P. 241 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 24001-24100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	0dgr 24001	A constant function has degree 0. (Contributed by Mario Carneiro, 24-Jul-2014.)
|- ( A e. CC -> (deg ` ( CC X. { A } ) ) = 0 )
Theorem	0dgrb 24002	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 24003	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 24004	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 24005	Lemma for coeadd 24007 and dgradd 24023. (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 24006*	Lemma for coemul 24008 and dgrmul 24026. (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 24007	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 24008*	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 24009	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 24010	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 24011	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 24012	The coefficients of the zero polynomial are zero. (Contributed by Mario Carneiro, 22-Jul-2014.)
|- (coeff ` 0p ) = ( NN0 X. { 0 } )
Theorem	coesub 24013	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 24014*	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 24015*	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 24016*	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 24017	A polynomial is a continuous function. (Contributed by Mario Carneiro, 23-Jul-2014.)
|- ( F e. (Poly ` S ) -> F e. ( CC -cn-> CC ) )
Theorem	dgr0 24018	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 23989, dgreq0 24021 and coeid 23994 without having to special-case zero, although plydivalg 24054 is a little more complicated as a result. (Contributed by Mario Carneiro, 22-Jul-2014.)
|- (deg ` 0p ) = 0
Theorem	coeidp 24019	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 24020	The degree of the identity function. (Contributed by Mario Carneiro, 26-Jul-2014.)
|- (deg ` Xp ) = 1
Theorem	dgreq0 24021	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 24022	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 24023	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 24024	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 24025	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 24026	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 24027	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 24028	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 24029*	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 24030*	Lemma for dgrco 24031. (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 24031	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 24032*	Lemma for plycj 24033 and coecj 24034. (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 24033*	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 24034	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 24035	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 24036	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 24037	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 24038	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 24039*	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 24040	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 24041	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 24042	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 24043	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 24044	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 24045	Extend class notation to include the quotient of a polynomial division.
class quot
Definition	df-quot 24046*	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 24047*	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 24048*	Lemma for plydivalg 24054. (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 24049*	Lemma for plydivalg 24054. (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 24050*	Lemma for plydivex 24052. 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 24051*	Lemma for plydivex 24052. 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 24052*	Lemma for plydivalg 24054. (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 24053*	Lemma for plydivalg 24054. (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 24054*	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 24055*	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 24056*	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 24057	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 24058	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 24059	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 24060	The polynomial remainder theorem, or little Bézout's theorem (by contrast to the regular Bézout's theorem bezout 15260). 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 24061	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 24062*	Lemma for fta1 24063. (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 24063	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 24064	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 24065*	Lemma for vieta1 24067. (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 24066*	Lemma for vieta1 24067: 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 24067*	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 24068*	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 24069	Extend class notation to include the set of algebraic numbers.
class AA
Definition	df-aa 24070	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 24071*	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 24072	An algebraic number is a complex number. (Contributed by Mario Carneiro, 23-Jul-2014.)
|- ( A e. AA -> A e. CC )
Theorem	aasscn 24073	The algebraic numbers are a subset of the complex numbers. (Contributed by Mario Carneiro, 23-Jul-2014.)
|- AA C_ CC
Theorem	elqaalem1 24074*	Lemma for elqaa 24077. 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 24075*	Lemma for elqaa 24077. (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 24076*	Lemma for elqaa 24077. (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	elqaa 24077*	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 24071 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 24078	Every rational number is algebraic. (Contributed by Mario Carneiro, 23-Jul-2014.)
|- ( A e. QQ -> A e. AA )
Theorem	qssaa 24079	The rational numbers are contained in the algebraic numbers. (Contributed by Mario Carneiro, 23-Jul-2014.)
|- QQ C_ AA
Theorem	iaa 24080	The imaginary unit is algebraic. (Contributed by Mario Carneiro, 23-Jul-2014.)
|- _i e. AA
Theorem	aareccl 24081	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 24082	The conjugate of an algebraic number is algebraic. (Contributed by Mario Carneiro, 24-Jul-2014.)
|- ( A e. AA -> ( * ` A ) e. AA )
Theorem	aannenlem1 24083*	Lemma for aannen 24086. (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 24084*	Lemma for aannen 24086. (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 24085*	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 24086	The algebraic numbers are countable. (Contributed by Stefan O'Rear, 16-Nov-2014.)
|- AA ~~ NN
14.1.6  Liouville's approximation theorem
Theorem	aalioulem1 24087	Lemma for aaliou 24093. 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 24088*	Lemma for aaliou 24093. (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 24089*	Lemma for aaliou 24093. (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 24090*	Lemma for aaliou 24093. (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 24091*	Lemma for aaliou 24093. (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 24092*	Lemma for aaliou 24093. (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 24093*	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 24094*	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 24095*	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 24096*	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 24097*	Lemma for aaliou3 24106. (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 24098*	Lemma for aaliou3 24106. (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 24099*	Lemma for aaliou3 24106. (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 24100*	Lemma for aaliou3 24106. (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 ) ) )