P. 88 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 8701-8800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	lbzbi 8701*	If a set of reals is bounded below, it is bounded below by an integer. (Contributed by Paul Chapman, 21-Mar-2011.)
|- ( A C_ RR -> ( E. x e. RR A. y e. A x <_ y <-> E. x e. ZZ A. y e. A x <_ y ) )
Theorem	nn01to3 8702	A (nonnegative) integer between 1 and 3 must be 1, 2 or 3. (Contributed by Alexander van der Vekens, 13-Sep-2018.)
|- ( ( N e. NN0 /\ 1 <_ N /\ N <_ 3 ) -> ( N = 1 \/ N = 2 \/ N = 3 ) )
Theorem	nn0ge2m1nnALT 8703	Alternate proof of nn0ge2m1nn 8348: If a nonnegative integer is greater than or equal to two, the integer decreased by 1 is a positive integer. This version is proved using eluz2 8625, a theorem for upper sets of integers, which are defined later than the positive and nonnegative integers. This proof is, however, much shorter than the proof of nn0ge2m1nn 8348. (Contributed by Alexander van der Vekens, 1-Aug-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
|- ( ( N e. NN0 /\ 2 <_ N ) -> ( N - 1 ) e. NN )
3.4.11  Rational numbers (as a subset of complex numbers)
Syntax	cq 8704	Extend class notation to include the class of rationals.
class QQ
Definition	df-q 8705	Define the set of rational numbers. Based on definition of rationals in [Apostol] p. 22. See elq 8707 for the relation "is rational." (Contributed by NM, 8-Jan-2002.)
|- QQ = ( / " ( ZZ X. NN ) )
Theorem	divfnzn 8706	Division restricted to ZZ X. NN is a function. Given excluded middle, it would be easy to prove this for CC X. ( CC \ { 0 } ). The key difference is that an element of NN is apart from zero, whereas being an element of CC \ { 0 } implies being not equal to zero. (Contributed by Jim Kingdon, 19-Mar-2020.)
|- ( / |` ( ZZ X. NN ) ) Fn ( ZZ X. NN )
Theorem	elq 8707*	Membership in the set of rationals. (Contributed by NM, 8-Jan-2002.) (Revised by Mario Carneiro, 28-Jan-2014.)
|- ( A e. QQ <-> E. x e. ZZ E. y e. NN A = ( x / y ) )
Theorem	qmulz 8708*	If A is rational, then some integer multiple of it is an integer. (Contributed by NM, 7-Nov-2008.) (Revised by Mario Carneiro, 22-Jul-2014.)
|- ( A e. QQ -> E. x e. NN ( A x. x ) e. ZZ )
Theorem	znq 8709	The ratio of an integer and a positive integer is a rational number. (Contributed by NM, 12-Jan-2002.)
|- ( ( A e. ZZ /\ B e. NN ) -> ( A / B ) e. QQ )
Theorem	qre 8710	A rational number is a real number. (Contributed by NM, 14-Nov-2002.)
|- ( A e. QQ -> A e. RR )
Theorem	zq 8711	An integer is a rational number. (Contributed by NM, 9-Jan-2002.)
|- ( A e. ZZ -> A e. QQ )
Theorem	zssq 8712	The integers are a subset of the rationals. (Contributed by NM, 9-Jan-2002.)
|- ZZ C_ QQ
Theorem	nn0ssq 8713	The nonnegative integers are a subset of the rationals. (Contributed by NM, 31-Jul-2004.)
|- NN0 C_ QQ
Theorem	nnssq 8714	The positive integers are a subset of the rationals. (Contributed by NM, 31-Jul-2004.)
|- NN C_ QQ
Theorem	qssre 8715	The rationals are a subset of the reals. (Contributed by NM, 9-Jan-2002.)
|- QQ C_ RR
Theorem	qsscn 8716	The rationals are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.)
|- QQ C_ CC
Theorem	qex 8717	The set of rational numbers exists. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)
|- QQ e. _V
Theorem	nnq 8718	A positive integer is rational. (Contributed by NM, 17-Nov-2004.)
|- ( A e. NN -> A e. QQ )
Theorem	qcn 8719	A rational number is a complex number. (Contributed by NM, 2-Aug-2004.)
|- ( A e. QQ -> A e. CC )
Theorem	qaddcl 8720	Closure of addition of rationals. (Contributed by NM, 1-Aug-2004.)
|- ( ( A e. QQ /\ B e. QQ ) -> ( A + B ) e. QQ )
Theorem	qnegcl 8721	Closure law for the negative of a rational. (Contributed by NM, 2-Aug-2004.) (Revised by Mario Carneiro, 15-Sep-2014.)
|- ( A e. QQ -> -u A e. QQ )
Theorem	qmulcl 8722	Closure of multiplication of rationals. (Contributed by NM, 1-Aug-2004.)
|- ( ( A e. QQ /\ B e. QQ ) -> ( A x. B ) e. QQ )
Theorem	qsubcl 8723	Closure of subtraction of rationals. (Contributed by NM, 2-Aug-2004.)
|- ( ( A e. QQ /\ B e. QQ ) -> ( A - B ) e. QQ )
Theorem	qapne 8724	Apartness is equivalent to not equal for rationals. (Contributed by Jim Kingdon, 20-Mar-2020.)
|- ( ( A e. QQ /\ B e. QQ ) -> ( A # B <-> A =/= B ) )
Theorem	qltlen 8725	Rational 'Less than' expressed in terms of 'less than or equal to'. Also see ltleap 7730 which is a similar result for real numbers. (Contributed by Jim Kingdon, 11-Oct-2021.)
|- ( ( A e. QQ /\ B e. QQ ) -> ( A < B <-> ( A <_ B /\ B =/= A ) ) )
Theorem	qlttri2 8726	Apartness is equivalent to not equal for rationals. (Contributed by Jim Kingdon, 9-Nov-2021.)
|- ( ( A e. QQ /\ B e. QQ ) -> ( A =/= B <-> ( A < B \/ B < A ) ) )
Theorem	qreccl 8727	Closure of reciprocal of rationals. (Contributed by NM, 3-Aug-2004.)
|- ( ( A e. QQ /\ A =/= 0 ) -> ( 1 / A ) e. QQ )
Theorem	qdivcl 8728	Closure of division of rationals. (Contributed by NM, 3-Aug-2004.)
|- ( ( A e. QQ /\ B e. QQ /\ B =/= 0 ) -> ( A / B ) e. QQ )
Theorem	qrevaddcl 8729	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 ) )
Theorem	nnrecq 8730	The reciprocal of a positive integer is rational. (Contributed by NM, 17-Nov-2004.)
|- ( A e. NN -> ( 1 / A ) e. QQ )
Theorem	irradd 8731	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 ) )
Theorem	irrmul 8732	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 ) )
3.4.12  Complex numbers as pairs of reals
Theorem	cnref1o 8733*	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 6987), 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
3.5  Order sets
3.5.1  Positive reals (as a subset of complex numbers)
Syntax	crp 8734	Extend class notation to include the class of positive reals.
class RR+
Definition	df-rp 8735	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 }
Theorem	elrp 8736	Membership in the set of positive reals. (Contributed by NM, 27-Oct-2007.)
|- ( A e. RR+ <-> ( A e. RR /\ 0 < A ) )
Theorem	elrpii 8737	Membership in the set of positive reals. (Contributed by NM, 23-Feb-2008.)
|- A e. RR   &    |- 0 < A   =>    |- A e. RR+
Theorem	1rp 8738	1 is a positive real. (Contributed by Jeff Hankins, 23-Nov-2008.)
|- 1 e. RR+
Theorem	2rp 8739	2 is a positive real. (Contributed by Mario Carneiro, 28-May-2016.)
|- 2 e. RR+
Theorem	rpre 8740	A positive real is a real. (Contributed by NM, 27-Oct-2007.)
|- ( A e. RR+ -> A e. RR )
Theorem	rpxr 8741	A positive real is an extended real. (Contributed by Mario Carneiro, 21-Aug-2015.)
|- ( A e. RR+ -> A e. RR* )
Theorem	rpcn 8742	A positive real is a complex number. (Contributed by NM, 11-Nov-2008.)
|- ( A e. RR+ -> A e. CC )
Theorem	nnrp 8743	A positive integer is a positive real. (Contributed by NM, 28-Nov-2008.)
|- ( A e. NN -> A e. RR+ )
Theorem	rpssre 8744	The positive reals are a subset of the reals. (Contributed by NM, 24-Feb-2008.)
|- RR+ C_ RR
Theorem	rpgt0 8745	A positive real is greater than zero. (Contributed by FL, 27-Dec-2007.)
|- ( A e. RR+ -> 0 < A )
Theorem	rpge0 8746	A positive real is greater than or equal to zero. (Contributed by NM, 22-Feb-2008.)
|- ( A e. RR+ -> 0 <_ A )
Theorem	rpregt0 8747	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 8748	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 8749	A positive real is nonzero. (Contributed by NM, 18-Jul-2008.)
|- ( A e. RR+ -> A =/= 0 )
Theorem	rpap0 8750	A positive real is apart from zero. (Contributed by Jim Kingdon, 22-Mar-2020.)
|- ( A e. RR+ -> A # 0 )
Theorem	rprene0 8751	A positive real is a nonzero real number. (Contributed by NM, 11-Nov-2008.)
|- ( A e. RR+ -> ( A e. RR /\ A =/= 0 ) )
Theorem	rpreap0 8752	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 8753	A positive real is a nonzero complex number. (Contributed by NM, 11-Nov-2008.)
|- ( A e. RR+ -> ( A e. CC /\ A =/= 0 ) )
Theorem	rpcnap0 8754	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 8755	Quantification over positive reals. (Contributed by NM, 12-Feb-2008.)
|- ( A. x e. RR+ ph <-> A. x e. RR ( 0 < x -> ph ) )
Theorem	rexrp 8756	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 8757	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 8758	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 8759	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 8760	Closure law for reciprocation of positive reals. (Contributed by Jeff Hankins, 23-Nov-2008.)
|- ( A e. RR+ -> ( 1 / A ) e. RR+ )
Theorem	rphalfcl 8761	Closure law for half of a positive real. (Contributed by Mario Carneiro, 31-Jan-2014.)
|- ( A e. RR+ -> ( A / 2 ) e. RR+ )
Theorem	rpgecl 8762	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 8763	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 8764	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 8765	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 8766	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	0nrp 8767	Zero is not a positive real. Axiom 9 of [Apostol] p. 20. (Contributed by NM, 27-Oct-2007.)
|- -. 0 e. RR+
Theorem	ltsubrp 8768	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 8769	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 8770	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 8771	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 8772	A positive integer is a positive real. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. NN )   =>    |- ( ph -> A e. RR+ )
Theorem	rpred 8773	A positive real is a real. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. RR+ )   =>    |- ( ph -> A e. RR )
Theorem	rpxrd 8774	A positive real is an extended real. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. RR+ )   =>    |- ( ph -> A e. RR* )
Theorem	rpcnd 8775	A positive real is a complex number. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. RR+ )   =>    |- ( ph -> A e. CC )
Theorem	rpgt0d 8776	A positive real is greater than zero. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. RR+ )   =>    |- ( ph -> 0 < A )
Theorem	rpge0d 8777	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 8778	A positive real is nonzero. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. RR+ )   =>    |- ( ph -> A =/= 0 )
Theorem	rpap0d 8779	A positive real is apart from zero. (Contributed by Jim Kingdon, 28-Jul-2021.)
|- ( ph -> A e. RR+ )   =>    |- ( ph -> A # 0 )
Theorem	rpregt0d 8780	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 8781	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 8782	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 8783	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 8784	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 8785	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 8786	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 8787	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 8788	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 8789	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 8790	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 8791	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 8792	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 8793	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 8794	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 8795	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 8796	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 8797	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 8798	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 8799	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 8800	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 ) )