢ + = (+g‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (coe1‘(𝐹 ✚ 𝐺)) = ((coe1‘𝐹) ∘f + (coe1‘𝐺)))
Theorem	coe1addfv 22158	A particular coefficient of an addition. (Contributed by Stefan O'Rear, 23-Mar-2015.)
⊢ 𝑌 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑌)    &   ⊢ ✚ = (+g‘𝑌)    &   ⊢ + = (+g‘𝑅)    ⇒   ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑋 ∈ ℕ0) → ((coe1‘(𝐹 ✚ 𝐺))‘𝑋) = (((coe1‘𝐹)‘𝑋) + ((coe1‘𝐺)‘𝑋)))
Theorem	coe1subfv 22159	A particular coefficient of a subtraction. (Contributed by Stefan O'Rear, 23-Mar-2015.)
⊢ 𝑌 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑌)    &   ⊢ − = (-g‘𝑌)    &   ⊢ 𝑁 = (-g‘𝑅)    ⇒   ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑋 ∈ ℕ0) → ((coe1‘(𝐹 − 𝐺))‘𝑋) = (((coe1‘𝐹)‘𝑋)𝑁((coe1‘𝐺)‘𝑋)))
Theorem	coe1mul2lem1 22160	An equivalence for coe1mul2 22162. (Contributed by Stefan O'Rear, 25-Mar-2015.)
⊢ ((𝐴 ∈ ℕ0 ∧ 𝑋 ∈ (ℕ0 ↑m 1o)) → (𝑋 ∘r ≤ (1o × {𝐴}) ↔ (𝑋‘∅) ∈ (0...𝐴)))
Theorem	coe1mul2lem2 22161*	An equivalence for coe1mul2 22162. (Contributed by Stefan O'Rear, 25-Mar-2015.)
⊢ 𝐻 = {𝑑 ∈ (ℕ0 ↑m 1o) ∣ 𝑑 ∘r ≤ (1o × {𝑘})}    ⇒   ⊢ (𝑘 ∈ ℕ0 → (𝑐 ∈ 𝐻 ↦ (𝑐‘∅)):𝐻–1-1-onto→(0...𝑘))
Theorem	coe1mul2 22162*	The coefficient vector of multiplication in the univariate power series ring. (Contributed by Stefan O'Rear, 25-Mar-2015.)
⊢ 𝑆 = (PwSer1‘𝑅)    &   ⊢ ∙ = (.r‘𝑆)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑆)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (coe1‘(𝐹 ∙ 𝐺)) = (𝑘 ∈ ℕ0 ↦ (𝑅 Σg (𝑥 ∈ (0...𝑘) ↦ (((coe1‘𝐹)‘𝑥) · ((coe1‘𝐺)‘(𝑘 − 𝑥)))))))
Theorem	coe1mul 22163*	The coefficient vector of multiplication in the univariate polynomial ring. (Contributed by Stefan O'Rear, 25-Mar-2015.)
⊢ 𝑌 = (Poly1‘𝑅)    &   ⊢ ∙ = (.r‘𝑌)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑌)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (coe1‘(𝐹 ∙ 𝐺)) = (𝑘 ∈ ℕ0 ↦ (𝑅 Σg (𝑥 ∈ (0...𝑘) ↦ (((coe1‘𝐹)‘𝑥) · ((coe1‘𝐺)‘(𝑘 − 𝑥)))))))
Theorem	ply1moncl 22164	Closure of the expression for a univariate primitive monomial. (Contributed by AV, 14-Aug-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ 𝑁 = (mulGrp‘𝑃)    &   ⊢ ↑ = (.g‘𝑁)    &   ⊢ 𝐵 = (Base‘𝑃)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐷 ∈ ℕ0) → (𝐷 ↑ 𝑋) ∈ 𝐵)
Theorem	ply1tmcl 22165	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 = (Poly1‘𝑅)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ · = ( ·𝑠 ‘𝑃)    &   ⊢ 𝑁 = (mulGrp‘𝑃)    &   ⊢ ↑ = (.g‘𝑁)    &   ⊢ 𝐵 = (Base‘𝑃)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) → (𝐶 · (𝐷 ↑ 𝑋)) ∈ 𝐵)
Theorem	coe1tm 22166*	Coefficient vector of a polynomial term. (Contributed by Stefan O'Rear, 27-Mar-2015.)
⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ · = ( ·𝑠 ‘𝑃)    &   ⊢ 𝑁 = (mulGrp‘𝑃)    &   ⊢ ↑ = (.g‘𝑁)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) → (coe1‘(𝐶 · (𝐷 ↑ 𝑋))) = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 𝐷, 𝐶, 0 )))
Theorem	coe1tmfv1 22167	Nonzero coefficient of a polynomial term. (Contributed by Stefan O'Rear, 27-Mar-2015.)
⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ · = ( ·𝑠 ‘𝑃)    &   ⊢ 𝑁 = (mulGrp‘𝑃)    &   ⊢ ↑ = (.g‘𝑁)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) → ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘𝐷) = 𝐶)
Theorem	coe1tmfv2 22168	Zero coefficient of a polynomial term. (Contributed by Stefan O'Rear, 27-Mar-2015.)
⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ · = ( ·𝑠 ‘𝑃)    &   ⊢ 𝑁 = (mulGrp‘𝑃)    &   ⊢ ↑ = (.g‘𝑁)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐾)    &   ⊢ (𝜑 → 𝐷 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐹 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐷 ≠ 𝐹)    ⇒   ⊢ (𝜑 → ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘𝐹) = 0 )
Theorem	coe1tmmul2 22169*	Coefficient vector of a polynomial multiplied on the right by a term. (Contributed by Stefan O'Rear, 27-Mar-2015.)
⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ · = ( ·𝑠 ‘𝑃)    &   ⊢ 𝑁 = (mulGrp‘𝑃)    &   ⊢ ↑ = (.g‘𝑁)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ ∙ = (.r‘𝑃)    &   ⊢ × = (.r‘𝑅)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐾)    &   ⊢ (𝜑 → 𝐷 ∈ ℕ0)    ⇒   ⊢ (𝜑 → (coe1‘(𝐴 ∙ (𝐶 · (𝐷 ↑ 𝑋)))) = (𝑥 ∈ ℕ0 ↦ if(𝐷 ≤ 𝑥, (((coe1‘𝐴)‘(𝑥 − 𝐷)) × 𝐶), 0 )))
Theorem	coe1tmmul 22170*	Coefficient vector of a polynomial multiplied on the left by a term. (Contributed by Stefan O'Rear, 29-Mar-2015.)
⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ · = ( ·𝑠 ‘𝑃)    &   ⊢ 𝑁 = (mulGrp‘𝑃)    &   ⊢ ↑ = (.g‘𝑁)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ ∙ = (.r‘𝑃)    &   ⊢ × = (.r‘𝑅)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐾)    &   ⊢ (𝜑 → 𝐷 ∈ ℕ0)    ⇒   ⊢ (𝜑 → (coe1‘((𝐶 · (𝐷 ↑ 𝑋)) ∙ 𝐴)) = (𝑥 ∈ ℕ0 ↦ if(𝐷 ≤ 𝑥, (𝐶 × ((coe1‘𝐴)‘(𝑥 − 𝐷))), 0 )))
Theorem	coe1tmmul2fv 22171	Function value of a right-multiplication by a term in the shifted domain. (Contributed by Stefan O'Rear, 27-Mar-2015.)
⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ · = ( ·𝑠 ‘𝑃)    &   ⊢ 𝑁 = (mulGrp‘𝑃)    &   ⊢ ↑ = (.g‘𝑁)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ ∙ = (.r‘𝑃)    &   ⊢ × = (.r‘𝑅)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐾)    &   ⊢ (𝜑 → 𝐷 ∈ ℕ0)    &   ⊢ (𝜑 → 𝑌 ∈ ℕ0)    ⇒   ⊢ (𝜑 → ((coe1‘(𝐴 ∙ (𝐶 · (𝐷 ↑ 𝑋))))‘(𝐷 + 𝑌)) = (((coe1‘𝐴)‘𝑌) × 𝐶))
Theorem	coe1pwmul 22172*	Coefficient vector of a polynomial multiplied on the left by a variable power. (Contributed by Stefan O'Rear, 1-Apr-2015.)
⊢ 0 = (0g‘𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ 𝑁 = (mulGrp‘𝑃)    &   ⊢ ↑ = (.g‘𝑁)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ · = (.r‘𝑃)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐷 ∈ ℕ0)    ⇒   ⊢ (𝜑 → (coe1‘((𝐷 ↑ 𝑋) · 𝐴)) = (𝑥 ∈ ℕ0 ↦ if(𝐷 ≤ 𝑥, ((coe1‘𝐴)‘(𝑥 − 𝐷)), 0 )))
Theorem	coe1pwmulfv 22173	Function value of a right-multiplication by a variable power in the shifted domain. (Contributed by Stefan O'Rear, 1-Apr-2015.)
⊢ 0 = (0g‘𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ 𝑁 = (mulGrp‘𝑃)    &   ⊢ ↑ = (.g‘𝑁)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ · = (.r‘𝑃)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐷 ∈ ℕ0)    &   ⊢ (𝜑 → 𝑌 ∈ ℕ0)    ⇒   ⊢ (𝜑 → ((coe1‘((𝐷 ↑ 𝑋) · 𝐴))‘(𝐷 + 𝑌)) = ((coe1‘𝐴)‘𝑌))
Theorem	ply1scltm 22174	A scalar is a term with zero exponent. (Contributed by Stefan O'Rear, 29-Mar-2015.)
⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ · = ( ·𝑠 ‘𝑃)    &   ⊢ 𝑁 = (mulGrp‘𝑃)    &   ⊢ ↑ = (.g‘𝑁)    &   ⊢ 𝐴 = (algSc‘𝑃)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐾) → (𝐴‘𝐹) = (𝐹 · (0 ↑ 𝑋)))
Theorem	coe1sclmul 22175	Coefficient vector of a polynomial multiplied on the left by a scalar. (Contributed by Stefan O'Rear, 29-Mar-2015.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 𝐴 = (algSc‘𝑃)    &   ⊢ ∙ = (.r‘𝑃)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → (coe1‘((𝐴‘𝑋) ∙ 𝑌)) = ((ℕ0 × {𝑋}) ∘f · (coe1‘𝑌)))
Theorem	coe1sclmulfv 22176	A single coefficient of a polynomial multiplied on the left by a scalar. (Contributed by Stefan O'Rear, 1-Apr-2015.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 𝐴 = (algSc‘𝑃)    &   ⊢ ∙ = (.r‘𝑃)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) ∧ 0 ∈ ℕ0) → ((coe1‘((𝐴‘𝑋) ∙ 𝑌))‘ 0 ) = (𝑋 · ((coe1‘𝑌)‘ 0 )))
Theorem	coe1sclmul2 22177	Coefficient vector of a polynomial multiplied on the right by a scalar. (Contributed by Stefan O'Rear, 29-Mar-2015.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 𝐴 = (algSc‘𝑃)    &   ⊢ ∙ = (.r‘𝑃)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → (coe1‘(𝑌 ∙ (𝐴‘𝑋))) = ((coe1‘𝑌) ∘f · (ℕ0 × {𝑋})))
Theorem	ply1sclf 22178	A scalar polynomial is a polynomial. (Contributed by Stefan O'Rear, 28-Mar-2015.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐴 = (algSc‘𝑃)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    ⇒   ⊢ (𝑅 ∈ Ring → 𝐴:𝐾⟶𝐵)
Theorem	ply1sclcl 22179	The value of the algebra scalar lifting function for (univariate) polynomials applied to a scalar results in a constant polynomial. (Contributed by AV, 27-Nov-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐴 = (algSc‘𝑃)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐾) → (𝐴‘𝑆) ∈ 𝐵)
Theorem	coe1scl 22180*	Coefficient vector of a scalar. (Contributed by Stefan O'Rear, 28-Mar-2015.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐴 = (algSc‘𝑃)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾) → (coe1‘(𝐴‘𝑋)) = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, 𝑋, 0 )))
Theorem	ply1sclid 22181	Recover the base scalar from a scalar polynomial. (Contributed by Stefan O'Rear, 28-Mar-2015.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐴 = (algSc‘𝑃)    &   ⊢ 𝐾 = (Base‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾) → 𝑋 = ((coe1‘(𝐴‘𝑋))‘0))
Theorem	ply1sclf1 22182	The polynomial scalar function is injective. (Contributed by Stefan O'Rear, 28-Mar-2015.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐴 = (algSc‘𝑃)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    ⇒   ⊢ (𝑅 ∈ Ring → 𝐴:𝐾–1-1→𝐵)
Theorem	ply1scl0 22183	The zero scalar is zero. (Contributed by Stefan O'Rear, 29-Mar-2015.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐴 = (algSc‘𝑃)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝑌 = (0g‘𝑃)    ⇒   ⊢ (𝑅 ∈ Ring → (𝐴‘ 0 ) = 𝑌)
Theorem	ply1scl0OLD 22184	Obsolete version of ply1scl1 22186 as of 12-Mar-2025. (Contributed by Stefan O'Rear, 29-Mar-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐴 = (algSc‘𝑃)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝑌 = (0g‘𝑃)    ⇒   ⊢ (𝑅 ∈ Ring → (𝐴‘ 0 ) = 𝑌)
Theorem	ply1scln0 22185	Nonzero scalars create nonzero polynomials. (Contributed by Stefan O'Rear, 29-Mar-2015.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐴 = (algSc‘𝑃)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝑌 = (0g‘𝑃)    &   ⊢ 𝐾 = (Base‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑋 ≠ 0 ) → (𝐴‘𝑋) ≠ 𝑌)
Theorem	ply1scl1 22186	The one scalar is the unit polynomial. (Contributed by Stefan O'Rear, 1-Apr-2015.) (Proof shortened by SN, 12-Mar-2025.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐴 = (algSc‘𝑃)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 𝑁 = (1r‘𝑃)    ⇒   ⊢ (𝑅 ∈ Ring → (𝐴‘ 1 ) = 𝑁)
Theorem	ply1scl1OLD 22187	Obsolete version of ply1scl1 22186 as of 12-Mar-2025. (Contributed by Stefan O'Rear, 1-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐴 = (algSc‘𝑃)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 𝑁 = (1r‘𝑃)    ⇒   ⊢ (𝑅 ∈ Ring → (𝐴‘ 1 ) = 𝑁)
Theorem	ply1idvr1 22188	The identity of a polynomial ring expressed as power of the polynomial variable. (Contributed by AV, 14-Aug-2019.) (Proof shortened by SN, 3-Jul-2025.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ 𝑁 = (mulGrp‘𝑃)    &   ⊢ ↑ = (.g‘𝑁)    ⇒   ⊢ (𝑅 ∈ Ring → (0 ↑ 𝑋) = (1r‘𝑃))
Theorem	ply1idvr1OLD 22189	Obsolete version of ply1idvr1 22188 as of 3-Jul-2025. (Contributed by AV, 14-Aug-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ 𝑁 = (mulGrp‘𝑃)    &   ⊢ ↑ = (.g‘𝑁)    ⇒   ⊢ (𝑅 ∈ Ring → (0 ↑ 𝑋) = (1r‘𝑃))
Theorem	cply1mul 22190*	The product of two constant polynomials is a constant polynomial. (Contributed by AV, 18-Nov-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ × = (.r‘𝑃)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ (𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵)) → (∀𝑐 ∈ ℕ (((coe1‘𝐹)‘𝑐) = 0 ∧ ((coe1‘𝐺)‘𝑐) = 0 ) → ∀𝑐 ∈ ℕ ((coe1‘(𝐹 × 𝐺))‘𝑐) = 0 ))
Theorem	ply1coefsupp 22191*	The decomposition of a univariate polynomial is finitely supported. Formerly part of proof for ply1coe 22192. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by AV, 8-Aug-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ · = ( ·𝑠 ‘𝑃)    &   ⊢ 𝑀 = (mulGrp‘𝑃)    &   ⊢ ↑ = (.g‘𝑀)    &   ⊢ 𝐴 = (coe1‘𝐾)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → (𝑘 ∈ ℕ0 ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑋))) finSupp (0g‘𝑃))
Theorem	ply1coe 22192*	Decompose a univariate polynomial as a sum of powers. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by AV, 7-Oct-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ · = ( ·𝑠 ‘𝑃)    &   ⊢ 𝑀 = (mulGrp‘𝑃)    &   ⊢ ↑ = (.g‘𝑀)    &   ⊢ 𝐴 = (coe1‘𝐾)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → 𝐾 = (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑋)))))
Theorem	eqcoe1ply1eq 22193*	Two polynomials over the same ring are equal if they have identical coefficients. (Contributed by AV, 7-Oct-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐴 = (coe1‘𝐾)    &   ⊢ 𝐶 = (coe1‘𝐿)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) → (∀𝑘 ∈ ℕ0 (𝐴‘𝑘) = (𝐶‘𝑘) → 𝐾 = 𝐿))
Theorem	ply1coe1eq 22194*	Two polynomials over the same ring are equal iff they have identical coefficients. (Contributed by AV, 13-Oct-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐴 = (coe1‘𝐾)    &   ⊢ 𝐶 = (coe1‘𝐿)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) → (∀𝑘 ∈ ℕ0 (𝐴‘𝑘) = (𝐶‘𝑘) ↔ 𝐾 = 𝐿))
Theorem	cply1coe0 22195*	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 = (0g‘𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐴 = (algSc‘𝑃)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐾) → ∀𝑛 ∈ ℕ ((coe1‘(𝐴‘𝑆))‘𝑛) = 0 )
Theorem	cply1coe0bi 22196*	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 = (0g‘𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐴 = (algSc‘𝑃)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (∃𝑠 ∈ 𝐾 𝑀 = (𝐴‘𝑠) ↔ ∀𝑛 ∈ ℕ ((coe1‘𝑀)‘𝑛) = 0 ))
Theorem	coe1fzgsumdlem 22197*	Lemma for coe1fzgsumd 22198 (induction step). (Contributed by AV, 8-Oct-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐾 ∈ ℕ0)    ⇒   ⊢ ((𝑚 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑚 ∧ 𝜑) → ((∀𝑥 ∈ 𝑚 𝑀 ∈ 𝐵 → ((coe1‘(𝑃 Σg (𝑥 ∈ 𝑚 ↦ 𝑀)))‘𝐾) = (𝑅 Σg (𝑥 ∈ 𝑚 ↦ ((coe1‘𝑀)‘𝐾)))) → (∀𝑥 ∈ (𝑚 ∪ {𝑎})𝑀 ∈ 𝐵 → ((coe1‘(𝑃 Σg (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ 𝑀)))‘𝐾) = (𝑅 Σg (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ ((coe1‘𝑀)‘𝐾))))))
Theorem	coe1fzgsumd 22198*	Value of an evaluated coefficient in a finite group sum of polynomials. (Contributed by AV, 8-Oct-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐾 ∈ ℕ0)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝑁 𝑀 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑁 ∈ Fin)    ⇒   ⊢ (𝜑 → ((coe1‘(𝑃 Σg (𝑥 ∈ 𝑁 ↦ 𝑀)))‘𝐾) = (𝑅 Σg (𝑥 ∈ 𝑁 ↦ ((coe1‘𝑀)‘𝐾))))
Theorem	ply1scleq 22199	Equality of a constant polynomial is the same as equality of the constant term. (Contributed by Thierry Arnoux, 24-Jul-2024.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐴 = (algSc‘𝑃)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐸 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ((𝐴‘𝐸) = (𝐴‘𝐹) ↔ 𝐸 = 𝐹))
Theorem	ply1chr 22200	The characteristic of a polynomial ring is the characteristic of the underlying ring. (Contributed by Thierry Arnoux, 24-Jul-2024.)
⊢ 𝑃 = (Poly1‘𝑅)    ⇒   ⊢ (𝑅 ∈ CRing → (chr‘𝑃) = (chr‘𝑅))