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Theorem List for Metamath Proof Explorer - 19501-19600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremevlsval2 19501* Characterizing properties of the polynomial evaluation map function. (Contributed by Stefan O'Rear, 12-Mar-2015.) (Revised by AV, 18-Sep-2021.)
𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)    &   𝑊 = (𝐼 mPoly 𝑈)    &   𝑉 = (𝐼 mVar 𝑈)    &   𝑈 = (𝑆s 𝑅)    &   𝑇 = (𝑆s (𝐵𝑚 𝐼))    &   𝐵 = (Base‘𝑆)    &   𝐴 = (algSc‘𝑊)    &   𝑋 = (𝑥𝑅 ↦ ((𝐵𝑚 𝐼) × {𝑥}))    &   𝑌 = (𝑥𝐼 ↦ (𝑔 ∈ (𝐵𝑚 𝐼) ↦ (𝑔𝑥)))       ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (𝑄 ∈ (𝑊 RingHom 𝑇) ∧ ((𝑄𝐴) = 𝑋 ∧ (𝑄𝑉) = 𝑌)))
 
Theoremevlsrhm 19502 Polynomial evaluation is a homomorphism (into the product ring). (Contributed by Stefan O'Rear, 12-Mar-2015.)
𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)    &   𝑊 = (𝐼 mPoly 𝑈)    &   𝑈 = (𝑆s 𝑅)    &   𝑇 = (𝑆s (𝐵𝑚 𝐼))    &   𝐵 = (Base‘𝑆)       ((𝐼𝑉𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 ∈ (𝑊 RingHom 𝑇))
 
Theoremevlssca 19503 Polynomial evaluation maps scalars to constant functions. (Contributed by Stefan O'Rear, 13-Mar-2015.) (Proof shortened by AV, 18-Sep-2021.)
𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)    &   𝑊 = (𝐼 mPoly 𝑈)    &   𝑈 = (𝑆s 𝑅)    &   𝐵 = (Base‘𝑆)    &   𝐴 = (algSc‘𝑊)    &   (𝜑𝐼𝑉)    &   (𝜑𝑆 ∈ CRing)    &   (𝜑𝑅 ∈ (SubRing‘𝑆))    &   (𝜑𝑋𝑅)       (𝜑 → (𝑄‘(𝐴𝑋)) = ((𝐵𝑚 𝐼) × {𝑋}))
 
Theoremevlsvar 19504* Polynomial evaluation maps variables to projections. (Contributed by Stefan O'Rear, 12-Mar-2015.) (Proof shortened by AV, 18-Sep-2021.)
𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)    &   𝑉 = (𝐼 mVar 𝑈)    &   𝑈 = (𝑆s 𝑅)    &   𝐵 = (Base‘𝑆)    &   (𝜑𝐼𝑊)    &   (𝜑𝑆 ∈ CRing)    &   (𝜑𝑅 ∈ (SubRing‘𝑆))    &   (𝜑𝑋𝐼)       (𝜑 → (𝑄‘(𝑉𝑋)) = (𝑔 ∈ (𝐵𝑚 𝐼) ↦ (𝑔𝑋)))
 
Theoremevlval 19505 Value of the simple/same ring evaluation map. (Contributed by Stefan O'Rear, 19-Mar-2015.) (Revised by Mario Carneiro, 12-Jun-2015.)
𝑄 = (𝐼 eval 𝑅)    &   𝐵 = (Base‘𝑅)       𝑄 = ((𝐼 evalSub 𝑅)‘𝐵)
 
Theoremevlrhm 19506 The simple evaluation map is a ring homomorphism. (Contributed by Mario Carneiro, 12-Jun-2015.)
𝑄 = (𝐼 eval 𝑅)    &   𝐵 = (Base‘𝑅)    &   𝑊 = (𝐼 mPoly 𝑅)    &   𝑇 = (𝑅s (𝐵𝑚 𝐼))       ((𝐼𝑉𝑅 ∈ CRing) → 𝑄 ∈ (𝑊 RingHom 𝑇))
 
Theoremevlsscasrng 19507 The evaluation of a scalar of a subring yields the same result as evaluated as a scalar over the ring itself. (Contributed by AV, 12-Sep-2019.)
𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)    &   𝑂 = (𝐼 eval 𝑆)    &   𝑊 = (𝐼 mPoly 𝑈)    &   𝑈 = (𝑆s 𝑅)    &   𝑃 = (𝐼 mPoly 𝑆)    &   𝐵 = (Base‘𝑆)    &   𝐴 = (algSc‘𝑊)    &   𝐶 = (algSc‘𝑃)    &   (𝜑𝐼𝑉)    &   (𝜑𝑆 ∈ CRing)    &   (𝜑𝑅 ∈ (SubRing‘𝑆))    &   (𝜑𝑋𝑅)       (𝜑 → (𝑄‘(𝐴𝑋)) = (𝑂‘(𝐶𝑋)))
 
Theoremevlsca 19508 Simple polynomial evaluation maps scalars to constant functions. (Contributed by AV, 12-Sep-2019.)
𝑄 = (𝐼 eval 𝑆)    &   𝑊 = (𝐼 mPoly 𝑆)    &   𝐵 = (Base‘𝑆)    &   𝐴 = (algSc‘𝑊)    &   (𝜑𝐼𝑉)    &   (𝜑𝑆 ∈ CRing)    &   (𝜑𝑋𝐵)       (𝜑 → (𝑄‘(𝐴𝑋)) = ((𝐵𝑚 𝐼) × {𝑋}))
 
Theoremevlsvarsrng 19509 The evaluation of the variable of 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 = ((𝐼 evalSub 𝑆)‘𝑅)    &   𝑂 = (𝐼 eval 𝑆)    &   𝑉 = (𝐼 mVar 𝑈)    &   𝑈 = (𝑆s 𝑅)    &   𝐵 = (Base‘𝑆)    &   (𝜑𝐼𝐴)    &   (𝜑𝑆 ∈ CRing)    &   (𝜑𝑅 ∈ (SubRing‘𝑆))    &   (𝜑𝑋𝐼)       (𝜑 → (𝑄‘(𝑉𝑋)) = (𝑂‘(𝑉𝑋)))
 
Theoremevlvar 19510* Simple polynomial evaluation maps variables to projections. (Contributed by AV, 12-Sep-2019.)
𝑄 = (𝐼 eval 𝑆)    &   𝑉 = (𝐼 mVar 𝑆)    &   𝐵 = (Base‘𝑆)    &   (𝜑𝐼𝑊)    &   (𝜑𝑆 ∈ CRing)    &   (𝜑𝑋𝐼)       (𝜑 → (𝑄‘(𝑉𝑋)) = (𝑔 ∈ (𝐵𝑚 𝐼) ↦ (𝑔𝑋)))
 
Theoremmpfconst 19511 Constants are multivariate polynomial functions. (Contributed by Mario Carneiro, 19-Mar-2015.)
𝐵 = (Base‘𝑆)    &   𝑄 = ran ((𝐼 evalSub 𝑆)‘𝑅)    &   (𝜑𝐼𝑉)    &   (𝜑𝑆 ∈ CRing)    &   (𝜑𝑅 ∈ (SubRing‘𝑆))    &   (𝜑𝑋𝑅)       (𝜑 → ((𝐵𝑚 𝐼) × {𝑋}) ∈ 𝑄)
 
Theoremmpfproj 19512* Projections are multivariate polynomial functions. (Contributed by Mario Carneiro, 20-Mar-2015.)
𝐵 = (Base‘𝑆)    &   𝑄 = ran ((𝐼 evalSub 𝑆)‘𝑅)    &   (𝜑𝐼𝑉)    &   (𝜑𝑆 ∈ CRing)    &   (𝜑𝑅 ∈ (SubRing‘𝑆))    &   (𝜑𝐽𝐼)       (𝜑 → (𝑓 ∈ (𝐵𝑚 𝐼) ↦ (𝑓𝐽)) ∈ 𝑄)
 
Theoremmpfsubrg 19513 Polynomial functions are a subring. (Contributed by Mario Carneiro, 19-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.) (Revised by AV, 19-Sep-2021.)
𝑄 = ran ((𝐼 evalSub 𝑆)‘𝑅)       ((𝐼𝑉𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 ∈ (SubRing‘(𝑆s ((Base‘𝑆) ↑𝑚 𝐼))))
 
Theoremmpff 19514 Polynomial functions are functions. (Contributed by Mario Carneiro, 19-Mar-2015.)
𝑄 = ran ((𝐼 evalSub 𝑆)‘𝑅)    &   𝐵 = (Base‘𝑆)       (𝐹𝑄𝐹:(𝐵𝑚 𝐼)⟶𝐵)
 
Theoremmpfaddcl 19515 The sum of multivariate polynomial functions. (Contributed by Mario Carneiro, 19-Mar-2015.)
𝑄 = ran ((𝐼 evalSub 𝑆)‘𝑅)    &    + = (+g𝑆)       ((𝐹𝑄𝐺𝑄) → (𝐹𝑓 + 𝐺) ∈ 𝑄)
 
Theoremmpfmulcl 19516 The product of multivariate polynomial functions. (Contributed by Mario Carneiro, 19-Mar-2015.)
𝑄 = ran ((𝐼 evalSub 𝑆)‘𝑅)    &    · = (.r𝑆)       ((𝐹𝑄𝐺𝑄) → (𝐹𝑓 · 𝐺) ∈ 𝑄)
 
Theoremmpfind 19517* Prove a property of polynomials by "structural" induction, under a simplified model of structure which loses the sum of products structure. (Contributed by Mario Carneiro, 19-Mar-2015.)
𝐵 = (Base‘𝑆)    &    + = (+g𝑆)    &    · = (.r𝑆)    &   𝑄 = ran ((𝐼 evalSub 𝑆)‘𝑅)    &   ((𝜑 ∧ ((𝑓𝑄𝜏) ∧ (𝑔𝑄𝜂))) → 𝜁)    &   ((𝜑 ∧ ((𝑓𝑄𝜏) ∧ (𝑔𝑄𝜂))) → 𝜎)    &   (𝑥 = ((𝐵𝑚 𝐼) × {𝑓}) → (𝜓𝜒))    &   (𝑥 = (𝑔 ∈ (𝐵𝑚 𝐼) ↦ (𝑔𝑓)) → (𝜓𝜃))    &   (𝑥 = 𝑓 → (𝜓𝜏))    &   (𝑥 = 𝑔 → (𝜓𝜂))    &   (𝑥 = (𝑓𝑓 + 𝑔) → (𝜓𝜁))    &   (𝑥 = (𝑓𝑓 · 𝑔) → (𝜓𝜎))    &   (𝑥 = 𝐴 → (𝜓𝜌))    &   ((𝜑𝑓𝑅) → 𝜒)    &   ((𝜑𝑓𝐼) → 𝜃)    &   (𝜑𝐴𝑄)       (𝜑𝜌)
 
10.10.3  Additional definitions for (multivariate) polynomials

Remark: There are no theorems using these definitions yet!

 
Syntaxcmhp 19518 Multivariate polynomials.
class mHomP
 
Syntaxcpsd 19519 Power series partial derivative function.
class mPSDer
 
Syntaxcslv 19520 Select a subset of variables in a multivariate polynomial.
class selectVars
 
Syntaxcai 19521 Algebraically independent.
class AlgInd
 
Definitiondf-mhp 19522* Define the subspaces of order- 𝑛 homogeneous polynomials. (Contributed by Mario Carneiro, 21-Mar-2015.)
mHomP = (𝑖 ∈ V, 𝑟 ∈ V ↦ (𝑛 ∈ ℕ0 ↦ {𝑓 ∈ (Base‘(𝑖 mPoly 𝑟)) ∣ (𝑓 supp (0g𝑟)) ⊆ {𝑔 ∈ { ∈ (ℕ0𝑚 𝑖) ∣ ( “ ℕ) ∈ Fin} ∣ Σ𝑗 ∈ ℕ0 (𝑔𝑗) = 𝑛}}))
 
Definitiondf-psd 19523* Define the differentiation operation on multivariate polynomials. (Contributed by Mario Carneiro, 21-Mar-2015.)
mPSDer = (𝑖 ∈ V, 𝑟 ∈ V ↦ (𝑥𝑖 ↦ (𝑓 ∈ (Base‘(𝑖 mPwSer 𝑟)) ↦ (𝑘 ∈ { ∈ (ℕ0𝑚 𝑖) ∣ ( “ ℕ) ∈ Fin} ↦ (((𝑘𝑥) + 1)(.g𝑟)(𝑓‘(𝑘𝑓 + (𝑦𝑖 ↦ if(𝑦 = 𝑥, 1, 0)))))))))
 
Definitiondf-selv 19524* Define the "variable selection" function. The function ((𝐼 selectVars 𝑅)‘𝐽) maps elements of (𝐼 mPoly 𝑅) bijectively onto (𝐽 mPoly ((𝐼𝐽) mPoly 𝑅)) in the natural way, for example if 𝐼 = {𝑥, 𝑦} and 𝐽 = {𝑦} it would map 1 + 𝑥 + 𝑦 + 𝑥𝑦 ∈ ({𝑥, 𝑦} mPoly ℤ) to (1 + 𝑥) + (1 + 𝑥)𝑦 ∈ ({𝑦} mPoly ({𝑥} mPoly ℤ)). This, for example, allows one to treat a multivariate polynomial as a univariate polynomial with coefficients in a polynomial ring with one less variable. (Contributed by Mario Carneiro, 21-Mar-2015.)
selectVars = (𝑖 ∈ V, 𝑟 ∈ V ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑓 ∈ (𝑖 mPoly 𝑟) ↦ ((𝑖𝑗) mPoly 𝑟) / 𝑠(𝑥 ∈ (Scalar‘𝑠) ↦ (𝑥( ·𝑠𝑠)(1r𝑠))) / 𝑐((((𝑖 evalSub 𝑠)‘(𝑐s 𝑟))‘(𝑐𝑓))‘(𝑥𝑖 ↦ if(𝑥𝑗, ((𝑗 mVar ((𝑖𝑗) mPoly 𝑟))‘𝑥), (𝑐 ∘ (((𝑖𝑗) mVar 𝑟)‘𝑥))))))))
 
Definitiondf-algind 19525* Define the predicate "the set 𝑣 is algebraically independent in the algebra 𝑤". A collection of vectors is algebraically independent if no nontrivial polynomial with elements from the subset evaluates to zero. (Contributed by Mario Carneiro, 21-Mar-2015.)
AlgInd = (𝑤 ∈ V, 𝑘 ∈ 𝒫 (Base‘𝑤) ↦ {𝑣 ∈ 𝒫 (Base‘𝑤) ∣ Fun (𝑓 ∈ (Base‘(𝑣 mPoly (𝑤s 𝑘))) ↦ ((((𝑣 evalSub 𝑤)‘𝑘)‘𝑓)‘( I ↾ 𝑣)))})
 
10.10.4  Univariate polynomials

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 19555.

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). 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".

 
Syntaxcps1 19526 Univariate power series.
class PwSer1
 
Syntaxcv1 19527 The base variable of a univariate power series.
class var1
 
Syntaxcpl1 19528 Univariate polynomials.
class Poly1
 
Syntaxcco1 19529 Coefficient function for a univariate polynomial.
class coe1
 
Syntaxctp1 19530 Convert a univariate polynomial representation to multivariate.
class toPoly1
 
Definitiondf-psr1 19531 Define the algebra of univariate power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
PwSer1 = (𝑟 ∈ V ↦ ((1𝑜 ordPwSer 𝑟)‘∅))
 
Definitiondf-vr1 19532 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)‘∅))
 
Definitiondf-ply1 19533 Define the algebra of univariate polynomials. (Contributed by Mario Carneiro, 9-Feb-2015.)
Poly1 = (𝑟 ∈ V ↦ ((PwSer1𝑟) ↾s (Base‘(1𝑜 mPoly 𝑟))))
 
Definitiondf-coe1 19534* Define the coefficient function for a univariate polynomial. (Contributed by Stefan O'Rear, 21-Mar-2015.)
coe1 = (𝑓 ∈ V ↦ (𝑛 ∈ ℕ0 ↦ (𝑓‘(1𝑜 × {𝑛}))))
 
Definitiondf-toply1 19535* 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 ∈ (ℕ0𝑚 1𝑜) ↦ (𝑓‘(𝑛‘∅))))
 
Theorempsr1baslem 19536 The set of finite bags on 1𝑜 is just the set of all functions from 1𝑜 to 0. (Contributed by Mario Carneiro, 9-Feb-2015.)
(ℕ0𝑚 1𝑜) = {𝑓 ∈ (ℕ0𝑚 1𝑜) ∣ (𝑓 “ ℕ) ∈ Fin}
 
Theorempsr1val 19537 Value of the ring of univariate power series. (Contributed by Mario Carneiro, 8-Feb-2015.)
𝑆 = (PwSer1𝑅)       𝑆 = ((1𝑜 ordPwSer 𝑅)‘∅)
 
Theorempsr1crng 19538 The ring of univariate power series is a commutative ring. (Contributed by Mario Carneiro, 8-Feb-2015.)
𝑆 = (PwSer1𝑅)       (𝑅 ∈ CRing → 𝑆 ∈ CRing)
 
Theorempsr1assa 19539 The ring of univariate power series is an associative algebra. (Contributed by Mario Carneiro, 8-Feb-2015.)
𝑆 = (PwSer1𝑅)       (𝑅 ∈ CRing → 𝑆 ∈ AssAlg)
 
Theorempsr1tos 19540 The ordered power series structure is a totally ordered set. (Contributed by Mario Carneiro, 2-Jun-2015.)
𝑆 = (PwSer1𝑅)       (𝑅 ∈ Toset → 𝑆 ∈ Toset)
 
Theorempsr1bas2 19541 The base set of the ring of univariate power series. (Contributed by Mario Carneiro, 3-Jul-2015.)
𝑆 = (PwSer1𝑅)    &   𝐵 = (Base‘𝑆)    &   𝑂 = (1𝑜 mPwSer 𝑅)       𝐵 = (Base‘𝑂)
 
Theorempsr1bas 19542 The base set of the ring of univariate power series. (Contributed by Mario Carneiro, 8-Feb-2015.)
𝑆 = (PwSer1𝑅)    &   𝐵 = (Base‘𝑆)    &   𝐾 = (Base‘𝑅)       𝐵 = (𝐾𝑚 (ℕ0𝑚 1𝑜))
 
Theoremvr1val 19543 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 = (var1𝑅)       𝑋 = ((1𝑜 mVar 𝑅)‘∅)
 
Theoremvr1cl2 19544 The variable 𝑋 is a member of the power series algebra 𝑅[[𝑋]]. (Contributed by Mario Carneiro, 8-Feb-2015.)
𝑋 = (var1𝑅)    &   𝑆 = (PwSer1𝑅)    &   𝐵 = (Base‘𝑆)       (𝑅 ∈ Ring → 𝑋𝐵)
 
Theoremply1val 19545 The value of the set of univariate polynomials. (Contributed by Mario Carneiro, 9-Feb-2015.)
𝑃 = (Poly1𝑅)    &   𝑆 = (PwSer1𝑅)       𝑃 = (𝑆s (Base‘(1𝑜 mPoly 𝑅)))
 
Theoremply1bas 19546 The value of the base set of univariate polynomials. (Contributed by Mario Carneiro, 9-Feb-2015.)
𝑃 = (Poly1𝑅)    &   𝑆 = (PwSer1𝑅)    &   𝑈 = (Base‘𝑃)       𝑈 = (Base‘(1𝑜 mPoly 𝑅))
 
Theoremply1lss 19547 Univariate polynomials form a linear subspace of the set of univariate power series. (Contributed by Mario Carneiro, 9-Feb-2015.)
𝑃 = (Poly1𝑅)    &   𝑆 = (PwSer1𝑅)    &   𝑈 = (Base‘𝑃)       (𝑅 ∈ Ring → 𝑈 ∈ (LSubSp‘𝑆))
 
Theoremply1subrg 19548 Univariate polynomials form a subring of the set of univariate power series. (Contributed by Mario Carneiro, 9-Feb-2015.)
𝑃 = (Poly1𝑅)    &   𝑆 = (PwSer1𝑅)    &   𝑈 = (Base‘𝑃)       (𝑅 ∈ Ring → 𝑈 ∈ (SubRing‘𝑆))
 
Theoremply1crng 19549 The ring of univariate polynomials is a commutative ring. (Contributed by Mario Carneiro, 9-Feb-2015.)
𝑃 = (Poly1𝑅)       (𝑅 ∈ CRing → 𝑃 ∈ CRing)
 
Theoremply1assa 19550 The ring of univariate polynomials is an associative algebra. (Contributed by Mario Carneiro, 9-Feb-2015.)
𝑃 = (Poly1𝑅)       (𝑅 ∈ CRing → 𝑃 ∈ AssAlg)
 
Theorempsr1bascl 19551 A univariate power series is a multivariate power series on one index. (Contributed by Stefan O'Rear, 25-Mar-2015.)
𝑃 = (PwSer1𝑅)    &   𝐵 = (Base‘𝑃)       (𝐹𝐵𝐹 ∈ (Base‘(1𝑜 mPwSer 𝑅)))
 
Theorempsr1basf 19552 Univariate power series base set elements are functions. (Contributed by Stefan O'Rear, 25-Mar-2015.)
𝑃 = (PwSer1𝑅)    &   𝐵 = (Base‘𝑃)    &   𝐾 = (Base‘𝑅)       (𝐹𝐵𝐹:(ℕ0𝑚 1𝑜)⟶𝐾)
 
Theoremply1basf 19553 Univariate polynomial base set elements are functions. (Contributed by Stefan O'Rear, 21-Mar-2015.)
𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &   𝐾 = (Base‘𝑅)       (𝐹𝐵𝐹:(ℕ0𝑚 1𝑜)⟶𝐾)
 
Theoremply1bascl 19554 A univariate polynomial is a univariate power series. (Contributed by Stefan O'Rear, 25-Mar-2015.)
𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)       (𝐹𝐵𝐹 ∈ (Base‘(PwSer1𝑅)))
 
Theoremply1bascl2 19555 A univariate polynomial is a multivariate polynomial on one index. (Contributed by Stefan O'Rear, 25-Mar-2015.)
𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)       (𝐹𝐵𝐹 ∈ (Base‘(1𝑜 mPoly 𝑅)))
 
Theoremcoe1fval 19556* Value of the univariate polynomial coefficient function. (Contributed by Stefan O'Rear, 21-Mar-2015.)
𝐴 = (coe1𝐹)       (𝐹𝑉𝐴 = (𝑛 ∈ ℕ0 ↦ (𝐹‘(1𝑜 × {𝑛}))))
 
Theoremcoe1fv 19557 Value of an evaluated coefficient in a polynomial coefficient vector. (Contributed by Stefan O'Rear, 21-Mar-2015.)
𝐴 = (coe1𝐹)       ((𝐹𝑉𝑁 ∈ ℕ0) → (𝐴𝑁) = (𝐹‘(1𝑜 × {𝑁})))
 
Theoremfvcoe1 19558 Value of a multivariate coefficient in terms of the coefficient vector. (Contributed by Stefan O'Rear, 21-Mar-2015.)
𝐴 = (coe1𝐹)       ((𝐹𝑉𝑋 ∈ (ℕ0𝑚 1𝑜)) → (𝐹𝑋) = (𝐴‘(𝑋‘∅)))
 
Theoremcoe1fval3 19559* Univariate power series coefficient vectors expressed as a function composition. (Contributed by Stefan O'Rear, 25-Mar-2015.)
𝐴 = (coe1𝐹)    &   𝐵 = (Base‘𝑃)    &   𝑃 = (PwSer1𝑅)    &   𝐺 = (𝑦 ∈ ℕ0 ↦ (1𝑜 × {𝑦}))       (𝐹𝐵𝐴 = (𝐹𝐺))
 
Theoremcoe1f2 19560 Functionality of univariate power series coefficient vectors. (Contributed by Stefan O'Rear, 25-Mar-2015.)
𝐴 = (coe1𝐹)    &   𝐵 = (Base‘𝑃)    &   𝑃 = (PwSer1𝑅)    &   𝐾 = (Base‘𝑅)       (𝐹𝐵𝐴:ℕ0𝐾)
 
Theoremcoe1fval2 19561* Univariate polynomial coefficient vectors expressed as a function composition. (Contributed by Stefan O'Rear, 21-Mar-2015.)
𝐴 = (coe1𝐹)    &   𝐵 = (Base‘𝑃)    &   𝑃 = (Poly1𝑅)    &   𝐺 = (𝑦 ∈ ℕ0 ↦ (1𝑜 × {𝑦}))       (𝐹𝐵𝐴 = (𝐹𝐺))
 
Theoremcoe1f 19562 Functionality of univariate polynomial coefficient vectors. (Contributed by Stefan O'Rear, 21-Mar-2015.)
𝐴 = (coe1𝐹)    &   𝐵 = (Base‘𝑃)    &   𝑃 = (Poly1𝑅)    &   𝐾 = (Base‘𝑅)       (𝐹𝐵𝐴:ℕ0𝐾)
 
Theoremcoe1fvalcl 19563 A coefficient of a univariate polynomial over a class/ring is an element of this class/ring. (Contributed by AV, 9-Oct-2019.)
𝐴 = (coe1𝐹)    &   𝐵 = (Base‘𝑃)    &   𝑃 = (Poly1𝑅)    &   𝐾 = (Base‘𝑅)       ((𝐹𝐵𝑁 ∈ ℕ0) → (𝐴𝑁) ∈ 𝐾)
 
Theoremcoe1sfi 19564 Finite support of univariate polynomial coefficient vectors. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by AV, 19-Jul-2019.)
𝐴 = (coe1𝐹)    &   𝐵 = (Base‘𝑃)    &   𝑃 = (Poly1𝑅)    &    0 = (0g𝑅)       (𝐹𝐵𝐴 finSupp 0 )
 
Theoremcoe1fsupp 19565* 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 19566* 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 19567* 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 19568 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 19569 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 19570 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 19571 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 19572 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 19573 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 19574 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 19575 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 19576 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 19577 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 19578 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 19579 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 19580 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 19581 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 19582 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 19583 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 19584 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 19585 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 19586 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 19587* 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 19588* 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 19589 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 19590 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 19591 Lemma for ply1basfvi 19592 and deg1fvi 23826. (Contributed by Stefan O'Rear, 28-Mar-2015.)
∅ = (Base‘(Poly1‘∅))
 
Theoremply1basfvi 19592 Protection compatibility of the univariate polynomial base set. (Contributed by Stefan O'Rear, 27-Mar-2015.)
(Base‘(Poly1𝑅)) = (Base‘(Poly1‘( I ‘𝑅)))
 
Theoremply1plusgfvi 19593 Protection compatibility of the univariate polynomial addition. (Contributed by Stefan O'Rear, 27-Mar-2015.)
(+g‘(Poly1𝑅)) = (+g‘(Poly1‘( I ‘𝑅)))
 
Theoremply1baspropd 19594* 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 19595* 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 19596 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 19597 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 19598 Univariate power series form a ring. (Contributed by Stefan O'Rear, 22-Mar-2015.)
𝑆 = (PwSer1𝑅)       (𝑅 ∈ Ring → 𝑆 ∈ Ring)
 
Theoremply1ring 19599 Univariate polynomials form a ring. (Contributed by Stefan O'Rear, 22-Mar-2015.)
𝑃 = (Poly1𝑅)       (𝑅 ∈ Ring → 𝑃 ∈ Ring)
 
Theorempsr1lmod 19600 Univariate power series form a left module. (Contributed by Stefan O'Rear, 26-Mar-2015.)
𝑃 = (PwSer1𝑅)       (𝑅 ∈ Ring → 𝑃 ∈ LMod)
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