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Type | Label | Description |
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Statement | ||
Definition | df-dgr 23201 | Define the degree of a polynomial. (Contributed by Mario Carneiro, 22-Jul-2014.) |
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Theorem | plyco0 23202* | Two ways to say that a function on the nonnegative integers has finite support. (Contributed by Mario Carneiro, 22-Jul-2014.) |
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Theorem | plyval 23203* | Value of the polynomial set function. (Contributed by Mario Carneiro, 17-Jul-2014.) |
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Theorem | plybss 23204 |
Reverse closure of the parameter ![]() |
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Theorem | elply 23205* |
Definition of a polynomial with coefficients in ![]() |
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Theorem | elply2 23206* |
The coefficient function can be assumed to have zeroes outside
![]() ![]() ![]() |
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Theorem | plyun0 23207 | The set of polynomials is unaffected by the addition of zero. (This is built into the definition because all higher powers of a polynomial are effectively zero, so we require that the coefficient field contain zero to simplify some of our closure theorems.) (Contributed by Mario Carneiro, 17-Jul-2014.) |
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Theorem | plyf 23208 | The polynomial is a function on the complex numbers. (Contributed by Mario Carneiro, 22-Jul-2014.) |
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Theorem | plyss 23209 | The polynomial set function preserves the subset relation. (Contributed by Mario Carneiro, 17-Jul-2014.) |
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Theorem | plyssc 23210 |
Every polynomial ring is contained in the ring of polynomials over
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Theorem | elplyr 23211* | Sufficient condition for elementhood in the set of polynomials. (Contributed by Mario Carneiro, 17-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.) |
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Theorem | elplyd 23212* | Sufficient condition for elementhood in the set of polynomials. (Contributed by Mario Carneiro, 17-Jul-2014.) |
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Theorem | ply1termlem 23213* | Lemma for ply1term 23214. (Contributed by Mario Carneiro, 26-Jul-2014.) |
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Theorem | ply1term 23214* | A one-term polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.) |
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Theorem | plypow 23215* | A power is a polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.) |
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Theorem | plyconst 23216 | A constant function is a polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.) |
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Theorem | ne0p 23217 | A test to show that a polynomial is nonzero. (Contributed by Mario Carneiro, 23-Jul-2014.) |
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Theorem | ply0 23218 | The zero function is a polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.) |
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Theorem | plyid 23219 | The identity function is a polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.) |
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Theorem | plyeq0lem 23220* |
Lemma for plyeq0 23221. If ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | plyeq0 23221* | If a polynomial is zero at every point (or even just zero at the positive integers), then all the coefficients must be zero. This is the basis for the method of equating coefficients of equal polynomials, and ensures that df-coe 23200 is well-defined. (Contributed by Mario Carneiro, 22-Jul-2014.) |
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Theorem | plypf1 23222 | Write the set of complex polynomials in a subring in terms of the abstract polynomial construction. (Contributed by Mario Carneiro, 3-Jul-2015.) (Proof shortened by AV, 29-Sep-2019.) |
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Theorem | plyaddlem1 23223* | Derive the coefficient function for the sum of two polynomials. (Contributed by Mario Carneiro, 23-Jul-2014.) |
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Theorem | plymullem1 23224* | Derive the coefficient function for the product of two polynomials. (Contributed by Mario Carneiro, 23-Jul-2014.) |
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Theorem | plyaddlem 23225* | Lemma for plyadd 23227. (Contributed by Mario Carneiro, 21-Jul-2014.) |
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Theorem | plymullem 23226* | Lemma for plymul 23228. (Contributed by Mario Carneiro, 21-Jul-2014.) |
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Theorem | plyadd 23227* | The sum of two polynomials is a polynomial. (Contributed by Mario Carneiro, 21-Jul-2014.) |
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Theorem | plymul 23228* | The product of two polynomials is a polynomial. (Contributed by Mario Carneiro, 21-Jul-2014.) |
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Theorem | plysub 23229* | The difference of two polynomials is a polynomial. (Contributed by Mario Carneiro, 21-Jul-2014.) |
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Theorem | plyaddcl 23230 | The sum of two polynomials is a polynomial. (Contributed by Mario Carneiro, 24-Jul-2014.) |
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Theorem | plymulcl 23231 | The product of two polynomials is a polynomial. (Contributed by Mario Carneiro, 24-Jul-2014.) |
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Theorem | plysubcl 23232 | The difference of two polynomials is a polynomial. (Contributed by Mario Carneiro, 24-Jul-2014.) |
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Theorem | coeval 23233* | Value of the coefficient function. (Contributed by Mario Carneiro, 22-Jul-2014.) |
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Theorem | coeeulem 23234* | Lemma for coeeu 23235. (Contributed by Mario Carneiro, 22-Jul-2014.) |
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Theorem | coeeu 23235* | Uniqueness of the coefficient function. (Contributed by Mario Carneiro, 22-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.) |
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Theorem | coelem 23236* | Lemma for properties of the coefficient function. (Contributed by Mario Carneiro, 22-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.) |
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Theorem | coeeq 23237* |
If ![]() |
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Theorem | dgrval 23238 | Value of the degree function. (Contributed by Mario Carneiro, 22-Jul-2014.) |
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Theorem | dgrlem 23239* | Lemma for dgrcl 23243 and similar theorems. (Contributed by Mario Carneiro, 22-Jul-2014.) |
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Theorem | coef 23240 | The domain and range of the coefficient function. (Contributed by Mario Carneiro, 22-Jul-2014.) |
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Theorem | coef2 23241 | The domain and range of the coefficient function. (Contributed by Mario Carneiro, 22-Jul-2014.) |
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Theorem | coef3 23242 | The domain and range of the coefficient function. (Contributed by Mario Carneiro, 22-Jul-2014.) |
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Theorem | dgrcl 23243 | The degree of any polynomial is a nonnegative integer. (Contributed by Mario Carneiro, 22-Jul-2014.) |
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Theorem | dgrub 23244 |
If the ![]() ![]() ![]() ![]() |
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Theorem | dgrub2 23245 |
All the coefficients above the degree of ![]() |
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Theorem | dgrlb 23246 |
If all the coefficients above ![]() ![]() ![]() |
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Theorem | coeidlem 23247* | Lemma for coeid 23248. (Contributed by Mario Carneiro, 22-Jul-2014.) |
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Theorem | coeid 23248* | Reconstruct a polynomial as an explicit sum of the coefficient function up to the degree of the polynomial. (Contributed by Mario Carneiro, 22-Jul-2014.) |
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Theorem | coeid2 23249* | Reconstruct a polynomial as an explicit sum of the coefficient function up to the degree of the polynomial. (Contributed by Mario Carneiro, 22-Jul-2014.) |
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Theorem | coeid3 23250* | Reconstruct a polynomial as an explicit sum of the coefficient function up to at least the degree of the polynomial. (Contributed by Mario Carneiro, 22-Jul-2014.) |
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Theorem | plyco 23251* | The composition of two polynomials is a polynomial. (Contributed by Mario Carneiro, 23-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.) |
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Theorem | coeeq2 23252* | Compute the coefficient function given a sum expression for the polynomial. (Contributed by Mario Carneiro, 24-Jul-2014.) |
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Theorem | dgrle 23253* | Given an explicit expression for a polynomial, the degree is at most the highest term in the sum. (Contributed by Mario Carneiro, 24-Jul-2014.) |
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Theorem | dgreq 23254* | If the highest term in a polynomial expression is nonzero, then the polynomial's degree is completely determined. (Contributed by Mario Carneiro, 24-Jul-2014.) |
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Theorem | 0dgr 23255 | A constant function has degree 0. (Contributed by Mario Carneiro, 24-Jul-2014.) |
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Theorem | 0dgrb 23256 | A function has degree zero iff it is a constant function. (Contributed by Mario Carneiro, 23-Jul-2014.) |
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Theorem | dgrnznn 23257 | A nonzero polynomial with a root has positive degree. (Contributed by Stefan O'Rear, 25-Nov-2014.) |
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Theorem | coefv0 23258 | The result of evaluating a polynomial at zero is the constant term. (Contributed by Mario Carneiro, 24-Jul-2014.) |
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Theorem | coeaddlem 23259 | Lemma for coeadd 23261 and dgradd 23277. (Contributed by Mario Carneiro, 24-Jul-2014.) |
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Theorem | coemullem 23260* | Lemma for coemul 23262 and dgrmul 23280. (Contributed by Mario Carneiro, 24-Jul-2014.) |
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Theorem | coeadd 23261 | The coefficient function of a sum is the sum of coefficients. (Contributed by Mario Carneiro, 24-Jul-2014.) |
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Theorem | coemul 23262* | A coefficient of a product of polynomials. (Contributed by Mario Carneiro, 24-Jul-2014.) |
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Theorem | coe11 23263 | The coefficient function is one-to-one, so if the coefficients are equal then the functions are equal and vice-versa. (Contributed by Mario Carneiro, 24-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.) |
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Theorem | coemulhi 23264 | The leading coefficient of a product of polynomials. (Contributed by Mario Carneiro, 24-Jul-2014.) |
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Theorem | coemulc 23265 | The coefficient function is linear under scalar multiplication. (Contributed by Mario Carneiro, 24-Jul-2014.) |
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Theorem | coe0 23266 | The coefficients of the zero polynomial are zero. (Contributed by Mario Carneiro, 22-Jul-2014.) |
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Theorem | coesub 23267 | The coefficient function of a sum is the sum of coefficients. (Contributed by Mario Carneiro, 24-Jul-2014.) |
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Theorem | coe1termlem 23268* | The coefficient function of a monomial. (Contributed by Mario Carneiro, 26-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.) |
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Theorem | coe1term 23269* | The coefficient function of a monomial. (Contributed by Mario Carneiro, 26-Jul-2014.) |
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Theorem | dgr1term 23270* | The degree of a monomial. (Contributed by Mario Carneiro, 26-Jul-2014.) |
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Theorem | plycn 23271 | A polynomial is a continuous function. (Contributed by Mario Carneiro, 23-Jul-2014.) |
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Theorem | dgr0 23272 |
The degree of the zero polynomial is zero. Note: this differs from some
other definitions of the degree of the zero polynomial, such as
![]() ![]() ![]() ![]() |
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Theorem | coeidp 23273 | The coefficients of the identity function. (Contributed by Mario Carneiro, 28-Jul-2014.) |
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Theorem | dgrid 23274 | The degree of the identity function. (Contributed by Mario Carneiro, 26-Jul-2014.) |
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Theorem | dgreq0 23275 | The leading coefficient of a polynomial is nonzero, unless the entire polynomial is zero. (Contributed by Mario Carneiro, 22-Jul-2014.) (Proof shortened by Fan Zheng, 21-Jun-2016.) |
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Theorem | dgrlt 23276 |
Two ways to say that the degree of ![]() ![]() |
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Theorem | dgradd 23277 | The degree of a sum of polynomials is at most the maximum of the degrees. (Contributed by Mario Carneiro, 24-Jul-2014.) |
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Theorem | dgradd2 23278 | The degree of a sum of polynomials of unequal degrees is the degree of the larger polynomial. (Contributed by Mario Carneiro, 24-Jul-2014.) |
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Theorem | dgrmul2 23279 | The degree of a product of polynomials is at most the sum of degrees. (Contributed by Mario Carneiro, 24-Jul-2014.) |
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Theorem | dgrmul 23280 | The degree of a product of nonzero polynomials is the sum of degrees. (Contributed by Mario Carneiro, 24-Jul-2014.) |
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Theorem | dgrmulc 23281 | Scalar multiplication by a nonzero constant does not change the degree of a function. (Contributed by Mario Carneiro, 24-Jul-2014.) |
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Theorem | dgrsub 23282 | The degree of a difference of polynomials is at most the maximum of the degrees. (Contributed by Mario Carneiro, 26-Jul-2014.) |
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Theorem | dgrcolem1 23283* | The degree of a composition of a monomial with a polynomial. (Contributed by Mario Carneiro, 15-Sep-2014.) |
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Theorem | dgrcolem2 23284* | Lemma for dgrco 23285. (Contributed by Mario Carneiro, 15-Sep-2014.) |
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Theorem | dgrco 23285 | The degree of a composition of two polynomials is the product of the degrees. (Contributed by Mario Carneiro, 15-Sep-2014.) |
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Theorem | plycjlem 23286* | Lemma for plycj 23287 and coecj 23288. (Contributed by Mario Carneiro, 24-Jul-2014.) |
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Theorem | plycj 23287* |
The double conjugation of a polynomial is a polynomial. (The single
conjugation is not because our definition of polynomial includes only
holomorphic functions, i.e. no dependence on ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | coecj 23288 | Double conjugation of a polynomial causes the coefficients to be conjugated. (Contributed by Mario Carneiro, 24-Jul-2014.) |
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Theorem | plyrecj 23289 | A polynomial with real coefficients distributes under conjugation. (Contributed by Mario Carneiro, 24-Jul-2014.) |
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Theorem | plymul0or 23290 | Polynomial multiplication has no zero divisors. (Contributed by Mario Carneiro, 26-Jul-2014.) |
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Theorem | ofmulrt 23291 | The set of roots of a product is the union of the roots of the terms. (Contributed by Mario Carneiro, 28-Jul-2014.) |
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Theorem | plyreres 23292 | Real-coefficient polynomials restrict to real functions. (Contributed by Stefan O'Rear, 16-Nov-2014.) |
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Theorem | dvply1 23293* | Derivative of a polynomial, explicit sum version. (Contributed by Stefan O'Rear, 13-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.) |
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Theorem | dvply2g 23294 | The derivative of a polynomial with coefficients in a subring is a polynomial with coefficients in the same ring. (Contributed by Mario Carneiro, 1-Jan-2017.) |
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Theorem | dvply2 23295 | The derivative of a polynomial is a polynomial. (Contributed by Stefan O'Rear, 14-Nov-2014.) (Proof shortened by Mario Carneiro, 1-Jan-2017.) |
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Theorem | dvnply2 23296 | Polynomials have polynomials as derivatives of all orders. (Contributed by Mario Carneiro, 1-Jan-2017.) |
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Theorem | dvnply 23297 | Polynomials have polynomials as derivatives of all orders. (Contributed by Stefan O'Rear, 15-Nov-2014.) (Revised by Mario Carneiro, 1-Jan-2017.) |
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Theorem | plycpn 23298 | Polynomials are smooth. (Contributed by Stefan O'Rear, 16-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.) |
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Syntax | cquot 23299 | Extend class notation to include the quotient of a polynomial division. |
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Definition | df-quot 23300* |
Define the quotient function on polynomials. This is the ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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