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Theorem List for Metamath Proof Explorer - 19801-19900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdvnf 19801 The N-times derivative is a function. (Contributed by Stefan O'Rear, 16-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)

Theoremdvnbss 19802 The set of N-times differentiable points is a subset of the domain of the function. (Contributed by Stefan O'Rear, 16-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)

Theoremdvnadd 19803 The -th derivative of the -th derivative of is the same as the -th derivative of . (Contributed by Mario Carneiro, 11-Feb-2015.)

Theoremdvn2bss 19804 An N-times differentiable point is an M-times differeentiable point, if . (Contributed by Mario Carneiro, 30-Dec-2016.)

Theoremdvnres 19805 Multiple derivative version of dvres3a 19789. (Contributed by Mario Carneiro, 11-Feb-2015.)

Theoremcpnfval 19806* Condition for n-times continuous differentiability. (Contributed by Stefan O'Rear, 15-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)

Theoremfncpn 19807 The object is a function. (Contributed by Stefan O'Rear, 16-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)

Theoremelcpn 19808 Condition for n-times continuous differentiability. (Contributed by Stefan O'Rear, 15-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)

Theoremcpnord 19809 conditions are ordered by strength. (Contributed by Stefan O'Rear, 16-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)

Theoremcpncn 19810 A function is continuous. (Contributed by Mario Carneiro, 11-Feb-2015.)

Theoremcpnres 19811 The restriction of a function is . (Contributed by Mario Carneiro, 11-Feb-2015.)

Theoremdvaddbr 19812 The sum rule for derivatives at a point. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 28-Dec-2016.)
fld

Theoremdvmulbr 19813 The product rule for derivatives at a point. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 28-Dec-2016.)
fld

Theoremdvadd 19814 The sum rule for derivatives at a point. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)

Theoremdvmul 19815 The product rule for derivatives at a point. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)

Theoremdvaddf 19816 The sum rule for everywhere-differentiable functions. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)

Theoremdvmulf 19817 The product rule for everywhere-differentiable functions. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)

Theoremdvcmul 19818 The product rule when one argument is a constant. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)

Theoremdvcmulf 19819 The product rule when one argument is a constant. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)

Theoremdvcobr 19820 The chain rule for derivatives at a point. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 28-Dec-2016.)
fld

Theoremdvco 19821 The chain rule for derivatives at a point. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)

Theoremdvcof 19822 The chain rule for everywhere-differentiable functions. (Contributed by Mario Carneiro, 10-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)

Theoremdvcjbr 19823 The derivative of the conjugate of a function. (This doesn't follow from dvcobr 19820 because is not a function on the reals, and even if we used complex derivatives, is not complex-differentiable.) (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)

Theoremdvcj 19824 The derivative of the conjugate of a function. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)

Theoremdvfre 19825 The derivative of a real function is real. (Contributed by Mario Carneiro, 1-Sep-2014.)

Theoremdvnfre 19826 The -th derivative of a real function is real. (Contributed by Mario Carneiro, 1-Jan-2017.)

Theoremdvexp 19827* Derivative of a power function. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)

Theoremdvexp2 19828* Derivative of an exponential, possibly zero power. (Contributed by Stefan O'Rear, 13-Nov-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)

Theoremdvrec 19829* Derivative of the reciprocal function. (Contributed by Mario Carneiro, 25-Feb-2015.) (Revised by Mario Carneiro, 28-Dec-2016.)

Theoremdvmptres3 19830* Function-builder for derivative: restrict a derivative to a subset. (Contributed by Mario Carneiro, 11-Feb-2015.)
fld

Theoremdvmptid 19831* Function-builder for derivative: derivative of the identity. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)

Theoremdvmptc 19832* Function-builder for derivative: derivative of a constant. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)

Theoremdvmptcl 19833* Closure lemma for dvmptcmul 19838 and other related theorems. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)

Theoremdvmptadd 19834* Function-builder for derivative, addition rule. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)

Theoremdvmptmul 19835* Function-builder for derivative, product rule. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)

Theoremdvmptres2 19836* Function-builder for derivative: restrict a derivative to a subset. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
t        fld

Theoremdvmptres 19837* Function-builder for derivative: restrict a derivative to an open subset. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
t        fld

Theoremdvmptcmul 19838* Function-builder for derivative, product rule for constant multiplier. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)

Theoremdvmptdivc 19839* Function-builder for derivative, division rule for constant divisor. (Contributed by Mario Carneiro, 18-May-2016.)

Theoremdvmptneg 19840* Function-builder for derivative, product rule for negatives. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)

Theoremdvmptsub 19841* Function-builder for derivative, subtraction rule. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)

Theoremdvmptcj 19842* Function-builder for derivative, conjugate rule. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)

Theoremdvmptre 19843* Function-builder for derivative, real part. (Contributed by Mario Carneiro, 1-Sep-2014.)

Theoremdvmptim 19844* Function-builder for derivative, imaginary part. (Contributed by Mario Carneiro, 1-Sep-2014.)

Theoremdvmptntr 19845* 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 19846* Function-builder for derivative, chain rule. (Contributed by Mario Carneiro, 1-Sep-2014.)

Theoremdvmptfsum 19847* Function-builder for derivative, finite sums rule. (Contributed by Stefan O'Rear, 12-Nov-2014.)
t        fld

Theoremdvcnvlem 19848 Lemma for dvcnvre 19891. (Contributed by Mario Carneiro, 25-Feb-2015.) (Revised by Mario Carneiro, 8-Sep-2015.)
fld       t

Theoremdvcnv 19849* A weak version of dvcnvre 19891, 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 19850* Derivative of an exponential of integer exponent. (Contributed by Mario Carneiro, 26-Feb-2015.)

Theoremdveflem 19851 Derivative of the exponential function at 0. The key step in the proof is eftlub 12698, to show that . (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 28-Dec-2016.)

Theoremdvef 19852 Derivative of the exponential function. (Contributed by Mario Carneiro, 9-Aug-2014.) (Proof shortened by Mario Carneiro, 10-Feb-2015.)

Theoremdvsincos 19853 Derivative of the sine and cosine functions. (Contributed by Mario Carneiro, 21-May-2016.)

Theoremdvsin 19854 Derivative of the sine function. (Contributed by Mario Carneiro, 21-May-2016.)

Theoremdvcos 19855 Derivative of the cosine function. (Contributed by Mario Carneiro, 21-May-2016.)

Theoremdvferm1lem 19856* Lemma for dvferm 19860. (Contributed by Mario Carneiro, 24-Feb-2015.)

Theoremdvferm1 19857* One-sided version of dvferm 19860. 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 19858* Lemma for dvferm 19860. (Contributed by Mario Carneiro, 24-Feb-2015.)

Theoremdvferm2 19859* One-sided version of dvferm 19860. 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 19860* 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 19861* Lemma for rolle 19862. (Contributed by Mario Carneiro, 1-Sep-2014.)

Theoremrolle 19862* 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 19863* 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 19864* 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 . (Contributed by Mario Carneiro, 1-Sep-2014.) (Proof shortened by Mario Carneiro, 29-Dec-2016.)

Theoremdvlip 19865* A function with derivative bounded by is Lipschitz continuous with Lipchitz constant equal to . (Contributed by Mario Carneiro, 3-Mar-2015.)

Theoremdvlipcn 19866* A complex function with derivative bounded by on an open ball is Lipschitz continuous with Lipchitz constant equal to . (Contributed by Mario Carneiro, 18-Mar-2015.)

Theoremdvlip2 19867* Combine the results of dvlip 19865 and dvlipcn 19866 into one. (Contributed by Mario Carneiro, 18-Mar-2015.) (Revised by Mario Carneiro, 8-Sep-2015.)

Theoremc1liplem1 19868* Lemma for c1lip1 19869. (Contributed by Stefan O'Rear, 15-Nov-2014.)

Theoremc1lip1 19869* C1 functions are Lipschitz continuous on closed intervals. (Contributed by Stefan O'Rear, 16-Nov-2014.)

Theoremc1lip2 19870* 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 19871* C1 functions are Lipschitz continuous on closed intervals. (Contributed by Stefan O'Rear, 16-Nov-2014.)

Theoremdveq0 19872 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 19873 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 19874 Lemma for dvgt0 19876 and dvlt0 19877. (Contributed by Mario Carneiro, 19-Feb-2015.)

Theoremdvgt0lem2 19875* Lemma for dvgt0 19876 and dvlt0 19877. (Contributed by Mario Carneiro, 19-Feb-2015.)

Theoremdvgt0 19876 A function on a closed interval with positive derivative is increasing. (Contributed by Mario Carneiro, 19-Feb-2015.)

Theoremdvlt0 19877 A function on a closed interval with negative derivative is decreasing. (Contributed by Mario Carneiro, 19-Feb-2015.)

Theoremdvge0 19878 A function on a closed interval with nonnegative derivative is weakly increasing. (Contributed by Mario Carneiro, 30-Apr-2016.)

Theoremdvle 19879* If are differentiable functions and , then for , . (Contributed by Mario Carneiro, 16-May-2016.)

Theoremdvivthlem1 19880* Lemma for dvivth 19882. (Contributed by Mario Carneiro, 24-Feb-2015.)

Theoremdvivthlem2 19881* Lemma for dvivth 19882. (Contributed by Mario Carneiro, 20-Feb-2015.)

Theoremdvivth 19882 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 19343 does not directly apply). (Contributed by Mario Carneiro, 24-Feb-2015.)

Theoremdvne0 19883 A function on a closed interval with nonzero derivative is either monotone increasing or monotone decreasing. (Contributed by Mario Carneiro, 19-Feb-2015.)

Theoremdvne0f1 19884 A function on a closed interval with nonzero derivative is one-to-one. (Contributed by Mario Carneiro, 19-Feb-2015.)

Theoremlhop1lem 19885* Lemma for lhop1 19886. (Contributed by Mario Carneiro, 29-Dec-2016.)
lim        lim                      lim

Theoremlhop1 19886* 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 19887* 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 19888* 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 19889 Lemma for dvcnvre 19891. (Contributed by Mario Carneiro, 24-Feb-2015.)

Theoremdvcnvrelem2 19890 Lemma for dvcnvre 19891. (Contributed by Mario Carneiro, 19-Feb-2015.) (Revised by Mario Carneiro, 8-Sep-2015.)
fld       t        t

Theoremdvcnvre 19891* 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 19892 A real function with strictly increasing derivative is strictly convex. (Contributed by Mario Carneiro, 20-Jun-2015.)

Theoremdvfsumle 19893* 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 19894* 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 19895* 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 19896* Real closure of a derivative. (Contributed by Mario Carneiro, 18-May-2016.)

Theoremdvfsumrlimf 19897* Lemma for dvfsumrlim 19903. (Contributed by Mario Carneiro, 18-May-2016.)

Theoremdvfsumlem1 19898* Lemma for dvfsumrlim 19903. (Contributed by Mario Carneiro, 17-May-2016.)

Theoremdvfsumlem2 19899* Lemma for dvfsumrlim 19903. (Contributed by Mario Carneiro, 17-May-2016.)

Theoremdvfsumlem3 19900* Lemma for dvfsumrlim 19903. (Contributed by Mario Carneiro, 17-May-2016.)

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