P. 99 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 9801-9900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	3rp 9801	3 is a positive real. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- 3 e. RR+
Theorem	rpre 9802	A positive real is a real. (Contributed by NM, 27-Oct-2007.)
|- ( A e. RR+ -> A e. RR )
Theorem	rpxr 9803	A positive real is an extended real. (Contributed by Mario Carneiro, 21-Aug-2015.)
|- ( A e. RR+ -> A e. RR* )
Theorem	rpcn 9804	A positive real is a complex number. (Contributed by NM, 11-Nov-2008.)
|- ( A e. RR+ -> A e. CC )
Theorem	nnrp 9805	A positive integer is a positive real. (Contributed by NM, 28-Nov-2008.)
|- ( A e. NN -> A e. RR+ )
Theorem	rpssre 9806	The positive reals are a subset of the reals. (Contributed by NM, 24-Feb-2008.)
|- RR+ C_ RR
Theorem	rpgt0 9807	A positive real is greater than zero. (Contributed by FL, 27-Dec-2007.)
|- ( A e. RR+ -> 0 < A )
Theorem	rpge0 9808	A positive real is greater than or equal to zero. (Contributed by NM, 22-Feb-2008.)
|- ( A e. RR+ -> 0 <_ A )
Theorem	rpregt0 9809	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 ) )
Theorem	rprege0 9810	A positive real is a nonnegative real number. (Contributed by Mario Carneiro, 31-Jan-2014.)
|- ( A e. RR+ -> ( A e. RR /\ 0 <_ A ) )
Theorem	rpne0 9811	A positive real is nonzero. (Contributed by NM, 18-Jul-2008.)
|- ( A e. RR+ -> A =/= 0 )
Theorem	rpap0 9812	A positive real is apart from zero. (Contributed by Jim Kingdon, 22-Mar-2020.)
|- ( A e. RR+ -> A # 0 )
Theorem	rprene0 9813	A positive real is a nonzero real number. (Contributed by NM, 11-Nov-2008.)
|- ( A e. RR+ -> ( A e. RR /\ A =/= 0 ) )
Theorem	rpreap0 9814	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 ) )
Theorem	rpcnne0 9815	A positive real is a nonzero complex number. (Contributed by NM, 11-Nov-2008.)
|- ( A e. RR+ -> ( A e. CC /\ A =/= 0 ) )
Theorem	rpcnap0 9816	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 ) )
Theorem	ralrp 9817	Quantification over positive reals. (Contributed by NM, 12-Feb-2008.)
|- ( A. x e. RR+ ph <-> A. x e. RR ( 0 < x -> ph ) )
Theorem	rexrp 9818	Quantification over positive reals. (Contributed by Mario Carneiro, 21-May-2014.)
|- ( E. x e. RR+ ph <-> E. x e. RR ( 0 < x /\ ph ) )
Theorem	rpaddcl 9819	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+ )
Theorem	rpmulcl 9820	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+ )
Theorem	rpdivcl 9821	Closure law for division of positive reals. (Contributed by FL, 27-Dec-2007.)
|- ( ( A e. RR+ /\ B e. RR+ ) -> ( A / B ) e. RR+ )
Theorem	rpreccl 9822	Closure law for reciprocation of positive reals. (Contributed by Jeff Hankins, 23-Nov-2008.)
|- ( A e. RR+ -> ( 1 / A ) e. RR+ )
Theorem	rphalfcl 9823	Closure law for half of a positive real. (Contributed by Mario Carneiro, 31-Jan-2014.)
|- ( A e. RR+ -> ( A / 2 ) e. RR+ )
Theorem	rpgecl 9824	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+ )
Theorem	rphalflt 9825	Half of a positive real is less than the original number. (Contributed by Mario Carneiro, 21-May-2014.)
|- ( A e. RR+ -> ( A / 2 ) < A )
Theorem	rerpdivcl 9826	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 )
Theorem	ge0p1rp 9827	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+ )
Theorem	rpnegap 9828	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+ ) )
Theorem	negelrp 9829	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 ) )
Theorem	negelrpd 9830	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+ )
Theorem	0nrp 9831	Zero is not a positive real. Axiom 9 of [Apostol] p. 20. (Contributed by NM, 27-Oct-2007.)
|- -. 0 e. RR+
Theorem	ltsubrp 9832	Subtracting a positive real from another number decreases it. (Contributed by FL, 27-Dec-2007.)
|- ( ( A e. RR /\ B e. RR+ ) -> ( A - B ) < A )
Theorem	ltaddrp 9833	Adding a positive number to another number increases it. (Contributed by FL, 27-Dec-2007.)
|- ( ( A e. RR /\ B e. RR+ ) -> A < ( A + B ) )
Theorem	difrp 9834	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+ ) )
Theorem	elrpd 9835	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+ )
Theorem	nnrpd 9836	A positive integer is a positive real. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. NN )   =>    |- ( ph -> A e. RR+ )
Theorem	zgt1rpn0n1 9837	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 ) )
Theorem	rpred 9838	A positive real is a real. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. RR+ )   =>    |- ( ph -> A e. RR )
Theorem	rpxrd 9839	A positive real is an extended real. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. RR+ )   =>    |- ( ph -> A e. RR* )
Theorem	rpcnd 9840	A positive real is a complex number. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. RR+ )   =>    |- ( ph -> A e. CC )
Theorem	rpgt0d 9841	A positive real is greater than zero. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. RR+ )   =>    |- ( ph -> 0 < A )
Theorem	rpge0d 9842	A positive real is greater than or equal to zero. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. RR+ )   =>    |- ( ph -> 0 <_ A )
Theorem	rpne0d 9843	A positive real is nonzero. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. RR+ )   =>    |- ( ph -> A =/= 0 )
Theorem	rpap0d 9844	A positive real is apart from zero. (Contributed by Jim Kingdon, 28-Jul-2021.)
|- ( ph -> A e. RR+ )   =>    |- ( ph -> A # 0 )
Theorem	rpregt0d 9845	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 ) )
Theorem	rprege0d 9846	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 ) )
Theorem	rprene0d 9847	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 ) )
Theorem	rpcnne0d 9848	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 ) )
Theorem	rpreccld 9849	Closure law for reciprocation of positive reals. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. RR+ )   =>    |- ( ph -> ( 1 / A ) e. RR+ )
Theorem	rprecred 9850	Closure law for reciprocation of positive reals. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. RR+ )   =>    |- ( ph -> ( 1 / A ) e. RR )
Theorem	rphalfcld 9851	Closure law for half of a positive real. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. RR+ )   =>    |- ( ph -> ( A / 2 ) e. RR+ )
Theorem	reclt1d 9852	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 ) ) )
Theorem	recgt1d 9853	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 ) )
Theorem	rpaddcld 9854	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+ )
Theorem	rpmulcld 9855	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+ )
Theorem	rpdivcld 9856	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+ )
Theorem	ltrecd 9857	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 ) ) )
Theorem	lerecd 9858	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 ) ) )
Theorem	ltrec1d 9859	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 )
Theorem	lerec2d 9860	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 ) )
Theorem	lediv2ad 9861	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 ) )
Theorem	ltdiv2d 9862	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 ) ) )
Theorem	lediv2d 9863	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 ) ) )
Theorem	ledivdivd 9864	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 ) )
Theorem	divge1 9865	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 ) )
Theorem	divlt1lt 9866	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 ) )
Theorem	divle1le 9867	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 ) )
Theorem	ledivge1le 9868	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 ) )
Theorem	ge0p1rpd 9869	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+ )
Theorem	rerpdivcld 9870	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 )
Theorem	ltsubrpd 9871	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 )
Theorem	ltaddrpd 9872	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 ) )
Theorem	ltaddrp2d 9873	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 ) )
Theorem	ltmulgt11d 9874	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 ) ) )
Theorem	ltmulgt12d 9875	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 ) ) )
Theorem	gt0divd 9876	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 ) ) )
Theorem	ge0divd 9877	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 ) ) )
Theorem	rpgecld 9878	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+ )
Theorem	divge0d 9879	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 ) )
Theorem	ltmul1d 9880	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 ) ) )
Theorem	ltmul2d 9881	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 ) ) )
Theorem	lemul1d 9882	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 ) ) )
Theorem	lemul2d 9883	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 ) ) )
Theorem	ltdiv1d 9884	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 ) ) )
Theorem	lediv1d 9885	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 ) ) )
Theorem	ltmuldivd 9886	'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 ) ) )
Theorem	ltmuldiv2d 9887	'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 ) ) )
Theorem	lemuldivd 9888	'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 ) ) )
Theorem	lemuldiv2d 9889	'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 ) ) )
Theorem	ltdivmuld 9890	'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 ) ) )
Theorem	ltdivmul2d 9891	'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 ) ) )
Theorem	ledivmuld 9892	'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 ) ) )
Theorem	ledivmul2d 9893	'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 ) ) )
Theorem	ltmul1dd 9894	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 ) )
Theorem	ltmul2dd 9895	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 ) )
Theorem	ltdiv1dd 9896	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 ) )
Theorem	lediv1dd 9897	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 ) )
Theorem	lediv12ad 9898	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 ) )
Theorem	ltdiv23d 9899	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 )
Theorem	lediv23d 9900	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 )