Home Intuitionistic Logic ExplorerTheorem List (p. 115 of 135) < Previous  Next > Browser slow? Try the Unicode version. Mirrors  >  Metamath Home Page  >  ILE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Theorem List for Intuitionistic Logic Explorer - 11401-11500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremiprodap 11401* Series product with an upper integer index set (i.e. an infinite product.) (Contributed by Scott Fenton, 5-Dec-2017.)
#

Theoremzprodap0 11402* Nonzero series product with index set a subset of the upper integers. (Contributed by Scott Fenton, 6-Dec-2017.)
#               DECID

Theoremiprodap0 11403* Nonzero series product with an upper integer index set (i.e. an infinite product.) (Contributed by Scott Fenton, 6-Dec-2017.)
#

4.8.10.4  Finite products

Theoremfprodseq 11404* The value of a product over a nonempty finite set. (Contributed by Scott Fenton, 6-Dec-2017.) (Revised by Jim Kingdon, 15-Jul-2024.)

4.9  Elementary trigonometry

4.9.1  The exponential, sine, and cosine functions

Syntaxce 11405 Extend class notation to include the exponential function.

Syntaxceu 11406 Extend class notation to include Euler's constant = 2.71828....

Syntaxcsin 11407 Extend class notation to include the sine function.

Syntaxccos 11408 Extend class notation to include the cosine function.

Syntaxctan 11409 Extend class notation to include the tangent function.

Syntaxcpi 11410 Extend class notation to include the constant pi, = 3.14159....

Definitiondf-ef 11411* Define the exponential function. Its value at the complex number is and is called the "exponential of "; see efval 11424. (Contributed by NM, 14-Mar-2005.)

Definitiondf-e 11412 Define Euler's constant = 2.71828.... (Contributed by NM, 14-Mar-2005.)

Definitiondf-sin 11413 Define the sine function. (Contributed by NM, 14-Mar-2005.)

Definitiondf-cos 11414 Define the cosine function. (Contributed by NM, 14-Mar-2005.)

Definitiondf-tan 11415 Define the tangent function. We define it this way for cmpt 3998, which requires the form . (Contributed by Mario Carneiro, 14-Mar-2014.)

Definitiondf-pi 11416 Define the constant pi, = 3.14159..., which is the smallest positive number whose sine is zero. Definition of in [Gleason] p. 311. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by AV, 14-Sep-2020.)
inf

Theoremeftcl 11417 Closure of a term in the series expansion of the exponential function. (Contributed by Paul Chapman, 11-Sep-2007.)

Theoremreeftcl 11418 The terms of the series expansion of the exponential function at a real number are real. (Contributed by Paul Chapman, 15-Jan-2008.)

Theoremeftabs 11419 The absolute value of a term in the series expansion of the exponential function. (Contributed by Paul Chapman, 23-Nov-2007.)

Theoremeftvalcn 11420* The value of a term in the series expansion of the exponential function. (Contributed by Paul Chapman, 21-Aug-2007.) (Revised by Jim Kingdon, 8-Dec-2022.)

Theoremefcllemp 11421* Lemma for efcl 11427. The series that defines the exponential function converges. The ratio test cvgratgt0 11354 is used to show convergence. (Contributed by NM, 26-Apr-2005.) (Revised by Jim Kingdon, 8-Dec-2022.)

Theoremefcllem 11422* Lemma for efcl 11427. The series that defines the exponential function converges. (Contributed by NM, 26-Apr-2005.) (Revised by Jim Kingdon, 8-Dec-2022.)

Theoremef0lem 11423* The series defining the exponential function converges in the (trivial) case of a zero argument. (Contributed by Steve Rodriguez, 7-Jun-2006.) (Revised by Mario Carneiro, 28-Apr-2014.)

Theoremefval 11424* Value of the exponential function. (Contributed by NM, 8-Jan-2006.) (Revised by Mario Carneiro, 10-Nov-2013.)

Theoremesum 11425 Value of Euler's constant = 2.71828.... (Contributed by Steve Rodriguez, 5-Mar-2006.)

Theoremeff 11426 Domain and codomain of the exponential function. (Contributed by Paul Chapman, 22-Oct-2007.) (Proof shortened by Mario Carneiro, 28-Apr-2014.)

Theoremefcl 11427 Closure law for the exponential function. (Contributed by NM, 8-Jan-2006.) (Revised by Mario Carneiro, 10-Nov-2013.)

Theoremefval2 11428* Value of the exponential function. (Contributed by Mario Carneiro, 29-Apr-2014.)

Theoremefcvg 11429* The series that defines the exponential function converges to it. (Contributed by NM, 9-Jan-2006.) (Revised by Mario Carneiro, 28-Apr-2014.)

Theoremefcvgfsum 11430* Exponential function convergence in terms of a sequence of partial finite sums. (Contributed by NM, 10-Jan-2006.) (Revised by Mario Carneiro, 28-Apr-2014.)

Theoremreefcl 11431 The exponential function is real if its argument is real. (Contributed by NM, 27-Apr-2005.) (Revised by Mario Carneiro, 28-Apr-2014.)

Theoremreefcld 11432 The exponential function is real if its argument is real. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremere 11433 Euler's constant = 2.71828... is a real number. (Contributed by NM, 19-Mar-2005.) (Revised by Steve Rodriguez, 8-Mar-2006.)

Theoremege2le3 11434 Euler's constant = 2.71828... is bounded by 2 and 3. (Contributed by NM, 20-Mar-2005.) (Proof shortened by Mario Carneiro, 28-Apr-2014.)

Theoremef0 11435 Value of the exponential function at 0. Equation 2 of [Gleason] p. 308. (Contributed by Steve Rodriguez, 27-Jun-2006.) (Revised by Mario Carneiro, 28-Apr-2014.)

Theoremefcj 11436 The exponential of a complex conjugate. Equation 3 of [Gleason] p. 308. (Contributed by NM, 29-Apr-2005.) (Revised by Mario Carneiro, 28-Apr-2014.)

Theoremefadd 11438 Sum of exponents law for exponential function. (Contributed by NM, 10-Jan-2006.) (Proof shortened by Mario Carneiro, 29-Apr-2014.)

Theoremefcan 11439 Cancellation law for exponential function. Equation 27 of [Rudin] p. 164. (Contributed by NM, 13-Jan-2006.)

Theoremefap0 11440 The exponential of a complex number is apart from zero. (Contributed by Jim Kingdon, 12-Dec-2022.)
#

Theoremefne0 11441 The exponential of a complex number is nonzero. Corollary 15-4.3 of [Gleason] p. 309. The same result also holds with not equal replaced by apart, as seen at efap0 11440 (which will be more useful in most contexts). (Contributed by NM, 13-Jan-2006.) (Revised by Mario Carneiro, 29-Apr-2014.)

Theoremefneg 11442 The exponential of the opposite is the inverse of the exponential. (Contributed by Mario Carneiro, 10-May-2014.)

Theoremeff2 11443 The exponential function maps the complex numbers to the nonzero complex numbers. (Contributed by Paul Chapman, 16-Apr-2008.)

Theoremefsub 11444 Difference of exponents law for exponential function. (Contributed by Steve Rodriguez, 25-Nov-2007.)

Theoremefexp 11445 The exponential of an integer power. Corollary 15-4.4 of [Gleason] p. 309, restricted to integers. (Contributed by NM, 13-Jan-2006.) (Revised by Mario Carneiro, 5-Jun-2014.)

Theoremefzval 11446 Value of the exponential function for integers. Special case of efval 11424. Equation 30 of [Rudin] p. 164. (Contributed by Steve Rodriguez, 15-Sep-2006.) (Revised by Mario Carneiro, 5-Jun-2014.)

Theoremefgt0 11447 The exponential of a real number is greater than 0. (Contributed by Paul Chapman, 21-Aug-2007.) (Revised by Mario Carneiro, 30-Apr-2014.)

Theoremrpefcl 11448 The exponential of a real number is a positive real. (Contributed by Mario Carneiro, 10-Nov-2013.)

Theoremrpefcld 11449 The exponential of a real number is a positive real. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremeftlcvg 11450* The tail series of the exponential function are convergent. (Contributed by Mario Carneiro, 29-Apr-2014.)

Theoremeftlcl 11451* Closure of the sum of an infinite tail of the series defining the exponential function. (Contributed by Paul Chapman, 17-Jan-2008.) (Revised by Mario Carneiro, 30-Apr-2014.)

Theoremreeftlcl 11452* Closure of the sum of an infinite tail of the series defining the exponential function. (Contributed by Paul Chapman, 17-Jan-2008.) (Revised by Mario Carneiro, 30-Apr-2014.)

Theoremeftlub 11453* An upper bound on the absolute value of the infinite tail of the series expansion of the exponential function on the closed unit disk. (Contributed by Paul Chapman, 19-Jan-2008.) (Proof shortened by Mario Carneiro, 29-Apr-2014.)

Theoremefsep 11454* Separate out the next term of the power series expansion of the exponential function. The last hypothesis allows the separated terms to be rearranged as desired. (Contributed by Paul Chapman, 23-Nov-2007.) (Revised by Mario Carneiro, 29-Apr-2014.)

Theoremeffsumlt 11455* The partial sums of the series expansion of the exponential function at a positive real number are bounded by the value of the function. (Contributed by Paul Chapman, 21-Aug-2007.) (Revised by Mario Carneiro, 29-Apr-2014.)

Theoremeft0val 11456 The value of the first term of the series expansion of the exponential function is 1. (Contributed by Paul Chapman, 21-Aug-2007.) (Revised by Mario Carneiro, 29-Apr-2014.)

Theoremef4p 11457* Separate out the first four terms of the infinite series expansion of the exponential function. (Contributed by Paul Chapman, 19-Jan-2008.) (Revised by Mario Carneiro, 29-Apr-2014.)

Theoremefgt1p2 11458 The exponential of a positive real number is greater than the sum of the first three terms of the series expansion. (Contributed by Mario Carneiro, 15-Sep-2014.)

Theoremefgt1p 11459 The exponential of a positive real number is greater than 1 plus that number. (Contributed by Mario Carneiro, 14-Mar-2014.) (Revised by Mario Carneiro, 30-Apr-2014.)

Theoremefgt1 11460 The exponential of a positive real number is greater than 1. (Contributed by Paul Chapman, 21-Aug-2007.) (Revised by Mario Carneiro, 30-Apr-2014.)

Theoremefltim 11461 The exponential function on the reals is strictly increasing. (Contributed by Paul Chapman, 21-Aug-2007.) (Revised by Jim Kingdon, 20-Dec-2022.)

Theoremreef11 11462 The exponential function on real numbers is one-to-one. (Contributed by NM, 21-Aug-2008.) (Revised by Jim Kingdon, 20-Dec-2022.)

Theoremreeff1 11463 The exponential function maps real arguments one-to-one to positive reals. (Contributed by Steve Rodriguez, 25-Aug-2007.) (Revised by Mario Carneiro, 10-Nov-2013.)

Theoremeflegeo 11464 The exponential function on the reals between 0 and 1 lies below the comparable geometric series sum. (Contributed by Paul Chapman, 11-Sep-2007.)

Theoremsinval 11465 Value of the sine function. (Contributed by NM, 14-Mar-2005.) (Revised by Mario Carneiro, 10-Nov-2013.)

Theoremcosval 11466 Value of the cosine function. (Contributed by NM, 14-Mar-2005.) (Revised by Mario Carneiro, 10-Nov-2013.)

Theoremsinf 11467 Domain and codomain of the sine function. (Contributed by Paul Chapman, 22-Oct-2007.) (Revised by Mario Carneiro, 30-Apr-2014.)

Theoremcosf 11468 Domain and codomain of the cosine function. (Contributed by Paul Chapman, 22-Oct-2007.) (Revised by Mario Carneiro, 30-Apr-2014.)

Theoremsincl 11469 Closure of the sine function. (Contributed by NM, 28-Apr-2005.) (Revised by Mario Carneiro, 30-Apr-2014.)

Theoremcoscl 11470 Closure of the cosine function with a complex argument. (Contributed by NM, 28-Apr-2005.) (Revised by Mario Carneiro, 30-Apr-2014.)

Theoremtanvalap 11471 Value of the tangent function. (Contributed by Mario Carneiro, 14-Mar-2014.) (Revised by Jim Kingdon, 21-Dec-2022.)
#

Theoremtanclap 11472 The closure of the tangent function with a complex argument. (Contributed by David A. Wheeler, 15-Mar-2014.) (Revised by Jim Kingdon, 21-Dec-2022.)
#

Theoremsincld 11473 Closure of the sine function. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremcoscld 11474 Closure of the cosine function. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremtanclapd 11475 Closure of the tangent function. (Contributed by Mario Carneiro, 29-May-2016.) (Revised by Jim Kingdon, 22-Dec-2022.)
#

Theoremtanval2ap 11476 Express the tangent function directly in terms of . (Contributed by Mario Carneiro, 25-Feb-2015.) (Revised by Jim Kingdon, 22-Dec-2022.)
#

Theoremtanval3ap 11477 Express the tangent function directly in terms of . (Contributed by Mario Carneiro, 25-Feb-2015.) (Revised by Jim Kingdon, 22-Dec-2022.)
#

Theoremresinval 11478 The sine of a real number in terms of the exponential function. (Contributed by NM, 30-Apr-2005.)

Theoremrecosval 11479 The cosine of a real number in terms of the exponential function. (Contributed by NM, 30-Apr-2005.)

Theoremefi4p 11480* Separate out the first four terms of the infinite series expansion of the exponential function. (Contributed by Paul Chapman, 19-Jan-2008.) (Revised by Mario Carneiro, 30-Apr-2014.)

Theoremresin4p 11481* Separate out the first four terms of the infinite series expansion of the sine of a real number. (Contributed by Paul Chapman, 19-Jan-2008.) (Revised by Mario Carneiro, 30-Apr-2014.)

Theoremrecos4p 11482* Separate out the first four terms of the infinite series expansion of the cosine of a real number. (Contributed by Paul Chapman, 19-Jan-2008.) (Revised by Mario Carneiro, 30-Apr-2014.)

Theoremresincl 11483 The sine of a real number is real. (Contributed by NM, 30-Apr-2005.)

Theoremrecoscl 11484 The cosine of a real number is real. (Contributed by NM, 30-Apr-2005.)

Theoremretanclap 11485 The closure of the tangent function with a real argument. (Contributed by David A. Wheeler, 15-Mar-2014.)
#

Theoremresincld 11486 Closure of the sine function. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremrecoscld 11487 Closure of the cosine function. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremretanclapd 11488 Closure of the tangent function. (Contributed by Mario Carneiro, 29-May-2016.)
#

Theoremsinneg 11489 The sine of a negative is the negative of the sine. (Contributed by NM, 30-Apr-2005.)

Theoremcosneg 11490 The cosines of a number and its negative are the same. (Contributed by NM, 30-Apr-2005.)

Theoremtannegap 11491 The tangent of a negative is the negative of the tangent. (Contributed by David A. Wheeler, 23-Mar-2014.)
#

Theoremsin0 11492 Value of the sine function at 0. (Contributed by Steve Rodriguez, 14-Mar-2005.)

Theoremcos0 11493 Value of the cosine function at 0. (Contributed by NM, 30-Apr-2005.)

Theoremtan0 11494 The value of the tangent function at zero is zero. (Contributed by David A. Wheeler, 16-Mar-2014.)

Theoremefival 11495 The exponential function in terms of sine and cosine. (Contributed by NM, 30-Apr-2005.)

Theoremefmival 11496 The exponential function in terms of sine and cosine. (Contributed by NM, 14-Jan-2006.)

Theoremefeul 11497 Eulerian representation of the complex exponential. (Suggested by Jeff Hankins, 3-Jul-2006.) (Contributed by NM, 4-Jul-2006.)

Theoremefieq 11498 The exponentials of two imaginary numbers are equal iff their sine and cosine components are equal. (Contributed by Paul Chapman, 15-Mar-2008.)

Theoremsinadd 11499 Addition formula for sine. Equation 14 of [Gleason] p. 310. (Contributed by Steve Rodriguez, 10-Nov-2006.) (Revised by Mario Carneiro, 30-Apr-2014.)

Theoremcosadd 11500 Addition formula for cosine. Equation 15 of [Gleason] p. 310. (Contributed by NM, 15-Jan-2006.) (Revised by Mario Carneiro, 30-Apr-2014.)

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13470
 Copyright terms: Public domain < Previous  Next >