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Theorem List for Metamath Proof Explorer - 19401-19500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremftc1lem3 19401* Lemma for ftc1 19405. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 8-Sep-2015.)
t        t        fld

Theoremftc1lem4 19402* Lemma for ftc1 19405. (Contributed by Mario Carneiro, 31-Aug-2014.)
t        t        fld

Theoremftc1lem5 19403* Lemma for ftc1 19405. (Contributed by Mario Carneiro, 14-Aug-2014.) (Revised by Mario Carneiro, 28-Dec-2016.)
t        t        fld

Theoremftc1lem6 19404* Lemma for ftc1 19405. (Contributed by Mario Carneiro, 14-Aug-2014.) (Proof shortened by Mario Carneiro, 28-Dec-2016.)
t        t        fld              lim

Theoremftc1 19405* The Fundamental Theorem of Calculus, part one. The function formed by varying the right endpoint of an integral is differentiable at with derivative if the original function is continuous at . (Contributed by Mario Carneiro, 1-Sep-2014.)
t        t        fld

Theoremftc1cn 19406* Strengthen the assumptions of ftc1 19405 to when the function is continuous on the entire interval ; in this case we can calculate exactly. (Contributed by Mario Carneiro, 1-Sep-2014.)

Theoremftc2 19407* The Fundamental Theorem of Calculus, part two. If is a function continuous on and continuously differentiable on , then the integral of the derivative of is equal to . (Contributed by Mario Carneiro, 2-Sep-2014.)

Theoremftc2ditglem 19408* Lemma for ftc2ditg 19409. (Contributed by Mario Carneiro, 3-Sep-2014.)
_

Theoremftc2ditg 19409* Directed integral analog of ftc2 19407. (Contributed by Mario Carneiro, 3-Sep-2014.)
_

Theoremitgparts 19410* Integration by parts. If is the derivative of and is the derivative of , and and , then under suitable integrability and differentiability assumptions, the integral of from to is equal to minus the integral of . (Contributed by Mario Carneiro, 3-Sep-2014.)

Theoremitgsubstlem 19411* Lemma for itgsubst 19412. (Contributed by Mario Carneiro, 12-Sep-2014.)
_ _

Theoremitgsubst 19412* Integration by -substitution. If is a continuous, differentiable function from to , whose derivative is continuous and integrable, and is a continuous function on , then the integral of from to is equal to the integral of from to . In this part of the proof we discharge the assumptions in itgsubstlem 19411, which use the fact that is open to shrink the interval a little to where - this is possible because is a continuous function on a closed interval, so its range is in fact a closed interval, and we have some wiggle room on the edges. (Contributed by Mario Carneiro, 7-Sep-2014.)
_ _

PART 13  BASIC REAL AND COMPLEX FUNCTIONS

13.1  Polynomials

13.1.1  Abstract polynomials, continued

Theoremevlslem6 19413* Lemma for evlseu 19416. Finiteness and consistency of the top-level sum. (Contributed by Stefan O'Rear, 9-Mar-2015.)
mPoly                                    mulGrp       .g              mVar        g g                             RingHom                      g g

Theoremevlslem3 19414* Lemma for evlseu 19416. Polynomial evaluation of a scaled monomial. (Contributed by Stefan O'Rear, 8-Mar-2015.)
mPoly                                    mulGrp       .g              mVar        g g                             RingHom                                    g

Theoremevlslem1 19415* Lemma for evlseu 19416, give a formula for (the unique) polynomial evaluation homomorphism. (Contributed by Stefan O'Rear, 9-Mar-2015.)
mPoly                                    mulGrp       .g              mVar        g g                             RingHom               algSc       RingHom

Theoremevlseu 19416* For a given intepretation of the variables and of the scalars , this extends to a homomorphic interpretation of the polynomial ring in exactly one way. (Contributed by Stefan O'Rear, 9-Mar-2015.)
mPoly               algSc       mVar                             RingHom               RingHom

Theoremreldmevls 19417 Well-behaved binary operation property of evalSub. (Contributed by Stefan O'Rear, 19-Mar-2015.)
evalSub

Theoremmpfrcl 19418 Reverse closure for the set of polynomial functions. (Contributed by Stefan O'Rear, 19-Mar-2015.)
evalSub        SubRing

Theoremevlsval 19419* Value of the polynomial evaluation map function. (Contributed by Stefan O'Rear, 11-Mar-2015.)
evalSub        mPoly        mVar        s        s               algSc                     SubRing RingHom

Theoremevlsval2 19420* Characterizing properties of the polynomial evaluation map function. (Contributed by Stefan O'Rear, 12-Mar-2015.)
evalSub        mPoly        mVar        s        s               algSc                     SubRing RingHom

Theoremevlsrhm 19421 Polynomial evaluation is a homomorphism (into the product ring). (Contributed by Stefan O'Rear, 12-Mar-2015.)
evalSub        mPoly        s        s               SubRing RingHom

Theoremevlssca 19422 Polynomial evaluation maps scalars to constant functions. (Contributed by Stefan O'Rear, 13-Mar-2015.)
evalSub        mPoly        s               algSc                     SubRing

Theoremevlsvar 19423* Polynomial evaluation maps variables to projections. (Contributed by Stefan O'Rear, 12-Mar-2015.)
evalSub        mVar        s                             SubRing

Theoremevlval 19424 Value of the simple/same ring evaluation map. (Contributed by Stefan O'Rear, 19-Mar-2015.) (Revised by Mario Carneiro, 12-Jun-2015.)
eval               evalSub

Theoremevlrhm 19425 The simple evaluation map is a ring homomorphism. (Contributed by Mario Carneiro, 12-Jun-2015.)
eval               mPoly        s        RingHom

Theoremevl1fval 19426* Value of the simple/same ring evalutation map. (Contributed by Mario Carneiro, 12-Jun-2015.)
eval1       eval

Theoremevl1val 19427* Value of the simple/same ring evalutation map. (Contributed by Mario Carneiro, 12-Jun-2015.)
eval1       eval               mPoly

Theoremevl1rhm 19428 Polynomial evaluation is a homomorphism (into the product ring). (Contributed by Mario Carneiro, 12-Jun-2015.)
eval1       Poly1       s               RingHom

Theoremevl1sca 19429 Polynomial evaluation maps scalars to constant functions. (Contributed by Mario Carneiro, 12-Jun-2015.)
eval1       Poly1              algSc

Theoremevl1scad 19430 Polynomial evaluation builder for scalars. (Contributed by Mario Carneiro, 4-Jul-2015.)
eval1       Poly1              algSc

Theoremevl1var 19431 Polynomial evaluation maps the variable to the identity function. (Contributed by Mario Carneiro, 12-Jun-2015.)
eval1       var1

Theoremevl1vard 19432 Polynomial evaluation builder for the variable. (Contributed by Mario Carneiro, 4-Jul-2015.)
eval1       var1              Poly1

Theoremevl1addd 19433 Polynomial evaluation builder for addition of polynomials. (Contributed by Mario Carneiro, 4-Jul-2015.)
eval1       Poly1

Theoremevl1subd 19434 Polynomial evaluation builder for subtraction of polynomials. (Contributed by Mario Carneiro, 4-Jul-2015.)
eval1       Poly1

Theoremevl1muld 19435 Polynomial evaluation builder for multiplication of polynomials. (Contributed by Mario Carneiro, 4-Jul-2015.)
eval1       Poly1

Theoremevl1vsd 19436 Polynomial evaluation builder for scalar multiplication of polynomials. (Contributed by Mario Carneiro, 4-Jul-2015.)
eval1       Poly1

Theoremevl1expd 19437 Polynomial evaluation builder for an exponential. (Contributed by Mario Carneiro, 12-Jun-2015.)
eval1       Poly1                                          .gmulGrp       .gmulGrp

Theoremmpfconst 19438 Constants are multivariate polynomial functions. (Contributed by Mario Carneiro, 19-Mar-2015.)
evalSub                      SubRing

Theoremmpfproj 19439* Projections are multivariate polynomial functions. (Contributed by Mario Carneiro, 20-Mar-2015.)
evalSub                      SubRing

Theoremmpfsubrg 19440 Polynomial functions are a subring. (Contributed by Mario Carneiro, 19-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.)
evalSub        SubRing SubRing s

Theoremmpff 19441 Polynomial functions are functions. (Contributed by Mario Carneiro, 19-Mar-2015.)
evalSub

Theoremmpfaddcl 19442 The sum of multivariate polynomial functions. (Contributed by Mario Carneiro, 19-Mar-2015.)
evalSub

Theoremmpfmulcl 19443 The product of multivariate polynomial functions. (Contributed by Mario Carneiro, 19-Mar-2015.)
evalSub

Theoremmpfind 19444* Prove a property of polynomials by "structural" induction, under a simplified model of structure which loses the sum of products structure. (Contributed by Mario Carneiro, 19-Mar-2015.)
evalSub

Theorempf1const 19445 Constants are polynomial functions. (Contributed by Mario Carneiro, 12-Jun-2015.)
eval1

Theorempf1id 19446 The identity is a polynomial function. (Contributed by Mario Carneiro, 20-Mar-2015.)
eval1

Theorempf1subrg 19447 Polynomial functions are a subring. (Contributed by Mario Carneiro, 19-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.)
eval1       SubRing s

Theorempf1rcl 19448 Reverse closure for the set of polynomial functions. (Contributed by Mario Carneiro, 12-Jun-2015.)
eval1

Theorempf1f 19449 Polynomial functions are functions. (Contributed by Mario Carneiro, 12-Jun-2015.)
eval1

Theoremmpfpf1 19450* Convert a multivariate polynomial function to univariate. (Contributed by Mario Carneiro, 12-Jun-2015.)
eval1              eval

Theorempf1mpf 19451* Convert a univariate polynomial function to multivariate. (Contributed by Mario Carneiro, 12-Jun-2015.)
eval1              eval

Theorempf1addcl 19452 The sum of multivariate polynomial functions. (Contributed by Mario Carneiro, 12-Jun-2015.)
eval1

Theorempf1mulcl 19453 The product of multivariate polynomial functions. (Contributed by Mario Carneiro, 12-Jun-2015.)
eval1

Theorempf1ind 19454* Prove a property of polynomials by "structural" induction, under a simplified model of structure which loses the sum of products structure. (Contributed by Mario Carneiro, 12-Jun-2015.)
eval1

13.1.2  Polynomial degrees

Syntaxcmdg 19455 Multivariate polynomial degree.
mDeg

Syntaxcdg1 19456 Univariate polynomial degree.
deg1

Definitiondf-mdeg 19457* Define the degree of a polynomial. Note (SO): as an experiment I am using a definition which makes the degree of the zero polynomial , contrary to the convention used in df-dgr 19589. (Contributed by Stefan O'Rear, 19-Mar-2015.)
mDeg mPoly fld g

Definitiondf-deg1 19458 Define the degree of a univariate polynomial. (Contributed by Stefan O'Rear, 23-Mar-2015.)
deg1 mDeg

Theoremreldmmdeg 19459 Multivariate degree is a binary operation. (Contributed by Stefan O'Rear, 28-Mar-2015.)
mDeg

Theoremtdeglem1 19460* Functionality of the total degree helper function. (Contributed by Stefan O'Rear, 19-Mar-2015.)
fld g

Theoremtdeglem3 19461* Additivity of the total degree helper function. (Contributed by Stefan O'Rear, 26-Mar-2015.)
fld g

Theoremtdeglem4 19462* There is only one multi-index with total degree 0. (Contributed by Stefan O'Rear, 29-Mar-2015.)
fld g

Theoremtdeglem2 19463 Simplification of total degree for the univariate case. (Contributed by Stefan O'Rear, 23-Mar-2015.)
fld g

Theoremmdegfval 19464* Value of the multivariate degree function. (Contributed by Stefan O'Rear, 19-Mar-2015.)
mDeg        mPoly                             fld g

Theoremmdegval 19465* Value of the multivariate degree function at some particular polynomial. (Contributed by Stefan O'Rear, 19-Mar-2015.)
mDeg        mPoly                             fld g

Theoremmdegleb 19466* Property of being of limited degree. (Contributed by Stefan O'Rear, 19-Mar-2015.)
mDeg        mPoly                             fld g

Theoremmdeglt 19467* If there is an upper limit on the degree of a polynomial that is lower than the degree of some exponent bag, then that exponent bag is unrepresented in the polynomial. (Contributed by Stefan O'Rear, 26-Mar-2015.)
mDeg        mPoly                             fld g

Theoremmdegldg 19468* A nonzero polynomial has some coefficient which witnesses its degree. (Contributed by Stefan O'Rear, 23-Mar-2015.)
mDeg        mPoly                             fld g

Theoremmdegxrcl 19469 Closure of polynomial degree in the extended reals. (Contributed by Stefan O'Rear, 19-Mar-2015.)
mDeg        mPoly

Theoremmdegxrf 19470 Functionality of polynomial degree in the extended reals. (Contributed by Stefan O'Rear, 19-Mar-2015.)
mDeg        mPoly

Theoremmdegcl 19471 Sharp closure for multivariate polynomials. (Contributed by Stefan O'Rear, 23-Mar-2015.)
mDeg        mPoly

Theoremmdeg0 19472 Degree of the zero polynomial. (Contributed by Stefan O'Rear, 20-Mar-2015.)
mDeg        mPoly

Theoremmdegnn0cl 19473 Degree of a nonzero polynomial. (Contributed by Stefan O'Rear, 23-Mar-2015.)
mDeg        mPoly

Theoremdegltlem1 19474 Theorem on arithmetic of extended reals useful for degrees. (Contributed by Stefan O'Rear, 23-Mar-2015.)

Theoremdegltp1le 19475 Theorem on arithmetic of extended reals useful for degrees. (Contributed by Stefan O'Rear, 1-Apr-2015.)

Theoremmdegaddle 19476 The degree of a sum is at most the maximum of the degrees of the factors. (Contributed by Stefan O'Rear, 26-Mar-2015.)
mPoly        mDeg

Theoremmdegvscale 19477 The degree of a scalar multiple of a polynomial is at most the degree of the original polynomial. (Contributed by Stefan O'Rear, 26-Mar-2015.)
mPoly        mDeg

Theoremmdegvsca 19478 The degree of a scalar multiple of a polynomial is exactly the degree of the original polynomial when the multiple is a non-zero-divisor. (Contributed by Stefan O'Rear, 28-Mar-2015.)
mPoly        mDeg                             RLReg

Theoremmdegle0 19479 A polynomial has nonpositive degree iff it is a constant. (Contributed by Stefan O'Rear, 29-Mar-2015.)
mPoly        mDeg                             algSc

Theoremmdegmullem 19480* Lemma for mdegmulle2 19481. (Contributed by Stefan O'Rear, 26-Mar-2015.)
mPoly        mDeg                                                                                     fld g

Theoremmdegmulle2 19481 The multivariate degree of a product of polynomials is at most the sum of the degrees of the polynomials. (Contributed by Stefan O'Rear, 26-Mar-2015.)
mPoly        mDeg

Theoremdeg1fval 19482 Relate univariate polynomial degree to multivariate. (Contributed by Stefan O'Rear, 23-Mar-2015.) (Revised by Mario Carneiro, 7-Oct-2015.)
deg1        mDeg

Theoremdeg1xrf 19483 Functionality of univariate polynomial degree, weak range. (Contributed by Stefan O'Rear, 23-Mar-2015.)
deg1        Poly1

Theoremdeg1xrcl 19484 Closure of univariate polynomial degree in extended reals. (Contributed by Stefan O'Rear, 23-Mar-2015.)
deg1        Poly1

Theoremdeg1cl 19485 Sharp closure of univariate polynomial degree. (Contributed by Stefan O'Rear, 23-Mar-2015.)
deg1        Poly1

Theoremmdegpropd 19486* Property deduction for polynomial degree. (Contributed by Stefan O'Rear, 28-Mar-2015.)
mDeg mDeg

Theoremdeg1fvi 19487 Univariate polynomial degree respects protection. (Contributed by Stefan O'Rear, 28-Mar-2015.)
deg1 deg1

Theoremdeg1propd 19488* Property deduction for polynomial degree. (Contributed by Stefan O'Rear, 28-Mar-2015.)
deg1 deg1

Theoremdeg1z 19489 Degree of the zero univariate polynomial. (Contributed by Stefan O'Rear, 23-Mar-2015.)
deg1        Poly1

Theoremdeg1nn0cl 19490 Degree of a nonzero univariate polynomial. (Contributed by Stefan O'Rear, 23-Mar-2015.) (Revised by Mario Carneiro, 7-Oct-2015.)
deg1        Poly1

Theoremdeg1n0ima 19491 Degree image of a set of polynomials whcih does not include zero. (Contributed by Stefan O'Rear, 28-Mar-2015.)
deg1        Poly1

Theoremdeg1nn0clb 19492 A polynomial is nonzero iff it has definite degree. (Contributed by Stefan O'Rear, 23-Mar-2015.)
deg1        Poly1

Theoremdeg1lt0 19493 A polynomial is zero iff it has negative degree. (Contributed by Stefan O'Rear, 1-Apr-2015.)
deg1        Poly1

Theoremdeg1ldg 19494 A nonzero univariate polynomial always has a nonzero leading coefficient. (Contributed by Stefan O'Rear, 23-Mar-2015.)
deg1        Poly1                            coe1

Theoremdeg1ldgn 19495 An index at which a polynomial is zero, cannot be its degree. (Contributed by Stefan O'Rear, 23-Mar-2015.)
deg1        Poly1                            coe1

Theoremdeg1ldgdomn 19496 A nonzero univariate polynomial over a domain always has a non-zero-divisor leading coefficient. (Contributed by Stefan O'Rear, 28-Mar-2015.)
deg1        Poly1                     RLReg       coe1       Domn

Theoremdeg1leb 19497* Property of being of limited degree. (Contributed by Stefan O'Rear, 23-Mar-2015.)
deg1        Poly1                     coe1

Theoremdeg1val 19498 Value of the univariate degree as a supremum. (Contributed by Stefan O'Rear, 29-Mar-2015.)
deg1        Poly1                     coe1

Theoremdeg1lt 19499 If the degree of a univariate polynomial is less than some index, then that coefficient must be zero. (Contributed by Stefan O'Rear, 23-Mar-2015.)
deg1        Poly1                     coe1

Theoremdeg1ge 19500 Conversely, a nonzero coefficient sets a lower bound on the degree. (Contributed by Stefan O'Rear, 23-Mar-2015.)
deg1        Poly1                     coe1

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