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Theorem List for Metamath Proof Explorer - 19701-19800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremditgsplit 19701* This theorem is the raison d'être for the directed integral, because unlike itgspliticc 19681, there is no constraint on the ordering of the points in the domain. (Contributed by Mario Carneiro, 13-Aug-2014.)
_ _ _

12.3  Derivatives

12.3.1  Real and complex differentiation

Syntaxclimc 19702 The limit operator.
lim

Syntaxcdv 19703 The derivative operator.

Syntaxcdvn 19704 The -th derivative operator.

Syntaxccpn 19705 The set of -times continuously differentiable functions.

Definitiondf-limc 19706* Define the set of limits of a complex function at a point. Under normal circumstances, this will be a singleton or empty, depending on whether the limit exists. (Contributed by Mario Carneiro, 24-Dec-2016.)
lim fld t

Definitiondf-dv 19707* Define the derivative operator on functions on the reals. This acts on functions to produce a function that is defined where the original function is differentiable, with value the derivative of the function at these points. The set here is the ambient topological space under which we are evaluating the continuity of the difference quotient. Although the definition is valid for any subset of and is well-behaved when contains no isolated points, we will restrict our attention to the cases or for the majority of the development, these corresponding respectively to real and complex differentiation. (Contributed by Mario Carneiro, 7-Aug-2014.)
fldt lim

Definitiondf-dvn 19708* Define the -th derivative operator on functions on the complexes. This just iterates the derivative operation according to the last argument. (Contributed by Mario Carneiro, 11-Feb-2015.)

Definitiondf-cpn 19709* Define the set of -times continuously differentiable functions. (Contributed by Stefan O'Rear, 15-Nov-2014.)

Theoremreldv 19710 The derivative function is a relation. (Contributed by Mario Carneiro, 7-Aug-2014.) (Revised by Mario Carneiro, 24-Dec-2016.)

Theoremlimcvallem 19711* Lemma for ellimc 19713. (Contributed by Mario Carneiro, 25-Dec-2016.)
t        fld

Theoremlimcfval 19712* Value and set bounds on the limit operator. (Contributed by Mario Carneiro, 25-Dec-2016.)
t        fld       lim lim

Theoremellimc 19713* Value of the limit predicate. is the limit of the function at if the function , formed by adding to the domain of and setting it to , is continuous at . (Contributed by Mario Carneiro, 25-Dec-2016.)
t        fld                                   lim

Theoremlimcrcl 19714 Reverse closure for the limit operator. (Contributed by Mario Carneiro, 28-Dec-2016.)
lim

Theoremlimccl 19715 Closure of the limit operator. (Contributed by Mario Carneiro, 25-Dec-2016.)
lim

Theoremlimcdif 19716 It suffices to consider functions which are not defined at to define the limit of a function. In particular, the value of the original function at does not affect the limit of . (Contributed by Mario Carneiro, 25-Dec-2016.)
lim lim

Theoremellimc2 19717* Write the definition of a limit directly in terms of open sets of the topology on the complexes. (Contributed by Mario Carneiro, 25-Dec-2016.)
fld       lim

Theoremlimcnlp 19718 If is not a limit point of the domain of the function , then every point is a limit of at . (Contributed by Mario Carneiro, 25-Dec-2016.)
fld              lim

Theoremellimc3 19719* Write the epsilon-delta definition of a limit. (Contributed by Mario Carneiro, 28-Dec-2016.)
lim

Theoremlimcflflem 19720 Lemma for limcflf 19721. (Contributed by Mario Carneiro, 25-Dec-2016.)
fld              t

Theoremlimcflf 19721 The limit operator can be expressed as a filter limit, from the filter of neighborhoods of restricted to , to the topology of the complexes. (If is not a limit point of , then it is still formally a filter limit, but the neighborhood filter is not a proper filter in this case.) (Contributed by Mario Carneiro, 25-Dec-2016.)
fld              t        lim

Theoremlimcmo 19722* If is a limit point of the domain of the function , then there is at most one limit value of at . (Contributed by Mario Carneiro, 25-Dec-2016.)
fld       lim

Theoremlimcmpt 19723* Express the limit operator for a function defined by a mapping. (Contributed by Mario Carneiro, 25-Dec-2016.)
t        fld       lim

Theoremlimcmpt2 19724* Express the limit operator for a function defined by a mapping. (Contributed by Mario Carneiro, 25-Dec-2016.)
t        fld       lim

Theoremlimcresi 19725 Any limit of is also a limit of the restriction of . (Contributed by Mario Carneiro, 28-Dec-2016.)
lim lim

Theoremlimcres 19726 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 19727 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 19728* 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 19729 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 19730* 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 19731 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 19732* 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 19733* Composition of two limits. (Contributed by Mario Carneiro, 29-Dec-2016.)
lim        lim                      lim

Theoremlimciun 19734* 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 19735 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 19736 Closure for a difference quotient. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 9-Feb-2015.)

Theoremdvfval 19737* 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 19738* 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 19739 The derivative function takes values in the complex numbers. (Contributed by Mario Carneiro, 7-Aug-2014.) (Revised by Mario Carneiro, 9-Feb-2015.)

Theoremdvbssntr 19740 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 19741 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 19742 The set of differentiable points is a subset of the ambient topology. (Contributed by Mario Carneiro, 18-Mar-2015.)

Theoremperfdvf 19743 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 19744 Both and are subsets of . (Contributed by Mario Carneiro, 10-Feb-2015.)

Theoremrecnperf 19745 Both and are perfect subsets of . (Contributed by Mario Carneiro, 28-Dec-2016.)
fld       t Perf

Theoremdvfg 19746 Explicitly write out the functionality condition on derivative for and . (Contributed by Mario Carneiro, 9-Feb-2015.)

Theoremdvf 19747 The derivative is a function. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Mario Carneiro, 9-Feb-2015.)

Theoremdvfcn 19748 The derivative is a function. (Contributed by Mario Carneiro, 9-Feb-2015.)

Theoremdvreslem 19749* Lemma for dvres 19751. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Mario Carneiro, 28-Dec-2016.)
fld       t

Theoremdvres2lem 19750* Lemma for dvres2 19752. (Contributed by Mario Carneiro, 9-Feb-2015.) (Revised by Mario Carneiro, 28-Dec-2016.)
fld       t

Theoremdvres 19751 Restriction of a derivative. Note that our definition of derivative df-dv 19707 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 19752 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 19751, 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 19753 Restriction of a complex differentiable function to the reals. (Contributed by Mario Carneiro, 10-Feb-2015.)

Theoremdvres3a 19754 Restriction of a complex differentiable function to the reals. This version of dvres3 19753 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 19755* Lemma for dvid 19757 and dvconst 19756. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Mario Carneiro, 9-Feb-2015.)

Theoremdvconst 19756 Derivative of a constant function. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Mario Carneiro, 9-Feb-2015.)

Theoremdvid 19757 Derivative of the identity function. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Mario Carneiro, 9-Feb-2015.)

Theoremdvcnp 19758* 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 19759 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 19760 A differentiable function is continuous. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by Mario Carneiro, 7-Sep-2015.)

Theoremdvnfval 19761* Value of the iterated derivative. (Contributed by Mario Carneiro, 11-Feb-2015.)

Theoremdvnff 19762 The iterated derivative is a function. (Contributed by Mario Carneiro, 11-Feb-2015.)

Theoremdvn0 19763 Zero times iterated derivative. (Contributed by Stefan O'Rear, 15-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)

Theoremdvnp1 19764 Successor iterated derivative. (Contributed by Stefan O'Rear, 15-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)

Theoremdvn1 19765 One times iterated derivative. (Contributed by Mario Carneiro, 1-Jan-2017.)

Theoremdvnf 19766 The N-times derivative is a function. (Contributed by Stefan O'Rear, 16-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)

Theoremdvnbss 19767 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 19768 The -th derivative of the -th derivative of is the same as the -th derivative of . (Contributed by Mario Carneiro, 11-Feb-2015.)

Theoremdvn2bss 19769 An N-times differentiable point is an M-times differeentiable point, if . (Contributed by Mario Carneiro, 30-Dec-2016.)

Theoremdvnres 19770 Multiple derivative version of dvres3a 19754. (Contributed by Mario Carneiro, 11-Feb-2015.)

Theoremcpnfval 19771* Condition for n-times continuous differentiability. (Contributed by Stefan O'Rear, 15-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)

Theoremfncpn 19772 The object is a function. (Contributed by Stefan O'Rear, 16-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)

Theoremelcpn 19773 Condition for n-times continuous differentiability. (Contributed by Stefan O'Rear, 15-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)

Theoremcpnord 19774 conditions are ordered by strength. (Contributed by Stefan O'Rear, 16-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)

Theoremcpncn 19775 A function is continuous. (Contributed by Mario Carneiro, 11-Feb-2015.)

Theoremcpnres 19776 The restriction of a function is . (Contributed by Mario Carneiro, 11-Feb-2015.)

Theoremdvaddbr 19777 The sum rule for derivatives at a point. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 28-Dec-2016.)
fld

Theoremdvmulbr 19778 The product rule for derivatives at a point. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 28-Dec-2016.)
fld

Theoremdvadd 19779 The sum rule for derivatives at a point. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)

Theoremdvmul 19780 The product rule for derivatives at a point. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)

Theoremdvaddf 19781 The sum rule for everywhere-differentiable functions. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)

Theoremdvmulf 19782 The product rule for everywhere-differentiable functions. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)

Theoremdvcmul 19783 The product rule when one argument is a constant. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)

Theoremdvcmulf 19784 The product rule when one argument is a constant. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)

Theoremdvcobr 19785 The chain rule for derivatives at a point. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 28-Dec-2016.)
fld

Theoremdvco 19786 The chain rule for derivatives at a point. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)

Theoremdvcof 19787 The chain rule for everywhere-differentiable functions. (Contributed by Mario Carneiro, 10-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)

Theoremdvcjbr 19788 The derivative of the conjugate of a function. (This doesn't follow from dvcobr 19785 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 19789 The derivative of the conjugate of a function. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)

Theoremdvfre 19790 The derivative of a real function is real. (Contributed by Mario Carneiro, 1-Sep-2014.)

Theoremdvnfre 19791 The -th derivative of a real function is real. (Contributed by Mario Carneiro, 1-Jan-2017.)

Theoremdvexp 19792* Derivative of a power function. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)

Theoremdvexp2 19793* Derivative of an exponential, possibly zero power. (Contributed by Stefan O'Rear, 13-Nov-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)

Theoremdvrec 19794* Derivative of the reciprocal function. (Contributed by Mario Carneiro, 25-Feb-2015.) (Revised by Mario Carneiro, 28-Dec-2016.)

Theoremdvmptres3 19795* Function-builder for derivative: restrict a derivative to a subset. (Contributed by Mario Carneiro, 11-Feb-2015.)
fld

Theoremdvmptid 19796* Function-builder for derivative: derivative of the identity. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)

Theoremdvmptc 19797* Function-builder for derivative: derivative of a constant. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)

Theoremdvmptcl 19798* Closure lemma for dvmptcmul 19803 and other related theorems. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)

Theoremdvmptadd 19799* Function-builder for derivative, addition rule. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)

Theoremdvmptmul 19800* Function-builder for derivative, product rule. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)

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