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**Theorem List for Intuitionistic Logic Explorer -** 9001-9100 *Has distinct variable group(s)
Type	Label	Description
Statement

Theorem	ztri3or 9001	Integer trichotomy. (Contributed by Jim Kingdon, 14-Mar-2020.)
$/\$ $\/$ $\/$

Theorem	zletric 9002	Trichotomy law. (Contributed by Jim Kingdon, 27-Mar-2020.)
$/\$ $\/$

Theorem	zlelttric 9003	Trichotomy law. (Contributed by Jim Kingdon, 17-Apr-2020.)
$/\$ $\/$

Theorem	zltnle 9004	'Less than' expressed in terms of 'less than or equal to'. (Contributed by Jim Kingdon, 14-Mar-2020.)
$/\$

Theorem	zleloe 9005	Integer 'Less than or equal to' expressed in terms of 'less than' or 'equals'. (Contributed by Jim Kingdon, 8-Apr-2020.)
$/\$ $\/$

Theorem	znnnlt1 9006	An integer is not a positive integer iff it is less than one. (Contributed by NM, 13-Jul-2005.)


Theorem	zletr 9007	Transitive law of ordering for integers. (Contributed by Alexander van der Vekens, 3-Apr-2018.)
$/\$ $/\$ $/\$

Theorem	zrevaddcl 9008	Reverse closure law for addition of integers. (Contributed by NM, 11-May-2004.)
$/\$

Theorem	znnsub 9009	The positive difference of unequal integers is a positive integer. (Generalization of nnsub 8669.) (Contributed by NM, 11-May-2004.)
$/\$

Theorem	nzadd 9010	The sum of a real number not being an integer and an integer is not an integer. Note that "not being an integer" in this case means "the negation of is an integer" rather than "is apart from any integer" (given excluded middle, those two would be equivalent). (Contributed by AV, 19-Jul-2021.)
$\$ $/\$ $\$

Theorem	zmulcl 9011	Closure of multiplication of integers. (Contributed by NM, 30-Jul-2004.)
$/\$

Theorem	zltp1le 9012	Integer ordering relation. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)
$/\$

Theorem	zleltp1 9013	Integer ordering relation. (Contributed by NM, 10-May-2004.)
$/\$

Theorem	zlem1lt 9014	Integer ordering relation. (Contributed by NM, 13-Nov-2004.)
$/\$

Theorem	zltlem1 9015	Integer ordering relation. (Contributed by NM, 13-Nov-2004.)
$/\$

Theorem	zgt0ge1 9016	An integer greater than is greater than or equal to . (Contributed by AV, 14-Oct-2018.)


Theorem	nnleltp1 9017	Positive integer ordering relation. (Contributed by NM, 13-Aug-2001.) (Proof shortened by Mario Carneiro, 16-May-2014.)
$/\$

Theorem	nnltp1le 9018	Positive integer ordering relation. (Contributed by NM, 19-Aug-2001.)
$/\$

Theorem	nnaddm1cl 9019	Closure of addition of positive integers minus one. (Contributed by NM, 6-Aug-2003.) (Proof shortened by Mario Carneiro, 16-May-2014.)
$/\$

Theorem	nn0ltp1le 9020	Nonnegative integer ordering relation. (Contributed by Raph Levien, 10-Dec-2002.) (Proof shortened by Mario Carneiro, 16-May-2014.)
$/\$

Theorem	nn0leltp1 9021	Nonnegative integer ordering relation. (Contributed by Raph Levien, 10-Apr-2004.)
$/\$

Theorem	nn0ltlem1 9022	Nonnegative integer ordering relation. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)
$/\$

Theorem	znn0sub 9023	The nonnegative difference of integers is a nonnegative integer. (Generalization of nn0sub 9024.) (Contributed by NM, 14-Jul-2005.)
$/\$

Theorem	nn0sub 9024	Subtraction of nonnegative integers. (Contributed by NM, 9-May-2004.)
$/\$

Theorem	nn0n0n1ge2 9025	A nonnegative integer which is neither 0 nor 1 is greater than or equal to 2. (Contributed by Alexander van der Vekens, 6-Dec-2017.)
$/\$ $/\$

Theorem	elz2 9026*	Membership in the set of integers. Commonly used in constructions of the integers as equivalence classes under subtraction of the positive integers. (Contributed by Mario Carneiro, 16-May-2014.)


Theorem	dfz2 9027	Alternate definition of the integers, based on elz2 9026. (Contributed by Mario Carneiro, 16-May-2014.)


Theorem	nn0sub2 9028	Subtraction of nonnegative integers. (Contributed by NM, 4-Sep-2005.)
$/\$ $/\$

Theorem	zapne 9029	Apartness is equivalent to not equal for integers. (Contributed by Jim Kingdon, 14-Mar-2020.)
$/\$ #

Theorem	zdceq 9030	Equality of integers is decidable. (Contributed by Jim Kingdon, 14-Mar-2020.)
$/\$ DECID

Theorem	zdcle 9031	Integer is decidable. (Contributed by Jim Kingdon, 7-Apr-2020.)
$/\$ DECID

Theorem	zdclt 9032	Integer is decidable. (Contributed by Jim Kingdon, 1-Jun-2020.)
$/\$ DECID

Theorem	zltlen 9033	Integer 'Less than' expressed in terms of 'less than or equal to'. Also see ltleap 8311 which is a similar result for real numbers. (Contributed by Jim Kingdon, 14-Mar-2020.)
$/\$ $/\$

Theorem	nn0n0n1ge2b 9034	A nonnegative integer is neither 0 nor 1 if and only if it is greater than or equal to 2. (Contributed by Alexander van der Vekens, 17-Jan-2018.)
$/\$

Theorem	nn0lt10b 9035	A nonnegative integer less than is . (Contributed by Paul Chapman, 22-Jun-2011.)


Theorem	nn0lt2 9036	A nonnegative integer less than 2 must be 0 or 1. (Contributed by Alexander van der Vekens, 16-Sep-2018.)
$/\$ $\/$

Theorem	nn0le2is012 9037	A nonnegative integer which is less than or equal to 2 is either 0 or 1 or 2. (Contributed by AV, 16-Mar-2019.)
$/\$ $\/$ $\/$

Theorem	nn0lem1lt 9038	Nonnegative integer ordering relation. (Contributed by NM, 21-Jun-2005.)
$/\$

Theorem	nnlem1lt 9039	Positive integer ordering relation. (Contributed by NM, 21-Jun-2005.)
$/\$

Theorem	nnltlem1 9040	Positive integer ordering relation. (Contributed by NM, 21-Jun-2005.)
$/\$

Theorem	nnm1ge0 9041	A positive integer decreased by 1 is greater than or equal to 0. (Contributed by AV, 30-Oct-2018.)


Theorem	nn0ge0div 9042	Division of a nonnegative integer by a positive number is not negative. (Contributed by Alexander van der Vekens, 14-Apr-2018.)
$/\$

Theorem	zdiv 9043*	Two ways to express " divides . (Contributed by NM, 3-Oct-2008.)
$/\$

Theorem	zdivadd 9044	Property of divisibility: if divides and then it divides . (Contributed by NM, 3-Oct-2008.)
$/\$ $/\$ $/\$ $/\$

Theorem	zdivmul 9045	Property of divisibility: if divides then it divides . (Contributed by NM, 3-Oct-2008.)
$/\$ $/\$ $/\$

Theorem	zextle 9046*	An extensionality-like property for integer ordering. (Contributed by NM, 29-Oct-2005.)
$/\$ $/\$

Theorem	zextlt 9047*	An extensionality-like property for integer ordering. (Contributed by NM, 29-Oct-2005.)
$/\$ $/\$

Theorem	recnz 9048	The reciprocal of a number greater than 1 is not an integer. (Contributed by NM, 3-May-2005.)
$/\$

Theorem	btwnnz 9049	A number between an integer and its successor is not an integer. (Contributed by NM, 3-May-2005.)
$/\$ $/\$

Theorem	gtndiv 9050	A larger number does not divide a smaller positive integer. (Contributed by NM, 3-May-2005.)
$/\$ $/\$

Theorem	halfnz 9051	One-half is not an integer. (Contributed by NM, 31-Jul-2004.)


Theorem	3halfnz 9052	Three halves is not an integer. (Contributed by AV, 2-Jun-2020.)


Theorem	suprzclex 9053*	The supremum of a set of integers is an element of the set. (Contributed by Jim Kingdon, 20-Dec-2021.)
$/\$

Theorem	prime 9054*	Two ways to express " is a prime number (or 1)." (Contributed by NM, 4-May-2005.)
$\/$ $/\$ $/\$

Theorem	msqznn 9055	The square of a nonzero integer is a positive integer. (Contributed by NM, 2-Aug-2004.)
$/\$

Theorem	zneo 9056	No even integer equals an odd integer (i.e. no integer can be both even and odd). Exercise 10(a) of [Apostol] p. 28. (Contributed by NM, 31-Jul-2004.) (Proof shortened by Mario Carneiro, 18-May-2014.)
$/\$

Theorem	nneoor 9057	A positive integer is even or odd. (Contributed by Jim Kingdon, 15-Mar-2020.)
$\/$

Theorem	nneo 9058	A positive integer is even or odd but not both. (Contributed by NM, 1-Jan-2006.) (Proof shortened by Mario Carneiro, 18-May-2014.)


Theorem	nneoi 9059	A positive integer is even or odd but not both. (Contributed by NM, 20-Aug-2001.)


Theorem	zeo 9060	An integer is even or odd. (Contributed by NM, 1-Jan-2006.)
$\/$

Theorem	zeo2 9061	An integer is even or odd but not both. (Contributed by Mario Carneiro, 12-Sep-2015.)


Theorem	peano2uz2 9062*	Second Peano postulate for upper integers. (Contributed by NM, 3-Oct-2004.)
$/\$

Theorem	peano5uzti 9063*	Peano's inductive postulate for upper integers. (Contributed by NM, 6-Jul-2005.) (Revised by Mario Carneiro, 25-Jul-2013.)
$/\$

Theorem	peano5uzi 9064*	Peano's inductive postulate for upper integers. (Contributed by NM, 6-Jul-2005.) (Revised by Mario Carneiro, 3-May-2014.)
$/\$

Theorem	dfuzi 9065*	An expression for the upper integers that start at that is analogous to dfnn2 8632 for positive integers. (Contributed by NM, 6-Jul-2005.) (Proof shortened by Mario Carneiro, 3-May-2014.)
$/\$

Theorem	uzind 9066*	Induction on the upper integers that start at . The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction step. (Contributed by NM, 5-Jul-2005.)
$/\$ $/\$ $/\$ $/\$

Theorem	uzind2 9067*	Induction on the upper integers that start after an integer . The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction step. (Contributed by NM, 25-Jul-2005.)
$/\$ $/\$ $/\$ $/\$

Theorem	uzind3 9068*	Induction on the upper integers that start at an integer . The first four hypotheses give us the substitution instances we need, and the last two are the basis and the induction step. (Contributed by NM, 26-Jul-2005.)
$/\$ $/\$

Theorem	nn0ind 9069*	Principle of Mathematical Induction (inference schema) on nonnegative integers. The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction step. (Contributed by NM, 13-May-2004.)


Theorem	fzind 9070*	Induction on the integers from to inclusive . The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction step. (Contributed by Paul Chapman, 31-Mar-2011.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	fnn0ind 9071*	Induction on the integers from to inclusive . The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction step. (Contributed by Paul Chapman, 31-Mar-2011.)
$/\$ $/\$ $/\$ $/\$

Theorem	nn0ind-raph 9072*	Principle of Mathematical Induction (inference schema) on nonnegative integers. The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction step. Raph Levien remarks: "This seems a bit painful. I wonder if an explicit substitution version would be easier." (Contributed by Raph Levien, 10-Apr-2004.)


Theorem	zindd 9073*	Principle of Mathematical Induction on all integers, deduction version. The first five hypotheses give the substitutions; the last three are the basis, the induction, and the extension to negative numbers. (Contributed by Paul Chapman, 17-Apr-2009.) (Proof shortened by Mario Carneiro, 4-Jan-2017.)


Theorem	btwnz 9074*	Any real number can be sandwiched between two integers. Exercise 2 of [Apostol] p. 28. (Contributed by NM, 10-Nov-2004.)
$/\$

Theorem	nn0zd 9075	A positive integer is an integer. (Contributed by Mario Carneiro, 28-May-2016.)


Theorem	nnzd 9076	A nonnegative integer is an integer. (Contributed by Mario Carneiro, 28-May-2016.)


Theorem	zred 9077	An integer is a real number. (Contributed by Mario Carneiro, 28-May-2016.)


Theorem	zcnd 9078	An integer is a complex number. (Contributed by Mario Carneiro, 28-May-2016.)


Theorem	znegcld 9079	Closure law for negative integers. (Contributed by Mario Carneiro, 28-May-2016.)


Theorem	peano2zd 9080	Deduction from second Peano postulate generalized to integers. (Contributed by Mario Carneiro, 28-May-2016.)


Theorem	zaddcld 9081	Closure of addition of integers. (Contributed by Mario Carneiro, 28-May-2016.)


Theorem	zsubcld 9082	Closure of subtraction of integers. (Contributed by Mario Carneiro, 28-May-2016.)


Theorem	zmulcld 9083	Closure of multiplication of integers. (Contributed by Mario Carneiro, 28-May-2016.)


Theorem	zadd2cl 9084	Increasing an integer by 2 results in an integer. (Contributed by Alexander van der Vekens, 16-Sep-2018.)


Theorem	btwnapz 9085	A number between an integer and its successor is apart from any integer. (Contributed by Jim Kingdon, 6-Jan-2023.)
#

4.4.10 Decimal arithmetic

Syntax	cdc 9086	Constant used for decimal constructor.
;

Definition	df-dec 9087	Define the "decimal constructor", which is used to build up "decimal integers" or "numeric terms" in base . For example, ;;; ;;; ;;; 1kp2ke3k 12629. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 1-Aug-2021.)
;

Theorem	9p1e10 9088	9 + 1 = 10. (Contributed by Mario Carneiro, 18-Apr-2015.) (Revised by Stanislas Polu, 7-Apr-2020.) (Revised by AV, 1-Aug-2021.)
;

Theorem	dfdec10 9089	Version of the definition of the "decimal constructor" using ; instead of the symbol 10. Of course, this statement cannot be used as definition, because it uses the "decimal constructor". (Contributed by AV, 1-Aug-2021.)
; ;

Theorem	deceq1 9090	Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
; ;

Theorem	deceq2 9091	Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
; ;

Theorem	deceq1i 9092	Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
; ;

Theorem	deceq2i 9093	Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
; ;

Theorem	deceq12i 9094	Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
; ;

Theorem	numnncl 9095	Closure for a numeral (with units place). (Contributed by Mario Carneiro, 18-Feb-2014.)


Theorem	num0u 9096	Add a zero in the units place. (Contributed by Mario Carneiro, 18-Feb-2014.)


Theorem	num0h 9097	Add a zero in the higher places. (Contributed by Mario Carneiro, 18-Feb-2014.)


Theorem	numcl 9098	Closure for a decimal integer (with units place). (Contributed by Mario Carneiro, 18-Feb-2014.)


Theorem	numsuc 9099	The successor of a decimal integer (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)


Theorem	deccl 9100	Closure for a numeral. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
;

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