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Theorem List for Intuitionistic Logic Explorer - 9001-9100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremeluz 9001 Membership in an upper set of integers. (Contributed by NM, 2-Oct-2005.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( N  e.  ( ZZ>= `  M )  <->  M 
 <_  N ) )
 
Theoremuzid 9002 Membership of the least member in an upper set of integers. (Contributed by NM, 2-Sep-2005.)
 |-  ( M  e.  ZZ  ->  M  e.  ( ZZ>= `  M ) )
 
Theoremuzn0 9003 The upper integers are all nonempty. (Contributed by Mario Carneiro, 16-Jan-2014.)
 |-  ( M  e.  ran  ZZ>=  ->  M  =/=  (/) )
 
Theoremuztrn 9004 Transitive law for sets of upper integers. (Contributed by NM, 20-Sep-2005.)
 |-  ( ( M  e.  ( ZZ>= `  K )  /\  K  e.  ( ZZ>= `  N ) )  ->  M  e.  ( ZZ>= `  N ) )
 
Theoremuztrn2 9005 Transitive law for sets of upper integers. (Contributed by Mario Carneiro, 26-Dec-2013.)
 |-  Z  =  ( ZZ>= `  K )   =>    |-  ( ( N  e.  Z  /\  M  e.  ( ZZ>=
 `  N ) ) 
 ->  M  e.  Z )
 
Theoremuzneg 9006 Contraposition law for upper integers. (Contributed by NM, 28-Nov-2005.)
 |-  ( N  e.  ( ZZ>=
 `  M )  ->  -u M  e.  ( ZZ>= `  -u N ) )
 
Theoremuzssz 9007 An upper set of integers is a subset of all integers. (Contributed by NM, 2-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)
 |-  ( ZZ>= `  M )  C_ 
 ZZ
 
Theoremuzss 9008 Subset relationship for two sets of upper integers. (Contributed by NM, 5-Sep-2005.)
 |-  ( N  e.  ( ZZ>=
 `  M )  ->  ( ZZ>= `  N )  C_  ( ZZ>= `  M )
 )
 
Theoremuztric 9009 Trichotomy of the ordering relation on integers, stated in terms of upper integers. (Contributed by NM, 6-Jul-2005.) (Revised by Mario Carneiro, 25-Jun-2013.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( N  e.  ( ZZ>= `  M )  \/  M  e.  ( ZZ>= `  N ) ) )
 
Theoremuz11 9010 The upper integers function is one-to-one. (Contributed by NM, 12-Dec-2005.)
 |-  ( M  e.  ZZ  ->  ( ( ZZ>= `  M )  =  ( ZZ>= `  N )  <->  M  =  N ) )
 
Theoremeluzp1m1 9011 Membership in the next upper set of integers. (Contributed by NM, 12-Sep-2005.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ( ZZ>=
 `  ( M  +  1 ) ) ) 
 ->  ( N  -  1
 )  e.  ( ZZ>= `  M ) )
 
Theoremeluzp1l 9012 Strict ordering implied by membership in the next upper set of integers. (Contributed by NM, 12-Sep-2005.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ( ZZ>=
 `  ( M  +  1 ) ) ) 
 ->  M  <  N )
 
Theoremeluzp1p1 9013 Membership in the next upper set of integers. (Contributed by NM, 5-Oct-2005.)
 |-  ( N  e.  ( ZZ>=
 `  M )  ->  ( N  +  1
 )  e.  ( ZZ>= `  ( M  +  1
 ) ) )
 
Theoremeluzaddi 9014 Membership in a later upper set of integers. (Contributed by Paul Chapman, 22-Nov-2007.)
 |-  M  e.  ZZ   &    |-  K  e.  ZZ   =>    |-  ( N  e.  ( ZZ>=
 `  M )  ->  ( N  +  K )  e.  ( ZZ>= `  ( M  +  K ) ) )
 
Theoremeluzsubi 9015 Membership in an earlier upper set of integers. (Contributed by Paul Chapman, 22-Nov-2007.)
 |-  M  e.  ZZ   &    |-  K  e.  ZZ   =>    |-  ( N  e.  ( ZZ>=
 `  ( M  +  K ) )  ->  ( N  -  K )  e.  ( ZZ>= `  M ) )
 
Theoremeluzadd 9016 Membership in a later upper set of integers. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( ( N  e.  ( ZZ>= `  M )  /\  K  e.  ZZ )  ->  ( N  +  K )  e.  ( ZZ>= `  ( M  +  K ) ) )
 
Theoremeluzsub 9017 Membership in an earlier upper set of integers. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( ( M  e.  ZZ  /\  K  e.  ZZ  /\  N  e.  ( ZZ>= `  ( M  +  K ) ) )  ->  ( N  -  K )  e.  ( ZZ>= `  M ) )
 
Theoremuzm1 9018 Choices for an element of an upper interval of integers. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( N  e.  ( ZZ>=
 `  M )  ->  ( N  =  M  \/  ( N  -  1
 )  e.  ( ZZ>= `  M ) ) )
 
Theoremuznn0sub 9019 The nonnegative difference of integers is a nonnegative integer. (Contributed by NM, 4-Sep-2005.)
 |-  ( N  e.  ( ZZ>=
 `  M )  ->  ( N  -  M )  e.  NN0 )
 
Theoremuzin 9020 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 9021 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 9022 Nonnegative integers expressed as an upper set of integers. (Contributed by NM, 2-Sep-2005.)
 |- 
 NN0  =  ( ZZ>= `  0 )
 
Theoremnnuz 9023 Positive integers expressed as an upper set of integers. (Contributed by NM, 2-Sep-2005.)
 |- 
 NN  =  ( ZZ>= `  1 )
 
Theoremelnnuz 9024 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 9025 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 9026 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 9027 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 9028 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 9029 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 9030 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 9031 1 is an integer greater than or equal to 0. (Contributed by Alexander van der Vekens, 8-Jun-2018.)
 |-  1  e.  ( ZZ>= `  0 )
 
Theorem2eluzge0 9032 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 )
 
Theorem2eluzge1 9033 2 is an integer greater than or equal to 1. (Contributed by Alexander van der Vekens, 8-Jun-2018.)
 |-  2  e.  ( ZZ>= `  1 )
 
Theoremuznnssnn 9034 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 9035* 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 9036* 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 9037* 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 9038* 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 9039* 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 9040 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 9041 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 9042 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 9043 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 9044 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 9045* 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 9046* 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 9045 or uzind4ALT 9046 may be used; see comment for nnind 8410. (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 9047* 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 9048* 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 9047 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 9049* 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 9050* 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 )
 
Theoreminfrenegsupex 9051* The infimum of a set of reals  A is the negative of the supremum of the negatives of its elements. (Contributed by Jim Kingdon, 14-Jan-2022.)
 |-  ( ph  ->  E. x  e.  RR  ( A. y  e.  A  -.  y  < 
 x  /\  A. y  e. 
 RR  ( x  < 
 y  ->  E. z  e.  A  z  <  y
 ) ) )   &    |-  ( ph  ->  A  C_  RR )   =>    |-  ( ph  -> inf ( A ,  RR ,  <  )  =  -u sup ( {
 z  e.  RR  |  -u z  e.  A } ,  RR ,  <  )
 )
 
Theoremsupinfneg 9052* If a set of real numbers has a least upper bound, the set of the negation of those numbers has a greatest lower bound. For a theorem which is similar but only for the boundedness part, see ublbneg 9067. (Contributed by Jim Kingdon, 15-Jan-2022.)
 |-  ( ph  ->  E. x  e.  RR  ( A. y  e.  A  -.  x  < 
 y  /\  A. y  e. 
 RR  ( y  < 
 x  ->  E. z  e.  A  y  <  z
 ) ) )   &    |-  ( ph  ->  A  C_  RR )   =>    |-  ( ph  ->  E. x  e.  RR  ( A. y  e.  { w  e.  RR  |  -u w  e.  A }  -.  y  <  x  /\  A. y  e.  RR  ( x  <  y  ->  E. z  e.  { w  e.  RR  |  -u w  e.  A } z  < 
 y ) ) )
 
Theoreminfsupneg 9053* If a set of real numbers has a greatest lower bound, the set of the negation of those numbers has a least upper bound. To go in the other direction see supinfneg 9052. (Contributed by Jim Kingdon, 15-Jan-2022.)
 |-  ( ph  ->  E. x  e.  RR  ( A. y  e.  A  -.  y  < 
 x  /\  A. y  e. 
 RR  ( x  < 
 y  ->  E. z  e.  A  z  <  y
 ) ) )   &    |-  ( ph  ->  A  C_  RR )   =>    |-  ( ph  ->  E. x  e.  RR  ( A. y  e.  { w  e.  RR  |  -u w  e.  A }  -.  x  <  y  /\  A. y  e.  RR  ( y  <  x  ->  E. z  e.  { w  e.  RR  |  -u w  e.  A } y  < 
 z ) ) )
 
Theoremsupminfex 9054* A supremum is the negation of the infimum of that set's image under negation. (Contributed by Jim Kingdon, 14-Jan-2022.)
 |-  ( ph  ->  E. x  e.  RR  ( A. y  e.  A  -.  x  < 
 y  /\  A. y  e. 
 RR  ( y  < 
 x  ->  E. z  e.  A  y  <  z
 ) ) )   &    |-  ( ph  ->  A  C_  RR )   =>    |-  ( ph  ->  sup ( A ,  RR ,  <  )  =  -uinf ( { w  e.  RR  |  -u w  e.  A } ,  RR ,  <  ) )
 
Theoremeluznn0 9055 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 9056 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 9057 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 9058 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 9059 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 9060 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 9061 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 9062 1 is not in  ( ZZ>= `  2
). (Contributed by Paul Chapman, 21-Nov-2012.)
 |- 
 -.  1  e.  ( ZZ>=
 `  2 )
 
Theoremelnn1uz2 9063 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 9064 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 9065* 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 9066 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 9067* The image under negation of a bounded-above set of reals is bounded below. For a theorem which is similar but also adds that the bounds need to be the tightest possible, see supinfneg 9052. (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 9068* 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 9069* 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 9070* 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 9071 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 9072 Alternate proof of nn0ge2m1nn 8703: 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 8994, 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 8703. (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 )
 
3.4.12  Rational numbers (as a subset of complex numbers)
 
Syntaxcq 9073 Extend class notation to include the class of rationals.
 class  QQ
 
Definitiondf-q 9074 Define the set of rational numbers. Based on definition of rationals in [Apostol] p. 22. See elq 9076 for the relation "is rational." (Contributed by NM, 8-Jan-2002.)
 |- 
 QQ  =  (  /  " ( ZZ  X.  NN ) )
 
Theoremdivfnzn 9075 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 9076* 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 9077* 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 9078 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 9079 A rational number is a real number. (Contributed by NM, 14-Nov-2002.)
 |-  ( A  e.  QQ  ->  A  e.  RR )
 
Theoremzq 9080 An integer is a rational number. (Contributed by NM, 9-Jan-2002.)
 |-  ( A  e.  ZZ  ->  A  e.  QQ )
 
Theoremzssq 9081 The integers are a subset of the rationals. (Contributed by NM, 9-Jan-2002.)
 |- 
 ZZ  C_  QQ
 
Theoremnn0ssq 9082 The nonnegative integers are a subset of the rationals. (Contributed by NM, 31-Jul-2004.)
 |- 
 NN0  C_  QQ
 
Theoremnnssq 9083 The positive integers are a subset of the rationals. (Contributed by NM, 31-Jul-2004.)
 |- 
 NN  C_  QQ
 
Theoremqssre 9084 The rationals are a subset of the reals. (Contributed by NM, 9-Jan-2002.)
 |- 
 QQ  C_  RR
 
Theoremqsscn 9085 The rationals are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.)
 |- 
 QQ  C_  CC
 
Theoremqex 9086 The set of rational numbers exists. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)
 |- 
 QQ  e.  _V
 
Theoremnnq 9087 A positive integer is rational. (Contributed by NM, 17-Nov-2004.)
 |-  ( A  e.  NN  ->  A  e.  QQ )
 
Theoremqcn 9088 A rational number is a complex number. (Contributed by NM, 2-Aug-2004.)
 |-  ( A  e.  QQ  ->  A  e.  CC )
 
Theoremqaddcl 9089 Closure of addition of rationals. (Contributed by NM, 1-Aug-2004.)
 |-  ( ( A  e.  QQ  /\  B  e.  QQ )  ->  ( A  +  B )  e.  QQ )
 
Theoremqnegcl 9090 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 9091 Closure of multiplication of rationals. (Contributed by NM, 1-Aug-2004.)
 |-  ( ( A  e.  QQ  /\  B  e.  QQ )  ->  ( A  x.  B )  e.  QQ )
 
Theoremqsubcl 9092 Closure of subtraction of rationals. (Contributed by NM, 2-Aug-2004.)
 |-  ( ( A  e.  QQ  /\  B  e.  QQ )  ->  ( A  -  B )  e.  QQ )
 
Theoremqapne 9093 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 9094 Rational 'Less than' expressed in terms of 'less than or equal to'. Also see ltleap 8083 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 9095 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 9096 Closure of reciprocal of rationals. (Contributed by NM, 3-Aug-2004.)
 |-  ( ( A  e.  QQ  /\  A  =/=  0
 )  ->  ( 1  /  A )  e.  QQ )
 
Theoremqdivcl 9097 Closure of division of rationals. (Contributed by NM, 3-Aug-2004.)
 |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  B  =/=  0 ) 
 ->  ( A  /  B )  e.  QQ )
 
Theoremqrevaddcl 9098 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 9099 The reciprocal of a positive integer is rational. (Contributed by NM, 17-Nov-2004.)
 |-  ( A  e.  NN  ->  ( 1  /  A )  e.  QQ )
 
Theoremirradd 9100 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 ) )
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