HomeHome Metamath Proof Explorer
Theorem List (p. 124 of 309)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-21328)
  Hilbert Space Explorer  Hilbert Space Explorer
(21329-22851)
  Users' Mathboxes  Users' Mathboxes
(22852-30843)
 

Theorem List for Metamath Proof Explorer - 12301-12400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcosneg 12301 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 ) )
 
Theoremtanneg 12302 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 12303 Value of the sine function at 0. (Contributed by Steve Rodriguez, 14-Mar-2005.)
 |-  ( sin `  0
 )  =  0
 
Theoremcos0 12304 Value of the cosine function at 0. (Contributed by NM, 30-Apr-2005.)
 |-  ( cos `  0
 )  =  1
 
Theoremtan0 12305 The value of the tangent function at zero is zero. (Contributed by David A. Wheeler, 16-Mar-2014.)
 |-  ( tan `  0
 )  =  0
 
Theoremefival 12306 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 12307 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 ) ) ) )
 
Theoremsinhval 12308 Value of the hyperbolic sine of a complex number. (Contributed by Mario Carneiro, 4-Apr-2015.)
 |-  ( A  e.  CC  ->  ( ( sin `  ( _i  x.  A ) ) 
 /  _i )  =  ( ( ( exp `  A )  -  ( exp `  -u A ) ) 
 /  2 ) )
 
Theoremcoshval 12309 Value of the hyperbolic cosine of a complex number. (Contributed by Mario Carneiro, 4-Apr-2015.)
 |-  ( A  e.  CC  ->  ( cos `  ( _i  x.  A ) )  =  ( ( ( exp `  A )  +  ( exp `  -u A ) )  /  2
 ) )
 
Theoremresinhcl 12310 The hyperbolic sine of a real number is real. (Contributed by Mario Carneiro, 4-Apr-2015.)
 |-  ( A  e.  RR  ->  ( ( sin `  ( _i  x.  A ) ) 
 /  _i )  e. 
 RR )
 
Theoremrpcoshcl 12311 The hyperbolic cosine of a real number is a positive real. (Contributed by Mario Carneiro, 4-Apr-2015.)
 |-  ( A  e.  RR  ->  ( cos `  ( _i  x.  A ) )  e.  RR+ )
 
Theoremrecoshcl 12312 The hyperbolic cosine of a real number is real. (Contributed by Mario Carneiro, 4-Apr-2015.)
 |-  ( A  e.  RR  ->  ( cos `  ( _i  x.  A ) )  e.  RR )
 
Theoremretanhcl 12313 The hyperbolic tangent of a real number is real. (Contributed by Mario Carneiro, 4-Apr-2015.)
 |-  ( A  e.  RR  ->  ( ( tan `  ( _i  x.  A ) ) 
 /  _i )  e. 
 RR )
 
Theoremtanhlt1 12314 The hyperbolic tangent of a real number is upper bounded by  1. (Contributed by Mario Carneiro, 4-Apr-2015.)
 |-  ( A  e.  RR  ->  ( ( tan `  ( _i  x.  A ) ) 
 /  _i )  < 
 1 )
 
Theoremtanhbnd 12315 The hyperbolic tangent of a real number is bounded by  1. (Contributed by Mario Carneiro, 4-Apr-2015.)
 |-  ( A  e.  RR  ->  ( ( tan `  ( _i  x.  A ) ) 
 /  _i )  e.  ( -u 1 (,) 1
 ) )
 
Theoremefeul 12316 Eulerian representation of the complex exponential. (Suggested by Jeffrey 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 12317 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 12318 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 12319 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 ) ) ) )
 
Theoremtanaddlem 12320 A useful intermediate step in tanadd 12321 when showing that the addition of tangents is well-defined. (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 )  x.  ( tan `  B ) )  =/=  1 ) )
 
Theoremtanadd 12321 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 12322 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 12323 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 12324 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 12325 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 12326 Product of sines can be rewritten as half the difference of certain cosines. This follows from cosadd 12319 and cossub 12323. (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 12327 Product of cosines can be rewritten as half the sum of certain cosines. This follows from cosadd 12319 and cossub 12323. (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 12328 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 12329 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 12330 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 12331 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 12332 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 12333 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 12334 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 12335 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 12336 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 12337 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 12338* Lemma for sin01bnd 12339 and cos01bnd 12340. (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 12339 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 12340 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 12341 Bounds on the cosine of 1. (Contributed by Paul Chapman, 19-Jan-2008.)
 |-  ( ( 1  / 
 3 )  <  ( cos `  1 )  /\  ( cos `  1 )  <  ( 2  /  3
 ) )
 
Theoremcos2bnd 12342 Bounds on the cosine of 2. (Contributed by Paul Chapman, 19-Jan-2008.)
 |-  ( -u ( 7  / 
 9 )  <  ( cos `  2 )  /\  ( cos `  2 )  < 
 -u ( 1  / 
 9 ) )
 
Theoremsinltx 12343 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 12344 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 12345 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 12346 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 12347 The signs of the sine and cosine of 1. (Contributed by Paul Chapman, 19-Jan-2008.)
 |-  ( 0  <  ( sin `  1 )  /\  0  <  ( cos `  1
 ) )
 
Theoremsincos2sgn 12348 The signs of the sine and cosine of 2. (Contributed by Paul Chapman, 19-Jan-2008.)
 |-  ( 0  <  ( sin `  2 )  /\  ( cos `  2 )  <  0 )
 
Theoremsin4lt0 12349 The sine of 4 is negative. (Contributed by Paul Chapman, 19-Jan-2008.)
 |-  ( sin `  4
 )  <  0
 
Theoremabsefi 12350 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 12351 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 12352 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 12353 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 12354 De Moivre's Formula. Shorter proof of demoivreALT 12355 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 12355 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 12356* Lemma for eirr 12357. (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 12357  _e is irrational. (Contributed by Paul Chapman, 9-Feb-2008.) (Proof shortened by Mario Carneiro, 29-Apr-2014.)
 |-  _e  e/  QQ
 
Theoremegt2lt3 12358 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 12359 Euler's constant  _e is greater than 0. (Contributed by Jeff Hankins, 22-Nov-2008.)
 |-  0  <  _e
 
Theoremepr 12360 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 12361 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 12362 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 11165 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 12363 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 6866 in xpnnen 12361). (Contributed by NM, 23-Jul-2004.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( om  X.  om )  ~~  om
 
Theoremznnenlem 12364 Lemma for znnen 12365. (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 12365 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 12366 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 12367* Lemma for rpnnen2 12378. (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 12368* Lemma for rpnnen2 12378. (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 12369* Lemma for rpnnen2 12378. (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 12370* Lemma for rpnnen2 12378. (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 12371* Lemma for rpnnen2 12378. (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 12372* Lemma for rpnnen2 12378. (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 12373* Lemma for rpnnen2 12378. (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 12374* Lemma for rpnnen2 12378. (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 12375* Lemma for rpnnen2 12378. (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 12376* Lemma for rpnnen2 12378. (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 12377* Lemma for rpnnen2 12378. (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 12378* The other half of rpnnen 12379, 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... 12211). 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 12367). This is an infinite sum of real numbers (rpnnen2lem2 12368), and since  A 
C_  B implies  ( F `  A )  <_  ( F `  B ) (rpnnen2lem4 12370) and  ( F `  NN ) converges to  1  /  2 (rpnnen2lem3 12369) by geoisum1 12209, the sum is convergent to some real (rpnnen2lem5 12371 and rpnnen2lem6 12372) by the comparison test for convergence cvgcmp 12151. The comparison test also tells us that  A  C_  B implies  sum_ ( F `  A )  <_ 
sum_ ( F `  B ) (rpnnen2lem7 12373).

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 12374) and cancel them (rpnnen2lem10 12376), 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 12375) and  sum_ ( F `  { m } )  =  ( 3 ^
-u m ), we establish that  sum_ ( F `
 y )  <  sum_ ( F `  x
) (rpnnen2lem11 12377) 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 12379 The cardinality of the continuum is the same as the powerset of  om. This is a stronger statement than ruc 12395, which only asserts that  RR is uncountable, i.e. has a cardinality larger than  om. The main proof is in two parts, rpnnen1 10226 and rpnnen2 12378, each showing an injection in one direction, and this last part uses sbth 6866 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 12380 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 7528 directly). (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 16-Jun-2013.)
 |-  ( RR  X.  RR )  ~~  RR
 
Theoremcpnnen 12381 The complex numbers are equinumerous to the powerset of the natural numbers. (Contributed by Mario Carneiro, 16-Jun-2013.)
 |- 
 CC  ~~  ~P NN
 
TheoremrucALT 12382 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 12379, which determines the exact cardinality of the reals. For an alternate proof discussed at http://us.metamath.org/mpegif/mmcomplex.html#uncountable, see ruc 12395. (Contributed by NM, 13-Oct-2004.) (Revised by Mario Carneiro, 13-May-2013.)
 |- 
 NN  ~<  RR
 
Theoremruclem1 12383* Lemma for ruc 12395 (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 12384* Lemma for ruc 12395. 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 12385* Lemma for ruc 12395. 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 12386* Lemma for ruc 12395. 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 12387* Lemma for ruc 12395. 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 12388* Lemma for ruc 12395. 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 12389* Lemma for ruc 12395. 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 12390* Lemma for ruc 12395. 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 12391* Lemma for ruc 12395. 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 12392* Lemma for ruc 12395. 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 12393* Lemma for ruc 12395. 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 12394 Lemma for ruc 12395. 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 12395 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 12383 through ruclem13 12394 and this final piece. Our proof is based on the proof of Theorem 5.18 of [Truss] p. 114. See ruclem13 12394 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 12382. (Contributed by NM, 13-Oct-2004.) (Proof modification is discouraged.)
 |- 
 NN  ~<  RR
 
Theoremresdomq 12396 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 12397 There are at least aleph-one real numbers. (Contributed by NM, 2-Feb-2005.)
 |-  ( aleph `  1o )  ~<_  RR
 
Theoremaleph1irr 12398 There are at least aleph-one irrationals. (Contributed by NM, 2-Feb-2005.)
 |-  ( aleph `  1o )  ~<_  ( RR  \  QQ )
 
Theoremcnso 12399 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 12400 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 ) )
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30843
  Copyright terms: Public domain < Previous  Next >