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Theorem List for Intuitionistic Logic Explorer - 8301-8400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem2rexuz 8301* Double existential quantification in an upper set of integers. (Contributed by NM, 3-Nov-2005.)
 m n  ZZ>= `  m  m  ZZ  n 
 ZZ  m  <_  n
 
Theorempeano2uz 8302 Second Peano postulate for an upper set of integers. (Contributed by NM, 7-Sep-2005.)
 N  ZZ>=
 `  M  N  +  1  ZZ>= `  M
 
Theorempeano2uzs 8303 Second Peano postulate for an upper set of integers. (Contributed by Mario Carneiro, 26-Dec-2013.)
 Z  ZZ>= `  M   =>     N  Z  N  +  1  Z
 
Theorempeano2uzr 8304 Reversed second Peano axiom for upper integers. (Contributed by NM, 2-Jan-2006.)
 M  ZZ  N  ZZ>=
 `  M  +  1 
 N  ZZ>= `  M
 
Theoremuzaddcl 8305 Addition closure law for an upper set of integers. (Contributed by NM, 4-Jun-2006.)
 N  ZZ>= `  M  K  NN0  N  +  K  ZZ>= `  M
 
Theoremnn0pzuz 8306 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  NN0  Z  ZZ  N  +  Z  ZZ>= `  Z
 
Theoremuzind4 8307* 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    &     j  k    &     j  k  +  1    &     j  N    &     M  ZZ    &     k  ZZ>=
 `  M    =>     N  ZZ>=
 `  M
 
Theoremuzind4ALT 8308* 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 8307 or uzind4ALT 8308 may be used; see comment for nnind 7711. (Contributed by NM, 7-Sep-2005.) (New usage is discouraged.) (Proof modification is discouraged.)
 M  ZZ    &     k  ZZ>=
 `  M    &     j  M    &     j  k    &     j  k  +  1    &     j  N    =>     N  ZZ>=
 `  M
 
Theoremuzind4s 8309* 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  ZZ  [. M  k ].   &     k  ZZ>=
 `  M  [. k  +  1  k ].   =>     N  ZZ>=
 `  M  [. N  k ].
 
Theoremuzind4s2 8310* 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 8309 when  j and  k must be distinct in  [. k  +  1  j ].. (Contributed by NM, 16-Nov-2005.)
 M  ZZ  [. M  j ].   &     k  ZZ>=
 `  M  [. k  j ].  [. k  +  1  j ].   =>     N  ZZ>=
 `  M  [. N  j ].
 
Theoremuzind4i 8311* 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  ZZ   &     j  M    &     j  k    &     j  k  +  1    &     j  N    &       &     k  ZZ>=
 `  M    =>     N  ZZ>=
 `  M
 
Theoremindstr 8312* Strong Mathematical Induction for positive integers (inference schema). (Contributed by NM, 17-Aug-2001.)
   &     NN 
 NN  <    =>     NN
 
Theoremeluznn0 8313 Membership in a nonnegative upper set of integers implies membership in  NN0. (Contributed by Paul Chapman, 22-Jun-2011.)
 N  NN0  M  ZZ>= `  N  M  NN0
 
Theoremeluznn 8314 Membership in a positive upper set of integers implies membership in  NN. (Contributed by JJ, 1-Oct-2018.)
 N  NN  M  ZZ>=
 `  N  M  NN
 
Theoremeluz2b1 8315 Two ways to say "an integer greater than or equal to 2." (Contributed by Paul Chapman, 23-Nov-2012.)
 N  ZZ>=
 `  2  N  ZZ  1  <  N
 
Theoremeluz2b2 8316 Two ways to say "an integer greater than or equal to 2." (Contributed by Paul Chapman, 23-Nov-2012.)
 N  ZZ>=
 `  2  N  NN  1  <  N
 
Theoremeluz2b3 8317 Two ways to say "an integer greater than or equal to 2." (Contributed by Paul Chapman, 23-Nov-2012.)
 N  ZZ>=
 `  2  N  NN  N  =/=  1
 
Theoremuz2m1nn 8318 One less than an integer greater than or equal to 2 is a positive integer. (Contributed by Paul Chapman, 17-Nov-2012.)
 N  ZZ>=
 `  2  N  -  1  NN
 
Theorem1nuz2 8319 1 is not in  ZZ>= `  2. (Contributed by Paul Chapman, 21-Nov-2012.)
 1  ZZ>=
 `  2
 
Theoremelnn1uz2 8320 A positive integer is either 1 or greater than or equal to 2. (Contributed by Paul Chapman, 17-Nov-2012.)
 N  NN  N  1  N  ZZ>= `  2
 
Theoremuz2mulcl 8321 Closure of multiplication of integers greater than or equal to 2. (Contributed by Paul Chapman, 26-Oct-2012.)
 M  ZZ>= `  2  N  ZZ>= `  2  M  x.  N  ZZ>= `  2
 
Theoremindstr2 8322* 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.)
 1    &       &       &     ZZ>=
 `  2  NN  <    =>     NN
 
Theoremeluzdc 8323 Membership of an integer in an upper set of integers is decidable. (Contributed by Jim Kingdon, 18-Apr-2020.)
 M  ZZ  N  ZZ DECID  N  ZZ>= `  M
 
Theoremublbneg 8324* The image under negation of a bounded-above set of reals is bounded below. (Contributed by Paul Chapman, 21-Mar-2011.)
 RR  <_  RR  {  RR  |  -u  }  <_
 
Theoremeqreznegel 8325* Two ways to express the image under negation of a set of integers. (Contributed by Paul Chapman, 21-Mar-2011.)
 C_  ZZ  {  RR  |  -u  }  {  ZZ  |  -u  }
 
Theoremnegm 8326* The image under negation of an inhabited set of reals is inhabited. (Contributed by Jim Kingdon, 10-Apr-2020.)
 C_  RR  {  RR  |  -u  }
 
Theoremlbzbi 8327* If a set of reals is bounded below, it is bounded below by an integer. (Contributed by Paul Chapman, 21-Mar-2011.)
 C_  RR  RR  <_  ZZ  <_
 
Theoremnn01to3 8328 A (nonnegative) integer between 1 and 3 must be 1, 2 or 3. (Contributed by Alexander van der Vekens, 13-Sep-2018.)
 N  NN0  1  <_  N  N  <_  3  N  1  N  2  N  3
 
Theoremnn0ge2m1nnALT 8329 Alternate proof of nn0ge2m1nn 8018: 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 8255, 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 8018. (Contributed by Alexander van der Vekens, 1-Aug-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
 N  NN0  2  <_  N  N  -  1  NN
 
3.4.11  Rational numbers (as a subset of complex numbers)
 
Syntaxcq 8330 Extend class notation to include the class of rationals.

 QQ
 
Definitiondf-q 8331 Define the set of rational numbers. Based on definition of rationals in [Apostol] p. 22. See elq 8333 for the relation "is rational." (Contributed by NM, 8-Jan-2002.)

 QQ  " ZZ  X.  NN
 
Theoremdivfnzn 8332 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 8333* Membership in the set of rationals. (Contributed by NM, 8-Jan-2002.) (Revised by Mario Carneiro, 28-Jan-2014.)
 QQ  ZZ  NN
 
Theoremqmulz 8334* If is rational, then some integer multiple of it is an integer. (Contributed by NM, 7-Nov-2008.) (Revised by Mario Carneiro, 22-Jul-2014.)
 QQ  NN  x.  ZZ
 
Theoremznq 8335 The ratio of an integer and a positive integer is a rational number. (Contributed by NM, 12-Jan-2002.)
 ZZ  NN  QQ
 
Theoremqre 8336 A rational number is a real number. (Contributed by NM, 14-Nov-2002.)
 QQ  RR
 
Theoremzq 8337 An integer is a rational number. (Contributed by NM, 9-Jan-2002.)
 ZZ  QQ
 
Theoremzssq 8338 The integers are a subset of the rationals. (Contributed by NM, 9-Jan-2002.)

 ZZ  C_  QQ
 
Theoremnn0ssq 8339 The nonnegative integers are a subset of the rationals. (Contributed by NM, 31-Jul-2004.)

 NN0  C_  QQ
 
Theoremnnssq 8340 The positive integers are a subset of the rationals. (Contributed by NM, 31-Jul-2004.)

 NN  C_  QQ
 
Theoremqssre 8341 The rationals are a subset of the reals. (Contributed by NM, 9-Jan-2002.)

 QQ  C_  RR
 
Theoremqsscn 8342 The rationals are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.)

 QQ  C_  CC
 
Theoremqex 8343 The set of rational numbers exists. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)

 QQ  _V
 
Theoremnnq 8344 A positive integer is rational. (Contributed by NM, 17-Nov-2004.)
 NN  QQ
 
Theoremqcn 8345 A rational number is a complex number. (Contributed by NM, 2-Aug-2004.)
 QQ  CC
 
Theoremqaddcl 8346 Closure of addition of rationals. (Contributed by NM, 1-Aug-2004.)
 QQ  QQ  +  QQ
 
Theoremqnegcl 8347 Closure law for the negative of a rational. (Contributed by NM, 2-Aug-2004.) (Revised by Mario Carneiro, 15-Sep-2014.)
 QQ  -u  QQ
 
Theoremqmulcl 8348 Closure of multiplication of rationals. (Contributed by NM, 1-Aug-2004.)
 QQ  QQ  x.  QQ
 
Theoremqsubcl 8349 Closure of subtraction of rationals. (Contributed by NM, 2-Aug-2004.)
 QQ  QQ  -  QQ
 
Theoremqapne 8350 Apartness is equivalent to not equal for rationals. (Contributed by Jim Kingdon, 20-Mar-2020.)
 QQ  QQ #  =/=
 
Theoremqreccl 8351 Closure of reciprocal of rationals. (Contributed by NM, 3-Aug-2004.)
 QQ  =/=  0  1  QQ
 
Theoremqdivcl 8352 Closure of division of rationals. (Contributed by NM, 3-Aug-2004.)
 QQ  QQ  =/=  0  QQ
 
Theoremqrevaddcl 8353 Reverse closure law for addition of rationals. (Contributed by NM, 2-Aug-2004.)
 QQ  CC  +  QQ  QQ
 
Theoremnnrecq 8354 The reciprocal of a positive integer is rational. (Contributed by NM, 17-Nov-2004.)
 NN  1  QQ
 
Theoremirradd 8355 The sum of an irrational number and a rational number is irrational. (Contributed by NM, 7-Nov-2008.)
 RR  \  QQ  QQ  +  RR  \  QQ
 
Theoremirrmul 8356 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.)
 RR  \  QQ  QQ  =/=  0  x.  RR  \  QQ
 
3.4.12  Complex numbers as pairs of reals
 
Theoremcnref1o 8357* There is a natural one-to-one mapping from  RR  X.  RR to  CC, where we map  <. , 
>. to  +  _i  x. . In our construction of the complex numbers, this is in fact our definition of  CC (see df-c 6717), 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  RR ,  RR  |->  +  _i  x.    =>     F : RR 
 X.  RR -1-1-onto-> CC
 
3.5  Order sets
 
3.5.1  Positive reals (as a subset of complex numbers)
 
Syntaxcrp 8358 Extend class notation to include the class of positive reals.
 RR+
 
Definitiondf-rp 8359 Define the set of positive reals. Definition of positive numbers in [Apostol] p. 20. (Contributed by NM, 27-Oct-2007.)
 RR+  {  RR  |  0  <  }
 
Theoremelrp 8360 Membership in the set of positive reals. (Contributed by NM, 27-Oct-2007.)
 RR+  RR  0  <
 
Theoremelrpii 8361 Membership in the set of positive reals. (Contributed by NM, 23-Feb-2008.)
 RR   &     0  <    =>     RR+
 
Theorem1rp 8362 1 is a positive real. (Contributed by Jeff Hankins, 23-Nov-2008.)
 1  RR+
 
Theorem2rp 8363 2 is a positive real. (Contributed by Mario Carneiro, 28-May-2016.)
 2  RR+
 
Theoremrpre 8364 A positive real is a real. (Contributed by NM, 27-Oct-2007.)
 RR+  RR
 
Theoremrpxr 8365 A positive real is an extended real. (Contributed by Mario Carneiro, 21-Aug-2015.)
 RR+  RR*
 
Theoremrpcn 8366 A positive real is a complex number. (Contributed by NM, 11-Nov-2008.)
 RR+  CC
 
Theoremnnrp 8367 A positive integer is a positive real. (Contributed by NM, 28-Nov-2008.)
 NN  RR+
 
Theoremrpssre 8368 The positive reals are a subset of the reals. (Contributed by NM, 24-Feb-2008.)
 RR+  C_  RR
 
Theoremrpgt0 8369 A positive real is greater than zero. (Contributed by FL, 27-Dec-2007.)
 RR+  0  <
 
Theoremrpge0 8370 A positive real is greater than or equal to zero. (Contributed by NM, 22-Feb-2008.)
 RR+  0  <_
 
Theoremrpregt0 8371 A positive real is a positive real number. (Contributed by NM, 11-Nov-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
 RR+  RR  0  <
 
Theoremrprege0 8372 A positive real is a nonnegative real number. (Contributed by Mario Carneiro, 31-Jan-2014.)
 RR+  RR  0  <_
 
Theoremrpne0 8373 A positive real is nonzero. (Contributed by NM, 18-Jul-2008.)
 RR+  =/=  0
 
Theoremrpap0 8374 A positive real is apart from zero. (Contributed by Jim Kingdon, 22-Mar-2020.)
 RR+ #  0
 
Theoremrprene0 8375 A positive real is a nonzero real number. (Contributed by NM, 11-Nov-2008.)
 RR+  RR  =/=  0
 
Theoremrpreap0 8376 A positive real is a real number apart from zero. (Contributed by Jim Kingdon, 22-Mar-2020.)
 RR+  RR #  0
 
Theoremrpcnne0 8377 A positive real is a nonzero complex number. (Contributed by NM, 11-Nov-2008.)
 RR+  CC  =/=  0
 
Theoremrpcnap0 8378 A positive real is a complex number apart from zero. (Contributed by Jim Kingdon, 22-Mar-2020.)
 RR+  CC #  0
 
Theoremralrp 8379 Quantification over positive reals. (Contributed by NM, 12-Feb-2008.)
 RR+  RR  0  <
 
Theoremrexrp 8380 Quantification over positive reals. (Contributed by Mario Carneiro, 21-May-2014.)
 RR+  RR  0  <
 
Theoremrpaddcl 8381 Closure law for addition of positive reals. Part of Axiom 7 of [Apostol] p. 20. (Contributed by NM, 27-Oct-2007.)
 RR+  RR+  +  RR+
 
Theoremrpmulcl 8382 Closure law for multiplication of positive reals. Part of Axiom 7 of [Apostol] p. 20. (Contributed by NM, 27-Oct-2007.)
 RR+  RR+  x.  RR+
 
Theoremrpdivcl 8383 Closure law for division of positive reals. (Contributed by FL, 27-Dec-2007.)
 RR+  RR+  RR+
 
Theoremrpreccl 8384 Closure law for reciprocation of positive reals. (Contributed by Jeff Hankins, 23-Nov-2008.)
 RR+  1  RR+
 
Theoremrphalfcl 8385 Closure law for half of a positive real. (Contributed by Mario Carneiro, 31-Jan-2014.)
 RR+  2  RR+
 
Theoremrpgecl 8386 A number greater or equal to a positive real is positive real. (Contributed by Mario Carneiro, 28-May-2016.)
 RR+  RR  <_  RR+
 
Theoremrphalflt 8387 Half of a positive real is less than the original number. (Contributed by Mario Carneiro, 21-May-2014.)
 RR+  2  <
 
Theoremrerpdivcl 8388 Closure law for division of a real by a positive real. (Contributed by NM, 10-Nov-2008.)
 RR  RR+  RR
 
Theoremge0p1rp 8389 A nonnegative number plus one is a positive number. (Contributed by Mario Carneiro, 5-Oct-2015.)
 RR  0  <_  +  1  RR+
 
Theoremrpnegap 8390 Either a real apart from zero or its negation is a positive real, but not both. (Contributed by Jim Kingdon, 23-Mar-2020.)
 RR #  0  RR+  \/_  -u  RR+
 
Theorem0nrp 8391 Zero is not a positive real. Axiom 9 of [Apostol] p. 20. (Contributed by NM, 27-Oct-2007.)
 0  RR+
 
Theoremltsubrp 8392 Subtracting a positive real from another number decreases it. (Contributed by FL, 27-Dec-2007.)
 RR  RR+  -  <
 
Theoremltaddrp 8393 Adding a positive number to another number increases it. (Contributed by FL, 27-Dec-2007.)
 RR  RR+  <  +
 
Theoremdifrp 8394 Two ways to say one number is less than another. (Contributed by Mario Carneiro, 21-May-2014.)
 RR  RR  <  -  RR+
 
Theoremelrpd 8395 Membership in the set of positive reals. (Contributed by Mario Carneiro, 28-May-2016.)
 RR   &     0  <    =>     RR+
 
Theoremnnrpd 8396 A positive integer is a positive real. (Contributed by Mario Carneiro, 28-May-2016.)
 NN   =>     RR+
 
Theoremrpred 8397 A positive real is a real. (Contributed by Mario Carneiro, 28-May-2016.)
 RR+   =>     RR
 
Theoremrpxrd 8398 A positive real is an extended real. (Contributed by Mario Carneiro, 28-May-2016.)
 RR+   =>     RR*
 
Theoremrpcnd 8399 A positive real is a complex number. (Contributed by Mario Carneiro, 28-May-2016.)
 RR+   =>     CC
 
Theoremrpgt0d 8400 A positive real is greater than zero. (Contributed by Mario Carneiro, 28-May-2016.)
 RR+   =>     0  <
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