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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | mhpsubg 20801 | Homogeneous polynomials form a subgroup of the polynomials. (Contributed by SN, 25-Sep-2023.) |
⊢ 𝐻 = (𝐼 mHomP 𝑅) & ⊢ 𝑃 = (𝐼 mPoly 𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ Grp) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) ⇒ ⊢ (𝜑 → (𝐻‘𝑁) ∈ (SubGrp‘𝑃)) | ||
Theorem | mhpvscacl 20802 | Homogeneous polynomials are closed under scalar multiplication. (Contributed by SN, 25-Sep-2023.) |
⊢ 𝐻 = (𝐼 mHomP 𝑅) & ⊢ 𝑃 = (𝐼 mPoly 𝑅) & ⊢ · = ( ·𝑠 ‘𝑃) & ⊢ 𝐾 = (Base‘𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → 𝑋 ∈ 𝐾) & ⊢ (𝜑 → 𝐹 ∈ (𝐻‘𝑁)) ⇒ ⊢ (𝜑 → (𝑋 · 𝐹) ∈ (𝐻‘𝑁)) | ||
Theorem | mhplss 20803 | Homogeneous polynomials form a linear subspace of the polynomials. (Contributed by SN, 25-Sep-2023.) |
⊢ 𝐻 = (𝐼 mHomP 𝑅) & ⊢ 𝑃 = (𝐼 mPoly 𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) ⇒ ⊢ (𝜑 → (𝐻‘𝑁) ∈ (LSubSp‘𝑃)) | ||
According to Wikipedia ("Polynomial", 23-Dec-2019, https://en.wikipedia.org/wiki/Polynomial) "A polynomial in one indeterminate is called a univariate polynomial, a polynomial in more than one indeterminate is called a multivariate polynomial." In this sense univariate polynomials are defined as multivariate polynomials restricted to one indeterminate/polynomial variable in the following, see ply1bascl2 20833. According to the definition in Wikipedia "a polynomial can either be zero or can be written as the sum of a finite number of nonzero terms. Each term consists of the product of a number - called the coefficient of the term - and a finite number of indeterminates, raised to nonnegative integer powers.". By this, a term of a univariate polynomial (often also called "polynomial term") is the product of a coefficient (usually a member of the underlying ring) and the variable, raised to a nonnegative integer power. A (univariate) polynomial which has only one term is called (univariate) monomial - therefore, the notions "term" and "monomial" are often used synonymously, see also the definition in [Lang] p. 102. Sometimes, however, a monomial is defined as power product, "a product of powers of variables with nonnegative integer exponents", see Wikipedia ("Monomial", 23-Dec-2019, https://en.wikipedia.org/wiki/Mononomial 20833). In [Lang] p. 101, such terms are called "primitive monomials". To avoid any ambiguity, the notion "primitive monomial" is used for such power products ("x^i") in the following, whereas the synonym for "term" ("ai x^i") will be "scaled monomial". | ||
Syntax | cps1 20804 | Univariate power series. |
class PwSer1 | ||
Syntax | cv1 20805 | The base variable of a univariate power series. |
class var1 | ||
Syntax | cpl1 20806 | Univariate polynomials. |
class Poly1 | ||
Syntax | cco1 20807 | Coefficient function for a univariate polynomial. |
class coe1 | ||
Syntax | ctp1 20808 | Convert a univariate polynomial representation to multivariate. |
class toPoly1 | ||
Definition | df-psr1 20809 | Define the algebra of univariate power series. (Contributed by Mario Carneiro, 29-Dec-2014.) |
⊢ PwSer1 = (𝑟 ∈ V ↦ ((1o ordPwSer 𝑟)‘∅)) | ||
Definition | df-vr1 20810 | Define the base element of a univariate power series (the 𝑋 element of the set 𝑅[𝑋] of polynomials and also the 𝑋 in the set 𝑅[[𝑋]] of power series). (Contributed by Mario Carneiro, 8-Feb-2015.) |
⊢ var1 = (𝑟 ∈ V ↦ ((1o mVar 𝑟)‘∅)) | ||
Definition | df-ply1 20811 | Define the algebra of univariate polynomials. (Contributed by Mario Carneiro, 9-Feb-2015.) |
⊢ Poly1 = (𝑟 ∈ V ↦ ((PwSer1‘𝑟) ↾s (Base‘(1o mPoly 𝑟)))) | ||
Definition | df-coe1 20812* | Define the coefficient function for a univariate polynomial. (Contributed by Stefan O'Rear, 21-Mar-2015.) |
⊢ coe1 = (𝑓 ∈ V ↦ (𝑛 ∈ ℕ0 ↦ (𝑓‘(1o × {𝑛})))) | ||
Definition | df-toply1 20813* | Define a function which maps a coefficient function for a univariate polynomial to the corresponding polynomial object. (Contributed by Mario Carneiro, 12-Jun-2015.) |
⊢ toPoly1 = (𝑓 ∈ V ↦ (𝑛 ∈ (ℕ0 ↑m 1o) ↦ (𝑓‘(𝑛‘∅)))) | ||
Theorem | psr1baslem 20814 | The set of finite bags on 1o is just the set of all functions from 1o to ℕ0. (Contributed by Mario Carneiro, 9-Feb-2015.) |
⊢ (ℕ0 ↑m 1o) = {𝑓 ∈ (ℕ0 ↑m 1o) ∣ (◡𝑓 “ ℕ) ∈ Fin} | ||
Theorem | psr1val 20815 | Value of the ring of univariate power series. (Contributed by Mario Carneiro, 8-Feb-2015.) |
⊢ 𝑆 = (PwSer1‘𝑅) ⇒ ⊢ 𝑆 = ((1o ordPwSer 𝑅)‘∅) | ||
Theorem | psr1crng 20816 | The ring of univariate power series is a commutative ring. (Contributed by Mario Carneiro, 8-Feb-2015.) |
⊢ 𝑆 = (PwSer1‘𝑅) ⇒ ⊢ (𝑅 ∈ CRing → 𝑆 ∈ CRing) | ||
Theorem | psr1assa 20817 | The ring of univariate power series is an associative algebra. (Contributed by Mario Carneiro, 8-Feb-2015.) |
⊢ 𝑆 = (PwSer1‘𝑅) ⇒ ⊢ (𝑅 ∈ CRing → 𝑆 ∈ AssAlg) | ||
Theorem | psr1tos 20818 | The ordered power series structure is a totally ordered set. (Contributed by Mario Carneiro, 2-Jun-2015.) |
⊢ 𝑆 = (PwSer1‘𝑅) ⇒ ⊢ (𝑅 ∈ Toset → 𝑆 ∈ Toset) | ||
Theorem | psr1bas2 20819 | The base set of the ring of univariate power series. (Contributed by Mario Carneiro, 3-Jul-2015.) |
⊢ 𝑆 = (PwSer1‘𝑅) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ 𝑂 = (1o mPwSer 𝑅) ⇒ ⊢ 𝐵 = (Base‘𝑂) | ||
Theorem | psr1bas 20820 | The base set of the ring of univariate power series. (Contributed by Mario Carneiro, 8-Feb-2015.) |
⊢ 𝑆 = (PwSer1‘𝑅) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ 𝐾 = (Base‘𝑅) ⇒ ⊢ 𝐵 = (𝐾 ↑m (ℕ0 ↑m 1o)) | ||
Theorem | vr1val 20821 | The value of the generator of the power series algebra (the 𝑋 in 𝑅[[𝑋]]). Since all univariate polynomial rings over a fixed base ring 𝑅 are isomorphic, we don't bother to pass this in as a parameter; internally we are actually using the empty set as this generator and 1o = {∅} is the index set (but for most purposes this choice should not be visible anyway). (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by Mario Carneiro, 12-Jun-2015.) |
⊢ 𝑋 = (var1‘𝑅) ⇒ ⊢ 𝑋 = ((1o mVar 𝑅)‘∅) | ||
Theorem | vr1cl2 20822 | The variable 𝑋 is a member of the power series algebra 𝑅[[𝑋]]. (Contributed by Mario Carneiro, 8-Feb-2015.) |
⊢ 𝑋 = (var1‘𝑅) & ⊢ 𝑆 = (PwSer1‘𝑅) & ⊢ 𝐵 = (Base‘𝑆) ⇒ ⊢ (𝑅 ∈ Ring → 𝑋 ∈ 𝐵) | ||
Theorem | ply1val 20823 | The value of the set of univariate polynomials. (Contributed by Mario Carneiro, 9-Feb-2015.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝑆 = (PwSer1‘𝑅) ⇒ ⊢ 𝑃 = (𝑆 ↾s (Base‘(1o mPoly 𝑅))) | ||
Theorem | ply1bas 20824 | The value of the base set of univariate polynomials. (Contributed by Mario Carneiro, 9-Feb-2015.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝑆 = (PwSer1‘𝑅) & ⊢ 𝑈 = (Base‘𝑃) ⇒ ⊢ 𝑈 = (Base‘(1o mPoly 𝑅)) | ||
Theorem | ply1lss 20825 | Univariate polynomials form a linear subspace of the set of univariate power series. (Contributed by Mario Carneiro, 9-Feb-2015.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝑆 = (PwSer1‘𝑅) & ⊢ 𝑈 = (Base‘𝑃) ⇒ ⊢ (𝑅 ∈ Ring → 𝑈 ∈ (LSubSp‘𝑆)) | ||
Theorem | ply1subrg 20826 | Univariate polynomials form a subring of the set of univariate power series. (Contributed by Mario Carneiro, 9-Feb-2015.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝑆 = (PwSer1‘𝑅) & ⊢ 𝑈 = (Base‘𝑃) ⇒ ⊢ (𝑅 ∈ Ring → 𝑈 ∈ (SubRing‘𝑆)) | ||
Theorem | ply1crng 20827 | The ring of univariate polynomials is a commutative ring. (Contributed by Mario Carneiro, 9-Feb-2015.) |
⊢ 𝑃 = (Poly1‘𝑅) ⇒ ⊢ (𝑅 ∈ CRing → 𝑃 ∈ CRing) | ||
Theorem | ply1assa 20828 | The ring of univariate polynomials is an associative algebra. (Contributed by Mario Carneiro, 9-Feb-2015.) |
⊢ 𝑃 = (Poly1‘𝑅) ⇒ ⊢ (𝑅 ∈ CRing → 𝑃 ∈ AssAlg) | ||
Theorem | psr1bascl 20829 | A univariate power series is a multivariate power series on one index. (Contributed by Stefan O'Rear, 25-Mar-2015.) |
⊢ 𝑃 = (PwSer1‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) ⇒ ⊢ (𝐹 ∈ 𝐵 → 𝐹 ∈ (Base‘(1o mPwSer 𝑅))) | ||
Theorem | psr1basf 20830 | Univariate power series base set elements are functions. (Contributed by Stefan O'Rear, 25-Mar-2015.) |
⊢ 𝑃 = (PwSer1‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝐾 = (Base‘𝑅) ⇒ ⊢ (𝐹 ∈ 𝐵 → 𝐹:(ℕ0 ↑m 1o)⟶𝐾) | ||
Theorem | ply1basf 20831 | Univariate polynomial base set elements are functions. (Contributed by Stefan O'Rear, 21-Mar-2015.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝐾 = (Base‘𝑅) ⇒ ⊢ (𝐹 ∈ 𝐵 → 𝐹:(ℕ0 ↑m 1o)⟶𝐾) | ||
Theorem | ply1bascl 20832 | A univariate polynomial is a univariate power series. (Contributed by Stefan O'Rear, 25-Mar-2015.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) ⇒ ⊢ (𝐹 ∈ 𝐵 → 𝐹 ∈ (Base‘(PwSer1‘𝑅))) | ||
Theorem | ply1bascl2 20833 | A univariate polynomial is a multivariate polynomial on one index. (Contributed by Stefan O'Rear, 25-Mar-2015.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) ⇒ ⊢ (𝐹 ∈ 𝐵 → 𝐹 ∈ (Base‘(1o mPoly 𝑅))) | ||
Theorem | coe1fval 20834* | Value of the univariate polynomial coefficient function. (Contributed by Stefan O'Rear, 21-Mar-2015.) |
⊢ 𝐴 = (coe1‘𝐹) ⇒ ⊢ (𝐹 ∈ 𝑉 → 𝐴 = (𝑛 ∈ ℕ0 ↦ (𝐹‘(1o × {𝑛})))) | ||
Theorem | coe1fv 20835 | Value of an evaluated coefficient in a polynomial coefficient vector. (Contributed by Stefan O'Rear, 21-Mar-2015.) |
⊢ 𝐴 = (coe1‘𝐹) ⇒ ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝐴‘𝑁) = (𝐹‘(1o × {𝑁}))) | ||
Theorem | fvcoe1 20836 | Value of a multivariate coefficient in terms of the coefficient vector. (Contributed by Stefan O'Rear, 21-Mar-2015.) |
⊢ 𝐴 = (coe1‘𝐹) ⇒ ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑋 ∈ (ℕ0 ↑m 1o)) → (𝐹‘𝑋) = (𝐴‘(𝑋‘∅))) | ||
Theorem | coe1fval3 20837* | Univariate power series coefficient vectors expressed as a function composition. (Contributed by Stefan O'Rear, 25-Mar-2015.) |
⊢ 𝐴 = (coe1‘𝐹) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝑃 = (PwSer1‘𝑅) & ⊢ 𝐺 = (𝑦 ∈ ℕ0 ↦ (1o × {𝑦})) ⇒ ⊢ (𝐹 ∈ 𝐵 → 𝐴 = (𝐹 ∘ 𝐺)) | ||
Theorem | coe1f2 20838 | 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 20839* | 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 20840 | 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 20841 | 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 20842 | 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 20843* | 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 20844* | 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 20845* | 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 20846 | 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 20847 | 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 20848 | 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 | mplplusg 20849 | 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) & ⊢ + = (+g‘𝑌) ⇒ ⊢ + = (+g‘𝑆) | ||
Theorem | mplmulr 20850 | 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) & ⊢ · = (.r‘𝑌) ⇒ ⊢ · = (.r‘𝑆) | ||
Theorem | psr1plusg 20851 | 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 20852 | 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 20853 | 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 20854 | 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 20855 | 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 20856 | 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 | ressply1bas2 20857 | 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 20858 | 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 20859 | 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 20860 | 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 20861 | 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 20862 | 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 20863 | 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 20864 | 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 20865* | 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 20866* | 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 20867 | 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 20868 | 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 20869 | Lemma for ply1basfvi 20870 and deg1fvi 24686. (Contributed by Stefan O'Rear, 28-Mar-2015.) |
⊢ ∅ = (Base‘(Poly1‘∅)) | ||
Theorem | ply1basfvi 20870 | 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 20871 | Protection compatibility of the univariate polynomial addition. (Contributed by Stefan O'Rear, 27-Mar-2015.) |
⊢ (+g‘(Poly1‘𝑅)) = (+g‘(Poly1‘( I ‘𝑅))) | ||
Theorem | ply1baspropd 20872* | 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 20873* | 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 20874 | 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 20875 | 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 20876 | Univariate power series form a ring. (Contributed by Stefan O'Rear, 22-Mar-2015.) |
⊢ 𝑆 = (PwSer1‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring → 𝑆 ∈ Ring) | ||
Theorem | ply1ring 20877 | Univariate polynomials form a ring. (Contributed by Stefan O'Rear, 22-Mar-2015.) |
⊢ 𝑃 = (Poly1‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring) | ||
Theorem | psr1lmod 20878 | Univariate power series form a left module. (Contributed by Stefan O'Rear, 26-Mar-2015.) |
⊢ 𝑃 = (PwSer1‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring → 𝑃 ∈ LMod) | ||
Theorem | psr1sca 20879 | 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 20880 | 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 20881 | Univariate polynomials form a left module. (Contributed by Stefan O'Rear, 26-Mar-2015.) |
⊢ 𝑃 = (Poly1‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring → 𝑃 ∈ LMod) | ||
Theorem | ply1sca 20882 | Scalars of a univariate polynomial ring. (Contributed by Stefan O'Rear, 26-Mar-2015.) |
⊢ 𝑃 = (Poly1‘𝑅) ⇒ ⊢ (𝑅 ∈ 𝑉 → 𝑅 = (Scalar‘𝑃)) | ||
Theorem | ply1sca2 20883 | Scalars of a univariate polynomial ring. (Contributed by Stefan O'Rear, 26-Mar-2015.) |
⊢ 𝑃 = (Poly1‘𝑅) ⇒ ⊢ ( I ‘𝑅) = (Scalar‘𝑃) | ||
Theorem | ply1mpl0 20884 | 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 20885 | Zero times a univariate polynomial is the zero polynomial (lmod0vs 19660 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 20886 | 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 20887 | 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 20888 | 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 20889 | 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 20890 | 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 20891 | 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 20892 | 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 20893 | 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 20894 | 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 20895 | 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 20896 | An equivalence for coe1mul2 20898. (Contributed by Stefan O'Rear, 25-Mar-2015.) |
⊢ ((𝐴 ∈ ℕ0 ∧ 𝑋 ∈ (ℕ0 ↑m 1o)) → (𝑋 ∘r ≤ (1o × {𝐴}) ↔ (𝑋‘∅) ∈ (0...𝐴))) | ||
Theorem | coe1mul2lem2 20897* | An equivalence for coe1mul2 20898. (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 20898* | 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 20899* | 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 20900 | 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) |
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