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Theorem List for Metamath Proof Explorer - 23001-23100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdvmptcj 23001* Function-builder for derivative, conjugate rule. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)

Theoremdvmptre 23002* Function-builder for derivative, real part. (Contributed by Mario Carneiro, 1-Sep-2014.)

Theoremdvmptim 23003* Function-builder for derivative, imaginary part. (Contributed by Mario Carneiro, 1-Sep-2014.)

Theoremdvmptntr 23004* Function-builder for derivative: expand the function from an open set to its closure. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
t        fld

Theoremdvmptco 23005* Function-builder for derivative, chain rule. (Contributed by Mario Carneiro, 1-Sep-2014.)

Theoremdvmptfsum 23006* Function-builder for derivative, finite sums rule. (Contributed by Stefan O'Rear, 12-Nov-2014.)
t        fld

Theoremdvcnvlem 23007 Lemma for dvcnvre 23050. (Contributed by Mario Carneiro, 25-Feb-2015.) (Revised by Mario Carneiro, 8-Sep-2015.)
fld       t

Theoremdvcnv 23008* A weak version of dvcnvre 23050, valid for both real and complex domains but under the hypothesis that the inverse function is already known to be continuous, and the image set is known to be open. A more advanced proof can show that these conditions are unnecessary. (Contributed by Mario Carneiro, 25-Feb-2015.) (Revised by Mario Carneiro, 8-Sep-2015.)
fld       t

Theoremdvexp3 23009* Derivative of an exponential of integer exponent. (Contributed by Mario Carneiro, 26-Feb-2015.)

Theoremdveflem 23010 Derivative of the exponential function at 0. The key step in the proof is eftlub 14240, to show that . (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 28-Dec-2016.)

Theoremdvef 23011 Derivative of the exponential function. (Contributed by Mario Carneiro, 9-Aug-2014.) (Proof shortened by Mario Carneiro, 10-Feb-2015.)

Theoremdvsincos 23012 Derivative of the sine and cosine functions. (Contributed by Mario Carneiro, 21-May-2016.)

Theoremdvsin 23013 Derivative of the sine function. (Contributed by Mario Carneiro, 21-May-2016.)

Theoremdvcos 23014 Derivative of the cosine function. (Contributed by Mario Carneiro, 21-May-2016.)

13.3.1.2  Results on real differentiation

Theoremdvferm1lem 23015* Lemma for dvferm 23019. (Contributed by Mario Carneiro, 24-Feb-2015.)

Theoremdvferm1 23016* One-sided version of dvferm 23019. A point which is the local maximum of its right neighborhood has derivative at most zero. (Contributed by Mario Carneiro, 24-Feb-2015.) (Proof shortened by Mario Carneiro, 28-Dec-2016.)

Theoremdvferm2lem 23017* Lemma for dvferm 23019. (Contributed by Mario Carneiro, 24-Feb-2015.)

Theoremdvferm2 23018* One-sided version of dvferm 23019. A point which is the local maximum of its left neighborhood has derivative at least zero. (Contributed by Mario Carneiro, 24-Feb-2015.) (Proof shortened by Mario Carneiro, 28-Dec-2016.)

Theoremdvferm 23019* Fermat's theorem on stationary points. A point which is a local maximum has derivative equal to zero. (Contributed by Mario Carneiro, 1-Sep-2014.)

Theoremrollelem 23020* Lemma for rolle 23021. (Contributed by Mario Carneiro, 1-Sep-2014.)

Theoremrolle 23021* Rolle's theorem. If is a real continuous function on which is differentiable on , and , then there is some such that . (Contributed by Mario Carneiro, 1-Sep-2014.)

Theoremcmvth 23022* Cauchy's Mean Value Theorem. If are real continuous functions on differentiable on , then there is some such that ' ' . (We express the condition without division, so that we need no nonzero constraints.) (Contributed by Mario Carneiro, 29-Dec-2016.)

Theoremmvth 23023* The Mean Value Theorem. If is a real continuous function on which is differentiable on , then there is some such that is equal to the average slope over . This is Metamath 100 proof #75. (Contributed by Mario Carneiro, 1-Sep-2014.) (Proof shortened by Mario Carneiro, 29-Dec-2016.)

Theoremdvlip 23024* A function with derivative bounded by is Lipschitz continuous with Lipschitz constant equal to . (Contributed by Mario Carneiro, 3-Mar-2015.)

Theoremdvlipcn 23025* A complex function with derivative bounded by on an open ball is Lipschitz continuous with Lipschitz constant equal to . (Contributed by Mario Carneiro, 18-Mar-2015.)

Theoremdvlip2 23026* Combine the results of dvlip 23024 and dvlipcn 23025 into one. (Contributed by Mario Carneiro, 18-Mar-2015.) (Revised by Mario Carneiro, 8-Sep-2015.)

Theoremc1liplem1 23027* Lemma for c1lip1 23028. (Contributed by Stefan O'Rear, 15-Nov-2014.)

Theoremc1lip1 23028* C1 functions are Lipschitz continuous on closed intervals. (Contributed by Stefan O'Rear, 16-Nov-2014.)

Theoremc1lip2 23029* C1 functions are Lipschitz continuous on closed intervals. (Contributed by Stefan O'Rear, 16-Nov-2014.) (Revised by Stefan O'Rear, 6-May-2015.)

Theoremc1lip3 23030* C1 functions are Lipschitz continuous on closed intervals. (Contributed by Stefan O'Rear, 16-Nov-2014.)

Theoremdveq0 23031 If a continuous function has zero derivative at all points on the interior of a closed interval, then it must be a constant function. (Contributed by Mario Carneiro, 2-Sep-2014.) (Proof shortened by Mario Carneiro, 3-Mar-2015.)

Theoremdv11cn 23032 Two functions defined on a ball whose derivatives are the same and which are equal at any given point in the ball must be equal everywhere. (Contributed by Mario Carneiro, 31-Mar-2015.)

Theoremdvgt0lem1 23033 Lemma for dvgt0 23035 and dvlt0 23036. (Contributed by Mario Carneiro, 19-Feb-2015.)

Theoremdvgt0lem2 23034* Lemma for dvgt0 23035 and dvlt0 23036. (Contributed by Mario Carneiro, 19-Feb-2015.)

Theoremdvgt0 23035 A function on a closed interval with positive derivative is increasing. (Contributed by Mario Carneiro, 19-Feb-2015.)

Theoremdvlt0 23036 A function on a closed interval with negative derivative is decreasing. (Contributed by Mario Carneiro, 19-Feb-2015.)

Theoremdvge0 23037 A function on a closed interval with nonnegative derivative is weakly increasing. (Contributed by Mario Carneiro, 30-Apr-2016.)

Theoremdvle 23038* If are differentiable functions and , then for , . (Contributed by Mario Carneiro, 16-May-2016.)

Theoremdvivthlem1 23039* Lemma for dvivth 23041. (Contributed by Mario Carneiro, 24-Feb-2015.)

Theoremdvivthlem2 23040* Lemma for dvivth 23041. (Contributed by Mario Carneiro, 20-Feb-2015.)

Theoremdvivth 23041 Darboux' theorem, or the intermediate value theorem for derivatives. A differentiable function's derivative satisfies the intermediate value property, even though it may not be continuous (so that ivthicc 22487 does not directly apply). (Contributed by Mario Carneiro, 24-Feb-2015.)

Theoremdvne0 23042 A function on a closed interval with nonzero derivative is either monotone increasing or monotone decreasing. (Contributed by Mario Carneiro, 19-Feb-2015.)

Theoremdvne0f1 23043 A function on a closed interval with nonzero derivative is one-to-one. (Contributed by Mario Carneiro, 19-Feb-2015.)

Theoremlhop1lem 23044* Lemma for lhop1 23045. (Contributed by Mario Carneiro, 29-Dec-2016.)
lim        lim                      lim

Theoremlhop1 23045* 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 23046* L'Hôpital's Rule for limits from the left. 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 23047* 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 . This is Metamath 100 proof #64. (Contributed by Mario Carneiro, 30-Dec-2016.)
lim        lim                      lim        lim

Theoremdvcnvrelem1 23048 Lemma for dvcnvre 23050. (Contributed by Mario Carneiro, 24-Feb-2015.)

Theoremdvcnvrelem2 23049 Lemma for dvcnvre 23050. (Contributed by Mario Carneiro, 19-Feb-2015.) (Revised by Mario Carneiro, 8-Sep-2015.)
fld       t        t

Theoremdvcnvre 23050* 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 23051 A real function with strictly increasing derivative is strictly convex. (Contributed by Mario Carneiro, 20-Jun-2015.)

Theoremdvfsumle 23052* 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 23053* 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 23054* 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 23055* Real closure of a derivative. (Contributed by Mario Carneiro, 18-May-2016.)

Theoremdvfsumrlimf 23056* Lemma for dvfsumrlim 23062. (Contributed by Mario Carneiro, 18-May-2016.)

Theoremdvfsumlem1 23057* Lemma for dvfsumrlim 23062. (Contributed by Mario Carneiro, 17-May-2016.)

Theoremdvfsumlem2 23058* Lemma for dvfsumrlim 23062. (Contributed by Mario Carneiro, 17-May-2016.)

Theoremdvfsumlem3 23059* Lemma for dvfsumrlim 23062. (Contributed by Mario Carneiro, 17-May-2016.)

Theoremdvfsumlem4 23060* Lemma for dvfsumrlim 23062. (Contributed by Mario Carneiro, 18-May-2016.)

Theoremdvfsumrlimge0 23061* Lemma for dvfsumrlim 23062. Satisfy the assumption of dvfsumlem4 23060. (Contributed by Mario Carneiro, 18-May-2016.)

Theoremdvfsumrlim 23062* 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 23063* 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 23064* Conjoin the statements of dvfsumrlim 23062 and dvfsumrlim2 23063. (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 23065* The reverse of dvfsumrlim 23062, 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 23066* Lemma for ftc1a 23068 and ftc1 23073. (Contributed by Mario Carneiro, 31-Aug-2014.)

Theoremftc1lem2 23067* Lemma for ftc1 23073. (Contributed by Mario Carneiro, 12-Aug-2014.)

Theoremftc1a 23068* 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 23069* Lemma for ftc1 23073. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 8-Sep-2015.)
t        t        fld

Theoremftc1lem4 23070* Lemma for ftc1 23073. (Contributed by Mario Carneiro, 31-Aug-2014.)
t        t        fld

Theoremftc1lem5 23071* Lemma for ftc1 23073. (Contributed by Mario Carneiro, 14-Aug-2014.) (Revised by Mario Carneiro, 28-Dec-2016.)
t        t        fld

Theoremftc1lem6 23072* Lemma for ftc1 23073. (Contributed by Mario Carneiro, 14-Aug-2014.) (Proof shortened by Mario Carneiro, 28-Dec-2016.)
t        t        fld              lim

Theoremftc1 23073* 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 . This is part of Metamath 100 proof #15. (Contributed by Mario Carneiro, 1-Sep-2014.)
t        t        fld

Theoremftc1cn 23074* Strengthen the assumptions of ftc1 23073 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 23075* 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 . This is part of Metamath 100 proof #15. (Contributed by Mario Carneiro, 2-Sep-2014.)

Theoremftc2ditglem 23076* Lemma for ftc2ditg 23077. (Contributed by Mario Carneiro, 3-Sep-2014.)
_

Theoremftc2ditg 23077* Directed integral analogue of ftc2 23075. (Contributed by Mario Carneiro, 3-Sep-2014.)
_

Theoremitgparts 23078* 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 23079* Lemma for itgsubst 23080. (Contributed by Mario Carneiro, 12-Sep-2014.)
_ _

Theoremitgsubst 23080* 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 23079, 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 14  BASIC REAL AND COMPLEX FUNCTIONS

14.1  Polynomials

14.1.1  Polynomial degrees

Syntaxcmdg 23081 Multivariate polynomial degree.
mDeg

Syntaxcdg1 23082 Univariate polynomial degree.
deg1

Definitiondf-mdeg 23083* 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 23224. (Contributed by Stefan O'Rear, 19-Mar-2015.) (Revised by AV, 25-Jun-2019.)
mDeg mPoly supp fld g

Definitiondf-deg1 23084 Define the degree of a univariate polynomial. (Contributed by Stefan O'Rear, 23-Mar-2015.)
deg1 mDeg

Theoremreldmmdeg 23085 Multivariate degree is a binary operation. (Contributed by Stefan O'Rear, 28-Mar-2015.)
mDeg

Theoremtdeglem1 23086* Functionality of the total degree helper function. (Contributed by Stefan O'Rear, 19-Mar-2015.) (Proof shortened by AV, 27-Jul-2019.)
fld g

Theoremtdeglem3 23087* Additivity of the total degree helper function. (Contributed by Stefan O'Rear, 26-Mar-2015.) (Proof shortened by AV, 27-Jul-2019.)
fld g

Theoremtdeglem4 23088* There is only one multi-index with total degree 0. (Contributed by Stefan O'Rear, 29-Mar-2015.)
fld g

Theoremtdeglem2 23089 Simplification of total degree for the univariate case. (Contributed by Stefan O'Rear, 23-Mar-2015.)
fld g

Theoremmdegfval 23090* Value of the multivariate degree function. (Contributed by Stefan O'Rear, 19-Mar-2015.) (Revised by AV, 25-Jun-2019.)
mDeg        mPoly                             fld g        supp

Theoremmdegval 23091* Value of the multivariate degree function at some particular polynomial. (Contributed by Stefan O'Rear, 19-Mar-2015.) (Revised by AV, 25-Jun-2019.)
mDeg        mPoly                             fld g        supp

Theoremmdegleb 23092* Property of being of limited degree. (Contributed by Stefan O'Rear, 19-Mar-2015.)
mDeg        mPoly                             fld g

Theoremmdeglt 23093* 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.) (Proof shortened by AV, 27-Jul-2019.)
mDeg        mPoly                             fld g

Theoremmdegldg 23094* A nonzero polynomial has some coefficient which witnesses its degree. (Contributed by Stefan O'Rear, 23-Mar-2015.)
mDeg        mPoly                             fld g

Theoremmdegxrcl 23095 Closure of polynomial degree in the extended reals. (Contributed by Stefan O'Rear, 19-Mar-2015.) (Proof shortened by AV, 27-Jul-2019.)
mDeg        mPoly

Theoremmdegxrf 23096 Functionality of polynomial degree in the extended reals. (Contributed by Stefan O'Rear, 19-Mar-2015.) (Proof shortened by AV, 27-Jul-2019.)
mDeg        mPoly

Theoremmdegcl 23097 Sharp closure for multivariate polynomials. (Contributed by Stefan O'Rear, 23-Mar-2015.)
mDeg        mPoly

Theoremmdeg0 23098 Degree of the zero polynomial. (Contributed by Stefan O'Rear, 20-Mar-2015.) (Proof shortened by AV, 27-Jul-2019.)
mDeg        mPoly

Theoremmdegnn0cl 23099 Degree of a nonzero polynomial. (Contributed by Stefan O'Rear, 23-Mar-2015.)
mDeg        mPoly

Theoremdegltlem1 23100 Theorem on arithmetic of extended reals useful for degrees. (Contributed by Stefan O'Rear, 23-Mar-2015.)

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330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41046
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