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Theorem List for Metamath Proof Explorer - 37301-37400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremaddlimc 37301* Sum of two limits. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
lim        lim        lim

Theorem0ellimcdiv 37302* If the numerator converges to 0 and the denominator converges to non zero then the fraction converges to 0. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
lim        lim               lim

Theoremclim2cf 37303* Express the predicate converges to . Similar to clim2 13546, but without the disjoint var constraint . (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Theoremlimclner 37304 For a limit point, both from the left and from the right, of the domain, the limit of the function exits only if the left and the right limits are equal. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
fld                                          lim        lim               lim

Theoremsublimc 37305* Subtraction of two limits. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
lim        lim        lim

Theoremreclimc 37306* Limit of the reciprocal of a function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
lim               lim

Theoremclim0cf 37307* Express the predicate converges to . Similar to clim 13536, but without the disjoint var constraint . (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Theoremlimclr 37308 For a limit point, both from the left and from the right, of the domain, the limit of the function exits only if the left and the right limits are equal. In this case, the three limits coincide. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
fld                                          lim        lim        lim lim

Theoremdivlimc 37309* Limit of the quotient of two funcions. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
lim        lim                      lim

Theoremexpfac 37310* Factorial grows faster than exponential. (Contributed by Glauco Siliprandi, 5-Apr-2020.)

21.30.8  Trigonometry

Theoremcoseq0 37311 A complex number whose cosine is zero. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Theoremsinmulcos 37312 Multiplication formula for sine and cosine. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Theoremcoskpi2 37313 The cosine of an integer multiple of negative is either or negative . (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Theoremcosnegpi 37314 The cosine of negative is negative . (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Theoremsinaover2ne0 37315 If in then is not . (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Theoremcosknegpi 37316 The cosine of an integer multiple of negative is either ore negative . (Contributed by Glauco Siliprandi, 11-Dec-2019.)

21.30.9  Continuous Functions

Theoremmulcncff 37317 The multiplication of two continuous complex functions is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Theoremsubcncf 37318* The addition of two continuous complex functions is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Theoremcncfmptssg 37319* A continuous complex function restricted to a subset is continuous, using "map to" notation. This theorem generalizes cncfmptss 37237 because it allows to establish a subset for the codomain also. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Theoremconstcncfg 37320* A constant function is a continuous function on . (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Theoremidcncfg 37321* The identity function is a continuous function on . (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Theoremaddcncf 37322* The addition of two continuous complex functions is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Theoremcncfshift 37323* A periodic continuous function stays continuous if the domain is shifted a period. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Theoremresincncf 37324 restricted to reals is continuous from reals to reals. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Theoremaddccncf2 37325* Adding a constant is a continuous function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Theorem0cnf 37326 The empty set is a continuous function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Theoremfsumcncf 37327* The finite sum of continuous complex function is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Theoremcncfperiod 37328* A periodic continuous function stays continuous if the domain is shifted a period. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Theoremsubcncff 37329 The subtraction of two continuous complex functions is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Theoremnegcncfg 37330* The opposite of a continuous function is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Theoremcnfdmsn 37331* A function with a singleton domain is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Theoremcncfcompt 37332* Composition of continuous functions. A generalization of cncfmpt1f 21841 to arbitrary domains. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Theoremdivcncf 37333* The quotient of two continuous complex functions is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Theoremaddcncff 37334 The addition of two continuous complex functions is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Theoremioccncflimc 37335 Limit at the upper bound, of a continuous function defined on a left open right closed interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
lim

Theoremcncfuni 37336* A function is continuous if it's domain is the union of sets over which the function is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
fldt

Theoremicccncfext 37337* A continuous function on a closed interval can be extended to a continuous function on the whole real line. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
t        t

Theoremcncficcgt0 37338* A the absolute value of a continuous function on a closed interval, that is never 0, has a strictly positive lower bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Theoremicocncflimc 37339 Limit at the lower bound, of a continuous function defined on a left closed right open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
lim

Theoremcncfdmsn 37340* A complex function with a singleton domain is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Theoremdivcncff 37341 The quotient of two continuous complex functions is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Theoremcncfshiftioo 37342* A periodic continuous function stays continuous if the domain is an open interval that is shifted a period. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Theoremcncfiooicclem1 37343* A continuous function on an open interval can be extended to a continuous function on the corresponding close interval, if it has a finite right limit in and a finite left limit in . can be complex valued. This lemma assumes , the invoking theorem drops this assumption. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
lim        lim

Theoremcncfiooicc 37344* A continuous function on an open interval can be extended to a continuous function on the corresponding close interval, if it has a finite right limit in and a finite left limit in . can be complex valued. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
lim        lim

Theoremcncfiooiccre 37345* A continuous function on an open interval can be extended to a continuous function on the corresponding close interval, if it has a finite right limit in and a finite left limit in . is assumed to be real valued. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
lim        lim

Theoremcncfioobdlem 37346* actually extends . (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Theoremcncfioobd 37347* A continuous function on an open interval with a finite right limit in and a finite left limit in is bounded. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
lim        lim

Theoremjumpncnp 37348 Jump discontinuity or discontinuity of the first kind: if the left and the right limit don't match, the function is discontinuous at the point. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
fld                                                 lim        lim               fld

Theoremcncfcompt2 37349* Composition of continuous functions. (Contributed by Glauco Siliprandi, 5-Apr-2020.)

Theoremcxpcncf2 37350* The complex power function is continuous with respect to its second argument. (Contributed by Glauco Siliprandi, 5-Apr-2020.)

Theoremfprodcncf 37351* The finite product of continuous complex functions is continuous. (Contributed by Glauco Siliprandi, 5-Apr-2020.)

21.30.10  Derivatives

Theoremdvsinexp 37352* The derivative of sin^N . (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Theoremdvcosre 37353 The real derivative of the cosine (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Theoremdvrecg 37354* Derivative of the reciprocal of a function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Theoremdvsinax 37355* Derivative exercise: the derivative with respect to y of sin(Ay), given a constant . (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Theoremdvsubf 37356 The subtraction rule for everywhere-differentiable functions. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Theoremdvmptconst 37357* Function-builder for derivative: derivative of a constant. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
fldt

Theoremdvcnre 37358 From compex differentiation to real differentiation. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Theoremdvmptidg 37359* Function-builder for derivative: derivative of the identity. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
fldt

Theoremdvresntr 37360 Function-builder for derivative: expand the function from an open set to its closure. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
t        fld

Theoremdvmptdiv 37361* Function-builder for derivative, quotient rule. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Theoremfperdvper 37362* The derivative of a periodic function is periodic (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Theoremdvmptresicc 37363* Derivative of a function restricted to a closed interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Theoremdvasinbx 37364* Derivative exercise: the derivative with respect to y of A x sin(By), given two constants and . (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Theoremdvresioo 37365 Restriction of a derivative to an open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Theoremdvdivf 37366 The quotient rule for everywhere-differentiable functions. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Theoremdvdivbd 37367* A sufficient condition for the derivative to be bounded, for the quotient of two functions. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Theoremdvsubcncf 37368 A sufficient condition for the derivative of a product to be continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Theoremdvmulcncf 37369 A sufficient condition for the derivative of a product to be continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Theoremdvcosax 37370* Derivative exercise: the derivative with respect to x of cos(Ax), given a constant . (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Theoremdvdivcncf 37371 A sufficient condition for the derivative of a quotient to be continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Theoremdvbdfbdioolem1 37372* Given a function with bounded derivative, on an open interval, here is an absolute bound to the difference of the image of two points in the interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Theoremdvbdfbdioolem2 37373* A function on an open interval, with bounded derivative, is bounded. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Theoremdvbdfbdioo 37374* A function on an open interval, with bounded derivative, is bounded. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Theoremioodvbdlimc1lem1 37375* If has bounded derivative on then a sequence of points in its image converges to its . (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Theoremioodvbdlimc1lem2 37376* Limit at the lower bound of an open interval, for a function with bounded derivative. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
lim

Theoremioodvbdlimc1 37377* A real function with bounded derivative, has a limit at the upper bound of an open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
lim

Theoremioodvbdlimc2lem 37378* Limit at the upper bound of an open interval, for a function with bounded derivative. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
lim

Theoremioodvbdlimc2 37379* A real function with bounded derivative, has a limit at the upper bound of an open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
lim

Theoremdvdmsscn 37380 is a subset of . This statement is very often used when computing derivatives. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
fldt

Theoremdvmptmulf 37381* Function-builder for derivative, product rule. A version of dvmptmul 22792 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.)

Theoremdvnmptdivc 37382* Function-builder for iterated derivative, division rule for constant divisor. (Contributed by Glauco Siliprandi, 5-Apr-2020.)

Theoremdvdsn1add 37383 If divides but does not divide , then does not divide . (Contributed by Glauco Siliprandi, 5-Apr-2020.)

Theoremdvxpaek 37384* Derivative of the polynomial . (Contributed by Glauco Siliprandi, 5-Apr-2020.)
fldt

Theoremdvnmptconst 37385* The -th derivative of a constant function. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
fldt

Theoremdvnxpaek 37386* The -th derivative of the polynomial (x+A)^K (Contributed by Glauco Siliprandi, 5-Apr-2020.)
fldt

Theoremdvnmul 37387* Function-builder for the -th derivative, product rule. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
fldt

Theoremdvmptfprodlem 37388* Induction step for dvmptfprod 37389. (Contributed by Glauco Siliprandi, 5-Apr-2020.)

Theoremdvmptfprod 37389* Function-builder for derivative, finite product rule. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
t        fld

Theoremdvnprodlem1 37390* is bijective. (Contributed by Glauco Siliprandi, 5-Apr-2020.)

Theoremdvnprodlem2 37391* Induction step for dvnprodlem2 37391. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
fldt

Theoremdvnprodlem3 37392* The multinomial formula for the -th derivative of a finite product. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
fldt

Theoremdvnprod 37393* The multinomial formula for the -th derivative of a finite product. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
fldt

21.30.11  Integrals

Theoremvolioo 37394 The measure of an open interval. (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Theoremitgsin0pilem1 37395* Calculation of the integral for sine on the (0,π) interval (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Theoremibliccsinexp 37396* sin^n on a closed interval is integrable. (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Theoremitgsin0pi 37397 Calculation of the integral for sine on the (0,π) interval (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Theoremiblioosinexp 37398* sin^n on an open integral is integrable (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Theoremitgsinexplem1 37399* Integration by parts is applied to integrate sin^(N+1). (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Theoremitgsinexp 37400* A recursive formula for the integral of sin^N on the interval (0,π) . (Contributed by Glauco Siliprandi, 29-Jun-2017.)

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