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Theorem List for Metamath Proof Explorer - 19201-19300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdvfsumge 19201* Compare a finite sum to an integral (the integral here is given as a function with a known derivative). (Contributed by Mario Carneiro, 14-May-2016.)
..^        ..^        ..^

Theoremdvfsumabs 19202* Compare a finite sum to an integral (the integral here is given as a function with a known derivative). (Contributed by Mario Carneiro, 14-May-2016.)
..^        ..^        ..^        ..^ ..^

Theoremdvmptrecl 19203* Real closure of a derivative. (Contributed by Mario Carneiro, 18-May-2016.)

Theoremdvfsumrlimf 19204* Lemma for dvfsumrlim 19210. (Contributed by Mario Carneiro, 18-May-2016.)

Theoremdvfsumlem1 19205* Lemma for dvfsumrlim 19210. (Contributed by Mario Carneiro, 17-May-2016.)

Theoremdvfsumlem2 19206* Lemma for dvfsumrlim 19210. (Contributed by Mario Carneiro, 17-May-2016.)

Theoremdvfsumlem3 19207* Lemma for dvfsumrlim 19210. (Contributed by Mario Carneiro, 17-May-2016.)

Theoremdvfsumlem4 19208* Lemma for dvfsumrlim 19210. (Contributed by Mario Carneiro, 18-May-2016.)

Theoremdvfsumrlimge0 19209* Lemma for dvfsumrlim 19210. Satisfy the assumption of dvfsumlem4 19208. (Contributed by Mario Carneiro, 18-May-2016.)

Theoremdvfsumrlim 19210* Compare a finite sum to an integral (the integral here is given as a function with a known derivative). The statement here says that if is a decreasing function with antiderivative converging to zero, then the difference between and converges to a constant limit value, with the remainder term bounded by . (Contributed by Mario Carneiro, 18-May-2016.)

Theoremdvfsumrlim2 19211* Compare a finite sum to an integral (the integral here is given as a function with a known derivative). The statement here says that if is a decreasing function with antiderivative converging to zero, then the difference between and converges to a constant limit value, with the remainder term bounded by . (Contributed by Mario Carneiro, 18-May-2016.)

Theoremdvfsumrlim3 19212* Conjoin the statements of dvfsumrlim 19210 and dvfsumrlim2 19211. (This is useful as a target for lemmas, because the hypotheses to this theorem are complex and we don't want to repeat ourselves.) (Contributed by Mario Carneiro, 18-May-2016.)

Theoremdvfsum2 19213* The reverse of dvfsumrlim 19210, when comparing a finite sum of increasing terms to an integral. In this case there is no point in stating the limit properties, because the terms of the sum aren't approaching zero, but there is nevertheless still a natural asymptotic statement that can be made. (Contributed by Mario Carneiro, 20-May-2016.)

Theoremftc1lem1 19214* Lemma for ftc1a 19216 and ftc1 19221. (Contributed by Mario Carneiro, 31-Aug-2014.)

Theoremftc1lem2 19215* Lemma for ftc1 19221. (Contributed by Mario Carneiro, 12-Aug-2014.)

Theoremftc1a 19216* The Fundamental Theorem of Calculus, part one. The function formed by varying the right endpoint of an integral of is continuous if is integrable. (Contributed by Mario Carneiro, 1-Sep-2014.)

Theoremftc1lem3 19217* Lemma for ftc1 19221. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 8-Sep-2015.)
t        t        fld

Theoremftc1lem4 19218* Lemma for ftc1 19221. (Contributed by Mario Carneiro, 31-Aug-2014.)
t        t        fld

Theoremftc1lem5 19219* Lemma for ftc1 19221. (Contributed by Mario Carneiro, 14-Aug-2014.) (Revised by Mario Carneiro, 28-Dec-2016.)
t        t        fld

Theoremftc1lem6 19220* Lemma for ftc1 19221. (Contributed by Mario Carneiro, 14-Aug-2014.) (Proof shortened by Mario Carneiro, 28-Dec-2016.)
t        t        fld              lim

Theoremftc1 19221* 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 19222* Strengthen the assumptions of ftc1 19221 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 19223* 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 19224* Lemma for ftc2ditg 19225. (Contributed by Mario Carneiro, 3-Sep-2014.)
_

Theoremftc2ditg 19225* Directed integral analog of ftc2 19223. (Contributed by Mario Carneiro, 3-Sep-2014.)
_

Theoremitgparts 19226* 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 19227* Lemma for itgsubst 19228. (Contributed by Mario Carneiro, 12-Sep-2014.)
_ _

Theoremitgsubst 19228* 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 19227, 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 19229* Lemma for evlseu 19232. 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 19230* Lemma for evlseu 19232. Polynomial evaluation of a scaled monomial. (Contributed by Stefan O'Rear, 8-Mar-2015.)
mPoly                                    mulGrp       .g              mVar        g g                             RingHom                                    g

Theoremevlslem1 19231* Lemma for evlseu 19232, 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 19232* 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 19233 Well-behaved binary operation property of evalSub. (Contributed by Stefan O'Rear, 19-Mar-2015.)
evalSub

Theoremmpfrcl 19234 Reverse closure for the set of polynomial functions. (Contributed by Stefan O'Rear, 19-Mar-2015.)
evalSub        SubRing

Theoremevlsval 19235* Value of the polynomial evaluation map function. (Contributed by Stefan O'Rear, 11-Mar-2015.)
evalSub        mPoly        mVar        s        s               algSc                     SubRing RingHom

Theoremevlsval2 19236* 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 19237 Polynomial evaluation is a homomorphism (into the product ring). (Contributed by Stefan O'Rear, 12-Mar-2015.)
evalSub        mPoly        s        s               SubRing RingHom

Theoremevlssca 19238 Polynomial evaluation maps scalars to constant functions. (Contributed by Stefan O'Rear, 13-Mar-2015.)
evalSub        mPoly        s               algSc                     SubRing

Theoremevlsvar 19239* Polynomial evaluation maps variables to projections. (Contributed by Stefan O'Rear, 12-Mar-2015.)
evalSub        mVar        s                             SubRing

Theoremevlval 19240 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 19241 The simple evaluation map is a ring homomorphism. (Contributed by Mario Carneiro, 12-Jun-2015.)
eval               mPoly        s        RingHom

Theoremevl1fval 19242* Value of the simple/same ring evalutation map. (Contributed by Mario Carneiro, 12-Jun-2015.)
eval1       eval

Theoremevl1val 19243* Value of the simple/same ring evalutation map. (Contributed by Mario Carneiro, 12-Jun-2015.)
eval1       eval               mPoly

Theoremevl1rhm 19244 Polynomial evaluation is a homomorphism (into the product ring). (Contributed by Mario Carneiro, 12-Jun-2015.)
eval1       Poly1       s               RingHom

Theoremevl1sca 19245 Polynomial evaluation maps scalars to constant functions. (Contributed by Mario Carneiro, 12-Jun-2015.)
eval1       Poly1              algSc

Theoremevl1scad 19246 Polynomial evaluation builder for scalars. (Contributed by Mario Carneiro, 4-Jul-2015.)
eval1       Poly1              algSc

Theoremevl1var 19247 Polynomial evaluation maps the variable to the identity function. (Contributed by Mario Carneiro, 12-Jun-2015.)
eval1       var1

Theoremevl1vard 19248 Polynomial evaluation builder for the variable. (Contributed by Mario Carneiro, 4-Jul-2015.)
eval1       var1              Poly1

Theoremevl1addd 19249 Polynomial evaluation builder for addition of polynomials. (Contributed by Mario Carneiro, 4-Jul-2015.)
eval1       Poly1

Theoremevl1subd 19250 Polynomial evaluation builder for subtraction of polynomials. (Contributed by Mario Carneiro, 4-Jul-2015.)
eval1       Poly1

Theoremevl1muld 19251 Polynomial evaluation builder for multiplication of polynomials. (Contributed by Mario Carneiro, 4-Jul-2015.)
eval1       Poly1

Theoremevl1vsd 19252 Polynomial evaluation builder for scalar multiplication of polynomials. (Contributed by Mario Carneiro, 4-Jul-2015.)
eval1       Poly1

Theoremevl1expd 19253 Polynomial evaluation builder for an exponential. (Contributed by Mario Carneiro, 12-Jun-2015.)
eval1       Poly1                                          .gmulGrp       .gmulGrp

Theoremmpfconst 19254 Constants are multivariate polynomial functions. (Contributed by Mario Carneiro, 19-Mar-2015.)
evalSub                      SubRing

Theoremmpfproj 19255* Projections are multivariate polynomial functions. (Contributed by Mario Carneiro, 20-Mar-2015.)
evalSub                      SubRing

Theoremmpfsubrg 19256 Polynomial functions are a subring. (Contributed by Mario Carneiro, 19-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.)
evalSub        SubRing SubRing s

Theoremmpff 19257 Polynomial functions are functions. (Contributed by Mario Carneiro, 19-Mar-2015.)
evalSub

Theoremmpfaddcl 19258 The sum of multivariate polynomial functions. (Contributed by Mario Carneiro, 19-Mar-2015.)
evalSub

Theoremmpfmulcl 19259 The product of multivariate polynomial functions. (Contributed by Mario Carneiro, 19-Mar-2015.)
evalSub

Theoremmpfind 19260* 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 19261 Constants are polynomial functions. (Contributed by Mario Carneiro, 12-Jun-2015.)
eval1

Theorempf1id 19262 The identity is a polynomial function. (Contributed by Mario Carneiro, 20-Mar-2015.)
eval1

Theorempf1subrg 19263 Polynomial functions are a subring. (Contributed by Mario Carneiro, 19-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.)
eval1       SubRing s

Theorempf1rcl 19264 Reverse closure for the set of polynomial functions. (Contributed by Mario Carneiro, 12-Jun-2015.)
eval1

Theorempf1f 19265 Polynomial functions are functions. (Contributed by Mario Carneiro, 12-Jun-2015.)
eval1

Theoremmpfpf1 19266* Convert a multivariate polynomial function to univariate. (Contributed by Mario Carneiro, 12-Jun-2015.)
eval1              eval

Theorempf1mpf 19267* Convert a univariate polynomial function to multivariate. (Contributed by Mario Carneiro, 12-Jun-2015.)
eval1              eval

Theorempf1addcl 19268 The sum of multivariate polynomial functions. (Contributed by Mario Carneiro, 12-Jun-2015.)
eval1

Theorempf1mulcl 19269 The product of multivariate polynomial functions. (Contributed by Mario Carneiro, 12-Jun-2015.)
eval1

Theorempf1ind 19270* 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 19271 Multivariate polynomial degree.
mDeg

Syntaxcdg1 19272 Univariate polynomial degree.
deg1

Definitiondf-mdeg 19273* 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 19405. (Contributed by Stefan O'Rear, 19-Mar-2015.)
mDeg mPoly fld g

Definitiondf-deg1 19274 Define the degree of a univariate polynomial. (Contributed by Stefan O'Rear, 23-Mar-2015.)
deg1 mDeg

Theoremreldmmdeg 19275 Multivariate degree is a binary operation. (Contributed by Stefan O'Rear, 28-Mar-2015.)
mDeg

Theoremtdeglem1 19276* Functionality of the total degree helper function. (Contributed by Stefan O'Rear, 19-Mar-2015.)
fld g

Theoremtdeglem3 19277* Additivity of the total degree helper function. (Contributed by Stefan O'Rear, 26-Mar-2015.)
fld g

Theoremtdeglem4 19278* There is only one multi-index with total degree 0. (Contributed by Stefan O'Rear, 29-Mar-2015.)
fld g

Theoremtdeglem2 19279 Simplification of total degree for the univariate case. (Contributed by Stefan O'Rear, 23-Mar-2015.)
fld g

Theoremmdegfval 19280* Value of the multivariate degree function. (Contributed by Stefan O'Rear, 19-Mar-2015.)
mDeg        mPoly                             fld g

Theoremmdegval 19281* Value of the multivariate degree function at some particular polynomial. (Contributed by Stefan O'Rear, 19-Mar-2015.)
mDeg        mPoly                             fld g

Theoremmdegleb 19282* Property of being of limited degree. (Contributed by Stefan O'Rear, 19-Mar-2015.)
mDeg        mPoly                             fld g

Theoremmdeglt 19283* 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 19284* A nonzero polynomial has some coefficient which witnesses its degree. (Contributed by Stefan O'Rear, 23-Mar-2015.)
mDeg        mPoly                             fld g

Theoremmdegxrcl 19285 Closure of polynomial degree in the extended reals. (Contributed by Stefan O'Rear, 19-Mar-2015.)
mDeg        mPoly

Theoremmdegxrf 19286 Functionality of polynomial degree in the extended reals. (Contributed by Stefan O'Rear, 19-Mar-2015.)
mDeg        mPoly

Theoremmdegcl 19287 Sharp closure for multivariate polynomials. (Contributed by Stefan O'Rear, 23-Mar-2015.)
mDeg        mPoly

Theoremmdeg0 19288 Degree of the zero polynomial. (Contributed by Stefan O'Rear, 20-Mar-2015.)
mDeg        mPoly

Theoremmdegnn0cl 19289 Degree of a nonzero polynomial. (Contributed by Stefan O'Rear, 23-Mar-2015.)
mDeg        mPoly

Theoremdegltlem1 19290 Theorem on arithmetic of extended reals useful for degrees. (Contributed by Stefan O'Rear, 23-Mar-2015.)

Theoremdegltp1le 19291 Theorem on arithmetic of extended reals useful for degrees. (Contributed by Stefan O'Rear, 1-Apr-2015.)

Theoremmdegaddle 19292 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 19293 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 19294 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 19295 A polynomial has nonpositive degree iff it is a constant. (Contributed by Stefan O'Rear, 29-Mar-2015.)
mPoly        mDeg                             algSc

Theoremmdegmullem 19296* Lemma for mdegmulle2 19297. (Contributed by Stefan O'Rear, 26-Mar-2015.)
mPoly        mDeg                                                                                     fld g

Theoremmdegmulle2 19297 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 19298 Relate univariate polynomial degree to multivariate. (Contributed by Stefan O'Rear, 23-Mar-2015.) (Revised by Mario Carneiro, 7-Oct-2015.)
deg1        mDeg

Theoremdeg1xrf 19299 Functionality of univariate polynomial degree, weak range. (Contributed by Stefan O'Rear, 23-Mar-2015.)
deg1        Poly1

Theoremdeg1xrcl 19300 Closure of univariate polynomial degree in extended reals. (Contributed by Stefan O'Rear, 23-Mar-2015.)
deg1        Poly1

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