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Theorem List for Metamath Proof Explorer - 19501-19600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremplycpn 19501 Polynomials are smooth. (Contributed by Stefan O'Rear, 16-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
Poly

13.1.5  The Division algorithm for polynomials

Syntaxcquot 19502 Extend class notation to include the quotient of a polynomial division.
quot

Definitiondf-quot 19503* Define the quotient function on polynomials. This is the of the expression in the division algorithm. (Contributed by Mario Carneiro, 23-Jul-2014.)
quot Poly Poly Poly deg deg

Theoremquotval 19504* Value of the quotient function. (Contributed by Mario Carneiro, 23-Jul-2014.)
Poly Poly quot Poly deg deg

Theoremplydivlem1 19505* Lemma for plydivalg 19511. (Contributed by Mario Carneiro, 24-Jul-2014.)

Theoremplydivlem2 19506* Lemma for plydivalg 19511. (Contributed by Mario Carneiro, 24-Jul-2014.)
Poly       Poly                     Poly Poly

Theoremplydivlem3 19507* Lemma for plydivex 19509. Base case of induction. (Contributed by Mario Carneiro, 24-Jul-2014.)
Poly       Poly                     deg deg        Poly deg deg

Theoremplydivlem4 19508* Lemma for plydivex 19509. Induction step. (Contributed by Mario Carneiro, 26-Jul-2014.)
Poly       Poly                                                        Poly deg Poly deg        coeff       coeff       deg       deg       Poly deg

Theoremplydivex 19509* Lemma for plydivalg 19511. (Contributed by Mario Carneiro, 24-Jul-2014.)
Poly       Poly                     Poly deg deg

Theoremplydiveu 19510* Lemma for plydivalg 19511. (Contributed by Mario Carneiro, 24-Jul-2014.)
Poly       Poly                     Poly       deg deg              Poly       deg deg

Theoremplydivalg 19511* The division algorithm on polynomials over a subfield of the complex numbers. If and are polynomials over , then there is a unique quotient polynomial such that the remainder is either zero or has degree less than . (Contributed by Mario Carneiro, 26-Jul-2014.)
Poly       Poly                     Poly deg deg

Theoremquotlem 19512* Lemma for properties of the polynomial quotient function. (Contributed by Mario Carneiro, 26-Jul-2014.)
Poly       Poly              quot        quot Poly deg deg

Theoremquotcl 19513* The quotient of two polynomials in a field is also in the field. (Contributed by Mario Carneiro, 26-Jul-2014.)
Poly       Poly              quot Poly

Theoremquotcl2 19514 Closure of the quotient function. (Contributed by Mario Carneiro, 26-Jul-2014.)
Poly Poly quot Poly

Theoremquotdgr 19515 Remainder property of the quotient function. (Contributed by Mario Carneiro, 26-Jul-2014.)
quot        Poly Poly deg deg

Theoremplyremlem 19516 Closure of a linear factor. (Contributed by Mario Carneiro, 26-Jul-2014.)
Poly deg

Theoremplyrem 19517 The polynomial remainder theorem, or little Bézout's theorem (by contrast to the regular Bézout's theorem bezout 12595). If a polynomial is divided by the linear factor , the remainder is equal to , the evaluation of the polynomial at (interpreted as a constant polynomial). (Contributed by Mario Carneiro, 26-Jul-2014.)
quot        Poly

Theoremfacth 19518 The factor theorem. If a polynomial has a root at , then is a factor of (and the other factor is quot ). (Contributed by Mario Carneiro, 26-Jul-2014.)
Poly quot

Theoremfta1lem 19519* Lemma for fta1 19520. (Contributed by Mario Carneiro, 26-Jul-2014.)
Poly        deg               Poly deg deg       deg

Theoremfta1 19520 The easy direction of the Fundamental Theorem of Algebra: A nonzero polynomial has at most deg roots. (Contributed by Mario Carneiro, 26-Jul-2014.)
Poly deg

Theoremquotcan 19521 Exact division with a multiple. (Contributed by Mario Carneiro, 26-Jul-2014.)
Poly Poly quot

Theoremvieta1lem1 19522* Lemma for vieta1 19524. (Contributed by Mario Carneiro, 28-Jul-2014.)
coeff       deg              Poly                            Poly deg deg coeffdeg coeffdeg       quot        Poly deg

Theoremvieta1lem2 19523* Lemma for vieta1 19524: inductive step. Let be a root of . Then for some by the factor theorem, and is a degree- polynomial, so by the induction hypothesis coeff coeff, so coeff coeff. Now the coefficients of are coeff and coeff coeff , which works out to coeff coeff , so putting it all together we have as we wanted to show. (Contributed by Mario Carneiro, 28-Jul-2014.)
coeff       deg              Poly                            Poly deg deg coeffdeg coeffdeg       quot

Theoremvieta1 19524* The first-order Vieta's formula (see http://en.wikipedia.org/wiki/Vieta%27s_formulas). If a polynomial of degree has distinct roots, then the sum over these roots can be calculated as . (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.)
coeff       deg              Poly

Theoremplyexmo 19525* An infinite set of values can be extended to a polynomial in at most one way. (Contributed by Stefan O'Rear, 14-Nov-2014.)
Poly

13.1.6  Algebraic numbers

Syntaxcaa 19526 Extend class notation to include the set of algebraic numbers.

Definitiondf-aa 19527 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 ) of all polynomials in Poly, except the zero polynomial . (Contributed by Mario Carneiro, 22-Jul-2014.)
Poly

Theoremelaa 19528* Elementhood in the set of algebraic numbers. (Contributed by Mario Carneiro, 22-Jul-2014.)
Poly

Theoremaacn 19529 An algebraic number is a complex number. (Contributed by Mario Carneiro, 23-Jul-2014.)

Theoremaasscn 19530 The algebraic numbers are a subset of the complex numbers. (Contributed by Mario Carneiro, 23-Jul-2014.)

Theoremelqaalem1 19531* Lemma for elqaa 19534. The function represents the denominators of the rational coefficients . By multiplying them all together to make , we get a number big enough to clear all the denominators and make an integer polynomial. (Contributed by Mario Carneiro, 23-Jul-2014.)
Poly               coeff              deg

Theoremelqaalem2 19532* Lemma for elqaa 19534. (Contributed by Mario Carneiro, 23-Jul-2014.)
Poly               coeff              deg              deg

Theoremelqaalem3 19533* Lemma for elqaa 19534. (Contributed by Mario Carneiro, 23-Jul-2014.)
Poly               coeff              deg

Theoremelqaa 19534* 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 19528 are defined only in terms of polynomials over the integers. (Contributed by Mario Carneiro, 23-Jul-2014.)
Poly

Theoremqaa 19535 Every rational number is algebraic. (Contributed by Mario Carneiro, 23-Jul-2014.)

Theoremqssaa 19536 The rational numbers are contained in the algebraic numbers. (Contributed by Mario Carneiro, 23-Jul-2014.)

Theoremiaa 19537 The imaginary unit is algebraic. (Contributed by Mario Carneiro, 23-Jul-2014.)

Theoremaareccl 19538 The reciprocal of an algebraic number is algebraic. (Contributed by Mario Carneiro, 24-Jul-2014.)

Theoremaacjcl 19539 The conjugate of an algebraic number is algebraic. (Contributed by Mario Carneiro, 24-Jul-2014.)

Theoremaannenlem1 19540* Lemma for aannen 19543. (Contributed by Stefan O'Rear, 16-Nov-2014.)
Poly deg coeff

Theoremaannenlem2 19541* Lemma for aannen 19543. (Contributed by Stefan O'Rear, 16-Nov-2014.)
Poly deg coeff

Theoremaannenlem3 19542* The algebraic numbers are countable. (Contributed by Stefan O'Rear, 16-Nov-2014.)
Poly deg coeff

Theoremaannen 19543 The algebraic numbers are countable. (Contributed by Stefan O'Rear, 16-Nov-2014.)

13.1.7  Liouville's approximation theorem

Theoremaalioulem1 19544 Lemma for aaliou 19550. An integer polynomial cannot inflate the denominator of a rational by more than its degree. (Contributed by Stefan O'Rear, 12-Nov-2014.)
Poly                     deg

Theoremaalioulem2 19545* Lemma for aaliou 19550. (Contributed by Stefan O'Rear, 15-Nov-2014.)
deg       Poly

Theoremaalioulem3 19546* Lemma for aaliou 19550. (Contributed by Stefan O'Rear, 15-Nov-2014.)
deg       Poly

Theoremaalioulem4 19547* Lemma for aaliou 19550. (Contributed by Stefan O'Rear, 16-Nov-2014.)
deg       Poly

Theoremaalioulem5 19548* Lemma for aaliou 19550. (Contributed by Stefan O'Rear, 16-Nov-2014.)
deg       Poly

Theoremaalioulem6 19549* Lemma for aaliou 19550. (Contributed by Stefan O'Rear, 16-Nov-2014.)
deg       Poly

Theoremaaliou 19550* Liouville's theorem on diophantine approximation: Any algebraic number, being a root of a polynomial in integer coefficients, is not approximable beyond order deg by rational numbers. In this form, it also applies to rational numbers themselves, which are not well approximable by other rational numbers. (Contributed by Stefan O'Rear, 16-Nov-2014.)
deg       Poly

Theoremgeolim3 19551* Geometric series convergence with arbitrary shift, radix, and multiplicative constant. (Contributed by Stefan O'Rear, 16-Nov-2014.)

Theoremaaliou2 19552* Liouville's approximation theorem for algebraic numbers per se. (Contributed by Stefan O'Rear, 16-Nov-2014.)

Theoremaaliou2b 19553* Liouville's approximation theorem extended to complex . (Contributed by Stefan O'Rear, 20-Nov-2014.)

Theoremaaliou3lem1 19554* Lemma for aaliou3 19563. (Contributed by Stefan O'Rear, 16-Nov-2014.)

Theoremaaliou3lem2 19555* Lemma for aaliou3 19563. (Contributed by Stefan O'Rear, 16-Nov-2014.)

Theoremaaliou3lem3 19556* Lemma for aaliou3 19563. (Contributed by Stefan O'Rear, 16-Nov-2014.)

Theoremaaliou3lem8 19557* Lemma for aaliou3 19563. (Contributed by Stefan O'Rear, 20-Nov-2014.)

Theoremaaliou3lem4 19558* Lemma for aaliou3 19563. (Contributed by Stefan O'Rear, 16-Nov-2014.)

Theoremaaliou3lem5 19559* Lemma for aaliou3 19563. (Contributed by Stefan O'Rear, 16-Nov-2014.)

Theoremaaliou3lem6 19560* Lemma for aaliou3 19563. (Contributed by Stefan O'Rear, 16-Nov-2014.)

Theoremaaliou3lem7 19561* Lemma for aaliou3 19563. (Contributed by Stefan O'Rear, 16-Nov-2014.)

Theoremaaliou3lem9 19562* Example of a "Liouville number", a very simple definable transcendental real. (Contributed by Stefan O'Rear, 20-Nov-2014.)

Theoremaaliou3 19563 Example of a "Liouville number", a very simple definable transcendental real. (Contributed by Stefan O'Rear, 23-Nov-2014.)

13.2  Sequences and series

13.2.1  Taylor polynomials and Taylor's theorem

Syntaxctayl 19564 Taylor polynomial of a function.
Tayl

Syntaxcana 19565 The class of analytic functions.
Ana

Definitiondf-tayl 19566* Define the Taylor polynomial or Taylor series of a function. (Contributed by Mario Carneiro, 30-Dec-2016.)
Tayl fld tsums

Definitiondf-ana 19567* Define the set of analytic functions, which are functions such that the Taylor series of the function at each point converges to the function in some neighborhood of the point. (Contributed by Mario Carneiro, 31-Dec-2016.)
Ana fldt Tayl

Theoremtaylfvallem1 19568* Lemma for taylfval 19570. (Contributed by Mario Carneiro, 30-Dec-2016.)

Theoremtaylfvallem 19569* Lemma for taylfval 19570. (Contributed by Mario Carneiro, 30-Dec-2016.)
fld tsums

Theoremtaylfval 19570* Define the Taylor polynomial of a function. The constant Tayl is a function of five arguments: is the base set with respect to evaluate the derivatives (generally or ), is the function we are approximating, at point , to order . The result is a polynomial function of .

This "extended" version of taylpfval 19576 additionally handles the case , in which case this is not a polynomial but an infinite series, the Taylor series of the function. (Contributed by Mario Carneiro, 30-Dec-2016.)

Tayl        fld tsums

Theoremeltayl 19571* Value of the Taylor series as a relation (elementhood in the domain here expresses that the series is convergent). (Contributed by Mario Carneiro, 30-Dec-2016.)
Tayl        fld tsums

Theoremtaylf 19572* The Taylor series defines a function on a subset of the complexes. (Contributed by Mario Carneiro, 30-Dec-2016.)
Tayl

Theoremtayl0 19573* The Taylor series is always defined at the basepoint, with value equal to the value of the function. (Contributed by Mario Carneiro, 30-Dec-2016.)
Tayl

Theoremtaylplem1 19574* Lemma for taylpfval 19576 and similar theorems. (Contributed by Mario Carneiro, 31-Dec-2016.)

Theoremtaylplem2 19575* Lemma for taylpfval 19576 and similar theorems. (Contributed by Mario Carneiro, 31-Dec-2016.)

Theoremtaylpfval 19576* Define the Taylor polynomial of a function. The constant Tayl is a function of five arguments: is the base set with respect to evaluate the derivatives (generally or ), is the function we are approximating, at point , to order . The result is a polynomial function of . (Contributed by Mario Carneiro, 31-Dec-2016.)
Tayl

Theoremtaylpf 19577 The Taylor polynomial is a function on the complexes (even if the base set of the original function is the reals). (Contributed by Mario Carneiro, 31-Dec-2016.)
Tayl

Theoremtaylpval 19578* Value of the Taylor polynomial. (Contributed by Mario Carneiro, 31-Dec-2016.)
Tayl

Theoremtaylply2 19579* The Taylor polynomial is a polynomial of degree (at most) . This version of taylply 19580 shows that the coefficients of are in a subring of the complexes. (Contributed by Mario Carneiro, 1-Jan-2017.)
Tayl        SubRingfld                     Poly deg

Theoremtaylply 19580 The Taylor polynomial is a polynomial of degree (at most) . (Contributed by Mario Carneiro, 31-Dec-2016.)
Tayl        Poly deg

Theoremdvtaylp 19581 The derivative of the Taylor polynomial is the Taylor polynomial of the derivative of the function. (Contributed by Mario Carneiro, 31-Dec-2016.)
Tayl Tayl

Theoremdvntaylp 19582 The -th derivative of the Taylor polynomial is the Taylor polynomial of the -th derivative of the function. (Contributed by Mario Carneiro, 1-Jan-2017.)
Tayl Tayl

Theoremdvntaylp0 19583 The first derivatives of the Taylor polynomial at match the derivatives of the function from which it is derived. (Contributed by Mario Carneiro, 1-Jan-2017.)
Tayl

Theoremtaylthlem1 19584* Lemma for taylth 19586. This is the main part of Taylor's theorem, except for the induction step, which is supposed to be proven using L'Hôpital's rule. However, since our proof of L'Hôpital assumes that , we can only do this part generically, and for taylth 19586 itself we must restrict to . (Contributed by Mario Carneiro, 1-Jan-2017.)
Tayl               ..^ lim lim        lim

Theoremtaylthlem2 19585* Lemma for taylth 19586. (Contributed by Mario Carneiro, 1-Jan-2017.)
Tayl        ..^       lim        lim

Theoremtaylth 19586* Taylor's theorem. The Taylor polynomial of a -times differentiable function is such that the error term goes to zero faster than . (Contributed by Mario Carneiro, 1-Jan-2017.)
Tayl               lim

13.2.2  Uniform convergence

Syntaxculm 19587 Extend class notation to include the uniform convergence predicate.

Definitiondf-ulm 19588* Define the uniform convergence of a sequence of functions. Here if is a sequence of functions defined on and is a function on , and for every there is a such that the functions for are all uniformly within of on the domain . Compare with df-clim 11839. (Contributed by Mario Carneiro, 26-Feb-2015.)

Theoremulmrel 19589 The uniform limit relation is a relation. (Contributed by Mario Carneiro, 26-Feb-2015.)

Theoremulmscl 19590 Closure of the base set in a uniform limit. (Contributed by Mario Carneiro, 26-Feb-2015.)

Theoremulmval 19591* Express the predicate: The sequence of functions converges uniformly to on . (Contributed by Mario Carneiro, 26-Feb-2015.)

Theoremulmcl 19592 Closure of a uniform limit of functions. (Contributed by Mario Carneiro, 26-Feb-2015.)

Theoremulmf 19593* Closure of a uniform limit of functions. (Contributed by Mario Carneiro, 26-Feb-2015.)

Theoremulmpm 19594 Closure of a uniform limit of functions. (Contributed by Mario Carneiro, 26-Feb-2015.)

Theoremulmf2 19595 Closure of a uniform limit of functions. (Contributed by Mario Carneiro, 18-Mar-2015.)

Theoremulm2 19596* Simplify ulmval 19591 when and are known to be functions. (Contributed by Mario Carneiro, 26-Feb-2015.)

Theoremulmi 19597* The uniform limit property. (Contributed by Mario Carneiro, 27-Feb-2015.)

Theoremulmclm 19598* A uniform limit of functions converges pointwise. (Contributed by Mario Carneiro, 27-Feb-2015.)

Theoremulmres 19599 A sequence of functions converges iff the tail of the sequence converges (for any finite cutoff). (Contributed by Mario Carneiro, 24-Mar-2015.)

Theoremulmshftlem 19600* Lemma for ulmshft 19601. (Contributed by Mario Carneiro, 24-Mar-2015.)

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