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Theorem List for Metamath Proof Explorer - 11201-11300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremuzneg 11201 Contraposition law for upper integers. (Contributed by NM, 28-Nov-2005.)
 |-  ( N  e.  ( ZZ>=
 `  M )  ->  -u M  e.  ( ZZ>= `  -u N ) )
 
Theoremuzssz 11202 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 11203 Subset relationship for two sets of upper integers. (Contributed by NM, 5-Sep-2005.)
 |-  ( N  e.  ( ZZ>=
 `  M )  ->  ( ZZ>= `  N )  C_  ( ZZ>= `  M )
 )
 
Theoremuztric 11204 Totality 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 11205 The upper integers function is one-to-one. (Contributed by NM, 12-Dec-2005.)
 |-  ( M  e.  ZZ  ->  ( ( ZZ>= `  M )  =  ( ZZ>= `  N )  <->  M  =  N ) )
 
Theoremeluzp1m1 11206 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 11207 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 11208 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 11209 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 11210 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 11211 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 11212 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 11213 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 11214 The nonnegative difference of integers is a nonnegative integer. (Contributed by NM, 4-Sep-2005.)
 |-  ( N  e.  ( ZZ>=
 `  M )  ->  ( N  -  M )  e.  NN0 )
 
Theoremuzin 11215 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 11216 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 11217 Nonnegative integers expressed as an upper set of integers. (Contributed by NM, 2-Sep-2005.)
 |- 
 NN0  =  ( ZZ>= `  0 )
 
Theoremnnuz 11218 Positive integers expressed as an upper set of integers. (Contributed by NM, 2-Sep-2005.)
 |- 
 NN  =  ( ZZ>= `  1 )
 
Theoremelnnuz 11219 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 11220 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 11221 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 11222 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 11223 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 11224 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 11225 An integer greater than or equal to 3 decreased by 2 is a positive integer, analogous to uz2m1nn 11256. (Contributed by Alexander van der Vekens, 17-Sep-2018.)
 |-  ( N  e.  ( ZZ>=
 `  3 )  ->  ( N  -  2
 )  e.  NN )
 
Theorem1eluzge0 11226 1 is an integer greater than or equal to 0. (Contributed by Alexander van der Vekens, 8-Jun-2018.)
 |-  1  e.  ( ZZ>= `  0 )
 
Theorem2eluzge0 11227 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 11228 2 is an integer greater than or equal to 1. (Contributed by Alexander van der Vekens, 8-Jun-2018.)
 |-  2  e.  ( ZZ>= `  1 )
 
Theoremuznnssnn 11229 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 11230* 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 11231* 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 11232* 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 11233* 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 11234* 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 11235 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 11236 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 11237 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 11238 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 11239 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 11240* 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 11241* 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 11240 or uzind4ALT 11241 may be used; see comment for nnind 10649. (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 11242* 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 11243* 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 11242 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 11244* 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 )
 
Theoremuzwo 11245* Well-ordering principle: any nonempty subset of an upper set of integers has the least element. (Contributed by NM, 8-Oct-2005.)
 |-  ( ( S  C_  ( ZZ>= `  M )  /\  S  =/=  (/) )  ->  E. j  e.  S  A. k  e.  S  j 
 <_  k )
 
Theoremuzwo2 11246* Well-ordering principle: any nonempty subset of an upper set of integers has a unique least element. (Contributed by NM, 8-Oct-2005.)
 |-  ( ( S  C_  ( ZZ>= `  M )  /\  S  =/=  (/) )  ->  E! j  e.  S  A. k  e.  S  j 
 <_  k )
 
Theoremnnwo 11247* Well-ordering principle: any nonempty set of positive integers has a least element. Theorem I.37 (well-ordering principle) of [Apostol] p. 34. (Contributed by NM, 17-Aug-2001.)
 |-  ( ( A  C_  NN  /\  A  =/=  (/) )  ->  E. x  e.  A  A. y  e.  A  x  <_  y )
 
Theoremnnwof 11248* Well-ordering principle: any nonempty set of positive integers has a least element. This version allows  x and  y to be present in  A as long as they are effectively not free. (Contributed by NM, 17-Aug-2001.) (Revised by Mario Carneiro, 15-Oct-2016.)
 |-  F/_ x A   &    |-  F/_ y A   =>    |-  ( ( A 
 C_  NN  /\  A  =/=  (/) )  ->  E. x  e.  A  A. y  e.  A  x  <_  y
 )
 
Theoremnnwos 11249* Well-ordering principle: any nonempty set of positive integers has a least element (schema form). (Contributed by NM, 17-Aug-2001.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   =>    |-  ( E. x  e.  NN  ph  ->  E. x  e.  NN  ( ph  /\  A. y  e.  NN  ( ps  ->  x  <_  y
 ) ) )
 
Theoremindstr 11250* 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 11251 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 11252 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 11253 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 ) )
 
Theoremeluz2b2 11254 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 11255 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 11256 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 11257 1 is not in  ( ZZ>= `  2
). (Contributed by Paul Chapman, 21-Nov-2012.)
 |- 
 -.  1  e.  ( ZZ>=
 `  2 )
 
Theoremelnn1uz2 11258 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 11259 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 11260* 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 )
 
Theoremuzinfi 11261 Extract the lower bound of an upper set of integers as its infimum. (Contributed by NM, 7-Oct-2005.) (Revised by AV, 4-Sep-2020.)
 |-  M  e.  ZZ   =>    |- inf ( ( ZZ>= `  M ) ,  RR ,  <  )  =  M
 
TheoremuzinfmiOLD 11262 Extract the lower bound of an upper set of integers as its infimum. Note that the " `'  < " argument turns supremum into infimum (for which we do not currently have a separate notation). (Contributed by NM, 7-Oct-2005.) Obsolete version of uzinfi 11261 as of 4-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.)
 |-  M  e.  ZZ   =>    |-  sup ( (
 ZZ>= `  M ) ,  RR ,  `'  <  )  =  M
 
Theoremnninf 11263 The infimum of the set of positive integers is one. (Contributed by NM, 16-Jun-2005.) (Revised by AV, 5-Sep-2020.)
 |- inf
 ( NN ,  RR ,  <  )  =  1
 
Theoremnn0inf 11264 The infimum of the set of nonnegative integers is zero. (Contributed by NM, 16-Jun-2005.) (Revised by AV, 5-Sep-2020.)
 |- inf
 ( NN0 ,  RR ,  <  )  =  0
 
TheoremnninfmOLD 11265 The infimum of the set of positive integers is one. Note that " `'  < " turns sup into inf. (Contributed by NM, 16-Jun-2005.) Obsolete version of nninf 11263 as of 5-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.)
 |- 
 sup ( NN ,  RR ,  `'  <  )  =  1
 
Theoremnn0infmOLD 11266 The infimum of the set of nonnegative integers is zero. Note that " `'  < " turns sup into inf. (Contributed by NM, 16-Jun-2005.) Obsolete version of nn0inf 11264 as of 5-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.)
 |- 
 sup ( NN0 ,  RR ,  `'  <  )  =  0
 
Theoreminfssuzle 11267 The infimum of a subset of an upper set of integers is less than or equal to all members of the subset. (Contributed by NM, 11-Oct-2005.) (Revised by AV, 5-Sep-2020.)
 |-  ( ( S  C_  ( ZZ>= `  M )  /\  A  e.  S ) 
 -> inf ( S ,  RR ,  <  )  <_  A )
 
Theoreminfssuzcl 11268 The infimum of a subset of an upper set of integers belongs to the subset. (Contributed by NM, 11-Oct-2005.) (Revised by AV, 5-Sep-2020.)
 |-  ( ( S  C_  ( ZZ>= `  M )  /\  S  =/=  (/) )  -> inf ( S ,  RR ,  <  )  e.  S )
 
TheoreminfmssuzleOLD 11269 The infimum of a subset of an upper set of integers is less than or equal to all members of the subset. Note that the " `'  < " argument turns supremum into infimum (for which we do not currently have a separate notation). (Contributed by NM, 11-Oct-2005.) Obsolete version of infssuzle 11267 as of 5-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( ( S  C_  ( ZZ>= `  M )  /\  A  e.  S ) 
 ->  sup ( S ,  RR ,  `'  <  )  <_  A )
 
TheoreminfmssuzclOLD 11270 The infimum of a subset of an upper set of integers belongs to the subset. (Contributed by NM, 11-Oct-2005.) Obsolete version of infssuzcl 11268 as of 5-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( ( S  C_  ( ZZ>= `  M )  /\  S  =/=  (/) )  ->  sup ( S ,  RR ,  `'  <  )  e.  S )
 
Theoremublbneg 11271* 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 11272* 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 } )
 
Theoremsupminf 11273* The supremum of a bounded-above set of reals is the negation of the supremum of that set's image under negation. (Contributed by Paul Chapman, 21-Mar-2011.) ( Revised by AV, 13-Sep-2020.)
 |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y  <_  x )  ->  sup ( A ,  RR ,  <  )  =  -uinf ( { z  e. 
 RR  |  -u z  e.  A } ,  RR ,  <  ) )
 
TheoremsupminfOLD 11274* The supremum of a bounded-above set of reals is the negation of the supremum of that set's image under negation. (Contributed by Paul Chapman, 21-Mar-2011.) Obsolete version of supminf 11273 as of 13-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y  <_  x )  ->  sup ( A ,  RR ,  <  )  =  -u sup ( { z  e.  RR  |  -u z  e.  A } ,  RR ,  `'  <  ) )
 
Theoremlbzbi 11275* 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 ) )
 
Theoremzsupss 11276* Any nonempty bounded subset of integers has a supremum in the set. (The proof does not use ax-pre-sup 9635.) (Contributed by Mario Carneiro, 21-Apr-2015.)
 |-  ( ( A  C_  ZZ  /\  A  =/=  (/)  /\  E. x  e.  ZZ  A. y  e.  A  y  <_  x )  ->  E. x  e.  A  ( A. y  e.  A  -.  x  <  y  /\  A. y  e.  B  ( y  <  x  ->  E. z  e.  A  y  <  z ) ) )
 
Theoremsuprzcl2 11277* The supremum of a bounded-above set of integers is a member of the set. (This version of suprzcl 11038 avoids ax-pre-sup 9635.) (Contributed by Mario Carneiro, 21-Apr-2015.) (Revised by Mario Carneiro, 24-Dec-2016.)
 |-  ( ( A  C_  ZZ  /\  A  =/=  (/)  /\  E. x  e.  ZZ  A. y  e.  A  y  <_  x )  ->  sup ( A ,  RR ,  <  )  e.  A )
 
Theoremsuprzub 11278* The supremum of a bounded-above set of integers is greater than any member of the set. (Contributed by Mario Carneiro, 21-Apr-2015.)
 |-  ( ( A  C_  ZZ  /\  E. x  e. 
 ZZ  A. y  e.  A  y  <_  x  /\  B  e.  A )  ->  B  <_  sup ( A ,  RR ,  <  ) )
 
Theoremuzsupss 11279* Any bounded subset of an upper set of integers has a supremum. (Contributed by Mario Carneiro, 22-Jul-2014.) (Revised by Mario Carneiro, 21-Apr-2015.)
 |-  Z  =  ( ZZ>= `  M )   =>    |-  ( ( M  e.  ZZ  /\  A  C_  Z  /\  E. x  e.  ZZ  A. y  e.  A  y 
 <_  x )  ->  E. x  e.  Z  ( A. y  e.  A  -.  x  < 
 y  /\  A. y  e.  Z  ( y  < 
 x  ->  E. z  e.  A  y  <  z
 ) ) )
 
Theoremnn01to3 11280 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 11281 Alternate proof of nn0ge2m1nn 10958: 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 11188, 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 10958. (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 )
 
5.4.11  Well-ordering principle for bounded-below sets of integers
 
Theoremuzwo3 11282* Well-ordering principle: any nonempty subset of an upper set of integers has a unique least element. This generalization of uzwo2 11246 allows the lower bound  B to be any real number. See also nnwo 11247 and nnwos 11249. (Contributed by NM, 12-Nov-2004.) (Proof shortened by Mario Carneiro, 2-Oct-2015.) (Proof shortened by AV, 27-Sep-2020.)
 |-  ( ( B  e.  RR  /\  ( A  C_  { z  e.  ZZ  |  B  <_  z }  /\  A  =/=  (/) ) )  ->  E! x  e.  A  A. y  e.  A  x  <_  y )
 
Theoremzmin 11283* There is a unique smallest integer greater than or equal to a given real number. (Contributed by NM, 12-Nov-2004.) (Revised by Mario Carneiro, 13-Jun-2014.)
 |-  ( A  e.  RR  ->  E! x  e.  ZZ  ( A  <_  x  /\  A. y  e.  ZZ  ( A  <_  y  ->  x  <_  y ) ) )
 
Theoremzmax 11284* There is a unique largest integer less than or equal to a given real number. (Contributed by NM, 15-Nov-2004.)
 |-  ( A  e.  RR  ->  E! x  e.  ZZ  ( x  <_  A  /\  A. y  e.  ZZ  (
 y  <_  A  ->  y 
 <_  x ) ) )
 
Theoremzbtwnre 11285* There is a unique integer between a real number and the number plus one. Exercise 5 of [Apostol] p. 28. (Contributed by NM, 13-Nov-2004.)
 |-  ( A  e.  RR  ->  E! x  e.  ZZ  ( A  <_  x  /\  x  <  ( A  +  1 ) ) )
 
Theoremrebtwnz 11286* There is a unique greatest integer less than or equal to a real number. Exercise 4 of [Apostol] p. 28. (Contributed by NM, 15-Nov-2004.)
 |-  ( A  e.  RR  ->  E! x  e.  ZZ  ( x  <_  A  /\  A  <  ( x  +  1 ) ) )
 
5.4.12  Rational numbers (as a subset of complex numbers)
 
Syntaxcq 11287 Extend class notation to include the class of rationals.
 class  QQ
 
Definitiondf-q 11288 Define the set of rational numbers. Based on definition of rationals in [Apostol] p. 22. See elq 11289 for the relation "is rational." (Contributed by NM, 8-Jan-2002.)
 |- 
 QQ  =  (  /  " ( ZZ  X.  NN ) )
 
Theoremelq 11289* 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 11290* 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 11291 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 11292 A rational number is a real number. (Contributed by NM, 14-Nov-2002.)
 |-  ( A  e.  QQ  ->  A  e.  RR )
 
Theoremzq 11293 An integer is a rational number. (Contributed by NM, 9-Jan-2002.)
 |-  ( A  e.  ZZ  ->  A  e.  QQ )
 
Theoremzssq 11294 The integers are a subset of the rationals. (Contributed by NM, 9-Jan-2002.)
 |- 
 ZZ  C_  QQ
 
Theoremnn0ssq 11295 The nonnegative integers are a subset of the rationals. (Contributed by NM, 31-Jul-2004.)
 |- 
 NN0  C_  QQ
 
Theoremnnssq 11296 The positive integers are a subset of the rationals. (Contributed by NM, 31-Jul-2004.)
 |- 
 NN  C_  QQ
 
Theoremqssre 11297 The rationals are a subset of the reals. (Contributed by NM, 9-Jan-2002.)
 |- 
 QQ  C_  RR
 
Theoremqsscn 11298 The rationals are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.)
 |- 
 QQ  C_  CC
 
Theoremqex 11299 The set of rational numbers exists. See also qexALT 11302. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)
 |- 
 QQ  e.  _V
 
Theoremnnq 11300 A positive integer is rational. (Contributed by NM, 17-Nov-2004.)
 |-  ( A  e.  NN  ->  A  e.  QQ )
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