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Theorem List for Metamath Proof Explorer - 12401-12500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcosmul 12401 Product of cosines can be rewritten as half the sum of certain cosines. This follows from cosadd 12393 and cossub 12397. (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 12402 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 12403 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 12404 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 12405 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 12406 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 12407 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 12408 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 12409 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 12410 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 12411 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 12412* Lemma for sin01bnd 12413 and cos01bnd 12414. (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 12413 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 12414 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 12415 Bounds on the cosine of 1. (Contributed by Paul Chapman, 19-Jan-2008.)
 |-  ( ( 1  / 
 3 )  <  ( cos `  1 )  /\  ( cos `  1 )  <  ( 2  /  3
 ) )
 
Theoremcos2bnd 12416 Bounds on the cosine of 2. (Contributed by Paul Chapman, 19-Jan-2008.)
 |-  ( -u ( 7  / 
 9 )  <  ( cos `  2 )  /\  ( cos `  2 )  < 
 -u ( 1  / 
 9 ) )
 
Theoremsinltx 12417 The sine of a positive real number is less than its argument. (Contributed by Mario Carneiro, 29-Jul-2014.)
 |-  ( A  e.  RR+  ->  ( sin `  A )  <  A )
 
Theoremsin01gt0 12418 The sine 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  <  ( sin `  A ) )
 
Theoremcos01gt0 12419 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 12420 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 12421 The signs of the sine and cosine of 1. (Contributed by Paul Chapman, 19-Jan-2008.)
 |-  ( 0  <  ( sin `  1 )  /\  0  <  ( cos `  1
 ) )
 
Theoremsincos2sgn 12422 The signs of the sine and cosine of 2. (Contributed by Paul Chapman, 19-Jan-2008.)
 |-  ( 0  <  ( sin `  2 )  /\  ( cos `  2 )  <  0 )
 
Theoremsin4lt0 12423 The sine of 4 is negative. (Contributed by Paul Chapman, 19-Jan-2008.)
 |-  ( sin `  4
 )  <  0
 
Theoremabsefi 12424 The absolute value of the exponential function 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 12425 The absolute value of the exponential function is the exponential function of the real part. (Contributed by Paul Chapman, 13-Sep-2007.)
 |-  ( A  e.  CC  ->  ( abs `  ( exp `  A ) )  =  ( exp `  ( Re `  A ) ) )
 
Theoremabsefib 12426 A number is real iff its imaginary exponential has absolute value one. (Contributed by NM, 21-Aug-2008.)
 |-  ( A  e.  CC  ->  ( A  e.  RR  <->  ( abs `  ( exp `  ( _i  x.  A ) ) )  =  1 ) )
 
Theoremefieq1re 12427 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 12428 De Moivre's Formula. Shorter proof of demoivreALT 12429 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 12429 De Moivre's Formula. Proof by induction given at http://en.wikipedia.org/wiki/De_Moivre's_formula, but restricted to nonnegative integer powers. (Contributed by Steve Rodriguez, 10-Nov-2006.) (Proof modification 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 ) ) ) ) )
 
5.9.2  _e is irrational
 
Theoremeirrlem 12430* Lemma for eirr 12431. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Mario Carneiro, 29-Apr-2014.)
 |-  F  =  ( n  e.  NN0  |->  ( 1 
 /  ( ! `  n ) ) )   &    |-  ( ph  ->  P  e.  ZZ )   &    |-  ( ph  ->  Q  e.  NN )   &    |-  ( ph  ->  _e  =  ( P  /  Q ) )   =>    |- 
 -.  ph
 
Theoremeirr 12431  _e is irrational. (Contributed by Paul Chapman, 9-Feb-2008.) (Proof shortened by Mario Carneiro, 29-Apr-2014.)
 |-  _e  e/  QQ
 
Theoremegt2lt3 12432 Euler's constant  _e = 2.71828... is bounded by 2 and 3. (Contributed by NM, 28-Nov-2008.) (Revised by Mario Carneiro, 29-Apr-2014.)
 |-  ( 2  <  _e  /\  _e  <  3 )
 
Theoremepos 12433 Euler's constant  _e is greater than 0. (Contributed by Jeff Hankins, 22-Nov-2008.)
 |-  0  <  _e
 
Theoremepr 12434 Euler's constant  _e is a positive real. (Contributed by Jeff Hankins, 22-Nov-2008.)
 |-  _e  e.  RR+
 
5.10  Cardinality of real and complex number subsets
 
5.10.1  Countability of integers and rationals
 
Theoremxpnnen 12435 The cross product of the set of natural numbers with itself is equinumerous to the set of natural numbers. (Contributed by NM, 1-Aug-2004.) (Revised by Mario Carneiro, 9-Mar-2013.)
 |-  ( NN  X.  NN )  ~~  NN
 
TheoremxpnnenOLD 12436 The cross product of the set of natural numbers with itself is equinumerous to the set of natural numbers. The key idea is to use nn0opth2 11239 to show that the mapping from natural numbers  z and  w to  ( ( z  +  w ) ^
2 )  +  w is one-to-one. (Contributed by NM, 1-Aug-2004.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( NN  X.  NN )  ~~  NN
 
TheoremxpomenOLD 12437 The cross product of omega (the set of ordinal natural numbers) with itself is equinumerous to omega. Exercise 1 of [Enderton] p. 133 (which proves this with a direct, but longer, proof; ours uses instead the Schroeder-Bernstein Theorem sbth 6935 in xpnnen 12435). (Contributed by NM, 23-Jul-2004.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( om  X.  om )  ~~  om
 
Theoremznnenlem 12438 Lemma for znnen 12439. (Contributed by NM, 31-Jul-2004.)
 |-  ( ( ( 0 
 <_  x  /\  -.  0  <_  y )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  ->  ( x  =  y  <->  ( 2  x.  x )  =  ( ( -u 2  x.  y )  +  1 ) ) )
 
Theoremznnen 12439 The set of integers and the set of natural numbers are equinumerous. Exercise 1 of [Gleason] p. 140. (Contributed by NM, 31-Jul-2004.) (Proof shortened by Mario Carneiro, 13-Jun-2014.)
 |- 
 ZZ  ~~  NN
 
Theoremqnnen 12440 The rational numbers are countable. This proof does not use the Axiom of Choice, even though it uses an onto function, because the base set  ( ZZ  X.  NN ) is numerable. Exercise 2 of [Enderton] p. 133. (Contributed by NM, 31-Jul-2004.) (Revised by Mario Carneiro, 3-Mar-2013.)
 |- 
 QQ  ~~  NN
 
5.10.2  The reals are uncountable
 
Theoremrpnnen2lem1 12441* Lemma for rpnnen2 12452. (Contributed by Mario Carneiro, 13-May-2013.)
 |-  F  =  ( x  e.  ~P NN  |->  ( n  e.  NN  |->  if ( n  e.  x ,  ( ( 1  / 
 3 ) ^ n ) ,  0 )
 ) )   =>    |-  ( ( A  C_  NN  /\  N  e.  NN )  ->  ( ( F `
  A ) `  N )  =  if ( N  e.  A ,  ( ( 1  / 
 3 ) ^ N ) ,  0 )
 )
 
Theoremrpnnen2lem2 12442* Lemma for rpnnen2 12452. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 23-Aug-2014.)
 |-  F  =  ( x  e.  ~P NN  |->  ( n  e.  NN  |->  if ( n  e.  x ,  ( ( 1  / 
 3 ) ^ n ) ,  0 )
 ) )   =>    |-  ( A  C_  NN  ->  ( F `  A ) : NN --> RR )
 
Theoremrpnnen2lem3 12443* Lemma for rpnnen2 12452. (Contributed by Mario Carneiro, 13-May-2013.)
 |-  F  =  ( x  e.  ~P NN  |->  ( n  e.  NN  |->  if ( n  e.  x ,  ( ( 1  / 
 3 ) ^ n ) ,  0 )
 ) )   =>    |- 
 seq  1 (  +  ,  ( F `  NN ) )  ~~>  ( 1  / 
 2 )
 
Theoremrpnnen2lem4 12444* Lemma for rpnnen2 12452. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 31-Aug-2014.)
 |-  F  =  ( x  e.  ~P NN  |->  ( n  e.  NN  |->  if ( n  e.  x ,  ( ( 1  / 
 3 ) ^ n ) ,  0 )
 ) )   =>    |-  ( ( A  C_  B  /\  B  C_  NN  /\  k  e.  NN )  ->  ( 0  <_  (
 ( F `  A ) `  k )  /\  ( ( F `  A ) `  k
 )  <_  ( ( F `  B ) `  k ) ) )
 
Theoremrpnnen2lem5 12445* Lemma for rpnnen2 12452. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 30-Apr-2014.)
 |-  F  =  ( x  e.  ~P NN  |->  ( n  e.  NN  |->  if ( n  e.  x ,  ( ( 1  / 
 3 ) ^ n ) ,  0 )
 ) )   =>    |-  ( ( A  C_  NN  /\  M  e.  NN )  ->  seq  M (  +  ,  ( F `  A ) )  e. 
 dom 
 ~~>  )
 
Theoremrpnnen2lem6 12446* Lemma for rpnnen2 12452. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 30-Apr-2014.)
 |-  F  =  ( x  e.  ~P NN  |->  ( n  e.  NN  |->  if ( n  e.  x ,  ( ( 1  / 
 3 ) ^ n ) ,  0 )
 ) )   =>    |-  ( ( A  C_  NN  /\  M  e.  NN )  ->  sum_ k  e.  ( ZZ>=
 `  M ) ( ( F `  A ) `  k )  e. 
 RR )
 
Theoremrpnnen2lem7 12447* Lemma for rpnnen2 12452. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 30-Apr-2014.)
 |-  F  =  ( x  e.  ~P NN  |->  ( n  e.  NN  |->  if ( n  e.  x ,  ( ( 1  / 
 3 ) ^ n ) ,  0 )
 ) )   =>    |-  ( ( A  C_  B  /\  B  C_  NN  /\  M  e.  NN )  -> 
 sum_ k  e.  ( ZZ>=
 `  M ) ( ( F `  A ) `  k )  <_  sum_ k  e.  ( ZZ>= `  M ) ( ( F `  B ) `
  k ) )
 
Theoremrpnnen2lem8 12448* Lemma for rpnnen2 12452. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 30-Apr-2014.)
 |-  F  =  ( x  e.  ~P NN  |->  ( n  e.  NN  |->  if ( n  e.  x ,  ( ( 1  / 
 3 ) ^ n ) ,  0 )
 ) )   =>    |-  ( ( A  C_  NN  /\  M  e.  NN )  ->  sum_ k  e.  NN  ( ( F `  A ) `  k
 )  =  ( sum_ k  e.  ( 1 ... ( M  -  1
 ) ) ( ( F `  A ) `
  k )  +  sum_
 k  e.  ( ZZ>= `  M ) ( ( F `  A ) `
  k ) ) )
 
Theoremrpnnen2lem9 12449* Lemma for rpnnen2 12452. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 30-Apr-2014.)
 |-  F  =  ( x  e.  ~P NN  |->  ( n  e.  NN  |->  if ( n  e.  x ,  ( ( 1  / 
 3 ) ^ n ) ,  0 )
 ) )   =>    |-  ( M  e.  NN  -> 
 sum_ k  e.  ( ZZ>=
 `  M ) ( ( F `  ( NN  \  { M }
 ) ) `  k
 )  =  ( 0  +  ( ( ( 1  /  3 ) ^ ( M  +  1 ) )  /  ( 1  -  (
 1  /  3 )
 ) ) ) )
 
Theoremrpnnen2lem10 12450* Lemma for rpnnen2 12452. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 30-Apr-2014.)
 |-  F  =  ( x  e.  ~P NN  |->  ( n  e.  NN  |->  if ( n  e.  x ,  ( ( 1  / 
 3 ) ^ n ) ,  0 )
 ) )   &    |-  ( ph  ->  A 
 C_  NN )   &    |-  ( ph  ->  B 
 C_  NN )   &    |-  ( ph  ->  m  e.  ( A  \  B ) )   &    |-  ( ph  ->  A. n  e.  NN  ( n  <  m  ->  ( n  e.  A  <->  n  e.  B ) ) )   &    |-  ( ps  <->  sum_ k  e.  NN  ( ( F `  A ) `  k
 )  =  sum_ k  e.  NN  ( ( F `
  B ) `  k ) )   =>    |-  ( ( ph  /\ 
 ps )  ->  sum_ k  e.  ( ZZ>= `  m )
 ( ( F `  A ) `  k
 )  =  sum_ k  e.  ( ZZ>= `  m )
 ( ( F `  B ) `  k
 ) )
 
Theoremrpnnen2lem11 12451* Lemma for rpnnen2 12452. (Contributed by Mario Carneiro, 13-May-2013.)
 |-  F  =  ( x  e.  ~P NN  |->  ( n  e.  NN  |->  if ( n  e.  x ,  ( ( 1  / 
 3 ) ^ n ) ,  0 )
 ) )   &    |-  ( ph  ->  A 
 C_  NN )   &    |-  ( ph  ->  B 
 C_  NN )   &    |-  ( ph  ->  m  e.  ( A  \  B ) )   &    |-  ( ph  ->  A. n  e.  NN  ( n  <  m  ->  ( n  e.  A  <->  n  e.  B ) ) )   &    |-  ( ps  <->  sum_ k  e.  NN  ( ( F `  A ) `  k
 )  =  sum_ k  e.  NN  ( ( F `
  B ) `  k ) )   =>    |-  ( ph  ->  -. 
 ps )
 
Theoremrpnnen2 12452* The other half of rpnnen 12453, where we show an injection from sets of natural numbers to real numbers. The obvious choice for this is binary expansion, but it has the unfortunate property that it does not produce an injection on numbers which end with all 0's or all 1's (the more well-known decimal version of this is 0.999... 12285). Instead, we opt for a ternary expansion, which produces (a scaled version of) the Cantor set. Since the Cantor set is riddled with gaps, we can show that any two sequences that are not equal must differ somewhere, and when they do, they are placed a finite distance apart, thus ensuring that the map is injective.

Our map assigns to each subset  A of the natural numbers the number  sum_ k  e.  A ( 3 ^
-u k )  = 
sum_ k  e.  NN ( ( F `  A ) `  k
), where  ( ( F `  A ) `  k )  =  if ( k  e.  A ,  ( 3 ^
-u k ) ,  0 ) ) (rpnnen2lem1 12441). This is an infinite sum of real numbers (rpnnen2lem2 12442), and since  A 
C_  B implies  ( F `  A )  <_  ( F `  B ) (rpnnen2lem4 12444) and  ( F `  NN ) converges to  1  /  2 (rpnnen2lem3 12443) by geoisum1 12283, the sum is convergent to some real (rpnnen2lem5 12445 and rpnnen2lem6 12446) by the comparison test for convergence cvgcmp 12225. The comparison test also tells us that  A  C_  B implies  sum_ ( F `  A )  <_ 
sum_ ( F `  B ) (rpnnen2lem7 12447).

Putting it all together, if we have two sets  x  =/=  y, there must differ somewhere, and so there must be an  m such that  A. n  < 
m ( n  e.  x  <->  n  e.  y
) but  m  e.  ( x  \  y ) or vice versa. In this case, we split off the first  m  -  1 terms (rpnnen2lem8 12448) and cancel them (rpnnen2lem10 12450), since these are the same for both sets. For the remaining terms, we use the subset property to establish that  sum_ ( F `
 y )  <_  sum_ ( F `  ( NN  \  { m }
) ) and  sum_ ( F `
 { m }
)  <_  sum_ ( F `
 x ) (where these sums are only over  ( ZZ>= `  m
)), and since  sum_ ( F `
 ( NN  \  { m } ) )  =  ( 3 ^ -u m )  /  2 (rpnnen2lem9 12449) and  sum_ ( F `  { m } )  =  ( 3 ^
-u m ), we establish that  sum_ ( F `
 y )  <  sum_ ( F `  x
) (rpnnen2lem11 12451) so that they must be different. By contraposition, we find that this map is an injection. (Contributed by Mario Carneiro, 13-May-2013.) (Proof shortened by Mario Carneiro, 30-Apr-2014.)

 |-  F  =  ( x  e.  ~P NN  |->  ( n  e.  NN  |->  if ( n  e.  x ,  ( ( 1  / 
 3 ) ^ n ) ,  0 )
 ) )   =>    |- 
 ~P NN  ~<_  ( 0 [,] 1 )
 
Theoremrpnnen 12453 The cardinality of the continuum is the same as the powerset of  om. This is a stronger statement than ruc 12469, which only asserts that  RR is uncountable, i.e. has a cardinality larger than  om. The main proof is in two parts, rpnnen1 10300 and rpnnen2 12452, each showing an injection in one direction, and this last part uses sbth 6935 to prove that the sets are equinumerous. By constructing explicit injections, we avoid the use of AC. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 23-Aug-2014.)
 |- 
 RR  ~~  ~P NN
 
Theoremrexpen 12454 The real numbers are equinumerous to their own cross product, even though it is not necessarily true that  RR is well-orderable (so we cannot use infxpidm2 7598 directly). (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 16-Jun-2013.)
 |-  ( RR  X.  RR )  ~~  RR
 
Theoremcpnnen 12455 The complex numbers are equinumerous to the powerset of the natural numbers. (Contributed by Mario Carneiro, 16-Jun-2013.)
 |- 
 CC  ~~  ~P NN
 
TheoremrucALT 12456 The set of natural numbers is strictly dominated by the set of real numbers, i.e. the real numbers are uncountable. This proof is a simple corollary of rpnnen 12453, which determines the exact cardinality of the reals. For an alternate proof discussed at http://us.metamath.org/mpegif/mmcomplex.html#uncountable, see ruc 12469. (Contributed by NM, 13-Oct-2004.) (Revised by Mario Carneiro, 13-May-2013.)
 |- 
 NN  ~<  RR
 
Theoremruclem1 12457* Lemma for ruc 12469 (the reals are uncountable). Substitutions for the function  D. (Contributed by Mario Carneiro, 28-May-2014.) (Revised by Fan Zheng, 6-Jun-2016.)
 |-  ( ph  ->  F : NN --> RR )   &    |-  ( ph  ->  D  =  ( x  e.  ( RR 
 X.  RR ) ,  y  e.  RR  |->  [_ ( ( ( 1st `  x )  +  ( 2nd `  x ) )  /  2
 )  /  m ]_ if ( m  <  y , 
 <. ( 1st `  x ) ,  m >. , 
 <. ( ( m  +  ( 2nd `  x )
 )  /  2 ) ,  ( 2nd `  x ) >. ) ) )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  M  e.  RR )   &    |-  X  =  ( 1st `  ( <. A ,  B >. D M ) )   &    |-  Y  =  ( 2nd `  ( <. A ,  B >. D M ) )   =>    |-  ( ph  ->  ( ( <. A ,  B >. D M )  e.  ( RR  X.  RR )  /\  X  =  if (
 ( ( A  +  B )  /  2
 )  <  M ,  A ,  ( (
 ( ( A  +  B )  /  2
 )  +  B ) 
 /  2 ) ) 
 /\  Y  =  if ( ( ( A  +  B )  / 
 2 )  <  M ,  ( ( A  +  B )  /  2
 ) ,  B ) ) )
 
Theoremruclem2 12458* Lemma for ruc 12469. Ordering property for the input to  D. (Contributed by Mario Carneiro, 28-May-2014.)
 |-  ( ph  ->  F : NN --> RR )   &    |-  ( ph  ->  D  =  ( x  e.  ( RR 
 X.  RR ) ,  y  e.  RR  |->  [_ ( ( ( 1st `  x )  +  ( 2nd `  x ) )  /  2
 )  /  m ]_ if ( m  <  y , 
 <. ( 1st `  x ) ,  m >. , 
 <. ( ( m  +  ( 2nd `  x )
 )  /  2 ) ,  ( 2nd `  x ) >. ) ) )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  M  e.  RR )   &    |-  X  =  ( 1st `  ( <. A ,  B >. D M ) )   &    |-  Y  =  ( 2nd `  ( <. A ,  B >. D M ) )   &    |-  ( ph  ->  A  <  B )   =>    |-  ( ph  ->  ( A  <_  X  /\  X  <  Y  /\  Y  <_  B ) )
 
Theoremruclem3 12459* Lemma for ruc 12469. The constructed interval  [ X ,  Y ] always excludes  M. (Contributed by Mario Carneiro, 28-May-2014.)
 |-  ( ph  ->  F : NN --> RR )   &    |-  ( ph  ->  D  =  ( x  e.  ( RR 
 X.  RR ) ,  y  e.  RR  |->  [_ ( ( ( 1st `  x )  +  ( 2nd `  x ) )  /  2
 )  /  m ]_ if ( m  <  y , 
 <. ( 1st `  x ) ,  m >. , 
 <. ( ( m  +  ( 2nd `  x )
 )  /  2 ) ,  ( 2nd `  x ) >. ) ) )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  M  e.  RR )   &    |-  X  =  ( 1st `  ( <. A ,  B >. D M ) )   &    |-  Y  =  ( 2nd `  ( <. A ,  B >. D M ) )   &    |-  ( ph  ->  A  <  B )   =>    |-  ( ph  ->  ( M  <  X  \/  Y  <  M ) )
 
Theoremruclem4 12460* Lemma for ruc 12469. Initial value of the interval sequence. (Contributed by Mario Carneiro, 28-May-2014.)
 |-  ( ph  ->  F : NN --> RR )   &    |-  ( ph  ->  D  =  ( x  e.  ( RR 
 X.  RR ) ,  y  e.  RR  |->  [_ ( ( ( 1st `  x )  +  ( 2nd `  x ) )  /  2
 )  /  m ]_ if ( m  <  y , 
 <. ( 1st `  x ) ,  m >. , 
 <. ( ( m  +  ( 2nd `  x )
 )  /  2 ) ,  ( 2nd `  x ) >. ) ) )   &    |-  C  =  ( { <. 0 ,  <. 0 ,  1 >. >. }  u.  F )   &    |-  G  =  seq  0 ( D ,  C )   =>    |-  ( ph  ->  ( G `  0 )  = 
 <. 0 ,  1 >.
 )
 
Theoremruclem6 12461* Lemma for ruc 12469. Domain and range of the interval sequence. (Contributed by Mario Carneiro, 28-May-2014.)
 |-  ( ph  ->  F : NN --> RR )   &    |-  ( ph  ->  D  =  ( x  e.  ( RR 
 X.  RR ) ,  y  e.  RR  |->  [_ ( ( ( 1st `  x )  +  ( 2nd `  x ) )  /  2
 )  /  m ]_ if ( m  <  y , 
 <. ( 1st `  x ) ,  m >. , 
 <. ( ( m  +  ( 2nd `  x )
 )  /  2 ) ,  ( 2nd `  x ) >. ) ) )   &    |-  C  =  ( { <. 0 ,  <. 0 ,  1 >. >. }  u.  F )   &    |-  G  =  seq  0 ( D ,  C )   =>    |-  ( ph  ->  G : NN0 --> ( RR  X.  RR ) )
 
Theoremruclem7 12462* Lemma for ruc 12469. Successor value for the interval sequence. (Contributed by Mario Carneiro, 28-May-2014.)
 |-  ( ph  ->  F : NN --> RR )   &    |-  ( ph  ->  D  =  ( x  e.  ( RR 
 X.  RR ) ,  y  e.  RR  |->  [_ ( ( ( 1st `  x )  +  ( 2nd `  x ) )  /  2
 )  /  m ]_ if ( m  <  y , 
 <. ( 1st `  x ) ,  m >. , 
 <. ( ( m  +  ( 2nd `  x )
 )  /  2 ) ,  ( 2nd `  x ) >. ) ) )   &    |-  C  =  ( { <. 0 ,  <. 0 ,  1 >. >. }  u.  F )   &    |-  G  =  seq  0 ( D ,  C )   =>    |-  ( ( ph  /\  N  e.  NN0 )  ->  ( G `  ( N  +  1 ) )  =  ( ( G `  N ) D ( F `  ( N  +  1 ) ) ) )
 
Theoremruclem8 12463* Lemma for ruc 12469. The intervals of the  G sequence are all nonempty. (Contributed by Mario Carneiro, 28-May-2014.)
 |-  ( ph  ->  F : NN --> RR )   &    |-  ( ph  ->  D  =  ( x  e.  ( RR 
 X.  RR ) ,  y  e.  RR  |->  [_ ( ( ( 1st `  x )  +  ( 2nd `  x ) )  /  2
 )  /  m ]_ if ( m  <  y , 
 <. ( 1st `  x ) ,  m >. , 
 <. ( ( m  +  ( 2nd `  x )
 )  /  2 ) ,  ( 2nd `  x ) >. ) ) )   &    |-  C  =  ( { <. 0 ,  <. 0 ,  1 >. >. }  u.  F )   &    |-  G  =  seq  0 ( D ,  C )   =>    |-  ( ( ph  /\  N  e.  NN0 )  ->  ( 1st `  ( G `  N ) )  < 
 ( 2nd `  ( G `  N ) ) )
 
Theoremruclem9 12464* Lemma for ruc 12469. The first components of the  G sequence are increasing, and the second components are decreasing. (Contributed by Mario Carneiro, 28-May-2014.)
 |-  ( ph  ->  F : NN --> RR )   &    |-  ( ph  ->  D  =  ( x  e.  ( RR 
 X.  RR ) ,  y  e.  RR  |->  [_ ( ( ( 1st `  x )  +  ( 2nd `  x ) )  /  2
 )  /  m ]_ if ( m  <  y , 
 <. ( 1st `  x ) ,  m >. , 
 <. ( ( m  +  ( 2nd `  x )
 )  /  2 ) ,  ( 2nd `  x ) >. ) ) )   &    |-  C  =  ( { <. 0 ,  <. 0 ,  1 >. >. }  u.  F )   &    |-  G  =  seq  0 ( D ,  C )   &    |-  ( ph  ->  M  e.  NN0 )   &    |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )   =>    |-  ( ph  ->  (
 ( 1st `  ( G `  M ) )  <_  ( 1st `  ( G `  N ) )  /\  ( 2nd `  ( G `  N ) )  <_  ( 2nd `  ( G `  M ) ) ) )
 
Theoremruclem10 12465* Lemma for ruc 12469. Every first component of the  G sequence is less than every second component. That is, the sequences form a chain a1  < a2 
<...  < b2  < b1, where ai are the first components and bi are the second components. (Contributed by Mario Carneiro, 28-May-2014.)
 |-  ( ph  ->  F : NN --> RR )   &    |-  ( ph  ->  D  =  ( x  e.  ( RR 
 X.  RR ) ,  y  e.  RR  |->  [_ ( ( ( 1st `  x )  +  ( 2nd `  x ) )  /  2
 )  /  m ]_ if ( m  <  y , 
 <. ( 1st `  x ) ,  m >. , 
 <. ( ( m  +  ( 2nd `  x )
 )  /  2 ) ,  ( 2nd `  x ) >. ) ) )   &    |-  C  =  ( { <. 0 ,  <. 0 ,  1 >. >. }  u.  F )   &    |-  G  =  seq  0 ( D ,  C )   &    |-  ( ph  ->  M  e.  NN0 )   &    |-  ( ph  ->  N  e.  NN0 )   =>    |-  ( ph  ->  ( 1st `  ( G `  M ) )  < 
 ( 2nd `  ( G `  N ) ) )
 
Theoremruclem11 12466* Lemma for ruc 12469. Closure lemmas for supremum. (Contributed by Mario Carneiro, 28-May-2014.)
 |-  ( ph  ->  F : NN --> RR )   &    |-  ( ph  ->  D  =  ( x  e.  ( RR 
 X.  RR ) ,  y  e.  RR  |->  [_ ( ( ( 1st `  x )  +  ( 2nd `  x ) )  /  2
 )  /  m ]_ if ( m  <  y , 
 <. ( 1st `  x ) ,  m >. , 
 <. ( ( m  +  ( 2nd `  x )
 )  /  2 ) ,  ( 2nd `  x ) >. ) ) )   &    |-  C  =  ( { <. 0 ,  <. 0 ,  1 >. >. }  u.  F )   &    |-  G  =  seq  0 ( D ,  C )   =>    |-  ( ph  ->  ( ran  ( 1st  o.  G )  C_  RR  /\  ran  ( 1st  o.  G )  =/=  (/)  /\  A. z  e. 
 ran  ( 1st  o.  G ) z  <_ 
 1 ) )
 
Theoremruclem12 12467* Lemma for ruc 12469. The supremum of the increasing sequence  1st  o.  G is a real number that is not in the range of  F. (Contributed by Mario Carneiro, 28-May-2014.)
 |-  ( ph  ->  F : NN --> RR )   &    |-  ( ph  ->  D  =  ( x  e.  ( RR 
 X.  RR ) ,  y  e.  RR  |->  [_ ( ( ( 1st `  x )  +  ( 2nd `  x ) )  /  2
 )  /  m ]_ if ( m  <  y , 
 <. ( 1st `  x ) ,  m >. , 
 <. ( ( m  +  ( 2nd `  x )
 )  /  2 ) ,  ( 2nd `  x ) >. ) ) )   &    |-  C  =  ( { <. 0 ,  <. 0 ,  1 >. >. }  u.  F )   &    |-  G  =  seq  0 ( D ,  C )   &    |-  S  =  sup ( ran  ( 1st  o.  G ) ,  RR ,  <  )   =>    |-  ( ph  ->  S  e.  ( RR  \  ran  F ) )
 
Theoremruclem13 12468 Lemma for ruc 12469. There is no function that maps  NN onto  RR. (Use nex 1587 if you want this in the form  -.  E. f
f : NN -onto-> RR.) (Contributed by NM, 14-Oct-2004.) (Proof shortened by Fan Zheng, 6-Jun-2016.)
 |- 
 -.  F : NN -onto-> RR
 
Theoremruc 12469 The set of natural numbers is strictly dominated by the set of real numbers, i.e. the real numbers are uncountable. The proof consists of lemmas ruclem1 12457 through ruclem13 12468 and this final piece. Our proof is based on the proof of Theorem 5.18 of [Truss] p. 114. See ruclem13 12468 for the function existence version of this theorem. For an informal discussion of this proof, see http://us.metamath.org/mpegif/mmcomplex.html#uncountable. For an alternate proof see rucALT 12456. (Contributed by NM, 13-Oct-2004.) (Proof modification is discouraged.)
 |- 
 NN  ~<  RR
 
Theoremresdomq 12470 The set of rationals is strictly less equinumerous than the set of reals ( RR strictly dominates  QQ). (Contributed by NM, 18-Dec-2004.)
 |- 
 QQ  ~<  RR
 
Theoremaleph1re 12471 There are at least aleph-one real numbers. (Contributed by NM, 2-Feb-2005.)
 |-  ( aleph `  1o )  ~<_  RR
 
Theoremaleph1irr 12472 There are at least aleph-one irrationals. (Contributed by NM, 2-Feb-2005.)
 |-  ( aleph `  1o )  ~<_  ( RR  \  QQ )
 
Theoremcnso 12473 The complex numbers can be linearly ordered. (Contributed by Stefan O'Rear, 16-Nov-2014.)
 |- 
 E. x  x  Or  CC
 
PART 6  ELEMENTARY NUMBER THEORY

Here we introduce elementary number theory, in particular the elementary properties of divisibility and elementary prime number theory.

 
6.1  Elementary properties of divisibility
 
6.1.1  Irrationality of square root of 2
 
Theoremsqr2irrlem 12474 Lemma for irrationality of square root of 2. The core of the proof - if  A  /  B  =  sqr ( 2 ), then 
A and  B are even, so  A  /  2 and  B  /  2 are smaller representatives, which is absurd by the method of infinite descent (here implemented by strong induction). (Contributed by NM, 20-Aug-2001.) (Revised by Mario Carneiro, 12-Sep-2015.)
 |-  ( ph  ->  A  e.  ZZ )   &    |-  ( ph  ->  B  e.  NN )   &    |-  ( ph  ->  ( sqr `  2
 )  =  ( A 
 /  B ) )   =>    |-  ( ph  ->  ( ( A  /  2 )  e. 
 ZZ  /\  ( B  /  2 )  e.  NN ) )
 
Theoremsqr2irr 12475 The square root of 2 is irrational. See zsqrelqelz 12777 for a generalization to all non-square integers. The proof's core is proven in sqr2irrlem 12474, which shows that if  A  /  B  =  sqr ( 2 ), then  A and  B are even, so  A  /  2 and  B  /  2 are smaller representatives, which is absurd. An older version of this proof was included in The Seventeen Provers of the World compiled by Freek Wiedijk. It is also the first "top 100" mathematical theorems whose formalization is tracked by Freek Wiedijk on his Formalizing 100 Theorems page at http://www.cs.ru.nl/~freek/100/. (Contributed by NM, 8-Jan-2002.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)
 |-  ( sqr `  2
 )  e/  QQ
 
Theoremsqr2re 12476 The square root of 2 exists and is a real number. (Contributed by NM, 3-Dec-2004.)
 |-  ( sqr `  2
 )  e.  RR
 
6.1.2  Some Number sets are chains of proper subsets
 
Theoremnthruc 12477 The sequence  NN,  ZZ,  QQ,  RR, and  CC forms a chain of proper subsets. In each case the proper subset relationship is shown by demonstrating a number that belongs to one set but not the other. We show that zero belongs to  ZZ but not  NN, one-half belongs to  QQ but not  ZZ, the square root of 2 belongs to  RR but not  QQ, and finally that the imaginary number  _i belongs to  CC but not  RR. See nthruz 12478 for a further refinement. (Contributed by NM, 12-Jan-2002.)
 |-  ( ( NN  C.  ZZ  /\  ZZ  C.  QQ )  /\  ( QQ  C.  RR  /\  RR  C.  CC ) )
 
Theoremnthruz 12478 The sequence  NN,  NN0, and  ZZ forms a chain of proper subsets. In each case the proper subset relationship is shown by demonstrating a number that belongs to one set but not the other. We show that zero belongs to 
NN0 but not  NN and minus one belongs to  ZZ but not  NN0. This theorem refines the chain of proper subsets nthruc 12477. (Contributed by NM, 9-May-2004.)
 |-  ( NN  C.  NN0  /\ 
 NN0  C.  ZZ )
 
6.1.3  The divides relation
 
Syntaxcdivides 12479 Extend the definition of a class to include the divides relation. See df-divides 12480.
 class  ||
 
Definitiondf-divides 12480* Define the divides relation. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ||  =  { <. x ,  y >.  |  ( ( x  e.  ZZ  /\  y  e.  ZZ )  /\  E. n  e.  ZZ  ( n  x.  x )  =  y ) }
 
Theoremdivides 12481* Define the divides relation.  M  ||  N means  M divides into  N with no remainder. For example,  3  ||  6 (ex-dvds 20764). As proven in divides3 12483, 
M  ||  N  <->  ( N  mod  M )  =  0. See divides 12481 and divides2 12482 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 )
 )
 
Theoremdivides2 12482 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 ) )
 
Theoremdivides3 12483 One nonzero integer divides another integer if and only if the remainder upon division is zero. (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 12484 Reverse closure for the divisibility relation. (Contributed by Stefan O'Rear, 5-Sep-2015.)
 |-  ( X  ||  Y  ->  ( X  e.  ZZ  /\  Y  e.  ZZ )
 )
 
Theoremnndivdivides 12485 Strong form of divides2 12482 for natural numbers. (Contributed by Stefan O'Rear, 13-Sep-2014.)
 |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( B  ||  A 
 <->  ( A  /  B )  e.  NN )
 )
 
Theoremmoddvds 12486 Two ways to say  A  ==  B
(  mod  N ). (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 ) ) )
 
Theoremdvds0lem 12487 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 12488* 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 12489* 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 12490 An integer divides itself. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( N  e.  ZZ  ->  N  ||  N )
 
Theorem1dvds 12491 1 divides any integer. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( N  e.  ZZ  ->  1  ||  N )
 
Theoremdvds0 12492 Any integer divides 0. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( N  e.  ZZ  ->  N  ||  0 )
 
Theoremnegdvdsb 12493 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 12494 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 12495 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 12496 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 12497 Only 0 is divisible by 0 . (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( N  e.  ZZ  ->  ( 0  ||  N  <->  N  =  0 ) )
 
Theoremdvdsmul1 12498 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 12499 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 12500 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 ) )
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