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Theorem List for Metamath Proof Explorer - 19901-20000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdvne0f1 19901 A function on a closed interval with nonzero derivative is one-to-one. (Contributed by Mario Carneiro, 19-Feb-2015.)

Theoremlhop1lem 19902* Lemma for lhop1 19903. (Contributed by Mario Carneiro, 29-Dec-2016.)
lim        lim                      lim

Theoremlhop1 19903* L'Hôpital's Rule for limits from the right. If and are differentiable real functions on , and and both approach 0 at , and and ' are not zero on , and the limit of ' ' at is , then the limit at also exists and equals . (Contributed by Mario Carneiro, 29-Dec-2016.)
lim        lim                      lim        lim

Theoremlhop2 19904* L'Hôpital's Rule for limits from the right. If and are differentiable real functions on , and and both approach 0 at , and and ' are not zero on , and the limit of ' ' at is , then the limit at also exists and equals . (Contributed by Mario Carneiro, 29-Dec-2016.)
lim        lim                      lim        lim

Theoremlhop 19905* L'Hôpital's Rule. If is an open set of the reals, and are real functions on containing all of except possibly , which are differentiable everywhere on , and both approach 0, and the limit of ' ' at is , then the limit at also exists and equals . (Contributed by Mario Carneiro, 30-Dec-2016.)
lim        lim                      lim        lim

Theoremdvcnvrelem1 19906 Lemma for dvcnvre 19908. (Contributed by Mario Carneiro, 24-Feb-2015.)

Theoremdvcnvrelem2 19907 Lemma for dvcnvre 19908. (Contributed by Mario Carneiro, 19-Feb-2015.) (Revised by Mario Carneiro, 8-Sep-2015.)
fld       t        t

Theoremdvcnvre 19908* The derivative rule for inverse functions. If is a continuous and differentiable bijective function from to which never has derivative , then is also differentiable, and its derivative is the reciprocal of the derivative of . (Contributed by Mario Carneiro, 24-Feb-2015.)

Theoremdvcvx 19909 A real function with strictly increasing derivative is strictly convex. (Contributed by Mario Carneiro, 20-Jun-2015.)

Theoremdvfsumle 19910* 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.)
..^        ..^        ..^

Theoremdvfsumge 19911* 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 19912* 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 19913* Real closure of a derivative. (Contributed by Mario Carneiro, 18-May-2016.)

Theoremdvfsumrlimf 19914* Lemma for dvfsumrlim 19920. (Contributed by Mario Carneiro, 18-May-2016.)

Theoremdvfsumlem1 19915* Lemma for dvfsumrlim 19920. (Contributed by Mario Carneiro, 17-May-2016.)

Theoremdvfsumlem2 19916* Lemma for dvfsumrlim 19920. (Contributed by Mario Carneiro, 17-May-2016.)

Theoremdvfsumlem3 19917* Lemma for dvfsumrlim 19920. (Contributed by Mario Carneiro, 17-May-2016.)

Theoremdvfsumlem4 19918* Lemma for dvfsumrlim 19920. (Contributed by Mario Carneiro, 18-May-2016.)

Theoremdvfsumrlimge0 19919* Lemma for dvfsumrlim 19920. Satisfy the assumption of dvfsumlem4 19918. (Contributed by Mario Carneiro, 18-May-2016.)

Theoremdvfsumrlim 19920* 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 19921* 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 19922* Conjoin the statements of dvfsumrlim 19920 and dvfsumrlim2 19921. (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 19923* The reverse of dvfsumrlim 19920, 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 19924* Lemma for ftc1a 19926 and ftc1 19931. (Contributed by Mario Carneiro, 31-Aug-2014.)

Theoremftc1lem2 19925* Lemma for ftc1 19931. (Contributed by Mario Carneiro, 12-Aug-2014.)

Theoremftc1a 19926* 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 19927* Lemma for ftc1 19931. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 8-Sep-2015.)
t        t        fld

Theoremftc1lem4 19928* Lemma for ftc1 19931. (Contributed by Mario Carneiro, 31-Aug-2014.)
t        t        fld

Theoremftc1lem5 19929* Lemma for ftc1 19931. (Contributed by Mario Carneiro, 14-Aug-2014.) (Revised by Mario Carneiro, 28-Dec-2016.)
t        t        fld

Theoremftc1lem6 19930* Lemma for ftc1 19931. (Contributed by Mario Carneiro, 14-Aug-2014.) (Proof shortened by Mario Carneiro, 28-Dec-2016.)
t        t        fld              lim

Theoremftc1 19931* 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 19932* Strengthen the assumptions of ftc1 19931 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 19933* 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 19934* Lemma for ftc2ditg 19935. (Contributed by Mario Carneiro, 3-Sep-2014.)
_

Theoremftc2ditg 19935* Directed integral analog of ftc2 19933. (Contributed by Mario Carneiro, 3-Sep-2014.)
_

Theoremitgparts 19936* 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 19937* Lemma for itgsubst 19938. (Contributed by Mario Carneiro, 12-Sep-2014.)
_ _

Theoremitgsubst 19938* 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 19937, 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 19939* Lemma for evlseu 19942. 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 19940* Lemma for evlseu 19942. Polynomial evaluation of a scaled monomial. (Contributed by Stefan O'Rear, 8-Mar-2015.)
mPoly                                    mulGrp       .g              mVar        g g                             RingHom                                    g

Theoremevlslem1 19941* Lemma for evlseu 19942, 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 19942* 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 19943 Well-behaved binary operation property of evalSub. (Contributed by Stefan O'Rear, 19-Mar-2015.)
evalSub

Theoremmpfrcl 19944 Reverse closure for the set of polynomial functions. (Contributed by Stefan O'Rear, 19-Mar-2015.)
evalSub        SubRing

Theoremevlsval 19945* Value of the polynomial evaluation map function. (Contributed by Stefan O'Rear, 11-Mar-2015.)
evalSub        mPoly        mVar        s        s               algSc                     SubRing RingHom

Theoremevlsval2 19946* 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 19947 Polynomial evaluation is a homomorphism (into the product ring). (Contributed by Stefan O'Rear, 12-Mar-2015.)
evalSub        mPoly        s        s               SubRing RingHom

Theoremevlssca 19948 Polynomial evaluation maps scalars to constant functions. (Contributed by Stefan O'Rear, 13-Mar-2015.)
evalSub        mPoly        s               algSc                     SubRing

Theoremevlsvar 19949* Polynomial evaluation maps variables to projections. (Contributed by Stefan O'Rear, 12-Mar-2015.)
evalSub        mVar        s                             SubRing

Theoremevlval 19950 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 19951 The simple evaluation map is a ring homomorphism. (Contributed by Mario Carneiro, 12-Jun-2015.)
eval               mPoly        s        RingHom

Theoremevl1fval 19952* Value of the simple/same ring evalutation map. (Contributed by Mario Carneiro, 12-Jun-2015.)
eval1       eval

Theoremevl1val 19953* Value of the simple/same ring evalutation map. (Contributed by Mario Carneiro, 12-Jun-2015.)
eval1       eval               mPoly

Theoremevl1rhm 19954 Polynomial evaluation is a homomorphism (into the product ring). (Contributed by Mario Carneiro, 12-Jun-2015.)
eval1       Poly1       s               RingHom

Theoremevl1sca 19955 Polynomial evaluation maps scalars to constant functions. (Contributed by Mario Carneiro, 12-Jun-2015.)
eval1       Poly1              algSc

Theoremevl1scad 19956 Polynomial evaluation builder for scalars. (Contributed by Mario Carneiro, 4-Jul-2015.)
eval1       Poly1              algSc

Theoremevl1var 19957 Polynomial evaluation maps the variable to the identity function. (Contributed by Mario Carneiro, 12-Jun-2015.)
eval1       var1

Theoremevl1vard 19958 Polynomial evaluation builder for the variable. (Contributed by Mario Carneiro, 4-Jul-2015.)
eval1       var1              Poly1

Theoremevl1addd 19959 Polynomial evaluation builder for addition of polynomials. (Contributed by Mario Carneiro, 4-Jul-2015.)
eval1       Poly1

Theoremevl1subd 19960 Polynomial evaluation builder for subtraction of polynomials. (Contributed by Mario Carneiro, 4-Jul-2015.)
eval1       Poly1

Theoremevl1muld 19961 Polynomial evaluation builder for multiplication of polynomials. (Contributed by Mario Carneiro, 4-Jul-2015.)
eval1       Poly1

Theoremevl1vsd 19962 Polynomial evaluation builder for scalar multiplication of polynomials. (Contributed by Mario Carneiro, 4-Jul-2015.)
eval1       Poly1

Theoremevl1expd 19963 Polynomial evaluation builder for an exponential. (Contributed by Mario Carneiro, 12-Jun-2015.)
eval1       Poly1                                          .gmulGrp       .gmulGrp

Theoremmpfconst 19964 Constants are multivariate polynomial functions. (Contributed by Mario Carneiro, 19-Mar-2015.)
evalSub                      SubRing

Theoremmpfproj 19965* Projections are multivariate polynomial functions. (Contributed by Mario Carneiro, 20-Mar-2015.)
evalSub                      SubRing

Theoremmpfsubrg 19966 Polynomial functions are a subring. (Contributed by Mario Carneiro, 19-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.)
evalSub        SubRing SubRing s

Theoremmpff 19967 Polynomial functions are functions. (Contributed by Mario Carneiro, 19-Mar-2015.)
evalSub

Theoremmpfaddcl 19968 The sum of multivariate polynomial functions. (Contributed by Mario Carneiro, 19-Mar-2015.)
evalSub

Theoremmpfmulcl 19969 The product of multivariate polynomial functions. (Contributed by Mario Carneiro, 19-Mar-2015.)
evalSub

Theoremmpfind 19970* 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 19971 Constants are polynomial functions. (Contributed by Mario Carneiro, 12-Jun-2015.)
eval1

Theorempf1id 19972 The identity is a polynomial function. (Contributed by Mario Carneiro, 20-Mar-2015.)
eval1

Theorempf1subrg 19973 Polynomial functions are a subring. (Contributed by Mario Carneiro, 19-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.)
eval1       SubRing s

Theorempf1rcl 19974 Reverse closure for the set of polynomial functions. (Contributed by Mario Carneiro, 12-Jun-2015.)
eval1

Theorempf1f 19975 Polynomial functions are functions. (Contributed by Mario Carneiro, 12-Jun-2015.)
eval1

Theoremmpfpf1 19976* Convert a multivariate polynomial function to univariate. (Contributed by Mario Carneiro, 12-Jun-2015.)
eval1              eval

Theorempf1mpf 19977* Convert a univariate polynomial function to multivariate. (Contributed by Mario Carneiro, 12-Jun-2015.)
eval1              eval

Theorempf1addcl 19978 The sum of multivariate polynomial functions. (Contributed by Mario Carneiro, 12-Jun-2015.)
eval1

Theorempf1mulcl 19979 The product of multivariate polynomial functions. (Contributed by Mario Carneiro, 12-Jun-2015.)
eval1

Theorempf1ind 19980* 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 19981 Multivariate polynomial degree.
mDeg

Syntaxcdg1 19982 Univariate polynomial degree.
deg1

Definitiondf-mdeg 19983* 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 20115. (Contributed by Stefan O'Rear, 19-Mar-2015.)
mDeg mPoly fld g

Definitiondf-deg1 19984 Define the degree of a univariate polynomial. (Contributed by Stefan O'Rear, 23-Mar-2015.)
deg1 mDeg

Theoremreldmmdeg 19985 Multivariate degree is a binary operation. (Contributed by Stefan O'Rear, 28-Mar-2015.)
mDeg

Theoremtdeglem1 19986* Functionality of the total degree helper function. (Contributed by Stefan O'Rear, 19-Mar-2015.)
fld g

Theoremtdeglem3 19987* Additivity of the total degree helper function. (Contributed by Stefan O'Rear, 26-Mar-2015.)
fld g

Theoremtdeglem4 19988* There is only one multi-index with total degree 0. (Contributed by Stefan O'Rear, 29-Mar-2015.)
fld g

Theoremtdeglem2 19989 Simplification of total degree for the univariate case. (Contributed by Stefan O'Rear, 23-Mar-2015.)
fld g

Theoremmdegfval 19990* Value of the multivariate degree function. (Contributed by Stefan O'Rear, 19-Mar-2015.)
mDeg        mPoly                             fld g

Theoremmdegval 19991* Value of the multivariate degree function at some particular polynomial. (Contributed by Stefan O'Rear, 19-Mar-2015.)
mDeg        mPoly                             fld g

Theoremmdegleb 19992* Property of being of limited degree. (Contributed by Stefan O'Rear, 19-Mar-2015.)
mDeg        mPoly                             fld g

Theoremmdeglt 19993* 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 19994* A nonzero polynomial has some coefficient which witnesses its degree. (Contributed by Stefan O'Rear, 23-Mar-2015.)
mDeg        mPoly                             fld g

Theoremmdegxrcl 19995 Closure of polynomial degree in the extended reals. (Contributed by Stefan O'Rear, 19-Mar-2015.)
mDeg        mPoly

Theoremmdegxrf 19996 Functionality of polynomial degree in the extended reals. (Contributed by Stefan O'Rear, 19-Mar-2015.)
mDeg        mPoly

Theoremmdegcl 19997 Sharp closure for multivariate polynomials. (Contributed by Stefan O'Rear, 23-Mar-2015.)
mDeg        mPoly

Theoremmdeg0 19998 Degree of the zero polynomial. (Contributed by Stefan O'Rear, 20-Mar-2015.)
mDeg        mPoly

Theoremmdegnn0cl 19999 Degree of a nonzero polynomial. (Contributed by Stefan O'Rear, 23-Mar-2015.)
mDeg        mPoly

Theoremdegltlem1 20000 Theorem on arithmetic of extended reals useful for degrees. (Contributed by Stefan O'Rear, 23-Mar-2015.)

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