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**Theorem List for Intuitionistic Logic Explorer -** 12101-12200 *Has distinct variable group(s)
Type	Label	Description
Statement

Theorem	resinval 12101	The sine of a real number in terms of the exponential function. (Contributed by NM, 30-Apr-2005.)


Theorem	recosval 12102	The cosine of a real number in terms of the exponential function. (Contributed by NM, 30-Apr-2005.)


Theorem	efi4p 12103*	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.)


Theorem	resin4p 12104*	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.)


Theorem	recos4p 12105*	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.)


Theorem	resincl 12106	The sine of a real number is real. (Contributed by NM, 30-Apr-2005.)


Theorem	recoscl 12107	The cosine of a real number is real. (Contributed by NM, 30-Apr-2005.)


Theorem	retanclap 12108	The closure of the tangent function with a real argument. (Contributed by David A. Wheeler, 15-Mar-2014.)
$/\$ #

Theorem	resincld 12109	Closure of the sine function. (Contributed by Mario Carneiro, 29-May-2016.)


Theorem	recoscld 12110	Closure of the cosine function. (Contributed by Mario Carneiro, 29-May-2016.)


Theorem	retanclapd 12111	Closure of the tangent function. (Contributed by Mario Carneiro, 29-May-2016.)
#

Theorem	sinneg 12112	The sine of a negative is the negative of the sine. (Contributed by NM, 30-Apr-2005.)


Theorem	cosneg 12113	The cosines of a number and its negative are the same. (Contributed by NM, 30-Apr-2005.)


Theorem	tannegap 12114	The tangent of a negative is the negative of the tangent. (Contributed by David A. Wheeler, 23-Mar-2014.)
$/\$ #

Theorem	sin0 12115	Value of the sine function at 0. (Contributed by Steve Rodriguez, 14-Mar-2005.)


Theorem	cos0 12116	Value of the cosine function at 0. (Contributed by NM, 30-Apr-2005.)


Theorem	tan0 12117	The value of the tangent function at zero is zero. (Contributed by David A. Wheeler, 16-Mar-2014.)


Theorem	efival 12118	The exponential function in terms of sine and cosine. (Contributed by NM, 30-Apr-2005.)


Theorem	efmival 12119	The exponential function in terms of sine and cosine. (Contributed by NM, 14-Jan-2006.)


Theorem	efeul 12120	Eulerian representation of the complex exponential. (Suggested by Jeff Hankins, 3-Jul-2006.) (Contributed by NM, 4-Jul-2006.)


Theorem	efieq 12121	The exponentials of two imaginary numbers are equal iff their sine and cosine components are equal. (Contributed by Paul Chapman, 15-Mar-2008.)
$/\$ $/\$

Theorem	sinadd 12122	Addition formula for sine. Equation 14 of [Gleason] p. 310. (Contributed by Steve Rodriguez, 10-Nov-2006.) (Revised by Mario Carneiro, 30-Apr-2014.)
$/\$

Theorem	cosadd 12123	Addition formula for cosine. Equation 15 of [Gleason] p. 310. (Contributed by NM, 15-Jan-2006.) (Revised by Mario Carneiro, 30-Apr-2014.)
$/\$

Theorem	tanaddaplem 12124	A useful intermediate step in tanaddap 12125 when showing that the addition of tangents is well-defined. (Contributed by Mario Carneiro, 4-Apr-2015.) (Revised by Jim Kingdon, 25-Dec-2022.)
$/\$ $/\$ # $/\$ # # #

Theorem	tanaddap 12125	Addition formula for tangent. (Contributed by Mario Carneiro, 4-Apr-2015.)
$/\$ $/\$ # $/\$ # $/\$ #

Theorem	sinsub 12126	Sine of difference. (Contributed by Paul Chapman, 12-Oct-2007.)
$/\$

Theorem	cossub 12127	Cosine of difference. (Contributed by Paul Chapman, 12-Oct-2007.)
$/\$

Theorem	addsin 12128	Sum of sines. (Contributed by Paul Chapman, 12-Oct-2007.)
$/\$

Theorem	subsin 12129	Difference of sines. (Contributed by Paul Chapman, 12-Oct-2007.)
$/\$

Theorem	sinmul 12130	Product of sines can be rewritten as half the difference of certain cosines. This follows from cosadd 12123 and cossub 12127. (Contributed by David A. Wheeler, 26-May-2015.)
$/\$

Theorem	cosmul 12131	Product of cosines can be rewritten as half the sum of certain cosines. This follows from cosadd 12123 and cossub 12127. (Contributed by David A. Wheeler, 26-May-2015.)
$/\$

Theorem	addcos 12132	Sum of cosines. (Contributed by Paul Chapman, 12-Oct-2007.)
$/\$

Theorem	subcos 12133	Difference of cosines. (Contributed by Paul Chapman, 12-Oct-2007.) (Revised by Mario Carneiro, 10-May-2014.)
$/\$

Theorem	sincossq 12134	Sine squared plus cosine squared is 1. Equation 17 of [Gleason] p. 311. Note that this holds for non-real arguments, even though individually each term is unbounded. (Contributed by NM, 15-Jan-2006.)


Theorem	sin2t 12135	Double-angle formula for sine. (Contributed by Paul Chapman, 17-Jan-2008.)


Theorem	cos2t 12136	Double-angle formula for cosine. (Contributed by Paul Chapman, 24-Jan-2008.)


Theorem	cos2tsin 12137	Double-angle formula for cosine in terms of sine. (Contributed by NM, 12-Sep-2008.)


Theorem	sinbnd 12138	The sine of a real number lies between -1 and 1. Equation 18 of [Gleason] p. 311. (Contributed by NM, 16-Jan-2006.)
$/\$

Theorem	cosbnd 12139	The cosine of a real number lies between -1 and 1. Equation 18 of [Gleason] p. 311. (Contributed by NM, 16-Jan-2006.)
$/\$

Theorem	sinbnd2 12140	The sine of a real number is in the closed interval from -1 to 1. (Contributed by Mario Carneiro, 12-May-2014.)


Theorem	cosbnd2 12141	The cosine of a real number is in the closed interval from -1 to 1. (Contributed by Mario Carneiro, 12-May-2014.)


Theorem	ef01bndlem 12142*	Lemma for sin01bnd 12143 and cos01bnd 12144. (Contributed by Paul Chapman, 19-Jan-2008.)


Theorem	sin01bnd 12143	Bounds on the sine of a positive real number less than or equal to 1. (Contributed by Paul Chapman, 19-Jan-2008.) (Revised by Mario Carneiro, 30-Apr-2014.)
$/\$

Theorem	cos01bnd 12144	Bounds on the cosine of a positive real number less than or equal to 1. (Contributed by Paul Chapman, 19-Jan-2008.) (Revised by Mario Carneiro, 30-Apr-2014.)
$/\$

Theorem	cos1bnd 12145	Bounds on the cosine of 1. (Contributed by Paul Chapman, 19-Jan-2008.)
$/\$

Theorem	cos2bnd 12146	Bounds on the cosine of 2. (Contributed by Paul Chapman, 19-Jan-2008.)
$/\$

Theorem	sinltxirr 12147*	The sine of a positive irrational number is less than its argument. Here irrational means apart from any rational number. (Contributed by Mario Carneiro, 29-Jul-2014.)
$/\$ #

Theorem	sin01gt0 12148	The sine of a positive real number less than or equal to 1 is positive. (Contributed by Paul Chapman, 19-Jan-2008.) (Revised by Wolf Lammen, 25-Sep-2020.)


Theorem	cos01gt0 12149	The cosine of a positive real number less than or equal to 1 is positive. (Contributed by Paul Chapman, 19-Jan-2008.)


Theorem	sin02gt0 12150	The sine of a positive real number less than or equal to 2 is positive. (Contributed by Paul Chapman, 19-Jan-2008.)


Theorem	sincos1sgn 12151	The signs of the sine and cosine of 1. (Contributed by Paul Chapman, 19-Jan-2008.)
$/\$

Theorem	sincos2sgn 12152	The signs of the sine and cosine of 2. (Contributed by Paul Chapman, 19-Jan-2008.)
$/\$

Theorem	sin4lt0 12153	The sine of 4 is negative. (Contributed by Paul Chapman, 19-Jan-2008.)


Theorem	cos12dec 12154	Cosine is decreasing from one to two. (Contributed by Mario Carneiro and Jim Kingdon, 6-Mar-2024.)
$/\$ $/\$

Theorem	absefi 12155	The absolute value of the exponential of an imaginary number is one. Equation 48 of [Rudin] p. 167. (Contributed by Jason Orendorff, 9-Feb-2007.)


Theorem	absef 12156	The absolute value of the exponential is the exponential of the real part. (Contributed by Paul Chapman, 13-Sep-2007.)


Theorem	absefib 12157	A complex number is real iff the exponential of its product with has absolute value one. (Contributed by NM, 21-Aug-2008.)


Theorem	efieq1re 12158	A number whose imaginary exponential is one is real. (Contributed by NM, 21-Aug-2008.)
$/\$

Theorem	demoivre 12159	De Moivre's Formula. Proof by induction given at http://en.wikipedia.org/wiki/De_Moivre's_formula, but restricted to nonnegative integer powers. See also demoivreALT 12160 for an alternate longer proof not using the exponential function. (Contributed by NM, 24-Jul-2007.)
$/\$

Theorem	demoivreALT 12160	Alternate proof of demoivre 12159. It is longer but does not use the exponential function. This is Metamath 100 proof #17. (Contributed by Steve Rodriguez, 10-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
$/\$

4.10.1.1 The circle constant (tau = 2 pi)

Syntax	ctau 12161	Extend class notation to include the constant tau, = 6.28318....


Definition	df-tau 12162	Define the circle constant tau, = 6.28318..., which is the smallest positive real number whose cosine is one. Various notations have been used or proposed for this number including , a three-legged variant of , or . Note the difference between this constant and the formula variable . Following our convention, the constant is displayed in upright font while the variable is in italic font; furthermore, the colors are different. (Contributed by Jim Kingdon, 9-Apr-2018.) (Revised by AV, 1-Oct-2020.)
inf

4.10.2 _e is irrational

Theorem	eirraplem 12163*	Lemma for eirrap 12164. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Jim Kingdon, 5-Jan-2022.)
#

Theorem	eirrap 12164	is irrational. That is, for any rational number, is apart from it. In the absence of excluded middle, we can distinguish between this and saying that is not rational, which is eirr 12165. (Contributed by Jim Kingdon, 6-Jan-2023.)
#

Theorem	eirr 12165	is not rational. In the absence of excluded middle, we can distinguish between this and saying that is irrational in the sense of being apart from any rational number, which is eirrap 12164. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Jim Kingdon, 6-Jan-2023.)


Theorem	egt2lt3 12166	Euler's constant = 2.71828... is bounded by 2 and 3. (Contributed by NM, 28-Nov-2008.) (Revised by Jim Kingdon, 7-Jan-2023.)
$/\$

Theorem	epos 12167	Euler's constant is greater than 0. (Contributed by Jeff Hankins, 22-Nov-2008.)


Theorem	epr 12168	Euler's constant is a positive real. (Contributed by Jeff Hankins, 22-Nov-2008.)


Theorem	ene0 12169	is not 0. (Contributed by David A. Wheeler, 17-Oct-2017.)


Theorem	eap0 12170	is apart from 0. (Contributed by Jim Kingdon, 7-Jan-2023.)
#

Theorem	ene1 12171	is not 1. (Contributed by David A. Wheeler, 17-Oct-2017.)


Theorem	eap1 12172	is apart from 1. (Contributed by Jim Kingdon, 7-Jan-2023.)
#

PART 5 ELEMENTARY NUMBER THEORY This part introduces elementary number theory, in particular the elementary properties of divisibility and elementary prime number theory.

5.1 Elementary properties of divisibility

5.1.1 The divides relation

Syntax	cdvds 12173	Extend the definition of a class to include the divides relation. See df-dvds 12174.


Definition	df-dvds 12174*	Define the divides relation, see definition in [ApostolNT] p. 14. (Contributed by Paul Chapman, 21-Mar-2011.)
$/\$ $/\$

Theorem	divides 12175*	Define the divides relation. means divides into with no remainder. For example, (ex-dvds 15805). As proven in dvdsval3 12177, . See divides 12175 and dvdsval2 12176 for other equivalent expressions. (Contributed by Paul Chapman, 21-Mar-2011.)
$/\$

Theorem	dvdsval2 12176	One nonzero integer divides another integer if and only if their quotient is an integer. (Contributed by Jeff Hankins, 29-Sep-2013.)
$/\$ $/\$

Theorem	dvdsval3 12177	One nonzero integer divides another integer if and only if the remainder upon division is zero, see remark in [ApostolNT] p. 106. (Contributed by Mario Carneiro, 22-Feb-2014.) (Revised by Mario Carneiro, 15-Jul-2014.)
$/\$

Theorem	dvdszrcl 12178	Reverse closure for the divisibility relation. (Contributed by Stefan O'Rear, 5-Sep-2015.)
$/\$

Theorem	dvdsmod0 12179	If a positive integer divides another integer, then the remainder upon division is zero. (Contributed by AV, 3-Mar-2022.)
$/\$

Theorem	p1modz1 12180	If a number greater than 1 divides another number, the second number increased by 1 is 1 modulo the first number. (Contributed by AV, 19-Mar-2022.)
$/\$

Theorem	dvdsmodexp 12181	If a positive integer divides another integer, this other integer is equal to its positive powers modulo the positive integer. (Formerly part of the proof for fermltl 12631). (Contributed by Mario Carneiro, 28-Feb-2014.) (Revised by AV, 19-Mar-2022.)
$/\$ $/\$

Theorem	nndivdvds 12182	Strong form of dvdsval2 12176 for positive integers. (Contributed by Stefan O'Rear, 13-Sep-2014.)
$/\$

Theorem	nndivides 12183*	Definition of the divides relation for positive integers. (Contributed by AV, 26-Jul-2021.)
$/\$

Theorem	dvdsdc 12184	Divisibility is decidable. (Contributed by Jim Kingdon, 14-Nov-2021.)
$/\$ DECID

Theorem	moddvds 12185	Two ways to say (mod ), see also definition in [ApostolNT] p. 106. (Contributed by Mario Carneiro, 18-Feb-2014.)
$/\$ $/\$

Theorem	modm1div 12186	An integer greater than one divides another integer minus one iff the second integer modulo the first integer is one. (Contributed by AV, 30-May-2023.)
$/\$

Theorem	dvds0lem 12187	A lemma to assist theorems of with no antecedents. (Contributed by Paul Chapman, 21-Mar-2011.)
$/\$ $/\$ $/\$

Theorem	dvds1lem 12188*	A lemma to assist theorems of with one antecedent. (Contributed by Paul Chapman, 21-Mar-2011.)
$/\$ $/\$ $/\$ $/\$

Theorem	dvds2lem 12189*	A lemma to assist theorems of with two antecedents. (Contributed by Paul Chapman, 21-Mar-2011.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	iddvds 12190	An integer divides itself. Theorem 1.1(a) in [ApostolNT] p. 14 (reflexive property of the divides relation). (Contributed by Paul Chapman, 21-Mar-2011.)


Theorem	1dvds 12191	1 divides any integer. Theorem 1.1(f) in [ApostolNT] p. 14. (Contributed by Paul Chapman, 21-Mar-2011.)


Theorem	dvds0 12192	Any integer divides 0. Theorem 1.1(g) in [ApostolNT] p. 14. (Contributed by Paul Chapman, 21-Mar-2011.)


Theorem	negdvdsb 12193	An integer divides another iff its negation does. (Contributed by Paul Chapman, 21-Mar-2011.)
$/\$

Theorem	dvdsnegb 12194	An integer divides another iff it divides its negation. (Contributed by Paul Chapman, 21-Mar-2011.)
$/\$

Theorem	absdvdsb 12195	An integer divides another iff its absolute value does. (Contributed by Paul Chapman, 21-Mar-2011.)
$/\$

Theorem	dvdsabsb 12196	An integer divides another iff it divides its absolute value. (Contributed by Paul Chapman, 21-Mar-2011.)
$/\$

Theorem	0dvds 12197	Only 0 is divisible by 0. Theorem 1.1(h) in [ApostolNT] p. 14. (Contributed by Paul Chapman, 21-Mar-2011.)


Theorem	zdvdsdc 12198	Divisibility of integers is decidable. (Contributed by Jim Kingdon, 17-Jan-2022.)
$/\$ DECID

Theorem	dvdsmul1 12199	An integer divides a multiple of itself. (Contributed by Paul Chapman, 21-Mar-2011.)
$/\$

Theorem	dvdsmul2 12200	An integer divides a multiple of itself. (Contributed by Paul Chapman, 21-Mar-2011.)
$/\$

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