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

Theoremsincosq4sgn 19701 The signs of the sine and cosine functions in the fourth quadrant. (Contributed by Paul Chapman, 24-Jan-2008.)

Theoremcoseq00topi 19702 Location of the zeroes of cosine in . (Contributed by David Moews, 28-Feb-2017.)

Theoremcoseq0negpitopi 19703 Location of the zeroes of cosine in . (Contributed by David Moews, 28-Feb-2017.)

Theoremtanrpcl 19704 Positive real closure of the tangent function. (Contributed by Mario Carneiro, 29-Jul-2014.)

Theoremtangtx 19705 The tangent function is greater than its argument on positive reals in its principal domain. (Contributed by Mario Carneiro, 29-Jul-2014.)

Theoremtanabsge 19706 The tangent function is greater than or equal to its argument in absolute value. (Contributed by Mario Carneiro, 25-Feb-2015.)

Theoremsinq12gt0 19707 The sine of a number strictly between and is positive. (Contributed by Paul Chapman, 15-Mar-2008.)

Theoremsinq12ge0 19708 The sine of a number between and is nonnegative. (Contributed by Mario Carneiro, 13-May-2014.)

Theoremsinq34lt0t 19709 The sine of a number strictly between and is negative. (Contributed by NM, 17-Aug-2008.)

Theoremcosq14gt0 19710 The cosine of a number strictly between and is positive. (Contributed by Mario Carneiro, 25-Feb-2015.)

Theoremcosq14ge0 19711 The cosine of a number between and is nonnegative. (Contributed by Mario Carneiro, 13-May-2014.)

Theoremsincosq1eq 19712 Complementarity of the sine and cosine functions in the first quadrant. (Contributed by Paul Chapman, 25-Jan-2008.)

Theoremsincos4thpi 19713 The sine and cosine of . (Contributed by Paul Chapman, 25-Jan-2008.)

Theoremtan4thpi 19714 The tangent of . (Contributed by Mario Carneiro, 5-Apr-2015.)

Theoremsincos6thpi 19715 The sine and cosine of . (Contributed by Paul Chapman, 25-Jan-2008.)

Theoremsincos3rdpi 19716 The sine and cosine of . (Contributed by Mario Carneiro, 21-May-2016.)

Theorempige3 19717 is greater or equal to 3. This proof is based on the geometric observation that a hexagon of unit side length has perimeter 6, which is less than the unit-radius circumcircle, of perimeter . We translate this to algebra by looking at the function as goes from to ; it moves at unit speed and travels distance , hence . (Contributed by Mario Carneiro, 21-May-2016.)

Theoremabssinper 19718 The absolute value of sine has period . (Contributed by NM, 17-Aug-2008.)

Theoremsinkpi 19719 The sine of an integer multiple of is 0. (Contributed by NM, 11-Aug-2008.)

Theoremcoskpi 19720 The absolute value of the cosine of an integer multiple of is 1. (Contributed by NM, 19-Aug-2008.)

Theoremsineq0 19721 A complex number whose sine is zero is an integer multiple of . (Contributed by NM, 17-Aug-2008.) (Revised by Mario Carneiro, 10-May-2014.)

Theoremcoseq1 19722 A complex number whose cosine is one is an integer multiple of . (Contributed by Mario Carneiro, 12-May-2014.)

Theoremefeq1 19723 A complex number whose exponential is one is an integer multiple of . (Contributed by NM, 17-Aug-2008.) (Revised by Mario Carneiro, 10-May-2014.)

Theoremcosne0 19724 The cosine function has no zeroes within the vertical strip of the complex plane between real part and . (Contributed by Mario Carneiro, 2-Apr-2015.)

Theoremcosordlem 19725 Lemma for cosord 19726. (Contributed by Mario Carneiro, 10-May-2014.)

Theoremcosord 19726 Cosine is decreasing over the closed interval from to . (Contributed by Paul Chapman, 16-Mar-2008.) (Proof shortened by Mario Carneiro, 10-May-2014.)

Theoremcos11 19727 Cosine is one-to-one over the closed interval from to . (Contributed by Paul Chapman, 16-Mar-2008.) (Proof shortened by Mario Carneiro, 10-May-2014.)

Theoremsinord 19728 Sine is increasing over the closed interval from to . (Contributed by Mario Carneiro, 29-Jul-2014.)

Theoremrecosf1o 19729 The cosine function is a bijection when restricted to its principal domain. (Contributed by Mario Carneiro, 12-May-2014.)

Theoremresinf1o 19730 The sine function is a bijection when restricted to its principal domain. (Contributed by Mario Carneiro, 12-May-2014.)

Theoremtanord1 19731 The tangent function is strictly increasing on the nonnegative part of its principal domain. (Lemma for tanord 19732.) (Contributed by Mario Carneiro, 29-Jul-2014.)

Theoremtanord 19732 The tangent function is strictly increasing on its principal domain. (Contributed by Mario Carneiro, 4-Apr-2015.)

Theoremtanregt0 19733 The positivity of extends to complex numbers with the same real part. (Contributed by Mario Carneiro, 5-Apr-2015.)

Theoremnegpitopissre 19734 is a subset of the reals. (Contributed by David Moews, 28-Feb-2017.)

13.3.3  Mapping of the exponential function

Theoremefgh 19735* The exponential function of a scaled complex number is a group homomorphism from the group of complex numbers under addition to the set of complex numbers under multiplication. (Contributed by Paul Chapman, 25-Apr-2008.) (Revised by Mario Carneiro, 11-May-2014.)

Theoremefif1olem1 19736* Lemma for efif1o 19740. (Contributed by Mario Carneiro, 13-May-2014.)

Theoremefif1olem2 19737* Lemma for efif1o 19740. (Contributed by Mario Carneiro, 13-May-2014.)

Theoremefif1olem3 19738* Lemma for efif1o 19740. (Contributed by Mario Carneiro, 8-May-2015.)

Theoremefif1olem4 19739* The exponential function of an imaginary number maps any interval of length one-to-one onto the unit circle. (Contributed by Paul Chapman, 16-Mar-2008.) (Proof shortened by Mario Carneiro, 13-May-2014.)

Theoremefif1o 19740* The exponential function of an imaginary number maps any open-below, closed-above interval of length one-to-one onto the unit circle. (Contributed by Paul Chapman, 16-Mar-2008.) (Revised by Mario Carneiro, 13-May-2014.)

Theoremefifo 19741* The exponential function of an imaginary number maps the reals onto the unit circle. (Contributed by Mario Carneiro, 13-May-2014.)

Theoremeff1olem 19742* The exponential function maps the set , of complex numbers with imaginary part in a real interval of length , one-to-one onto the nonzero complex numbers. (Contributed by Paul Chapman, 16-Apr-2008.) (Proof shortened by Mario Carneiro, 13-May-2014.)

Theoremeff1o 19743 The exponential function maps the set , of complex numbers with imaginary part in the closed-above, open-below interval from to one-to-one onto the nonzero complex numbers. (Contributed by Paul Chapman, 16-Apr-2008.) (Revised by Mario Carneiro, 13-May-2014.)

13.3.4  The natural logarithm on complex numbers

Syntaxclog 19744 Extend class notation with the natural logarithm function on complex numbers.

Syntaxccxp 19745 Extend class notation with the complex power function.

Definitiondf-log 19746 Define the natural logarithm function on complex numbers. See http://en.wikipedia.org/wiki/Natural_logarithm ("The natural logarithm function can also be defined as the inverse function of the exponential function"). (Contributed by Paul Chapman, 21-Apr-2008.)

Definitiondf-cxp 19747* Define the power function on complex numbers. Note that the value of this function when and should properly be undefined, but defining it by convention this way simplifies the domain. (Contributed by Mario Carneiro, 2-Aug-2014.)

Theoremlogrn 19748 The range of the natural logarithm function, also the principal domain of the exponential function. This allows us to write the longer class expression as simply . (Contributed by Paul Chapman, 21-Apr-2008.) (Revised by Mario Carneiro, 13-May-2014.)

Theoremellogrn 19749 Write out the property explicitly. (Contributed by Mario Carneiro, 1-Apr-2015.)

Theoremdflog2 19750 The natural logarithm function in terms of the exponential function restricted to its principal domain. (Contributed by Paul Chapman, 21-Apr-2008.)

Theoremrelogrn 19751 The range of the natural logarithm function includes the real numbers. (Contributed by Paul Chapman, 21-Apr-2008.) (Proof shortened by Mario Carneiro, 1-Apr-2015.)

Theoremlogrncn 19752 The range of the natural logarithm function is a subset of the complex numbers. (Contributed by Mario Carneiro, 13-May-2014.)

Theoremeff1o2 19753 The exponential function restricted to its principal domain maps one-to-one onto the nonzero complex numbers. (Contributed by Paul Chapman, 21-Apr-2008.) (Revised by Mario Carneiro, 13-May-2014.)

Theoremlogf1o 19754 The natural logarithm function maps the nonzero complex numbers one-to-one onto its range. (Contributed by Paul Chapman, 21-Apr-2008.)

Theoremdfrelog 19755 The natural logarithm function on the positive reals in terms of the real exponential function. (Contributed by Paul Chapman, 21-Apr-2008.)

Theoremrelogf1o 19756 The natural logarithm function maps the positive reals one-to-one onto the real numbers. (Contributed by Paul Chapman, 21-Apr-2008.)

Theoremlogrncl 19757 Closure of the natural logarithm function. (Contributed by Paul Chapman, 21-Apr-2008.)

Theoremlogcl 19758 Closure of the natural logarithm function. (Contributed by NM, 21-Apr-2008.) (Revised by Mario Carneiro, 23-Sep-2014.)

Theoremlogimcl 19759 Closure of the imaginary part of the logarithm function. (Contributed by Mario Carneiro, 23-Sep-2014.) (Revised by Mario Carneiro, 1-Apr-2015.)

Theoremlogcld 19760 The logarithm of a nonzero complex number is a complex number. Deduction form of logcl 19758. (Contributed by David Moews, 28-Feb-2017.)

Theoremlogimcld 19761 The imaginary part of the logarithm is in . Deduction form of logimcl 19759. Compare logimclad 19762. (Contributed by David Moews, 28-Feb-2017.)

Theoremlogimclad 19762 The imaginary part of the logarithm is in . Alternate form of logimcld 19761. (Contributed by David Moews, 28-Feb-2017.)

Theoremlogrnaddcl 19763 The range of the natural logarithm is closed under addition with reals. (Contributed by Mario Carneiro, 3-Apr-2015.)

Theoremrelogcl 19764 Closure of the natural logarithm function on positive reals. (Contributed by Steve Rodriguez, 25-Nov-2007.)

Theoremeflog 19765 Relationship between the natural logarithm function and the exponential function. (Contributed by Paul Chapman, 21-Apr-2008.)

Theoremreeflog 19766 Relationship between the natural logarithm function and the exponential function. (Contributed by Steve Rodriguez, 25-Nov-2007.)

Theoremlogef 19767 Relationship between the natural logarithm function and the exponential function. (Contributed by Paul Chapman, 21-Apr-2008.)

Theoremrelogef 19768 Relationship between the natural logarithm function and the exponential function. (Contributed by Steve Rodriguez, 25-Nov-2007.)

Theoremlogeftb 19769 Relationship between the natural logarithm function and the exponential function. (Contributed by Paul Chapman, 21-Apr-2008.)

Theoremrelogeftb 19770 Relationship between the natural logarithm function and the exponential function. (Contributed by Steve Rodriguez, 25-Nov-2007.)

Theoremlog1 19771 The natural logarithm of . One case of Property 1a of [Cohen] p. 301. (Contributed by Steve Rodriguez, 25-Nov-2007.)

Theoremloge 19772 The natural logarithm of . One case of Property 1b of [Cohen] p. 301. (Contributed by Steve Rodriguez, 25-Nov-2007.)

Theoremlogneg 19773 The natural logarithm of a negative real number. (Contributed by Mario Carneiro, 13-May-2014.) (Revised by Mario Carneiro, 3-Apr-2015.)

Theoremlogm1 19774 The natural logarithm of negative . (Contributed by Paul Chapman, 21-Apr-2008.) (Revised by Mario Carneiro, 13-May-2014.)

Theoremlognegb 19775 If a number has imaginary part equal to , then it is on the negative real axis and vice-versa. (Contributed by Mario Carneiro, 23-Sep-2014.)

Theoremrelogoprlem 19776 Lemma for relogmul 19777 and relogdiv 19778. Remark of [Cohen] p. 301 ("The proof of Property 3 is quite similar to the proof given for Property 2"). (Contributed by Steve Rodriguez, 25-Nov-2007.)

Theoremrelogmul 19777 The natural logarithm of the product of two positive real numbers is the sum of natural logarithms. Property 2 of [Cohen] p. 301, restricted to natural logarithms. (Contributed by Steve Rodriguez, 25-Nov-2007.)

Theoremrelogdiv 19778 The natural logarithm of the quotient of two positive real numbers is the difference of natural logarithms. Exercise 72(a) and Property 3 of [Cohen] p. 301, restricted to natural logarithms. (Contributed by Steve Rodriguez, 25-Nov-2007.)

Theoremexplog 19779 Exponentiation of a nonzero complex number to an integer power. (Contributed by Paul Chapman, 21-Apr-2008.)

Theoremreexplog 19780 Exponentiation of a positive real number to an integer power. (Contributed by Steve Rodriguez, 25-Nov-2007.)

Theoremrelogexp 19781 The natural logarithm of positive raised to an integer power. Property 4 of [Cohen] p. 301-302, restricted to natural logarithms and integer powers . (Contributed by Steve Rodriguez, 25-Nov-2007.)

Theoremrelog 19782 Real part of a logarithm. (Contributed by Mario Carneiro, 15-Sep-2014.)

Theoremrelogiso 19783 The natural logarithm function on positive reals determines an isomorphism from the positive reals onto the reals. (Contributed by Steve Rodriguez, 25-Nov-2007.)

Theoremreloggim 19784 The natural logarithm is a group isomorphism from the group of positive reals under multiplication to the group of reals under addition. (Contributed by Mario Carneiro, 21-Jun-2015.)
flds        mulGrpflds        GrpIso

Theoremlogltb 19785 The natural logarithm function on positive reals is strictly monotonic. (Contributed by Steve Rodriguez, 25-Nov-2007.)

Theoremlogfac 19786* The logarithm of a factorial can be expressed as a finite sum of logs. (Contributed by Mario Carneiro, 17-Apr-2015.)

Theoremeflogeq 19787* Solve an equation involving an exponential. (Contributed by Mario Carneiro, 23-Apr-2015.)

Theoremlogne0 19788 Logarithm of a non-1 number is not zero and thus suitable as a divisor. (Contributed by Stefan O'Rear, 19-Sep-2014.)

Theoremlogleb 19789 Natural logarithm preserves . (Contributed by Stefan O'Rear, 19-Sep-2014.)

Theoremrplogcl 19790 Closure of the logarithm function in the positive reals. (Contributed by Mario Carneiro, 21-Sep-2014.)

Theoremlogge0 19791 The logarithm of a number greater than 1 is nonnegative. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremlogcj 19792 The natural logarithm distributes under conjugation away from the branch cut. (Contributed by Mario Carneiro, 25-Feb-2015.)

Theoremefiarg 19793 The exponential of the "arg" function . (Contributed by Mario Carneiro, 25-Feb-2015.)

Theoremcosargd 19794 The cosine of the argument is the quotient of the real part and the absolute value. Compare to efiarg 19793. (Contributed by David Moews, 28-Feb-2017.)

Theoremcosarg0d 19795 The cosine of the argument is zero precisely on the imaginary axis. (Contributed by David Moews, 28-Feb-2017.)

Theoremargregt0 19796 Closure of the argument of a complex number with positive real part. (Contributed by Mario Carneiro, 25-Feb-2015.)

Theoremargrege0 19797 Closure of the argument of a complex number with nonnegative real part. (Contributed by Mario Carneiro, 2-Apr-2015.)

Theoremargimgt0 19798 Closure of the argument of a complex number with positive imaginary part. (Contributed by Mario Carneiro, 25-Feb-2015.)

Theoremargimlt0 19799 Closure of the argument of a complex number with negative imaginary part. (Contributed by Mario Carneiro, 25-Feb-2015.)

Theoremlogimul 19800 Multiplying a number by increases the logarithm of the number by . (Contributed by Mario Carneiro, 4-Apr-2015.)

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