P. 85 of Theorem List - Intuitionistic Logic Explorer

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**Theorem List for Intuitionistic Logic Explorer -** 8401-8500 *Has distinct variable group(s)
Type	Label	Description
Statement

Theorem	zrevaddcl 8401	Reverse closure law for addition of integers. (Contributed by NM, 11-May-2004.)
$/\$

Theorem	znnsub 8402	The positive difference of unequal integers is a positive integer. (Generalization of nnsub 8077.) (Contributed by NM, 11-May-2004.)
$/\$

Theorem	nzadd 8403	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 8404	Closure of multiplication of integers. (Contributed by NM, 30-Jul-2004.)
$/\$

Theorem	zltp1le 8405	Integer ordering relation. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)
$/\$

Theorem	zleltp1 8406	Integer ordering relation. (Contributed by NM, 10-May-2004.)
$/\$

Theorem	zlem1lt 8407	Integer ordering relation. (Contributed by NM, 13-Nov-2004.)
$/\$

Theorem	zltlem1 8408	Integer ordering relation. (Contributed by NM, 13-Nov-2004.)
$/\$

Theorem	zgt0ge1 8409	An integer greater than is greater than or equal to . (Contributed by AV, 14-Oct-2018.)


Theorem	nnleltp1 8410	Positive integer ordering relation. (Contributed by NM, 13-Aug-2001.) (Proof shortened by Mario Carneiro, 16-May-2014.)
$/\$

Theorem	nnltp1le 8411	Positive integer ordering relation. (Contributed by NM, 19-Aug-2001.)
$/\$

Theorem	nnaddm1cl 8412	Closure of addition of positive integers minus one. (Contributed by NM, 6-Aug-2003.) (Proof shortened by Mario Carneiro, 16-May-2014.)
$/\$

Theorem	nn0ltp1le 8413	Nonnegative integer ordering relation. (Contributed by Raph Levien, 10-Dec-2002.) (Proof shortened by Mario Carneiro, 16-May-2014.)
$/\$

Theorem	nn0leltp1 8414	Nonnegative integer ordering relation. (Contributed by Raph Levien, 10-Apr-2004.)
$/\$

Theorem	nn0ltlem1 8415	Nonnegative integer ordering relation. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)
$/\$

Theorem	znn0sub 8416	The nonnegative difference of integers is a nonnegative integer. (Generalization of nn0sub 8417.) (Contributed by NM, 14-Jul-2005.)
$/\$

Theorem	nn0sub 8417	Subtraction of nonnegative integers. (Contributed by NM, 9-May-2004.)
$/\$

Theorem	nn0n0n1ge2 8418	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 8419*	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 8420	Alternate definition of the integers, based on elz2 8419. (Contributed by Mario Carneiro, 16-May-2014.)


Theorem	nn0sub2 8421	Subtraction of nonnegative integers. (Contributed by NM, 4-Sep-2005.)
$/\$ $/\$

Theorem	zapne 8422	Apartness is equivalent to not equal for integers. (Contributed by Jim Kingdon, 14-Mar-2020.)
$/\$ #

Theorem	zdceq 8423	Equality of integers is decidable. (Contributed by Jim Kingdon, 14-Mar-2020.)
$/\$ DECID

Theorem	zdcle 8424	Integer is decidable. (Contributed by Jim Kingdon, 7-Apr-2020.)
$/\$ DECID

Theorem	zdclt 8425	Integer is decidable. (Contributed by Jim Kingdon, 1-Jun-2020.)
$/\$ DECID

Theorem	zltlen 8426	Integer 'Less than' expressed in terms of 'less than or equal to'. Also see ltleap 7730 which is a similar result for real numbers. (Contributed by Jim Kingdon, 14-Mar-2020.)
$/\$ $/\$

Theorem	nn0n0n1ge2b 8427	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 8428	A nonnegative integer less than is . (Contributed by Paul Chapman, 22-Jun-2011.)


Theorem	nn0lt2 8429	A nonnegative integer less than 2 must be 0 or 1. (Contributed by Alexander van der Vekens, 16-Sep-2018.)
$/\$ $\/$

Theorem	nn0lem1lt 8430	Nonnegative integer ordering relation. (Contributed by NM, 21-Jun-2005.)
$/\$

Theorem	nnlem1lt 8431	Positive integer ordering relation. (Contributed by NM, 21-Jun-2005.)
$/\$

Theorem	nnltlem1 8432	Positive integer ordering relation. (Contributed by NM, 21-Jun-2005.)
$/\$

Theorem	nnm1ge0 8433	A positive integer decreased by 1 is greater than or equal to 0. (Contributed by AV, 30-Oct-2018.)


Theorem	nn0ge0div 8434	Division of a nonnegative integer by a positive number is not negative. (Contributed by Alexander van der Vekens, 14-Apr-2018.)
$/\$

Theorem	zdiv 8435*	Two ways to express " divides . (Contributed by NM, 3-Oct-2008.)
$/\$

Theorem	zdivadd 8436	Property of divisibility: if divides and then it divides . (Contributed by NM, 3-Oct-2008.)
$/\$ $/\$ $/\$ $/\$

Theorem	zdivmul 8437	Property of divisibility: if divides then it divides . (Contributed by NM, 3-Oct-2008.)
$/\$ $/\$ $/\$

Theorem	zextle 8438*	An extensionality-like property for integer ordering. (Contributed by NM, 29-Oct-2005.)
$/\$ $/\$

Theorem	zextlt 8439*	An extensionality-like property for integer ordering. (Contributed by NM, 29-Oct-2005.)
$/\$ $/\$

Theorem	recnz 8440	The reciprocal of a number greater than 1 is not an integer. (Contributed by NM, 3-May-2005.)
$/\$

Theorem	btwnnz 8441	A number between an integer and its successor is not an integer. (Contributed by NM, 3-May-2005.)
$/\$ $/\$

Theorem	gtndiv 8442	A larger number does not divide a smaller positive integer. (Contributed by NM, 3-May-2005.)
$/\$ $/\$

Theorem	halfnz 8443	One-half is not an integer. (Contributed by NM, 31-Jul-2004.)


Theorem	3halfnz 8444	Three halves is not an integer. (Contributed by AV, 2-Jun-2020.)


Theorem	suprzclex 8445*	The supremum of a set of integers is an element of the set. (Contributed by Jim Kingdon, 20-Dec-2021.)
$/\$

Theorem	prime 8446*	Two ways to express " is a prime number (or 1)." (Contributed by NM, 4-May-2005.)
$\/$ $/\$ $/\$

Theorem	msqznn 8447	The square of a nonzero integer is a positive integer. (Contributed by NM, 2-Aug-2004.)
$/\$

Theorem	zneo 8448	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 8449	A positive integer is even or odd. (Contributed by Jim Kingdon, 15-Mar-2020.)
$\/$

Theorem	nneo 8450	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 8451	A positive integer is even or odd but not both. (Contributed by NM, 20-Aug-2001.)


Theorem	zeo 8452	An integer is even or odd. (Contributed by NM, 1-Jan-2006.)
$\/$

Theorem	zeo2 8453	An integer is even or odd but not both. (Contributed by Mario Carneiro, 12-Sep-2015.)


Theorem	peano2uz2 8454*	Second Peano postulate for upper integers. (Contributed by NM, 3-Oct-2004.)
$/\$

Theorem	peano5uzti 8455*	Peano's inductive postulate for upper integers. (Contributed by NM, 6-Jul-2005.) (Revised by Mario Carneiro, 25-Jul-2013.)
$/\$

Theorem	peano5uzi 8456*	Peano's inductive postulate for upper integers. (Contributed by NM, 6-Jul-2005.) (Revised by Mario Carneiro, 3-May-2014.)
$/\$

Theorem	dfuzi 8457*	An expression for the upper integers that start at that is analogous to dfnn2 8041 for positive integers. (Contributed by NM, 6-Jul-2005.) (Proof shortened by Mario Carneiro, 3-May-2014.)
$/\$

Theorem	uzind 8458*	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 8459*	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 8460*	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 8461*	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 8462*	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 8463*	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 8464*	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 8465*	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 8466*	Any real number can be sandwiched between two integers. Exercise 2 of [Apostol] p. 28. (Contributed by NM, 10-Nov-2004.)
$/\$

Theorem	nn0zd 8467	A positive integer is an integer. (Contributed by Mario Carneiro, 28-May-2016.)


Theorem	nnzd 8468	A nonnegative integer is an integer. (Contributed by Mario Carneiro, 28-May-2016.)


Theorem	zred 8469	An integer is a real number. (Contributed by Mario Carneiro, 28-May-2016.)


Theorem	zcnd 8470	An integer is a complex number. (Contributed by Mario Carneiro, 28-May-2016.)


Theorem	znegcld 8471	Closure law for negative integers. (Contributed by Mario Carneiro, 28-May-2016.)


Theorem	peano2zd 8472	Deduction from second Peano postulate generalized to integers. (Contributed by Mario Carneiro, 28-May-2016.)


Theorem	zaddcld 8473	Closure of addition of integers. (Contributed by Mario Carneiro, 28-May-2016.)


Theorem	zsubcld 8474	Closure of subtraction of integers. (Contributed by Mario Carneiro, 28-May-2016.)


Theorem	zmulcld 8475	Closure of multiplication of integers. (Contributed by Mario Carneiro, 28-May-2016.)


Theorem	zadd2cl 8476	Increasing an integer by 2 results in an integer. (Contributed by Alexander van der Vekens, 16-Sep-2018.)


3.4.9 Decimal arithmetic

Syntax	cdc 8477	Constant used for decimal constructor.
;

Definition	df-dec 8478	Define the "decimal constructor", which is used to build up "decimal integers" or "numeric terms" in base . For example, ;;; ;;; ;;; 1kp2ke3k 10562. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 1-Aug-2021.)
;

Theorem	9p1e10 8479	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 8480	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 8481	Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
; ;

Theorem	deceq2 8482	Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
; ;

Theorem	deceq1i 8483	Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
; ;

Theorem	deceq2i 8484	Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
; ;

Theorem	deceq12i 8485	Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
; ;

Theorem	numnncl 8486	Closure for a numeral (with units place). (Contributed by Mario Carneiro, 18-Feb-2014.)


Theorem	num0u 8487	Add a zero in the units place. (Contributed by Mario Carneiro, 18-Feb-2014.)


Theorem	num0h 8488	Add a zero in the higher places. (Contributed by Mario Carneiro, 18-Feb-2014.)


Theorem	numcl 8489	Closure for a decimal integer (with units place). (Contributed by Mario Carneiro, 18-Feb-2014.)


Theorem	numsuc 8490	The successor of a decimal integer (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)


Theorem	deccl 8491	Closure for a numeral. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
;

Theorem	10nn 8492	10 is a positive integer. (Contributed by NM, 8-Nov-2012.) (Revised by AV, 6-Sep-2021.)
;

Theorem	10pos 8493	The number 10 is positive. (Contributed by NM, 5-Feb-2007.) (Revised by AV, 8-Sep-2021.)
;

Theorem	10nn0 8494	10 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
;

Theorem	10re 8495	The number 10 is real. (Contributed by NM, 5-Feb-2007.) (Revised by AV, 8-Sep-2021.)
;

Theorem	decnncl 8496	Closure for a numeral. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
;

Theorem	dec0u 8497	Add a zero in the units place. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
; ;

Theorem	dec0h 8498	Add a zero in the higher places. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
;

Theorem	numnncl2 8499	Closure for a decimal integer (zero units place). (Contributed by Mario Carneiro, 9-Mar-2015.)


Theorem	decnncl2 8500	Closure for a decimal integer (zero units place). (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
;

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