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Theorem List for Metamath Proof Explorer - 22501-22600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdvmptco 22501* Function-builder for derivative, chain rule. (Contributed by Mario Carneiro, 1-Sep-2014.)

Theoremdvmptfsum 22502* Function-builder for derivative, finite sums rule. (Contributed by Stefan O'Rear, 12-Nov-2014.)
t        fld

Theoremdvcnvlem 22503 Lemma for dvcnvre 22546. (Contributed by Mario Carneiro, 25-Feb-2015.) (Revised by Mario Carneiro, 8-Sep-2015.)
fld       t

Theoremdvcnv 22504* A weak version of dvcnvre 22546, 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 22505* Derivative of an exponential of integer exponent. (Contributed by Mario Carneiro, 26-Feb-2015.)

Theoremdveflem 22506 Derivative of the exponential function at 0. The key step in the proof is eftlub 13856, to show that . (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 28-Dec-2016.)

Theoremdvef 22507 Derivative of the exponential function. (Contributed by Mario Carneiro, 9-Aug-2014.) (Proof shortened by Mario Carneiro, 10-Feb-2015.)

Theoremdvsincos 22508 Derivative of the sine and cosine functions. (Contributed by Mario Carneiro, 21-May-2016.)

Theoremdvsin 22509 Derivative of the sine function. (Contributed by Mario Carneiro, 21-May-2016.)

Theoremdvcos 22510 Derivative of the cosine function. (Contributed by Mario Carneiro, 21-May-2016.)

13.3.1.2  Results on real differentiation

Theoremdvferm1lem 22511* Lemma for dvferm 22515. (Contributed by Mario Carneiro, 24-Feb-2015.)

Theoremdvferm1 22512* One-sided version of dvferm 22515. 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 22513* Lemma for dvferm 22515. (Contributed by Mario Carneiro, 24-Feb-2015.)

Theoremdvferm2 22514* One-sided version of dvferm 22515. 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 22515* 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 22516* Lemma for rolle 22517. (Contributed by Mario Carneiro, 1-Sep-2014.)

Theoremrolle 22517* 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 22518* 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 22519* 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 22520* A function with derivative bounded by is Lipschitz continuous with Lipschitz constant equal to . (Contributed by Mario Carneiro, 3-Mar-2015.)

Theoremdvlipcn 22521* 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 22522* Combine the results of dvlip 22520 and dvlipcn 22521 into one. (Contributed by Mario Carneiro, 18-Mar-2015.) (Revised by Mario Carneiro, 8-Sep-2015.)

Theoremc1liplem1 22523* Lemma for c1lip1 22524. (Contributed by Stefan O'Rear, 15-Nov-2014.)

Theoremc1lip1 22524* C1 functions are Lipschitz continuous on closed intervals. (Contributed by Stefan O'Rear, 16-Nov-2014.)

Theoremc1lip2 22525* 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 22526* C1 functions are Lipschitz continuous on closed intervals. (Contributed by Stefan O'Rear, 16-Nov-2014.)

Theoremdveq0 22527 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 22528 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 22529 Lemma for dvgt0 22531 and dvlt0 22532. (Contributed by Mario Carneiro, 19-Feb-2015.)

Theoremdvgt0lem2 22530* Lemma for dvgt0 22531 and dvlt0 22532. (Contributed by Mario Carneiro, 19-Feb-2015.)

Theoremdvgt0 22531 A function on a closed interval with positive derivative is increasing. (Contributed by Mario Carneiro, 19-Feb-2015.)

Theoremdvlt0 22532 A function on a closed interval with negative derivative is decreasing. (Contributed by Mario Carneiro, 19-Feb-2015.)

Theoremdvge0 22533 A function on a closed interval with nonnegative derivative is weakly increasing. (Contributed by Mario Carneiro, 30-Apr-2016.)

Theoremdvle 22534* If are differentiable functions and , then for , . (Contributed by Mario Carneiro, 16-May-2016.)

Theoremdvivthlem1 22535* Lemma for dvivth 22537. (Contributed by Mario Carneiro, 24-Feb-2015.)

Theoremdvivthlem2 22536* Lemma for dvivth 22537. (Contributed by Mario Carneiro, 20-Feb-2015.)

Theoremdvivth 22537 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 21996 does not directly apply). (Contributed by Mario Carneiro, 24-Feb-2015.)

Theoremdvne0 22538 A function on a closed interval with nonzero derivative is either monotone increasing or monotone decreasing. (Contributed by Mario Carneiro, 19-Feb-2015.)

Theoremdvne0f1 22539 A function on a closed interval with nonzero derivative is one-to-one. (Contributed by Mario Carneiro, 19-Feb-2015.)

Theoremlhop1lem 22540* Lemma for lhop1 22541. (Contributed by Mario Carneiro, 29-Dec-2016.)
lim        lim                      lim

Theoremlhop1 22541* 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 22542* 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 22543* 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 22544 Lemma for dvcnvre 22546. (Contributed by Mario Carneiro, 24-Feb-2015.)

Theoremdvcnvrelem2 22545 Lemma for dvcnvre 22546. (Contributed by Mario Carneiro, 19-Feb-2015.) (Revised by Mario Carneiro, 8-Sep-2015.)
fld       t        t

Theoremdvcnvre 22546* 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 22547 A real function with strictly increasing derivative is strictly convex. (Contributed by Mario Carneiro, 20-Jun-2015.)

Theoremdvfsumle 22548* 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 22549* 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 22550* 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 22551* Real closure of a derivative. (Contributed by Mario Carneiro, 18-May-2016.)

Theoremdvfsumrlimf 22552* Lemma for dvfsumrlim 22558. (Contributed by Mario Carneiro, 18-May-2016.)

Theoremdvfsumlem1 22553* Lemma for dvfsumrlim 22558. (Contributed by Mario Carneiro, 17-May-2016.)

Theoremdvfsumlem2 22554* Lemma for dvfsumrlim 22558. (Contributed by Mario Carneiro, 17-May-2016.)

Theoremdvfsumlem3 22555* Lemma for dvfsumrlim 22558. (Contributed by Mario Carneiro, 17-May-2016.)

Theoremdvfsumlem4 22556* Lemma for dvfsumrlim 22558. (Contributed by Mario Carneiro, 18-May-2016.)

Theoremdvfsumrlimge0 22557* Lemma for dvfsumrlim 22558. Satisfy the assumption of dvfsumlem4 22556. (Contributed by Mario Carneiro, 18-May-2016.)

Theoremdvfsumrlim 22558* 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 22559* 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 22560* Conjoin the statements of dvfsumrlim 22558 and dvfsumrlim2 22559. (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 22561* The reverse of dvfsumrlim 22558, 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 22562* Lemma for ftc1a 22564 and ftc1 22569. (Contributed by Mario Carneiro, 31-Aug-2014.)

Theoremftc1lem2 22563* Lemma for ftc1 22569. (Contributed by Mario Carneiro, 12-Aug-2014.)

Theoremftc1a 22564* 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 22565* Lemma for ftc1 22569. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 8-Sep-2015.)
t        t        fld

Theoremftc1lem4 22566* Lemma for ftc1 22569. (Contributed by Mario Carneiro, 31-Aug-2014.)
t        t        fld

Theoremftc1lem5 22567* Lemma for ftc1 22569. (Contributed by Mario Carneiro, 14-Aug-2014.) (Revised by Mario Carneiro, 28-Dec-2016.)
t        t        fld

Theoremftc1lem6 22568* Lemma for ftc1 22569. (Contributed by Mario Carneiro, 14-Aug-2014.) (Proof shortened by Mario Carneiro, 28-Dec-2016.)
t        t        fld              lim

Theoremftc1 22569* 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 22570* Strengthen the assumptions of ftc1 22569 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 22571* 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 22572* Lemma for ftc2ditg 22573. (Contributed by Mario Carneiro, 3-Sep-2014.)
_

Theoremftc2ditg 22573* Directed integral analog of ftc2 22571. (Contributed by Mario Carneiro, 3-Sep-2014.)
_

Theoremitgparts 22574* 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 22575* Lemma for itgsubst 22576. (Contributed by Mario Carneiro, 12-Sep-2014.)
_ _

Theoremitgsubst 22576* 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 22575, 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 22577 Multivariate polynomial degree.
mDeg

Syntaxcdg1 22578 Univariate polynomial degree.
deg1

Definitiondf-mdeg 22579* 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 22714. (Contributed by Stefan O'Rear, 19-Mar-2015.) (Revised by AV, 25-Jun-2019.)
mDeg mPoly supp fld g

Definitiondf-deg1 22580 Define the degree of a univariate polynomial. (Contributed by Stefan O'Rear, 23-Mar-2015.)
deg1 mDeg

Theoremreldmmdeg 22581 Multivariate degree is a binary operation. (Contributed by Stefan O'Rear, 28-Mar-2015.)
mDeg

Theoremtdeglem1 22582* 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 22583* 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 22584* There is only one multi-index with total degree 0. (Contributed by Stefan O'Rear, 29-Mar-2015.)
fld g

Theoremtdeglem2 22585 Simplification of total degree for the univariate case. (Contributed by Stefan O'Rear, 23-Mar-2015.)
fld g

Theoremmdegfval 22586* 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

TheoremmdegfvalOLD 22587* Value of the multivariate degree function. (Contributed by Stefan O'Rear, 19-Mar-2015.) Obsolete version of mdegfval 22586 as of 25-Jun-2019. Proof modified to avoid an old version of definition df-mdeg 22579. (New usage is discouraged.) (Proof modification is discouraged.)
mDeg        mPoly                             fld g

Theoremmdegval 22588* 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

TheoremmdegvalOLD 22589* Value of the multivariate degree function at some particular polynomial. (Contributed by Stefan O'Rear, 19-Mar-2015.) Obsolete version of mdegval 22588 as of 25-Jun-2019. (New usage is discouraged.) (Proof modification is discouraged.)
mDeg        mPoly                             fld g

Theoremmdegleb 22590* Property of being of limited degree. (Contributed by Stefan O'Rear, 19-Mar-2015.)
mDeg        mPoly                             fld g

Theoremmdeglt 22591* 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 22592* A nonzero polynomial has some coefficient which witnesses its degree. (Contributed by Stefan O'Rear, 23-Mar-2015.)
mDeg        mPoly                             fld g

Theoremmdegxrcl 22593 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 22594 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 22595 Sharp closure for multivariate polynomials. (Contributed by Stefan O'Rear, 23-Mar-2015.)
mDeg        mPoly

Theoremmdeg0 22596 Degree of the zero polynomial. (Contributed by Stefan O'Rear, 20-Mar-2015.) (Proof shortened by AV, 27-Jul-2019.)
mDeg        mPoly

Theoremmdegnn0cl 22597 Degree of a nonzero polynomial. (Contributed by Stefan O'Rear, 23-Mar-2015.)
mDeg        mPoly

Theoremdegltlem1 22598 Theorem on arithmetic of extended reals useful for degrees. (Contributed by Stefan O'Rear, 23-Mar-2015.)

Theoremdegltp1le 22599 Theorem on arithmetic of extended reals useful for degrees. (Contributed by Stefan O'Rear, 1-Apr-2015.)

Theoremmdegaddle 22600 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

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206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 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 38101-38143
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