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

Theoremdvres3a 19801 Restriction of a complex differentiable function to the reals. This version of dvres3 19800 assumes that is differentiable on its domain, but does not require to be differentiable on the whole real line. (Contributed by Mario Carneiro, 11-Feb-2015.)
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Theoremdvidlem 19802* Lemma for dvid 19804 and dvconst 19803. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Mario Carneiro, 9-Feb-2015.)

Theoremdvconst 19803 Derivative of a constant function. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Mario Carneiro, 9-Feb-2015.)

Theoremdvid 19804 Derivative of the identity function. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Mario Carneiro, 9-Feb-2015.)

Theoremdvcnp 19805* The difference quotient is continuous at when the original function is differentiable at . (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Mario Carneiro, 28-Dec-2016.)
t        fld

Theoremdvcnp2 19806 A function is continuous at each point for which it is differentiable. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 28-Dec-2016.)
t        fld

Theoremdvcn 19807 A differentiable function is continuous. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by Mario Carneiro, 7-Sep-2015.)

Theoremdvnfval 19808* Value of the iterated derivative. (Contributed by Mario Carneiro, 11-Feb-2015.)

Theoremdvnff 19809 The iterated derivative is a function. (Contributed by Mario Carneiro, 11-Feb-2015.)

Theoremdvn0 19810 Zero times iterated derivative. (Contributed by Stefan O'Rear, 15-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)

Theoremdvnp1 19811 Successor iterated derivative. (Contributed by Stefan O'Rear, 15-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)

Theoremdvn1 19812 One times iterated derivative. (Contributed by Mario Carneiro, 1-Jan-2017.)

Theoremdvnf 19813 The N-times derivative is a function. (Contributed by Stefan O'Rear, 16-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)

Theoremdvnbss 19814 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 19815 The -th derivative of the -th derivative of is the same as the -th derivative of . (Contributed by Mario Carneiro, 11-Feb-2015.)

Theoremdvn2bss 19816 An N-times differentiable point is an M-times differeentiable point, if . (Contributed by Mario Carneiro, 30-Dec-2016.)

Theoremdvnres 19817 Multiple derivative version of dvres3a 19801. (Contributed by Mario Carneiro, 11-Feb-2015.)

Theoremcpnfval 19818* Condition for n-times continuous differentiability. (Contributed by Stefan O'Rear, 15-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)

Theoremfncpn 19819 The object is a function. (Contributed by Stefan O'Rear, 16-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)

Theoremelcpn 19820 Condition for n-times continuous differentiability. (Contributed by Stefan O'Rear, 15-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)

Theoremcpnord 19821 conditions are ordered by strength. (Contributed by Stefan O'Rear, 16-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)

Theoremcpncn 19822 A function is continuous. (Contributed by Mario Carneiro, 11-Feb-2015.)

Theoremcpnres 19823 The restriction of a function is . (Contributed by Mario Carneiro, 11-Feb-2015.)

Theoremdvaddbr 19824 The sum rule for derivatives at a point. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 28-Dec-2016.)
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Theoremdvmulbr 19825 The product rule for derivatives at a point. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 28-Dec-2016.)
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Theoremdvadd 19826 The sum rule for derivatives at a point. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)

Theoremdvmul 19827 The product rule for derivatives at a point. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)

Theoremdvaddf 19828 The sum rule for everywhere-differentiable functions. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)

Theoremdvmulf 19829 The product rule for everywhere-differentiable functions. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)

Theoremdvcmul 19830 The product rule when one argument is a constant. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)

Theoremdvcmulf 19831 The product rule when one argument is a constant. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)

Theoremdvcobr 19832 The chain rule for derivatives at a point. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 28-Dec-2016.)
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Theoremdvco 19833 The chain rule for derivatives at a point. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)

Theoremdvcof 19834 The chain rule for everywhere-differentiable functions. (Contributed by Mario Carneiro, 10-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)

Theoremdvcjbr 19835 The derivative of the conjugate of a function. (This doesn't follow from dvcobr 19832 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 19836 The derivative of the conjugate of a function. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)

Theoremdvfre 19837 The derivative of a real function is real. (Contributed by Mario Carneiro, 1-Sep-2014.)

Theoremdvnfre 19838 The -th derivative of a real function is real. (Contributed by Mario Carneiro, 1-Jan-2017.)

Theoremdvexp 19839* Derivative of a power function. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)

Theoremdvexp2 19840* Derivative of an exponential, possibly zero power. (Contributed by Stefan O'Rear, 13-Nov-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)

Theoremdvrec 19841* Derivative of the reciprocal function. (Contributed by Mario Carneiro, 25-Feb-2015.) (Revised by Mario Carneiro, 28-Dec-2016.)

Theoremdvmptres3 19842* Function-builder for derivative: restrict a derivative to a subset. (Contributed by Mario Carneiro, 11-Feb-2015.)
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Theoremdvmptid 19843* Function-builder for derivative: derivative of the identity. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)

Theoremdvmptc 19844* Function-builder for derivative: derivative of a constant. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)

Theoremdvmptcl 19845* Closure lemma for dvmptcmul 19850 and other related theorems. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)

Theoremdvmptadd 19846* Function-builder for derivative, addition rule. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)

Theoremdvmptmul 19847* Function-builder for derivative, product rule. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)

Theoremdvmptres2 19848* 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 19849* 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 19850* Function-builder for derivative, product rule for constant multiplier. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)

Theoremdvmptdivc 19851* Function-builder for derivative, division rule for constant divisor. (Contributed by Mario Carneiro, 18-May-2016.)

Theoremdvmptneg 19852* Function-builder for derivative, product rule for negatives. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)

Theoremdvmptsub 19853* Function-builder for derivative, subtraction rule. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)

Theoremdvmptcj 19854* Function-builder for derivative, conjugate rule. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)

Theoremdvmptre 19855* Function-builder for derivative, real part. (Contributed by Mario Carneiro, 1-Sep-2014.)

Theoremdvmptim 19856* Function-builder for derivative, imaginary part. (Contributed by Mario Carneiro, 1-Sep-2014.)

Theoremdvmptntr 19857* 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 19858* Function-builder for derivative, chain rule. (Contributed by Mario Carneiro, 1-Sep-2014.)

Theoremdvmptfsum 19859* Function-builder for derivative, finite sums rule. (Contributed by Stefan O'Rear, 12-Nov-2014.)
t        fld

Theoremdvcnvlem 19860 Lemma for dvcnvre 19903. (Contributed by Mario Carneiro, 25-Feb-2015.) (Revised by Mario Carneiro, 8-Sep-2015.)
fld       t

Theoremdvcnv 19861* A weak version of dvcnvre 19903, 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 19862* Derivative of an exponential of integer exponent. (Contributed by Mario Carneiro, 26-Feb-2015.)

Theoremdveflem 19863 Derivative of the exponential function at 0. The key step in the proof is eftlub 12710, to show that . (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 28-Dec-2016.)

Theoremdvef 19864 Derivative of the exponential function. (Contributed by Mario Carneiro, 9-Aug-2014.) (Proof shortened by Mario Carneiro, 10-Feb-2015.)

Theoremdvsincos 19865 Derivative of the sine and cosine functions. (Contributed by Mario Carneiro, 21-May-2016.)

Theoremdvsin 19866 Derivative of the sine function. (Contributed by Mario Carneiro, 21-May-2016.)

Theoremdvcos 19867 Derivative of the cosine function. (Contributed by Mario Carneiro, 21-May-2016.)

Theoremdvferm1lem 19868* Lemma for dvferm 19872. (Contributed by Mario Carneiro, 24-Feb-2015.)

Theoremdvferm1 19869* One-sided version of dvferm 19872. 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 19870* Lemma for dvferm 19872. (Contributed by Mario Carneiro, 24-Feb-2015.)

Theoremdvferm2 19871* One-sided version of dvferm 19872. 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 19872* 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 19873* Lemma for rolle 19874. (Contributed by Mario Carneiro, 1-Sep-2014.)

Theoremrolle 19874* 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 19875* 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 19876* 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 19877* A function with derivative bounded by is Lipschitz continuous with Lipchitz constant equal to . (Contributed by Mario Carneiro, 3-Mar-2015.)

Theoremdvlipcn 19878* 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 19879* Combine the results of dvlip 19877 and dvlipcn 19878 into one. (Contributed by Mario Carneiro, 18-Mar-2015.) (Revised by Mario Carneiro, 8-Sep-2015.)

Theoremc1liplem1 19880* Lemma for c1lip1 19881. (Contributed by Stefan O'Rear, 15-Nov-2014.)

Theoremc1lip1 19881* C1 functions are Lipschitz continuous on closed intervals. (Contributed by Stefan O'Rear, 16-Nov-2014.)

Theoremc1lip2 19882* 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 19883* C1 functions are Lipschitz continuous on closed intervals. (Contributed by Stefan O'Rear, 16-Nov-2014.)

Theoremdveq0 19884 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 19885 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 19886 Lemma for dvgt0 19888 and dvlt0 19889. (Contributed by Mario Carneiro, 19-Feb-2015.)

Theoremdvgt0lem2 19887* Lemma for dvgt0 19888 and dvlt0 19889. (Contributed by Mario Carneiro, 19-Feb-2015.)

Theoremdvgt0 19888 A function on a closed interval with positive derivative is increasing. (Contributed by Mario Carneiro, 19-Feb-2015.)

Theoremdvlt0 19889 A function on a closed interval with negative derivative is decreasing. (Contributed by Mario Carneiro, 19-Feb-2015.)

Theoremdvge0 19890 A function on a closed interval with nonnegative derivative is weakly increasing. (Contributed by Mario Carneiro, 30-Apr-2016.)

Theoremdvle 19891* If are differentiable functions and , then for , . (Contributed by Mario Carneiro, 16-May-2016.)

Theoremdvivthlem1 19892* Lemma for dvivth 19894. (Contributed by Mario Carneiro, 24-Feb-2015.)

Theoremdvivthlem2 19893* Lemma for dvivth 19894. (Contributed by Mario Carneiro, 20-Feb-2015.)

Theoremdvivth 19894 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 19355 does not directly apply). (Contributed by Mario Carneiro, 24-Feb-2015.)

Theoremdvne0 19895 A function on a closed interval with nonzero derivative is either monotone increasing or monotone decreasing. (Contributed by Mario Carneiro, 19-Feb-2015.)

Theoremdvne0f1 19896 A function on a closed interval with nonzero derivative is one-to-one. (Contributed by Mario Carneiro, 19-Feb-2015.)

Theoremlhop1lem 19897* Lemma for lhop1 19898. (Contributed by Mario Carneiro, 29-Dec-2016.)
lim        lim                      lim

Theoremlhop1 19898* 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 19899* 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 19900* 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

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