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Theorem List for Intuitionistic Logic Explorer - 9601-9700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremqreccl 9601 Closure of reciprocal of rationals. (Contributed by NM, 3-Aug-2004.)
 |-  ( ( A  e.  QQ  /\  A  =/=  0
 )  ->  ( 1  /  A )  e.  QQ )
 
Theoremqdivcl 9602 Closure of division of rationals. (Contributed by NM, 3-Aug-2004.)
 |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  B  =/=  0 ) 
 ->  ( A  /  B )  e.  QQ )
 
Theoremqrevaddcl 9603 Reverse closure law for addition of rationals. (Contributed by NM, 2-Aug-2004.)
 |-  ( B  e.  QQ  ->  ( ( A  e.  CC  /\  ( A  +  B )  e.  QQ ) 
 <->  A  e.  QQ )
 )
 
Theoremnnrecq 9604 The reciprocal of a positive integer is rational. (Contributed by NM, 17-Nov-2004.)
 |-  ( A  e.  NN  ->  ( 1  /  A )  e.  QQ )
 
Theoremirradd 9605 The sum of an irrational number and a rational number is irrational. (Contributed by NM, 7-Nov-2008.)
 |-  ( ( A  e.  ( RR  \  QQ )  /\  B  e.  QQ )  ->  ( A  +  B )  e.  ( RR  \  QQ ) )
 
Theoremirrmul 9606 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.)
 |-  ( ( A  e.  ( RR  \  QQ )  /\  B  e.  QQ  /\  B  =/=  0 )  ->  ( A  x.  B )  e.  ( RR  \  QQ ) )
 
Theoremelpq 9607* A positive rational is the quotient of two positive integers. (Contributed by AV, 29-Dec-2022.)
 |-  ( ( A  e.  QQ  /\  0  <  A )  ->  E. x  e.  NN  E. y  e.  NN  A  =  ( x  /  y
 ) )
 
Theoremelpqb 9608* A class is a positive rational iff it is the quotient of two positive integers. (Contributed by AV, 30-Dec-2022.)
 |-  ( ( A  e.  QQ  /\  0  <  A ) 
 <-> 
 E. x  e.  NN  E. y  e.  NN  A  =  ( x  /  y
 ) )
 
4.4.13  Complex numbers as pairs of reals
 
Theoremcnref1o 9609* There is a natural one-to-one mapping from  ( RR  X.  RR ) to  CC, where we map  <. x ,  y
>. to  ( x  +  ( _i  x.  y ) ). In our construction of the complex numbers, this is in fact our definition of  CC (see df-c 7780), 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  =  ( x  e.  RR ,  y  e.  RR  |->  ( x  +  ( _i  x.  y
 ) ) )   =>    |-  F : ( RR  X.  RR ) -1-1-onto-> CC
 
4.5  Order sets
 
4.5.1  Positive reals (as a subset of complex numbers)
 
Syntaxcrp 9610 Extend class notation to include the class of positive reals.
 class  RR+
 
Definitiondf-rp 9611 Define the set of positive reals. Definition of positive numbers in [Apostol] p. 20. (Contributed by NM, 27-Oct-2007.)
 |-  RR+  =  { x  e. 
 RR  |  0  < 
 x }
 
Theoremelrp 9612 Membership in the set of positive reals. (Contributed by NM, 27-Oct-2007.)
 |-  ( A  e.  RR+  <->  ( A  e.  RR  /\  0  <  A ) )
 
Theoremelrpii 9613 Membership in the set of positive reals. (Contributed by NM, 23-Feb-2008.)
 |-  A  e.  RR   &    |-  0  <  A   =>    |-  A  e.  RR+
 
Theorem1rp 9614 1 is a positive real. (Contributed by Jeff Hankins, 23-Nov-2008.)
 |-  1  e.  RR+
 
Theorem2rp 9615 2 is a positive real. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  2  e.  RR+
 
Theorem3rp 9616 3 is a positive real. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  3  e.  RR+
 
Theoremrpre 9617 A positive real is a real. (Contributed by NM, 27-Oct-2007.)
 |-  ( A  e.  RR+  ->  A  e.  RR )
 
Theoremrpxr 9618 A positive real is an extended real. (Contributed by Mario Carneiro, 21-Aug-2015.)
 |-  ( A  e.  RR+  ->  A  e.  RR* )
 
Theoremrpcn 9619 A positive real is a complex number. (Contributed by NM, 11-Nov-2008.)
 |-  ( A  e.  RR+  ->  A  e.  CC )
 
Theoremnnrp 9620 A positive integer is a positive real. (Contributed by NM, 28-Nov-2008.)
 |-  ( A  e.  NN  ->  A  e.  RR+ )
 
Theoremrpssre 9621 The positive reals are a subset of the reals. (Contributed by NM, 24-Feb-2008.)
 |-  RR+  C_  RR
 
Theoremrpgt0 9622 A positive real is greater than zero. (Contributed by FL, 27-Dec-2007.)
 |-  ( A  e.  RR+  -> 
 0  <  A )
 
Theoremrpge0 9623 A positive real is greater than or equal to zero. (Contributed by NM, 22-Feb-2008.)
 |-  ( A  e.  RR+  -> 
 0  <_  A )
 
Theoremrpregt0 9624 A positive real is a positive real number. (Contributed by NM, 11-Nov-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
 |-  ( A  e.  RR+  ->  ( A  e.  RR  /\  0  <  A ) )
 
Theoremrprege0 9625 A positive real is a nonnegative real number. (Contributed by Mario Carneiro, 31-Jan-2014.)
 |-  ( A  e.  RR+  ->  ( A  e.  RR  /\  0  <_  A )
 )
 
Theoremrpne0 9626 A positive real is nonzero. (Contributed by NM, 18-Jul-2008.)
 |-  ( A  e.  RR+  ->  A  =/=  0 )
 
Theoremrpap0 9627 A positive real is apart from zero. (Contributed by Jim Kingdon, 22-Mar-2020.)
 |-  ( A  e.  RR+  ->  A #  0 )
 
Theoremrprene0 9628 A positive real is a nonzero real number. (Contributed by NM, 11-Nov-2008.)
 |-  ( A  e.  RR+  ->  ( A  e.  RR  /\  A  =/=  0 ) )
 
Theoremrpreap0 9629 A positive real is a real number apart from zero. (Contributed by Jim Kingdon, 22-Mar-2020.)
 |-  ( A  e.  RR+  ->  ( A  e.  RR  /\  A #  0 ) )
 
Theoremrpcnne0 9630 A positive real is a nonzero complex number. (Contributed by NM, 11-Nov-2008.)
 |-  ( A  e.  RR+  ->  ( A  e.  CC  /\  A  =/=  0 ) )
 
Theoremrpcnap0 9631 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 9632 Quantification over positive reals. (Contributed by NM, 12-Feb-2008.)
 |-  ( A. x  e.  RR+  ph  <->  A. x  e.  RR  ( 0  <  x  -> 
 ph ) )
 
Theoremrexrp 9633 Quantification over positive reals. (Contributed by Mario Carneiro, 21-May-2014.)
 |-  ( E. x  e.  RR+  ph  <->  E. x  e.  RR  ( 0  <  x  /\  ph ) )
 
Theoremrpaddcl 9634 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 9635 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 9636 Closure law for division of positive reals. (Contributed by FL, 27-Dec-2007.)
 |-  ( ( A  e.  RR+  /\  B  e.  RR+ )  ->  ( A  /  B )  e.  RR+ )
 
Theoremrpreccl 9637 Closure law for reciprocation of positive reals. (Contributed by Jeff Hankins, 23-Nov-2008.)
 |-  ( A  e.  RR+  ->  ( 1  /  A )  e.  RR+ )
 
Theoremrphalfcl 9638 Closure law for half of a positive real. (Contributed by Mario Carneiro, 31-Jan-2014.)
 |-  ( A  e.  RR+  ->  ( A  /  2
 )  e.  RR+ )
 
Theoremrpgecl 9639 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 9640 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 9641 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 9642 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 9643 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+ )
 )
 
Theoremnegelrp 9644 Elementhood of a negation in the positive real numbers. (Contributed by Thierry Arnoux, 19-Sep-2018.)
 |-  ( A  e.  RR  ->  ( -u A  e.  RR+  <->  A  <  0 ) )
 
Theoremnegelrpd 9645 The negation of a negative number is in the positive real numbers. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  A  <  0 )   =>    |-  ( ph  ->  -u A  e.  RR+ )
 
Theorem0nrp 9646 Zero is not a positive real. Axiom 9 of [Apostol] p. 20. (Contributed by NM, 27-Oct-2007.)
 |- 
 -.  0  e.  RR+
 
Theoremltsubrp 9647 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 9648 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 9649 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 9650 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 9651 A positive integer is a positive real. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  NN )   =>    |-  ( ph  ->  A  e.  RR+ )
 
Theoremzgt1rpn0n1 9652 An integer greater than 1 is a positive real number not equal to 0 or 1. Useful for working with integer logarithm bases (which is a common case, e.g., base 2, base 3, or base 10). (Contributed by Thierry Arnoux, 26-Sep-2017.) (Proof shortened by AV, 9-Jul-2022.)
 |-  ( B  e.  ( ZZ>=
 `  2 )  ->  ( B  e.  RR+  /\  B  =/=  0  /\  B  =/=  1 ) )
 
Theoremrpred 9653 A positive real is a real. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   =>    |-  ( ph  ->  A  e.  RR )
 
Theoremrpxrd 9654 A positive real is an extended real. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   =>    |-  ( ph  ->  A  e.  RR* )
 
Theoremrpcnd 9655 A positive real is a complex number. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   =>    |-  ( ph  ->  A  e.  CC )
 
Theoremrpgt0d 9656 A positive real is greater than zero. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   =>    |-  ( ph  ->  0  <  A )
 
Theoremrpge0d 9657 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 9658 A positive real is nonzero. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   =>    |-  ( ph  ->  A  =/=  0 )
 
Theoremrpap0d 9659 A positive real is apart from zero. (Contributed by Jim Kingdon, 28-Jul-2021.)
 |-  ( ph  ->  A  e.  RR+ )   =>    |-  ( ph  ->  A #  0 )
 
Theoremrpregt0d 9660 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 9661 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 9662 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 9663 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 9664 Closure law for reciprocation of positive reals. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   =>    |-  ( ph  ->  (
 1  /  A )  e.  RR+ )
 
Theoremrprecred 9665 Closure law for reciprocation of positive reals. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR+ )   =>    |-  ( ph  ->  (
 1  /  A )  e.  RR )
 
Theoremrphalfcld 9666 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 9667 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 9668 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 9669 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 9670 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 9671 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 9672 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 9673 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 9674 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 9675 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 9676 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 9677 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 9678 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 9679 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 9680 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 9681 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 9682 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 9683 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 9684 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 9685 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 9686 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 9687 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 9688 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 9689 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 9690 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 9691 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 9692 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 9693 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 9694 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 9695 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 9696 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 9697 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 9698 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 9699 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 9700 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 ) ) )
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