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Theorem List for Intuitionistic Logic Explorer - 8301-8400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremqssre 8301 The rationals are a subset of the reals. (Contributed by NM, 9-Jan-2002.)

 QQ  C_  RR
 
Theoremqsscn 8302 The rationals are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.)

 QQ  C_  CC
 
Theoremqex 8303 The set of rational numbers exists. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)

 QQ  _V
 
Theoremnnq 8304 A positive integer is rational. (Contributed by NM, 17-Nov-2004.)
 NN  QQ
 
Theoremqcn 8305 A rational number is a complex number. (Contributed by NM, 2-Aug-2004.)
 QQ  CC
 
Theoremqaddcl 8306 Closure of addition of rationals. (Contributed by NM, 1-Aug-2004.)
 QQ  QQ  +  QQ
 
Theoremqnegcl 8307 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 8308 Closure of multiplication of rationals. (Contributed by NM, 1-Aug-2004.)
 QQ  QQ  x.  QQ
 
Theoremqsubcl 8309 Closure of subtraction of rationals. (Contributed by NM, 2-Aug-2004.)
 QQ  QQ  -  QQ
 
Theoremqapne 8310 Apartness is equivalent to not equal for rationals. (Contributed by Jim Kingdon, 20-Mar-2020.)
 QQ  QQ #  =/=
 
Theoremqreccl 8311 Closure of reciprocal of rationals. (Contributed by NM, 3-Aug-2004.)
 QQ  =/=  0  1  QQ
 
Theoremqdivcl 8312 Closure of division of rationals. (Contributed by NM, 3-Aug-2004.)
 QQ  QQ  =/=  0  QQ
 
Theoremqrevaddcl 8313 Reverse closure law for addition of rationals. (Contributed by NM, 2-Aug-2004.)
 QQ  CC  +  QQ  QQ
 
Theoremnnrecq 8314 The reciprocal of a positive integer is rational. (Contributed by NM, 17-Nov-2004.)
 NN  1  QQ
 
Theoremirradd 8315 The sum of an irrational number and a rational number is irrational. (Contributed by NM, 7-Nov-2008.)
 RR  \  QQ  QQ  +  RR  \  QQ
 
Theoremirrmul 8316 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 8317* 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 6677), 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 8318 Extend class notation to include the class of positive reals.
 RR+
 
Definitiondf-rp 8319 Define the set of positive reals. Definition of positive numbers in [Apostol] p. 20. (Contributed by NM, 27-Oct-2007.)
 RR+  {  RR  |  0  <  }
 
Theoremelrp 8320 Membership in the set of positive reals. (Contributed by NM, 27-Oct-2007.)
 RR+  RR  0  <
 
Theoremelrpii 8321 Membership in the set of positive reals. (Contributed by NM, 23-Feb-2008.)
 RR   &     0  <    =>     RR+
 
Theorem1rp 8322 1 is a positive real. (Contributed by Jeff Hankins, 23-Nov-2008.)
 1  RR+
 
Theorem2rp 8323 2 is a positive real. (Contributed by Mario Carneiro, 28-May-2016.)
 2  RR+
 
Theoremrpre 8324 A positive real is a real. (Contributed by NM, 27-Oct-2007.)
 RR+  RR
 
Theoremrpxr 8325 A positive real is an extended real. (Contributed by Mario Carneiro, 21-Aug-2015.)
 RR+  RR*
 
Theoremrpcn 8326 A positive real is a complex number. (Contributed by NM, 11-Nov-2008.)
 RR+  CC
 
Theoremnnrp 8327 A positive integer is a positive real. (Contributed by NM, 28-Nov-2008.)
 NN  RR+
 
Theoremrpssre 8328 The positive reals are a subset of the reals. (Contributed by NM, 24-Feb-2008.)
 RR+  C_  RR
 
Theoremrpgt0 8329 A positive real is greater than zero. (Contributed by FL, 27-Dec-2007.)
 RR+  0  <
 
Theoremrpge0 8330 A positive real is greater than or equal to zero. (Contributed by NM, 22-Feb-2008.)
 RR+  0  <_
 
Theoremrpregt0 8331 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 8332 A positive real is a nonnegative real number. (Contributed by Mario Carneiro, 31-Jan-2014.)
 RR+  RR  0  <_
 
Theoremrpne0 8333 A positive real is nonzero. (Contributed by NM, 18-Jul-2008.)
 RR+  =/=  0
 
Theoremrpap0 8334 A positive real is apart from zero. (Contributed by Jim Kingdon, 22-Mar-2020.)
 RR+ #  0
 
Theoremrprene0 8335 A positive real is a nonzero real number. (Contributed by NM, 11-Nov-2008.)
 RR+  RR  =/=  0
 
Theoremrpreap0 8336 A positive real is a real number apart from zero. (Contributed by Jim Kingdon, 22-Mar-2020.)
 RR+  RR #  0
 
Theoremrpcnne0 8337 A positive real is a nonzero complex number. (Contributed by NM, 11-Nov-2008.)
 RR+  CC  =/=  0
 
Theoremrpcnap0 8338 A positive real is a complex number apart from zero. (Contributed by Jim Kingdon, 22-Mar-2020.)
 RR+  CC #  0
 
Theoremralrp 8339 Quantification over positive reals. (Contributed by NM, 12-Feb-2008.)
 RR+  RR  0  <
 
Theoremrexrp 8340 Quantification over positive reals. (Contributed by Mario Carneiro, 21-May-2014.)
 RR+  RR  0  <
 
Theoremrpaddcl 8341 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 8342 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 8343 Closure law for division of positive reals. (Contributed by FL, 27-Dec-2007.)
 RR+  RR+  RR+
 
Theoremrpreccl 8344 Closure law for reciprocation of positive reals. (Contributed by Jeff Hankins, 23-Nov-2008.)
 RR+  1  RR+
 
Theoremrphalfcl 8345 Closure law for half of a positive real. (Contributed by Mario Carneiro, 31-Jan-2014.)
 RR+  2  RR+
 
Theoremrpgecl 8346 A number greater or equal to a positive real is positive real. (Contributed by Mario Carneiro, 28-May-2016.)
 RR+  RR  <_  RR+
 
Theoremrphalflt 8347 Half of a positive real is less than the original number. (Contributed by Mario Carneiro, 21-May-2014.)
 RR+  2  <
 
Theoremrerpdivcl 8348 Closure law for division of a real by a positive real. (Contributed by NM, 10-Nov-2008.)
 RR  RR+  RR
 
Theoremge0p1rp 8349 A nonnegative number plus one is a positive number. (Contributed by Mario Carneiro, 5-Oct-2015.)
 RR  0  <_  +  1  RR+
 
Theoremrpnegap 8350 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 8351 Zero is not a positive real. Axiom 9 of [Apostol] p. 20. (Contributed by NM, 27-Oct-2007.)
 0  RR+
 
Theoremltsubrp 8352 Subtracting a positive real from another number decreases it. (Contributed by FL, 27-Dec-2007.)
 RR  RR+  -  <
 
Theoremltaddrp 8353 Adding a positive number to another number increases it. (Contributed by FL, 27-Dec-2007.)
 RR  RR+  <  +
 
Theoremdifrp 8354 Two ways to say one number is less than another. (Contributed by Mario Carneiro, 21-May-2014.)
 RR  RR  <  -  RR+
 
Theoremelrpd 8355 Membership in the set of positive reals. (Contributed by Mario Carneiro, 28-May-2016.)
 RR   &     0  <    =>     RR+
 
Theoremnnrpd 8356 A positive integer is a positive real. (Contributed by Mario Carneiro, 28-May-2016.)
 NN   =>     RR+
 
Theoremrpred 8357 A positive real is a real. (Contributed by Mario Carneiro, 28-May-2016.)
 RR+   =>     RR
 
Theoremrpxrd 8358 A positive real is an extended real. (Contributed by Mario Carneiro, 28-May-2016.)
 RR+   =>     RR*
 
Theoremrpcnd 8359 A positive real is a complex number. (Contributed by Mario Carneiro, 28-May-2016.)
 RR+   =>     CC
 
Theoremrpgt0d 8360 A positive real is greater than zero. (Contributed by Mario Carneiro, 28-May-2016.)
 RR+   =>     0  <
 
Theoremrpge0d 8361 A positive real is greater than or equal to zero. (Contributed by Mario Carneiro, 28-May-2016.)
 RR+   =>     0  <_
 
Theoremrpne0d 8362 A positive real is nonzero. (Contributed by Mario Carneiro, 28-May-2016.)
 RR+   =>     =/=  0
 
Theoremrpregt0d 8363 A positive real is real and greater than zero. (Contributed by Mario Carneiro, 28-May-2016.)
 RR+   =>     RR  0  <
 
Theoremrprege0d 8364 A positive real is real and greater or equal to zero. (Contributed by Mario Carneiro, 28-May-2016.)
 RR+   =>     RR  0  <_
 
Theoremrprene0d 8365 A positive real is a nonzero real number. (Contributed by Mario Carneiro, 28-May-2016.)
 RR+   =>     RR  =/=  0
 
Theoremrpcnne0d 8366 A positive real is a nonzero complex number. (Contributed by Mario Carneiro, 28-May-2016.)
 RR+   =>     CC  =/=  0
 
Theoremrpreccld 8367 Closure law for reciprocation of positive reals. (Contributed by Mario Carneiro, 28-May-2016.)
 RR+   =>    
 1  RR+
 
Theoremrprecred 8368 Closure law for reciprocation of positive reals. (Contributed by Mario Carneiro, 28-May-2016.)
 RR+   =>    
 1  RR
 
Theoremrphalfcld 8369 Closure law for half of a positive real. (Contributed by Mario Carneiro, 28-May-2016.)
 RR+   =>     2  RR+
 
Theoremreclt1d 8370 The reciprocal of a positive number less than 1 is greater than 1. (Contributed by Mario Carneiro, 28-May-2016.)
 RR+   =>     <  1  1  <  1
 
Theoremrecgt1d 8371 The reciprocal of a positive number greater than 1 is less than 1. (Contributed by Mario Carneiro, 28-May-2016.)
 RR+   =>    
 1  <  1  <  1
 
Theoremrpaddcld 8372 Closure law for addition of positive reals. Part of Axiom 7 of [Apostol] p. 20. (Contributed by Mario Carneiro, 28-May-2016.)
 RR+   &     RR+   =>     +  RR+
 
Theoremrpmulcld 8373 Closure law for multiplication of positive reals. Part of Axiom 7 of [Apostol] p. 20. (Contributed by Mario Carneiro, 28-May-2016.)
 RR+   &     RR+   =>     x.  RR+
 
Theoremrpdivcld 8374 Closure law for division of positive reals. (Contributed by Mario Carneiro, 28-May-2016.)
 RR+   &     RR+   =>     RR+
 
Theoremltrecd 8375 The reciprocal of both sides of 'less than'. (Contributed by Mario Carneiro, 28-May-2016.)
 RR+   &     RR+   =>     <  1  < 
 1
 
Theoremlerecd 8376 The reciprocal of both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 28-May-2016.)
 RR+   &     RR+   =>     <_  1  <_ 
 1
 
Theoremltrec1d 8377 Reciprocal swap in a 'less than' relation. (Contributed by Mario Carneiro, 28-May-2016.)
 RR+   &     RR+   &    
 1  <    =>    
 1  <
 
Theoremlerec2d 8378 Reciprocal swap in a 'less than or equal to' relation. (Contributed by Mario Carneiro, 28-May-2016.)
 RR+   &     RR+   &     <_  1    =>     <_  1
 
Theoremlediv2ad 8379 Division of both sides of 'less than or equal to' into a nonnegative number. (Contributed by Mario Carneiro, 28-May-2016.)
 RR+   &     RR+   &     C  RR   &     0  <_  C   &     <_    =>     C  <_  C
 
Theoremltdiv2d 8380 Division of a positive number by both sides of 'less than'. (Contributed by Mario Carneiro, 28-May-2016.)
 RR+   &     RR+   &     C  RR+   =>     <  C  <  C
 
Theoremlediv2d 8381 Division of a positive number by both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 28-May-2016.)
 RR+   &     RR+   &     C  RR+   =>     <_  C  <_  C
 
Theoremledivdivd 8382 Invert ratios of positive numbers and swap their ordering. (Contributed by Mario Carneiro, 28-May-2016.)
 RR+   &     RR+   &     C  RR+   &     D  RR+   &     <_  C  D   =>     D  C  <_
 
Theoremge0p1rpd 8383 A nonnegative number plus one is a positive number. (Contributed by Mario Carneiro, 28-May-2016.)
 RR   &     0  <_    =>     +  1  RR+
 
Theoremrerpdivcld 8384 Closure law for division of a real by a positive real. (Contributed by Mario Carneiro, 28-May-2016.)
 RR   &     RR+   =>    
 RR
 
Theoremltsubrpd 8385 Subtracting a positive real from another number decreases it. (Contributed by Mario Carneiro, 28-May-2016.)
 RR   &     RR+   =>     -  <
 
Theoremltaddrpd 8386 Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 28-May-2016.)
 RR   &     RR+   =>     <  +
 
Theoremltaddrp2d 8387 Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 28-May-2016.)
 RR   &     RR+   =>     <  +
 
Theoremltmulgt11d 8388 Multiplication by a number greater than 1. (Contributed by Mario Carneiro, 28-May-2016.)
 RR   &     RR+   =>    
 1  <  <  x.
 
Theoremltmulgt12d 8389 Multiplication by a number greater than 1. (Contributed by Mario Carneiro, 28-May-2016.)
 RR   &     RR+   =>    
 1  <  <  x.
 
Theoremgt0divd 8390 Division of a positive number by a positive number. (Contributed by Mario Carneiro, 28-May-2016.)
 RR   &     RR+   =>    
 0  <  0  <
 
Theoremge0divd 8391 Division of a nonnegative number by a positive number. (Contributed by Mario Carneiro, 28-May-2016.)
 RR   &     RR+   =>    
 0  <_  0  <_
 
Theoremrpgecld 8392 A number greater or equal to a positive real is positive real. (Contributed by Mario Carneiro, 28-May-2016.)
 RR   &     RR+   &     <_    =>     RR+
 
Theoremdivge0d 8393 The ratio of nonnegative and positive numbers is nonnegative. (Contributed by Mario Carneiro, 28-May-2016.)
 RR   &     RR+   &     0  <_    =>     0  <_
 
Theoremltmul1d 8394 The ratio of nonnegative and positive numbers is nonnegative. (Contributed by Mario Carneiro, 28-May-2016.)
 RR   &     RR   &     C  RR+   =>     <  x.  C  <  x.  C
 
Theoremltmul2d 8395 Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by Mario Carneiro, 28-May-2016.)
 RR   &     RR   &     C  RR+   =>     <  C  x.  <  C  x.
 
Theoremlemul1d 8396 Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by Mario Carneiro, 28-May-2016.)
 RR   &     RR   &     C  RR+   =>     <_  x.  C  <_  x.  C
 
Theoremlemul2d 8397 Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by Mario Carneiro, 28-May-2016.)
 RR   &     RR   &     C  RR+   =>     <_  C  x.  <_  C  x.
 
Theoremltdiv1d 8398 Division of both sides of 'less than' by a positive number. (Contributed by Mario Carneiro, 28-May-2016.)
 RR   &     RR   &     C  RR+   =>     <  C  <  C
 
Theoremlediv1d 8399 Division of both sides of a less than or equal to relation by a positive number. (Contributed by Mario Carneiro, 28-May-2016.)
 RR   &     RR   &     C  RR+   =>     <_  C  <_  C
 
Theoremltmuldivd 8400 'Less than' relationship between division and multiplication. (Contributed by Mario Carneiro, 28-May-2016.)
 RR   &     RR   &     C  RR+   =>     x.  C  <  <  C
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