P. 233 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 23201-23300   *Has distinct variable group(s)

Type	Label	Description
Statement
Definition	df-dgr 23201	Define the degree of a polynomial. (Contributed by Mario Carneiro, 22-Jul-2014.)
|- deg = ( f e. (Poly ` CC ) |-> sup ( ( `' (coeff ` f ) " ( CC \ { 0 } ) ) , NN0 , < ) )
Theorem	plyco0 23202*	Two ways to say that a function on the nonnegative integers has finite support. (Contributed by Mario Carneiro, 22-Jul-2014.)
|- ( ( N e. NN0 /\ A : NN0 --> CC ) -> ( ( A " ( ZZ>= ` ( N + 1 ) ) ) = { 0 } <-> A. k e. NN0 ( ( A ` k ) =/= 0 -> k <_ N ) ) )
Theorem	plyval 23203*	Value of the polynomial set function. (Contributed by Mario Carneiro, 17-Jul-2014.)
|- ( S C_ CC -> (Poly ` S ) = { f | E. n e. NN0 E. a e. ( ( S u. { 0 } ) ^m NN0 ) f = ( z e. CC |-> sum_ k e. ( 0 ... n ) ( ( a ` k ) x. ( z ^ k ) ) ) } )
Theorem	plybss 23204	Reverse closure of the parameter S of the polynomial set function. (Contributed by Mario Carneiro, 22-Jul-2014.)
|- ( F e. (Poly ` S ) -> S C_ CC )
Theorem	elply 23205*	Definition of a polynomial with coefficients in S. (Contributed by Mario Carneiro, 17-Jul-2014.)
|- ( F e. (Poly ` S ) <-> ( S C_ CC /\ E. n e. NN0 E. a e. ( ( S u. { 0 } ) ^m NN0 ) F = ( z e. CC |-> sum_ k e. ( 0 ... n ) ( ( a ` k ) x. ( z ^ k ) ) ) ) )
Theorem	elply2 23206*	The coefficient function can be assumed to have zeroes outside 0 ... n. (Contributed by Mario Carneiro, 20-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
|- ( F e. (Poly ` S ) <-> ( S C_ CC /\ E. n e. NN0 E. a e. ( ( S u. { 0 } ) ^m NN0 ) ( ( a " ( ZZ>= ` ( n + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... n ) ( ( a ` k ) x. ( z ^ k ) ) ) ) ) )
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.)
|- (Poly ` ( S u. { 0 } ) ) = (Poly ` S )
Theorem	plyf 23208	The polynomial is a function on the complex numbers. (Contributed by Mario Carneiro, 22-Jul-2014.)
|- ( F e. (Poly ` S ) -> F : CC --> CC )
Theorem	plyss 23209	The polynomial set function preserves the subset relation. (Contributed by Mario Carneiro, 17-Jul-2014.)
|- ( ( S C_ T /\ T C_ CC ) -> (Poly ` S ) C_ (Poly ` T ) )
Theorem	plyssc 23210	Every polynomial ring is contained in the ring of polynomials over CC. (Contributed by Mario Carneiro, 22-Jul-2014.)
|- (Poly ` S ) C_ (Poly ` CC )
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.)
|- ( ( S C_ CC /\ N e. NN0 /\ A : NN0 --> S ) -> ( z e. CC |-> sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( z ^ k ) ) ) e. (Poly ` S ) )
Theorem	elplyd 23212*	Sufficient condition for elementhood in the set of polynomials. (Contributed by Mario Carneiro, 17-Jul-2014.)
|- ( ph -> S C_ CC )   &    |- ( ph -> N e. NN0 )   &    |- ( ( ph /\ k e. ( 0 ... N ) ) -> A e. S )   =>    |- ( ph -> ( z e. CC |-> sum_ k e. ( 0 ... N ) ( A x. ( z ^ k ) ) ) e. (Poly ` S ) )
Theorem	ply1termlem 23213*	Lemma for ply1term 23214. (Contributed by Mario Carneiro, 26-Jul-2014.)
|- F = ( z e. CC |-> ( A x. ( z ^ N ) ) )   =>    |- ( ( A e. CC /\ N e. NN0 ) -> F = ( z e. CC |-> sum_ k e. ( 0 ... N ) ( if ( k = N , A , 0 ) x. ( z ^ k ) ) ) )
Theorem	ply1term 23214*	A one-term polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)
|- F = ( z e. CC |-> ( A x. ( z ^ N ) ) )   =>    |- ( ( S C_ CC /\ A e. S /\ N e. NN0 ) -> F e. (Poly ` S ) )
Theorem	plypow 23215*	A power is a polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)
|- ( ( S C_ CC /\ 1 e. S /\ N e. NN0 ) -> ( z e. CC |-> ( z ^ N ) ) e. (Poly ` S ) )
Theorem	plyconst 23216	A constant function is a polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)
|- ( ( S C_ CC /\ A e. S ) -> ( CC X. { A } ) e. (Poly ` S ) )
Theorem	ne0p 23217	A test to show that a polynomial is nonzero. (Contributed by Mario Carneiro, 23-Jul-2014.)
|- ( ( A e. CC /\ ( F ` A ) =/= 0 ) -> F =/= 0p )
Theorem	ply0 23218	The zero function is a polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)
|- ( S C_ CC -> 0p e. (Poly ` S ) )
Theorem	plyid 23219	The identity function is a polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)
|- ( ( S C_ CC /\ 1 e. S ) -> Xp e. (Poly ` S ) )
Theorem	plyeq0lem 23220*	Lemma for plyeq0 23221. If A is the coefficient function for a nonzero polynomial such that P ( z ) = sum_ k e. NN0 A ( k ) x. z ^ k = 0 for every z e. CC and A ( M ) is the nonzero leading coefficient, then the function F ( z ) = P ( z ) / z ^ M is a sum of powers of 1 / z, and so the limit of this function as z ~~> +oo is the constant term, A ( M ). But F ( z ) = 0 everywhere, so this limit is also equal to zero so that A ( M ) = 0, a contradiction. (Contributed by Mario Carneiro, 22-Jul-2014.)
|- ( ph -> S C_ CC )   &    |- ( ph -> N e. NN0 )   &    |- ( ph -> A e. ( ( S u. { 0 } ) ^m NN0 ) )   &    |- ( ph -> ( A " ( ZZ>= ` ( N + 1 ) ) ) = { 0 } )   &    |- ( ph -> 0p = ( z e. CC |-> sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( z ^ k ) ) ) )   &    |- M = sup ( ( `' A " ( S \ { 0 } ) ) , RR , < )   &    |- ( ph -> ( `' A " ( S \ { 0 } ) ) =/= (/) )   =>    |- -. ph
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.)
|- ( ph -> S C_ CC )   &    |- ( ph -> N e. NN0 )   &    |- ( ph -> A e. ( ( S u. { 0 } ) ^m NN0 ) )   &    |- ( ph -> ( A " ( ZZ>= ` ( N + 1 ) ) ) = { 0 } )   &    |- ( ph -> 0p = ( z e. CC |-> sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( z ^ k ) ) ) )   =>    |- ( ph -> A = ( NN0 X. { 0 } ) )
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.)
|- R = (ℂfld ↾s S )   &    |- P = (Poly1 ` R )   &    |- A = ( Base ` P )   &    |- E = (eval1 ` ℂfld )   =>    |- ( S e. (SubRing ` ℂfld ) -> (Poly ` S ) = ( E " A ) )
Theorem	plyaddlem1 23223*	Derive the coefficient function for the sum of two polynomials. (Contributed by Mario Carneiro, 23-Jul-2014.)
|- ( ph -> F e. (Poly ` S ) )   &    |- ( ph -> G e. (Poly ` S ) )   &    |- ( ph -> M e. NN0 )   &    |- ( ph -> N e. NN0 )   &    |- ( ph -> A : NN0 --> CC )   &    |- ( ph -> B : NN0 --> CC )   &    |- ( ph -> ( A " ( ZZ>= ` ( M + 1 ) ) ) = { 0 } )   &    |- ( ph -> ( B " ( ZZ>= ` ( N + 1 ) ) ) = { 0 } )   &    |- ( ph -> F = ( z e. CC |-> sum_ k e. ( 0 ... M ) ( ( A ` k ) x. ( z ^ k ) ) ) )   &    |- ( ph -> G = ( z e. CC |-> sum_ k e. ( 0 ... N ) ( ( B ` k ) x. ( z ^ k ) ) ) )   =>    |- ( ph -> ( F oF + G ) = ( z e. CC |-> sum_ k e. ( 0 ... if ( M <_ N , N , M ) ) ( ( ( A oF + B ) ` k ) x. ( z ^ k ) ) ) )
Theorem	plymullem1 23224*	Derive the coefficient function for the product of two polynomials. (Contributed by Mario Carneiro, 23-Jul-2014.)
|- ( ph -> F e. (Poly ` S ) )   &    |- ( ph -> G e. (Poly ` S ) )   &    |- ( ph -> M e. NN0 )   &    |- ( ph -> N e. NN0 )   &    |- ( ph -> A : NN0 --> CC )   &    |- ( ph -> B : NN0 --> CC )   &    |- ( ph -> ( A " ( ZZ>= ` ( M + 1 ) ) ) = { 0 } )   &    |- ( ph -> ( B " ( ZZ>= ` ( N + 1 ) ) ) = { 0 } )   &    |- ( ph -> F = ( z e. CC |-> sum_ k e. ( 0 ... M ) ( ( A ` k ) x. ( z ^ k ) ) ) )   &    |- ( ph -> G = ( z e. CC |-> sum_ k e. ( 0 ... N ) ( ( B ` k ) x. ( z ^ k ) ) ) )   =>    |- ( ph -> ( F oF x. G ) = ( z e. CC |-> sum_ n e. ( 0 ... ( M + N ) ) ( sum_ k e. ( 0 ... n ) ( ( A ` k ) x. ( B ` ( n - k ) ) ) x. ( z ^ n ) ) ) )
Theorem	plyaddlem 23225*	Lemma for plyadd 23227. (Contributed by Mario Carneiro, 21-Jul-2014.)
|- ( ph -> F e. (Poly ` S ) )   &    |- ( ph -> G e. (Poly ` S ) )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x + y ) e. S )   &    |- ( ph -> M e. NN0 )   &    |- ( ph -> N e. NN0 )   &    |- ( ph -> A e. ( ( S u. { 0 } ) ^m NN0 ) )   &    |- ( ph -> B e. ( ( S u. { 0 } ) ^m NN0 ) )   &    |- ( ph -> ( A " ( ZZ>= ` ( M + 1 ) ) ) = { 0 } )   &    |- ( ph -> ( B " ( ZZ>= ` ( N + 1 ) ) ) = { 0 } )   &    |- ( ph -> F = ( z e. CC |-> sum_ k e. ( 0 ... M ) ( ( A ` k ) x. ( z ^ k ) ) ) )   &    |- ( ph -> G = ( z e. CC |-> sum_ k e. ( 0 ... N ) ( ( B ` k ) x. ( z ^ k ) ) ) )   =>    |- ( ph -> ( F oF + G ) e. (Poly ` S ) )
Theorem	plymullem 23226*	Lemma for plymul 23228. (Contributed by Mario Carneiro, 21-Jul-2014.)
|- ( ph -> F e. (Poly ` S ) )   &    |- ( ph -> G e. (Poly ` S ) )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x + y ) e. S )   &    |- ( ph -> M e. NN0 )   &    |- ( ph -> N e. NN0 )   &    |- ( ph -> A e. ( ( S u. { 0 } ) ^m NN0 ) )   &    |- ( ph -> B e. ( ( S u. { 0 } ) ^m NN0 ) )   &    |- ( ph -> ( A " ( ZZ>= ` ( M + 1 ) ) ) = { 0 } )   &    |- ( ph -> ( B " ( ZZ>= ` ( N + 1 ) ) ) = { 0 } )   &    |- ( ph -> F = ( z e. CC |-> sum_ k e. ( 0 ... M ) ( ( A ` k ) x. ( z ^ k ) ) ) )   &    |- ( ph -> G = ( z e. CC |-> sum_ k e. ( 0 ... N ) ( ( B ` k ) x. ( z ^ k ) ) ) )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x x. y ) e. S )   =>    |- ( ph -> ( F oF x. G ) e. (Poly ` S ) )
Theorem	plyadd 23227*	The sum of two polynomials is a polynomial. (Contributed by Mario Carneiro, 21-Jul-2014.)
|- ( ph -> F e. (Poly ` S ) )   &    |- ( ph -> G e. (Poly ` S ) )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x + y ) e. S )   =>    |- ( ph -> ( F oF + G ) e. (Poly ` S ) )
Theorem	plymul 23228*	The product of two polynomials is a polynomial. (Contributed by Mario Carneiro, 21-Jul-2014.)
|- ( ph -> F e. (Poly ` S ) )   &    |- ( ph -> G e. (Poly ` S ) )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x + y ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x x. y ) e. S )   =>    |- ( ph -> ( F oF x. G ) e. (Poly ` S ) )
Theorem	plysub 23229*	The difference of two polynomials is a polynomial. (Contributed by Mario Carneiro, 21-Jul-2014.)
|- ( ph -> F e. (Poly ` S ) )   &    |- ( ph -> G e. (Poly ` S ) )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x + y ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x x. y ) e. S )   &    |- ( ph -> -u 1 e. S )   =>    |- ( ph -> ( F oF - G ) e. (Poly ` S ) )
Theorem	plyaddcl 23230	The sum of two polynomials is a polynomial. (Contributed by Mario Carneiro, 24-Jul-2014.)
|- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> ( F oF + G ) e. (Poly ` CC ) )
Theorem	plymulcl 23231	The product of two polynomials is a polynomial. (Contributed by Mario Carneiro, 24-Jul-2014.)
|- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> ( F oF x. G ) e. (Poly ` CC ) )
Theorem	plysubcl 23232	The difference of two polynomials is a polynomial. (Contributed by Mario Carneiro, 24-Jul-2014.)
|- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> ( F oF - G ) e. (Poly ` CC ) )
Theorem	coeval 23233*	Value of the coefficient function. (Contributed by Mario Carneiro, 22-Jul-2014.)
|- ( F e. (Poly ` S ) -> (coeff ` F ) = ( iota_ a e. ( CC ^m NN0 ) E. n e. NN0 ( ( a " ( ZZ>= ` ( n + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... n ) ( ( a ` k ) x. ( z ^ k ) ) ) ) ) )
Theorem	coeeulem 23234*	Lemma for coeeu 23235. (Contributed by Mario Carneiro, 22-Jul-2014.)
|- ( ph -> F e. (Poly ` S ) )   &    |- ( ph -> A e. ( CC ^m NN0 ) )   &    |- ( ph -> B e. ( CC ^m NN0 ) )   &    |- ( ph -> M e. NN0 )   &    |- ( ph -> N e. NN0 )   &    |- ( ph -> ( A " ( ZZ>= ` ( M + 1 ) ) ) = { 0 } )   &    |- ( ph -> ( B " ( ZZ>= ` ( N + 1 ) ) ) = { 0 } )   &    |- ( ph -> F = ( z e. CC |-> sum_ k e. ( 0 ... M ) ( ( A ` k ) x. ( z ^ k ) ) ) )   &    |- ( ph -> F = ( z e. CC |-> sum_ k e. ( 0 ... N ) ( ( B ` k ) x. ( z ^ k ) ) ) )   =>    |- ( ph -> A = B )
Theorem	coeeu 23235*	Uniqueness of the coefficient function. (Contributed by Mario Carneiro, 22-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
|- ( F e. (Poly ` S ) -> E! a e. ( CC ^m NN0 ) E. n e. NN0 ( ( a " ( ZZ>= ` ( n + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... n ) ( ( a ` k ) x. ( z ^ k ) ) ) ) )
Theorem	coelem 23236*	Lemma for properties of the coefficient function. (Contributed by Mario Carneiro, 22-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
|- ( F e. (Poly ` S ) -> ( (coeff ` F ) e. ( CC ^m NN0 ) /\ E. n e. NN0 ( ( (coeff ` F ) " ( ZZ>= ` ( n + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... n ) ( ( (coeff ` F ) ` k ) x. ( z ^ k ) ) ) ) ) )
Theorem	coeeq 23237*	If A satisfies the properties of the coefficient function, it must be equal to the coefficient function. (Contributed by Mario Carneiro, 22-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
|- ( ph -> F e. (Poly ` S ) )   &    |- ( ph -> N e. NN0 )   &    |- ( ph -> A : NN0 --> CC )   &    |- ( ph -> ( A " ( ZZ>= ` ( N + 1 ) ) ) = { 0 } )   &    |- ( ph -> F = ( z e. CC |-> sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( z ^ k ) ) ) )   =>    |- ( ph -> (coeff ` F ) = A )
Theorem	dgrval 23238	Value of the degree function. (Contributed by Mario Carneiro, 22-Jul-2014.)
|- A = (coeff ` F )   =>    |- ( F e. (Poly ` S ) -> (deg ` F ) = sup ( ( `' A " ( CC \ { 0 } ) ) , NN0 , < ) )
Theorem	dgrlem 23239*	Lemma for dgrcl 23243 and similar theorems. (Contributed by Mario Carneiro, 22-Jul-2014.)
|- A = (coeff ` F )   =>    |- ( F e. (Poly ` S ) -> ( A : NN0 --> ( S u. { 0 } ) /\ E. n e. ZZ A. x e. ( `' A " ( CC \ { 0 } ) ) x <_ n ) )
Theorem	coef 23240	The domain and range of the coefficient function. (Contributed by Mario Carneiro, 22-Jul-2014.)
|- A = (coeff ` F )   =>    |- ( F e. (Poly ` S ) -> A : NN0 --> ( S u. { 0 } ) )
Theorem	coef2 23241	The domain and range of the coefficient function. (Contributed by Mario Carneiro, 22-Jul-2014.)
|- A = (coeff ` F )   =>    |- ( ( F e. (Poly ` S ) /\ 0 e. S ) -> A : NN0 --> S )
Theorem	coef3 23242	The domain and range of the coefficient function. (Contributed by Mario Carneiro, 22-Jul-2014.)
|- A = (coeff ` F )   =>    |- ( F e. (Poly ` S ) -> A : NN0 --> CC )
Theorem	dgrcl 23243	The degree of any polynomial is a nonnegative integer. (Contributed by Mario Carneiro, 22-Jul-2014.)
|- ( F e. (Poly ` S ) -> (deg ` F ) e. NN0 )
Theorem	dgrub 23244	If the M-th coefficient of F is nonzero, then the degree of F is at least M. (Contributed by Mario Carneiro, 22-Jul-2014.)
|- A = (coeff ` F )   &    |- N = (deg ` F )   =>    |- ( ( F e. (Poly ` S ) /\ M e. NN0 /\ ( A ` M ) =/= 0 ) -> M <_ N )
Theorem	dgrub2 23245	All the coefficients above the degree of F are zero. (Contributed by Mario Carneiro, 23-Jul-2014.)
|- A = (coeff ` F )   &    |- N = (deg ` F )   =>    |- ( F e. (Poly ` S ) -> ( A " ( ZZ>= ` ( N + 1 ) ) ) = { 0 } )
Theorem	dgrlb 23246	If all the coefficients above M are zero, then the degree of F is at most M. (Contributed by Mario Carneiro, 22-Jul-2014.)
|- A = (coeff ` F )   &    |- N = (deg ` F )   =>    |- ( ( F e. (Poly ` S ) /\ M e. NN0 /\ ( A " ( ZZ>= ` ( M + 1 ) ) ) = { 0 } ) -> N <_ M )
Theorem	coeidlem 23247*	Lemma for coeid 23248. (Contributed by Mario Carneiro, 22-Jul-2014.)
|- A = (coeff ` F )   &    |- N = (deg ` F )   &    |- ( ph -> F e. (Poly ` S ) )   &    |- ( ph -> M e. NN0 )   &    |- ( ph -> B e. ( ( S u. { 0 } ) ^m NN0 ) )   &    |- ( ph -> ( B " ( ZZ>= ` ( M + 1 ) ) ) = { 0 } )   &    |- ( ph -> F = ( z e. CC |-> sum_ k e. ( 0 ... M ) ( ( B ` k ) x. ( z ^ k ) ) ) )   =>    |- ( ph -> F = ( z e. CC |-> sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( z ^ k ) ) ) )
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.)
|- A = (coeff ` F )   &    |- N = (deg ` F )   =>    |- ( F e. (Poly ` S ) -> F = ( z e. CC |-> sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( z ^ k ) ) ) )
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.)
|- A = (coeff ` F )   &    |- N = (deg ` F )   =>    |- ( ( F e. (Poly ` S ) /\ X e. CC ) -> ( F ` X ) = sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( X ^ k ) ) )
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.)
|- A = (coeff ` F )   &    |- N = (deg ` F )   =>    |- ( ( F e. (Poly ` S ) /\ M e. ( ZZ>= ` N ) /\ X e. CC ) -> ( F ` X ) = sum_ k e. ( 0 ... M ) ( ( A ` k ) x. ( X ^ k ) ) )
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.)
|- ( ph -> F e. (Poly ` S ) )   &    |- ( ph -> G e. (Poly ` S ) )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x + y ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x x. y ) e. S )   =>    |- ( ph -> ( F o. G ) e. (Poly ` S ) )
Theorem	coeeq2 23252*	Compute the coefficient function given a sum expression for the polynomial. (Contributed by Mario Carneiro, 24-Jul-2014.)
|- ( ph -> F e. (Poly ` S ) )   &    |- ( ph -> N e. NN0 )   &    |- ( ( ph /\ k e. ( 0 ... N ) ) -> A e. CC )   &    |- ( ph -> F = ( z e. CC |-> sum_ k e. ( 0 ... N ) ( A x. ( z ^ k ) ) ) )   =>    |- ( ph -> (coeff ` F ) = ( k e. NN0 |-> if ( k <_ N , A , 0 ) ) )
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.)
|- ( ph -> F e. (Poly ` S ) )   &    |- ( ph -> N e. NN0 )   &    |- ( ( ph /\ k e. ( 0 ... N ) ) -> A e. CC )   &    |- ( ph -> F = ( z e. CC |-> sum_ k e. ( 0 ... N ) ( A x. ( z ^ k ) ) ) )   =>    |- ( ph -> (deg ` F ) <_ N )
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.)
|- ( ph -> F e. (Poly ` S ) )   &    |- ( ph -> N e. NN0 )   &    |- ( ph -> A : NN0 --> CC )   &    |- ( ph -> ( A " ( ZZ>= ` ( N + 1 ) ) ) = { 0 } )   &    |- ( ph -> F = ( z e. CC |-> sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( z ^ k ) ) ) )   &    |- ( ph -> ( A ` N ) =/= 0 )   =>    |- ( ph -> (deg ` F ) = N )
Theorem	0dgr 23255	A constant function has degree 0. (Contributed by Mario Carneiro, 24-Jul-2014.)
|- ( A e. CC -> (deg ` ( CC X. { A } ) ) = 0 )
Theorem	0dgrb 23256	A function has degree zero iff it is a constant function. (Contributed by Mario Carneiro, 23-Jul-2014.)
|- ( F e. (Poly ` S ) -> ( (deg ` F ) = 0 <-> F = ( CC X. { ( F ` 0 ) } ) ) )
Theorem	dgrnznn 23257	A nonzero polynomial with a root has positive degree. (Contributed by Stefan O'Rear, 25-Nov-2014.)
|- ( ( ( P e. (Poly ` S ) /\ P =/= 0p ) /\ ( A e. CC /\ ( P ` A ) = 0 ) ) -> (deg ` P ) e. NN )
Theorem	coefv0 23258	The result of evaluating a polynomial at zero is the constant term. (Contributed by Mario Carneiro, 24-Jul-2014.)
|- A = (coeff ` F )   =>    |- ( F e. (Poly ` S ) -> ( F ` 0 ) = ( A ` 0 ) )
Theorem	coeaddlem 23259	Lemma for coeadd 23261 and dgradd 23277. (Contributed by Mario Carneiro, 24-Jul-2014.)
|- A = (coeff ` F )   &    |- B = (coeff ` G )   &    |- M = (deg ` F )   &    |- N = (deg ` G )   =>    |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> ( (coeff ` ( F oF + G ) ) = ( A oF + B ) /\ (deg ` ( F oF + G ) ) <_ if ( M <_ N , N , M ) ) )
Theorem	coemullem 23260*	Lemma for coemul 23262 and dgrmul 23280. (Contributed by Mario Carneiro, 24-Jul-2014.)
|- A = (coeff ` F )   &    |- B = (coeff ` G )   &    |- M = (deg ` F )   &    |- N = (deg ` G )   =>    |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> ( (coeff ` ( F oF x. G ) ) = ( n e. NN0 |-> sum_ k e. ( 0 ... n ) ( ( A ` k ) x. ( B ` ( n - k ) ) ) ) /\ (deg ` ( F oF x. G ) ) <_ ( M + N ) ) )
Theorem	coeadd 23261	The coefficient function of a sum is the sum of coefficients. (Contributed by Mario Carneiro, 24-Jul-2014.)
|- A = (coeff ` F )   &    |- B = (coeff ` G )   =>    |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> (coeff ` ( F oF + G ) ) = ( A oF + B ) )
Theorem	coemul 23262*	A coefficient of a product of polynomials. (Contributed by Mario Carneiro, 24-Jul-2014.)
|- A = (coeff ` F )   &    |- B = (coeff ` G )   =>    |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ N e. NN0 ) -> ( (coeff ` ( F oF x. G ) ) ` N ) = sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( B ` ( N - k ) ) ) )
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.)
|- A = (coeff ` F )   &    |- B = (coeff ` G )   =>    |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> ( F = G <-> A = B ) )
Theorem	coemulhi 23264	The leading coefficient of a product of polynomials. (Contributed by Mario Carneiro, 24-Jul-2014.)
|- A = (coeff ` F )   &    |- B = (coeff ` G )   &    |- M = (deg ` F )   &    |- N = (deg ` G )   =>    |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> ( (coeff ` ( F oF x. G ) ) ` ( M + N ) ) = ( ( A ` M ) x. ( B ` N ) ) )
Theorem	coemulc 23265	The coefficient function is linear under scalar multiplication. (Contributed by Mario Carneiro, 24-Jul-2014.)
|- ( ( A e. CC /\ F e. (Poly ` S ) ) -> (coeff ` ( ( CC X. { A } ) oF x. F ) ) = ( ( NN0 X. { A } ) oF x. (coeff ` F ) ) )
Theorem	coe0 23266	The coefficients of the zero polynomial are zero. (Contributed by Mario Carneiro, 22-Jul-2014.)
|- (coeff ` 0p ) = ( NN0 X. { 0 } )
Theorem	coesub 23267	The coefficient function of a sum is the sum of coefficients. (Contributed by Mario Carneiro, 24-Jul-2014.)
|- A = (coeff ` F )   &    |- B = (coeff ` G )   =>    |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> (coeff ` ( F oF - G ) ) = ( A oF - B ) )
Theorem	coe1termlem 23268*	The coefficient function of a monomial. (Contributed by Mario Carneiro, 26-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
|- F = ( z e. CC |-> ( A x. ( z ^ N ) ) )   =>    |- ( ( A e. CC /\ N e. NN0 ) -> ( (coeff ` F ) = ( n e. NN0 |-> if ( n = N , A , 0 ) ) /\ ( A =/= 0 -> (deg ` F ) = N ) ) )
Theorem	coe1term 23269*	The coefficient function of a monomial. (Contributed by Mario Carneiro, 26-Jul-2014.)
|- F = ( z e. CC |-> ( A x. ( z ^ N ) ) )   =>    |- ( ( A e. CC /\ N e. NN0 /\ M e. NN0 ) -> ( (coeff ` F ) ` M ) = if ( M = N , A , 0 ) )
Theorem	dgr1term 23270*	The degree of a monomial. (Contributed by Mario Carneiro, 26-Jul-2014.)
|- F = ( z e. CC |-> ( A x. ( z ^ N ) ) )   =>    |- ( ( A e. CC /\ A =/= 0 /\ N e. NN0 ) -> (deg ` F ) = N )
Theorem	plycn 23271	A polynomial is a continuous function. (Contributed by Mario Carneiro, 23-Jul-2014.)
|- ( F e. (Poly ` S ) -> F e. ( CC -cn-> CC ) )
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 -u 1 , -oo or undefined. But it is convenient for us to define it this way, so that we have dgrcl 23243, dgreq0 23275 and coeid 23248 without having to special-case zero, although plydivalg 23308 is a little more complicated as a result. (Contributed by Mario Carneiro, 22-Jul-2014.)
|- (deg ` 0p ) = 0
Theorem	coeidp 23273	The coefficients of the identity function. (Contributed by Mario Carneiro, 28-Jul-2014.)
|- ( A e. NN0 -> ( (coeff ` Xp ) ` A ) = if ( A = 1 , 1 , 0 ) )
Theorem	dgrid 23274	The degree of the identity function. (Contributed by Mario Carneiro, 26-Jul-2014.)
|- (deg ` Xp ) = 1
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.)
|- N = (deg ` F )   &    |- A = (coeff ` F )   =>    |- ( F e. (Poly ` S ) -> ( F = 0p <-> ( A ` N ) = 0 ) )
Theorem	dgrlt 23276	Two ways to say that the degree of F is strictly less than N. (Contributed by Mario Carneiro, 25-Jul-2014.)
|- N = (deg ` F )   &    |- A = (coeff ` F )   =>    |- ( ( F e. (Poly ` S ) /\ M e. NN0 ) -> ( ( F = 0p \/ N < M ) <-> ( N <_ M /\ ( A ` M ) = 0 ) ) )
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.)
|- M = (deg ` F )   &    |- N = (deg ` G )   =>    |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> (deg ` ( F oF + G ) ) <_ if ( M <_ N , N , M ) )
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.)
|- M = (deg ` F )   &    |- N = (deg ` G )   =>    |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ M < N ) -> (deg ` ( F oF + G ) ) = N )
Theorem	dgrmul2 23279	The degree of a product of polynomials is at most the sum of degrees. (Contributed by Mario Carneiro, 24-Jul-2014.)
|- M = (deg ` F )   &    |- N = (deg ` G )   =>    |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> (deg ` ( F oF x. G ) ) <_ ( M + N ) )
Theorem	dgrmul 23280	The degree of a product of nonzero polynomials is the sum of degrees. (Contributed by Mario Carneiro, 24-Jul-2014.)
|- M = (deg ` F )   &    |- N = (deg ` G )   =>    |- ( ( ( F e. (Poly ` S ) /\ F =/= 0p ) /\ ( G e. (Poly ` S ) /\ G =/= 0p ) ) -> (deg ` ( F oF x. G ) ) = ( M + N ) )
Theorem	dgrmulc 23281	Scalar multiplication by a nonzero constant does not change the degree of a function. (Contributed by Mario Carneiro, 24-Jul-2014.)
|- ( ( A e. CC /\ A =/= 0 /\ F e. (Poly ` S ) ) -> (deg ` ( ( CC X. { A } ) oF x. F ) ) = (deg ` F ) )
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.)
|- M = (deg ` F )   &    |- N = (deg ` G )   =>    |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> (deg ` ( F oF - G ) ) <_ if ( M <_ N , N , M ) )
Theorem	dgrcolem1 23283*	The degree of a composition of a monomial with a polynomial. (Contributed by Mario Carneiro, 15-Sep-2014.)
|- N = (deg ` G )   &    |- ( ph -> M e. NN )   &    |- ( ph -> N e. NN )   &    |- ( ph -> G e. (Poly ` S ) )   =>    |- ( ph -> (deg ` ( x e. CC |-> ( ( G ` x ) ^ M ) ) ) = ( M x. N ) )
Theorem	dgrcolem2 23284*	Lemma for dgrco 23285. (Contributed by Mario Carneiro, 15-Sep-2014.)
|- M = (deg ` F )   &    |- N = (deg ` G )   &    |- ( ph -> F e. (Poly ` S ) )   &    |- ( ph -> G e. (Poly ` S ) )   &    |- A = (coeff ` F )   &    |- ( ph -> D e. NN0 )   &    |- ( ph -> M = ( D + 1 ) )   &    |- ( ph -> A. f e. (Poly ` CC ) ( (deg ` f ) <_ D -> (deg ` ( f o. G ) ) = ( (deg ` f ) x. N ) ) )   =>    |- ( ph -> (deg ` ( F o. G ) ) = ( M x. N ) )
Theorem	dgrco 23285	The degree of a composition of two polynomials is the product of the degrees. (Contributed by Mario Carneiro, 15-Sep-2014.)
|- M = (deg ` F )   &    |- N = (deg ` G )   &    |- ( ph -> F e. (Poly ` S ) )   &    |- ( ph -> G e. (Poly ` S ) )   =>    |- ( ph -> (deg ` ( F o. G ) ) = ( M x. N ) )
Theorem	plycjlem 23286*	Lemma for plycj 23287 and coecj 23288. (Contributed by Mario Carneiro, 24-Jul-2014.)
|- N = (deg ` F )   &    |- G = ( ( * o. F ) o. * )   &    |- A = (coeff ` F )   =>    |- ( F e. (Poly ` S ) -> G = ( z e. CC |-> sum_ k e. ( 0 ... N ) ( ( ( * o. A ) ` k ) x. ( z ^ k ) ) ) )
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 ( * ` z ) independently of z.) (Contributed by Mario Carneiro, 24-Jul-2014.)
|- N = (deg ` F )   &    |- G = ( ( * o. F ) o. * )   &    |- ( ( ph /\ x e. S ) -> ( * ` x ) e. S )   &    |- ( ph -> F e. (Poly ` S ) )   =>    |- ( ph -> G e. (Poly ` S ) )
Theorem	coecj 23288	Double conjugation of a polynomial causes the coefficients to be conjugated. (Contributed by Mario Carneiro, 24-Jul-2014.)
|- N = (deg ` F )   &    |- G = ( ( * o. F ) o. * )   &    |- A = (coeff ` F )   =>    |- ( F e. (Poly ` S ) -> (coeff ` G ) = ( * o. A ) )
Theorem	plyrecj 23289	A polynomial with real coefficients distributes under conjugation. (Contributed by Mario Carneiro, 24-Jul-2014.)
|- ( ( F e. (Poly ` RR ) /\ A e. CC ) -> ( * ` ( F ` A ) ) = ( F ` ( * ` A ) ) )
Theorem	plymul0or 23290	Polynomial multiplication has no zero divisors. (Contributed by Mario Carneiro, 26-Jul-2014.)
|- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> ( ( F oF x. G ) = 0p <-> ( F = 0p \/ G = 0p ) ) )
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.)
|- ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) -> ( `' ( F oF x. G ) " { 0 } ) = ( ( `' F " { 0 } ) u. ( `' G " { 0 } ) ) )
Theorem	plyreres 23292	Real-coefficient polynomials restrict to real functions. (Contributed by Stefan O'Rear, 16-Nov-2014.)
|- ( F e. (Poly ` RR ) -> ( F |` RR ) : RR --> RR )
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.)
|- ( ph -> F = ( z e. CC |-> sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( z ^ k ) ) ) )   &    |- ( ph -> G = ( z e. CC |-> sum_ k e. ( 0 ... ( N - 1 ) ) ( ( B ` k ) x. ( z ^ k ) ) ) )   &    |- ( ph -> A : NN0 --> CC )   &    |- B = ( k e. NN0 |-> ( ( k + 1 ) x. ( A ` ( k + 1 ) ) ) )   &    |- ( ph -> N e. NN0 )   =>    |- ( ph -> ( CC _D F ) = G )
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.)
|- ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) -> ( CC _D F ) e. (Poly ` S ) )
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.)
|- ( F e. (Poly ` S ) -> ( CC _D F ) e. (Poly ` CC ) )
Theorem	dvnply2 23296	Polynomials have polynomials as derivatives of all orders. (Contributed by Mario Carneiro, 1-Jan-2017.)
|- ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) /\ N e. NN0 ) -> ( ( CC Dn F ) ` N ) e. (Poly ` S ) )
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.)
|- ( ( F e. (Poly ` S ) /\ N e. NN0 ) -> ( ( CC Dn F ) ` N ) e. (Poly ` CC ) )
Theorem	plycpn 23298	Polynomials are smooth. (Contributed by Stefan O'Rear, 16-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
|- ( F e. (Poly ` S ) -> F e. |^| ran ( C^n ` CC ) )
14.1.4  The division algorithm for polynomials
Syntax	cquot 23299	Extend class notation to include the quotient of a polynomial division.
class quot
Definition	df-quot 23300*	Define the quotient function on polynomials. This is the q of the expression f = g x. q + r in the division algorithm. (Contributed by Mario Carneiro, 23-Jul-2014.)
|- quot = ( f e. (Poly ` CC ) , g e. ( (Poly ` CC ) \ { 0p } ) |-> ( iota_ q e. (Poly ` CC ) [. ( f oF - ( g oF x. q ) ) / r ]. ( r = 0p \/ (deg ` r ) < (deg ` g ) ) ) )