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Theorem List for Metamath Proof Explorer - 19601-19700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremmplbaspropd 19601* 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 19602 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 19603 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 19604 Lemma for ply1basfvi 19605 and deg1fvi 23839. (Contributed by Stefan O'Rear, 28-Mar-2015.)
∅ = (Base‘(Poly1‘∅))

Theoremply1basfvi 19605 Protection compatibility of the univariate polynomial base set. (Contributed by Stefan O'Rear, 27-Mar-2015.)
(Base‘(Poly1𝑅)) = (Base‘(Poly1‘( I ‘𝑅)))

Theoremply1plusgfvi 19606 Protection compatibility of the univariate polynomial addition. (Contributed by Stefan O'Rear, 27-Mar-2015.)
(+g‘(Poly1𝑅)) = (+g‘(Poly1‘( I ‘𝑅)))

Theoremply1baspropd 19607* 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 19608* 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 19609 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 19610 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 19611 Univariate power series form a ring. (Contributed by Stefan O'Rear, 22-Mar-2015.)
𝑆 = (PwSer1𝑅)       (𝑅 ∈ Ring → 𝑆 ∈ Ring)

Theoremply1ring 19612 Univariate polynomials form a ring. (Contributed by Stefan O'Rear, 22-Mar-2015.)
𝑃 = (Poly1𝑅)       (𝑅 ∈ Ring → 𝑃 ∈ Ring)

Theorempsr1lmod 19613 Univariate power series form a left module. (Contributed by Stefan O'Rear, 26-Mar-2015.)
𝑃 = (PwSer1𝑅)       (𝑅 ∈ Ring → 𝑃 ∈ LMod)

Theorempsr1sca 19614 Scalars of a univariate power series ring. (Contributed by Stefan O'Rear, 4-Jul-2015.)
𝑃 = (PwSer1𝑅)       (𝑅𝑉𝑅 = (Scalar‘𝑃))

Theorempsr1sca2 19615 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 19616 Univariate polynomials form a left module. (Contributed by Stefan O'Rear, 26-Mar-2015.)
𝑃 = (Poly1𝑅)       (𝑅 ∈ Ring → 𝑃 ∈ LMod)

Theoremply1sca 19617 Scalars of a univariate polynomial ring. (Contributed by Stefan O'Rear, 26-Mar-2015.)
𝑃 = (Poly1𝑅)       (𝑅𝑉𝑅 = (Scalar‘𝑃))

Theoremply1sca2 19618 Scalars of a univariate polynomial ring. (Contributed by Stefan O'Rear, 26-Mar-2015.)
𝑃 = (Poly1𝑅)       ( I ‘𝑅) = (Scalar‘𝑃)

Theoremply1mpl0 19619 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 19620 Zero times a univariate polynomial is the zero polynomial (lmod0vs 18890 analog.) (Contributed by AV, 2-Dec-2019.)
𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &    = ( ·𝑠𝑃)    &    0 = (0g𝑅)       ((𝑅 ∈ Ring ∧ 𝑀𝐵) → ( 0 𝑀) = (0g𝑃))

Theoremply1mpl1 19621 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 19622 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 19623 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 19624 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 19625 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 19626 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 19627 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 19628 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 19629 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 19630 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 19631 An equivalence for coe1mul2 19633. (Contributed by Stefan O'Rear, 25-Mar-2015.)
((𝐴 ∈ ℕ0𝑋 ∈ (ℕ0𝑚 1𝑜)) → (𝑋𝑟 ≤ (1𝑜 × {𝐴}) ↔ (𝑋‘∅) ∈ (0...𝐴)))

Theoremcoe1mul2lem2 19632* An equivalence for coe1mul2 19633. (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 19633* 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 19634* 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 19635 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 19636 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 19637* 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 19638 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 19639 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 19640* 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 19641* 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 19642 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 19643* 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 19644 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 19645 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 19646 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 19647 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 19648 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 19649 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 19650 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 19651* 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 19652 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 19653 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 19654 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 19655 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 19656 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 19657 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 19658* 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 19659* The decomposition of a univariate polynomial is finitely supported. Formerly part of proof for ply1coe 19660. (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 19660* 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 19661* 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 19662* 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 19663* 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 19664* 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 19665* Lemma for coe1fzgsumd 19666 (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 19666* 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 19667* 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 19668* 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 19669* 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 19670* 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 19671* 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))))))

10.10.5  Univariate polynomial evaluation

Syntaxces1 19672 Evaluation of a univariate polynomial in a subring.
class evalSub1

Syntaxce1 19673 Evaluation of a univariate polynomial.
class eval1

Definitiondf-evls1 19674* 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 19675* 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 19676 Well-behaved binary operation property of evalSub1. (Contributed by AV, 7-Sep-2019.)
Rel dom evalSub1

Theoremply1frcl 19677 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)))

Theoremevls1fval 19678* Value of the univariate polynomial evaluation map function. (Contributed by AV, 7-Sep-2019.)
𝑄 = (𝑆 evalSub1 𝑅)    &   𝐸 = (1𝑜 evalSub 𝑆)    &   𝐵 = (Base‘𝑆)       ((𝑆𝑉𝑅 ∈ 𝒫 𝐵) → 𝑄 = ((𝑥 ∈ (𝐵𝑚 (𝐵𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦})))) ∘ (𝐸𝑅)))

Theoremevls1val 19679* Value of the univariate polynomial evaluation map. (Contributed by AV, 10-Sep-2019.)
𝑄 = (𝑆 evalSub1 𝑅)    &   𝐸 = (1𝑜 evalSub 𝑆)    &   𝐵 = (Base‘𝑆)    &   𝑀 = (1𝑜 mPoly (𝑆s 𝑅))    &   𝐾 = (Base‘𝑀)       ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆) ∧ 𝐴𝐾) → (𝑄𝐴) = (((𝐸𝑅)‘𝐴) ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦}))))

Theoremevls1rhmlem 19680* Lemma for evl1rhm 19690 and evls1rhm 19681 (formerly part of the proof of evl1rhm 19690): The first function of the composition forming the univariate polynomial evaluation map function for a (sub)ring is a ring homomorphism. (Contributed by AV, 11-Sep-2019.)
𝐵 = (Base‘𝑅)    &   𝑇 = (𝑅s 𝐵)    &   𝐹 = (𝑥 ∈ (𝐵𝑚 (𝐵𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦}))))       (𝑅 ∈ CRing → 𝐹 ∈ ((𝑅s (𝐵𝑚 1𝑜)) RingHom 𝑇))

Theoremevls1rhm 19681 Polynomial evaluation is a homomorphism (into the product ring). (Contributed by AV, 11-Sep-2019.)
𝑄 = (𝑆 evalSub1 𝑅)    &   𝐵 = (Base‘𝑆)    &   𝑇 = (𝑆s 𝐵)    &   𝑈 = (𝑆s 𝑅)    &   𝑊 = (Poly1𝑈)       ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 ∈ (𝑊 RingHom 𝑇))

Theoremevls1sca 19682 Univariate polynomial evaluation maps scalars to constant functions. (Contributed by AV, 8-Sep-2019.)
𝑄 = (𝑆 evalSub1 𝑅)    &   𝑊 = (Poly1𝑈)    &   𝑈 = (𝑆s 𝑅)    &   𝐵 = (Base‘𝑆)    &   𝐴 = (algSc‘𝑊)    &   (𝜑𝑆 ∈ CRing)    &   (𝜑𝑅 ∈ (SubRing‘𝑆))    &   (𝜑𝑋𝑅)       (𝜑 → (𝑄‘(𝐴𝑋)) = (𝐵 × {𝑋}))

Theoremevls1gsumadd 19683* Univariate polynomial evaluation maps (additive) group sums to group sums. (Contributed by AV, 14-Sep-2019.)
𝑄 = (𝑆 evalSub1 𝑅)    &   𝐾 = (Base‘𝑆)    &   𝑊 = (Poly1𝑈)    &    0 = (0g𝑊)    &   𝑈 = (𝑆s 𝑅)    &   𝑃 = (𝑆s 𝐾)    &   𝐵 = (Base‘𝑊)    &   (𝜑𝑆 ∈ CRing)    &   (𝜑𝑅 ∈ (SubRing‘𝑆))    &   ((𝜑𝑥𝑁) → 𝑌𝐵)    &   (𝜑𝑁 ⊆ ℕ0)    &   (𝜑 → (𝑥𝑁𝑌) finSupp 0 )       (𝜑 → (𝑄‘(𝑊 Σg (𝑥𝑁𝑌))) = (𝑃 Σg (𝑥𝑁 ↦ (𝑄𝑌))))

Theoremevls1gsummul 19684* Univariate polynomial evaluation maps (multiplicative) group sums to group sums. (Contributed by AV, 14-Sep-2019.)
𝑄 = (𝑆 evalSub1 𝑅)    &   𝐾 = (Base‘𝑆)    &   𝑊 = (Poly1𝑈)    &   𝐺 = (mulGrp‘𝑊)    &    1 = (1r𝑊)    &   𝑈 = (𝑆s 𝑅)    &   𝑃 = (𝑆s 𝐾)    &   𝐻 = (mulGrp‘𝑃)    &   𝐵 = (Base‘𝑊)    &   (𝜑𝑆 ∈ CRing)    &   (𝜑𝑅 ∈ (SubRing‘𝑆))    &   ((𝜑𝑥𝑁) → 𝑌𝐵)    &   (𝜑𝑁 ⊆ ℕ0)    &   (𝜑 → (𝑥𝑁𝑌) finSupp 1 )       (𝜑 → (𝑄‘(𝐺 Σg (𝑥𝑁𝑌))) = (𝐻 Σg (𝑥𝑁 ↦ (𝑄𝑌))))

Theoremevls1varpw 19685 Univariate polynomial evaluation for subrings maps the exponentiation of a variable to the exponentiation of the evaluated variable. (Contributed by AV, 14-Sep-2019.)
𝑄 = (𝑆 evalSub1 𝑅)    &   𝑈 = (𝑆s 𝑅)    &   𝑊 = (Poly1𝑈)    &   𝐺 = (mulGrp‘𝑊)    &   𝑋 = (var1𝑈)    &   𝐵 = (Base‘𝑆)    &    = (.g𝐺)    &   (𝜑𝑆 ∈ CRing)    &   (𝜑𝑅 ∈ (SubRing‘𝑆))    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → (𝑄‘(𝑁 𝑋)) = (𝑁(.g‘(mulGrp‘(𝑆s 𝐵)))(𝑄𝑋)))

Theoremevl1fval 19686* Value of the simple/same ring evaluation map. (Contributed by Mario Carneiro, 12-Jun-2015.)
𝑂 = (eval1𝑅)    &   𝑄 = (1𝑜 eval 𝑅)    &   𝐵 = (Base‘𝑅)       𝑂 = ((𝑥 ∈ (𝐵𝑚 (𝐵𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦})))) ∘ 𝑄)

Theoremevl1val 19687* Value of the simple/same ring evaluation map. (Contributed by Mario Carneiro, 12-Jun-2015.)
𝑂 = (eval1𝑅)    &   𝑄 = (1𝑜 eval 𝑅)    &   𝐵 = (Base‘𝑅)    &   𝑀 = (1𝑜 mPoly 𝑅)    &   𝐾 = (Base‘𝑀)       ((𝑅 ∈ CRing ∧ 𝐴𝐾) → (𝑂𝐴) = ((𝑄𝐴) ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦}))))

Theoremevl1fval1lem 19688 Lemma for evl1fval1 19689. (Contributed by AV, 11-Sep-2019.)
𝑄 = (eval1𝑅)    &   𝐵 = (Base‘𝑅)       (𝑅𝑉𝑄 = (𝑅 evalSub1 𝐵))

Theoremevl1fval1 19689 Value of the simple/same ring evaluation map function for univariate polynomials. (Contributed by AV, 11-Sep-2019.)
𝑄 = (eval1𝑅)    &   𝐵 = (Base‘𝑅)       𝑄 = (𝑅 evalSub1 𝐵)

Theoremevl1rhm 19690 Polynomial evaluation is a homomorphism (into the product ring). (Contributed by Mario Carneiro, 12-Jun-2015.) (Proof shortened by AV, 13-Sep-2019.)
𝑂 = (eval1𝑅)    &   𝑃 = (Poly1𝑅)    &   𝑇 = (𝑅s 𝐵)    &   𝐵 = (Base‘𝑅)       (𝑅 ∈ CRing → 𝑂 ∈ (𝑃 RingHom 𝑇))

Theoremfveval1fvcl 19691 The function value of the evaluation function of a polynomial is an element of the underlying ring. (Contributed by AV, 17-Sep-2019.)
𝑂 = (eval1𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑅)    &   𝑈 = (Base‘𝑃)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑌𝐵)    &   (𝜑𝑀𝑈)       (𝜑 → ((𝑂𝑀)‘𝑌) ∈ 𝐵)

Theoremevl1sca 19692 Polynomial evaluation maps scalars to constant functions. (Contributed by Mario Carneiro, 12-Jun-2015.)
𝑂 = (eval1𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑅)    &   𝐴 = (algSc‘𝑃)       ((𝑅 ∈ CRing ∧ 𝑋𝐵) → (𝑂‘(𝐴𝑋)) = (𝐵 × {𝑋}))

Theoremevl1scad 19693 Polynomial evaluation builder for scalars. (Contributed by Mario Carneiro, 4-Jul-2015.)
𝑂 = (eval1𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑅)    &   𝐴 = (algSc‘𝑃)    &   𝑈 = (Base‘𝑃)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → ((𝐴𝑋) ∈ 𝑈 ∧ ((𝑂‘(𝐴𝑋))‘𝑌) = 𝑋))

Theoremevl1var 19694 Polynomial evaluation maps the variable to the identity function. (Contributed by Mario Carneiro, 12-Jun-2015.)
𝑂 = (eval1𝑅)    &   𝑋 = (var1𝑅)    &   𝐵 = (Base‘𝑅)       (𝑅 ∈ CRing → (𝑂𝑋) = ( I ↾ 𝐵))

Theoremevl1vard 19695 Polynomial evaluation builder for the variable. (Contributed by Mario Carneiro, 4-Jul-2015.)
𝑂 = (eval1𝑅)    &   𝑋 = (var1𝑅)    &   𝐵 = (Base‘𝑅)    &   𝑃 = (Poly1𝑅)    &   𝑈 = (Base‘𝑃)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑌𝐵)       (𝜑 → (𝑋𝑈 ∧ ((𝑂𝑋)‘𝑌) = 𝑌))

Theoremevls1var 19696 Univariate polynomial evaluation for subrings maps the variable to the identity function. (Contributed by AV, 13-Sep-2019.)
𝑄 = (𝑆 evalSub1 𝑅)    &   𝑋 = (var1𝑈)    &   𝑈 = (𝑆s 𝑅)    &   𝐵 = (Base‘𝑆)    &   (𝜑𝑆 ∈ CRing)    &   (𝜑𝑅 ∈ (SubRing‘𝑆))       (𝜑 → (𝑄𝑋) = ( I ↾ 𝐵))

Theoremevls1scasrng 19697 The evaluation of a scalar of a subring yields the same result as evaluated as a scalar over the ring itself. (Contributed by AV, 13-Sep-2019.)
𝑄 = (𝑆 evalSub1 𝑅)    &   𝑂 = (eval1𝑆)    &   𝑊 = (Poly1𝑈)    &   𝑈 = (𝑆s 𝑅)    &   𝑃 = (Poly1𝑆)    &   𝐵 = (Base‘𝑆)    &   𝐴 = (algSc‘𝑊)    &   𝐶 = (algSc‘𝑃)    &   (𝜑𝑆 ∈ CRing)    &   (𝜑𝑅 ∈ (SubRing‘𝑆))    &   (𝜑𝑋𝑅)       (𝜑 → (𝑄‘(𝐴𝑋)) = (𝑂‘(𝐶𝑋)))

Theoremevls1varsrng 19698 The evaluation of the variable of univariate polynomials over subring yields the same result as evaluated as variable of the polynomials over the ring itself. (Contributed by AV, 12-Sep-2019.)
𝑄 = (𝑆 evalSub1 𝑅)    &   𝑂 = (eval1𝑆)    &   𝑉 = (var1𝑈)    &   𝑈 = (𝑆s 𝑅)    &   𝐵 = (Base‘𝑆)    &   (𝜑𝑆 ∈ CRing)    &   (𝜑𝑅 ∈ (SubRing‘𝑆))       (𝜑 → (𝑄𝑉) = (𝑂𝑉))

Theoremevl1addd 19699 Polynomial evaluation builder for addition of polynomials. (Contributed by Mario Carneiro, 4-Jul-2015.)
𝑂 = (eval1𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑅)    &   𝑈 = (Base‘𝑃)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑌𝐵)    &   (𝜑 → (𝑀𝑈 ∧ ((𝑂𝑀)‘𝑌) = 𝑉))    &   (𝜑 → (𝑁𝑈 ∧ ((𝑂𝑁)‘𝑌) = 𝑊))    &    = (+g𝑃)    &    + = (+g𝑅)       (𝜑 → ((𝑀 𝑁) ∈ 𝑈 ∧ ((𝑂‘(𝑀 𝑁))‘𝑌) = (𝑉 + 𝑊)))

Theoremevl1subd 19700 Polynomial evaluation builder for subtraction of polynomials. (Contributed by Mario Carneiro, 4-Jul-2015.)
𝑂 = (eval1𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑅)    &   𝑈 = (Base‘𝑃)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑌𝐵)    &   (𝜑 → (𝑀𝑈 ∧ ((𝑂𝑀)‘𝑌) = 𝑉))    &   (𝜑 → (𝑁𝑈 ∧ ((𝑂𝑁)‘𝑌) = 𝑊))    &    = (-g𝑃)    &   𝐷 = (-g𝑅)       (𝜑 → ((𝑀 𝑁) ∈ 𝑈 ∧ ((𝑂‘(𝑀 𝑁))‘𝑌) = (𝑉𝐷𝑊)))

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