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**Theorem List for Intuitionistic Logic Explorer -** 9701-9800 *Has distinct variable group(s)
Type	Label	Description
Statement

Theorem	zltnle 9701	'Less than' expressed in terms of 'less than or equal to'. (Contributed by Jim Kingdon, 14-Mar-2020.)
$/\$

Theorem	zleloe 9702	Integer 'Less than or equal to' expressed in terms of 'less than' or 'equals'. (Contributed by Jim Kingdon, 8-Apr-2020.)
$/\$ $\/$

Theorem	znnnlt1 9703	An integer is not a positive integer iff it is less than one. (Contributed by NM, 13-Jul-2005.)


Theorem	nnnle0 9704	A positive integer is not less than or equal to zero. (Contributed by AV, 13-May-2020.)


Theorem	zletr 9705	Transitive law of ordering for integers. (Contributed by Alexander van der Vekens, 3-Apr-2018.)
$/\$ $/\$ $/\$

Theorem	zrevaddcl 9706	Reverse closure law for addition of integers. (Contributed by NM, 11-May-2004.)
$/\$

Theorem	znnsub 9707	The positive difference of unequal integers is a positive integer. (Generalization of nnsub 9352.) (Contributed by NM, 11-May-2004.)
$/\$

Theorem	nzadd 9708	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 9709	Closure of multiplication of integers. (Contributed by NM, 30-Jul-2004.)
$/\$

Theorem	zltp1le 9710	Integer ordering relation. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)
$/\$

Theorem	zleltp1 9711	Integer ordering relation. (Contributed by NM, 10-May-2004.)
$/\$

Theorem	zlem1lt 9712	Integer ordering relation. (Contributed by NM, 13-Nov-2004.)
$/\$

Theorem	zltlem1 9713	Integer ordering relation. (Contributed by NM, 13-Nov-2004.)
$/\$

Theorem	zgt0ge1 9714	An integer greater than is greater than or equal to . (Contributed by AV, 14-Oct-2018.)


Theorem	nnleltp1 9715	Positive integer ordering relation. (Contributed by NM, 13-Aug-2001.) (Proof shortened by Mario Carneiro, 16-May-2014.)
$/\$

Theorem	nnltp1le 9716	Positive integer ordering relation. (Contributed by NM, 19-Aug-2001.)
$/\$

Theorem	nnaddm1cl 9717	Closure of addition of positive integers minus one. (Contributed by NM, 6-Aug-2003.) (Proof shortened by Mario Carneiro, 16-May-2014.)
$/\$

Theorem	nn0ltp1le 9718	Nonnegative integer ordering relation. (Contributed by Raph Levien, 10-Dec-2002.) (Proof shortened by Mario Carneiro, 16-May-2014.)
$/\$

Theorem	nn0leltp1 9719	Nonnegative integer ordering relation. (Contributed by Raph Levien, 10-Apr-2004.)
$/\$

Theorem	nn0ltlem1 9720	Nonnegative integer ordering relation. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)
$/\$

Theorem	znn0sub 9721	The nonnegative difference of integers is a nonnegative integer. (Generalization of nn0sub 9722.) (Contributed by NM, 14-Jul-2005.)
$/\$

Theorem	nn0sub 9722	Subtraction of nonnegative integers. (Contributed by NM, 9-May-2004.)
$/\$

Theorem	ltsubnn0 9723	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 9724	A nonnegative integer is greater than or equal to its negative. (Contributed by AV, 13-Aug-2021.)


Theorem	difgtsumgt 9725	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 9726	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 9727*	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 9728	Alternate definition of the integers, based on elz2 9727. (Contributed by Mario Carneiro, 16-May-2014.)


Theorem	nn0sub2 9729	Subtraction of nonnegative integers. (Contributed by NM, 4-Sep-2005.)
$/\$ $/\$

Theorem	zapne 9730	Apartness is equivalent to not equal for integers. (Contributed by Jim Kingdon, 14-Mar-2020.)
$/\$ #

Theorem	zdceq 9731	Equality of integers is decidable. (Contributed by Jim Kingdon, 14-Mar-2020.)
$/\$ DECID

Theorem	zdcle 9732	Integer is decidable. (Contributed by Jim Kingdon, 7-Apr-2020.)
$/\$ DECID

Theorem	zdclt 9733	Integer is decidable. (Contributed by Jim Kingdon, 1-Jun-2020.)
$/\$ DECID

Theorem	zfidc 9734	Whether an integer is an element of a finite set of integers is decidable. (Contributed by Jim Kingdon, 8-Jun-2026.)
$/\$ $/\$ DECID

Theorem	zltlen 9735	Integer 'Less than' expressed in terms of 'less than or equal to'. Also see ltleap 8968 which is a similar result for real numbers. (Contributed by Jim Kingdon, 14-Mar-2020.)
$/\$ $/\$

Theorem	nn0n0n1ge2b 9736	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 9737	A nonnegative integer less than is . (Contributed by Paul Chapman, 22-Jun-2011.)


Theorem	nn0lt2 9738	A nonnegative integer less than 2 must be 0 or 1. (Contributed by Alexander van der Vekens, 16-Sep-2018.)
$/\$ $\/$

Theorem	nn0le2is012 9739	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 9740	Nonnegative integer ordering relation. (Contributed by NM, 21-Jun-2005.)
$/\$

Theorem	nnlem1lt 9741	Positive integer ordering relation. (Contributed by NM, 21-Jun-2005.)
$/\$

Theorem	nnltlem1 9742	Positive integer ordering relation. (Contributed by NM, 21-Jun-2005.)
$/\$

Theorem	nnm1ge0 9743	A positive integer decreased by 1 is greater than or equal to 0. (Contributed by AV, 30-Oct-2018.)


Theorem	nn0ge0div 9744	Division of a nonnegative integer by a positive number is not negative. (Contributed by Alexander van der Vekens, 14-Apr-2018.)
$/\$

Theorem	zdiv 9745*	Two ways to express " divides . (Contributed by NM, 3-Oct-2008.)
$/\$

Theorem	zdivadd 9746	Property of divisibility: if divides and then it divides . (Contributed by NM, 3-Oct-2008.)
$/\$ $/\$ $/\$ $/\$

Theorem	zdivmul 9747	Property of divisibility: if divides then it divides . (Contributed by NM, 3-Oct-2008.)
$/\$ $/\$ $/\$

Theorem	zextle 9748*	An extensionality-like property for integer ordering. (Contributed by NM, 29-Oct-2005.)
$/\$ $/\$

Theorem	zextlt 9749*	An extensionality-like property for integer ordering. (Contributed by NM, 29-Oct-2005.)
$/\$ $/\$

Theorem	recnz 9750	The reciprocal of a number greater than 1 is not an integer. (Contributed by NM, 3-May-2005.)
$/\$

Theorem	btwnnz 9751	A number between an integer and its successor is not an integer. (Contributed by NM, 3-May-2005.)
$/\$ $/\$

Theorem	gtndiv 9752	A larger number does not divide a smaller positive integer. (Contributed by NM, 3-May-2005.)
$/\$ $/\$

Theorem	halfnz 9753	One-half is not an integer. (Contributed by NM, 31-Jul-2004.)


Theorem	3halfnz 9754	Three halves is not an integer. (Contributed by AV, 2-Jun-2020.)


Theorem	suprzclex 9755*	The supremum of a set of integers is an element of the set. (Contributed by Jim Kingdon, 20-Dec-2021.)
$/\$

Theorem	prime 9756*	Two ways to express " is a prime number (or 1)". (Contributed by NM, 4-May-2005.)
$\/$ $/\$ $/\$

Theorem	msqznn 9757	The square of a nonzero integer is a positive integer. (Contributed by NM, 2-Aug-2004.)
$/\$

Theorem	zneo 9758	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 9759	A positive integer is even or odd. (Contributed by Jim Kingdon, 15-Mar-2020.)
$\/$

Theorem	nneo 9760	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 9761	A positive integer is even or odd but not both. (Contributed by NM, 20-Aug-2001.)


Theorem	zeo 9762	An integer is even or odd. (Contributed by NM, 1-Jan-2006.)
$\/$

Theorem	zeo2 9763	An integer is even or odd but not both. (Contributed by Mario Carneiro, 12-Sep-2015.)


Theorem	peano2uz2 9764*	Second Peano postulate for upper integers. (Contributed by NM, 3-Oct-2004.)
$/\$

Theorem	peano5uzti 9765*	Peano's inductive postulate for upper integers. (Contributed by NM, 6-Jul-2005.) (Revised by Mario Carneiro, 25-Jul-2013.)
$/\$

Theorem	peano5uzi 9766*	Peano's inductive postulate for upper integers. (Contributed by NM, 6-Jul-2005.) (Revised by Mario Carneiro, 3-May-2014.)
$/\$

Theorem	dfuzi 9767*	An expression for the upper integers that start at that is analogous to dfnn2 9315 for positive integers. (Contributed by NM, 6-Jul-2005.) (Proof shortened by Mario Carneiro, 3-May-2014.)
$/\$

Theorem	uzind 9768*	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 9769*	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 9770*	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 9771*	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 9772*	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 9773*	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 9774*	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 9775*	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 9776*	Any real number can be sandwiched between two integers. Exercise 2 of [Apostol] p. 28. (Contributed by NM, 10-Nov-2004.)
$/\$

Theorem	nn0zd 9777	A positive integer is an integer. (Contributed by Mario Carneiro, 28-May-2016.)


Theorem	nnzd 9778	A nonnegative integer is an integer. (Contributed by Mario Carneiro, 28-May-2016.)


Theorem	zred 9779	An integer is a real number. (Contributed by Mario Carneiro, 28-May-2016.)


Theorem	zcnd 9780	An integer is a complex number. (Contributed by Mario Carneiro, 28-May-2016.)


Theorem	znegcld 9781	Closure law for negative integers. (Contributed by Mario Carneiro, 28-May-2016.)


Theorem	peano2zd 9782	Deduction from second Peano postulate generalized to integers. (Contributed by Mario Carneiro, 28-May-2016.)


Theorem	zaddcld 9783	Closure of addition of integers. (Contributed by Mario Carneiro, 28-May-2016.)


Theorem	zsubcld 9784	Closure of subtraction of integers. (Contributed by Mario Carneiro, 28-May-2016.)


Theorem	zmulcld 9785	Closure of multiplication of integers. (Contributed by Mario Carneiro, 28-May-2016.)


Theorem	zadd2cl 9786	Increasing an integer by 2 results in an integer. (Contributed by Alexander van der Vekens, 16-Sep-2018.)


Theorem	btwnapz 9787	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 9788	Constant used for decimal constructor.
;

Definition	df-dec 9789	Define the "decimal constructor", which is used to build up "decimal integers" or "numeric terms" in base . For example, ;;; ;;; ;;; 1kp2ke3k 16943. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 1-Aug-2021.)
;

Theorem	9p1e10 9790	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 9791	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 9792	Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
; ;

Theorem	deceq2 9793	Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
; ;

Theorem	deceq1i 9794	Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
; ;

Theorem	deceq2i 9795	Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
; ;

Theorem	deceq12i 9796	Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
; ;

Theorem	numnncl 9797	Closure for a numeral (with units place). (Contributed by Mario Carneiro, 18-Feb-2014.)


Theorem	num0u 9798	Add a zero in the units place. (Contributed by Mario Carneiro, 18-Feb-2014.)


Theorem	num0h 9799	Add a zero in the higher places. (Contributed by Mario Carneiro, 18-Feb-2014.)


Theorem	numcl 9800	Closure for a decimal integer (with units place). (Contributed by Mario Carneiro, 18-Feb-2014.)

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