P. 101 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 10001-10100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	eluznn0 10001	Membership in a nonnegative upper set of integers implies membership in NN0. (Contributed by Paul Chapman, 22-Jun-2011.)
|- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) ) -> M e. NN0 )
Theorem	eluznn 10002	Membership in a positive upper set of integers implies membership in NN. (Contributed by JJ, 1-Oct-2018.)
|- ( ( N e. NN /\ M e. ( ZZ>= ` N ) ) -> M e. NN )
Theorem	eluz2b1 10003	Two ways to say "an integer greater than or equal to 2". (Contributed by Paul Chapman, 23-Nov-2012.)
|- ( N e. ( ZZ>= ` 2 ) <-> ( N e. ZZ /\ 1 < N ) )
Theorem	eluz2gt1 10004	An integer greater than or equal to 2 is greater than 1. (Contributed by AV, 24-May-2020.)
|- ( N e. ( ZZ>= ` 2 ) -> 1 < N )
Theorem	eluz2b2 10005	Two ways to say "an integer greater than or equal to 2". (Contributed by Paul Chapman, 23-Nov-2012.)
|- ( N e. ( ZZ>= ` 2 ) <-> ( N e. NN /\ 1 < N ) )
Theorem	eluz2b3 10006	Two ways to say "an integer greater than or equal to 2". (Contributed by Paul Chapman, 23-Nov-2012.)
|- ( N e. ( ZZ>= ` 2 ) <-> ( N e. NN /\ N =/= 1 ) )
Theorem	uz2m1nn 10007	One less than an integer greater than or equal to 2 is a positive integer. (Contributed by Paul Chapman, 17-Nov-2012.)
|- ( N e. ( ZZ>= ` 2 ) -> ( N - 1 ) e. NN )
Theorem	1nuz2 10008	1 is not in ( ZZ>= ` 2 ). (Contributed by Paul Chapman, 21-Nov-2012.)
|- -. 1 e. ( ZZ>= ` 2 )
Theorem	elnn1uz2 10009	A positive integer is either 1 or greater than or equal to 2. (Contributed by Paul Chapman, 17-Nov-2012.)
|- ( N e. NN <-> ( N = 1 \/ N e. ( ZZ>= ` 2 ) ) )
Theorem	uz2mulcl 10010	Closure of multiplication of integers greater than or equal to 2. (Contributed by Paul Chapman, 26-Oct-2012.)
|- ( ( M e. ( ZZ>= ` 2 ) /\ N e. ( ZZ>= ` 2 ) ) -> ( M x. N ) e. ( ZZ>= ` 2 ) )
Theorem	indstr2 10011*	Strong Mathematical Induction for positive integers (inference schema). The first two hypotheses give us the substitution instances we need; the last two are the basis and the induction step. (Contributed by Paul Chapman, 21-Nov-2012.)
|- ( x = 1 -> ( ph <-> ch ) )   &    |- ( x = y -> ( ph <-> ps ) )   &    |- ch   &    |- ( x e. ( ZZ>= ` 2 ) -> ( A. y e. NN ( y < x -> ps ) -> ph ) )   =>    |- ( x e. NN -> ph )
Theorem	eluzdc 10012	Membership of an integer in an upper set of integers is decidable. (Contributed by Jim Kingdon, 18-Apr-2020.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> DECID N e. ( ZZ>= ` M ) )
Theorem	elnn0dc 10013	Membership of an integer in NN0 is decidable. (Contributed by Jim Kingdon, 8-Oct-2024.)
|- ( N e. ZZ -> DECID N e. NN0 )
Theorem	elnndc 10014	Membership of an integer in NN is decidable. (Contributed by Jim Kingdon, 17-Oct-2024.)
|- ( N e. ZZ -> DECID N e. NN )
Theorem	ublbneg 10015*	The image under negation of a bounded-above set of reals is bounded below. For a theorem which is similar but also adds that the bounds need to be the tightest possible, see supinfneg 9997. (Contributed by Paul Chapman, 21-Mar-2011.)
|- ( E. x e. RR A. y e. A y <_ x -> E. x e. RR A. y e. { z e. RR | -u z e. A } x <_ y )
Theorem	eqreznegel 10016*	Two ways to express the image under negation of a set of integers. (Contributed by Paul Chapman, 21-Mar-2011.)
|- ( A C_ ZZ -> { z e. RR | -u z e. A } = { z e. ZZ | -u z e. A } )
Theorem	negm 10017*	The image under negation of an inhabited set of reals is inhabited. (Contributed by Jim Kingdon, 10-Apr-2020.)
|- ( ( A C_ RR /\ E. x x e. A ) -> E. y y e. { z e. RR | -u z e. A } )
Theorem	lbzbi 10018*	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 10019	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 10020	Alternate proof of nn0ge2m1nn 9629: 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 9929, 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 9629. (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 )
4.4.12  Rational numbers (as a subset of complex numbers)
Syntax	cq 10021	Extend class notation to include the class of rationals.
class QQ
Definition	df-q 10022	Define the set of rational numbers. Based on definition of rationals in [Apostol] p. 22. See elq 10024 for the relation "is rational". (Contributed by NM, 8-Jan-2002.)
|- QQ = ( / " ( ZZ X. NN ) )
Theorem	divfnzn 10023	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 10024*	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 10025*	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 10026	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 10027	A rational number is a real number. (Contributed by NM, 14-Nov-2002.)
|- ( A e. QQ -> A e. RR )
Theorem	zq 10028	An integer is a rational number. (Contributed by NM, 9-Jan-2002.)
|- ( A e. ZZ -> A e. QQ )
Theorem	zssq 10029	The integers are a subset of the rationals. (Contributed by NM, 9-Jan-2002.)
|- ZZ C_ QQ
Theorem	nn0ssq 10030	The nonnegative integers are a subset of the rationals. (Contributed by NM, 31-Jul-2004.)
|- NN0 C_ QQ
Theorem	nnssq 10031	The positive integers are a subset of the rationals. (Contributed by NM, 31-Jul-2004.)
|- NN C_ QQ
Theorem	qssre 10032	The rationals are a subset of the reals. (Contributed by NM, 9-Jan-2002.)
|- QQ C_ RR
Theorem	qsscn 10033	The rationals are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.)
|- QQ C_ CC
Theorem	qex 10034	The set of rational numbers exists. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)
|- QQ e. _V
Theorem	nnq 10035	A positive integer is rational. (Contributed by NM, 17-Nov-2004.)
|- ( A e. NN -> A e. QQ )
Theorem	qcn 10036	A rational number is a complex number. (Contributed by NM, 2-Aug-2004.)
|- ( A e. QQ -> A e. CC )
Theorem	qaddcl 10037	Closure of addition of rationals. (Contributed by NM, 1-Aug-2004.)
|- ( ( A e. QQ /\ B e. QQ ) -> ( A + B ) e. QQ )
Theorem	qnegcl 10038	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 10039	Closure of multiplication of rationals. (Contributed by NM, 1-Aug-2004.)
|- ( ( A e. QQ /\ B e. QQ ) -> ( A x. B ) e. QQ )
Theorem	qsubcl 10040	Closure of subtraction of rationals. (Contributed by NM, 2-Aug-2004.)
|- ( ( A e. QQ /\ B e. QQ ) -> ( A - B ) e. QQ )
Theorem	qapne 10041	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 10042	Rational 'Less than' expressed in terms of 'less than or equal to'. Also see ltleap 8961 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 10043	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 10044	Closure of reciprocal of rationals. (Contributed by NM, 3-Aug-2004.)
|- ( ( A e. QQ /\ A =/= 0 ) -> ( 1 / A ) e. QQ )
Theorem	qdivcl 10045	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 10046	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 10047	The reciprocal of a positive integer is rational. (Contributed by NM, 17-Nov-2004.)
|- ( A e. NN -> ( 1 / A ) e. QQ )
Theorem	irradd 10048	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 10049	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). For a similar theorem with irrational in place of not rational, see irrmulap 10050. (Contributed by NM, 7-Nov-2008.)
|- ( ( A e. ( RR \ QQ ) /\ B e. QQ /\ B =/= 0 ) -> ( A x. B ) e. ( RR \ QQ ) )
Theorem	irrmulap 10050*	The product of an irrational with a nonzero rational is irrational. By irrational we mean apart from any rational number. For a similar theorem with not rational in place of irrational, see irrmul 10049. (Contributed by Jim Kingdon, 25-Aug-2025.)
|- ( ph -> A e. RR )   &    |- ( ph -> A. q e. QQ A # q )   &    |- ( ph -> B e. QQ )   &    |- ( ph -> B =/= 0 )   &    |- ( ph -> Q e. QQ )   =>    |- ( ph -> ( A x. B ) # Q )
Theorem	elpq 10051*	A positive rational is the quotient of two positive integers. (Contributed by AV, 29-Dec-2022.)
|- ( ( A e. QQ /\ 0 < A ) -> E. x e. NN E. y e. NN A = ( x / y ) )
Theorem	elpqb 10052*	A class is a positive rational iff it is the quotient of two positive integers. (Contributed by AV, 30-Dec-2022.)
|- ( ( A e. QQ /\ 0 < A ) <-> E. x e. NN E. y e. NN A = ( x / y ) )
4.4.13  Complex numbers as pairs of reals
Theorem	cnref1o 10053*	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 8185), 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
Theorem	addex 10054	The addition operation is a set. (Contributed by NM, 19-Oct-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)
|- + e. _V
Theorem	mulex 10055	The multiplication operation is a set. (Contributed by NM, 19-Oct-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)
|- x. e. _V
4.5  Order sets
4.5.1  Positive reals (as a subset of complex numbers)
Syntax	crp 10056	Extend class notation to include the class of positive reals.
class RR+
Definition	df-rp 10057	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 10058	Membership in the set of positive reals. (Contributed by NM, 27-Oct-2007.)
|- ( A e. RR+ <-> ( A e. RR /\ 0 < A ) )
Theorem	elrpii 10059	Membership in the set of positive reals. (Contributed by NM, 23-Feb-2008.)
|- A e. RR   &    |- 0 < A   =>    |- A e. RR+
Theorem	1rp 10060	1 is a positive real. (Contributed by Jeff Hankins, 23-Nov-2008.)
|- 1 e. RR+
Theorem	2rp 10061	2 is a positive real. (Contributed by Mario Carneiro, 28-May-2016.)
|- 2 e. RR+
Theorem	3rp 10062	3 is a positive real. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- 3 e. RR+
Theorem	rpre 10063	A positive real is a real. (Contributed by NM, 27-Oct-2007.)
|- ( A e. RR+ -> A e. RR )
Theorem	rpxr 10064	A positive real is an extended real. (Contributed by Mario Carneiro, 21-Aug-2015.)
|- ( A e. RR+ -> A e. RR* )
Theorem	rpcn 10065	A positive real is a complex number. (Contributed by NM, 11-Nov-2008.)
|- ( A e. RR+ -> A e. CC )
Theorem	nnrp 10066	A positive integer is a positive real. (Contributed by NM, 28-Nov-2008.)
|- ( A e. NN -> A e. RR+ )
Theorem	rpssre 10067	The positive reals are a subset of the reals. (Contributed by NM, 24-Feb-2008.)
|- RR+ C_ RR
Theorem	rpgt0 10068	A positive real is greater than zero. (Contributed by FL, 27-Dec-2007.)
|- ( A e. RR+ -> 0 < A )
Theorem	rpge0 10069	A positive real is greater than or equal to zero. (Contributed by NM, 22-Feb-2008.)
|- ( A e. RR+ -> 0 <_ A )
Theorem	rpregt0 10070	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 10071	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 10072	A positive real is nonzero. (Contributed by NM, 18-Jul-2008.)
|- ( A e. RR+ -> A =/= 0 )
Theorem	rpap0 10073	A positive real is apart from zero. (Contributed by Jim Kingdon, 22-Mar-2020.)
|- ( A e. RR+ -> A # 0 )
Theorem	rprene0 10074	A positive real is a nonzero real number. (Contributed by NM, 11-Nov-2008.)
|- ( A e. RR+ -> ( A e. RR /\ A =/= 0 ) )
Theorem	rpreap0 10075	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 10076	A positive real is a nonzero complex number. (Contributed by NM, 11-Nov-2008.)
|- ( A e. RR+ -> ( A e. CC /\ A =/= 0 ) )
Theorem	rpcnap0 10077	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 10078	Quantification over positive reals. (Contributed by NM, 12-Feb-2008.)
|- ( A. x e. RR+ ph <-> A. x e. RR ( 0 < x -> ph ) )
Theorem	rexrp 10079	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 10080	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 10081	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 10082	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 10083	Closure law for reciprocation of positive reals. (Contributed by Jeff Hankins, 23-Nov-2008.)
|- ( A e. RR+ -> ( 1 / A ) e. RR+ )
Theorem	rphalfcl 10084	Closure law for half of a positive real. (Contributed by Mario Carneiro, 31-Jan-2014.)
|- ( A e. RR+ -> ( A / 2 ) e. RR+ )
Theorem	rpgecl 10085	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 10086	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 10087	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 10088	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 10089	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 10090	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 10091	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 10092	Zero is not a positive real. Axiom 9 of [Apostol] p. 20. (Contributed by NM, 27-Oct-2007.)
|- -. 0 e. RR+
Theorem	ltsubrp 10093	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 10094	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 10095	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 10096	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 10097	A positive integer is a positive real. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. NN )   =>    |- ( ph -> A e. RR+ )
Theorem	zgt1rpn0n1 10098	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 10099	A positive real is a real. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. RR+ )   =>    |- ( ph -> A e. RR )
Theorem	rpxrd 10100	A positive real is an extended real. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. RR+ )   =>    |- ( ph -> A e. RR* )