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Theorem List for Intuitionistic Logic Explorer - 8601-8700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremuzin 8601 Intersection of two upper intervals of integers. (Contributed by Mario Carneiro, 24-Dec-2013.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( ZZ>= `  M )  i^i  ( ZZ>= `  N ) )  =  ( ZZ>= `  if ( M  <_  N ,  N ,  M ) ) )
 
Theoremuzp1 8602 Choices for an element of an upper interval of integers. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( N  e.  ( ZZ>=
 `  M )  ->  ( N  =  M  \/  N  e.  ( ZZ>= `  ( M  +  1
 ) ) ) )
 
Theoremnn0uz 8603 Nonnegative integers expressed as an upper set of integers. (Contributed by NM, 2-Sep-2005.)
 |- 
 NN0  =  ( ZZ>= `  0 )
 
Theoremnnuz 8604 Positive integers expressed as an upper set of integers. (Contributed by NM, 2-Sep-2005.)
 |- 
 NN  =  ( ZZ>= `  1 )
 
Theoremelnnuz 8605 A positive integer expressed as a member of an upper set of integers. (Contributed by NM, 6-Jun-2006.)
 |-  ( N  e.  NN  <->  N  e.  ( ZZ>= `  1 )
 )
 
Theoremelnn0uz 8606 A nonnegative integer expressed as a member an upper set of integers. (Contributed by NM, 6-Jun-2006.)
 |-  ( N  e.  NN0  <->  N  e.  ( ZZ>= `  0 )
 )
 
Theoremeluz2nn 8607 An integer is greater than or equal to 2 is a positive integer. (Contributed by AV, 3-Nov-2018.)
 |-  ( A  e.  ( ZZ>=
 `  2 )  ->  A  e.  NN )
 
Theoremeluzge2nn0 8608 If an integer is greater than or equal to 2, then it is a nonnegative integer. (Contributed by AV, 27-Aug-2018.) (Proof shortened by AV, 3-Nov-2018.)
 |-  ( N  e.  ( ZZ>=
 `  2 )  ->  N  e.  NN0 )
 
Theoremuzuzle23 8609 An integer in the upper set of integers starting at 3 is element of the upper set of integers starting at 2. (Contributed by Alexander van der Vekens, 17-Sep-2018.)
 |-  ( A  e.  ( ZZ>=
 `  3 )  ->  A  e.  ( ZZ>= `  2 ) )
 
Theoremeluzge3nn 8610 If an integer is greater than 3, then it is a positive integer. (Contributed by Alexander van der Vekens, 17-Sep-2018.)
 |-  ( N  e.  ( ZZ>=
 `  3 )  ->  N  e.  NN )
 
Theoremuz3m2nn 8611 An integer greater than or equal to 3 decreased by 2 is a positive integer. (Contributed by Alexander van der Vekens, 17-Sep-2018.)
 |-  ( N  e.  ( ZZ>=
 `  3 )  ->  ( N  -  2
 )  e.  NN )
 
Theorem1eluzge0 8612 1 is an integer greater than or equal to 0. (Contributed by Alexander van der Vekens, 8-Jun-2018.)
 |-  1  e.  ( ZZ>= `  0 )
 
Theorem2eluzge0 8613 2 is an integer greater than or equal to 0. (Contributed by Alexander van der Vekens, 8-Jun-2018.) (Proof shortened by OpenAI, 25-Mar-2020.)
 |-  2  e.  ( ZZ>= `  0 )
 
Theorem2eluzge0OLD 8614 2 is an integer greater than or equal to 0. (Contributed by Alexander van der Vekens, 8-Jun-2018.) Obsolete version of 2eluzge0 8613 as of 25-Mar-2020. (New usage is discouraged.) (Proof modification is discouraged.)
 |-  2  e.  ( ZZ>= `  0 )
 
Theorem2eluzge1 8615 2 is an integer greater than or equal to 1. (Contributed by Alexander van der Vekens, 8-Jun-2018.)
 |-  2  e.  ( ZZ>= `  1 )
 
Theoremuznnssnn 8616 The upper integers starting from a natural are a subset of the naturals. (Contributed by Scott Fenton, 29-Jun-2013.)
 |-  ( N  e.  NN  ->  ( ZZ>= `  N )  C_ 
 NN )
 
Theoremraluz 8617* Restricted universal quantification in an upper set of integers. (Contributed by NM, 9-Sep-2005.)
 |-  ( M  e.  ZZ  ->  ( A. n  e.  ( ZZ>= `  M ) ph 
 <-> 
 A. n  e.  ZZ  ( M  <_  n  ->  ph ) ) )
 
Theoremraluz2 8618* Restricted universal quantification in an upper set of integers. (Contributed by NM, 9-Sep-2005.)
 |-  ( A. n  e.  ( ZZ>= `  M ) ph 
 <->  ( M  e.  ZZ  ->  A. n  e.  ZZ  ( M  <_  n  ->  ph ) ) )
 
Theoremrexuz 8619* Restricted existential quantification in an upper set of integers. (Contributed by NM, 9-Sep-2005.)
 |-  ( M  e.  ZZ  ->  ( E. n  e.  ( ZZ>= `  M ) ph 
 <-> 
 E. n  e.  ZZ  ( M  <_  n  /\  ph ) ) )
 
Theoremrexuz2 8620* Restricted existential quantification in an upper set of integers. (Contributed by NM, 9-Sep-2005.)
 |-  ( E. n  e.  ( ZZ>= `  M ) ph 
 <->  ( M  e.  ZZ  /\ 
 E. n  e.  ZZ  ( M  <_  n  /\  ph ) ) )
 
Theorem2rexuz 8621* Double existential quantification in an upper set of integers. (Contributed by NM, 3-Nov-2005.)
 |-  ( E. m E. n  e.  ( ZZ>= `  m ) ph  <->  E. m  e.  ZZ  E. n  e.  ZZ  ( m  <_  n  /\  ph )
 )
 
Theorempeano2uz 8622 Second Peano postulate for an upper set of integers. (Contributed by NM, 7-Sep-2005.)
 |-  ( N  e.  ( ZZ>=
 `  M )  ->  ( N  +  1
 )  e.  ( ZZ>= `  M ) )
 
Theorempeano2uzs 8623 Second Peano postulate for an upper set of integers. (Contributed by Mario Carneiro, 26-Dec-2013.)
 |-  Z  =  ( ZZ>= `  M )   =>    |-  ( N  e.  Z  ->  ( N  +  1 )  e.  Z )
 
Theorempeano2uzr 8624 Reversed second Peano axiom for upper integers. (Contributed by NM, 2-Jan-2006.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ( ZZ>=
 `  ( M  +  1 ) ) ) 
 ->  N  e.  ( ZZ>= `  M ) )
 
Theoremuzaddcl 8625 Addition closure law for an upper set of integers. (Contributed by NM, 4-Jun-2006.)
 |-  ( ( N  e.  ( ZZ>= `  M )  /\  K  e.  NN0 )  ->  ( N  +  K )  e.  ( ZZ>= `  M ) )
 
Theoremnn0pzuz 8626 The sum of a nonnegative integer and an integer is an integer greater than or equal to that integer. (Contributed by Alexander van der Vekens, 3-Oct-2018.)
 |-  ( ( N  e.  NN0  /\  Z  e.  ZZ )  ->  ( N  +  Z )  e.  ( ZZ>= `  Z ) )
 
Theoremuzind4 8627* Induction on the upper set of integers that starts at an integer  M. The first four hypotheses give us the substitution instances we need, and the last two are the basis and the induction step. (Contributed by NM, 7-Sep-2005.)
 |-  ( j  =  M  ->  ( ph  <->  ps ) )   &    |-  (
 j  =  k  ->  ( ph  <->  ch ) )   &    |-  (
 j  =  ( k  +  1 )  ->  ( ph  <->  th ) )   &    |-  (
 j  =  N  ->  (
 ph 
 <->  ta ) )   &    |-  ( M  e.  ZZ  ->  ps )   &    |-  ( k  e.  ( ZZ>= `  M )  ->  ( ch  ->  th )
 )   =>    |-  ( N  e.  ( ZZ>=
 `  M )  ->  ta )
 
Theoremuzind4ALT 8628* Induction on the upper set of integers that starts at an integer  M. The last four hypotheses give us the substitution instances we need; the first two are the basis and the induction step. Either uzind4 8627 or uzind4ALT 8628 may be used; see comment for nnind 8006. (Contributed by NM, 7-Sep-2005.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( M  e.  ZZ  ->  ps )   &    |-  ( k  e.  ( ZZ>= `  M )  ->  ( ch  ->  th )
 )   &    |-  ( j  =  M  ->  ( ph  <->  ps ) )   &    |-  (
 j  =  k  ->  ( ph  <->  ch ) )   &    |-  (
 j  =  ( k  +  1 )  ->  ( ph  <->  th ) )   &    |-  (
 j  =  N  ->  (
 ph 
 <->  ta ) )   =>    |-  ( N  e.  ( ZZ>= `  M )  ->  ta )
 
Theoremuzind4s 8629* Induction on the upper set of integers that starts at an integer  M, using explicit substitution. The hypotheses are the basis and the induction step. (Contributed by NM, 4-Nov-2005.)
 |-  ( M  e.  ZZ  -> 
 [. M  /  k ]. ph )   &    |-  ( k  e.  ( ZZ>= `  M )  ->  ( ph  ->  [. (
 k  +  1 ) 
 /  k ]. ph )
 )   =>    |-  ( N  e.  ( ZZ>=
 `  M )  ->  [. N  /  k ]. ph )
 
Theoremuzind4s2 8630* Induction on the upper set of integers that starts at an integer  M, using explicit substitution. The hypotheses are the basis and the induction step. Use this instead of uzind4s 8629 when  j and  k must be distinct in  [. ( k  +  1 )  /  j ]. ph. (Contributed by NM, 16-Nov-2005.)
 |-  ( M  e.  ZZ  -> 
 [. M  /  j ]. ph )   &    |-  ( k  e.  ( ZZ>= `  M )  ->  ( [. k  /  j ]. ph  ->  [. (
 k  +  1 ) 
 /  j ]. ph )
 )   =>    |-  ( N  e.  ( ZZ>=
 `  M )  ->  [. N  /  j ]. ph )
 
Theoremuzind4i 8631* Induction on the upper integers that start at  M. The first hypothesis specifies the lower bound, the next four give us the substitution instances we need, and the last two are the basis and the induction step. (Contributed by NM, 4-Sep-2005.)
 |-  M  e.  ZZ   &    |-  (
 j  =  M  ->  (
 ph 
 <->  ps ) )   &    |-  (
 j  =  k  ->  ( ph  <->  ch ) )   &    |-  (
 j  =  ( k  +  1 )  ->  ( ph  <->  th ) )   &    |-  (
 j  =  N  ->  (
 ph 
 <->  ta ) )   &    |-  ps   &    |-  (
 k  e.  ( ZZ>= `  M )  ->  ( ch 
 ->  th ) )   =>    |-  ( N  e.  ( ZZ>= `  M )  ->  ta )
 
Theoremindstr 8632* Strong Mathematical Induction for positive integers (inference schema). (Contributed by NM, 17-Aug-2001.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   &    |-  ( x  e.  NN  ->  (
 A. y  e.  NN  ( y  <  x  ->  ps )  ->  ph )
 )   =>    |-  ( x  e.  NN  -> 
 ph )
 
Theoremeluznn0 8633 Membership in a nonnegative upper set of integers implies membership in  NN0. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N ) )  ->  M  e.  NN0 )
 
Theoremeluznn 8634 Membership in a positive upper set of integers implies membership in  NN. (Contributed by JJ, 1-Oct-2018.)
 |-  ( ( N  e.  NN  /\  M  e.  ( ZZ>=
 `  N ) ) 
 ->  M  e.  NN )
 
Theoremeluz2b1 8635 Two ways to say "an integer greater than or equal to 2." (Contributed by Paul Chapman, 23-Nov-2012.)
 |-  ( N  e.  ( ZZ>=
 `  2 )  <->  ( N  e.  ZZ  /\  1  <  N ) )
 
Theoremeluz2gt1 8636 An integer greater than or equal to 2 is greater than 1. (Contributed by AV, 24-May-2020.)
 |-  ( N  e.  ( ZZ>=
 `  2 )  -> 
 1  <  N )
 
Theoremeluz2b2 8637 Two ways to say "an integer greater than or equal to 2." (Contributed by Paul Chapman, 23-Nov-2012.)
 |-  ( N  e.  ( ZZ>=
 `  2 )  <->  ( N  e.  NN  /\  1  <  N ) )
 
Theoremeluz2b3 8638 Two ways to say "an integer greater than or equal to 2." (Contributed by Paul Chapman, 23-Nov-2012.)
 |-  ( N  e.  ( ZZ>=
 `  2 )  <->  ( N  e.  NN  /\  N  =/=  1
 ) )
 
Theoremuz2m1nn 8639 One less than an integer greater than or equal to 2 is a positive integer. (Contributed by Paul Chapman, 17-Nov-2012.)
 |-  ( N  e.  ( ZZ>=
 `  2 )  ->  ( N  -  1
 )  e.  NN )
 
Theorem1nuz2 8640 1 is not in  ( ZZ>= `  2
). (Contributed by Paul Chapman, 21-Nov-2012.)
 |- 
 -.  1  e.  ( ZZ>=
 `  2 )
 
Theoremelnn1uz2 8641 A positive integer is either 1 or greater than or equal to 2. (Contributed by Paul Chapman, 17-Nov-2012.)
 |-  ( N  e.  NN  <->  ( N  =  1  \/  N  e.  ( ZZ>= `  2 ) ) )
 
Theoremuz2mulcl 8642 Closure of multiplication of integers greater than or equal to 2. (Contributed by Paul Chapman, 26-Oct-2012.)
 |-  ( ( M  e.  ( ZZ>= `  2 )  /\  N  e.  ( ZZ>= `  2 ) )  ->  ( M  x.  N )  e.  ( ZZ>= `  2 ) )
 
Theoremindstr2 8643* Strong Mathematical Induction for positive integers (inference schema). The first two hypotheses give us the substitution instances we need; the last two are the basis and the induction step. (Contributed by Paul Chapman, 21-Nov-2012.)
 |-  ( x  =  1 
 ->  ( ph  <->  ch ) )   &    |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   &    |-  ch   &    |-  ( x  e.  ( ZZ>= `  2 )  ->  ( A. y  e.  NN  (
 y  <  x  ->  ps )  ->  ph ) )   =>    |-  ( x  e.  NN  -> 
 ph )
 
Theoremeluzdc 8644 Membership of an integer in an upper set of integers is decidable. (Contributed by Jim Kingdon, 18-Apr-2020.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  -> DECID  N  e.  ( ZZ>= `  M ) )
 
Theoremublbneg 8645* The image under negation of a bounded-above set of reals is bounded below. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( E. x  e. 
 RR  A. y  e.  A  y  <_  x  ->  E. x  e.  RR  A. y  e. 
 { z  e.  RR  |  -u z  e.  A } x  <_  y )
 
Theoremeqreznegel 8646* Two ways to express the image under negation of a set of integers. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( A  C_  ZZ  ->  { z  e.  RR  |  -u z  e.  A }  =  { z  e.  ZZ  |  -u z  e.  A } )
 
Theoremnegm 8647* The image under negation of an inhabited set of reals is inhabited. (Contributed by Jim Kingdon, 10-Apr-2020.)
 |-  ( ( A  C_  RR  /\  E. x  x  e.  A )  ->  E. y  y  e.  { z  e.  RR  |  -u z  e.  A }
 )
 
Theoremlbzbi 8648* If a set of reals is bounded below, it is bounded below by an integer. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( A  C_  RR  ->  ( E. x  e. 
 RR  A. y  e.  A  x  <_  y  <->  E. x  e.  ZZ  A. y  e.  A  x  <_  y ) )
 
Theoremnn01to3 8649 A (nonnegative) integer between 1 and 3 must be 1, 2 or 3. (Contributed by Alexander van der Vekens, 13-Sep-2018.)
 |-  ( ( N  e.  NN0  /\  1  <_  N  /\  N  <_  3 )  ->  ( N  =  1  \/  N  =  2  \/  N  =  3 ) )
 
Theoremnn0ge2m1nnALT 8650 Alternate proof of nn0ge2m1nn 8299: If a nonnegative integer is greater than or equal to two, the integer decreased by 1 is a positive integer. This version is proved using eluz2 8575, a theorem for upper sets of integers, which are defined later than the positive and nonnegative integers. This proof is, however, much shorter than the proof of nn0ge2m1nn 8299. (Contributed by Alexander van der Vekens, 1-Aug-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( ( N  e.  NN0  /\  2  <_  N ) 
 ->  ( N  -  1
 )  e.  NN )
 
3.4.11  Rational numbers (as a subset of complex numbers)
 
Syntaxcq 8651 Extend class notation to include the class of rationals.
 class  QQ
 
Definitiondf-q 8652 Define the set of rational numbers. Based on definition of rationals in [Apostol] p. 22. See elq 8654 for the relation "is rational." (Contributed by NM, 8-Jan-2002.)
 |- 
 QQ  =  (  /  " ( ZZ  X.  NN ) )
 
Theoremdivfnzn 8653 Division restricted to  ZZ  X.  NN is a function. Given excluded middle, it would be easy to prove this for  CC 
X.  ( CC  \  { 0 } ). The key difference is that an element of  NN is apart from zero, whereas being an element of 
CC  \  { 0 } implies being not equal to zero. (Contributed by Jim Kingdon, 19-Mar-2020.)
 |-  (  /  |`  ( ZZ 
 X.  NN ) )  Fn  ( ZZ  X.  NN )
 
Theoremelq 8654* Membership in the set of rationals. (Contributed by NM, 8-Jan-2002.) (Revised by Mario Carneiro, 28-Jan-2014.)
 |-  ( A  e.  QQ  <->  E. x  e.  ZZ  E. y  e.  NN  A  =  ( x  /  y ) )
 
Theoremqmulz 8655* If  A is rational, then some integer multiple of it is an integer. (Contributed by NM, 7-Nov-2008.) (Revised by Mario Carneiro, 22-Jul-2014.)
 |-  ( A  e.  QQ  ->  E. x  e.  NN  ( A  x.  x )  e.  ZZ )
 
Theoremznq 8656 The ratio of an integer and a positive integer is a rational number. (Contributed by NM, 12-Jan-2002.)
 |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( A  /  B )  e.  QQ )
 
Theoremqre 8657 A rational number is a real number. (Contributed by NM, 14-Nov-2002.)
 |-  ( A  e.  QQ  ->  A  e.  RR )
 
Theoremzq 8658 An integer is a rational number. (Contributed by NM, 9-Jan-2002.)
 |-  ( A  e.  ZZ  ->  A  e.  QQ )
 
Theoremzssq 8659 The integers are a subset of the rationals. (Contributed by NM, 9-Jan-2002.)
 |- 
 ZZ  C_  QQ
 
Theoremnn0ssq 8660 The nonnegative integers are a subset of the rationals. (Contributed by NM, 31-Jul-2004.)
 |- 
 NN0  C_  QQ
 
Theoremnnssq 8661 The positive integers are a subset of the rationals. (Contributed by NM, 31-Jul-2004.)
 |- 
 NN  C_  QQ
 
Theoremqssre 8662 The rationals are a subset of the reals. (Contributed by NM, 9-Jan-2002.)
 |- 
 QQ  C_  RR
 
Theoremqsscn 8663 The rationals are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.)
 |- 
 QQ  C_  CC
 
Theoremqex 8664 The set of rational numbers exists. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)
 |- 
 QQ  e.  _V
 
Theoremnnq 8665 A positive integer is rational. (Contributed by NM, 17-Nov-2004.)
 |-  ( A  e.  NN  ->  A  e.  QQ )
 
Theoremqcn 8666 A rational number is a complex number. (Contributed by NM, 2-Aug-2004.)
 |-  ( A  e.  QQ  ->  A  e.  CC )
 
Theoremqaddcl 8667 Closure of addition of rationals. (Contributed by NM, 1-Aug-2004.)
 |-  ( ( A  e.  QQ  /\  B  e.  QQ )  ->  ( A  +  B )  e.  QQ )
 
Theoremqnegcl 8668 Closure law for the negative of a rational. (Contributed by NM, 2-Aug-2004.) (Revised by Mario Carneiro, 15-Sep-2014.)
 |-  ( A  e.  QQ  -> 
 -u A  e.  QQ )
 
Theoremqmulcl 8669 Closure of multiplication of rationals. (Contributed by NM, 1-Aug-2004.)
 |-  ( ( A  e.  QQ  /\  B  e.  QQ )  ->  ( A  x.  B )  e.  QQ )
 
Theoremqsubcl 8670 Closure of subtraction of rationals. (Contributed by NM, 2-Aug-2004.)
 |-  ( ( A  e.  QQ  /\  B  e.  QQ )  ->  ( A  -  B )  e.  QQ )
 
Theoremqapne 8671 Apartness is equivalent to not equal for rationals. (Contributed by Jim Kingdon, 20-Mar-2020.)
 |-  ( ( A  e.  QQ  /\  B  e.  QQ )  ->  ( A #  B  <->  A  =/=  B ) )
 
Theoremqltlen 8672 Rational 'Less than' expressed in terms of 'less than or equal to'. Also see ltleap 7695 which is a similar result for real numbers. (Contributed by Jim Kingdon, 11-Oct-2021.)
 |-  ( ( A  e.  QQ  /\  B  e.  QQ )  ->  ( A  <  B  <-> 
 ( A  <_  B  /\  B  =/=  A ) ) )
 
Theoremqlttri2 8673 Apartness is equivalent to not equal for rationals. (Contributed by Jim Kingdon, 9-Nov-2021.)
 |-  ( ( A  e.  QQ  /\  B  e.  QQ )  ->  ( A  =/=  B  <-> 
 ( A  <  B  \/  B  <  A ) ) )
 
Theoremqreccl 8674 Closure of reciprocal of rationals. (Contributed by NM, 3-Aug-2004.)
 |-  ( ( A  e.  QQ  /\  A  =/=  0
 )  ->  ( 1  /  A )  e.  QQ )
 
Theoremqdivcl 8675 Closure of division of rationals. (Contributed by NM, 3-Aug-2004.)
 |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  B  =/=  0 ) 
 ->  ( A  /  B )  e.  QQ )
 
Theoremqrevaddcl 8676 Reverse closure law for addition of rationals. (Contributed by NM, 2-Aug-2004.)
 |-  ( B  e.  QQ  ->  ( ( A  e.  CC  /\  ( A  +  B )  e.  QQ ) 
 <->  A  e.  QQ )
 )
 
Theoremnnrecq 8677 The reciprocal of a positive integer is rational. (Contributed by NM, 17-Nov-2004.)
 |-  ( A  e.  NN  ->  ( 1  /  A )  e.  QQ )
 
Theoremirradd 8678 The sum of an irrational number and a rational number is irrational. (Contributed by NM, 7-Nov-2008.)
 |-  ( ( A  e.  ( RR  \  QQ )  /\  B  e.  QQ )  ->  ( A  +  B )  e.  ( RR  \  QQ ) )
 
Theoremirrmul 8679 The product of a real which is not rational with a nonzero rational is not rational. Note that by "not rational" we mean the negation of "is rational" (whereas "irrational" is often defined to mean apart from any rational number - given excluded middle these two definitions would be equivalent). (Contributed by NM, 7-Nov-2008.)
 |-  ( ( A  e.  ( RR  \  QQ )  /\  B  e.  QQ  /\  B  =/=  0 )  ->  ( A  x.  B )  e.  ( RR  \  QQ ) )
 
3.4.12  Complex numbers as pairs of reals
 
Theoremcnref1o 8680* There is a natural one-to-one mapping from  ( RR  X.  RR ) to  CC, where we map  <. x ,  y
>. to  ( x  +  ( _i  x.  y ) ). In our construction of the complex numbers, this is in fact our definition of  CC (see df-c 6953), but in the axiomatic treatment we can only show that there is the expected mapping between these two sets. (Contributed by Mario Carneiro, 16-Jun-2013.) (Revised by Mario Carneiro, 17-Feb-2014.)
 |-  F  =  ( x  e.  RR ,  y  e.  RR  |->  ( x  +  ( _i  x.  y
 ) ) )   =>    |-  F : ( RR  X.  RR ) -1-1-onto-> CC
 
3.5  Order sets
 
3.5.1  Positive reals (as a subset of complex numbers)
 
Syntaxcrp 8681 Extend class notation to include the class of positive reals.
 class  RR+
 
Definitiondf-rp 8682 Define the set of positive reals. Definition of positive numbers in [Apostol] p. 20. (Contributed by NM, 27-Oct-2007.)
 |-  RR+  =  { x  e. 
 RR  |  0  < 
 x }
 
Theoremelrp 8683 Membership in the set of positive reals. (Contributed by NM, 27-Oct-2007.)
 |-  ( A  e.  RR+  <->  ( A  e.  RR  /\  0  <  A ) )
 
Theoremelrpii 8684 Membership in the set of positive reals. (Contributed by NM, 23-Feb-2008.)
 |-  A  e.  RR   &    |-  0  <  A   =>    |-  A  e.  RR+
 
Theorem1rp 8685 1 is a positive real. (Contributed by Jeff Hankins, 23-Nov-2008.)
 |-  1  e.  RR+
 
Theorem2rp 8686 2 is a positive real. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  2  e.  RR+
 
Theoremrpre 8687 A positive real is a real. (Contributed by NM, 27-Oct-2007.)
 |-  ( A  e.  RR+  ->  A  e.  RR )
 
Theoremrpxr 8688 A positive real is an extended real. (Contributed by Mario Carneiro, 21-Aug-2015.)
 |-  ( A  e.  RR+  ->  A  e.  RR* )
 
Theoremrpcn 8689 A positive real is a complex number. (Contributed by NM, 11-Nov-2008.)
 |-  ( A  e.  RR+  ->  A  e.  CC )
 
Theoremnnrp 8690 A positive integer is a positive real. (Contributed by NM, 28-Nov-2008.)
 |-  ( A  e.  NN  ->  A  e.  RR+ )
 
Theoremrpssre 8691 The positive reals are a subset of the reals. (Contributed by NM, 24-Feb-2008.)
 |-  RR+  C_  RR
 
Theoremrpgt0 8692 A positive real is greater than zero. (Contributed by FL, 27-Dec-2007.)
 |-  ( A  e.  RR+  -> 
 0  <  A )
 
Theoremrpge0 8693 A positive real is greater than or equal to zero. (Contributed by NM, 22-Feb-2008.)
 |-  ( A  e.  RR+  -> 
 0  <_  A )
 
Theoremrpregt0 8694 A positive real is a positive real number. (Contributed by NM, 11-Nov-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
 |-  ( A  e.  RR+  ->  ( A  e.  RR  /\  0  <  A ) )
 
Theoremrprege0 8695 A positive real is a nonnegative real number. (Contributed by Mario Carneiro, 31-Jan-2014.)
 |-  ( A  e.  RR+  ->  ( A  e.  RR  /\  0  <_  A )
 )
 
Theoremrpne0 8696 A positive real is nonzero. (Contributed by NM, 18-Jul-2008.)
 |-  ( A  e.  RR+  ->  A  =/=  0 )
 
Theoremrpap0 8697 A positive real is apart from zero. (Contributed by Jim Kingdon, 22-Mar-2020.)
 |-  ( A  e.  RR+  ->  A #  0 )
 
Theoremrprene0 8698 A positive real is a nonzero real number. (Contributed by NM, 11-Nov-2008.)
 |-  ( A  e.  RR+  ->  ( A  e.  RR  /\  A  =/=  0 ) )
 
Theoremrpreap0 8699 A positive real is a real number apart from zero. (Contributed by Jim Kingdon, 22-Mar-2020.)
 |-  ( A  e.  RR+  ->  ( A  e.  RR  /\  A #  0 ) )
 
Theoremrpcnne0 8700 A positive real is a nonzero complex number. (Contributed by NM, 11-Nov-2008.)
 |-  ( A  e.  RR+  ->  ( A  e.  CC  /\  A  =/=  0 ) )
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