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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.)
 |-  ( A  e.  RR+  ->  ( A  e.  CC  /\  A #  0 ) )
 
Theoremralrp 8702 Quantification over positive reals. (Contributed by NM, 12-Feb-2008.)
 |-  ( A. x  e.  RR+  ph  <->  A. x  e.  RR  ( 0  <  x  -> 
 ph ) )
 
Theoremrexrp 8703 Quantification over positive reals. (Contributed by Mario Carneiro, 21-May-2014.)
 |-  ( E. x  e.  RR+  ph  <->  E. x  e.  RR  ( 0  <  x  /\  ph ) )
 
Theoremrpaddcl 8704 Closure law for addition of positive reals. Part of Axiom 7 of [Apostol] p. 20. (Contributed by NM, 27-Oct-2007.)
 |-  ( ( A  e.  RR+  /\  B  e.  RR+ )  ->  ( A  +  B )  e.  RR+ )
 
Theoremrpmulcl 8705 Closure law for multiplication of positive reals. Part of Axiom 7 of [Apostol] p. 20. (Contributed by NM, 27-Oct-2007.)
 |-  ( ( A  e.  RR+  /\  B  e.  RR+ )  ->  ( A  x.  B )  e.  RR+ )
 
Theoremrpdivcl 8706 Closure law for division of positive reals. (Contributed by FL, 27-Dec-2007.)
 |-  ( ( A  e.  RR+  /\  B  e.  RR+ )  ->  ( A  /  B )  e.  RR+ )
 
Theoremrpreccl 8707 Closure law for reciprocation of positive reals. (Contributed by Jeff Hankins, 23-Nov-2008.)
 |-  ( A  e.  RR+  ->  ( 1  /  A )  e.  RR+ )
 
Theoremrphalfcl 8708 Closure law for half of a positive real. (Contributed by Mario Carneiro, 31-Jan-2014.)
 |-  ( A  e.  RR+  ->  ( A  /  2
 )  e.  RR+ )
 
Theoremrpgecl 8709 A number greater or equal to a positive real is positive real. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ( A  e.  RR+  /\  B  e.  RR  /\  A  <_  B )  ->  B  e.  RR+ )
 
Theoremrphalflt 8710 Half of a positive real is less than the original number. (Contributed by Mario Carneiro, 21-May-2014.)
 |-  ( A  e.  RR+  ->  ( A  /  2
 )  <  A )
 
Theoremrerpdivcl 8711 Closure law for division of a real by a positive real. (Contributed by NM, 10-Nov-2008.)
 |-  ( ( A  e.  RR  /\  B  e.  RR+ )  ->  ( A  /  B )  e.  RR )
 
Theoremge0p1rp 8712 A nonnegative number plus one is a positive number. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( ( A  e.  RR  /\  0  <_  A )  ->  ( A  +  1 )  e.  RR+ )
 
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.)
 |-  ( ( A  e.  RR  /\  A #  0 ) 
 ->  ( A  e.  RR+  \/_  -u A  e.  RR+ )
 )
 
Theorem0nrp 8714 Zero is not a positive real. Axiom 9 of [Apostol] p. 20. (Contributed by NM, 27-Oct-2007.)
 |- 
 -.  0  e.  RR+
 
Theoremltsubrp 8715 Subtracting a positive real from another number decreases it. (Contributed by FL, 27-Dec-2007.)
 |-  ( ( A  e.  RR  /\  B  e.  RR+ )  ->  ( A  -  B )  <  A )
 
Theoremltaddrp 8716 Adding a positive number to another number increases it. (Contributed by FL, 27-Dec-2007.)
 |-  ( ( A  e.  RR  /\  B  e.  RR+ )  ->  A  <  ( A  +  B )
 )
 
Theoremdifrp 8717 Two ways to say one number is less than another. (Contributed by Mario Carneiro, 21-May-2014.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <-> 
 ( B  -  A )  e.  RR+ ) )
 
Theoremelrpd 8718 Membership in the set of positive reals. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0  <  A )   =>    |-  ( ph  ->  A  e.  RR+ )
 
Theoremnnrpd 8719 A positive integer is a positive real. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  NN )   =>    |-  ( ph  ->  A  e.  RR+ )
 
Theoremrpred 8720 A positive real is a real. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   =>    |-  ( ph  ->  A  e.  RR )
 
Theoremrpxrd 8721 A positive real is an extended real. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   =>    |-  ( ph  ->  A  e.  RR* )
 
Theoremrpcnd 8722 A positive real is a complex number. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   =>    |-  ( ph  ->  A  e.  CC )
 
Theoremrpgt0d 8723 A positive real is greater than zero. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   =>    |-  ( ph  ->  0  <  A )
 
Theoremrpge0d 8724 A positive real is greater than or equal to zero. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   =>    |-  ( ph  ->  0  <_  A )
 
Theoremrpne0d 8725 A positive real is nonzero. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   =>    |-  ( ph  ->  A  =/=  0 )
 
Theoremrpap0d 8726 A positive real is apart from zero. (Contributed by Jim Kingdon, 28-Jul-2021.)
 |-  ( ph  ->  A  e.  RR+ )   =>    |-  ( ph  ->  A #  0 )
 
Theoremrpregt0d 8727 A positive real is real and greater than zero. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   =>    |-  ( ph  ->  ( A  e.  RR  /\  0  <  A ) )
 
Theoremrprege0d 8728 A positive real is real and greater or equal to zero. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   =>    |-  ( ph  ->  ( A  e.  RR  /\  0  <_  A ) )
 
Theoremrprene0d 8729 A positive real is a nonzero real number. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   =>    |-  ( ph  ->  ( A  e.  RR  /\  A  =/=  0 ) )
 
Theoremrpcnne0d 8730 A positive real is a nonzero complex number. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   =>    |-  ( ph  ->  ( A  e.  CC  /\  A  =/=  0 ) )
 
Theoremrpreccld 8731 Closure law for reciprocation of positive reals. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   =>    |-  ( ph  ->  (
 1  /  A )  e.  RR+ )
 
Theoremrprecred 8732 Closure law for reciprocation of positive reals. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   =>    |-  ( ph  ->  (
 1  /  A )  e.  RR )
 
Theoremrphalfcld 8733 Closure law for half of a positive real. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   =>    |-  ( ph  ->  ( A  /  2 )  e.  RR+ )
 
Theoremreclt1d 8734 The reciprocal of a positive number less than 1 is greater than 1. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   =>    |-  ( ph  ->  ( A  <  1  <->  1  <  (
 1  /  A )
 ) )
 
Theoremrecgt1d 8735 The reciprocal of a positive number greater than 1 is less than 1. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   =>    |-  ( ph  ->  (
 1  <  A  <->  ( 1  /  A )  <  1 ) )
 
Theoremrpaddcld 8736 Closure law for addition of positive reals. Part of Axiom 7 of [Apostol] p. 20. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR+ )   =>    |-  ( ph  ->  ( A  +  B )  e.  RR+ )
 
Theoremrpmulcld 8737 Closure law for multiplication of positive reals. Part of Axiom 7 of [Apostol] p. 20. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR+ )   =>    |-  ( ph  ->  ( A  x.  B )  e.  RR+ )
 
Theoremrpdivcld 8738 Closure law for division of positive reals. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR+ )   =>    |-  ( ph  ->  ( A  /  B )  e.  RR+ )
 
Theoremltrecd 8739 The reciprocal of both sides of 'less than'. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR+ )   =>    |-  ( ph  ->  ( A  <  B  <->  ( 1  /  B )  <  ( 1 
 /  A ) ) )
 
Theoremlerecd 8740 The reciprocal of both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR+ )   =>    |-  ( ph  ->  ( A  <_  B  <->  ( 1  /  B )  <_  ( 1 
 /  A ) ) )
 
Theoremltrec1d 8741 Reciprocal swap in a 'less than' relation. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  ( 1  /  A )  <  B )   =>    |-  ( ph  ->  ( 1  /  B )  <  A )
 
Theoremlerec2d 8742 Reciprocal swap in a 'less than or equal to' relation. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  A 
 <_  ( 1  /  B ) )   =>    |-  ( ph  ->  B  <_  ( 1  /  A ) )
 
Theoremlediv2ad 8743 Division of both sides of 'less than or equal to' into a nonnegative number. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  0  <_  C )   &    |-  ( ph  ->  A  <_  B )   =>    |-  ( ph  ->  ( C  /  B )  <_  ( C  /  A ) )
 
Theoremltdiv2d 8744 Division of a positive number by both sides of 'less than'. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  C  e.  RR+ )   =>    |-  ( ph  ->  ( A  <  B  <->  ( C  /  B )  <  ( C 
 /  A ) ) )
 
Theoremlediv2d 8745 Division of a positive number by both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  C  e.  RR+ )   =>    |-  ( ph  ->  ( A  <_  B  <->  ( C  /  B )  <_  ( C 
 /  A ) ) )
 
Theoremledivdivd 8746 Invert ratios of positive numbers and swap their ordering. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  ( ph  ->  D  e.  RR+ )   &    |-  ( ph  ->  ( A  /  B ) 
 <_  ( C  /  D ) )   =>    |-  ( ph  ->  ( D  /  C )  <_  ( B  /  A ) )
 
Theoremdivge1 8747 The ratio of a number over a smaller positive number is larger than 1. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
 |-  ( ( A  e.  RR+  /\  B  e.  RR  /\  A  <_  B )  -> 
 1  <_  ( B  /  A ) )
 
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.)
 |-  ( ( A  e.  RR  /\  B  e.  RR+ )  ->  ( ( A 
 /  B )  < 
 1 
 <->  A  <  B ) )
 
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.)
 |-  ( ( A  e.  RR  /\  B  e.  RR+ )  ->  ( ( A 
 /  B )  <_ 
 1 
 <->  A  <_  B )
 )
 
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.)
 |-  ( ( A  e.  RR  /\  B  e.  RR+  /\  ( C  e.  RR+  /\  1  <_  C )
 )  ->  ( A  <_  B  ->  ( A  /  C )  <_  B ) )
 
Theoremge0p1rpd 8751 A nonnegative number plus one is a positive number. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0 
 <_  A )   =>    |-  ( ph  ->  ( A  +  1 )  e.  RR+ )
 
Theoremrerpdivcld 8752 Closure law for division of a real by a positive real. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR+ )   =>    |-  ( ph  ->  ( A  /  B )  e. 
 RR )
 
Theoremltsubrpd 8753 Subtracting a positive real from another number decreases it. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR+ )   =>    |-  ( ph  ->  ( A  -  B )  <  A )
 
Theoremltaddrpd 8754 Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR+ )   =>    |-  ( ph  ->  A  <  ( A  +  B ) )
 
Theoremltaddrp2d 8755 Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR+ )   =>    |-  ( ph  ->  A  <  ( B  +  A ) )
 
Theoremltmulgt11d 8756 Multiplication by a number greater than 1. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR+ )   =>    |-  ( ph  ->  (
 1  <  A  <->  B  <  ( B  x.  A ) ) )
 
Theoremltmulgt12d 8757 Multiplication by a number greater than 1. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR+ )   =>    |-  ( ph  ->  (
 1  <  A  <->  B  <  ( A  x.  B ) ) )
 
Theoremgt0divd 8758 Division of a positive number by a positive number. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR+ )   =>    |-  ( ph  ->  (
 0  <  A  <->  0  <  ( A  /  B ) ) )
 
Theoremge0divd 8759 Division of a nonnegative number by a positive number. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR+ )   =>    |-  ( ph  ->  (
 0  <_  A  <->  0  <_  ( A  /  B ) ) )
 
Theoremrpgecld 8760 A number greater or equal to a positive real is positive real. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  B 
 <_  A )   =>    |-  ( ph  ->  A  e.  RR+ )
 
Theoremdivge0d 8761 The ratio of nonnegative and positive numbers is nonnegative. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  0 
 <_  A )   =>    |-  ( ph  ->  0  <_  ( A  /  B ) )
 
Theoremltmul1d 8762 The ratio of nonnegative and positive numbers is nonnegative. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR+ )   =>    |-  ( ph  ->  ( A  <  B  <->  ( A  x.  C )  <  ( B  x.  C ) ) )
 
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.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR+ )   =>    |-  ( ph  ->  ( A  <  B  <->  ( C  x.  A )  <  ( C  x.  B ) ) )
 
Theoremlemul1d 8764 Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR+ )   =>    |-  ( ph  ->  ( A  <_  B  <->  ( A  x.  C )  <_  ( B  x.  C ) ) )
 
Theoremlemul2d 8765 Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR+ )   =>    |-  ( ph  ->  ( A  <_  B  <->  ( C  x.  A )  <_  ( C  x.  B ) ) )
 
Theoremltdiv1d 8766 Division of both sides of 'less than' by a positive number. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR+ )   =>    |-  ( ph  ->  ( A  <  B  <->  ( A  /  C )  <  ( B 
 /  C ) ) )
 
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.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR+ )   =>    |-  ( ph  ->  ( A  <_  B  <->  ( A  /  C )  <_  ( B 
 /  C ) ) )
 
Theoremltmuldivd 8768 'Less than' relationship between division and multiplication. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR+ )   =>    |-  ( ph  ->  (
 ( A  x.  C )  <  B  <->  A  <  ( B 
 /  C ) ) )
 
Theoremltmuldiv2d 8769 'Less than' relationship between division and multiplication. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR+ )   =>    |-  ( ph  ->  (
 ( C  x.  A )  <  B  <->  A  <  ( B 
 /  C ) ) )
 
Theoremlemuldivd 8770 'Less than or equal to' relationship between division and multiplication. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR+ )   =>    |-  ( ph  ->  (
 ( A  x.  C )  <_  B  <->  A  <_  ( B 
 /  C ) ) )
 
Theoremlemuldiv2d 8771 'Less than or equal to' relationship between division and multiplication. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR+ )   =>    |-  ( ph  ->  (
 ( C  x.  A )  <_  B  <->  A  <_  ( B 
 /  C ) ) )
 
Theoremltdivmuld 8772 'Less than' relationship between division and multiplication. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR+ )   =>    |-  ( ph  ->  (
 ( A  /  C )  <  B  <->  A  <  ( C  x.  B ) ) )
 
Theoremltdivmul2d 8773 'Less than' relationship between division and multiplication. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR+ )   =>    |-  ( ph  ->  (
 ( A  /  C )  <  B  <->  A  <  ( B  x.  C ) ) )
 
Theoremledivmuld 8774 'Less than or equal to' relationship between division and multiplication. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR+ )   =>    |-  ( ph  ->  (
 ( A  /  C )  <_  B  <->  A  <_  ( C  x.  B ) ) )
 
Theoremledivmul2d 8775 'Less than or equal to' relationship between division and multiplication. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR+ )   =>    |-  ( ph  ->  (
 ( A  /  C )  <_  B  <->  A  <_  ( B  x.  C ) ) )
 
Theoremltmul1dd 8776 The ratio of nonnegative and positive numbers is nonnegative. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  ( ph  ->  A  <  B )   =>    |-  ( ph  ->  ( A  x.  C )  < 
 ( B  x.  C ) )
 
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.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  ( ph  ->  A  <  B )   =>    |-  ( ph  ->  ( C  x.  A )  < 
 ( C  x.  B ) )
 
Theoremltdiv1dd 8778 Division of both sides of 'less than' by a positive number. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  ( ph  ->  A  <  B )   =>    |-  ( ph  ->  ( A  /  C )  < 
 ( B  /  C ) )
 
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.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  ( ph  ->  A  <_  B )   =>    |-  ( ph  ->  ( A  /  C )  <_  ( B  /  C ) )
 
Theoremlediv12ad 8780 Comparison of ratio of two nonnegative numbers. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  ( ph  ->  D  e.  RR )   &    |-  ( ph  ->  0 
 <_  A )   &    |-  ( ph  ->  A 
 <_  B )   &    |-  ( ph  ->  C 
 <_  D )   =>    |-  ( ph  ->  ( A  /  D )  <_  ( B  /  C ) )
 
Theoremltdiv23d 8781 Swap denominator with other side of 'less than'. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  ( ph  ->  ( A  /  B )  <  C )   =>    |-  ( ph  ->  ( A  /  C )  <  B )
 
Theoremlediv23d 8782 Swap denominator with other side of 'less than or equal to'. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  ( ph  ->  ( A  /  B ) 
 <_  C )   =>    |-  ( ph  ->  ( A  /  C )  <_  B )
 
Theoremlt2mul2divd 8783 The ratio of nonnegative and positive numbers is nonnegative. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  D  e.  RR+ )   =>    |-  ( ph  ->  (
 ( A  x.  B )  <  ( C  x.  D )  <->  ( A  /  D )  <  ( C 
 /  B ) ) )
 
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.)
 |-  ( ( A  e.  NN  /\  B  e.  RR+ )  ->  ( 1  <_  B 
 <->  ( A  /  B )  <_  A ) )
 
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.)
 |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  ( A  /  B )  <_  A )
 
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.)
 |-  ( ( M  e.  RR  /\  N  e.  RR  /\  A  e.  RR+ )  ->  ( ( M  +  A )  <_  N  ->  M  <  N ) )
 
3.5.2  Infinity and the extended real number system (cont.)
 
Syntaxcxne 8787 Extend class notation to include the negative of an extended real.
 class  -e A
 
Syntaxcxad 8788 Extend class notation to include addition of extended reals.
 class  +e
 
Syntaxcxmu 8789 Extend class notation to include multiplication of extended reals.
 class  xe
 
Definitiondf-xneg 8790 Define the negative of an extended real number. (Contributed by FL, 26-Dec-2011.)
 |-  -e A  =  if ( A  = +oo , -oo ,  if ( A  = -oo , +oo ,  -u A ) )
 
Definitiondf-xadd 8791* Define addition over extended real numbers. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |- 
 +e  =  ( x  e.  RR* ,  y  e.  RR*  |->  if ( x  = +oo ,  if ( y  = -oo ,  0 , +oo ) ,  if ( x  = -oo ,  if ( y  = +oo ,  0 , -oo ) ,  if ( y  = +oo , +oo ,  if ( y  = -oo , -oo ,  ( x  +  y ) ) ) ) ) )
 
Definitiondf-xmul 8792* Define multiplication over extended real numbers. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  xe  =  ( x  e.  RR* ,  y  e.  RR*  |->  if ( ( x  =  0  \/  y  =  0 ) ,  0 ,  if (
 ( ( ( 0  <  y  /\  x  = +oo )  \/  (
 y  <  0  /\  x  = -oo ) )  \/  ( ( 0  <  x  /\  y  = +oo )  \/  ( x  <  0  /\  y  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  <  y  /\  x  = -oo )  \/  ( y  <  0  /\  x  = +oo ) )  \/  (
 ( 0  <  x  /\  y  = -oo )  \/  ( x  < 
 0  /\  y  = +oo ) ) ) , -oo ,  ( x  x.  y ) ) ) ) )
 
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.)
 |- +oo  e.  RR*
 
Theorempnfex 8794 Plus infinity exists (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |- +oo  e.  _V
 
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.)
 |- -oo  e.  RR*
 
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.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  <->  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <RR  B )  \/  ( A  = -oo  /\  B  = +oo ) )  \/  ( ( A  e.  RR  /\  B  = +oo )  \/  ( A  = -oo  /\  B  e.  RR ) ) ) ) )
 
Theoremelxr 8797 Membership in the set of extended reals. (Contributed by NM, 14-Oct-2005.)
 |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
 
Theorempnfnemnf 8798 Plus and minus infinity are different elements of  RR*. (Contributed by NM, 14-Oct-2005.)
 |- +oo  =/= -oo
 
Theoremmnfnepnf 8799 Minus and plus infinity are different (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |- -oo  =/= +oo
 
Theoremxrnemnf 8800 An extended real other than minus infinity is real or positive infinite. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ( A  e.  RR*  /\  A  =/= -oo )  <->  ( A  e.  RR  \/  A  = +oo ) )
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