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

Theoremlimcmpt 19801* Express the limit operator for a function defined by a mapping. (Contributed by Mario Carneiro, 25-Dec-2016.)
t        fld       lim

Theoremlimcmpt2 19802* Express the limit operator for a function defined by a mapping. (Contributed by Mario Carneiro, 25-Dec-2016.)
t        fld       lim

Theoremlimcresi 19803 Any limit of is also a limit of the restriction of . (Contributed by Mario Carneiro, 28-Dec-2016.)
lim lim

Theoremlimcres 19804 If is an interior point of relative to the domain , then a limit point of extends to a limit of . (Contributed by Mario Carneiro, 27-Dec-2016.)
fld       t               lim lim

Theoremcnplimc 19805 A function is continuous at iff its limit at equals the value of the function there. (Contributed by Mario Carneiro, 28-Dec-2016.)
fld       t        lim

Theoremcnlimc 19806* is a continuous function iff the limit of the function at each point equals the value of the function. (Contributed by Mario Carneiro, 28-Dec-2016.)
lim

Theoremcnlimci 19807 If is a continuous function, then the limit of the function at any point equals its value. (Contributed by Mario Carneiro, 28-Dec-2016.)
lim

Theoremcnmptlimc 19808* If is a continuous function, then the limit of the function at any point equals its value. (Contributed by Mario Carneiro, 28-Dec-2016.)
lim

Theoremlimccnp 19809 If the limit of at is and is continuous at , then the limit of at is . (Contributed by Mario Carneiro, 28-Dec-2016.)
fld       t        lim               lim

Theoremlimccnp2 19810* The image of a convergent sequence under a continuous map is convergent to the image of the original point. Binary operation version. (Contributed by Mario Carneiro, 28-Dec-2016.)
fld       t        lim        lim               lim

Theoremlimcco 19811* Composition of two limits. (Contributed by Mario Carneiro, 29-Dec-2016.)
lim        lim                      lim

Theoremlimciun 19812* A point is a limit of on the finite union iff it is the limit of the restriction of to each . (Contributed by Mario Carneiro, 30-Dec-2016.)
lim lim

Theoremlimcun 19813 A point is a limit of on iff it is the limit of the restriction of to and to . (Contributed by Mario Carneiro, 30-Dec-2016.)
lim lim lim

Theoremdvlem 19814 Closure for a difference quotient. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 9-Feb-2015.)

Theoremdvfval 19815* Value and set bounds on the derivative operator. (Contributed by Mario Carneiro, 7-Aug-2014.) (Revised by Mario Carneiro, 25-Dec-2016.)
t        fld       lim

Theoremeldv 19816* The differentiable predicate. A function is differentiable at with derivative iff is defined in a neighborhood of and the difference quotient has limit at . (Contributed by Mario Carneiro, 7-Aug-2014.) (Revised by Mario Carneiro, 25-Dec-2016.)
t        fld                                   lim

Theoremdvcl 19817 The derivative function takes values in the complex numbers. (Contributed by Mario Carneiro, 7-Aug-2014.) (Revised by Mario Carneiro, 9-Feb-2015.)

Theoremdvbssntr 19818 The set of differentiable points is a subset of the interior of the domain of the function. (Contributed by Mario Carneiro, 7-Aug-2014.) (Revised by Mario Carneiro, 9-Feb-2015.)
t        fld

Theoremdvbss 19819 The set of differentiable points is a subset of the domain of the function. (Contributed by Mario Carneiro, 6-Aug-2014.) (Revised by Mario Carneiro, 9-Feb-2015.)

Theoremdvbsss 19820 The set of differentiable points is a subset of the ambient topology. (Contributed by Mario Carneiro, 18-Mar-2015.)

Theoremperfdvf 19821 The derivative is a function, whenever it is defined relative to a perfect subset of the complex numbers. (Contributed by Mario Carneiro, 25-Dec-2016.)
fld       t Perf

Theoremrecnprss 19822 Both and are subsets of . (Contributed by Mario Carneiro, 10-Feb-2015.)

Theoremrecnperf 19823 Both and are perfect subsets of . (Contributed by Mario Carneiro, 28-Dec-2016.)
fld       t Perf

Theoremdvfg 19824 Explicitly write out the functionality condition on derivative for and . (Contributed by Mario Carneiro, 9-Feb-2015.)

Theoremdvf 19825 The derivative is a function. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Mario Carneiro, 9-Feb-2015.)

Theoremdvfcn 19826 The derivative is a function. (Contributed by Mario Carneiro, 9-Feb-2015.)

Theoremdvreslem 19827* Lemma for dvres 19829. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Mario Carneiro, 28-Dec-2016.)
fld       t

Theoremdvres2lem 19828* Lemma for dvres2 19830. (Contributed by Mario Carneiro, 9-Feb-2015.) (Revised by Mario Carneiro, 28-Dec-2016.)
fld       t

Theoremdvres 19829 Restriction of a derivative. Note that our definition of derivative df-dv 19785 would still make sense if we demanded that be an element of the domain instead of an interior point of the domain, but then it is possible for a non-differentiable function to have two different derivatives at a single point when restricted to different subsets containing ; a classic example is the absolute value function restricted to and . (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Mario Carneiro, 9-Feb-2015.)
fld       t

Theoremdvres2 19830 Restriction of the base set of a derivative. The primary application of this theorem says that if a function is complex differentiable then it is also real differentiable. Unlike dvres 19829, there is no simple reverse relation relating real differentiable functions to complex differentiability, and indeed there are functions like which are everywhere real-differentiable but nowhere complex-differentiable.) (Contributed by Mario Carneiro, 9-Feb-2015.)

Theoremdvres3 19831 Restriction of a complex differentiable function to the reals. (Contributed by Mario Carneiro, 10-Feb-2015.)

Theoremdvres3a 19832 Restriction of a complex differentiable function to the reals. This version of dvres3 19831 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.)
fld

Theoremdvidlem 19833* Lemma for dvid 19835 and dvconst 19834. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Mario Carneiro, 9-Feb-2015.)

Theoremdvconst 19834 Derivative of a constant function. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Mario Carneiro, 9-Feb-2015.)

Theoremdvid 19835 Derivative of the identity function. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Mario Carneiro, 9-Feb-2015.)

Theoremdvcnp 19836* 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 19837 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 19838 A differentiable function is continuous. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by Mario Carneiro, 7-Sep-2015.)

Theoremdvnfval 19839* Value of the iterated derivative. (Contributed by Mario Carneiro, 11-Feb-2015.)

Theoremdvnff 19840 The iterated derivative is a function. (Contributed by Mario Carneiro, 11-Feb-2015.)

Theoremdvn0 19841 Zero times iterated derivative. (Contributed by Stefan O'Rear, 15-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)

Theoremdvnp1 19842 Successor iterated derivative. (Contributed by Stefan O'Rear, 15-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)

Theoremdvn1 19843 One times iterated derivative. (Contributed by Mario Carneiro, 1-Jan-2017.)

Theoremdvnf 19844 The N-times derivative is a function. (Contributed by Stefan O'Rear, 16-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)

Theoremdvnbss 19845 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 19846 The -th derivative of the -th derivative of is the same as the -th derivative of . (Contributed by Mario Carneiro, 11-Feb-2015.)

Theoremdvn2bss 19847 An N-times differentiable point is an M-times differeentiable point, if . (Contributed by Mario Carneiro, 30-Dec-2016.)

Theoremdvnres 19848 Multiple derivative version of dvres3a 19832. (Contributed by Mario Carneiro, 11-Feb-2015.)

Theoremcpnfval 19849* Condition for n-times continuous differentiability. (Contributed by Stefan O'Rear, 15-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)

Theoremfncpn 19850 The object is a function. (Contributed by Stefan O'Rear, 16-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)

Theoremelcpn 19851 Condition for n-times continuous differentiability. (Contributed by Stefan O'Rear, 15-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)

Theoremcpnord 19852 conditions are ordered by strength. (Contributed by Stefan O'Rear, 16-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)

Theoremcpncn 19853 A function is continuous. (Contributed by Mario Carneiro, 11-Feb-2015.)

Theoremcpnres 19854 The restriction of a function is . (Contributed by Mario Carneiro, 11-Feb-2015.)

Theoremdvaddbr 19855 The sum rule for derivatives at a point. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 28-Dec-2016.)
fld

Theoremdvmulbr 19856 The product rule for derivatives at a point. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 28-Dec-2016.)
fld

Theoremdvadd 19857 The sum rule for derivatives at a point. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)

Theoremdvmul 19858 The product rule for derivatives at a point. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)

Theoremdvaddf 19859 The sum rule for everywhere-differentiable functions. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)

Theoremdvmulf 19860 The product rule for everywhere-differentiable functions. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)

Theoremdvcmul 19861 The product rule when one argument is a constant. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)

Theoremdvcmulf 19862 The product rule when one argument is a constant. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)

Theoremdvcobr 19863 The chain rule for derivatives at a point. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 28-Dec-2016.)
fld

Theoremdvco 19864 The chain rule for derivatives at a point. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)

Theoremdvcof 19865 The chain rule for everywhere-differentiable functions. (Contributed by Mario Carneiro, 10-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)

Theoremdvcjbr 19866 The derivative of the conjugate of a function. (This doesn't follow from dvcobr 19863 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 19867 The derivative of the conjugate of a function. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)

Theoremdvfre 19868 The derivative of a real function is real. (Contributed by Mario Carneiro, 1-Sep-2014.)

Theoremdvnfre 19869 The -th derivative of a real function is real. (Contributed by Mario Carneiro, 1-Jan-2017.)

Theoremdvexp 19870* Derivative of a power function. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)

Theoremdvexp2 19871* Derivative of an exponential, possibly zero power. (Contributed by Stefan O'Rear, 13-Nov-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)

Theoremdvrec 19872* Derivative of the reciprocal function. (Contributed by Mario Carneiro, 25-Feb-2015.) (Revised by Mario Carneiro, 28-Dec-2016.)

Theoremdvmptres3 19873* Function-builder for derivative: restrict a derivative to a subset. (Contributed by Mario Carneiro, 11-Feb-2015.)
fld

Theoremdvmptid 19874* Function-builder for derivative: derivative of the identity. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)

Theoremdvmptc 19875* Function-builder for derivative: derivative of a constant. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)

Theoremdvmptcl 19876* Closure lemma for dvmptcmul 19881 and other related theorems. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)

Theoremdvmptadd 19877* Function-builder for derivative, addition rule. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)

Theoremdvmptmul 19878* Function-builder for derivative, product rule. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)

Theoremdvmptres2 19879* 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 19880* 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 19881* Function-builder for derivative, product rule for constant multiplier. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)

Theoremdvmptdivc 19882* Function-builder for derivative, division rule for constant divisor. (Contributed by Mario Carneiro, 18-May-2016.)

Theoremdvmptneg 19883* Function-builder for derivative, product rule for negatives. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)

Theoremdvmptsub 19884* Function-builder for derivative, subtraction rule. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)

Theoremdvmptcj 19885* Function-builder for derivative, conjugate rule. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)

Theoremdvmptre 19886* Function-builder for derivative, real part. (Contributed by Mario Carneiro, 1-Sep-2014.)

Theoremdvmptim 19887* Function-builder for derivative, imaginary part. (Contributed by Mario Carneiro, 1-Sep-2014.)

Theoremdvmptntr 19888* 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 19889* Function-builder for derivative, chain rule. (Contributed by Mario Carneiro, 1-Sep-2014.)

Theoremdvmptfsum 19890* Function-builder for derivative, finite sums rule. (Contributed by Stefan O'Rear, 12-Nov-2014.)
t        fld

Theoremdvcnvlem 19891 Lemma for dvcnvre 19934. (Contributed by Mario Carneiro, 25-Feb-2015.) (Revised by Mario Carneiro, 8-Sep-2015.)
fld       t

Theoremdvcnv 19892* A weak version of dvcnvre 19934, 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 19893* Derivative of an exponential of integer exponent. (Contributed by Mario Carneiro, 26-Feb-2015.)

Theoremdveflem 19894 Derivative of the exponential function at 0. The key step in the proof is eftlub 12741, to show that . (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 28-Dec-2016.)

Theoremdvef 19895 Derivative of the exponential function. (Contributed by Mario Carneiro, 9-Aug-2014.) (Proof shortened by Mario Carneiro, 10-Feb-2015.)

Theoremdvsincos 19896 Derivative of the sine and cosine functions. (Contributed by Mario Carneiro, 21-May-2016.)

Theoremdvsin 19897 Derivative of the sine function. (Contributed by Mario Carneiro, 21-May-2016.)

Theoremdvcos 19898 Derivative of the cosine function. (Contributed by Mario Carneiro, 21-May-2016.)

12.3.1.2  Results on real differentiation

Theoremdvferm1lem 19899* Lemma for dvferm 19903. (Contributed by Mario Carneiro, 24-Feb-2015.)

Theoremdvferm1 19900* One-sided version of dvferm 19903. 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.)

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