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**Theorem List for Intuitionistic Logic Explorer -** 9501-9600 *Has distinct variable group(s)
Type	Label	Description
Statement

Theorem	zletric 9501	Trichotomy law. (Contributed by Jim Kingdon, 27-Mar-2020.)
$/\$ $\/$

Theorem	zlelttric 9502	Trichotomy law. (Contributed by Jim Kingdon, 17-Apr-2020.)
$/\$ $\/$

Theorem	zltnle 9503	'Less than' expressed in terms of 'less than or equal to'. (Contributed by Jim Kingdon, 14-Mar-2020.)
$/\$

Theorem	zleloe 9504	Integer 'Less than or equal to' expressed in terms of 'less than' or 'equals'. (Contributed by Jim Kingdon, 8-Apr-2020.)
$/\$ $\/$

Theorem	znnnlt1 9505	An integer is not a positive integer iff it is less than one. (Contributed by NM, 13-Jul-2005.)


Theorem	nnnle0 9506	A positive integer is not less than or equal to zero. (Contributed by AV, 13-May-2020.)


Theorem	zletr 9507	Transitive law of ordering for integers. (Contributed by Alexander van der Vekens, 3-Apr-2018.)
$/\$ $/\$ $/\$

Theorem	zrevaddcl 9508	Reverse closure law for addition of integers. (Contributed by NM, 11-May-2004.)
$/\$

Theorem	znnsub 9509	The positive difference of unequal integers is a positive integer. (Generalization of nnsub 9160.) (Contributed by NM, 11-May-2004.)
$/\$

Theorem	nzadd 9510	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 9511	Closure of multiplication of integers. (Contributed by NM, 30-Jul-2004.)
$/\$

Theorem	zltp1le 9512	Integer ordering relation. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)
$/\$

Theorem	zleltp1 9513	Integer ordering relation. (Contributed by NM, 10-May-2004.)
$/\$

Theorem	zlem1lt 9514	Integer ordering relation. (Contributed by NM, 13-Nov-2004.)
$/\$

Theorem	zltlem1 9515	Integer ordering relation. (Contributed by NM, 13-Nov-2004.)
$/\$

Theorem	zgt0ge1 9516	An integer greater than is greater than or equal to . (Contributed by AV, 14-Oct-2018.)


Theorem	nnleltp1 9517	Positive integer ordering relation. (Contributed by NM, 13-Aug-2001.) (Proof shortened by Mario Carneiro, 16-May-2014.)
$/\$

Theorem	nnltp1le 9518	Positive integer ordering relation. (Contributed by NM, 19-Aug-2001.)
$/\$

Theorem	nnaddm1cl 9519	Closure of addition of positive integers minus one. (Contributed by NM, 6-Aug-2003.) (Proof shortened by Mario Carneiro, 16-May-2014.)
$/\$

Theorem	nn0ltp1le 9520	Nonnegative integer ordering relation. (Contributed by Raph Levien, 10-Dec-2002.) (Proof shortened by Mario Carneiro, 16-May-2014.)
$/\$

Theorem	nn0leltp1 9521	Nonnegative integer ordering relation. (Contributed by Raph Levien, 10-Apr-2004.)
$/\$

Theorem	nn0ltlem1 9522	Nonnegative integer ordering relation. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)
$/\$

Theorem	znn0sub 9523	The nonnegative difference of integers is a nonnegative integer. (Generalization of nn0sub 9524.) (Contributed by NM, 14-Jul-2005.)
$/\$

Theorem	nn0sub 9524	Subtraction of nonnegative integers. (Contributed by NM, 9-May-2004.)
$/\$

Theorem	ltsubnn0 9525	Subtracting a nonnegative integer from a nonnegative integer which is greater than the first one results in a nonnegative integer. (Contributed by Alexander van der Vekens, 6-Apr-2018.)
$/\$

Theorem	nn0negleid 9526	A nonnegative integer is greater than or equal to its negative. (Contributed by AV, 13-Aug-2021.)


Theorem	difgtsumgt 9527	If the difference of a real number and a nonnegative integer is greater than another real number, the sum of the real number and the nonnegative integer is also greater than the other real number. (Contributed by AV, 13-Aug-2021.)
$/\$ $/\$

Theorem	nn0n0n1ge2 9528	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 9529*	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 9530	Alternate definition of the integers, based on elz2 9529. (Contributed by Mario Carneiro, 16-May-2014.)


Theorem	nn0sub2 9531	Subtraction of nonnegative integers. (Contributed by NM, 4-Sep-2005.)
$/\$ $/\$

Theorem	zapne 9532	Apartness is equivalent to not equal for integers. (Contributed by Jim Kingdon, 14-Mar-2020.)
$/\$ #

Theorem	zdceq 9533	Equality of integers is decidable. (Contributed by Jim Kingdon, 14-Mar-2020.)
$/\$ DECID

Theorem	zdcle 9534	Integer is decidable. (Contributed by Jim Kingdon, 7-Apr-2020.)
$/\$ DECID

Theorem	zdclt 9535	Integer is decidable. (Contributed by Jim Kingdon, 1-Jun-2020.)
$/\$ DECID

Theorem	zltlen 9536	Integer 'Less than' expressed in terms of 'less than or equal to'. Also see ltleap 8790 which is a similar result for real numbers. (Contributed by Jim Kingdon, 14-Mar-2020.)
$/\$ $/\$

Theorem	nn0n0n1ge2b 9537	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 9538	A nonnegative integer less than is . (Contributed by Paul Chapman, 22-Jun-2011.)


Theorem	nn0lt2 9539	A nonnegative integer less than 2 must be 0 or 1. (Contributed by Alexander van der Vekens, 16-Sep-2018.)
$/\$ $\/$

Theorem	nn0le2is012 9540	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 9541	Nonnegative integer ordering relation. (Contributed by NM, 21-Jun-2005.)
$/\$

Theorem	nnlem1lt 9542	Positive integer ordering relation. (Contributed by NM, 21-Jun-2005.)
$/\$

Theorem	nnltlem1 9543	Positive integer ordering relation. (Contributed by NM, 21-Jun-2005.)
$/\$

Theorem	nnm1ge0 9544	A positive integer decreased by 1 is greater than or equal to 0. (Contributed by AV, 30-Oct-2018.)


Theorem	nn0ge0div 9545	Division of a nonnegative integer by a positive number is not negative. (Contributed by Alexander van der Vekens, 14-Apr-2018.)
$/\$

Theorem	zdiv 9546*	Two ways to express " divides . (Contributed by NM, 3-Oct-2008.)
$/\$

Theorem	zdivadd 9547	Property of divisibility: if divides and then it divides . (Contributed by NM, 3-Oct-2008.)
$/\$ $/\$ $/\$ $/\$

Theorem	zdivmul 9548	Property of divisibility: if divides then it divides . (Contributed by NM, 3-Oct-2008.)
$/\$ $/\$ $/\$

Theorem	zextle 9549*	An extensionality-like property for integer ordering. (Contributed by NM, 29-Oct-2005.)
$/\$ $/\$

Theorem	zextlt 9550*	An extensionality-like property for integer ordering. (Contributed by NM, 29-Oct-2005.)
$/\$ $/\$

Theorem	recnz 9551	The reciprocal of a number greater than 1 is not an integer. (Contributed by NM, 3-May-2005.)
$/\$

Theorem	btwnnz 9552	A number between an integer and its successor is not an integer. (Contributed by NM, 3-May-2005.)
$/\$ $/\$

Theorem	gtndiv 9553	A larger number does not divide a smaller positive integer. (Contributed by NM, 3-May-2005.)
$/\$ $/\$

Theorem	halfnz 9554	One-half is not an integer. (Contributed by NM, 31-Jul-2004.)


Theorem	3halfnz 9555	Three halves is not an integer. (Contributed by AV, 2-Jun-2020.)


Theorem	suprzclex 9556*	The supremum of a set of integers is an element of the set. (Contributed by Jim Kingdon, 20-Dec-2021.)
$/\$

Theorem	prime 9557*	Two ways to express " is a prime number (or 1)". (Contributed by NM, 4-May-2005.)
$\/$ $/\$ $/\$

Theorem	msqznn 9558	The square of a nonzero integer is a positive integer. (Contributed by NM, 2-Aug-2004.)
$/\$

Theorem	zneo 9559	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 9560	A positive integer is even or odd. (Contributed by Jim Kingdon, 15-Mar-2020.)
$\/$

Theorem	nneo 9561	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 9562	A positive integer is even or odd but not both. (Contributed by NM, 20-Aug-2001.)


Theorem	zeo 9563	An integer is even or odd. (Contributed by NM, 1-Jan-2006.)
$\/$

Theorem	zeo2 9564	An integer is even or odd but not both. (Contributed by Mario Carneiro, 12-Sep-2015.)


Theorem	peano2uz2 9565*	Second Peano postulate for upper integers. (Contributed by NM, 3-Oct-2004.)
$/\$

Theorem	peano5uzti 9566*	Peano's inductive postulate for upper integers. (Contributed by NM, 6-Jul-2005.) (Revised by Mario Carneiro, 25-Jul-2013.)
$/\$

Theorem	peano5uzi 9567*	Peano's inductive postulate for upper integers. (Contributed by NM, 6-Jul-2005.) (Revised by Mario Carneiro, 3-May-2014.)
$/\$

Theorem	dfuzi 9568*	An expression for the upper integers that start at that is analogous to dfnn2 9123 for positive integers. (Contributed by NM, 6-Jul-2005.) (Proof shortened by Mario Carneiro, 3-May-2014.)
$/\$

Theorem	uzind 9569*	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 9570*	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 9571*	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 9572*	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 9573*	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 9574*	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 9575*	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 9576*	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 9577*	Any real number can be sandwiched between two integers. Exercise 2 of [Apostol] p. 28. (Contributed by NM, 10-Nov-2004.)
$/\$

Theorem	nn0zd 9578	A positive integer is an integer. (Contributed by Mario Carneiro, 28-May-2016.)


Theorem	nnzd 9579	A nonnegative integer is an integer. (Contributed by Mario Carneiro, 28-May-2016.)


Theorem	zred 9580	An integer is a real number. (Contributed by Mario Carneiro, 28-May-2016.)


Theorem	zcnd 9581	An integer is a complex number. (Contributed by Mario Carneiro, 28-May-2016.)


Theorem	znegcld 9582	Closure law for negative integers. (Contributed by Mario Carneiro, 28-May-2016.)


Theorem	peano2zd 9583	Deduction from second Peano postulate generalized to integers. (Contributed by Mario Carneiro, 28-May-2016.)


Theorem	zaddcld 9584	Closure of addition of integers. (Contributed by Mario Carneiro, 28-May-2016.)


Theorem	zsubcld 9585	Closure of subtraction of integers. (Contributed by Mario Carneiro, 28-May-2016.)


Theorem	zmulcld 9586	Closure of multiplication of integers. (Contributed by Mario Carneiro, 28-May-2016.)


Theorem	zadd2cl 9587	Increasing an integer by 2 results in an integer. (Contributed by Alexander van der Vekens, 16-Sep-2018.)


Theorem	btwnapz 9588	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 9589	Constant used for decimal constructor.
;

Definition	df-dec 9590	Define the "decimal constructor", which is used to build up "decimal integers" or "numeric terms" in base . For example, ;;; ;;; ;;; 1kp2ke3k 16143. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 1-Aug-2021.)
;

Theorem	9p1e10 9591	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 9592	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 9593	Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
; ;

Theorem	deceq2 9594	Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
; ;

Theorem	deceq1i 9595	Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
; ;

Theorem	deceq2i 9596	Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
; ;

Theorem	deceq12i 9597	Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
; ;

Theorem	numnncl 9598	Closure for a numeral (with units place). (Contributed by Mario Carneiro, 18-Feb-2014.)


Theorem	num0u 9599	Add a zero in the units place. (Contributed by Mario Carneiro, 18-Feb-2014.)


Theorem	num0h 9600	Add a zero in the higher places. (Contributed by Mario Carneiro, 18-Feb-2014.)

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