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Theorem List for Metamath Proof Explorer - 19101-19200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdvcnp2 19101 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 19102 A differentiable function is continuous. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by Mario Carneiro, 7-Sep-2015.)

Theoremdvnfval 19103* Value of the iterated derivative. (Contributed by Mario Carneiro, 11-Feb-2015.)

Theoremdvnff 19104 The iterated derivative is a function. (Contributed by Mario Carneiro, 11-Feb-2015.)

Theoremdvn0 19105 Zero times iterated derivative. (Contributed by Stefan O'Rear, 15-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)

Theoremdvnp1 19106 Successor iterated derivative. (Contributed by Stefan O'Rear, 15-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)

Theoremdvn1 19107 One times iterated derivative. (Contributed by Mario Carneiro, 1-Jan-2017.)

Theoremdvnf 19108 The N-times derivative is a function. (Contributed by Stefan O'Rear, 16-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)

Theoremdvnbss 19109 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 19110 The -th derivative of the -th derivative of is the same as the -th derivative of . (Contributed by Mario Carneiro, 11-Feb-2015.)

Theoremdvn2bss 19111 An N-times differentiable point is an M-times differeentiable point, if . (Contributed by Mario Carneiro, 30-Dec-2016.)

Theoremdvnres 19112 Multiple derivative version of dvres3a 19096. (Contributed by Mario Carneiro, 11-Feb-2015.)

Theoremcpnfval 19113* Condition for n-times continuous differentiability. (Contributed by Stefan O'Rear, 15-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)

Theoremfncpn 19114 The object is a function. (Contributed by Stefan O'Rear, 16-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)

Theoremelcpn 19115 Condition for n-times continuous differentiability. (Contributed by Stefan O'Rear, 15-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)

Theoremcpnord 19116 conditions are ordered by strength. (Contributed by Stefan O'Rear, 16-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)

Theoremcpncn 19117 A function is continuous. (Contributed by Mario Carneiro, 11-Feb-2015.)

Theoremcpnres 19118 The restriction of a function is . (Contributed by Mario Carneiro, 11-Feb-2015.)

Theoremdvaddbr 19119 The sum rule for derivatives at a point. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 28-Dec-2016.)
fld

Theoremdvmulbr 19120 The product rule for derivatives at a point. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 28-Dec-2016.)
fld

Theoremdvadd 19121 The sum rule for derivatives at a point. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)

Theoremdvmul 19122 The product rule for derivatives at a point. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)

Theoremdvaddf 19123 The sum rule for everywhere-differentiable functions. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)

Theoremdvmulf 19124 The product rule for everywhere-differentiable functions. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)

Theoremdvcmul 19125 The product rule when one argument is a constant. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)

Theoremdvcmulf 19126 The product rule when one argument is a constant. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)

Theoremdvcobr 19127 The chain rule for derivatives at a point. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 28-Dec-2016.)
fld

Theoremdvco 19128 The chain rule for derivatives at a point. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)

Theoremdvcof 19129 The chain rule for everywhere-differentiable functions. (Contributed by Mario Carneiro, 10-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)

Theoremdvcjbr 19130 The derivative of the conjugate of a function. (This doesn't follow from dvcobr 19127 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 19131 The derivative of the conjugate of a function. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)

Theoremdvfre 19132 The derivative of a real function is real. (Contributed by Mario Carneiro, 1-Sep-2014.)

Theoremdvnfre 19133 The -th derivative of a real function is real. (Contributed by Mario Carneiro, 1-Jan-2017.)

Theoremdvexp 19134* Derivative of a power function. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)

Theoremdvexp2 19135* Derivative of an exponential, possibly zero power. (Contributed by Stefan O'Rear, 13-Nov-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)

Theoremdvrec 19136* Derivative of the reciprocal function. (Contributed by Mario Carneiro, 25-Feb-2015.) (Revised by Mario Carneiro, 28-Dec-2016.)

Theoremdvmptres3 19137* Function-builder for derivative: restrict a derivative to a subset. (Contributed by Mario Carneiro, 11-Feb-2015.)
fld

Theoremdvmptid 19138* Function-builder for derivative: derivative of the identity. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)

Theoremdvmptc 19139* Function-builder for derivative: derivative of a constant. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)

Theoremdvmptcl 19140* Closure lemma for dvmptcmul 19145 and other related theorems. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)

Theoremdvmptadd 19141* Function-builder for derivative, addition rule. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)

Theoremdvmptmul 19142* Function-builder for derivative, product rule. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)

Theoremdvmptres2 19143* 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 19144* 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 19145* Function-builder for derivative, product rule for constant multiplier. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)

Theoremdvmptdivc 19146* Function-builder for derivative, division rule for constant divisor. (Contributed by Mario Carneiro, 18-May-2016.)

Theoremdvmptneg 19147* Function-builder for derivative, product rule for negatives. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)

Theoremdvmptsub 19148* Function-builder for derivative, subtraction rule. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)

Theoremdvmptcj 19149* Function-builder for derivative, conjugate rule. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)

Theoremdvmptre 19150* Function-builder for derivative, real part. (Contributed by Mario Carneiro, 1-Sep-2014.)

Theoremdvmptim 19151* Function-builder for derivative, imaginary part. (Contributed by Mario Carneiro, 1-Sep-2014.)

Theoremdvmptntr 19152* 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 19153* Function-builder for derivative, chain rule. (Contributed by Mario Carneiro, 1-Sep-2014.)

Theoremdvmptfsum 19154* Function-builder for derivative, finite sums rule. (Contributed by Stefan O'Rear, 12-Nov-2014.)
t        fld

Theoremdvcnvlem 19155 Lemma for dvcnvre 19198. (Contributed by Mario Carneiro, 25-Feb-2015.) (Revised by Mario Carneiro, 8-Sep-2015.)
fld       t

Theoremdvcnv 19156* A weak version of dvcnvre 19198, 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 19157* Derivative of an exponential of integer exponent. (Contributed by Mario Carneiro, 26-Feb-2015.)

Theoremdveflem 19158 Derivative of the exponential function at 0. The key step in the proof is eftlub 12263, to show that . (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 28-Dec-2016.)

Theoremdvef 19159 Derivative of the exponential function. (Contributed by Mario Carneiro, 9-Aug-2014.) (Proof shortened by Mario Carneiro, 10-Feb-2015.)

Theoremdvsincos 19160 Derivative of the sine and cosine functions. (Contributed by Mario Carneiro, 21-May-2016.)

Theoremdvsin 19161 Derivative of the sine function. (Contributed by Mario Carneiro, 21-May-2016.)

Theoremdvcos 19162 Derivative of the cosine function. (Contributed by Mario Carneiro, 21-May-2016.)

Theoremdvferm1lem 19163* Lemma for dvferm 19167. (Contributed by Mario Carneiro, 24-Feb-2015.)

Theoremdvferm1 19164* One-sided version of dvferm 19167. 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 19165* Lemma for dvferm 19167. (Contributed by Mario Carneiro, 24-Feb-2015.)

Theoremdvferm2 19166* One-sided version of dvferm 19167. 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 19167* 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 19168* Lemma for rolle 19169. (Contributed by Mario Carneiro, 1-Sep-2014.)

Theoremrolle 19169* 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 19170* 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 19171* 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 19172* A function with derivative bounded by is Lipschitz continuous with Lipchitz constant equal to . (Contributed by Mario Carneiro, 3-Mar-2015.)

Theoremdvlipcn 19173* 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 19174* Combine the results of dvlip 19172 and dvlipcn 19173 into one. (Contributed by Mario Carneiro, 18-Mar-2015.) (Revised by Mario Carneiro, 8-Sep-2015.)

Theoremc1liplem1 19175* Lemma for c1lip1 19176. (Contributed by Stefan O'Rear, 15-Nov-2014.)

Theoremc1lip1 19176* C1 functions are Lipschitz continuous on closed intervals. (Contributed by Stefan O'Rear, 16-Nov-2014.)

Theoremc1lip2 19177* 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 19178* C1 functions are Lipschitz continuous on closed intervals. (Contributed by Stefan O'Rear, 16-Nov-2014.)

Theoremdveq0 19179 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 19180 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 19181 Lemma for dvgt0 19183 and dvlt0 19184. (Contributed by Mario Carneiro, 19-Feb-2015.)

Theoremdvgt0lem2 19182* Lemma for dvgt0 19183 and dvlt0 19184. (Contributed by Mario Carneiro, 19-Feb-2015.)

Theoremdvgt0 19183 A function on a closed interval with positive derivative is increasing. (Contributed by Mario Carneiro, 19-Feb-2015.)

Theoremdvlt0 19184 A function on a closed interval with negative derivative is decreasing. (Contributed by Mario Carneiro, 19-Feb-2015.)

Theoremdvge0 19185 A function on a closed interval with nonnegative derivative is weakly increasing. (Contributed by Mario Carneiro, 30-Apr-2016.)

Theoremdvle 19186* If are differentiable functions and , then for , . (Contributed by Mario Carneiro, 16-May-2016.)

Theoremdvivthlem1 19187* Lemma for dvivth 19189. (Contributed by Mario Carneiro, 24-Feb-2015.)

Theoremdvivthlem2 19188* Lemma for dvivth 19189. (Contributed by Mario Carneiro, 20-Feb-2015.)

Theoremdvivth 19189 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 18650 does not directly apply). (Contributed by Mario Carneiro, 24-Feb-2015.)

Theoremdvne0 19190 A function on a closed interval with nonzero derivative is either monotone increasing or monotone decreasing. (Contributed by Mario Carneiro, 19-Feb-2015.)

Theoremdvne0f1 19191 A function on a closed interval with nonzero derivative is one-to-one. (Contributed by Mario Carneiro, 19-Feb-2015.)

Theoremlhop1lem 19192* Lemma for lhop1 19193. (Contributed by Mario Carneiro, 29-Dec-2016.)
lim        lim                      lim

Theoremlhop1 19193* 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 19194* 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 19195* 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 19196 Lemma for dvcnvre 19198. (Contributed by Mario Carneiro, 24-Feb-2015.)

Theoremdvcnvrelem2 19197 Lemma for dvcnvre 19198. (Contributed by Mario Carneiro, 19-Feb-2015.) (Revised by Mario Carneiro, 8-Sep-2015.)
fld       t        t

Theoremdvcnvre 19198* 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 19199 A real function with strictly increasing derivative is strictly convex. (Contributed by Mario Carneiro, 20-Jun-2015.)

Theoremdvfsumle 19200* 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.)
..^        ..^        ..^

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