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Theorem List for Intuitionistic Logic Explorer - 8701-8800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremrpcnap0 8701 A positive real is a complex number apart from zero. (Contributed by Jim Kingdon, 22-Mar-2020.)
#

Theoremralrp 8702 Quantification over positive reals. (Contributed by NM, 12-Feb-2008.)

Theoremrexrp 8703 Quantification over positive reals. (Contributed by Mario Carneiro, 21-May-2014.)

Theoremrpaddcl 8704 Closure law for addition of positive reals. Part of Axiom 7 of [Apostol] p. 20. (Contributed by NM, 27-Oct-2007.)

Theoremrpmulcl 8705 Closure law for multiplication of positive reals. Part of Axiom 7 of [Apostol] p. 20. (Contributed by NM, 27-Oct-2007.)

Theoremrpdivcl 8706 Closure law for division of positive reals. (Contributed by FL, 27-Dec-2007.)

Theoremrpreccl 8707 Closure law for reciprocation of positive reals. (Contributed by Jeff Hankins, 23-Nov-2008.)

Theoremrphalfcl 8708 Closure law for half of a positive real. (Contributed by Mario Carneiro, 31-Jan-2014.)

Theoremrpgecl 8709 A number greater or equal to a positive real is positive real. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrphalflt 8710 Half of a positive real is less than the original number. (Contributed by Mario Carneiro, 21-May-2014.)

Theoremrerpdivcl 8711 Closure law for division of a real by a positive real. (Contributed by NM, 10-Nov-2008.)

Theoremge0p1rp 8712 A nonnegative number plus one is a positive number. (Contributed by Mario Carneiro, 5-Oct-2015.)

Theoremrpnegap 8713 Either a real apart from zero or its negation is a positive real, but not both. (Contributed by Jim Kingdon, 23-Mar-2020.)
#

Theorem0nrp 8714 Zero is not a positive real. Axiom 9 of [Apostol] p. 20. (Contributed by NM, 27-Oct-2007.)

Theoremltsubrp 8715 Subtracting a positive real from another number decreases it. (Contributed by FL, 27-Dec-2007.)

Theoremltaddrp 8716 Adding a positive number to another number increases it. (Contributed by FL, 27-Dec-2007.)

Theoremdifrp 8717 Two ways to say one number is less than another. (Contributed by Mario Carneiro, 21-May-2014.)

Theoremelrpd 8718 Membership in the set of positive reals. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremnnrpd 8719 A positive integer is a positive real. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrpred 8720 A positive real is a real. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrpxrd 8721 A positive real is an extended real. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrpcnd 8722 A positive real is a complex number. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrpgt0d 8723 A positive real is greater than zero. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrpge0d 8724 A positive real is greater than or equal to zero. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrpne0d 8725 A positive real is nonzero. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrpap0d 8726 A positive real is apart from zero. (Contributed by Jim Kingdon, 28-Jul-2021.)
#

Theoremrpregt0d 8727 A positive real is real and greater than zero. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrprege0d 8728 A positive real is real and greater or equal to zero. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrprene0d 8729 A positive real is a nonzero real number. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrpcnne0d 8730 A positive real is a nonzero complex number. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrpreccld 8731 Closure law for reciprocation of positive reals. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrprecred 8732 Closure law for reciprocation of positive reals. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrphalfcld 8733 Closure law for half of a positive real. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremreclt1d 8734 The reciprocal of a positive number less than 1 is greater than 1. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrecgt1d 8735 The reciprocal of a positive number greater than 1 is less than 1. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrpaddcld 8736 Closure law for addition of positive reals. Part of Axiom 7 of [Apostol] p. 20. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrpmulcld 8737 Closure law for multiplication of positive reals. Part of Axiom 7 of [Apostol] p. 20. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrpdivcld 8738 Closure law for division of positive reals. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltrecd 8739 The reciprocal of both sides of 'less than'. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremlerecd 8740 The reciprocal of both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltrec1d 8741 Reciprocal swap in a 'less than' relation. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremlerec2d 8742 Reciprocal swap in a 'less than or equal to' relation. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremlediv2ad 8743 Division of both sides of 'less than or equal to' into a nonnegative number. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltdiv2d 8744 Division of a positive number by both sides of 'less than'. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremlediv2d 8745 Division of a positive number by both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremledivdivd 8746 Invert ratios of positive numbers and swap their ordering. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremdivge1 8747 The ratio of a number over a smaller positive number is larger than 1. (Contributed by Glauco Siliprandi, 5-Apr-2020.)

Theoremdivlt1lt 8748 A real number divided by a positive real number is less than 1 iff the real number is less than the positive real number. (Contributed by AV, 25-May-2020.)

Theoremdivle1le 8749 A real number divided by a positive real number is less than or equal to 1 iff the real number is less than or equal to the positive real number. (Contributed by AV, 29-Jun-2021.)

Theoremledivge1le 8750 If a number is less than or equal to another number, the number divided by a positive number greater than or equal to one is less than or equal to the other number. (Contributed by AV, 29-Jun-2021.)

Theoremge0p1rpd 8751 A nonnegative number plus one is a positive number. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrerpdivcld 8752 Closure law for division of a real by a positive real. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltsubrpd 8753 Subtracting a positive real from another number decreases it. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltaddrpd 8754 Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltaddrp2d 8755 Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltmulgt11d 8756 Multiplication by a number greater than 1. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltmulgt12d 8757 Multiplication by a number greater than 1. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremgt0divd 8758 Division of a positive number by a positive number. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremge0divd 8759 Division of a nonnegative number by a positive number. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrpgecld 8760 A number greater or equal to a positive real is positive real. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremdivge0d 8761 The ratio of nonnegative and positive numbers is nonnegative. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltmul1d 8762 The ratio of nonnegative and positive numbers is nonnegative. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltmul2d 8763 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.)

Theoremlemul1d 8764 Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremlemul2d 8765 Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltdiv1d 8766 Division of both sides of 'less than' by a positive number. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremlediv1d 8767 Division of both sides of a less than or equal to relation by a positive number. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltmuldivd 8768 'Less than' relationship between division and multiplication. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltmuldiv2d 8769 'Less than' relationship between division and multiplication. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremlemuldivd 8770 'Less than or equal to' relationship between division and multiplication. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremlemuldiv2d 8771 'Less than or equal to' relationship between division and multiplication. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremltdivmuld 8772 'Less than' relationship between division and multiplication. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltdivmul2d 8773 'Less than' relationship between division and multiplication. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremledivmuld 8774 'Less than or equal to' relationship between division and multiplication. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremledivmul2d 8775 'Less than or equal to' relationship between division and multiplication. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltmul1dd 8776 The ratio of nonnegative and positive numbers is nonnegative. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremltmul2dd 8777 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.)

Theoremltdiv1dd 8778 Division of both sides of 'less than' by a positive number. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremlediv1dd 8779 Division of both sides of a less than or equal to relation by a positive number. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremlediv12ad 8780 Comparison of ratio of two nonnegative numbers. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltdiv23d 8781 Swap denominator with other side of 'less than'. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremlediv23d 8782 Swap denominator with other side of 'less than or equal to'. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremlt2mul2divd 8783 The ratio of nonnegative and positive numbers is nonnegative. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremnnledivrp 8784 Division of a positive integer by a positive number is less than or equal to the integer iff the number is greater than or equal to 1. (Contributed by AV, 19-Jun-2021.)

Theoremnn0ledivnn 8785 Division of a nonnegative integer by a positive integer is less than or equal to the integer. (Contributed by AV, 19-Jun-2021.)

Theoremaddlelt 8786 If the sum of a real number and a positive real number is less than or equal to a third real number, the first real number is less than the third real number. (Contributed by AV, 1-Jul-2021.)

3.5.2  Infinity and the extended real number system (cont.)

Syntaxcxne 8787 Extend class notation to include the negative of an extended real.

Syntaxcxmu 8789 Extend class notation to include multiplication of extended reals.

Definitiondf-xneg 8790 Define the negative of an extended real number. (Contributed by FL, 26-Dec-2011.)

Definitiondf-xadd 8791* Define addition over extended real numbers. (Contributed by Mario Carneiro, 20-Aug-2015.)

Definitiondf-xmul 8792* Define multiplication over extended real numbers. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theorempnfxr 8793 Plus infinity belongs to the set of extended reals. (Contributed by NM, 13-Oct-2005.) (Proof shortened by Anthony Hart, 29-Aug-2011.)

Theorempnfex 8794 Plus infinity exists (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)

Theoremmnfxr 8795 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.)

Theoremltxr 8796 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.)

Theoremelxr 8797 Membership in the set of extended reals. (Contributed by NM, 14-Oct-2005.)

Theorempnfnemnf 8798 Plus and minus infinity are different elements of . (Contributed by NM, 14-Oct-2005.)

Theoremmnfnepnf 8799 Minus and plus infinity are different (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)

Theoremxrnemnf 8800 An extended real other than minus infinity is real or positive infinite. (Contributed by Mario Carneiro, 20-Aug-2015.)

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