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Theorem List for Intuitionistic Logic Explorer - 7901-8000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremle2add 7901 Adding both sides of two 'less than or equal to' relations. (Contributed by NM, 17-Apr-2005.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  ( ( A  <_  C 
 /\  B  <_  D )  ->  ( A  +  B )  <_  ( C  +  D ) ) )
 
Theoremlt2add 7902 Adding both sides of two 'less than' relations. Theorem I.25 of [Apostol] p. 20. (Contributed by NM, 15-Aug-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  ( ( A  <  C 
 /\  B  <  D )  ->  ( A  +  B )  <  ( C  +  D ) ) )
 
Theoremltleadd 7903 Adding both sides of two orderings. (Contributed by NM, 23-Dec-2007.)
 |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  ( ( A  <  C 
 /\  B  <_  D )  ->  ( A  +  B )  <  ( C  +  D ) ) )
 
Theoremleltadd 7904 Adding both sides of two orderings. (Contributed by NM, 15-Aug-2008.)
 |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  ( ( A  <_  C 
 /\  B  <  D )  ->  ( A  +  B )  <  ( C  +  D ) ) )
 
Theoremaddgt0 7905 The sum of 2 positive numbers is positive. (Contributed by NM, 1-Jun-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
 |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
 0  <  A  /\  0  <  B ) ) 
 ->  0  <  ( A  +  B ) )
 
Theoremaddgegt0 7906 The sum of nonnegative and positive numbers is positive. (Contributed by NM, 28-Dec-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
 |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
 0  <_  A  /\  0  <  B ) ) 
 ->  0  <  ( A  +  B ) )
 
Theoremaddgtge0 7907 The sum of nonnegative and positive numbers is positive. (Contributed by NM, 28-Dec-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
 |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
 0  <  A  /\  0  <_  B ) ) 
 ->  0  <  ( A  +  B ) )
 
Theoremaddge0 7908 The sum of 2 nonnegative numbers is nonnegative. (Contributed by NM, 17-Mar-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
 |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
 0  <_  A  /\  0  <_  B ) ) 
 ->  0  <_  ( A  +  B ) )
 
Theoremltaddpos 7909 Adding a positive number to another number increases it. (Contributed by NM, 17-Nov-2004.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 0  <  A 
 <->  B  <  ( B  +  A ) ) )
 
Theoremltaddpos2 7910 Adding a positive number to another number increases it. (Contributed by NM, 8-Apr-2005.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 0  <  A 
 <->  B  <  ( A  +  B ) ) )
 
Theoremltsubpos 7911 Subtracting a positive number from another number decreases it. (Contributed by NM, 17-Nov-2004.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 0  <  A 
 <->  ( B  -  A )  <  B ) )
 
Theoremposdif 7912 Comparison of two numbers whose difference is positive. (Contributed by NM, 17-Nov-2004.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <-> 
 0  <  ( B  -  A ) ) )
 
Theoremlesub1 7913 Subtraction from both sides of 'less than or equal to'. (Contributed by NM, 13-May-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <_  B  <->  ( A  -  C ) 
 <_  ( B  -  C ) ) )
 
Theoremlesub2 7914 Subtraction of both sides of 'less than or equal to'. (Contributed by NM, 29-Sep-2005.) (Revised by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <_  B  <->  ( C  -  B ) 
 <_  ( C  -  A ) ) )
 
Theoremltsub1 7915 Subtraction from both sides of 'less than'. (Contributed by FL, 3-Jan-2008.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <  B  <->  ( A  -  C )  <  ( B  -  C ) ) )
 
Theoremltsub2 7916 Subtraction of both sides of 'less than'. (Contributed by NM, 29-Sep-2005.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <  B  <->  ( C  -  B )  <  ( C  -  A ) ) )
 
Theoremlt2sub 7917 Subtracting both sides of two 'less than' relations. (Contributed by Mario Carneiro, 14-Apr-2016.)
 |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  ( ( A  <  C 
 /\  D  <  B )  ->  ( A  -  B )  <  ( C  -  D ) ) )
 
Theoremle2sub 7918 Subtracting both sides of two 'less than or equal to' relations. (Contributed by Mario Carneiro, 14-Apr-2016.)
 |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  ( ( A  <_  C 
 /\  D  <_  B )  ->  ( A  -  B )  <_  ( C  -  D ) ) )
 
Theoremltneg 7919 Negative of both sides of 'less than'. Theorem I.23 of [Apostol] p. 20. (Contributed by NM, 27-Aug-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <->  -u B  <  -u A ) )
 
Theoremltnegcon1 7920 Contraposition of negative in 'less than'. (Contributed by NM, 8-Nov-2004.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( -u A  <  B  <->  -u B  <  A ) )
 
Theoremltnegcon2 7921 Contraposition of negative in 'less than'. (Contributed by Mario Carneiro, 25-Feb-2015.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  -u B  <->  B  <  -u A ) )
 
Theoremleneg 7922 Negative of both sides of 'less than or equal to'. (Contributed by NM, 12-Sep-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <_  B  <->  -u B  <_  -u A ) )
 
Theoremlenegcon1 7923 Contraposition of negative in 'less than or equal to'. (Contributed by NM, 10-May-2004.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( -u A  <_  B  <->  -u B  <_  A ) )
 
Theoremlenegcon2 7924 Contraposition of negative in 'less than or equal to'. (Contributed by NM, 8-Oct-2005.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <_  -u B 
 <->  B  <_  -u A ) )
 
Theoremlt0neg1 7925 Comparison of a number and its negative to zero. Theorem I.23 of [Apostol] p. 20. (Contributed by NM, 14-May-1999.)
 |-  ( A  e.  RR  ->  ( A  <  0  <->  0  <  -u A ) )
 
Theoremlt0neg2 7926 Comparison of a number and its negative to zero. (Contributed by NM, 10-May-2004.)
 |-  ( A  e.  RR  ->  ( 0  <  A  <->  -u A  <  0 ) )
 
Theoremle0neg1 7927 Comparison of a number and its negative to zero. (Contributed by NM, 10-May-2004.)
 |-  ( A  e.  RR  ->  ( A  <_  0  <->  0 
 <_  -u A ) )
 
Theoremle0neg2 7928 Comparison of a number and its negative to zero. (Contributed by NM, 24-Aug-1999.)
 |-  ( A  e.  RR  ->  ( 0  <_  A  <->  -u A  <_  0 ) )
 
Theoremaddge01 7929 A number is less than or equal to itself plus a nonnegative number. (Contributed by NM, 21-Feb-2005.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 0  <_  B 
 <->  A  <_  ( A  +  B ) ) )
 
Theoremaddge02 7930 A number is less than or equal to itself plus a nonnegative number. (Contributed by NM, 27-Jul-2005.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 0  <_  B 
 <->  A  <_  ( B  +  A ) ) )
 
Theoremadd20 7931 Two nonnegative numbers are zero iff their sum is zero. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  ->  ( ( A  +  B )  =  0  <->  ( A  =  0  /\  B  =  0 )
 ) )
 
Theoremsubge0 7932 Nonnegative subtraction. (Contributed by NM, 14-Mar-2005.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 0  <_  ( A  -  B ) 
 <->  B  <_  A )
 )
 
Theoremsuble0 7933 Nonpositive subtraction. (Contributed by NM, 20-Mar-2008.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  -  B )  <_ 
 0 
 <->  A  <_  B )
 )
 
Theoremleaddle0 7934 The sum of a real number and a second real number is less then the real number iff the second real number is negative. (Contributed by Alexander van der Vekens, 30-May-2018.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  +  B )  <_  A 
 <->  B  <_  0 )
 )
 
Theoremsubge02 7935 Nonnegative subtraction. (Contributed by NM, 27-Jul-2005.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 0  <_  B 
 <->  ( A  -  B )  <_  A ) )
 
Theoremlesub0 7936 Lemma to show a nonnegative number is zero. (Contributed by NM, 8-Oct-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 0 
 <_  A  /\  B  <_  ( B  -  A ) )  <->  A  =  0
 ) )
 
Theoremmullt0 7937 The product of two negative numbers is positive. (Contributed by Jeff Hankins, 8-Jun-2009.)
 |-  ( ( ( A  e.  RR  /\  A  <  0 )  /\  ( B  e.  RR  /\  B  <  0 ) )  -> 
 0  <  ( A  x.  B ) )
 
Theorem0le1 7938 0 is less than or equal to 1. (Contributed by Mario Carneiro, 29-Apr-2015.)
 |-  0  <_  1
 
Theoremleidi 7939 'Less than or equal to' is reflexive. (Contributed by NM, 18-Aug-1999.)
 |-  A  e.  RR   =>    |-  A  <_  A
 
Theoremgt0ne0i 7940 Positive means nonzero (useful for ordering theorems involving division). (Contributed by NM, 16-Sep-1999.)
 |-  A  e.  RR   =>    |-  ( 0  <  A  ->  A  =/=  0
 )
 
Theoremgt0ne0ii 7941 Positive implies nonzero. (Contributed by NM, 15-May-1999.)
 |-  A  e.  RR   &    |-  0  <  A   =>    |-  A  =/=  0
 
Theoremaddgt0i 7942 Addition of 2 positive numbers is positive. (Contributed by NM, 16-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( ( 0  <  A  /\  0  <  B )  ->  0  <  ( A  +  B )
 )
 
Theoremaddge0i 7943 Addition of 2 nonnegative numbers is nonnegative. (Contributed by NM, 28-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( ( 0  <_  A  /\  0  <_  B )  ->  0  <_  ( A  +  B )
 )
 
Theoremaddgegt0i 7944 Addition of nonnegative and positive numbers is positive. (Contributed by NM, 25-Sep-1999.) (Revised by Mario Carneiro, 27-May-2016.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( ( 0  <_  A  /\  0  <  B )  ->  0  <  ( A  +  B )
 )
 
Theoremaddgt0ii 7945 Addition of 2 positive numbers is positive. (Contributed by NM, 18-May-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   &    |-  0  <  A   &    |-  0  <  B   =>    |-  0  <  ( A  +  B )
 
Theoremadd20i 7946 Two nonnegative numbers are zero iff their sum is zero. (Contributed by NM, 28-Jul-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( ( 0  <_  A  /\  0  <_  B )  ->  ( ( A  +  B )  =  0  <->  ( A  =  0  /\  B  =  0 ) ) )
 
Theoremltnegi 7947 Negative of both sides of 'less than'. Theorem I.23 of [Apostol] p. 20. (Contributed by NM, 21-Jan-1997.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( A  <  B  <->  -u B  <  -u A )
 
Theoremlenegi 7948 Negative of both sides of 'less than or equal to'. (Contributed by NM, 1-Aug-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( A  <_  B  <->  -u B  <_  -u A )
 
Theoremltnegcon2i 7949 Contraposition of negative in 'less than'. (Contributed by NM, 14-May-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( A  <  -u B  <->  B  <  -u A )
 
Theoremlesub0i 7950 Lemma to show a nonnegative number is zero. (Contributed by NM, 8-Oct-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( ( 0  <_  A  /\  B  <_  ( B  -  A ) )  <->  A  =  0 )
 
Theoremltaddposi 7951 Adding a positive number to another number increases it. (Contributed by NM, 25-Aug-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( 0  <  A  <->  B  <  ( B  +  A ) )
 
Theoremposdifi 7952 Comparison of two numbers whose difference is positive. (Contributed by NM, 19-Aug-2001.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( A  <  B  <->  0  <  ( B  -  A ) )
 
Theoremltnegcon1i 7953 Contraposition of negative in 'less than'. (Contributed by NM, 14-May-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( -u A  <  B  <->  -u B  <  A )
 
Theoremlenegcon1i 7954 Contraposition of negative in 'less than or equal to'. (Contributed by NM, 6-Apr-2005.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( -u A  <_  B  <->  -u B  <_  A )
 
Theoremsubge0i 7955 Nonnegative subtraction. (Contributed by NM, 13-Aug-2000.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( 0  <_  ( A  -  B )  <->  B  <_  A )
 
Theoremltadd1i 7956 Addition to both sides of 'less than'. Theorem I.18 of [Apostol] p. 20. (Contributed by NM, 21-Jan-1997.)
 |-  A  e.  RR   &    |-  B  e.  RR   &    |-  C  e.  RR   =>    |-  ( A  <  B  <->  ( A  +  C )  <  ( B  +  C ) )
 
Theoremleadd1i 7957 Addition to both sides of 'less than or equal to'. (Contributed by NM, 11-Aug-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   &    |-  C  e.  RR   =>    |-  ( A  <_  B  <->  ( A  +  C )  <_  ( B  +  C ) )
 
Theoremleadd2i 7958 Addition to both sides of 'less than or equal to'. (Contributed by NM, 11-Aug-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   &    |-  C  e.  RR   =>    |-  ( A  <_  B  <->  ( C  +  A )  <_  ( C  +  B ) )
 
Theoremltsubaddi 7959 'Less than' relationship between subtraction and addition. (Contributed by NM, 21-Jan-1997.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
 |-  A  e.  RR   &    |-  B  e.  RR   &    |-  C  e.  RR   =>    |-  (
 ( A  -  B )  <  C  <->  A  <  ( C  +  B ) )
 
Theoremlesubaddi 7960 'Less than or equal to' relationship between subtraction and addition. (Contributed by NM, 30-Sep-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
 |-  A  e.  RR   &    |-  B  e.  RR   &    |-  C  e.  RR   =>    |-  (
 ( A  -  B )  <_  C  <->  A  <_  ( C  +  B ) )
 
Theoremltsubadd2i 7961 'Less than' relationship between subtraction and addition. (Contributed by NM, 21-Jan-1997.)
 |-  A  e.  RR   &    |-  B  e.  RR   &    |-  C  e.  RR   =>    |-  (
 ( A  -  B )  <  C  <->  A  <  ( B  +  C ) )
 
Theoremlesubadd2i 7962 'Less than or equal to' relationship between subtraction and addition. (Contributed by NM, 3-Aug-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   &    |-  C  e.  RR   =>    |-  (
 ( A  -  B )  <_  C  <->  A  <_  ( B  +  C ) )
 
Theoremltaddsubi 7963 'Less than' relationship between subtraction and addition. (Contributed by NM, 14-May-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   &    |-  C  e.  RR   =>    |-  (
 ( A  +  B )  <  C  <->  A  <  ( C  -  B ) )
 
Theoremlt2addi 7964 Adding both side of two inequalities. Theorem I.25 of [Apostol] p. 20. (Contributed by NM, 14-May-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   &    |-  C  e.  RR   &    |-  D  e.  RR   =>    |-  ( ( A  <  C 
 /\  B  <  D )  ->  ( A  +  B )  <  ( C  +  D ) )
 
Theoremle2addi 7965 Adding both side of two inequalities. (Contributed by NM, 16-Sep-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   &    |-  C  e.  RR   &    |-  D  e.  RR   =>    |-  ( ( A  <_  C 
 /\  B  <_  D )  ->  ( A  +  B )  <_  ( C  +  D ) )
 
Theoremgt0ne0d 7966 Positive implies nonzero. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  0  <  A )   =>    |-  ( ph  ->  A  =/=  0 )
 
Theoremlt0ne0d 7967 Something less than zero is not zero. Deduction form. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  <  0 )   =>    |-  ( ph  ->  A  =/=  0 )
 
Theoremleidd 7968 'Less than or equal to' is reflexive. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  A  <_  A )
 
Theoremlt0neg1d 7969 Comparison of a number and its negative to zero. Theorem I.23 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  ( A  <  0  <->  0  <  -u A ) )
 
Theoremlt0neg2d 7970 Comparison of a number and its negative to zero. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  (
 0  <  A  <->  -u A  <  0
 ) )
 
Theoremle0neg1d 7971 Comparison of a number and its negative to zero. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  ( A  <_  0  <->  0  <_  -u A ) )
 
Theoremle0neg2d 7972 Comparison of a number and its negative to zero. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  (
 0  <_  A  <->  -u A  <_  0
 ) )
 
Theoremaddgegt0d 7973 Addition of nonnegative and positive numbers is positive. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  0  <_  A )   &    |-  ( ph  ->  0  <  B )   =>    |-  ( ph  ->  0  <  ( A  +  B ) )
 
Theoremaddgt0d 7974 Addition of 2 positive numbers is positive. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  0  <  A )   &    |-  ( ph  ->  0  <  B )   =>    |-  ( ph  ->  0  <  ( A  +  B ) )
 
Theoremaddge0d 7975 Addition of 2 nonnegative numbers is nonnegative. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  0  <_  A )   &    |-  ( ph  ->  0  <_  B )   =>    |-  ( ph  ->  0  <_  ( A  +  B ) )
 
Theoremltnegd 7976 Negative of both sides of 'less than'. Theorem I.23 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( A  <  B  <->  -u B  <  -u A ) )
 
Theoremlenegd 7977 Negative of both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( A  <_  B  <->  -u B  <_  -u A ) )
 
Theoremltnegcon1d 7978 Contraposition of negative in 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  -u A  <  B )   =>    |-  ( ph  ->  -u B  <  A )
 
Theoremltnegcon2d 7979 Contraposition of negative in 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <  -u B )   =>    |-  ( ph  ->  B  < 
 -u A )
 
Theoremlenegcon1d 7980 Contraposition of negative in 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  -u A  <_  B )   =>    |-  ( ph  ->  -u B  <_  A )
 
Theoremlenegcon2d 7981 Contraposition of negative in 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <_  -u B )   =>    |-  ( ph  ->  B  <_ 
 -u A )
 
Theoremltaddposd 7982 Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( 0  <  A  <->  B  <  ( B  +  A ) ) )
 
Theoremltaddpos2d 7983 Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( 0  <  A  <->  B  <  ( A  +  B ) ) )
 
Theoremltsubposd 7984 Subtracting a positive number from another number decreases it. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( 0  <  A  <->  ( B  -  A )  <  B ) )
 
Theoremposdifd 7985 Comparison of two numbers whose difference is positive. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( A  <  B  <->  0  <  ( B  -  A ) ) )
 
Theoremaddge01d 7986 A number is less than or equal to itself plus a nonnegative number. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( 0  <_  B  <->  A  <_  ( A  +  B ) ) )
 
Theoremaddge02d 7987 A number is less than or equal to itself plus a nonnegative number. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( 0  <_  B  <->  A  <_  ( B  +  A ) ) )
 
Theoremsubge0d 7988 Nonnegative subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( 0  <_  ( A  -  B )  <->  B  <_  A ) )
 
Theoremsuble0d 7989 Nonpositive subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( ( A  -  B )  <_  0  <->  A  <_  B ) )
 
Theoremsubge02d 7990 Nonnegative subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( 0  <_  B  <->  ( A  -  B )  <_  A ) )
 
Theoremltadd1d 7991 Addition to both sides of 'less than'. Theorem I.18 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   =>    |-  ( ph  ->  ( A  <  B  <->  ( A  +  C )  <  ( B  +  C ) ) )
 
Theoremleadd1d 7992 Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   =>    |-  ( ph  ->  ( A  <_  B  <->  ( A  +  C )  <_  ( B  +  C ) ) )
 
Theoremleadd2d 7993 Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   =>    |-  ( ph  ->  ( A  <_  B  <->  ( C  +  A )  <_  ( C  +  B ) ) )
 
Theoremltsubaddd 7994 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   =>    |-  ( ph  ->  (
 ( A  -  B )  <  C  <->  A  <  ( C  +  B ) ) )
 
Theoremlesubaddd 7995 'Less than or equal to' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   =>    |-  ( ph  ->  (
 ( A  -  B )  <_  C  <->  A  <_  ( C  +  B ) ) )
 
Theoremltsubadd2d 7996 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   =>    |-  ( ph  ->  (
 ( A  -  B )  <  C  <->  A  <  ( B  +  C ) ) )
 
Theoremlesubadd2d 7997 'Less than or equal to' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   =>    |-  ( ph  ->  (
 ( A  -  B )  <_  C  <->  A  <_  ( B  +  C ) ) )
 
Theoremltaddsubd 7998 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   =>    |-  ( ph  ->  (
 ( A  +  B )  <  C  <->  A  <  ( C  -  B ) ) )
 
Theoremltaddsub2d 7999 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 29-Dec-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   =>    |-  ( ph  ->  (
 ( A  +  B )  <  C  <->  B  <  ( C  -  A ) ) )
 
Theoremleaddsub2d 8000 'Less than or equal to' relationship between and addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   =>    |-  ( ph  ->  (
 ( A  +  B )  <_  C  <->  B  <_  ( C  -  A ) ) )
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