P. 126 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 12501-12600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	tanclap 12501	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.)
|- ( ( A e. CC /\ ( cos ` A ) # 0 ) -> ( tan ` A ) e. CC )
Theorem	sincld 12502	Closure of the sine function. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( sin ` A ) e. CC )
Theorem	coscld 12503	Closure of the cosine function. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( cos ` A ) e. CC )
Theorem	tanclapd 12504	Closure of the tangent function. (Contributed by Mario Carneiro, 29-May-2016.) (Revised by Jim Kingdon, 22-Dec-2022.)
|- ( ph -> A e. CC )   &    |- ( ph -> ( cos ` A ) # 0 )   =>    |- ( ph -> ( tan ` A ) e. CC )
Theorem	tanval2ap 12505	Express the tangent function directly in terms of exp. (Contributed by Mario Carneiro, 25-Feb-2015.) (Revised by Jim Kingdon, 22-Dec-2022.)
|- ( ( A e. CC /\ ( cos ` A ) # 0 ) -> ( tan ` A ) = ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) / ( _i x. ( ( exp ` ( _i x. A ) ) + ( exp ` ( -u _i x. A ) ) ) ) ) )
Theorem	tanval3ap 12506	Express the tangent function directly in terms of exp. (Contributed by Mario Carneiro, 25-Feb-2015.) (Revised by Jim Kingdon, 22-Dec-2022.)
|- ( ( A e. CC /\ ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) # 0 ) -> ( tan ` A ) = ( ( ( exp ` ( 2 x. ( _i x. A ) ) ) - 1 ) / ( _i x. ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) ) ) )
Theorem	resinval 12507	The sine of a real number in terms of the exponential function. (Contributed by NM, 30-Apr-2005.)
|- ( A e. RR -> ( sin ` A ) = ( Im ` ( exp ` ( _i x. A ) ) ) )
Theorem	recosval 12508	The cosine of a real number in terms of the exponential function. (Contributed by NM, 30-Apr-2005.)
|- ( A e. RR -> ( cos ` A ) = ( Re ` ( exp ` ( _i x. A ) ) ) )
Theorem	efi4p 12509*	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.)
|- F = ( n e. NN0 |-> ( ( ( _i x. A ) ^ n ) / ( ! ` n ) ) )   =>    |- ( A e. CC -> ( exp ` ( _i x. A ) ) = ( ( ( 1 - ( ( A ^ 2 ) / 2 ) ) + ( _i x. ( A - ( ( A ^ 3 ) / 6 ) ) ) ) + sum_ k e. ( ZZ>= ` 4 ) ( F ` k ) ) )
Theorem	resin4p 12510*	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.)
|- F = ( n e. NN0 |-> ( ( ( _i x. A ) ^ n ) / ( ! ` n ) ) )   =>    |- ( A e. RR -> ( sin ` A ) = ( ( A - ( ( A ^ 3 ) / 6 ) ) + ( Im ` sum_ k e. ( ZZ>= ` 4 ) ( F ` k ) ) ) )
Theorem	recos4p 12511*	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.)
|- F = ( n e. NN0 |-> ( ( ( _i x. A ) ^ n ) / ( ! ` n ) ) )   =>    |- ( A e. RR -> ( cos ` A ) = ( ( 1 - ( ( A ^ 2 ) / 2 ) ) + ( Re ` sum_ k e. ( ZZ>= ` 4 ) ( F ` k ) ) ) )
Theorem	resincl 12512	The sine of a real number is real. (Contributed by NM, 30-Apr-2005.)
|- ( A e. RR -> ( sin ` A ) e. RR )
Theorem	recoscl 12513	The cosine of a real number is real. (Contributed by NM, 30-Apr-2005.)
|- ( A e. RR -> ( cos ` A ) e. RR )
Theorem	retanclap 12514	The closure of the tangent function with a real argument. (Contributed by David A. Wheeler, 15-Mar-2014.)
|- ( ( A e. RR /\ ( cos ` A ) # 0 ) -> ( tan ` A ) e. RR )
Theorem	resincld 12515	Closure of the sine function. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. RR )   =>    |- ( ph -> ( sin ` A ) e. RR )
Theorem	recoscld 12516	Closure of the cosine function. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. RR )   =>    |- ( ph -> ( cos ` A ) e. RR )
Theorem	retanclapd 12517	Closure of the tangent function. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> ( cos ` A ) # 0 )   =>    |- ( ph -> ( tan ` A ) e. RR )
Theorem	sinneg 12518	The sine of a negative is the negative of the sine. (Contributed by NM, 30-Apr-2005.)
|- ( A e. CC -> ( sin ` -u A ) = -u ( sin ` A ) )
Theorem	cosneg 12519	The cosines of a number and its negative are the same. (Contributed by NM, 30-Apr-2005.)
|- ( A e. CC -> ( cos ` -u A ) = ( cos ` A ) )
Theorem	tannegap 12520	The tangent of a negative is the negative of the tangent. (Contributed by David A. Wheeler, 23-Mar-2014.)
|- ( ( A e. CC /\ ( cos ` A ) # 0 ) -> ( tan ` -u A ) = -u ( tan ` A ) )
Theorem	sin0 12521	Value of the sine function at 0. (Contributed by Steve Rodriguez, 14-Mar-2005.)
|- ( sin ` 0 ) = 0
Theorem	cos0 12522	Value of the cosine function at 0. (Contributed by NM, 30-Apr-2005.)
|- ( cos ` 0 ) = 1
Theorem	tan0 12523	The value of the tangent function at zero is zero. (Contributed by David A. Wheeler, 16-Mar-2014.)
|- ( tan ` 0 ) = 0
Theorem	efival 12524	The exponential function in terms of sine and cosine. (Contributed by NM, 30-Apr-2005.)
|- ( A e. CC -> ( exp ` ( _i x. A ) ) = ( ( cos ` A ) + ( _i x. ( sin ` A ) ) ) )
Theorem	efmival 12525	The exponential function in terms of sine and cosine. (Contributed by NM, 14-Jan-2006.)
|- ( A e. CC -> ( exp ` ( -u _i x. A ) ) = ( ( cos ` A ) - ( _i x. ( sin ` A ) ) ) )
Theorem	efeul 12526	Eulerian representation of the complex exponential. (Suggested by Jeff Hankins, 3-Jul-2006.) (Contributed by NM, 4-Jul-2006.)
|- ( A e. CC -> ( exp ` A ) = ( ( exp ` ( Re ` A ) ) x. ( ( cos ` ( Im ` A ) ) + ( _i x. ( sin ` ( Im ` A ) ) ) ) ) )
Theorem	efieq 12527	The exponentials of two imaginary numbers are equal iff their sine and cosine components are equal. (Contributed by Paul Chapman, 15-Mar-2008.)
|- ( ( A e. RR /\ B e. RR ) -> ( ( exp ` ( _i x. A ) ) = ( exp ` ( _i x. B ) ) <-> ( ( cos ` A ) = ( cos ` B ) /\ ( sin ` A ) = ( sin ` B ) ) ) )
Theorem	sinadd 12528	Addition formula for sine. Equation 14 of [Gleason] p. 310. (Contributed by Steve Rodriguez, 10-Nov-2006.) (Revised by Mario Carneiro, 30-Apr-2014.)
|- ( ( A e. CC /\ B e. CC ) -> ( sin ` ( A + B ) ) = ( ( ( sin ` A ) x. ( cos ` B ) ) + ( ( cos ` A ) x. ( sin ` B ) ) ) )
Theorem	cosadd 12529	Addition formula for cosine. Equation 15 of [Gleason] p. 310. (Contributed by NM, 15-Jan-2006.) (Revised by Mario Carneiro, 30-Apr-2014.)
|- ( ( A e. CC /\ B e. CC ) -> ( cos ` ( A + B ) ) = ( ( ( cos ` A ) x. ( cos ` B ) ) - ( ( sin ` A ) x. ( sin ` B ) ) ) )
Theorem	tanaddaplem 12530	A useful intermediate step in tanaddap 12531 when showing that the addition of tangents is well-defined. (Contributed by Mario Carneiro, 4-Apr-2015.) (Revised by Jim Kingdon, 25-Dec-2022.)
|- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) # 0 /\ ( cos ` B ) # 0 ) ) -> ( ( cos ` ( A + B ) ) # 0 <-> ( ( tan ` A ) x. ( tan ` B ) ) # 1 ) )
Theorem	tanaddap 12531	Addition formula for tangent. (Contributed by Mario Carneiro, 4-Apr-2015.)
|- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) # 0 /\ ( cos ` B ) # 0 /\ ( cos ` ( A + B ) ) # 0 ) ) -> ( tan ` ( A + B ) ) = ( ( ( tan ` A ) + ( tan ` B ) ) / ( 1 - ( ( tan ` A ) x. ( tan ` B ) ) ) ) )
Theorem	sinsub 12532	Sine of difference. (Contributed by Paul Chapman, 12-Oct-2007.)
|- ( ( A e. CC /\ B e. CC ) -> ( sin ` ( A - B ) ) = ( ( ( sin ` A ) x. ( cos ` B ) ) - ( ( cos ` A ) x. ( sin ` B ) ) ) )
Theorem	cossub 12533	Cosine of difference. (Contributed by Paul Chapman, 12-Oct-2007.)
|- ( ( A e. CC /\ B e. CC ) -> ( cos ` ( A - B ) ) = ( ( ( cos ` A ) x. ( cos ` B ) ) + ( ( sin ` A ) x. ( sin ` B ) ) ) )
Theorem	addsin 12534	Sum of sines. (Contributed by Paul Chapman, 12-Oct-2007.)
|- ( ( A e. CC /\ B e. CC ) -> ( ( sin ` A ) + ( sin ` B ) ) = ( 2 x. ( ( sin ` ( ( A + B ) / 2 ) ) x. ( cos ` ( ( A - B ) / 2 ) ) ) ) )
Theorem	subsin 12535	Difference of sines. (Contributed by Paul Chapman, 12-Oct-2007.)
|- ( ( A e. CC /\ B e. CC ) -> ( ( sin ` A ) - ( sin ` B ) ) = ( 2 x. ( ( cos ` ( ( A + B ) / 2 ) ) x. ( sin ` ( ( A - B ) / 2 ) ) ) ) )
Theorem	sinmul 12536	Product of sines can be rewritten as half the difference of certain cosines. This follows from cosadd 12529 and cossub 12533. (Contributed by David A. Wheeler, 26-May-2015.)
|- ( ( A e. CC /\ B e. CC ) -> ( ( sin ` A ) x. ( sin ` B ) ) = ( ( ( cos ` ( A - B ) ) - ( cos ` ( A + B ) ) ) / 2 ) )
Theorem	cosmul 12537	Product of cosines can be rewritten as half the sum of certain cosines. This follows from cosadd 12529 and cossub 12533. (Contributed by David A. Wheeler, 26-May-2015.)
|- ( ( A e. CC /\ B e. CC ) -> ( ( cos ` A ) x. ( cos ` B ) ) = ( ( ( cos ` ( A - B ) ) + ( cos ` ( A + B ) ) ) / 2 ) )
Theorem	addcos 12538	Sum of cosines. (Contributed by Paul Chapman, 12-Oct-2007.)
|- ( ( A e. CC /\ B e. CC ) -> ( ( cos ` A ) + ( cos ` B ) ) = ( 2 x. ( ( cos ` ( ( A + B ) / 2 ) ) x. ( cos ` ( ( A - B ) / 2 ) ) ) ) )
Theorem	subcos 12539	Difference of cosines. (Contributed by Paul Chapman, 12-Oct-2007.) (Revised by Mario Carneiro, 10-May-2014.)
|- ( ( A e. CC /\ B e. CC ) -> ( ( cos ` B ) - ( cos ` A ) ) = ( 2 x. ( ( sin ` ( ( A + B ) / 2 ) ) x. ( sin ` ( ( A - B ) / 2 ) ) ) ) )
Theorem	sincossq 12540	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.)
|- ( A e. CC -> ( ( ( sin ` A ) ^ 2 ) + ( ( cos ` A ) ^ 2 ) ) = 1 )
Theorem	sin2t 12541	Double-angle formula for sine. (Contributed by Paul Chapman, 17-Jan-2008.)
|- ( A e. CC -> ( sin ` ( 2 x. A ) ) = ( 2 x. ( ( sin ` A ) x. ( cos ` A ) ) ) )
Theorem	cos2t 12542	Double-angle formula for cosine. (Contributed by Paul Chapman, 24-Jan-2008.)
|- ( A e. CC -> ( cos ` ( 2 x. A ) ) = ( ( 2 x. ( ( cos ` A ) ^ 2 ) ) - 1 ) )
Theorem	cos2tsin 12543	Double-angle formula for cosine in terms of sine. (Contributed by NM, 12-Sep-2008.)
|- ( A e. CC -> ( cos ` ( 2 x. A ) ) = ( 1 - ( 2 x. ( ( sin ` A ) ^ 2 ) ) ) )
Theorem	sinbnd 12544	The sine of a real number lies between -1 and 1. Equation 18 of [Gleason] p. 311. (Contributed by NM, 16-Jan-2006.)
|- ( A e. RR -> ( -u 1 <_ ( sin ` A ) /\ ( sin ` A ) <_ 1 ) )
Theorem	cosbnd 12545	The cosine of a real number lies between -1 and 1. Equation 18 of [Gleason] p. 311. (Contributed by NM, 16-Jan-2006.)
|- ( A e. RR -> ( -u 1 <_ ( cos ` A ) /\ ( cos ` A ) <_ 1 ) )
Theorem	sinbnd2 12546	The sine of a real number is in the closed interval from -1 to 1. (Contributed by Mario Carneiro, 12-May-2014.)
|- ( A e. RR -> ( sin ` A ) e. ( -u 1 [,] 1 ) )
Theorem	cosbnd2 12547	The cosine of a real number is in the closed interval from -1 to 1. (Contributed by Mario Carneiro, 12-May-2014.)
|- ( A e. RR -> ( cos ` A ) e. ( -u 1 [,] 1 ) )
Theorem	ef01bndlem 12548*	Lemma for sin01bnd 12549 and cos01bnd 12550. (Contributed by Paul Chapman, 19-Jan-2008.)
|- F = ( n e. NN0 |-> ( ( ( _i x. A ) ^ n ) / ( ! ` n ) ) )   =>    |- ( A e. ( 0 (,] 1 ) -> ( abs ` sum_ k e. ( ZZ>= ` 4 ) ( F ` k ) ) < ( ( A ^ 4 ) / 6 ) )
Theorem	sin01bnd 12549	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.)
|- ( A e. ( 0 (,] 1 ) -> ( ( A - ( ( A ^ 3 ) / 3 ) ) < ( sin ` A ) /\ ( sin ` A ) < A ) )
Theorem	cos01bnd 12550	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.)
|- ( A e. ( 0 (,] 1 ) -> ( ( 1 - ( 2 x. ( ( A ^ 2 ) / 3 ) ) ) < ( cos ` A ) /\ ( cos ` A ) < ( 1 - ( ( A ^ 2 ) / 3 ) ) ) )
Theorem	cos1bnd 12551	Bounds on the cosine of 1. (Contributed by Paul Chapman, 19-Jan-2008.)
|- ( ( 1 / 3 ) < ( cos ` 1 ) /\ ( cos ` 1 ) < ( 2 / 3 ) )
Theorem	cos2bnd 12552	Bounds on the cosine of 2. (Contributed by Paul Chapman, 19-Jan-2008.)
|- ( -u ( 7 / 9 ) < ( cos ` 2 ) /\ ( cos ` 2 ) < -u ( 1 / 9 ) )
Theorem	sinltxirr 12553*	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.)
|- ( ( A e. RR+ /\ A. q e. QQ A # q ) -> ( sin ` A ) < A )
Theorem	sin01gt0 12554	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.)
|- ( A e. ( 0 (,] 1 ) -> 0 < ( sin ` A ) )
Theorem	cos01gt0 12555	The cosine of a positive real number less than or equal to 1 is positive. (Contributed by Paul Chapman, 19-Jan-2008.)
|- ( A e. ( 0 (,] 1 ) -> 0 < ( cos ` A ) )
Theorem	sin02gt0 12556	The sine of a positive real number less than or equal to 2 is positive. (Contributed by Paul Chapman, 19-Jan-2008.)
|- ( A e. ( 0 (,] 2 ) -> 0 < ( sin ` A ) )
Theorem	sincos1sgn 12557	The signs of the sine and cosine of 1. (Contributed by Paul Chapman, 19-Jan-2008.)
|- ( 0 < ( sin ` 1 ) /\ 0 < ( cos ` 1 ) )
Theorem	sincos2sgn 12558	The signs of the sine and cosine of 2. (Contributed by Paul Chapman, 19-Jan-2008.)
|- ( 0 < ( sin ` 2 ) /\ ( cos ` 2 ) < 0 )
Theorem	sin4lt0 12559	The sine of 4 is negative. (Contributed by Paul Chapman, 19-Jan-2008.)
|- ( sin ` 4 ) < 0
Theorem	cos12dec 12560	Cosine is decreasing from one to two. (Contributed by Mario Carneiro and Jim Kingdon, 6-Mar-2024.)
|- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( cos ` B ) < ( cos ` A ) )
Theorem	absefi 12561	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.)
|- ( A e. RR -> ( abs ` ( exp ` ( _i x. A ) ) ) = 1 )
Theorem	absef 12562	The absolute value of the exponential is the exponential of the real part. (Contributed by Paul Chapman, 13-Sep-2007.)
|- ( A e. CC -> ( abs ` ( exp ` A ) ) = ( exp ` ( Re ` A ) ) )
Theorem	absefib 12563	A complex number is real iff the exponential of its product with _i has absolute value one. (Contributed by NM, 21-Aug-2008.)
|- ( A e. CC -> ( A e. RR <-> ( abs ` ( exp ` ( _i x. A ) ) ) = 1 ) )
Theorem	efieq1re 12564	A number whose imaginary exponential is one is real. (Contributed by NM, 21-Aug-2008.)
|- ( ( A e. CC /\ ( exp ` ( _i x. A ) ) = 1 ) -> A e. RR )
Theorem	demoivre 12565	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 12566 for an alternate longer proof not using the exponential function. (Contributed by NM, 24-Jul-2007.)
|- ( ( A e. CC /\ N e. ZZ ) -> ( ( ( cos ` A ) + ( _i x. ( sin ` A ) ) ) ^ N ) = ( ( cos ` ( N x. A ) ) + ( _i x. ( sin ` ( N x. A ) ) ) ) )
Theorem	demoivreALT 12566	Alternate proof of demoivre 12565. 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.)
|- ( ( A e. CC /\ N e. NN0 ) -> ( ( ( cos ` A ) + ( _i x. ( sin ` A ) ) ) ^ N ) = ( ( cos ` ( N x. A ) ) + ( _i x. ( sin ` ( N x. A ) ) ) ) )
4.10.1.1  The circle constant (tau = 2 pi)
Syntax	ctau 12567	Extend class notation to include the constant tau, tau = 6.28318....
class tau
Definition	df-tau 12568	Define the circle constant tau, 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 tau, a three-legged variant of pi, or 2 pi. Note the difference between this constant tau and the formula variable ta. 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.)
|- tau = inf ( ( RR+ i^i ( `' cos " { 1 } ) ) , RR , < )
4.10.2  _e is irrational
Theorem	eirraplem 12569*	Lemma for eirrap 12570. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Jim Kingdon, 5-Jan-2022.)
|- F = ( n e. NN0 |-> ( 1 / ( ! ` n ) ) )   &    |- ( ph -> P e. ZZ )   &    |- ( ph -> Q e. NN )   =>    |- ( ph -> _e # ( P / Q ) )
Theorem	eirrap 12570	_e is irrational. That is, for any rational number, _e is apart from it. In the absence of excluded middle, we can distinguish between this and saying that _e is not rational, which is eirr 12571. (Contributed by Jim Kingdon, 6-Jan-2023.)
|- ( Q e. QQ -> _e # Q )
Theorem	eirr 12571	_e is not rational. In the absence of excluded middle, we can distinguish between this and saying that _e is irrational in the sense of being apart from any rational number, which is eirrap 12570. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Jim Kingdon, 6-Jan-2023.)
|- _e e/ QQ
Theorem	egt2lt3 12572	Euler's constant _e = 2.71828... is bounded by 2 and 3. (Contributed by NM, 28-Nov-2008.) (Revised by Jim Kingdon, 7-Jan-2023.)
|- ( 2 < _e /\ _e < 3 )
Theorem	epos 12573	Euler's constant _e is greater than 0. (Contributed by Jeff Hankins, 22-Nov-2008.)
|- 0 < _e
Theorem	epr 12574	Euler's constant _e is a positive real. (Contributed by Jeff Hankins, 22-Nov-2008.)
|- _e e. RR+
Theorem	ene0 12575	_e is not 0. (Contributed by David A. Wheeler, 17-Oct-2017.)
|- _e =/= 0
Theorem	eap0 12576	_e is apart from 0. (Contributed by Jim Kingdon, 7-Jan-2023.)
|- _e # 0
Theorem	ene1 12577	_e is not 1. (Contributed by David A. Wheeler, 17-Oct-2017.)
|- _e =/= 1
Theorem	eap1 12578	_e is apart from 1. (Contributed by Jim Kingdon, 7-Jan-2023.)
|- _e # 1
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 12579	Extend the definition of a class to include the divides relation. See df-dvds 12580.
class ||
Definition	df-dvds 12580*	Define the divides relation, see definition in [ApostolNT] p. 14. (Contributed by Paul Chapman, 21-Mar-2011.)
|- || = { <. x , y >. | ( ( x e. ZZ /\ y e. ZZ ) /\ E. n e. ZZ ( n x. x ) = y ) }
Theorem	divides 12581*	Define the divides relation. M || N means M divides into N with no remainder. For example, 3 || 6 (ex-dvds 16949). As proven in dvdsval3 12583, M || N <-> ( N mod M ) = 0. See divides 12581 and dvdsval2 12582 for other equivalent expressions. (Contributed by Paul Chapman, 21-Mar-2011.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || N <-> E. n e. ZZ ( n x. M ) = N ) )
Theorem	dvdsval2 12582	One nonzero integer divides another integer if and only if their quotient is an integer. (Contributed by Jeff Hankins, 29-Sep-2013.)
|- ( ( M e. ZZ /\ M =/= 0 /\ N e. ZZ ) -> ( M || N <-> ( N / M ) e. ZZ ) )
Theorem	dvdsval3 12583	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.)
|- ( ( M e. NN /\ N e. ZZ ) -> ( M || N <-> ( N mod M ) = 0 ) )
Theorem	dvdszrcl 12584	Reverse closure for the divisibility relation. (Contributed by Stefan O'Rear, 5-Sep-2015.)
|- ( X || Y -> ( X e. ZZ /\ Y e. ZZ ) )
Theorem	dvdsmod0 12585	If a positive integer divides another integer, then the remainder upon division is zero. (Contributed by AV, 3-Mar-2022.)
|- ( ( M e. NN /\ M || N ) -> ( N mod M ) = 0 )
Theorem	p1modz1 12586	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.)
|- ( ( M || A /\ 1 < M ) -> ( ( A + 1 ) mod M ) = 1 )
Theorem	dvdsmodexp 12587	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 13041). (Contributed by Mario Carneiro, 28-Feb-2014.) (Revised by AV, 19-Mar-2022.)
|- ( ( N e. NN /\ B e. NN /\ N || A ) -> ( ( A ^ B ) mod N ) = ( A mod N ) )
Theorem	nndivdvds 12588	Strong form of dvdsval2 12582 for positive integers. (Contributed by Stefan O'Rear, 13-Sep-2014.)
|- ( ( A e. NN /\ B e. NN ) -> ( B || A <-> ( A / B ) e. NN ) )
Theorem	nndivides 12589*	Definition of the divides relation for positive integers. (Contributed by AV, 26-Jul-2021.)
|- ( ( M e. NN /\ N e. NN ) -> ( M || N <-> E. n e. NN ( n x. M ) = N ) )
Theorem	dvdsdc 12590	Divisibility is decidable. (Contributed by Jim Kingdon, 14-Nov-2021.)
|- ( ( M e. NN /\ N e. ZZ ) -> DECID M || N )
Theorem	moddvds 12591	Two ways to say A == B (mod N), see also definition in [ApostolNT] p. 106. (Contributed by Mario Carneiro, 18-Feb-2014.)
|- ( ( N e. NN /\ A e. ZZ /\ B e. ZZ ) -> ( ( A mod N ) = ( B mod N ) <-> N || ( A - B ) ) )
Theorem	modm1div 12592	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.)
|- ( ( N e. ( ZZ>= ` 2 ) /\ A e. ZZ ) -> ( ( A mod N ) = 1 <-> N || ( A - 1 ) ) )
Theorem	dvds0lem 12593	A lemma to assist theorems of || with no antecedents. (Contributed by Paul Chapman, 21-Mar-2011.)
|- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( K x. M ) = N ) -> M || N )
Theorem	dvds1lem 12594*	A lemma to assist theorems of || with one antecedent. (Contributed by Paul Chapman, 21-Mar-2011.)
|- ( ph -> ( J e. ZZ /\ K e. ZZ ) )   &    |- ( ph -> ( M e. ZZ /\ N e. ZZ ) )   &    |- ( ( ph /\ x e. ZZ ) -> Z e. ZZ )   &    |- ( ( ph /\ x e. ZZ ) -> ( ( x x. J ) = K -> ( Z x. M ) = N ) )   =>    |- ( ph -> ( J || K -> M || N ) )
Theorem	dvds2lem 12595*	A lemma to assist theorems of || with two antecedents. (Contributed by Paul Chapman, 21-Mar-2011.)
|- ( ph -> ( I e. ZZ /\ J e. ZZ ) )   &    |- ( ph -> ( K e. ZZ /\ L e. ZZ ) )   &    |- ( ph -> ( M e. ZZ /\ N e. ZZ ) )   &    |- ( ( ph /\ ( x e. ZZ /\ y e. ZZ ) ) -> Z e. ZZ )   &    |- ( ( ph /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( ( ( x x. I ) = J /\ ( y x. K ) = L ) -> ( Z x. M ) = N ) )   =>    |- ( ph -> ( ( I || J /\ K || L ) -> M || N ) )
Theorem	iddvds 12596	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.)
|- ( N e. ZZ -> N || N )
Theorem	1dvds 12597	1 divides any integer. Theorem 1.1(f) in [ApostolNT] p. 14. (Contributed by Paul Chapman, 21-Mar-2011.)
|- ( N e. ZZ -> 1 || N )
Theorem	dvds0 12598	Any integer divides 0. Theorem 1.1(g) in [ApostolNT] p. 14. (Contributed by Paul Chapman, 21-Mar-2011.)
|- ( N e. ZZ -> N || 0 )
Theorem	negdvdsb 12599	An integer divides another iff its negation does. (Contributed by Paul Chapman, 21-Mar-2011.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || N <-> -u M || N ) )
Theorem	dvdsnegb 12600	An integer divides another iff it divides its negation. (Contributed by Paul Chapman, 21-Mar-2011.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || N <-> M || -u N ) )