P. 80 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 7901-8000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	le2add 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 ) ) )
Theorem	lt2add 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 ) ) )
Theorem	ltleadd 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 ) ) )
Theorem	leltadd 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 ) ) )
Theorem	addgt0 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 ) )
Theorem	addgegt0 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 ) )
Theorem	addgtge0 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 ) )
Theorem	addge0 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 ) )
Theorem	ltaddpos 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 ) ) )
Theorem	ltaddpos2 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 ) ) )
Theorem	ltsubpos 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 ) )
Theorem	posdif 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 ) ) )
Theorem	lesub1 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 ) ) )
Theorem	lesub2 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 ) ) )
Theorem	ltsub1 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 ) ) )
Theorem	ltsub2 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 ) ) )
Theorem	lt2sub 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 ) ) )
Theorem	le2sub 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 ) ) )
Theorem	ltneg 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 ) )
Theorem	ltnegcon1 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 ) )
Theorem	ltnegcon2 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 ) )
Theorem	leneg 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 ) )
Theorem	lenegcon1 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 ) )
Theorem	lenegcon2 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 ) )
Theorem	lt0neg1 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 ) )
Theorem	lt0neg2 7926	Comparison of a number and its negative to zero. (Contributed by NM, 10-May-2004.)
|- ( A e. RR -> ( 0 < A <-> -u A < 0 ) )
Theorem	le0neg1 7927	Comparison of a number and its negative to zero. (Contributed by NM, 10-May-2004.)
|- ( A e. RR -> ( A <_ 0 <-> 0 <_ -u A ) )
Theorem	le0neg2 7928	Comparison of a number and its negative to zero. (Contributed by NM, 24-Aug-1999.)
|- ( A e. RR -> ( 0 <_ A <-> -u A <_ 0 ) )
Theorem	addge01 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 ) ) )
Theorem	addge02 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 ) ) )
Theorem	add20 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 ) ) )
Theorem	subge0 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 ) )
Theorem	suble0 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 ) )
Theorem	leaddle0 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 ) )
Theorem	subge02 7935	Nonnegative subtraction. (Contributed by NM, 27-Jul-2005.)
|- ( ( A e. RR /\ B e. RR ) -> ( 0 <_ B <-> ( A - B ) <_ A ) )
Theorem	lesub0 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 ) )
Theorem	mullt0 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 ) )
Theorem	0le1 7938	0 is less than or equal to 1. (Contributed by Mario Carneiro, 29-Apr-2015.)
|- 0 <_ 1
Theorem	leidi 7939	'Less than or equal to' is reflexive. (Contributed by NM, 18-Aug-1999.)
|- A e. RR   =>    |- A <_ A
Theorem	gt0ne0i 7940	Positive means nonzero (useful for ordering theorems involving division). (Contributed by NM, 16-Sep-1999.)
|- A e. RR   =>    |- ( 0 < A -> A =/= 0 )
Theorem	gt0ne0ii 7941	Positive implies nonzero. (Contributed by NM, 15-May-1999.)
|- A e. RR   &    |- 0 < A   =>    |- A =/= 0
Theorem	addgt0i 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 ) )
Theorem	addge0i 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 ) )
Theorem	addgegt0i 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 ) )
Theorem	addgt0ii 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 )
Theorem	add20i 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 ) ) )
Theorem	ltnegi 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 )
Theorem	lenegi 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 )
Theorem	ltnegcon2i 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 )
Theorem	lesub0i 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 )
Theorem	ltaddposi 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 ) )
Theorem	posdifi 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 ) )
Theorem	ltnegcon1i 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 )
Theorem	lenegcon1i 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 )
Theorem	subge0i 7955	Nonnegative subtraction. (Contributed by NM, 13-Aug-2000.)
|- A e. RR   &    |- B e. RR   =>    |- ( 0 <_ ( A - B ) <-> B <_ A )
Theorem	ltadd1i 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 ) )
Theorem	leadd1i 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 ) )
Theorem	leadd2i 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 ) )
Theorem	ltsubaddi 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 ) )
Theorem	lesubaddi 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 ) )
Theorem	ltsubadd2i 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 ) )
Theorem	lesubadd2i 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 ) )
Theorem	ltaddsubi 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 ) )
Theorem	lt2addi 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 ) )
Theorem	le2addi 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 ) )
Theorem	gt0ne0d 7966	Positive implies nonzero. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> 0 < A )   =>    |- ( ph -> A =/= 0 )
Theorem	lt0ne0d 7967	Something less than zero is not zero. Deduction form. (Contributed by David Moews, 28-Feb-2017.)
|- ( ph -> A < 0 )   =>    |- ( ph -> A =/= 0 )
Theorem	leidd 7968	'Less than or equal to' is reflexive. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   =>    |- ( ph -> A <_ A )
Theorem	lt0neg1d 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 ) )
Theorem	lt0neg2d 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 ) )
Theorem	le0neg1d 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 ) )
Theorem	le0neg2d 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 ) )
Theorem	addgegt0d 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 ) )
Theorem	addgt0d 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 ) )
Theorem	addge0d 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 ) )
Theorem	ltnegd 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 ) )
Theorem	lenegd 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 ) )
Theorem	ltnegcon1d 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 )
Theorem	ltnegcon2d 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 )
Theorem	lenegcon1d 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 )
Theorem	lenegcon2d 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 )
Theorem	ltaddposd 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 ) ) )
Theorem	ltaddpos2d 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 ) ) )
Theorem	ltsubposd 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 ) )
Theorem	posdifd 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 ) ) )
Theorem	addge01d 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 ) ) )
Theorem	addge02d 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 ) ) )
Theorem	subge0d 7988	Nonnegative subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   =>    |- ( ph -> ( 0 <_ ( A - B ) <-> B <_ A ) )
Theorem	suble0d 7989	Nonpositive subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   =>    |- ( ph -> ( ( A - B ) <_ 0 <-> A <_ B ) )
Theorem	subge02d 7990	Nonnegative subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   =>    |- ( ph -> ( 0 <_ B <-> ( A - B ) <_ A ) )
Theorem	ltadd1d 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 ) ) )
Theorem	leadd1d 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 ) ) )
Theorem	leadd2d 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 ) ) )
Theorem	ltsubaddd 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 ) ) )
Theorem	lesubaddd 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 ) ) )
Theorem	ltsubadd2d 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 ) ) )
Theorem	lesubadd2d 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 ) ) )
Theorem	ltaddsubd 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 ) ) )
Theorem	ltaddsub2d 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 ) ) )
Theorem	leaddsub2d 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 ) ) )