P. 102 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 10101-10200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	lediv2d 10101	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 10102	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 10103	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 10104	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 10105	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 10106	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 10107	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 10108	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 10109	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 10110	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 10111	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 10112	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 10113	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 10114	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 10115	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 10116	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 10117	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 10118	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 10119	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 10120	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 10121	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 10122	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 10123	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 10124	'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 10125	'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 10126	'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 10127	'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 10128	'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 10129	'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 10130	'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 10131	'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 10132	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 10133	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 10134	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 10135	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 10136	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 10137	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 10138	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 )
Theorem	mul2lt0rlt0 10139	If the result of a multiplication is strictly negative, then multiplicands are of different signs. (Contributed by Thierry Arnoux, 19-Sep-2018.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> ( A x. B ) < 0 )   =>    |- ( ( ph /\ B < 0 ) -> 0 < A )
Theorem	mul2lt0rgt0 10140	If the result of a multiplication is strictly negative, then multiplicands are of different signs. (Contributed by Thierry Arnoux, 19-Sep-2018.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> ( A x. B ) < 0 )   =>    |- ( ( ph /\ 0 < B ) -> A < 0 )
Theorem	mul2lt0llt0 10141	If the result of a multiplication is strictly negative, then multiplicands are of different signs. (Contributed by Thierry Arnoux, 19-Sep-2018.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> ( A x. B ) < 0 )   =>    |- ( ( ph /\ A < 0 ) -> 0 < B )
Theorem	mul2lt0lgt0 10142	If the result of a multiplication is strictly negative, then multiplicands are of different signs. (Contributed by Thierry Arnoux, 2-Oct-2018.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> ( A x. B ) < 0 )   =>    |- ( ( ph /\ 0 < A ) -> B < 0 )
Theorem	mul2lt0np 10143	The product of multiplicands of different signs is negative. (Contributed by Jim Kingdon, 25-Feb-2024.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> A < 0 )   &    |- ( ph -> 0 < B )   =>    |- ( ph -> ( A x. B ) < 0 )
Theorem	mul2lt0pn 10144	The product of multiplicands of different signs is negative. (Contributed by Jim Kingdon, 25-Feb-2024.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> A < 0 )   &    |- ( ph -> 0 < B )   =>    |- ( ph -> ( B x. A ) < 0 )
Theorem	lt2mul2divd 10145	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 ) ) )
Theorem	nnledivrp 10146	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 ) )
Theorem	nn0ledivnn 10147	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 )
Theorem	addlelt 10148	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 ) )
Theorem	ltesubnnd 10149	Subtracting an integer number from another number decreases it. See ltsubrpd 10109. (Contributed by Thierry Arnoux, 18-Apr-2017.)
|- ( ph -> M e. ZZ )   &    |- ( ph -> N e. NN )   =>    |- ( ph -> ( ( M + 1 ) - N ) <_ M )
4.5.2  Infinity and the extended real number system (cont.)
Syntax	cxne 10150	Extend class notation to include the negative of an extended real.
class -e A
Syntax	cxad 10151	Extend class notation to include addition of extended reals.
class +e
Syntax	cxmu 10152	Extend class notation to include multiplication of extended reals.
class xe
Definition	df-xneg 10153	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 ) )
Definition	df-xadd 10154*	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 ) ) ) ) ) )
Definition	df-xmul 10155*	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 ) ) ) ) )
Theorem	ltxr 10156	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 ) ) ) ) )
Theorem	elxr 10157	Membership in the set of extended reals. (Contributed by NM, 14-Oct-2005.)
|- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
Theorem	xrnemnf 10158	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 ) )
Theorem	xrnepnf 10159	An extended real other than plus infinity is real or negative infinite. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( ( A e. RR* /\ A =/= +oo ) <-> ( A e. RR \/ A = -oo ) )
Theorem	xrltnr 10160	The extended real 'less than' is irreflexive. (Contributed by NM, 14-Oct-2005.)
|- ( A e. RR* -> -. A < A )
Theorem	ltpnf 10161	Any (finite) real is less than plus infinity. (Contributed by NM, 14-Oct-2005.)
|- ( A e. RR -> A < +oo )
Theorem	ltpnfd 10162	Any (finite) real is less than plus infinity. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ph -> A e. RR )   =>    |- ( ph -> A < +oo )
Theorem	0ltpnf 10163	Zero is less than plus infinity (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- 0 < +oo
Theorem	mnflt 10164	Minus infinity is less than any (finite) real. (Contributed by NM, 14-Oct-2005.)
|- ( A e. RR -> -oo < A )
Theorem	mnflt0 10165	Minus infinity is less than 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- -oo < 0
Theorem	mnfltpnf 10166	Minus infinity is less than plus infinity. (Contributed by NM, 14-Oct-2005.)
|- -oo < +oo
Theorem	mnfltxr 10167	Minus infinity is less than an extended real that is either real or plus infinity. (Contributed by NM, 2-Feb-2006.)
|- ( ( A e. RR \/ A = +oo ) -> -oo < A )
Theorem	pnfnlt 10168	No extended real is greater than plus infinity. (Contributed by NM, 15-Oct-2005.)
|- ( A e. RR* -> -. +oo < A )
Theorem	nltmnf 10169	No extended real is less than minus infinity. (Contributed by NM, 15-Oct-2005.)
|- ( A e. RR* -> -. A < -oo )
Theorem	pnfge 10170	Plus infinity is an upper bound for extended reals. (Contributed by NM, 30-Jan-2006.)
|- ( A e. RR* -> A <_ +oo )
Theorem	0lepnf 10171	0 less than or equal to positive infinity. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- 0 <_ +oo
Theorem	nn0pnfge0 10172	If a number is a nonnegative integer or positive infinity, it is greater than or equal to 0. (Contributed by Alexander van der Vekens, 6-Jan-2018.)
|- ( ( N e. NN0 \/ N = +oo ) -> 0 <_ N )
Theorem	mnfle 10173	Minus infinity is less than or equal to any extended real. (Contributed by NM, 19-Jan-2006.)
|- ( A e. RR* -> -oo <_ A )
Theorem	xrltnsym 10174	Ordering on the extended reals is not symmetric. (Contributed by NM, 15-Oct-2005.)
|- ( ( A e. RR* /\ B e. RR* ) -> ( A < B -> -. B < A ) )
Theorem	xrltnsym2 10175	'Less than' is antisymmetric and irreflexive for extended reals. (Contributed by NM, 6-Feb-2007.)
|- ( ( A e. RR* /\ B e. RR* ) -> -. ( A < B /\ B < A ) )
Theorem	xrlttr 10176	Ordering on the extended reals is transitive. (Contributed by NM, 15-Oct-2005.)
|- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( ( A < B /\ B < C ) -> A < C ) )
Theorem	xrltso 10177	'Less than' is a weakly linear ordering on the extended reals. (Contributed by NM, 15-Oct-2005.)
|- < Or RR*
Theorem	xrlttri3 10178	Extended real version of lttri3 8395. (Contributed by NM, 9-Feb-2006.)
|- ( ( A e. RR* /\ B e. RR* ) -> ( A = B <-> ( -. A < B /\ -. B < A ) ) )
Theorem	xrltle 10179	'Less than' implies 'less than or equal' for extended reals. (Contributed by NM, 19-Jan-2006.)
|- ( ( A e. RR* /\ B e. RR* ) -> ( A < B -> A <_ B ) )
Theorem	xrltled 10180	'Less than' implies 'less than or equal to' for extended reals. Deduction form of xrltle 10179. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ph -> A e. RR* )   &    |- ( ph -> B e. RR* )   &    |- ( ph -> A < B )   =>    |- ( ph -> A <_ B )
Theorem	xrleid 10181	'Less than or equal to' is reflexive for extended reals. (Contributed by NM, 7-Feb-2007.)
|- ( A e. RR* -> A <_ A )
Theorem	xrleidd 10182	'Less than or equal to' is reflexive for extended reals. Deduction form of xrleid 10181. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
|- ( ph -> A e. RR* )   =>    |- ( ph -> A <_ A )
Theorem	xnn0dcle 10183	Decidability of <_ for extended nonnegative integers. (Contributed by Jim Kingdon, 13-Oct-2024.)
|- ( ( A e. NN0* /\ B e. NN0* ) -> DECID A <_ B )
Theorem	xnn0letri 10184	Dichotomy for extended nonnegative integers. (Contributed by Jim Kingdon, 13-Oct-2024.)
|- ( ( A e. NN0* /\ B e. NN0* ) -> ( A <_ B \/ B <_ A ) )
Theorem	xrletri3 10185	Trichotomy law for extended reals. (Contributed by FL, 2-Aug-2009.)
|- ( ( A e. RR* /\ B e. RR* ) -> ( A = B <-> ( A <_ B /\ B <_ A ) ) )
Theorem	xrletrid 10186	Trichotomy law for extended reals. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( ph -> A e. RR* )   &    |- ( ph -> B e. RR* )   &    |- ( ph -> A <_ B )   &    |- ( ph -> B <_ A )   =>    |- ( ph -> A = B )
Theorem	xrlelttr 10187	Transitive law for ordering on extended reals. (Contributed by NM, 19-Jan-2006.)
|- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( ( A <_ B /\ B < C ) -> A < C ) )
Theorem	xrltletr 10188	Transitive law for ordering on extended reals. (Contributed by NM, 19-Jan-2006.)
|- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( ( A < B /\ B <_ C ) -> A < C ) )
Theorem	xrletr 10189	Transitive law for ordering on extended reals. (Contributed by NM, 9-Feb-2006.)
|- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( ( A <_ B /\ B <_ C ) -> A <_ C ) )
Theorem	xrlttrd 10190	Transitive law for ordering on extended reals. (Contributed by Mario Carneiro, 23-Aug-2015.)
|- ( ph -> A e. RR* )   &    |- ( ph -> B e. RR* )   &    |- ( ph -> C e. RR* )   &    |- ( ph -> A < B )   &    |- ( ph -> B < C )   =>    |- ( ph -> A < C )
Theorem	xrlelttrd 10191	Transitive law for ordering on extended reals. (Contributed by Mario Carneiro, 23-Aug-2015.)
|- ( ph -> A e. RR* )   &    |- ( ph -> B e. RR* )   &    |- ( ph -> C e. RR* )   &    |- ( ph -> A <_ B )   &    |- ( ph -> B < C )   =>    |- ( ph -> A < C )
Theorem	xrltletrd 10192	Transitive law for ordering on extended reals. (Contributed by Mario Carneiro, 23-Aug-2015.)
|- ( ph -> A e. RR* )   &    |- ( ph -> B e. RR* )   &    |- ( ph -> C e. RR* )   &    |- ( ph -> A < B )   &    |- ( ph -> B <_ C )   =>    |- ( ph -> A < C )
Theorem	xrletrd 10193	Transitive law for ordering on extended reals. (Contributed by Mario Carneiro, 23-Aug-2015.)
|- ( ph -> A e. RR* )   &    |- ( ph -> B e. RR* )   &    |- ( ph -> C e. RR* )   &    |- ( ph -> A <_ B )   &    |- ( ph -> B <_ C )   =>    |- ( ph -> A <_ C )
Theorem	xrltne 10194	'Less than' implies not equal for extended reals. (Contributed by NM, 20-Jan-2006.)
|- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> B =/= A )
Theorem	nltpnft 10195	An extended real is not less than plus infinity iff they are equal. (Contributed by NM, 30-Jan-2006.)
|- ( A e. RR* -> ( A = +oo <-> -. A < +oo ) )
Theorem	npnflt 10196	An extended real is less than plus infinity iff they are not equal. (Contributed by Jim Kingdon, 17-Apr-2023.)
|- ( A e. RR* -> ( A < +oo <-> A =/= +oo ) )
Theorem	xgepnf 10197	An extended real which is greater than plus infinity is plus infinity. (Contributed by Thierry Arnoux, 18-Dec-2016.)
|- ( A e. RR* -> ( +oo <_ A <-> A = +oo ) )
Theorem	ngtmnft 10198	An extended real is not greater than minus infinity iff they are equal. (Contributed by NM, 2-Feb-2006.)
|- ( A e. RR* -> ( A = -oo <-> -. -oo < A ) )
Theorem	nmnfgt 10199	An extended real is greater than minus infinite iff they are not equal. (Contributed by Jim Kingdon, 17-Apr-2023.)
|- ( A e. RR* -> ( -oo < A <-> A =/= -oo ) )
Theorem	xrrebnd 10200	An extended real is real iff it is strictly bounded by infinities. (Contributed by NM, 2-Feb-2006.)
|- ( A e. RR* -> ( A e. RR <-> ( -oo < A /\ A < +oo ) ) )