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Theorem List for Intuitionistic Logic Explorer - 8401-8500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremrpge0d 8401 A positive real is greater than or equal to zero. (Contributed by Mario Carneiro, 28-May-2016.)
 RR+   =>     0  <_
 
Theoremrpne0d 8402 A positive real is nonzero. (Contributed by Mario Carneiro, 28-May-2016.)
 RR+   =>     =/=  0
 
Theoremrpregt0d 8403 A positive real is real and greater than zero. (Contributed by Mario Carneiro, 28-May-2016.)
 RR+   =>     RR  0  <
 
Theoremrprege0d 8404 A positive real is real and greater or equal to zero. (Contributed by Mario Carneiro, 28-May-2016.)
 RR+   =>     RR  0  <_
 
Theoremrprene0d 8405 A positive real is a nonzero real number. (Contributed by Mario Carneiro, 28-May-2016.)
 RR+   =>     RR  =/=  0
 
Theoremrpcnne0d 8406 A positive real is a nonzero complex number. (Contributed by Mario Carneiro, 28-May-2016.)
 RR+   =>     CC  =/=  0
 
Theoremrpreccld 8407 Closure law for reciprocation of positive reals. (Contributed by Mario Carneiro, 28-May-2016.)
 RR+   =>    
 1  RR+
 
Theoremrprecred 8408 Closure law for reciprocation of positive reals. (Contributed by Mario Carneiro, 28-May-2016.)
 RR+   =>    
 1  RR
 
Theoremrphalfcld 8409 Closure law for half of a positive real. (Contributed by Mario Carneiro, 28-May-2016.)
 RR+   =>     2  RR+
 
Theoremreclt1d 8410 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 8411 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 8412 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 8413 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 8414 Closure law for division of positive reals. (Contributed by Mario Carneiro, 28-May-2016.)
 RR+   &     RR+   =>     RR+
 
Theoremltrecd 8415 The reciprocal of both sides of 'less than'. (Contributed by Mario Carneiro, 28-May-2016.)
 RR+   &     RR+   =>     <  1  < 
 1
 
Theoremlerecd 8416 The reciprocal of both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 28-May-2016.)
 RR+   &     RR+   =>     <_  1  <_ 
 1
 
Theoremltrec1d 8417 Reciprocal swap in a 'less than' relation. (Contributed by Mario Carneiro, 28-May-2016.)
 RR+   &     RR+   &    
 1  <    =>    
 1  <
 
Theoremlerec2d 8418 Reciprocal swap in a 'less than or equal to' relation. (Contributed by Mario Carneiro, 28-May-2016.)
 RR+   &     RR+   &     <_  1    =>     <_  1
 
Theoremlediv2ad 8419 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 8420 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 8421 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 8422 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 8423 A nonnegative number plus one is a positive number. (Contributed by Mario Carneiro, 28-May-2016.)
 RR   &     0  <_    =>     +  1  RR+
 
Theoremrerpdivcld 8424 Closure law for division of a real by a positive real. (Contributed by Mario Carneiro, 28-May-2016.)
 RR   &     RR+   =>    
 RR
 
Theoremltsubrpd 8425 Subtracting a positive real from another number decreases it. (Contributed by Mario Carneiro, 28-May-2016.)
 RR   &     RR+   =>     -  <
 
Theoremltaddrpd 8426 Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 28-May-2016.)
 RR   &     RR+   =>     <  +
 
Theoremltaddrp2d 8427 Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 28-May-2016.)
 RR   &     RR+   =>     <  +
 
Theoremltmulgt11d 8428 Multiplication by a number greater than 1. (Contributed by Mario Carneiro, 28-May-2016.)
 RR   &     RR+   =>    
 1  <  <  x.
 
Theoremltmulgt12d 8429 Multiplication by a number greater than 1. (Contributed by Mario Carneiro, 28-May-2016.)
 RR   &     RR+   =>    
 1  <  <  x.
 
Theoremgt0divd 8430 Division of a positive number by a positive number. (Contributed by Mario Carneiro, 28-May-2016.)
 RR   &     RR+   =>    
 0  <  0  <
 
Theoremge0divd 8431 Division of a nonnegative number by a positive number. (Contributed by Mario Carneiro, 28-May-2016.)
 RR   &     RR+   =>    
 0  <_  0  <_
 
Theoremrpgecld 8432 A number greater or equal to a positive real is positive real. (Contributed by Mario Carneiro, 28-May-2016.)
 RR   &     RR+   &     <_    =>     RR+
 
Theoremdivge0d 8433 The ratio of nonnegative and positive numbers is nonnegative. (Contributed by Mario Carneiro, 28-May-2016.)
 RR   &     RR+   &     0  <_    =>     0  <_
 
Theoremltmul1d 8434 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 8435 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 8436 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 8437 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 8438 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 8439 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 8440 'Less than' relationship between division and multiplication. (Contributed by Mario Carneiro, 28-May-2016.)
 RR   &     RR   &     C  RR+   =>     x.  C  <  <  C
 
Theoremltmuldiv2d 8441 'Less than' relationship between division and multiplication. (Contributed by Mario Carneiro, 28-May-2016.)
 RR   &     RR   &     C  RR+   =>     C  x.  <  <  C
 
Theoremlemuldivd 8442 'Less than or equal to' relationship between division and multiplication. (Contributed by Mario Carneiro, 30-May-2016.)
 RR   &     RR   &     C  RR+   =>     x.  C  <_  <_  C
 
Theoremlemuldiv2d 8443 'Less than or equal to' relationship between division and multiplication. (Contributed by Mario Carneiro, 30-May-2016.)
 RR   &     RR   &     C  RR+   =>     C  x.  <_  <_  C
 
Theoremltdivmuld 8444 'Less than' relationship between division and multiplication. (Contributed by Mario Carneiro, 28-May-2016.)
 RR   &     RR   &     C  RR+   =>     C 
 <  <  C  x.
 
Theoremltdivmul2d 8445 'Less than' relationship between division and multiplication. (Contributed by Mario Carneiro, 28-May-2016.)
 RR   &     RR   &     C  RR+   =>     C 
 <  <  x.  C
 
Theoremledivmuld 8446 'Less than or equal to' relationship between division and multiplication. (Contributed by Mario Carneiro, 28-May-2016.)
 RR   &     RR   &     C  RR+   =>     C 
 <_  <_  C  x.
 
Theoremledivmul2d 8447 'Less than or equal to' relationship between division and multiplication. (Contributed by Mario Carneiro, 28-May-2016.)
 RR   &     RR   &     C  RR+   =>     C 
 <_  <_  x.  C
 
Theoremltmul1dd 8448 The ratio of nonnegative and positive numbers is nonnegative. (Contributed by Mario Carneiro, 30-May-2016.)
 RR   &     RR   &     C  RR+   &     <    =>     x.  C  <  x.  C
 
Theoremltmul2dd 8449 Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by Mario Carneiro, 30-May-2016.)
 RR   &     RR   &     C  RR+   &     <    =>     C  x.  <  C  x.
 
Theoremltdiv1dd 8450 Division of both sides of 'less than' by a positive number. (Contributed by Mario Carneiro, 30-May-2016.)
 RR   &     RR   &     C  RR+   &     <    =>     C  <  C
 
Theoremlediv1dd 8451 Division of both sides of a less than or equal to relation by a positive number. (Contributed by Mario Carneiro, 30-May-2016.)
 RR   &     RR   &     C  RR+   &     <_    =>     C  <_  C
 
Theoremlediv12ad 8452 Comparison of ratio of two nonnegative numbers. (Contributed by Mario Carneiro, 28-May-2016.)
 RR   &     RR   &     C  RR+   &     D  RR   &     0  <_    &     <_    &     C  <_  D   =>     D  <_  C
 
Theoremltdiv23d 8453 Swap denominator with other side of 'less than'. (Contributed by Mario Carneiro, 28-May-2016.)
 RR   &     RR+   &     C  RR+   &     <  C   =>     C  <
 
Theoremlediv23d 8454 Swap denominator with other side of 'less than or equal to'. (Contributed by Mario Carneiro, 28-May-2016.)
 RR   &     RR+   &     C  RR+   &     <_  C   =>     C  <_
 
Theoremlt2mul2divd 8455 The ratio of nonnegative and positive numbers is nonnegative. (Contributed by Mario Carneiro, 28-May-2016.)
 RR   &     RR+   &     C  RR   &     D  RR+   =>     x.  <  C  x.  D  D  <  C
 
3.5.2  Infinity and the extended real number system (cont.)
 
Syntaxcxne 8456 Extend class notation to include the negative of an extended real.
 -e
 
Syntaxcxad 8457 Extend class notation to include addition of extended reals.
 +e
 
Syntaxcxmu 8458 Extend class notation to include multiplication of extended reals.

 xe
 
Definitiondf-xneg 8459 Define the negative of an extended real number. (Contributed by FL, 26-Dec-2011.)
 -e  if +oo , -oo ,  if -oo , +oo ,  -u
 
Definitiondf-xadd 8460* Define addition over extended real numbers. (Contributed by Mario Carneiro, 20-Aug-2015.)

 +e  RR* ,  RR*  |->  if +oo
 ,  if -oo ,  0 , +oo ,  if -oo ,  if +oo ,  0 , -oo ,  if +oo , +oo ,  if -oo , -oo ,  +
 
Definitiondf-xmul 8461* Define multiplication over extended real numbers. (Contributed by Mario Carneiro, 20-Aug-2015.)
 xe  RR* ,  RR*  |->  if  0  0 ,  0 ,  if 0  < +oo  < 
 0 -oo  0  < +oo  <  0 -oo , +oo ,  if 0  < -oo  <  0 +oo  0  < -oo  <  0 +oo , -oo ,  x.
 
Theorempnfxr 8462 Plus infinity belongs to the set of extended reals. (Contributed by NM, 13-Oct-2005.) (Proof shortened by Anthony Hart, 29-Aug-2011.)
+oo  RR*
 
Theorempnfex 8463 Plus infinity exists (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
+oo  _V
 
Theoremmnfxr 8464 Minus infinity belongs to the set of extended reals. (Contributed by NM, 13-Oct-2005.) (Proof shortened by Anthony Hart, 29-Aug-2011.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
-oo  RR*
 
Theoremltxr 8465 The 'less than' binary relation on the set of extended reals. Definition 12-3.1 of [Gleason] p. 173. (Contributed by NM, 14-Oct-2005.)
 RR*  RR*  <  RR  RR  <RR -oo +oo  RR +oo -oo  RR
 
Theoremelxr 8466 Membership in the set of extended reals. (Contributed by NM, 14-Oct-2005.)
 RR*  RR +oo -oo
 
Theorempnfnemnf 8467 Plus and minus infinity are different elements of  RR*. (Contributed by NM, 14-Oct-2005.)
+oo  =/= -oo
 
Theoremmnfnepnf 8468 Minus and plus infinity are different (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
-oo  =/= +oo
 
Theoremxrnemnf 8469 An extended real other than minus infinity is real or positive infinite. (Contributed by Mario Carneiro, 20-Aug-2015.)
 RR*  =/= -oo  RR +oo
 
Theoremxrnepnf 8470 An extended real other than plus infinity is real or negative infinite. (Contributed by Mario Carneiro, 20-Aug-2015.)
 RR*  =/= +oo  RR -oo
 
Theoremxrltnr 8471 The extended real 'less than' is irreflexive. (Contributed by NM, 14-Oct-2005.)
 RR*  <
 
Theoremltpnf 8472 Any (finite) real is less than plus infinity. (Contributed by NM, 14-Oct-2005.)
 RR  < +oo
 
Theorem0ltpnf 8473 Zero is less than plus infinity (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 0  < +oo
 
Theoremmnflt 8474 Minus infinity is less than any (finite) real. (Contributed by NM, 14-Oct-2005.)
 RR -oo  <
 
Theoremmnflt0 8475 Minus infinity is less than 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
-oo  <  0
 
Theoremmnfltpnf 8476 Minus infinity is less than plus infinity. (Contributed by NM, 14-Oct-2005.)
-oo  < +oo
 
Theoremmnfltxr 8477 Minus infinity is less than an extended real that is either real or plus infinity. (Contributed by NM, 2-Feb-2006.)
 RR +oo -oo  <
 
Theorempnfnlt 8478 No extended real is greater than plus infinity. (Contributed by NM, 15-Oct-2005.)
 RR* +oo  <
 
Theoremnltmnf 8479 No extended real is less than minus infinity. (Contributed by NM, 15-Oct-2005.)
 RR*  < -oo
 
Theorempnfge 8480 Plus infinity is an upper bound for extended reals. (Contributed by NM, 30-Jan-2006.)
 RR*  <_ +oo
 
Theorem0lepnf 8481 0 less than or equal to positive infinity. (Contributed by David A. Wheeler, 8-Dec-2018.)
 0  <_ +oo
 
Theoremnn0pnfge0 8482 If a number is a nonnegative integer or positive infinity, it is greater than or equal to 0. (Contributed by Alexander van der Vekens, 6-Jan-2018.)
 N  NN0  N +oo  0  <_  N
 
Theoremmnfle 8483 Minus infinity is less than or equal to any extended real. (Contributed by NM, 19-Jan-2006.)
 RR* -oo  <_
 
Theoremxrltnsym 8484 Ordering on the extended reals is not symmetric. (Contributed by NM, 15-Oct-2005.)
 RR*  RR*  <  <
 
Theoremxrltnsym2 8485 'Less than' is antisymmetric and irreflexive for extended reals. (Contributed by NM, 6-Feb-2007.)
 RR*  RR*  <  <
 
Theoremxrlttr 8486 Ordering on the extended reals is transitive. (Contributed by NM, 15-Oct-2005.)
 RR*  RR*  C  RR*  <  <  C  <  C
 
Theoremxrltso 8487 'Less than' is a weakly linear ordering on the extended reals. (Contributed by NM, 15-Oct-2005.)

 <  Or  RR*
 
Theoremxrlttri3 8488 Extended real version of lttri3 6895. (Contributed by NM, 9-Feb-2006.)
 RR*  RR*  <  <
 
Theoremxrltle 8489 'Less than' implies 'less than or equal' for extended reals. (Contributed by NM, 19-Jan-2006.)
 RR*  RR*  <  <_
 
Theoremxrleid 8490 'Less than or equal to' is reflexive for extended reals. (Contributed by NM, 7-Feb-2007.)
 RR*  <_
 
Theoremxrletri3 8491 Trichotomy law for extended reals. (Contributed by FL, 2-Aug-2009.)
 RR*  RR*  <_  <_
 
Theoremxrlelttr 8492 Transitive law for ordering on extended reals. (Contributed by NM, 19-Jan-2006.)
 RR*  RR*  C  RR*  <_  <  C  <  C
 
Theoremxrltletr 8493 Transitive law for ordering on extended reals. (Contributed by NM, 19-Jan-2006.)
 RR*  RR*  C  RR*  <  <_  C  <  C
 
Theoremxrletr 8494 Transitive law for ordering on extended reals. (Contributed by NM, 9-Feb-2006.)
 RR*  RR*  C  RR*  <_  <_  C  <_  C
 
Theoremxrlttrd 8495 Transitive law for ordering on extended reals. (Contributed by Mario Carneiro, 23-Aug-2015.)
 RR*   &     RR*   &     C  RR*   &     <    &     <  C   =>     <  C
 
Theoremxrlelttrd 8496 Transitive law for ordering on extended reals. (Contributed by Mario Carneiro, 23-Aug-2015.)
 RR*   &     RR*   &     C  RR*   &     <_    &     <  C   =>     <  C
 
Theoremxrltletrd 8497 Transitive law for ordering on extended reals. (Contributed by Mario Carneiro, 23-Aug-2015.)
 RR*   &     RR*   &     C  RR*   &     <    &     <_  C   =>     <  C
 
Theoremxrletrd 8498 Transitive law for ordering on extended reals. (Contributed by Mario Carneiro, 23-Aug-2015.)
 RR*   &     RR*   &     C  RR*   &     <_    &     <_  C   =>     <_  C
 
Theoremxrltne 8499 'Less than' implies not equal for extended reals. (Contributed by NM, 20-Jan-2006.)
 RR*  RR*  <  =/=
 
Theoremnltpnft 8500 An extended real is not less than plus infinity iff they are equal. (Contributed by NM, 30-Jan-2006.)
 RR* +oo  < +oo
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