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Theorem List for Metamath Proof Explorer - 11201-11300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnnuz 11201 Positive integers expressed as an upper set of integers. (Contributed by NM, 2-Sep-2005.)
 |- 
 NN  =  ( ZZ>= `  1 )
 
Theoremelnnuz 11202 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 11203 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 11204 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 11205 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 11206 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 11207 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 11208 An integer greater than or equal to 3 decreased by 2 is a positive integer, analogous to uz2m1nn 11240. (Contributed by Alexander van der Vekens, 17-Sep-2018.)
 |-  ( N  e.  ( ZZ>=
 `  3 )  ->  ( N  -  2
 )  e.  NN )
 
Theorem1eluzge0 11209 1 is an integer greater than or equal to 0. (Contributed by Alexander van der Vekens, 8-Jun-2018.)
 |-  1  e.  ( ZZ>= `  0 )
 
Theorem2eluzge0 11210 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 11211 2 is an integer greater than or equal to 0. (Contributed by Alexander van der Vekens, 8-Jun-2018.) Obsolete version of 2eluzge0 11210 as of 25-Mar-2020. (New usage is discouraged.) (Proof modification is discouraged.)
 |-  2  e.  ( ZZ>= `  0 )
 
Theorem2eluzge1 11212 2 is an integer greater than or equal to 1. (Contributed by Alexander van der Vekens, 8-Jun-2018.)
 |-  2  e.  ( ZZ>= `  1 )
 
Theoremuznnssnn 11213 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 11214* 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 11215* 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 11216* 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 11217* 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 11218* 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 11219 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 11220 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 11221 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 11222 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 11223 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 11224* 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 11225* 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 11224 or uzind4ALT 11225 may be used; see comment for nnind 10634. (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 11226* 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 11227* 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 11226 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 11228* 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 11229* 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 11230* 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 11231* 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 11232* 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 11233* 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 11234* 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 11235 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 11236 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 11237 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 11238 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 11239 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 11240 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 11241 1 is not in  ( ZZ>= `  2
). (Contributed by Paul Chapman, 21-Nov-2012.)
 |- 
 -.  1  e.  ( ZZ>=
 `  2 )
 
Theoremelnn1uz2 11242 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 11243 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 11244* 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 11245 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 11246 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 11245 as of 4-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.)
 |-  M  e.  ZZ   =>    |-  sup ( (
 ZZ>= `  M ) ,  RR ,  `'  <  )  =  M
 
Theoremnninf 11247 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 11248 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 11249 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 11247 as of 5-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.)
 |- 
 sup ( NN ,  RR ,  `'  <  )  =  1
 
Theoremnn0infmOLD 11250 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 11248 as of 5-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.)
 |- 
 sup ( NN0 ,  RR ,  `'  <  )  =  0
 
Theoreminfssuzle 11251 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 11252 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 11253 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 11251 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 11254 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 11252 as of 5-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( ( S  C_  ( ZZ>= `  M )  /\  S  =/=  (/) )  ->  sup ( S ,  RR ,  `'  <  )  e.  S )
 
Theoremublbneg 11255* 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 11256* 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 11257* 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 11258* 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 11257 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 11259* 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 11260* Any nonempty bounded subset of integers has a supremum in the set. (The proof does not use ax-pre-sup 9624.) (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 11261* The supremum of a bounded-above set of integers is a member of the set. (This version of suprzcl 11022 avoids ax-pre-sup 9624.) (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 11262* 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 11263* 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 11264 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 11265 Alternate proof of nn0ge2m1nn 10941: 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 11172, 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 10941. (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 11266* Well-ordering principle: any nonempty subset of an upper set of integers has a unique least element. This generalization of uzwo2 11230 allows the lower bound  B to be any real number. See also nnwo 11231 and nnwos 11233. (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 11267* 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 11268* 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 11269* 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 11270* 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 11271 Extend class notation to include the class of rationals.
 class  QQ
 
Definitiondf-q 11272 Define the set of rational numbers. Based on definition of rationals in [Apostol] p. 22. See elq 11273 for the relation "is rational." (Contributed by NM, 8-Jan-2002.)
 |- 
 QQ  =  (  /  " ( ZZ  X.  NN ) )
 
Theoremelq 11273* 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 11274* 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 11275 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 11276 A rational number is a real number. (Contributed by NM, 14-Nov-2002.)
 |-  ( A  e.  QQ  ->  A  e.  RR )
 
Theoremzq 11277 An integer is a rational number. (Contributed by NM, 9-Jan-2002.)
 |-  ( A  e.  ZZ  ->  A  e.  QQ )
 
Theoremzssq 11278 The integers are a subset of the rationals. (Contributed by NM, 9-Jan-2002.)
 |- 
 ZZ  C_  QQ
 
Theoremnn0ssq 11279 The nonnegative integers are a subset of the rationals. (Contributed by NM, 31-Jul-2004.)
 |- 
 NN0  C_  QQ
 
Theoremnnssq 11280 The positive integers are a subset of the rationals. (Contributed by NM, 31-Jul-2004.)
 |- 
 NN  C_  QQ
 
Theoremqssre 11281 The rationals are a subset of the reals. (Contributed by NM, 9-Jan-2002.)
 |- 
 QQ  C_  RR
 
Theoremqsscn 11282 The rationals are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.)
 |- 
 QQ  C_  CC
 
Theoremqex 11283 The set of rational numbers exists. See also qexALT 11286. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)
 |- 
 QQ  e.  _V
 
Theoremnnq 11284 A positive integer is rational. (Contributed by NM, 17-Nov-2004.)
 |-  ( A  e.  NN  ->  A  e.  QQ )
 
Theoremqcn 11285 A rational number is a complex number. (Contributed by NM, 2-Aug-2004.)
 |-  ( A  e.  QQ  ->  A  e.  CC )
 
TheoremqexALT 11286 Alternate proof of qex 11283. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 16-Jun-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
 |- 
 QQ  e.  _V
 
Theoremqaddcl 11287 Closure of addition of rationals. (Contributed by NM, 1-Aug-2004.)
 |-  ( ( A  e.  QQ  /\  B  e.  QQ )  ->  ( A  +  B )  e.  QQ )
 
Theoremqnegcl 11288 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 11289 Closure of multiplication of rationals. (Contributed by NM, 1-Aug-2004.)
 |-  ( ( A  e.  QQ  /\  B  e.  QQ )  ->  ( A  x.  B )  e.  QQ )
 
Theoremqsubcl 11290 Closure of subtraction of rationals. (Contributed by NM, 2-Aug-2004.)
 |-  ( ( A  e.  QQ  /\  B  e.  QQ )  ->  ( A  -  B )  e.  QQ )
 
Theoremqreccl 11291 Closure of reciprocal of rationals. (Contributed by NM, 3-Aug-2004.)
 |-  ( ( A  e.  QQ  /\  A  =/=  0
 )  ->  ( 1  /  A )  e.  QQ )
 
Theoremqdivcl 11292 Closure of division of rationals. (Contributed by NM, 3-Aug-2004.)
 |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  B  =/=  0 ) 
 ->  ( A  /  B )  e.  QQ )
 
Theoremqrevaddcl 11293 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 11294 The reciprocal of a positive integer is rational. (Contributed by NM, 17-Nov-2004.)
 |-  ( A  e.  NN  ->  ( 1  /  A )  e.  QQ )
 
Theoremirradd 11295 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 11296 The product of an irrational with a nonzero rational is irrational. (Contributed by NM, 7-Nov-2008.)
 |-  ( ( A  e.  ( RR  \  QQ )  /\  B  e.  QQ  /\  B  =/=  0 )  ->  ( A  x.  B )  e.  ( RR  \  QQ ) )
 
5.4.13  Existence of the set of complex numbers
 
Theoremrpnnen1lem1 11297* Lemma for rpnnen1 11302. (Contributed by Mario Carneiro, 12-May-2013.)
 |-  T  =  { n  e.  ZZ  |  ( n 
 /  k )  < 
 x }   &    |-  F  =  ( x  e.  RR  |->  ( k  e.  NN  |->  ( sup ( T ,  RR ,  <  )  /  k ) ) )   =>    |-  ( x  e.  RR  ->  ( F `  x )  e.  ( QQ  ^m 
 NN ) )
 
Theoremrpnnen1lem2 11298* Lemma for rpnnen1 11302. (Contributed by Mario Carneiro, 12-May-2013.)
 |-  T  =  { n  e.  ZZ  |  ( n 
 /  k )  < 
 x }   &    |-  F  =  ( x  e.  RR  |->  ( k  e.  NN  |->  ( sup ( T ,  RR ,  <  )  /  k ) ) )   =>    |-  ( ( x  e. 
 RR  /\  k  e.  NN )  ->  sup ( T ,  RR ,  <  )  e.  ZZ )
 
Theoremrpnnen1lem3 11299* Lemma for rpnnen1 11302. (Contributed by Mario Carneiro, 12-May-2013.)
 |-  T  =  { n  e.  ZZ  |  ( n 
 /  k )  < 
 x }   &    |-  F  =  ( x  e.  RR  |->  ( k  e.  NN  |->  ( sup ( T ,  RR ,  <  )  /  k ) ) )   =>    |-  ( x  e.  RR  ->  A. n  e.  ran  ( F `  x ) n  <_  x )
 
Theoremrpnnen1lem4 11300* Lemma for rpnnen1 11302. (Contributed by Mario Carneiro, 12-May-2013.)
 |-  T  =  { n  e.  ZZ  |  ( n 
 /  k )  < 
 x }   &    |-  F  =  ( x  e.  RR  |->  ( k  e.  NN  |->  ( sup ( T ,  RR ,  <  )  /  k ) ) )   =>    |-  ( x  e.  RR  ->  sup ( ran  ( F `  x ) ,  RR ,  <  )  e.  RR )
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