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Theorem List for Intuitionistic Logic Explorer - 11701-11800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremretanclap 11701 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 )
 
Theoremresincld 11702 Closure of the sine function. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  ( sin `  A )  e. 
 RR )
 
Theoremrecoscld 11703 Closure of the cosine function. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  ( cos `  A )  e. 
 RR )
 
Theoremretanclapd 11704 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 )
 
Theoremsinneg 11705 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 ) )
 
Theoremcosneg 11706 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 ) )
 
Theoremtannegap 11707 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 ) )
 
Theoremsin0 11708 Value of the sine function at 0. (Contributed by Steve Rodriguez, 14-Mar-2005.)
 |-  ( sin `  0
 )  =  0
 
Theoremcos0 11709 Value of the cosine function at 0. (Contributed by NM, 30-Apr-2005.)
 |-  ( cos `  0
 )  =  1
 
Theoremtan0 11710 The value of the tangent function at zero is zero. (Contributed by David A. Wheeler, 16-Mar-2014.)
 |-  ( tan `  0
 )  =  0
 
Theoremefival 11711 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 ) ) ) )
 
Theoremefmival 11712 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 ) ) ) )
 
Theoremefeul 11713 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 ) ) ) ) ) )
 
Theoremefieq 11714 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 ) ) ) )
 
Theoremsinadd 11715 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 ) ) ) )
 
Theoremcosadd 11716 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 ) ) ) )
 
Theoremtanaddaplem 11717 A useful intermediate step in tanaddap 11718 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 ) )
 
Theoremtanaddap 11718 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 ) ) ) ) )
 
Theoremsinsub 11719 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 ) ) ) )
 
Theoremcossub 11720 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 ) ) ) )
 
Theoremaddsin 11721 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 ) ) ) ) )
 
Theoremsubsin 11722 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 ) ) ) ) )
 
Theoremsinmul 11723 Product of sines can be rewritten as half the difference of certain cosines. This follows from cosadd 11716 and cossub 11720. (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 ) )
 
Theoremcosmul 11724 Product of cosines can be rewritten as half the sum of certain cosines. This follows from cosadd 11716 and cossub 11720. (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 ) )
 
Theoremaddcos 11725 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 ) ) ) ) )
 
Theoremsubcos 11726 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 ) ) ) ) )
 
Theoremsincossq 11727 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 )
 
Theoremsin2t 11728 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 ) ) ) )
 
Theoremcos2t 11729 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 ) )
 
Theoremcos2tsin 11730 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 ) ) ) )
 
Theoremsinbnd 11731 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 ) )
 
Theoremcosbnd 11732 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 ) )
 
Theoremsinbnd2 11733 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 ) )
 
Theoremcosbnd2 11734 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 ) )
 
Theoremef01bndlem 11735* Lemma for sin01bnd 11736 and cos01bnd 11737. (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 )
 )
 
Theoremsin01bnd 11736 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 ) )
 
Theoremcos01bnd 11737 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 )
 ) ) )
 
Theoremcos1bnd 11738 Bounds on the cosine of 1. (Contributed by Paul Chapman, 19-Jan-2008.)
 |-  ( ( 1  / 
 3 )  <  ( cos `  1 )  /\  ( cos `  1 )  <  ( 2  /  3
 ) )
 
Theoremcos2bnd 11739 Bounds on the cosine of 2. (Contributed by Paul Chapman, 19-Jan-2008.)
 |-  ( -u ( 7  / 
 9 )  <  ( cos `  2 )  /\  ( cos `  2 )  < 
 -u ( 1  / 
 9 ) )
 
Theoremsin01gt0 11740 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 ) )
 
Theoremcos01gt0 11741 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 ) )
 
Theoremsin02gt0 11742 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 ) )
 
Theoremsincos1sgn 11743 The signs of the sine and cosine of 1. (Contributed by Paul Chapman, 19-Jan-2008.)
 |-  ( 0  <  ( sin `  1 )  /\  0  <  ( cos `  1
 ) )
 
Theoremsincos2sgn 11744 The signs of the sine and cosine of 2. (Contributed by Paul Chapman, 19-Jan-2008.)
 |-  ( 0  <  ( sin `  2 )  /\  ( cos `  2 )  <  0 )
 
Theoremsin4lt0 11745 The sine of 4 is negative. (Contributed by Paul Chapman, 19-Jan-2008.)
 |-  ( sin `  4
 )  <  0
 
Theoremcos12dec 11746 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 ) )
 
Theoremabsefi 11747 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 )
 
Theoremabsef 11748 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 ) ) )
 
Theoremabsefib 11749 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 ) )
 
Theoremefieq1re 11750 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 )
 
Theoremdemoivre 11751 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 11752 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 ) ) ) ) )
 
TheoremdemoivreALT 11752 Alternate proof of demoivre 11751. 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 ) ) ) ) )
 
4.9.1.1  The circle constant (tau = 2 pi)
 
Syntaxctau 11753 Extend class notation to include the constant tau,  tau = 6.28318....
 class  tau
 
Definitiondf-tau 11754 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 ,  <  )
 
4.9.2  _e is irrational
 
Theoremeirraplem 11755* Lemma for eirrap 11756. (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 ) )
 
Theoremeirrap 11756  _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 11757. (Contributed by Jim Kingdon, 6-Jan-2023.)
 |-  ( Q  e.  QQ  ->  _e #  Q )
 
Theoremeirr 11757  _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 11756. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Jim Kingdon, 6-Jan-2023.)
 |-  _e  e/  QQ
 
Theoremegt2lt3 11758 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 )
 
Theoremepos 11759 Euler's constant  _e is greater than 0. (Contributed by Jeff Hankins, 22-Nov-2008.)
 |-  0  <  _e
 
Theoremepr 11760 Euler's constant  _e is a positive real. (Contributed by Jeff Hankins, 22-Nov-2008.)
 |-  _e  e.  RR+
 
Theoremene0 11761  _e is not 0. (Contributed by David A. Wheeler, 17-Oct-2017.)
 |-  _e  =/=  0
 
Theoremeap0 11762  _e is apart from 0. (Contributed by Jim Kingdon, 7-Jan-2023.)
 |-  _e #  0
 
Theoremene1 11763  _e is not 1. (Contributed by David A. Wheeler, 17-Oct-2017.)
 |-  _e  =/=  1
 
Theoremeap1 11764  _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
 
Syntaxcdvds 11765 Extend the definition of a class to include the divides relation. See df-dvds 11766.
 class  ||
 
Definitiondf-dvds 11766* 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 ) }
 
Theoremdivides 11767* Define the divides relation.  M  ||  N means  M divides into  N with no remainder. For example,  3  ||  6 (ex-dvds 14104). As proven in dvdsval3 11769, 
M  ||  N  <->  ( N  mod  M )  =  0. See divides 11767 and dvdsval2 11768 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 )
 )
 
Theoremdvdsval2 11768 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 ) )
 
Theoremdvdsval3 11769 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 )
 )
 
Theoremdvdszrcl 11770 Reverse closure for the divisibility relation. (Contributed by Stefan O'Rear, 5-Sep-2015.)
 |-  ( X  ||  Y  ->  ( X  e.  ZZ  /\  Y  e.  ZZ )
 )
 
Theoremdvdsmod0 11771 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 )
 
Theoremp1modz1 11772 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 )
 
Theoremdvdsmodexp 11773 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 12204). (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 ) )
 
Theoremnndivdvds 11774 Strong form of dvdsval2 11768 for positive integers. (Contributed by Stefan O'Rear, 13-Sep-2014.)
 |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( B  ||  A 
 <->  ( A  /  B )  e.  NN )
 )
 
Theoremnndivides 11775* 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 )
 )
 
Theoremdvdsdc 11776 Divisibility is decidable. (Contributed by Jim Kingdon, 14-Nov-2021.)
 |-  ( ( M  e.  NN  /\  N  e.  ZZ )  -> DECID  M  ||  N )
 
Theoremmoddvds 11777 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 ) ) )
 
Theoremmodm1div 11778 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 ) ) )
 
Theoremdvds0lem 11779 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 )
 
Theoremdvds1lem 11780* 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 ) )
 
Theoremdvds2lem 11781* 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 )
 )
 
Theoremiddvds 11782 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 )
 
Theorem1dvds 11783 1 divides any integer. Theorem 1.1(f) in [ApostolNT] p. 14. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( N  e.  ZZ  ->  1  ||  N )
 
Theoremdvds0 11784 Any integer divides 0. Theorem 1.1(g) in [ApostolNT] p. 14. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( N  e.  ZZ  ->  N  ||  0 )
 
Theoremnegdvdsb 11785 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 ) )
 
Theoremdvdsnegb 11786 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 ) )
 
Theoremabsdvdsb 11787 An integer divides another iff its absolute value does. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  N 
 <->  ( abs `  M )  ||  N ) )
 
Theoremdvdsabsb 11788 An integer divides another iff it divides its absolute value. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  N 
 <->  M  ||  ( abs `  N ) ) )
 
Theorem0dvds 11789 Only 0 is divisible by 0. Theorem 1.1(h) in [ApostolNT] p. 14. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( N  e.  ZZ  ->  ( 0  ||  N  <->  N  =  0 ) )
 
Theoremzdvdsdc 11790 Divisibility of integers is decidable. (Contributed by Jim Kingdon, 17-Jan-2022.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  -> DECID  M  ||  N )
 
Theoremdvdsmul1 11791 An integer divides a multiple of itself. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  M  ||  ( M  x.  N ) )
 
Theoremdvdsmul2 11792 An integer divides a multiple of itself. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  N  ||  ( M  x.  N ) )
 
Theoremiddvdsexp 11793 An integer divides a positive integer power of itself. (Contributed by Paul Chapman, 26-Oct-2012.)
 |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  M  ||  ( M ^ N ) )
 
Theoremmuldvds1 11794 If a product divides an integer, so does one of its factors. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( K  x.  M )  ||  N  ->  K 
 ||  N ) )
 
Theoremmuldvds2 11795 If a product divides an integer, so does one of its factors. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( K  x.  M )  ||  N  ->  M 
 ||  N ) )
 
Theoremdvdscmul 11796 Multiplication by a constant maintains the divides relation. Theorem 1.1(d) in [ApostolNT] p. 14 (multiplication property of the divides relation). (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  ->  ( M  ||  N  ->  ( K  x.  M )  ||  ( K  x.  N ) ) )
 
Theoremdvdsmulc 11797 Multiplication by a constant maintains the divides relation. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  ->  ( M  ||  N  ->  ( M  x.  K )  ||  ( N  x.  K ) ) )
 
Theoremdvdscmulr 11798 Cancellation law for the divides relation. Theorem 1.1(e) in [ApostolNT] p. 14. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  ( K  e.  ZZ  /\  K  =/=  0 ) )  ->  ( ( K  x.  M )  ||  ( K  x.  N ) 
 <->  M  ||  N )
 )
 
Theoremdvdsmulcr 11799 Cancellation law for the divides relation. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  ( K  e.  ZZ  /\  K  =/=  0 ) )  ->  ( ( M  x.  K )  ||  ( N  x.  K ) 
 <->  M  ||  N )
 )
 
Theoremsummodnegmod 11800 The sum of two integers modulo a positive integer equals zero iff the first of the two integers equals the negative of the other integer modulo the positive integer. (Contributed by AV, 25-Jul-2021.)
 |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  ->  ( ( ( A  +  B )  mod  N )  =  0  <->  ( A  mod  N )  =  ( -u B  mod  N ) ) )
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