P. 95 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 9401-9500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	elnn1uz2 9401	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 9402	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 9403*	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 9404	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	ublbneg 9405*	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 9390. (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 9406*	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 9407*	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 9408*	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 9409	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 9410	Alternate proof of nn0ge2m1nn 9037: 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 9332, 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 9037. (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 9411	Extend class notation to include the class of rationals.
class QQ
Definition	df-q 9412	Define the set of rational numbers. Based on definition of rationals in [Apostol] p. 22. See elq 9414 for the relation "is rational." (Contributed by NM, 8-Jan-2002.)
|- QQ = ( / " ( ZZ X. NN ) )
Theorem	divfnzn 9413	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 9414*	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 9415*	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 9416	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 9417	A rational number is a real number. (Contributed by NM, 14-Nov-2002.)
|- ( A e. QQ -> A e. RR )
Theorem	zq 9418	An integer is a rational number. (Contributed by NM, 9-Jan-2002.)
|- ( A e. ZZ -> A e. QQ )
Theorem	zssq 9419	The integers are a subset of the rationals. (Contributed by NM, 9-Jan-2002.)
|- ZZ C_ QQ
Theorem	nn0ssq 9420	The nonnegative integers are a subset of the rationals. (Contributed by NM, 31-Jul-2004.)
|- NN0 C_ QQ
Theorem	nnssq 9421	The positive integers are a subset of the rationals. (Contributed by NM, 31-Jul-2004.)
|- NN C_ QQ
Theorem	qssre 9422	The rationals are a subset of the reals. (Contributed by NM, 9-Jan-2002.)
|- QQ C_ RR
Theorem	qsscn 9423	The rationals are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.)
|- QQ C_ CC
Theorem	qex 9424	The set of rational numbers exists. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)
|- QQ e. _V
Theorem	nnq 9425	A positive integer is rational. (Contributed by NM, 17-Nov-2004.)
|- ( A e. NN -> A e. QQ )
Theorem	qcn 9426	A rational number is a complex number. (Contributed by NM, 2-Aug-2004.)
|- ( A e. QQ -> A e. CC )
Theorem	qaddcl 9427	Closure of addition of rationals. (Contributed by NM, 1-Aug-2004.)
|- ( ( A e. QQ /\ B e. QQ ) -> ( A + B ) e. QQ )
Theorem	qnegcl 9428	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 9429	Closure of multiplication of rationals. (Contributed by NM, 1-Aug-2004.)
|- ( ( A e. QQ /\ B e. QQ ) -> ( A x. B ) e. QQ )
Theorem	qsubcl 9430	Closure of subtraction of rationals. (Contributed by NM, 2-Aug-2004.)
|- ( ( A e. QQ /\ B e. QQ ) -> ( A - B ) e. QQ )
Theorem	qapne 9431	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 9432	Rational 'Less than' expressed in terms of 'less than or equal to'. Also see ltleap 8394 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 9433	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 9434	Closure of reciprocal of rationals. (Contributed by NM, 3-Aug-2004.)
|- ( ( A e. QQ /\ A =/= 0 ) -> ( 1 / A ) e. QQ )
Theorem	qdivcl 9435	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 9436	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 9437	The reciprocal of a positive integer is rational. (Contributed by NM, 17-Nov-2004.)
|- ( A e. NN -> ( 1 / A ) e. QQ )
Theorem	irradd 9438	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 9439	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 ) )
4.4.13  Complex numbers as pairs of reals
Theorem	cnref1o 9440*	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 7626), 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
4.5  Order sets
4.5.1  Positive reals (as a subset of complex numbers)
Syntax	crp 9441	Extend class notation to include the class of positive reals.
class RR+
Definition	df-rp 9442	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 9443	Membership in the set of positive reals. (Contributed by NM, 27-Oct-2007.)
|- ( A e. RR+ <-> ( A e. RR /\ 0 < A ) )
Theorem	elrpii 9444	Membership in the set of positive reals. (Contributed by NM, 23-Feb-2008.)
|- A e. RR   &    |- 0 < A   =>    |- A e. RR+
Theorem	1rp 9445	1 is a positive real. (Contributed by Jeff Hankins, 23-Nov-2008.)
|- 1 e. RR+
Theorem	2rp 9446	2 is a positive real. (Contributed by Mario Carneiro, 28-May-2016.)
|- 2 e. RR+
Theorem	3rp 9447	3 is a positive real. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- 3 e. RR+
Theorem	rpre 9448	A positive real is a real. (Contributed by NM, 27-Oct-2007.)
|- ( A e. RR+ -> A e. RR )
Theorem	rpxr 9449	A positive real is an extended real. (Contributed by Mario Carneiro, 21-Aug-2015.)
|- ( A e. RR+ -> A e. RR* )
Theorem	rpcn 9450	A positive real is a complex number. (Contributed by NM, 11-Nov-2008.)
|- ( A e. RR+ -> A e. CC )
Theorem	nnrp 9451	A positive integer is a positive real. (Contributed by NM, 28-Nov-2008.)
|- ( A e. NN -> A e. RR+ )
Theorem	rpssre 9452	The positive reals are a subset of the reals. (Contributed by NM, 24-Feb-2008.)
|- RR+ C_ RR
Theorem	rpgt0 9453	A positive real is greater than zero. (Contributed by FL, 27-Dec-2007.)
|- ( A e. RR+ -> 0 < A )
Theorem	rpge0 9454	A positive real is greater than or equal to zero. (Contributed by NM, 22-Feb-2008.)
|- ( A e. RR+ -> 0 <_ A )
Theorem	rpregt0 9455	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 9456	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 9457	A positive real is nonzero. (Contributed by NM, 18-Jul-2008.)
|- ( A e. RR+ -> A =/= 0 )
Theorem	rpap0 9458	A positive real is apart from zero. (Contributed by Jim Kingdon, 22-Mar-2020.)
|- ( A e. RR+ -> A # 0 )
Theorem	rprene0 9459	A positive real is a nonzero real number. (Contributed by NM, 11-Nov-2008.)
|- ( A e. RR+ -> ( A e. RR /\ A =/= 0 ) )
Theorem	rpreap0 9460	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 9461	A positive real is a nonzero complex number. (Contributed by NM, 11-Nov-2008.)
|- ( A e. RR+ -> ( A e. CC /\ A =/= 0 ) )
Theorem	rpcnap0 9462	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 9463	Quantification over positive reals. (Contributed by NM, 12-Feb-2008.)
|- ( A. x e. RR+ ph <-> A. x e. RR ( 0 < x -> ph ) )
Theorem	rexrp 9464	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 9465	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 9466	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 9467	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 9468	Closure law for reciprocation of positive reals. (Contributed by Jeff Hankins, 23-Nov-2008.)
|- ( A e. RR+ -> ( 1 / A ) e. RR+ )
Theorem	rphalfcl 9469	Closure law for half of a positive real. (Contributed by Mario Carneiro, 31-Jan-2014.)
|- ( A e. RR+ -> ( A / 2 ) e. RR+ )
Theorem	rpgecl 9470	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 9471	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 9472	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 9473	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 9474	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 9475	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 9476	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 9477	Zero is not a positive real. Axiom 9 of [Apostol] p. 20. (Contributed by NM, 27-Oct-2007.)
|- -. 0 e. RR+
Theorem	ltsubrp 9478	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 9479	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 9480	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 9481	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 9482	A positive integer is a positive real. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. NN )   =>    |- ( ph -> A e. RR+ )
Theorem	rpred 9483	A positive real is a real. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. RR+ )   =>    |- ( ph -> A e. RR )
Theorem	rpxrd 9484	A positive real is an extended real. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. RR+ )   =>    |- ( ph -> A e. RR* )
Theorem	rpcnd 9485	A positive real is a complex number. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. RR+ )   =>    |- ( ph -> A e. CC )
Theorem	rpgt0d 9486	A positive real is greater than zero. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. RR+ )   =>    |- ( ph -> 0 < A )
Theorem	rpge0d 9487	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 9488	A positive real is nonzero. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. RR+ )   =>    |- ( ph -> A =/= 0 )
Theorem	rpap0d 9489	A positive real is apart from zero. (Contributed by Jim Kingdon, 28-Jul-2021.)
|- ( ph -> A e. RR+ )   =>    |- ( ph -> A # 0 )
Theorem	rpregt0d 9490	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 9491	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 9492	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 9493	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 9494	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 9495	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 9496	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 9497	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 9498	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 9499	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 9500	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+ )