Theorem List for Intuitionistic Logic Explorer - 9101-9200 *Has distinct variable
group(s)
Type | Label | Description |
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Theorem | elnnz1 9101 |
Positive integer property expressed in terms of integers. (Contributed by
NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)
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Theorem | nnzrab 9102 |
Positive integers expressed as a subset of integers. (Contributed by NM,
3-Oct-2004.)
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Theorem | nn0zrab 9103 |
Nonnegative integers expressed as a subset of integers. (Contributed by
NM, 3-Oct-2004.)
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Theorem | 1z 9104 |
One is an integer. (Contributed by NM, 10-May-2004.)
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Theorem | 1zzd 9105 |
1 is an integer, deductive form (common case). (Contributed by David A.
Wheeler, 6-Dec-2018.)
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Theorem | 2z 9106 |
Two is an integer. (Contributed by NM, 10-May-2004.)
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Theorem | 3z 9107 |
3 is an integer. (Contributed by David A. Wheeler, 8-Dec-2018.)
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Theorem | 4z 9108 |
4 is an integer. (Contributed by BJ, 26-Mar-2020.)
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Theorem | znegcl 9109 |
Closure law for negative integers. (Contributed by NM, 9-May-2004.)
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Theorem | neg1z 9110 |
-1 is an integer (common case). (Contributed by David A. Wheeler,
5-Dec-2018.)
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Theorem | znegclb 9111 |
A number is an integer iff its negative is. (Contributed by Stefan
O'Rear, 13-Sep-2014.)
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Theorem | nn0negz 9112 |
The negative of a nonnegative integer is an integer. (Contributed by NM,
9-May-2004.)
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Theorem | nn0negzi 9113 |
The negative of a nonnegative integer is an integer. (Contributed by
Mario Carneiro, 18-Feb-2014.)
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Theorem | peano2z 9114 |
Second Peano postulate generalized to integers. (Contributed by NM,
13-Feb-2005.)
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Theorem | zaddcllempos 9115 |
Lemma for zaddcl 9118. Special case in which is a positive integer.
(Contributed by Jim Kingdon, 14-Mar-2020.)
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Theorem | peano2zm 9116 |
"Reverse" second Peano postulate for integers. (Contributed by NM,
12-Sep-2005.)
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Theorem | zaddcllemneg 9117 |
Lemma for zaddcl 9118. Special case in which is a positive
integer. (Contributed by Jim Kingdon, 14-Mar-2020.)
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Theorem | zaddcl 9118 |
Closure of addition of integers. (Contributed by NM, 9-May-2004.) (Proof
shortened by Mario Carneiro, 16-May-2014.)
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Theorem | zsubcl 9119 |
Closure of subtraction of integers. (Contributed by NM, 11-May-2004.)
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Theorem | ztri3or0 9120 |
Integer trichotomy (with zero). (Contributed by Jim Kingdon,
14-Mar-2020.)
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Theorem | ztri3or 9121 |
Integer trichotomy. (Contributed by Jim Kingdon, 14-Mar-2020.)
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Theorem | zletric 9122 |
Trichotomy law. (Contributed by Jim Kingdon, 27-Mar-2020.)
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Theorem | zlelttric 9123 |
Trichotomy law. (Contributed by Jim Kingdon, 17-Apr-2020.)
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Theorem | zltnle 9124 |
'Less than' expressed in terms of 'less than or equal to'. (Contributed
by Jim Kingdon, 14-Mar-2020.)
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Theorem | zleloe 9125 |
Integer 'Less than or equal to' expressed in terms of 'less than' or
'equals'. (Contributed by Jim Kingdon, 8-Apr-2020.)
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Theorem | znnnlt1 9126 |
An integer is not a positive integer iff it is less than one.
(Contributed by NM, 13-Jul-2005.)
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Theorem | zletr 9127 |
Transitive law of ordering for integers. (Contributed by Alexander van
der Vekens, 3-Apr-2018.)
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Theorem | zrevaddcl 9128 |
Reverse closure law for addition of integers. (Contributed by NM,
11-May-2004.)
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Theorem | znnsub 9129 |
The positive difference of unequal integers is a positive integer.
(Generalization of nnsub 8783.) (Contributed by NM, 11-May-2004.)
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Theorem | nzadd 9130 |
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.)
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Theorem | zmulcl 9131 |
Closure of multiplication of integers. (Contributed by NM,
30-Jul-2004.)
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Theorem | zltp1le 9132 |
Integer ordering relation. (Contributed by NM, 10-May-2004.) (Proof
shortened by Mario Carneiro, 16-May-2014.)
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Theorem | zleltp1 9133 |
Integer ordering relation. (Contributed by NM, 10-May-2004.)
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Theorem | zlem1lt 9134 |
Integer ordering relation. (Contributed by NM, 13-Nov-2004.)
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Theorem | zltlem1 9135 |
Integer ordering relation. (Contributed by NM, 13-Nov-2004.)
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Theorem | zgt0ge1 9136 |
An integer greater than
is greater than or equal to .
(Contributed by AV, 14-Oct-2018.)
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Theorem | nnleltp1 9137 |
Positive integer ordering relation. (Contributed by NM, 13-Aug-2001.)
(Proof shortened by Mario Carneiro, 16-May-2014.)
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Theorem | nnltp1le 9138 |
Positive integer ordering relation. (Contributed by NM, 19-Aug-2001.)
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Theorem | nnaddm1cl 9139 |
Closure of addition of positive integers minus one. (Contributed by NM,
6-Aug-2003.) (Proof shortened by Mario Carneiro, 16-May-2014.)
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Theorem | nn0ltp1le 9140 |
Nonnegative integer ordering relation. (Contributed by Raph Levien,
10-Dec-2002.) (Proof shortened by Mario Carneiro, 16-May-2014.)
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Theorem | nn0leltp1 9141 |
Nonnegative integer ordering relation. (Contributed by Raph Levien,
10-Apr-2004.)
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Theorem | nn0ltlem1 9142 |
Nonnegative integer ordering relation. (Contributed by NM, 10-May-2004.)
(Proof shortened by Mario Carneiro, 16-May-2014.)
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Theorem | znn0sub 9143 |
The nonnegative difference of integers is a nonnegative integer.
(Generalization of nn0sub 9144.) (Contributed by NM, 14-Jul-2005.)
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Theorem | nn0sub 9144 |
Subtraction of nonnegative integers. (Contributed by NM, 9-May-2004.)
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Theorem | nn0n0n1ge2 9145 |
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.)
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Theorem | elz2 9146* |
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.)
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Theorem | dfz2 9147 |
Alternate definition of the integers, based on elz2 9146.
(Contributed by
Mario Carneiro, 16-May-2014.)
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Theorem | nn0sub2 9148 |
Subtraction of nonnegative integers. (Contributed by NM, 4-Sep-2005.)
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Theorem | zapne 9149 |
Apartness is equivalent to not equal for integers. (Contributed by Jim
Kingdon, 14-Mar-2020.)
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Theorem | zdceq 9150 |
Equality of integers is decidable. (Contributed by Jim Kingdon,
14-Mar-2020.)
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DECID
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Theorem | zdcle 9151 |
Integer is
decidable. (Contributed by Jim Kingdon, 7-Apr-2020.)
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DECID |
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Theorem | zdclt 9152 |
Integer is
decidable. (Contributed by Jim Kingdon, 1-Jun-2020.)
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DECID |
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Theorem | zltlen 9153 |
Integer 'Less than' expressed in terms of 'less than or equal to'. Also
see ltleap 8418 which is a similar result for real numbers.
(Contributed by
Jim Kingdon, 14-Mar-2020.)
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Theorem | nn0n0n1ge2b 9154 |
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.)
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Theorem | nn0lt10b 9155 |
A nonnegative integer less than is .
(Contributed by Paul
Chapman, 22-Jun-2011.)
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Theorem | nn0lt2 9156 |
A nonnegative integer less than 2 must be 0 or 1. (Contributed by
Alexander van der Vekens, 16-Sep-2018.)
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Theorem | nn0le2is012 9157 |
A nonnegative integer which is less than or equal to 2 is either 0 or 1 or
2. (Contributed by AV, 16-Mar-2019.)
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Theorem | nn0lem1lt 9158 |
Nonnegative integer ordering relation. (Contributed by NM,
21-Jun-2005.)
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Theorem | nnlem1lt 9159 |
Positive integer ordering relation. (Contributed by NM, 21-Jun-2005.)
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Theorem | nnltlem1 9160 |
Positive integer ordering relation. (Contributed by NM, 21-Jun-2005.)
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Theorem | nnm1ge0 9161 |
A positive integer decreased by 1 is greater than or equal to 0.
(Contributed by AV, 30-Oct-2018.)
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Theorem | nn0ge0div 9162 |
Division of a nonnegative integer by a positive number is not negative.
(Contributed by Alexander van der Vekens, 14-Apr-2018.)
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Theorem | zdiv 9163* |
Two ways to express " divides .
(Contributed by NM,
3-Oct-2008.)
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Theorem | zdivadd 9164 |
Property of divisibility: if divides
and then it divides
. (Contributed by NM, 3-Oct-2008.)
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Theorem | zdivmul 9165 |
Property of divisibility: if divides
then it divides
. (Contributed by NM, 3-Oct-2008.)
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Theorem | zextle 9166* |
An extensionality-like property for integer ordering. (Contributed by
NM, 29-Oct-2005.)
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Theorem | zextlt 9167* |
An extensionality-like property for integer ordering. (Contributed by
NM, 29-Oct-2005.)
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Theorem | recnz 9168 |
The reciprocal of a number greater than 1 is not an integer. (Contributed
by NM, 3-May-2005.)
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Theorem | btwnnz 9169 |
A number between an integer and its successor is not an integer.
(Contributed by NM, 3-May-2005.)
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Theorem | gtndiv 9170 |
A larger number does not divide a smaller positive integer. (Contributed
by NM, 3-May-2005.)
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Theorem | halfnz 9171 |
One-half is not an integer. (Contributed by NM, 31-Jul-2004.)
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Theorem | 3halfnz 9172 |
Three halves is not an integer. (Contributed by AV, 2-Jun-2020.)
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Theorem | suprzclex 9173* |
The supremum of a set of integers is an element of the set.
(Contributed by Jim Kingdon, 20-Dec-2021.)
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Theorem | prime 9174* |
Two ways to express " is a prime number (or 1)." (Contributed by
NM, 4-May-2005.)
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Theorem | msqznn 9175 |
The square of a nonzero integer is a positive integer. (Contributed by
NM, 2-Aug-2004.)
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Theorem | zneo 9176 |
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.)
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Theorem | nneoor 9177 |
A positive integer is even or odd. (Contributed by Jim Kingdon,
15-Mar-2020.)
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Theorem | nneo 9178 |
A positive integer is even or odd but not both. (Contributed by NM,
1-Jan-2006.) (Proof shortened by Mario Carneiro, 18-May-2014.)
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Theorem | nneoi 9179 |
A positive integer is even or odd but not both. (Contributed by NM,
20-Aug-2001.)
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Theorem | zeo 9180 |
An integer is even or odd. (Contributed by NM, 1-Jan-2006.)
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Theorem | zeo2 9181 |
An integer is even or odd but not both. (Contributed by Mario Carneiro,
12-Sep-2015.)
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Theorem | peano2uz2 9182* |
Second Peano postulate for upper integers. (Contributed by NM,
3-Oct-2004.)
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Theorem | peano5uzti 9183* |
Peano's inductive postulate for upper integers. (Contributed by NM,
6-Jul-2005.) (Revised by Mario Carneiro, 25-Jul-2013.)
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Theorem | peano5uzi 9184* |
Peano's inductive postulate for upper integers. (Contributed by NM,
6-Jul-2005.) (Revised by Mario Carneiro, 3-May-2014.)
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Theorem | dfuzi 9185* |
An expression for the upper integers that start at that is
analogous to dfnn2 8746 for positive integers. (Contributed by NM,
6-Jul-2005.) (Proof shortened by Mario Carneiro, 3-May-2014.)
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Theorem | uzind 9186* |
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.)
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Theorem | uzind2 9187* |
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.)
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Theorem | uzind3 9188* |
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.)
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Theorem | nn0ind 9189* |
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.)
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Theorem | fzind 9190* |
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.)
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Theorem | fnn0ind 9191* |
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.)
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Theorem | nn0ind-raph 9192* |
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.)
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Theorem | zindd 9193* |
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.)
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Theorem | btwnz 9194* |
Any real number can be sandwiched between two integers. Exercise 2 of
[Apostol] p. 28. (Contributed by NM,
10-Nov-2004.)
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Theorem | nn0zd 9195 |
A positive integer is an integer. (Contributed by Mario Carneiro,
28-May-2016.)
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Theorem | nnzd 9196 |
A nonnegative integer is an integer. (Contributed by Mario Carneiro,
28-May-2016.)
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Theorem | zred 9197 |
An integer is a real number. (Contributed by Mario Carneiro,
28-May-2016.)
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Theorem | zcnd 9198 |
An integer is a complex number. (Contributed by Mario Carneiro,
28-May-2016.)
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Theorem | znegcld 9199 |
Closure law for negative integers. (Contributed by Mario Carneiro,
28-May-2016.)
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Theorem | peano2zd 9200 |
Deduction from second Peano postulate generalized to integers.
(Contributed by Mario Carneiro, 28-May-2016.)
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