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
Theorem	coefv0 23201	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 23202	Lemma for coeadd 23204 and dgradd 23220. (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 23203*	Lemma for coemul 23205 and dgrmul 23223. (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 23204	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 23205*	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 23206	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 23207	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 23208	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 23209	The coefficients of the zero polynomial are zero. (Contributed by Mario Carneiro, 22-Jul-2014.)
|- (coeff ` 0p ) = ( NN0 X. { 0 } )
Theorem	coesub 23210	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 23211*	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 23212*	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 23213*	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 23214	A polynomial is a continuous function. (Contributed by Mario Carneiro, 23-Jul-2014.)
|- ( F e. (Poly ` S ) -> F e. ( CC -cn-> CC ) )
Theorem	dgr0 23215	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 23186, dgreq0 23218 and coeid 23191 without having to special-case zero, although plydivalg 23251 is a little more complicated as a result. (Contributed by Mario Carneiro, 22-Jul-2014.)
|- (deg ` 0p ) = 0
Theorem	coeidp 23216	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 23217	The degree of the identity function. (Contributed by Mario Carneiro, 26-Jul-2014.)
|- (deg ` Xp ) = 1
Theorem	dgreq0 23218	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 23219	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 23220	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 23221	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 23222	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 23223	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 23224	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 23225	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 23226*	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 23227*	Lemma for dgrco 23228. (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 23228	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 23229*	Lemma for plycj 23230 and coecj 23231. (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 23230*	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 23231	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 23232	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 23233	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 23234	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 23235	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 23236*	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 23237	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 23238	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 23239	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 23240	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 23241	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 23242	Extend class notation to include the quotient of a polynomial division.
class quot
Definition	df-quot 23243*	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 ) ) ) )
Theorem	quotval 23244*	Value of the quotient function. (Contributed by Mario Carneiro, 23-Jul-2014.)
|- R = ( F oF - ( G oF x. q ) )   =>    |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ G =/= 0p ) -> ( F quot G ) = ( iota_ q e. (Poly ` CC ) ( R = 0p \/ (deg ` R ) < (deg ` G ) ) ) )
Theorem	plydivlem1 23245*	Lemma for plydivalg 23251. (Contributed by Mario Carneiro, 24-Jul-2014.)
|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x + y ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x x. y ) e. S )   &    |- ( ( ph /\ ( x e. S /\ x =/= 0 ) ) -> ( 1 / x ) e. S )   &    |- ( ph -> -u 1 e. S )   =>    |- ( ph -> 0 e. S )
Theorem	plydivlem2 23246*	Lemma for plydivalg 23251. (Contributed by Mario Carneiro, 24-Jul-2014.)
|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x + y ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x x. y ) e. S )   &    |- ( ( ph /\ ( x e. S /\ x =/= 0 ) ) -> ( 1 / x ) e. S )   &    |- ( ph -> -u 1 e. S )   &    |- ( ph -> F e. (Poly ` S ) )   &    |- ( ph -> G e. (Poly ` S ) )   &    |- ( ph -> G =/= 0p )   &    |- R = ( F oF - ( G oF x. q ) )   =>    |- ( ( ph /\ q e. (Poly ` S ) ) -> R e. (Poly ` S ) )
Theorem	plydivlem3 23247*	Lemma for plydivex 23249. Base case of induction. (Contributed by Mario Carneiro, 24-Jul-2014.)
|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x + y ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x x. y ) e. S )   &    |- ( ( ph /\ ( x e. S /\ x =/= 0 ) ) -> ( 1 / x ) e. S )   &    |- ( ph -> -u 1 e. S )   &    |- ( ph -> F e. (Poly ` S ) )   &    |- ( ph -> G e. (Poly ` S ) )   &    |- ( ph -> G =/= 0p )   &    |- R = ( F oF - ( G oF x. q ) )   &    |- ( ph -> ( F = 0p \/ ( (deg ` F ) - (deg ` G ) ) < 0 ) )   =>    |- ( ph -> E. q e. (Poly ` S ) ( R = 0p \/ (deg ` R ) < (deg ` G ) ) )
Theorem	plydivlem4 23248*	Lemma for plydivex 23249. Induction step. (Contributed by Mario Carneiro, 26-Jul-2014.)
|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x + y ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x x. y ) e. S )   &    |- ( ( ph /\ ( x e. S /\ x =/= 0 ) ) -> ( 1 / x ) e. S )   &    |- ( ph -> -u 1 e. S )   &    |- ( ph -> F e. (Poly ` S ) )   &    |- ( ph -> G e. (Poly ` S ) )   &    |- ( ph -> G =/= 0p )   &    |- R = ( F oF - ( G oF x. q ) )   &    |- ( ph -> D e. NN0 )   &    |- ( ph -> ( M - N ) = D )   &    |- ( ph -> F =/= 0p )   &    |- U = ( f oF - ( G oF x. p ) )   &    |- H = ( z e. CC |-> ( ( ( A ` M ) / ( B ` N ) ) x. ( z ^ D ) ) )   &    |- ( ph -> A. f e. (Poly ` S ) ( ( f = 0p \/ ( (deg ` f ) - N ) < D ) -> E. p e. (Poly ` S ) ( U = 0p \/ (deg ` U ) < N ) ) )   &    |- A = (coeff ` F )   &    |- B = (coeff ` G )   &    |- M = (deg ` F )   &    |- N = (deg ` G )   =>    |- ( ph -> E. q e. (Poly ` S ) ( R = 0p \/ (deg ` R ) < N ) )
Theorem	plydivex 23249*	Lemma for plydivalg 23251. (Contributed by Mario Carneiro, 24-Jul-2014.)
|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x + y ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x x. y ) e. S )   &    |- ( ( ph /\ ( x e. S /\ x =/= 0 ) ) -> ( 1 / x ) e. S )   &    |- ( ph -> -u 1 e. S )   &    |- ( ph -> F e. (Poly ` S ) )   &    |- ( ph -> G e. (Poly ` S ) )   &    |- ( ph -> G =/= 0p )   &    |- R = ( F oF - ( G oF x. q ) )   =>    |- ( ph -> E. q e. (Poly ` S ) ( R = 0p \/ (deg ` R ) < (deg ` G ) ) )
Theorem	plydiveu 23250*	Lemma for plydivalg 23251. (Contributed by Mario Carneiro, 24-Jul-2014.)
|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x + y ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x x. y ) e. S )   &    |- ( ( ph /\ ( x e. S /\ x =/= 0 ) ) -> ( 1 / x ) e. S )   &    |- ( ph -> -u 1 e. S )   &    |- ( ph -> F e. (Poly ` S ) )   &    |- ( ph -> G e. (Poly ` S ) )   &    |- ( ph -> G =/= 0p )   &    |- R = ( F oF - ( G oF x. q ) )   &    |- ( ph -> q e. (Poly ` S ) )   &    |- ( ph -> ( R = 0p \/ (deg ` R ) < (deg ` G ) ) )   &    |- T = ( F oF - ( G oF x. p ) )   &    |- ( ph -> p e. (Poly ` S ) )   &    |- ( ph -> ( T = 0p \/ (deg ` T ) < (deg ` G ) ) )   =>    |- ( ph -> p = q )
Theorem	plydivalg 23251*	The division algorithm on polynomials over a subfield S of the complex numbers. If F and G =/= 0 are polynomials over S, then there is a unique quotient polynomial q such that the remainder F - G x. q is either zero or has degree less than G. (Contributed by Mario Carneiro, 26-Jul-2014.)
|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x + y ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x x. y ) e. S )   &    |- ( ( ph /\ ( x e. S /\ x =/= 0 ) ) -> ( 1 / x ) e. S )   &    |- ( ph -> -u 1 e. S )   &    |- ( ph -> F e. (Poly ` S ) )   &    |- ( ph -> G e. (Poly ` S ) )   &    |- ( ph -> G =/= 0p )   &    |- R = ( F oF - ( G oF x. q ) )   =>    |- ( ph -> E! q e. (Poly ` S ) ( R = 0p \/ (deg ` R ) < (deg ` G ) ) )
Theorem	quotlem 23252*	Lemma for properties of the polynomial quotient function. (Contributed by Mario Carneiro, 26-Jul-2014.)
|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x + y ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x x. y ) e. S )   &    |- ( ( ph /\ ( x e. S /\ x =/= 0 ) ) -> ( 1 / x ) e. S )   &    |- ( ph -> -u 1 e. S )   &    |- ( ph -> F e. (Poly ` S ) )   &    |- ( ph -> G e. (Poly ` S ) )   &    |- ( ph -> G =/= 0p )   &    |- R = ( F oF - ( G oF x. ( F quot G ) ) )   =>    |- ( ph -> ( ( F quot G ) e. (Poly ` S ) /\ ( R = 0p \/ (deg ` R ) < (deg ` G ) ) ) )
Theorem	quotcl 23253*	The quotient of two polynomials in a field S is also in the field. (Contributed by Mario Carneiro, 26-Jul-2014.)
|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x + y ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x x. y ) e. S )   &    |- ( ( ph /\ ( x e. S /\ x =/= 0 ) ) -> ( 1 / x ) e. S )   &    |- ( ph -> -u 1 e. S )   &    |- ( ph -> F e. (Poly ` S ) )   &    |- ( ph -> G e. (Poly ` S ) )   &    |- ( ph -> G =/= 0p )   =>    |- ( ph -> ( F quot G ) e. (Poly ` S ) )
Theorem	quotcl2 23254	Closure of the quotient function. (Contributed by Mario Carneiro, 26-Jul-2014.)
|- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ G =/= 0p ) -> ( F quot G ) e. (Poly ` CC ) )
Theorem	quotdgr 23255	Remainder property of the quotient function. (Contributed by Mario Carneiro, 26-Jul-2014.)
|- R = ( F oF - ( G oF x. ( F quot G ) ) )   =>    |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ G =/= 0p ) -> ( R = 0p \/ (deg ` R ) < (deg ` G ) ) )
Theorem	plyremlem 23256	Closure of a linear factor. (Contributed by Mario Carneiro, 26-Jul-2014.)
|- G = ( Xp oF - ( CC X. { A } ) )   =>    |- ( A e. CC -> ( G e. (Poly ` CC ) /\ (deg ` G ) = 1 /\ ( `' G " { 0 } ) = { A } ) )
Theorem	plyrem 23257	The polynomial remainder theorem, or little Bézout's theorem (by contrast to the regular Bézout's theorem bezout 14510). If a polynomial F is divided by the linear factor x - A, the remainder is equal to F ( A ), the evaluation of the polynomial at A (interpreted as a constant polynomial). This is part of Metamath 100 proof #89. (Contributed by Mario Carneiro, 26-Jul-2014.)
|- G = ( Xp oF - ( CC X. { A } ) )   &    |- R = ( F oF - ( G oF x. ( F quot G ) ) )   =>    |- ( ( F e. (Poly ` S ) /\ A e. CC ) -> R = ( CC X. { ( F ` A ) } ) )
Theorem	facth 23258	The factor theorem. If a polynomial F has a root at A, then G = x - A is a factor of F (and the other factor is F quot G). This is part of Metamath 100 proof #89. (Contributed by Mario Carneiro, 26-Jul-2014.)
|- G = ( Xp oF - ( CC X. { A } ) )   =>    |- ( ( F e. (Poly ` S ) /\ A e. CC /\ ( F ` A ) = 0 ) -> F = ( G oF x. ( F quot G ) ) )
Theorem	fta1lem 23259*	Lemma for fta1 23260. (Contributed by Mario Carneiro, 26-Jul-2014.)
|- R = ( `' F " { 0 } )   &    |- ( ph -> D e. NN0 )   &    |- ( ph -> F e. ( (Poly ` CC ) \ { 0p } ) )   &    |- ( ph -> (deg ` F ) = ( D + 1 ) )   &    |- ( ph -> A e. ( `' F " { 0 } ) )   &    |- ( ph -> A. g e. ( (Poly ` CC ) \ { 0p } ) ( (deg ` g ) = D -> ( ( `' g " { 0 } ) e. Fin /\ ( # ` ( `' g " { 0 } ) ) <_ (deg ` g ) ) ) )   =>    |- ( ph -> ( R e. Fin /\ ( # ` R ) <_ (deg ` F ) ) )
Theorem	fta1 23260	The easy direction of the Fundamental Theorem of Algebra: A nonzero polynomial has at most deg ( F ) roots. (Contributed by Mario Carneiro, 26-Jul-2014.)
|- R = ( `' F " { 0 } )   =>    |- ( ( F e. (Poly ` S ) /\ F =/= 0p ) -> ( R e. Fin /\ ( # ` R ) <_ (deg ` F ) ) )
Theorem	quotcan 23261	Exact division with a multiple. (Contributed by Mario Carneiro, 26-Jul-2014.)
|- H = ( F oF x. G )   =>    |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ G =/= 0p ) -> ( H quot G ) = F )
Theorem	vieta1lem1 23262*	Lemma for vieta1 23264. (Contributed by Mario Carneiro, 28-Jul-2014.)
|- A = (coeff ` F )   &    |- N = (deg ` F )   &    |- R = ( `' F " { 0 } )   &    |- ( ph -> F e. (Poly ` S ) )   &    |- ( ph -> ( # ` R ) = N )   &    |- ( ph -> D e. NN )   &    |- ( ph -> ( D + 1 ) = N )   &    |- ( ph -> A. f e. (Poly ` CC ) ( ( D = (deg ` f ) /\ ( # ` ( `' f " { 0 } ) ) = (deg ` f ) ) -> sum_ x e. ( `' f " { 0 } ) x = -u ( ( (coeff ` f ) ` ( (deg ` f ) - 1 ) ) / ( (coeff ` f ) ` (deg ` f ) ) ) ) )   &    |- Q = ( F quot ( Xp oF - ( CC X. { z } ) ) )   =>    |- ( ( ph /\ z e. R ) -> ( Q e. (Poly ` CC ) /\ D = (deg ` Q ) ) )
Theorem	vieta1lem2 23263*	Lemma for vieta1 23264: inductive step. Let z be a root of F. Then F = ( Xp - z ) x. Q for some Q by the factor theorem, and Q is a degree- D polynomial, so by the induction hypothesis sum_ x e. ( `' Q " 0 ) x = -u (coeff ` Q ) ` ( D - 1 ) / (coeff ` Q ) ` D, so sum_ x e. R x = z - (coeff ` Q ) ` ( D - 1 ) / (coeff ` Q ) ` D. Now the coefficients of F are A ` ( D + 1 ) = (coeff ` Q ) ` D and A ` D = sum_ k e. ( 0 ... D ) (coeff ` Xp - z ) ` k x. (coeff ` Q ) ` ( D - k ), which works out to -u z x. (coeff ` Q ) ` D + (coeff ` Q ) ` ( D - 1 ), so putting it all together we have sum_ x e. R x = -u A ` D / A ` ( D + 1 ) as we wanted to show. (Contributed by Mario Carneiro, 28-Jul-2014.)
|- A = (coeff ` F )   &    |- N = (deg ` F )   &    |- R = ( `' F " { 0 } )   &    |- ( ph -> F e. (Poly ` S ) )   &    |- ( ph -> ( # ` R ) = N )   &    |- ( ph -> D e. NN )   &    |- ( ph -> ( D + 1 ) = N )   &    |- ( ph -> A. f e. (Poly ` CC ) ( ( D = (deg ` f ) /\ ( # ` ( `' f " { 0 } ) ) = (deg ` f ) ) -> sum_ x e. ( `' f " { 0 } ) x = -u ( ( (coeff ` f ) ` ( (deg ` f ) - 1 ) ) / ( (coeff ` f ) ` (deg ` f ) ) ) ) )   &    |- Q = ( F quot ( Xp oF - ( CC X. { z } ) ) )   =>    |- ( ph -> sum_ x e. R x = -u ( ( A ` ( N - 1 ) ) / ( A ` N ) ) )
Theorem	vieta1 23264*	The first-order Vieta's formula (see http://en.wikipedia.org/wiki/Vieta%27s_formulas). If a polynomial of degree N has N distinct roots, then the sum over these roots can be calculated as -u A ( N - 1 ) / A ( N ). (If the roots are not distinct, then this formula is still true but must double-count some of the roots according to their multiplicities.) (Contributed by Mario Carneiro, 28-Jul-2014.)
|- A = (coeff ` F )   &    |- N = (deg ` F )   &    |- R = ( `' F " { 0 } )   &    |- ( ph -> F e. (Poly ` S ) )   &    |- ( ph -> ( # ` R ) = N )   &    |- ( ph -> N e. NN )   =>    |- ( ph -> sum_ x e. R x = -u ( ( A ` ( N - 1 ) ) / ( A ` N ) ) )
Theorem	plyexmo 23265*	An infinite set of values can be extended to a polynomial in at most one way. (Contributed by Stefan O'Rear, 14-Nov-2014.)
|- ( ( D C_ CC /\ -. D e. Fin ) -> E* p ( p e. (Poly ` S ) /\ ( p |` D ) = F ) )
14.1.5  Algebraic numbers
Syntax	caa 23266	Extend class notation to include the set of algebraic numbers.
class AA
Definition	df-aa 23267	Define the set of algebraic numbers. An algebraic number is a root of a nonzero polynomial over the integers. Here we construct it as the union of all kernels (preimages of { 0 }) of all polynomials in (Poly ` ZZ ), except the zero polynomial 0p. (Contributed by Mario Carneiro, 22-Jul-2014.)
|- AA = U_ f e. ( (Poly ` ZZ ) \ { 0p } ) ( `' f " { 0 } )
Theorem	elaa 23268*	Elementhood in the set of algebraic numbers. (Contributed by Mario Carneiro, 22-Jul-2014.)
|- ( A e. AA <-> ( A e. CC /\ E. f e. ( (Poly ` ZZ ) \ { 0p } ) ( f ` A ) = 0 ) )
Theorem	aacn 23269	An algebraic number is a complex number. (Contributed by Mario Carneiro, 23-Jul-2014.)
|- ( A e. AA -> A e. CC )
Theorem	aasscn 23270	The algebraic numbers are a subset of the complex numbers. (Contributed by Mario Carneiro, 23-Jul-2014.)
|- AA C_ CC
Theorem	elqaalem1 23271*	Lemma for elqaa 23277. The function N represents the denominators of the rational coefficients B. By multiplying them all together to make R, we get a number big enough to clear all the denominators and make R x. F an integer polynomial. (Contributed by Mario Carneiro, 23-Jul-2014.) (Revised by AV, 3-Oct-2020.)
|- ( ph -> A e. CC )   &    |- ( ph -> F e. ( (Poly ` QQ ) \ { 0p } ) )   &    |- ( ph -> ( F ` A ) = 0 )   &    |- B = (coeff ` F )   &    |- N = ( k e. NN0 |-> inf ( { n e. NN | ( ( B ` k ) x. n ) e. ZZ } , RR , < ) )   &    |- R = ( seq 0 ( x. , N ) ` (deg ` F ) )   =>    |- ( ( ph /\ K e. NN0 ) -> ( ( N ` K ) e. NN /\ ( ( B ` K ) x. ( N ` K ) ) e. ZZ ) )
Theorem	elqaalem2 23272*	Lemma for elqaa 23277. (Contributed by Mario Carneiro, 23-Jul-2014.) (Revised by AV, 3-Oct-2020.)
|- ( ph -> A e. CC )   &    |- ( ph -> F e. ( (Poly ` QQ ) \ { 0p } ) )   &    |- ( ph -> ( F ` A ) = 0 )   &    |- B = (coeff ` F )   &    |- N = ( k e. NN0 |-> inf ( { n e. NN | ( ( B ` k ) x. n ) e. ZZ } , RR , < ) )   &    |- R = ( seq 0 ( x. , N ) ` (deg ` F ) )   &    |- P = ( x e. _V , y e. _V |-> ( ( x x. y ) mod ( N ` K ) ) )   =>    |- ( ( ph /\ K e. ( 0 ... (deg ` F ) ) ) -> ( R mod ( N ` K ) ) = 0 )
Theorem	elqaalem3 23273*	Lemma for elqaa 23277. (Contributed by Mario Carneiro, 23-Jul-2014.) (Revised by AV, 3-Oct-2020.)
|- ( ph -> A e. CC )   &    |- ( ph -> F e. ( (Poly ` QQ ) \ { 0p } ) )   &    |- ( ph -> ( F ` A ) = 0 )   &    |- B = (coeff ` F )   &    |- N = ( k e. NN0 |-> inf ( { n e. NN | ( ( B ` k ) x. n ) e. ZZ } , RR , < ) )   &    |- R = ( seq 0 ( x. , N ) ` (deg ` F ) )   =>    |- ( ph -> A e. AA )
Theorem	elqaalem1OLD 23274*	Lemma for elqaa 23277. The function N represents the denominators of the rational coefficients B. By multiplying them all together to make R, we get a number big enough to clear all the denominators and make R x. F an integer polynomial. (Contributed by Mario Carneiro, 23-Jul-2014.) Obsolete version of elqaalem1 23271 as of 3-Oct-2020. (New usage is discouraged.) (Proof modification is discouraged.)
|- ( ph -> A e. CC )   &    |- ( ph -> F e. ( (Poly ` QQ ) \ { 0p } ) )   &    |- ( ph -> ( F ` A ) = 0 )   &    |- B = (coeff ` F )   &    |- N = ( k e. NN0 |-> sup ( { n e. NN | ( ( B ` k ) x. n ) e. ZZ } , RR , `' < ) )   &    |- R = ( seq 0 ( x. , N ) ` (deg ` F ) )   =>    |- ( ( ph /\ K e. NN0 ) -> ( ( N ` K ) e. NN /\ ( ( B ` K ) x. ( N ` K ) ) e. ZZ ) )
Theorem	elqaalem2OLD 23275*	Lemma for elqaa 23277. (Contributed by Mario Carneiro, 23-Jul-2014.) Obsolete version of elqaalem2 23272 as of 3-Oct-2020. (New usage is discouraged.) (Proof modification is discouraged.)
|- ( ph -> A e. CC )   &    |- ( ph -> F e. ( (Poly ` QQ ) \ { 0p } ) )   &    |- ( ph -> ( F ` A ) = 0 )   &    |- B = (coeff ` F )   &    |- N = ( k e. NN0 |-> sup ( { n e. NN | ( ( B ` k ) x. n ) e. ZZ } , RR , `' < ) )   &    |- R = ( seq 0 ( x. , N ) ` (deg ` F ) )   &    |- P = ( x e. _V , y e. _V |-> ( ( x x. y ) mod ( N ` K ) ) )   =>    |- ( ( ph /\ K e. ( 0 ... (deg ` F ) ) ) -> ( R mod ( N ` K ) ) = 0 )
Theorem	elqaalem3OLD 23276*	Lemma for elqaa 23277. (Contributed by Mario Carneiro, 23-Jul-2014.) Obsolete version of elqaalem1 23271 as of 3-Oct-2020. (New usage is discouraged.) (Proof modification is discouraged.)
|- ( ph -> A e. CC )   &    |- ( ph -> F e. ( (Poly ` QQ ) \ { 0p } ) )   &    |- ( ph -> ( F ` A ) = 0 )   &    |- B = (coeff ` F )   &    |- N = ( k e. NN0 |-> sup ( { n e. NN | ( ( B ` k ) x. n ) e. ZZ } , RR , `' < ) )   &    |- R = ( seq 0 ( x. , N ) ` (deg ` F ) )   =>    |- ( ph -> A e. AA )
Theorem	elqaa 23277*	The set of numbers generated by the roots of polynomials in the rational numbers is the same as the set of algebraic numbers, which by elaa 23268 are defined only in terms of polynomials over the integers. (Contributed by Mario Carneiro, 23-Jul-2014.) (Proof shortened by AV, 3-Oct-2020.)
|- ( A e. AA <-> ( A e. CC /\ E. f e. ( (Poly ` QQ ) \ { 0p } ) ( f ` A ) = 0 ) )
Theorem	qaa 23278	Every rational number is algebraic. (Contributed by Mario Carneiro, 23-Jul-2014.)
|- ( A e. QQ -> A e. AA )
Theorem	qssaa 23279	The rational numbers are contained in the algebraic numbers. (Contributed by Mario Carneiro, 23-Jul-2014.)
|- QQ C_ AA
Theorem	iaa 23280	The imaginary unit is algebraic. (Contributed by Mario Carneiro, 23-Jul-2014.)
|- _i e. AA
Theorem	aareccl 23281	The reciprocal of an algebraic number is algebraic. (Contributed by Mario Carneiro, 24-Jul-2014.)
|- ( ( A e. AA /\ A =/= 0 ) -> ( 1 / A ) e. AA )
Theorem	aacjcl 23282	The conjugate of an algebraic number is algebraic. (Contributed by Mario Carneiro, 24-Jul-2014.)
|- ( A e. AA -> ( * ` A ) e. AA )
Theorem	aannenlem1 23283*	Lemma for aannen 23286. (Contributed by Stefan O'Rear, 16-Nov-2014.)
|- H = ( a e. NN0 |-> { b e. CC | E. c e. { d e. (Poly ` ZZ ) | ( d =/= 0p /\ (deg ` d ) <_ a /\ A. e e. NN0 ( abs ` ( (coeff ` d ) ` e ) ) <_ a ) } ( c ` b ) = 0 } )   =>    |- ( A e. NN0 -> ( H ` A ) e. Fin )
Theorem	aannenlem2 23284*	Lemma for aannen 23286. (Contributed by Stefan O'Rear, 16-Nov-2014.)
|- H = ( a e. NN0 |-> { b e. CC | E. c e. { d e. (Poly ` ZZ ) | ( d =/= 0p /\ (deg ` d ) <_ a /\ A. e e. NN0 ( abs ` ( (coeff ` d ) ` e ) ) <_ a ) } ( c ` b ) = 0 } )   =>    |- AA = U. ran H
Theorem	aannenlem3 23285*	The algebraic numbers are countable. (Contributed by Stefan O'Rear, 16-Nov-2014.)
|- H = ( a e. NN0 |-> { b e. CC | E. c e. { d e. (Poly ` ZZ ) | ( d =/= 0p /\ (deg ` d ) <_ a /\ A. e e. NN0 ( abs ` ( (coeff ` d ) ` e ) ) <_ a ) } ( c ` b ) = 0 } )   =>    |- AA ~~ NN
Theorem	aannen 23286	The algebraic numbers are countable. (Contributed by Stefan O'Rear, 16-Nov-2014.)
|- AA ~~ NN
14.1.6  Liouville's approximation theorem
Theorem	aalioulem1 23287	Lemma for aaliou 23293. An integer polynomial cannot inflate the denominator of a rational by more than its degree. (Contributed by Stefan O'Rear, 12-Nov-2014.)
|- ( ph -> F e. (Poly ` ZZ ) )   &    |- ( ph -> X e. ZZ )   &    |- ( ph -> Y e. NN )   =>    |- ( ph -> ( ( F ` ( X / Y ) ) x. ( Y ^ (deg ` F ) ) ) e. ZZ )
Theorem	aalioulem2 23288*	Lemma for aaliou 23293. (Contributed by Stefan O'Rear, 15-Nov-2014.) (Proof shortened by AV, 28-Sep-2020.)
|- N = (deg ` F )   &    |- ( ph -> F e. (Poly ` ZZ ) )   &    |- ( ph -> N e. NN )   &    |- ( ph -> A e. RR )   =>    |- ( ph -> E. x e. RR+ A. p e. ZZ A. q e. NN ( ( F ` ( p / q ) ) = 0 -> ( A = ( p / q ) \/ ( x / ( q ^ N ) ) <_ ( abs ` ( A - ( p / q ) ) ) ) ) )
Theorem	aalioulem3 23289*	Lemma for aaliou 23293. (Contributed by Stefan O'Rear, 15-Nov-2014.)
|- N = (deg ` F )   &    |- ( ph -> F e. (Poly ` ZZ ) )   &    |- ( ph -> N e. NN )   &    |- ( ph -> A e. RR )   &    |- ( ph -> ( F ` A ) = 0 )   =>    |- ( ph -> E. x e. RR+ A. r e. RR ( ( abs ` ( A - r ) ) <_ 1 -> ( x x. ( abs ` ( F ` r ) ) ) <_ ( abs ` ( A - r ) ) ) )
Theorem	aalioulem4 23290*	Lemma for aaliou 23293. (Contributed by Stefan O'Rear, 16-Nov-2014.)
|- N = (deg ` F )   &    |- ( ph -> F e. (Poly ` ZZ ) )   &    |- ( ph -> N e. NN )   &    |- ( ph -> A e. RR )   &    |- ( ph -> ( F ` A ) = 0 )   =>    |- ( ph -> E. x e. RR+ A. p e. ZZ A. q e. NN ( ( ( F ` ( p / q ) ) =/= 0 /\ ( abs ` ( A - ( p / q ) ) ) <_ 1 ) -> ( A = ( p / q ) \/ ( x / ( q ^ N ) ) <_ ( abs ` ( A - ( p / q ) ) ) ) ) )
Theorem	aalioulem5 23291*	Lemma for aaliou 23293. (Contributed by Stefan O'Rear, 16-Nov-2014.)
|- N = (deg ` F )   &    |- ( ph -> F e. (Poly ` ZZ ) )   &    |- ( ph -> N e. NN )   &    |- ( ph -> A e. RR )   &    |- ( ph -> ( F ` A ) = 0 )   =>    |- ( ph -> E. x e. RR+ A. p e. ZZ A. q e. NN ( ( F ` ( p / q ) ) =/= 0 -> ( A = ( p / q ) \/ ( x / ( q ^ N ) ) <_ ( abs ` ( A - ( p / q ) ) ) ) ) )
Theorem	aalioulem6 23292*	Lemma for aaliou 23293. (Contributed by Stefan O'Rear, 16-Nov-2014.)
|- N = (deg ` F )   &    |- ( ph -> F e. (Poly ` ZZ ) )   &    |- ( ph -> N e. NN )   &    |- ( ph -> A e. RR )   &    |- ( ph -> ( F ` A ) = 0 )   =>    |- ( ph -> E. x e. RR+ A. p e. ZZ A. q e. NN ( A = ( p / q ) \/ ( x / ( q ^ N ) ) <_ ( abs ` ( A - ( p / q ) ) ) ) )
Theorem	aaliou 23293*	Liouville's theorem on diophantine approximation: Any algebraic number, being a root of a polynomial F in integer coefficients, is not approximable beyond order N = deg ( F ) by rational numbers. In this form, it also applies to rational numbers themselves, which are not well approximable by other rational numbers. This is Metamath 100 proof #18. (Contributed by Stefan O'Rear, 16-Nov-2014.)
|- N = (deg ` F )   &    |- ( ph -> F e. (Poly ` ZZ ) )   &    |- ( ph -> N e. NN )   &    |- ( ph -> A e. RR )   &    |- ( ph -> ( F ` A ) = 0 )   =>    |- ( ph -> E. x e. RR+ A. p e. ZZ A. q e. NN ( A = ( p / q ) \/ ( x / ( q ^ N ) ) < ( abs ` ( A - ( p / q ) ) ) ) )
Theorem	geolim3 23294*	Geometric series convergence with arbitrary shift, radix, and multiplicative constant. (Contributed by Stefan O'Rear, 16-Nov-2014.)
|- ( ph -> A e. ZZ )   &    |- ( ph -> B e. CC )   &    |- ( ph -> ( abs ` B ) < 1 )   &    |- ( ph -> C e. CC )   &    |- F = ( k e. ( ZZ>= ` A ) |-> ( C x. ( B ^ ( k - A ) ) ) )   =>    |- ( ph -> seq A ( + , F ) ~~> ( C / ( 1 - B ) ) )
Theorem	aaliou2 23295*	Liouville's approximation theorem for algebraic numbers per se. (Contributed by Stefan O'Rear, 16-Nov-2014.)
|- ( A e. ( AA i^i RR ) -> E. k e. NN E. x e. RR+ A. p e. ZZ A. q e. NN ( A = ( p / q ) \/ ( x / ( q ^ k ) ) < ( abs ` ( A - ( p / q ) ) ) ) )
Theorem	aaliou2b 23296*	Liouville's approximation theorem extended to complex A. (Contributed by Stefan O'Rear, 20-Nov-2014.)
|- ( A e. AA -> E. k e. NN E. x e. RR+ A. p e. ZZ A. q e. NN ( A = ( p / q ) \/ ( x / ( q ^ k ) ) < ( abs ` ( A - ( p / q ) ) ) ) )
Theorem	aaliou3lem1 23297*	Lemma for aaliou3 23306. (Contributed by Stefan O'Rear, 16-Nov-2014.)
|- G = ( c e. ( ZZ>= ` A ) |-> ( ( 2 ^ -u ( ! ` A ) ) x. ( ( 1 / 2 ) ^ ( c - A ) ) ) )   =>    |- ( ( A e. NN /\ B e. ( ZZ>= ` A ) ) -> ( G ` B ) e. RR )
Theorem	aaliou3lem2 23298*	Lemma for aaliou3 23306. (Contributed by Stefan O'Rear, 16-Nov-2014.)
|- G = ( c e. ( ZZ>= ` A ) |-> ( ( 2 ^ -u ( ! ` A ) ) x. ( ( 1 / 2 ) ^ ( c - A ) ) ) )   &    |- F = ( a e. NN |-> ( 2 ^ -u ( ! ` a ) ) )   =>    |- ( ( A e. NN /\ B e. ( ZZ>= ` A ) ) -> ( F ` B ) e. ( 0 (,] ( G ` B ) ) )
Theorem	aaliou3lem3 23299*	Lemma for aaliou3 23306. (Contributed by Stefan O'Rear, 16-Nov-2014.)
|- G = ( c e. ( ZZ>= ` A ) |-> ( ( 2 ^ -u ( ! ` A ) ) x. ( ( 1 / 2 ) ^ ( c - A ) ) ) )   &    |- F = ( a e. NN |-> ( 2 ^ -u ( ! ` a ) ) )   =>    |- ( A e. NN -> ( seq A ( + , F ) e. dom ~~> /\ sum_ b e. ( ZZ>= ` A ) ( F ` b ) e. RR+ /\ sum_ b e. ( ZZ>= ` A ) ( F ` b ) <_ ( 2 x. ( 2 ^ -u ( ! ` A ) ) ) ) )
Theorem	aaliou3lem8 23300*	Lemma for aaliou3 23306. (Contributed by Stefan O'Rear, 20-Nov-2014.)
|- ( ( A e. NN /\ B e. RR+ ) -> E. x e. NN ( 2 x. ( 2 ^ -u ( ! ` ( x + 1 ) ) ) ) <_ ( B / ( ( 2 ^ ( ! ` x ) ) ^ A ) ) )