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Theorem List for Intuitionistic Logic Explorer - 8601-8700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremrpreap0 8601 A positive real is a real number apart from zero. (Contributed by Jim Kingdon, 22-Mar-2020.)
#

Theoremrpcnne0 8602 A positive real is a nonzero complex number. (Contributed by NM, 11-Nov-2008.)

Theoremrpcnap0 8603 A positive real is a complex number apart from zero. (Contributed by Jim Kingdon, 22-Mar-2020.)
#

Theoremralrp 8604 Quantification over positive reals. (Contributed by NM, 12-Feb-2008.)

Theoremrexrp 8605 Quantification over positive reals. (Contributed by Mario Carneiro, 21-May-2014.)

Theoremrpaddcl 8606 Closure law for addition of positive reals. Part of Axiom 7 of [Apostol] p. 20. (Contributed by NM, 27-Oct-2007.)

Theoremrpmulcl 8607 Closure law for multiplication of positive reals. Part of Axiom 7 of [Apostol] p. 20. (Contributed by NM, 27-Oct-2007.)

Theoremrpdivcl 8608 Closure law for division of positive reals. (Contributed by FL, 27-Dec-2007.)

Theoremrpreccl 8609 Closure law for reciprocation of positive reals. (Contributed by Jeff Hankins, 23-Nov-2008.)

Theoremrphalfcl 8610 Closure law for half of a positive real. (Contributed by Mario Carneiro, 31-Jan-2014.)

Theoremrpgecl 8611 A number greater or equal to a positive real is positive real. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrphalflt 8612 Half of a positive real is less than the original number. (Contributed by Mario Carneiro, 21-May-2014.)

Theoremrerpdivcl 8613 Closure law for division of a real by a positive real. (Contributed by NM, 10-Nov-2008.)

Theoremge0p1rp 8614 A nonnegative number plus one is a positive number. (Contributed by Mario Carneiro, 5-Oct-2015.)

Theoremrpnegap 8615 Either a real apart from zero or its negation is a positive real, but not both. (Contributed by Jim Kingdon, 23-Mar-2020.)
#

Theorem0nrp 8616 Zero is not a positive real. Axiom 9 of [Apostol] p. 20. (Contributed by NM, 27-Oct-2007.)

Theoremltsubrp 8617 Subtracting a positive real from another number decreases it. (Contributed by FL, 27-Dec-2007.)

Theoremltaddrp 8618 Adding a positive number to another number increases it. (Contributed by FL, 27-Dec-2007.)

Theoremdifrp 8619 Two ways to say one number is less than another. (Contributed by Mario Carneiro, 21-May-2014.)

Theoremelrpd 8620 Membership in the set of positive reals. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremnnrpd 8621 A positive integer is a positive real. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrpred 8622 A positive real is a real. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrpxrd 8623 A positive real is an extended real. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrpcnd 8624 A positive real is a complex number. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrpgt0d 8625 A positive real is greater than zero. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrpge0d 8626 A positive real is greater than or equal to zero. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrpne0d 8627 A positive real is nonzero. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrpap0d 8628 A positive real is apart from zero. (Contributed by Jim Kingdon, 28-Jul-2021.)
#

Theoremrpregt0d 8629 A positive real is real and greater than zero. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrprege0d 8630 A positive real is real and greater or equal to zero. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrprene0d 8631 A positive real is a nonzero real number. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrpcnne0d 8632 A positive real is a nonzero complex number. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrpreccld 8633 Closure law for reciprocation of positive reals. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrprecred 8634 Closure law for reciprocation of positive reals. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrphalfcld 8635 Closure law for half of a positive real. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremreclt1d 8636 The reciprocal of a positive number less than 1 is greater than 1. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrecgt1d 8637 The reciprocal of a positive number greater than 1 is less than 1. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrpaddcld 8638 Closure law for addition of positive reals. Part of Axiom 7 of [Apostol] p. 20. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrpmulcld 8639 Closure law for multiplication of positive reals. Part of Axiom 7 of [Apostol] p. 20. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrpdivcld 8640 Closure law for division of positive reals. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltrecd 8641 The reciprocal of both sides of 'less than'. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremlerecd 8642 The reciprocal of both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltrec1d 8643 Reciprocal swap in a 'less than' relation. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremlerec2d 8644 Reciprocal swap in a 'less than or equal to' relation. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremlediv2ad 8645 Division of both sides of 'less than or equal to' into a nonnegative number. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltdiv2d 8646 Division of a positive number by both sides of 'less than'. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremlediv2d 8647 Division of a positive number by both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremledivdivd 8648 Invert ratios of positive numbers and swap their ordering. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremdivge1 8649 The ratio of a number over a smaller positive number is larger than 1. (Contributed by Glauco Siliprandi, 5-Apr-2020.)

Theoremdivlt1lt 8650 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 8651 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 8652 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 8653 A nonnegative number plus one is a positive number. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrerpdivcld 8654 Closure law for division of a real by a positive real. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltsubrpd 8655 Subtracting a positive real from another number decreases it. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltaddrpd 8656 Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltaddrp2d 8657 Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltmulgt11d 8658 Multiplication by a number greater than 1. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltmulgt12d 8659 Multiplication by a number greater than 1. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremgt0divd 8660 Division of a positive number by a positive number. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremge0divd 8661 Division of a nonnegative number by a positive number. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrpgecld 8662 A number greater or equal to a positive real is positive real. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremdivge0d 8663 The ratio of nonnegative and positive numbers is nonnegative. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltmul1d 8664 The ratio of nonnegative and positive numbers is nonnegative. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltmul2d 8665 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 8666 Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremlemul2d 8667 Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltdiv1d 8668 Division of both sides of 'less than' by a positive number. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremlediv1d 8669 Division of both sides of a less than or equal to relation by a positive number. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltmuldivd 8670 'Less than' relationship between division and multiplication. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltmuldiv2d 8671 'Less than' relationship between division and multiplication. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremlemuldivd 8672 'Less than or equal to' relationship between division and multiplication. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremlemuldiv2d 8673 'Less than or equal to' relationship between division and multiplication. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremltdivmuld 8674 'Less than' relationship between division and multiplication. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltdivmul2d 8675 'Less than' relationship between division and multiplication. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremledivmuld 8676 'Less than or equal to' relationship between division and multiplication. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremledivmul2d 8677 'Less than or equal to' relationship between division and multiplication. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltmul1dd 8678 The ratio of nonnegative and positive numbers is nonnegative. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremltmul2dd 8679 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 8680 Division of both sides of 'less than' by a positive number. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremlediv1dd 8681 Division of both sides of a less than or equal to relation by a positive number. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremlediv12ad 8682 Comparison of ratio of two nonnegative numbers. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltdiv23d 8683 Swap denominator with other side of 'less than'. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremlediv23d 8684 Swap denominator with other side of 'less than or equal to'. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremlt2mul2divd 8685 The ratio of nonnegative and positive numbers is nonnegative. (Contributed by Mario Carneiro, 28-May-2016.)

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

Syntaxcxne 8686 Extend class notation to include the negative of an extended real.

Syntaxcxmu 8688 Extend class notation to include multiplication of extended reals.

Definitiondf-xneg 8689 Define the negative of an extended real number. (Contributed by FL, 26-Dec-2011.)

Definitiondf-xadd 8690* Define addition over extended real numbers. (Contributed by Mario Carneiro, 20-Aug-2015.)

Definitiondf-xmul 8691* Define multiplication over extended real numbers. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theorempnfxr 8692 Plus infinity belongs to the set of extended reals. (Contributed by NM, 13-Oct-2005.) (Proof shortened by Anthony Hart, 29-Aug-2011.)

Theorempnfex 8693 Plus infinity exists (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)

Theoremmnfxr 8694 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 8695 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 8696 Membership in the set of extended reals. (Contributed by NM, 14-Oct-2005.)

Theorempnfnemnf 8697 Plus and minus infinity are different elements of . (Contributed by NM, 14-Oct-2005.)

Theoremmnfnepnf 8698 Minus and plus infinity are different (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)

Theoremxrnemnf 8699 An extended real other than minus infinity is real or positive infinite. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxrnepnf 8700 An extended real other than plus infinity is real or negative infinite. (Contributed by Mario Carneiro, 20-Aug-2015.)

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