P. 94 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 9301-9400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	zrevaddcl 9301	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 9302	The positive difference of unequal integers is a positive integer. (Generalization of nnsub 8956.) (Contributed by NM, 11-May-2004.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M < N <-> ( N - M ) e. NN ) )
Theorem	nzadd 9303	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 9304	Closure of multiplication of integers. (Contributed by NM, 30-Jul-2004.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M x. N ) e. ZZ )
Theorem	zltp1le 9305	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 9306	Integer ordering relation. (Contributed by NM, 10-May-2004.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M <_ N <-> M < ( N + 1 ) ) )
Theorem	zlem1lt 9307	Integer ordering relation. (Contributed by NM, 13-Nov-2004.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M <_ N <-> ( M - 1 ) < N ) )
Theorem	zltlem1 9308	Integer ordering relation. (Contributed by NM, 13-Nov-2004.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M < N <-> M <_ ( N - 1 ) ) )
Theorem	zgt0ge1 9309	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 9310	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 9311	Positive integer ordering relation. (Contributed by NM, 19-Aug-2001.)
|- ( ( A e. NN /\ B e. NN ) -> ( A < B <-> ( A + 1 ) <_ B ) )
Theorem	nnaddm1cl 9312	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 9313	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 9314	Nonnegative integer ordering relation. (Contributed by Raph Levien, 10-Apr-2004.)
|- ( ( M e. NN0 /\ N e. NN0 ) -> ( M <_ N <-> M < ( N + 1 ) ) )
Theorem	nn0ltlem1 9315	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 9316	The nonnegative difference of integers is a nonnegative integer. (Generalization of nn0sub 9317.) (Contributed by NM, 14-Jul-2005.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M <_ N <-> ( N - M ) e. NN0 ) )
Theorem	nn0sub 9317	Subtraction of nonnegative integers. (Contributed by NM, 9-May-2004.)
|- ( ( M e. NN0 /\ N e. NN0 ) -> ( M <_ N <-> ( N - M ) e. NN0 ) )
Theorem	ltsubnn0 9318	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 9319	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 9320	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 9321	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 9322*	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 9323	Alternate definition of the integers, based on elz2 9322. (Contributed by Mario Carneiro, 16-May-2014.)
|- ZZ = ( - " ( NN X. NN ) )
Theorem	nn0sub2 9324	Subtraction of nonnegative integers. (Contributed by NM, 4-Sep-2005.)
|- ( ( M e. NN0 /\ N e. NN0 /\ M <_ N ) -> ( N - M ) e. NN0 )
Theorem	zapne 9325	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 9326	Equality of integers is decidable. (Contributed by Jim Kingdon, 14-Mar-2020.)
|- ( ( A e. ZZ /\ B e. ZZ ) -> DECID A = B )
Theorem	zdcle 9327	Integer <_ is decidable. (Contributed by Jim Kingdon, 7-Apr-2020.)
|- ( ( A e. ZZ /\ B e. ZZ ) -> DECID A <_ B )
Theorem	zdclt 9328	Integer < is decidable. (Contributed by Jim Kingdon, 1-Jun-2020.)
|- ( ( A e. ZZ /\ B e. ZZ ) -> DECID A < B )
Theorem	zltlen 9329	Integer 'Less than' expressed in terms of 'less than or equal to'. Also see ltleap 8587 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 9330	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 9331	A nonnegative integer less than 1 is 0. (Contributed by Paul Chapman, 22-Jun-2011.)
|- ( N e. NN0 -> ( N < 1 <-> N = 0 ) )
Theorem	nn0lt2 9332	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 9333	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 9334	Nonnegative integer ordering relation. (Contributed by NM, 21-Jun-2005.)
|- ( ( M e. NN0 /\ N e. NN0 ) -> ( M <_ N <-> ( M - 1 ) < N ) )
Theorem	nnlem1lt 9335	Positive integer ordering relation. (Contributed by NM, 21-Jun-2005.)
|- ( ( M e. NN /\ N e. NN ) -> ( M <_ N <-> ( M - 1 ) < N ) )
Theorem	nnltlem1 9336	Positive integer ordering relation. (Contributed by NM, 21-Jun-2005.)
|- ( ( M e. NN /\ N e. NN ) -> ( M < N <-> M <_ ( N - 1 ) ) )
Theorem	nnm1ge0 9337	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 9338	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 9339*	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 9340	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 9341	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 9342*	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 9343*	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 9344	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 9345	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 9346	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 9347	One-half is not an integer. (Contributed by NM, 31-Jul-2004.)
|- -. ( 1 / 2 ) e. ZZ
Theorem	3halfnz 9348	Three halves is not an integer. (Contributed by AV, 2-Jun-2020.)
|- -. ( 3 / 2 ) e. ZZ
Theorem	suprzclex 9349*	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 9350*	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 9351	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 9352	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 9353	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 9354	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 9355	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 9356	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 9357	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 9358*	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 9359*	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 9360*	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 9361*	An expression for the upper integers that start at N that is analogous to dfnn2 8919 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 9362*	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 9363*	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 9364*	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 9365*	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 9366*	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 9367*	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 9368*	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 9369*	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 9370*	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 9371	A positive integer is an integer. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. NN0 )   =>    |- ( ph -> A e. ZZ )
Theorem	nnzd 9372	A nonnegative integer is an integer. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. NN )   =>    |- ( ph -> A e. ZZ )
Theorem	zred 9373	An integer is a real number. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. ZZ )   =>    |- ( ph -> A e. RR )
Theorem	zcnd 9374	An integer is a complex number. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. ZZ )   =>    |- ( ph -> A e. CC )
Theorem	znegcld 9375	Closure law for negative integers. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. ZZ )   =>    |- ( ph -> -u A e. ZZ )
Theorem	peano2zd 9376	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 9377	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 9378	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 9379	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 9380	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 9381	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 )
4.4.10  Decimal arithmetic
Syntax	cdc 9382	Constant used for decimal constructor.
class ; A B
Definition	df-dec 9383	Define the "decimal constructor", which is used to build up "decimal integers" or "numeric terms" in base 1 0. For example, (;;; 1 0 0 0 + ;;; 2 0 0 0 ) = ;;; 3 0 0 0 1kp2ke3k 14358. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 1-Aug-2021.)
|- ; A B = ( ( ( 9 + 1 ) x. A ) + B )
Theorem	9p1e10 9384	9 + 1 = 10. (Contributed by Mario Carneiro, 18-Apr-2015.) (Revised by Stanislas Polu, 7-Apr-2020.) (Revised by AV, 1-Aug-2021.)
|- ( 9 + 1 ) = ; 1 0
Theorem	dfdec10 9385	Version of the definition of the "decimal constructor" using ; 1 0 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.)
|- ; A B = ( (; 1 0 x. A ) + B )
Theorem	deceq1 9386	Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
|- ( A = B -> ; A C = ; B C )
Theorem	deceq2 9387	Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
|- ( A = B -> ; C A = ; C B )
Theorem	deceq1i 9388	Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
|- A = B   =>    |- ; A C = ; B C
Theorem	deceq2i 9389	Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
|- A = B   =>    |- ; C A = ; C B
Theorem	deceq12i 9390	Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
|- A = B   &    |- C = D   =>    |- ; A C = ; B D
Theorem	numnncl 9391	Closure for a numeral (with units place). (Contributed by Mario Carneiro, 18-Feb-2014.)
|- T e. NN0   &    |- A e. NN0   &    |- B e. NN   =>    |- ( ( T x. A ) + B ) e. NN
Theorem	num0u 9392	Add a zero in the units place. (Contributed by Mario Carneiro, 18-Feb-2014.)
|- T e. NN0   &    |- A e. NN0   =>    |- ( T x. A ) = ( ( T x. A ) + 0 )
Theorem	num0h 9393	Add a zero in the higher places. (Contributed by Mario Carneiro, 18-Feb-2014.)
|- T e. NN0   &    |- A e. NN0   =>    |- A = ( ( T x. 0 ) + A )
Theorem	numcl 9394	Closure for a decimal integer (with units place). (Contributed by Mario Carneiro, 18-Feb-2014.)
|- T e. NN0   &    |- A e. NN0   &    |- B e. NN0   =>    |- ( ( T x. A ) + B ) e. NN0
Theorem	numsuc 9395	The successor of a decimal integer (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
|- T e. NN0   &    |- A e. NN0   &    |- B e. NN0   &    |- ( B + 1 ) = C   &    |- N = ( ( T x. A ) + B )   =>    |- ( N + 1 ) = ( ( T x. A ) + C )
Theorem	deccl 9396	Closure for a numeral. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
|- A e. NN0   &    |- B e. NN0   =>    |- ; A B e. NN0
Theorem	10nn 9397	10 is a positive integer. (Contributed by NM, 8-Nov-2012.) (Revised by AV, 6-Sep-2021.)
|- ; 1 0 e. NN
Theorem	10pos 9398	The number 10 is positive. (Contributed by NM, 5-Feb-2007.) (Revised by AV, 8-Sep-2021.)
|- 0 < ; 1 0
Theorem	10nn0 9399	10 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
|- ; 1 0 e. NN0
Theorem	10re 9400	The number 10 is real. (Contributed by NM, 5-Feb-2007.) (Revised by AV, 8-Sep-2021.)
|- ; 1 0 e. RR