P. 96 of Theorem List - Intuitionistic Logic Explorer

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

Type	Label	Description
Statement
Theorem	neg1z 9501	-1 is an integer (common case). (Contributed by David A. Wheeler, 5-Dec-2018.)
|- -u 1 e. ZZ
Theorem	znegclb 9502	A number is an integer iff its negative is. (Contributed by Stefan O'Rear, 13-Sep-2014.)
|- ( A e. CC -> ( A e. ZZ <-> -u A e. ZZ ) )
Theorem	nn0negz 9503	The negative of a nonnegative integer is an integer. (Contributed by NM, 9-May-2004.)
|- ( N e. NN0 -> -u N e. ZZ )
Theorem	nn0negzi 9504	The negative of a nonnegative integer is an integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
|- N e. NN0   =>    |- -u N e. ZZ
Theorem	peano2z 9505	Second Peano postulate generalized to integers. (Contributed by NM, 13-Feb-2005.)
|- ( N e. ZZ -> ( N + 1 ) e. ZZ )
Theorem	zaddcllempos 9506	Lemma for zaddcl 9509. Special case in which N is a positive integer. (Contributed by Jim Kingdon, 14-Mar-2020.)
|- ( ( M e. ZZ /\ N e. NN ) -> ( M + N ) e. ZZ )
Theorem	peano2zm 9507	"Reverse" second Peano postulate for integers. (Contributed by NM, 12-Sep-2005.)
|- ( N e. ZZ -> ( N - 1 ) e. ZZ )
Theorem	zaddcllemneg 9508	Lemma for zaddcl 9509. Special case in which -u N is a positive integer. (Contributed by Jim Kingdon, 14-Mar-2020.)
|- ( ( M e. ZZ /\ N e. RR /\ -u N e. NN ) -> ( M + N ) e. ZZ )
Theorem	zaddcl 9509	Closure of addition of integers. (Contributed by NM, 9-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M + N ) e. ZZ )
Theorem	zsubcl 9510	Closure of subtraction of integers. (Contributed by NM, 11-May-2004.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M - N ) e. ZZ )
Theorem	ztri3or0 9511	Integer trichotomy (with zero). (Contributed by Jim Kingdon, 14-Mar-2020.)
|- ( N e. ZZ -> ( N < 0 \/ N = 0 \/ 0 < N ) )
Theorem	ztri3or 9512	Integer trichotomy. (Contributed by Jim Kingdon, 14-Mar-2020.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M < N \/ M = N \/ N < M ) )
Theorem	zletric 9513	Trichotomy law. (Contributed by Jim Kingdon, 27-Mar-2020.)
|- ( ( A e. ZZ /\ B e. ZZ ) -> ( A <_ B \/ B <_ A ) )
Theorem	zlelttric 9514	Trichotomy law. (Contributed by Jim Kingdon, 17-Apr-2020.)
|- ( ( A e. ZZ /\ B e. ZZ ) -> ( A <_ B \/ B < A ) )
Theorem	zltnle 9515	'Less than' expressed in terms of 'less than or equal to'. (Contributed by Jim Kingdon, 14-Mar-2020.)
|- ( ( A e. ZZ /\ B e. ZZ ) -> ( A < B <-> -. B <_ A ) )
Theorem	zleloe 9516	Integer 'Less than or equal to' expressed in terms of 'less than' or 'equals'. (Contributed by Jim Kingdon, 8-Apr-2020.)
|- ( ( A e. ZZ /\ B e. ZZ ) -> ( A <_ B <-> ( A < B \/ A = B ) ) )
Theorem	znnnlt1 9517	An integer is not a positive integer iff it is less than one. (Contributed by NM, 13-Jul-2005.)
|- ( N e. ZZ -> ( -. N e. NN <-> N < 1 ) )
Theorem	nnnle0 9518	A positive integer is not less than or equal to zero. (Contributed by AV, 13-May-2020.)
|- ( A e. NN -> -. A <_ 0 )
Theorem	zletr 9519	Transitive law of ordering for integers. (Contributed by Alexander van der Vekens, 3-Apr-2018.)
|- ( ( J e. ZZ /\ K e. ZZ /\ L e. ZZ ) -> ( ( J <_ K /\ K <_ L ) -> J <_ L ) )
Theorem	zrevaddcl 9520	Reverse closure law for addition of integers. (Contributed by NM, 11-May-2004.)
|- ( N e. ZZ -> ( ( M e. CC /\ ( M + N ) e. ZZ ) <-> M e. ZZ ) )
Theorem	znnsub 9521	The positive difference of unequal integers is a positive integer. (Generalization of nnsub 9172.) (Contributed by NM, 11-May-2004.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M < N <-> ( N - M ) e. NN ) )
Theorem	nzadd 9522	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.)
|- ( ( A e. ( RR \ ZZ ) /\ B e. ZZ ) -> ( A + B ) e. ( RR \ ZZ ) )
Theorem	zmulcl 9523	Closure of multiplication of integers. (Contributed by NM, 30-Jul-2004.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M x. N ) e. ZZ )
Theorem	zltp1le 9524	Integer ordering relation. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M < N <-> ( M + 1 ) <_ N ) )
Theorem	zleltp1 9525	Integer ordering relation. (Contributed by NM, 10-May-2004.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M <_ N <-> M < ( N + 1 ) ) )
Theorem	zlem1lt 9526	Integer ordering relation. (Contributed by NM, 13-Nov-2004.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M <_ N <-> ( M - 1 ) < N ) )
Theorem	zltlem1 9527	Integer ordering relation. (Contributed by NM, 13-Nov-2004.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M < N <-> M <_ ( N - 1 ) ) )
Theorem	zgt0ge1 9528	An integer greater than 0 is greater than or equal to 1. (Contributed by AV, 14-Oct-2018.)
|- ( Z e. ZZ -> ( 0 < Z <-> 1 <_ Z ) )
Theorem	nnleltp1 9529	Positive integer ordering relation. (Contributed by NM, 13-Aug-2001.) (Proof shortened by Mario Carneiro, 16-May-2014.)
|- ( ( A e. NN /\ B e. NN ) -> ( A <_ B <-> A < ( B + 1 ) ) )
Theorem	nnltp1le 9530	Positive integer ordering relation. (Contributed by NM, 19-Aug-2001.)
|- ( ( A e. NN /\ B e. NN ) -> ( A < B <-> ( A + 1 ) <_ B ) )
Theorem	nnaddm1cl 9531	Closure of addition of positive integers minus one. (Contributed by NM, 6-Aug-2003.) (Proof shortened by Mario Carneiro, 16-May-2014.)
|- ( ( A e. NN /\ B e. NN ) -> ( ( A + B ) - 1 ) e. NN )
Theorem	nn0ltp1le 9532	Nonnegative integer ordering relation. (Contributed by Raph Levien, 10-Dec-2002.) (Proof shortened by Mario Carneiro, 16-May-2014.)
|- ( ( M e. NN0 /\ N e. NN0 ) -> ( M < N <-> ( M + 1 ) <_ N ) )
Theorem	nn0leltp1 9533	Nonnegative integer ordering relation. (Contributed by Raph Levien, 10-Apr-2004.)
|- ( ( M e. NN0 /\ N e. NN0 ) -> ( M <_ N <-> M < ( N + 1 ) ) )
Theorem	nn0ltlem1 9534	Nonnegative integer ordering relation. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)
|- ( ( M e. NN0 /\ N e. NN0 ) -> ( M < N <-> M <_ ( N - 1 ) ) )
Theorem	znn0sub 9535	The nonnegative difference of integers is a nonnegative integer. (Generalization of nn0sub 9536.) (Contributed by NM, 14-Jul-2005.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M <_ N <-> ( N - M ) e. NN0 ) )
Theorem	nn0sub 9536	Subtraction of nonnegative integers. (Contributed by NM, 9-May-2004.)
|- ( ( M e. NN0 /\ N e. NN0 ) -> ( M <_ N <-> ( N - M ) e. NN0 ) )
Theorem	ltsubnn0 9537	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.)
|- ( ( A e. NN0 /\ B e. NN0 ) -> ( B < A -> ( A - B ) e. NN0 ) )
Theorem	nn0negleid 9538	A nonnegative integer is greater than or equal to its negative. (Contributed by AV, 13-Aug-2021.)
|- ( A e. NN0 -> -u A <_ A )
Theorem	difgtsumgt 9539	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.)
|- ( ( A e. RR /\ B e. NN0 /\ C e. RR ) -> ( C < ( A - B ) -> C < ( A + B ) ) )
Theorem	nn0n0n1ge2 9540	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.)
|- ( ( N e. NN0 /\ N =/= 0 /\ N =/= 1 ) -> 2 <_ N )
Theorem	elz2 9541*	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.)
|- ( N e. ZZ <-> E. x e. NN E. y e. NN N = ( x - y ) )
Theorem	dfz2 9542	Alternate definition of the integers, based on elz2 9541. (Contributed by Mario Carneiro, 16-May-2014.)
|- ZZ = ( - " ( NN X. NN ) )
Theorem	nn0sub2 9543	Subtraction of nonnegative integers. (Contributed by NM, 4-Sep-2005.)
|- ( ( M e. NN0 /\ N e. NN0 /\ M <_ N ) -> ( N - M ) e. NN0 )
Theorem	zapne 9544	Apartness is equivalent to not equal for integers. (Contributed by Jim Kingdon, 14-Mar-2020.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M # N <-> M =/= N ) )
Theorem	zdceq 9545	Equality of integers is decidable. (Contributed by Jim Kingdon, 14-Mar-2020.)
|- ( ( A e. ZZ /\ B e. ZZ ) -> DECID A = B )
Theorem	zdcle 9546	Integer <_ is decidable. (Contributed by Jim Kingdon, 7-Apr-2020.)
|- ( ( A e. ZZ /\ B e. ZZ ) -> DECID A <_ B )
Theorem	zdclt 9547	Integer < is decidable. (Contributed by Jim Kingdon, 1-Jun-2020.)
|- ( ( A e. ZZ /\ B e. ZZ ) -> DECID A < B )
Theorem	zltlen 9548	Integer 'Less than' expressed in terms of 'less than or equal to'. Also see ltleap 8802 which is a similar result for real numbers. (Contributed by Jim Kingdon, 14-Mar-2020.)
|- ( ( A e. ZZ /\ B e. ZZ ) -> ( A < B <-> ( A <_ B /\ B =/= A ) ) )
Theorem	nn0n0n1ge2b 9549	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.)
|- ( N e. NN0 -> ( ( N =/= 0 /\ N =/= 1 ) <-> 2 <_ N ) )
Theorem	nn0lt10b 9550	A nonnegative integer less than 1 is 0. (Contributed by Paul Chapman, 22-Jun-2011.)
|- ( N e. NN0 -> ( N < 1 <-> N = 0 ) )
Theorem	nn0lt2 9551	A nonnegative integer less than 2 must be 0 or 1. (Contributed by Alexander van der Vekens, 16-Sep-2018.)
|- ( ( N e. NN0 /\ N < 2 ) -> ( N = 0 \/ N = 1 ) )
Theorem	nn0le2is012 9552	A nonnegative integer which is less than or equal to 2 is either 0 or 1 or 2. (Contributed by AV, 16-Mar-2019.)
|- ( ( N e. NN0 /\ N <_ 2 ) -> ( N = 0 \/ N = 1 \/ N = 2 ) )
Theorem	nn0lem1lt 9553	Nonnegative integer ordering relation. (Contributed by NM, 21-Jun-2005.)
|- ( ( M e. NN0 /\ N e. NN0 ) -> ( M <_ N <-> ( M - 1 ) < N ) )
Theorem	nnlem1lt 9554	Positive integer ordering relation. (Contributed by NM, 21-Jun-2005.)
|- ( ( M e. NN /\ N e. NN ) -> ( M <_ N <-> ( M - 1 ) < N ) )
Theorem	nnltlem1 9555	Positive integer ordering relation. (Contributed by NM, 21-Jun-2005.)
|- ( ( M e. NN /\ N e. NN ) -> ( M < N <-> M <_ ( N - 1 ) ) )
Theorem	nnm1ge0 9556	A positive integer decreased by 1 is greater than or equal to 0. (Contributed by AV, 30-Oct-2018.)
|- ( N e. NN -> 0 <_ ( N - 1 ) )
Theorem	nn0ge0div 9557	Division of a nonnegative integer by a positive number is not negative. (Contributed by Alexander van der Vekens, 14-Apr-2018.)
|- ( ( K e. NN0 /\ L e. NN ) -> 0 <_ ( K / L ) )
Theorem	zdiv 9558*	Two ways to express " M divides N. (Contributed by NM, 3-Oct-2008.)
|- ( ( M e. NN /\ N e. ZZ ) -> ( E. k e. ZZ ( M x. k ) = N <-> ( N / M ) e. ZZ ) )
Theorem	zdivadd 9559	Property of divisibility: if D divides A and B then it divides A + B. (Contributed by NM, 3-Oct-2008.)
|- ( ( ( D e. NN /\ A e. ZZ /\ B e. ZZ ) /\ ( ( A / D ) e. ZZ /\ ( B / D ) e. ZZ ) ) -> ( ( A + B ) / D ) e. ZZ )
Theorem	zdivmul 9560	Property of divisibility: if D divides A then it divides B x. A. (Contributed by NM, 3-Oct-2008.)
|- ( ( ( D e. NN /\ A e. ZZ /\ B e. ZZ ) /\ ( A / D ) e. ZZ ) -> ( ( B x. A ) / D ) e. ZZ )
Theorem	zextle 9561*	An extensionality-like property for integer ordering. (Contributed by NM, 29-Oct-2005.)
|- ( ( M e. ZZ /\ N e. ZZ /\ A. k e. ZZ ( k <_ M <-> k <_ N ) ) -> M = N )
Theorem	zextlt 9562*	An extensionality-like property for integer ordering. (Contributed by NM, 29-Oct-2005.)
|- ( ( M e. ZZ /\ N e. ZZ /\ A. k e. ZZ ( k < M <-> k < N ) ) -> M = N )
Theorem	recnz 9563	The reciprocal of a number greater than 1 is not an integer. (Contributed by NM, 3-May-2005.)
|- ( ( A e. RR /\ 1 < A ) -> -. ( 1 / A ) e. ZZ )
Theorem	btwnnz 9564	A number between an integer and its successor is not an integer. (Contributed by NM, 3-May-2005.)
|- ( ( A e. ZZ /\ A < B /\ B < ( A + 1 ) ) -> -. B e. ZZ )
Theorem	gtndiv 9565	A larger number does not divide a smaller positive integer. (Contributed by NM, 3-May-2005.)
|- ( ( A e. RR /\ B e. NN /\ B < A ) -> -. ( B / A ) e. ZZ )
Theorem	halfnz 9566	One-half is not an integer. (Contributed by NM, 31-Jul-2004.)
|- -. ( 1 / 2 ) e. ZZ
Theorem	3halfnz 9567	Three halves is not an integer. (Contributed by AV, 2-Jun-2020.)
|- -. ( 3 / 2 ) e. ZZ
Theorem	suprzclex 9568*	The supremum of a set of integers is an element of the set. (Contributed by Jim Kingdon, 20-Dec-2021.)
|- ( ph -> E. x e. RR ( A. y e. A -. x < y /\ A. y e. RR ( y < x -> E. z e. A y < z ) ) )   &    |- ( ph -> A C_ ZZ )   =>    |- ( ph -> sup ( A , RR , < ) e. A )
Theorem	prime 9569*	Two ways to express " A is a prime number (or 1)". (Contributed by NM, 4-May-2005.)
|- ( A e. NN -> ( A. x e. NN ( ( A / x ) e. NN -> ( x = 1 \/ x = A ) ) <-> A. x e. NN ( ( 1 < x /\ x <_ A /\ ( A / x ) e. NN ) -> x = A ) ) )
Theorem	msqznn 9570	The square of a nonzero integer is a positive integer. (Contributed by NM, 2-Aug-2004.)
|- ( ( A e. ZZ /\ A =/= 0 ) -> ( A x. A ) e. NN )
Theorem	zneo 9571	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.)
|- ( ( A e. ZZ /\ B e. ZZ ) -> ( 2 x. A ) =/= ( ( 2 x. B ) + 1 ) )
Theorem	nneoor 9572	A positive integer is even or odd. (Contributed by Jim Kingdon, 15-Mar-2020.)
|- ( N e. NN -> ( ( N / 2 ) e. NN \/ ( ( N + 1 ) / 2 ) e. NN ) )
Theorem	nneo 9573	A positive integer is even or odd but not both. (Contributed by NM, 1-Jan-2006.) (Proof shortened by Mario Carneiro, 18-May-2014.)
|- ( N e. NN -> ( ( N / 2 ) e. NN <-> -. ( ( N + 1 ) / 2 ) e. NN ) )
Theorem	nneoi 9574	A positive integer is even or odd but not both. (Contributed by NM, 20-Aug-2001.)
|- N e. NN   =>    |- ( ( N / 2 ) e. NN <-> -. ( ( N + 1 ) / 2 ) e. NN )
Theorem	zeo 9575	An integer is even or odd. (Contributed by NM, 1-Jan-2006.)
|- ( N e. ZZ -> ( ( N / 2 ) e. ZZ \/ ( ( N + 1 ) / 2 ) e. ZZ ) )
Theorem	zeo2 9576	An integer is even or odd but not both. (Contributed by Mario Carneiro, 12-Sep-2015.)
|- ( N e. ZZ -> ( ( N / 2 ) e. ZZ <-> -. ( ( N + 1 ) / 2 ) e. ZZ ) )
Theorem	peano2uz2 9577*	Second Peano postulate for upper integers. (Contributed by NM, 3-Oct-2004.)
|- ( ( A e. ZZ /\ B e. { x e. ZZ | A <_ x } ) -> ( B + 1 ) e. { x e. ZZ | A <_ x } )
Theorem	peano5uzti 9578*	Peano's inductive postulate for upper integers. (Contributed by NM, 6-Jul-2005.) (Revised by Mario Carneiro, 25-Jul-2013.)
|- ( N e. ZZ -> ( ( N e. A /\ A. x e. A ( x + 1 ) e. A ) -> { k e. ZZ | N <_ k } C_ A ) )
Theorem	peano5uzi 9579*	Peano's inductive postulate for upper integers. (Contributed by NM, 6-Jul-2005.) (Revised by Mario Carneiro, 3-May-2014.)
|- N e. ZZ   =>    |- ( ( N e. A /\ A. x e. A ( x + 1 ) e. A ) -> { k e. ZZ | N <_ k } C_ A )
Theorem	dfuzi 9580*	An expression for the upper integers that start at N that is analogous to dfnn2 9135 for positive integers. (Contributed by NM, 6-Jul-2005.) (Proof shortened by Mario Carneiro, 3-May-2014.)
|- N e. ZZ   =>    |- { z e. ZZ | N <_ z } = |^| { x | ( N e. x /\ A. y e. x ( y + 1 ) e. x ) }
Theorem	uzind 9581*	Induction on the upper integers that start at M. 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.)
|- ( j = M -> ( ph <-> ps ) )   &    |- ( j = k -> ( ph <-> ch ) )   &    |- ( j = ( k + 1 ) -> ( ph <-> th ) )   &    |- ( j = N -> ( ph <-> ta ) )   &    |- ( M e. ZZ -> ps )   &    |- ( ( M e. ZZ /\ k e. ZZ /\ M <_ k ) -> ( ch -> th ) )   =>    |- ( ( M e. ZZ /\ N e. ZZ /\ M <_ N ) -> ta )
Theorem	uzind2 9582*	Induction on the upper integers that start after an integer M. 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.)
|- ( j = ( M + 1 ) -> ( ph <-> ps ) )   &    |- ( j = k -> ( ph <-> ch ) )   &    |- ( j = ( k + 1 ) -> ( ph <-> th ) )   &    |- ( j = N -> ( ph <-> ta ) )   &    |- ( M e. ZZ -> ps )   &    |- ( ( M e. ZZ /\ k e. ZZ /\ M < k ) -> ( ch -> th ) )   =>    |- ( ( M e. ZZ /\ N e. ZZ /\ M < N ) -> ta )
Theorem	uzind3 9583*	Induction on the upper integers that start at an integer M. 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.)
|- ( j = M -> ( ph <-> ps ) )   &    |- ( j = m -> ( ph <-> ch ) )   &    |- ( j = ( m + 1 ) -> ( ph <-> th ) )   &    |- ( j = N -> ( ph <-> ta ) )   &    |- ( M e. ZZ -> ps )   &    |- ( ( M e. ZZ /\ m e. { k e. ZZ | M <_ k } ) -> ( ch -> th ) )   =>    |- ( ( M e. ZZ /\ N e. { k e. ZZ | M <_ k } ) -> ta )
Theorem	nn0ind 9584*	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.)
|- ( x = 0 -> ( ph <-> ps ) )   &    |- ( x = y -> ( ph <-> ch ) )   &    |- ( x = ( y + 1 ) -> ( ph <-> th ) )   &    |- ( x = A -> ( ph <-> ta ) )   &    |- ps   &    |- ( y e. NN0 -> ( ch -> th ) )   =>    |- ( A e. NN0 -> ta )
Theorem	fzind 9585*	Induction on the integers from M to N 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.)
|- ( x = M -> ( ph <-> ps ) )   &    |- ( x = y -> ( ph <-> ch ) )   &    |- ( x = ( y + 1 ) -> ( ph <-> th ) )   &    |- ( x = K -> ( ph <-> ta ) )   &    |- ( ( M e. ZZ /\ N e. ZZ /\ M <_ N ) -> ps )   &    |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( y e. ZZ /\ M <_ y /\ y < N ) ) -> ( ch -> th ) )   =>    |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( K e. ZZ /\ M <_ K /\ K <_ N ) ) -> ta )
Theorem	fnn0ind 9586*	Induction on the integers from 0 to N 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.)
|- ( x = 0 -> ( ph <-> ps ) )   &    |- ( x = y -> ( ph <-> ch ) )   &    |- ( x = ( y + 1 ) -> ( ph <-> th ) )   &    |- ( x = K -> ( ph <-> ta ) )   &    |- ( N e. NN0 -> ps )   &    |- ( ( N e. NN0 /\ y e. NN0 /\ y < N ) -> ( ch -> th ) )   =>    |- ( ( N e. NN0 /\ K e. NN0 /\ K <_ N ) -> ta )
Theorem	nn0ind-raph 9587*	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.)
|- ( x = 0 -> ( ph <-> ps ) )   &    |- ( x = y -> ( ph <-> ch ) )   &    |- ( x = ( y + 1 ) -> ( ph <-> th ) )   &    |- ( x = A -> ( ph <-> ta ) )   &    |- ps   &    |- ( y e. NN0 -> ( ch -> th ) )   =>    |- ( A e. NN0 -> ta )
Theorem	zindd 9588*	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.)
|- ( x = 0 -> ( ph <-> ps ) )   &    |- ( x = y -> ( ph <-> ch ) )   &    |- ( x = ( y + 1 ) -> ( ph <-> ta ) )   &    |- ( x = -u y -> ( ph <-> th ) )   &    |- ( x = A -> ( ph <-> et ) )   &    |- ( ze -> ps )   &    |- ( ze -> ( y e. NN0 -> ( ch -> ta ) ) )   &    |- ( ze -> ( y e. NN -> ( ch -> th ) ) )   =>    |- ( ze -> ( A e. ZZ -> et ) )
Theorem	btwnz 9589*	Any real number can be sandwiched between two integers. Exercise 2 of [Apostol] p. 28. (Contributed by NM, 10-Nov-2004.)
|- ( A e. RR -> ( E. x e. ZZ x < A /\ E. y e. ZZ A < y ) )
Theorem	nn0zd 9590	A positive integer is an integer. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. NN0 )   =>    |- ( ph -> A e. ZZ )
Theorem	nnzd 9591	A nonnegative integer is an integer. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. NN )   =>    |- ( ph -> A e. ZZ )
Theorem	zred 9592	An integer is a real number. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. ZZ )   =>    |- ( ph -> A e. RR )
Theorem	zcnd 9593	An integer is a complex number. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. ZZ )   =>    |- ( ph -> A e. CC )
Theorem	znegcld 9594	Closure law for negative integers. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. ZZ )   =>    |- ( ph -> -u A e. ZZ )
Theorem	peano2zd 9595	Deduction from second Peano postulate generalized to integers. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. ZZ )   =>    |- ( ph -> ( A + 1 ) e. ZZ )
Theorem	zaddcld 9596	Closure of addition of integers. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. ZZ )   &    |- ( ph -> B e. ZZ )   =>    |- ( ph -> ( A + B ) e. ZZ )
Theorem	zsubcld 9597	Closure of subtraction of integers. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. ZZ )   &    |- ( ph -> B e. ZZ )   =>    |- ( ph -> ( A - B ) e. ZZ )
Theorem	zmulcld 9598	Closure of multiplication of integers. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. ZZ )   &    |- ( ph -> B e. ZZ )   =>    |- ( ph -> ( A x. B ) e. ZZ )
Theorem	zadd2cl 9599	Increasing an integer by 2 results in an integer. (Contributed by Alexander van der Vekens, 16-Sep-2018.)
|- ( N e. ZZ -> ( N + 2 ) e. ZZ )
Theorem	btwnapz 9600	A number between an integer and its successor is apart from any integer. (Contributed by Jim Kingdon, 6-Jan-2023.)
|- ( ph -> A e. ZZ )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. ZZ )   &    |- ( ph -> A < B )   &    |- ( ph -> B < ( A + 1 ) )   =>    |- ( ph -> B # C )