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Theorem List for Intuitionistic Logic Explorer - 8001-8100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnn0z 8001 A nonnegative integer is an integer. (Contributed by NM, 9-May-2004.)
(𝑁 0𝑁 ℤ)
 
Theoremnnzi 8002 A positive integer is an integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
𝑁        𝑁
 
Theoremnn0zi 8003 A nonnegative integer is an integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
𝑁 0       𝑁
 
Theoremelnnz1 8004 Positive integer property expressed in terms of integers. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)
(𝑁 ℕ ↔ (𝑁 1 ≤ 𝑁))
 
Theoremnnzrab 8005 Positive integers expressed as a subset of integers. (Contributed by NM, 3-Oct-2004.)
ℕ = {x ℤ ∣ 1 ≤ x}
 
Theoremnn0zrab 8006 Nonnegative integers expressed as a subset of integers. (Contributed by NM, 3-Oct-2004.)
0 = {x ℤ ∣ 0 ≤ x}
 
Theorem1z 8007 One is an integer. (Contributed by NM, 10-May-2004.)
1
 
Theorem1zzd 8008 1 is an integer, deductive form (common case). (Contributed by David A. Wheeler, 6-Dec-2018.)
(φ → 1 ℤ)
 
Theorem2z 8009 Two is an integer. (Contributed by NM, 10-May-2004.)
2
 
Theorem3z 8010 3 is an integer. (Contributed by David A. Wheeler, 8-Dec-2018.)
3
 
Theorem4z 8011 4 is an integer. (Contributed by BJ, 26-Mar-2020.)
4
 
Theoremznegcl 8012 Closure law for negative integers. (Contributed by NM, 9-May-2004.)
(𝑁 ℤ → -𝑁 ℤ)
 
Theoremneg1z 8013 -1 is an integer (common case). (Contributed by David A. Wheeler, 5-Dec-2018.)
-1
 
Theoremznegclb 8014 A number is an integer iff its negative is. (Contributed by Stefan O'Rear, 13-Sep-2014.)
(A ℂ → (A ℤ ↔ -A ℤ))
 
Theoremnn0negz 8015 The negative of a nonnegative integer is an integer. (Contributed by NM, 9-May-2004.)
(𝑁 0 → -𝑁 ℤ)
 
Theoremnn0negzi 8016 The negative of a nonnegative integer is an integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
𝑁 0       -𝑁
 
Theorempeano2z 8017 Second Peano postulate generalized to integers. (Contributed by NM, 13-Feb-2005.)
(𝑁 ℤ → (𝑁 + 1) ℤ)
 
Theoremzaddcllempos 8018 Lemma for zaddcl 8021. Special case in which 𝑁 is a positive integer. (Contributed by Jim Kingdon, 14-Mar-2020.)
((𝑀 𝑁 ℕ) → (𝑀 + 𝑁) ℤ)
 
Theorempeano2zm 8019 "Reverse" second Peano postulate for integers. (Contributed by NM, 12-Sep-2005.)
(𝑁 ℤ → (𝑁 − 1) ℤ)
 
Theoremzaddcllemneg 8020 Lemma for zaddcl 8021. Special case in which -𝑁 is a positive integer. (Contributed by Jim Kingdon, 14-Mar-2020.)
((𝑀 𝑁 -𝑁 ℕ) → (𝑀 + 𝑁) ℤ)
 
Theoremzaddcl 8021 Closure of addition of integers. (Contributed by NM, 9-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)
((𝑀 𝑁 ℤ) → (𝑀 + 𝑁) ℤ)
 
Theoremzsubcl 8022 Closure of subtraction of integers. (Contributed by NM, 11-May-2004.)
((𝑀 𝑁 ℤ) → (𝑀𝑁) ℤ)
 
Theoremztri3or0 8023 Integer trichotomy (with zero). (Contributed by Jim Kingdon, 14-Mar-2020.)
(𝑁 ℤ → (𝑁 < 0 𝑁 = 0 0 < 𝑁))
 
Theoremztri3or 8024 Integer trichotomy. (Contributed by Jim Kingdon, 14-Mar-2020.)
((𝑀 𝑁 ℤ) → (𝑀 < 𝑁 𝑀 = 𝑁 𝑁 < 𝑀))
 
Theoremzletric 8025 Trichotomy law. (Contributed by Jim Kingdon, 27-Mar-2020.)
((A B ℤ) → (AB BA))
 
Theoremzlelttric 8026 Trichotomy law. (Contributed by Jim Kingdon, 17-Apr-2020.)
((A B ℤ) → (AB B < A))
 
Theoremzltnle 8027 'Less than' expressed in terms of 'less than or equal to'. (Contributed by Jim Kingdon, 14-Mar-2020.)
((A B ℤ) → (A < B ↔ ¬ BA))
 
Theoremzleloe 8028 Integer 'Less than or equal to' expressed in terms of 'less than' or 'equals'. (Contributed by Jim Kingdon, 8-Apr-2020.)
((A B ℤ) → (AB ↔ (A < B A = B)))
 
Theoremznnnlt1 8029 An integer is not a positive integer iff it is less than one. (Contributed by NM, 13-Jul-2005.)
(𝑁 ℤ → (¬ 𝑁 ℕ ↔ 𝑁 < 1))
 
Theoremzletr 8030 Transitive law of ordering for integers. (Contributed by Alexander van der Vekens, 3-Apr-2018.)
((𝐽 𝐾 𝐿 ℤ) → ((𝐽𝐾 𝐾𝐿) → 𝐽𝐿))
 
Theoremzrevaddcl 8031 Reverse closure law for addition of integers. (Contributed by NM, 11-May-2004.)
(𝑁 ℤ → ((𝑀 (𝑀 + 𝑁) ℤ) ↔ 𝑀 ℤ))
 
Theoremznnsub 8032 The positive difference of unequal integers is a positive integer. (Generalization of nnsub 7693.) (Contributed by NM, 11-May-2004.)
((𝑀 𝑁 ℤ) → (𝑀 < 𝑁 ↔ (𝑁𝑀) ℕ))
 
Theoremzmulcl 8033 Closure of multiplication of integers. (Contributed by NM, 30-Jul-2004.)
((𝑀 𝑁 ℤ) → (𝑀 · 𝑁) ℤ)
 
Theoremzltp1le 8034 Integer ordering relation. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)
((𝑀 𝑁 ℤ) → (𝑀 < 𝑁 ↔ (𝑀 + 1) ≤ 𝑁))
 
Theoremzleltp1 8035 Integer ordering relation. (Contributed by NM, 10-May-2004.)
((𝑀 𝑁 ℤ) → (𝑀𝑁𝑀 < (𝑁 + 1)))
 
Theoremzlem1lt 8036 Integer ordering relation. (Contributed by NM, 13-Nov-2004.)
((𝑀 𝑁 ℤ) → (𝑀𝑁 ↔ (𝑀 − 1) < 𝑁))
 
Theoremzltlem1 8037 Integer ordering relation. (Contributed by NM, 13-Nov-2004.)
((𝑀 𝑁 ℤ) → (𝑀 < 𝑁𝑀 ≤ (𝑁 − 1)))
 
Theoremzgt0ge1 8038 An integer greater than 0 is greater than or equal to 1. (Contributed by AV, 14-Oct-2018.)
(𝑍 ℤ → (0 < 𝑍 ↔ 1 ≤ 𝑍))
 
Theoremnnleltp1 8039 Positive integer ordering relation. (Contributed by NM, 13-Aug-2001.) (Proof shortened by Mario Carneiro, 16-May-2014.)
((A B ℕ) → (ABA < (B + 1)))
 
Theoremnnltp1le 8040 Positive integer ordering relation. (Contributed by NM, 19-Aug-2001.)
((A B ℕ) → (A < B ↔ (A + 1) ≤ B))
 
Theoremnnaddm1cl 8041 Closure of addition of positive integers minus one. (Contributed by NM, 6-Aug-2003.) (Proof shortened by Mario Carneiro, 16-May-2014.)
((A B ℕ) → ((A + B) − 1) ℕ)
 
Theoremnn0ltp1le 8042 Nonnegative integer ordering relation. (Contributed by Raph Levien, 10-Dec-2002.) (Proof shortened by Mario Carneiro, 16-May-2014.)
((𝑀 0 𝑁 0) → (𝑀 < 𝑁 ↔ (𝑀 + 1) ≤ 𝑁))
 
Theoremnn0leltp1 8043 Nonnegative integer ordering relation. (Contributed by Raph Levien, 10-Apr-2004.)
((𝑀 0 𝑁 0) → (𝑀𝑁𝑀 < (𝑁 + 1)))
 
Theoremnn0ltlem1 8044 Nonnegative integer ordering relation. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)
((𝑀 0 𝑁 0) → (𝑀 < 𝑁𝑀 ≤ (𝑁 − 1)))
 
Theoremznn0sub 8045 The nonnegative difference of integers is a nonnegative integer. (Generalization of nn0sub 8046.) (Contributed by NM, 14-Jul-2005.)
((𝑀 𝑁 ℤ) → (𝑀𝑁 ↔ (𝑁𝑀) 0))
 
Theoremnn0sub 8046 Subtraction of nonnegative integers. (Contributed by NM, 9-May-2004.)
((𝑀 0 𝑁 0) → (𝑀𝑁 ↔ (𝑁𝑀) 0))
 
Theoremnn0n0n1ge2 8047 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 0 𝑁 ≠ 0 𝑁 ≠ 1) → 2 ≤ 𝑁)
 
Theoremelz2 8048* 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 ℤ ↔ x y 𝑁 = (xy))
 
Theoremdfz2 8049 Alternative definition of the integers, based on elz2 8048. (Contributed by Mario Carneiro, 16-May-2014.)
ℤ = ( − “ (ℕ × ℕ))
 
Theoremnn0sub2 8050 Subtraction of nonnegative integers. (Contributed by NM, 4-Sep-2005.)
((𝑀 0 𝑁 0 𝑀𝑁) → (𝑁𝑀) 0)
 
Theoremzapne 8051 Apartness is equivalent to not equal for integers. (Contributed by Jim Kingdon, 14-Mar-2020.)
((𝑀 𝑁 ℤ) → (𝑀 # 𝑁𝑀𝑁))
 
Theoremzdceq 8052 Equality of integers is decidable. (Contributed by Jim Kingdon, 14-Mar-2020.)
((A B ℤ) → DECID A = B)
 
Theoremzdcle 8053 Integer is decidable. (Contributed by Jim Kingdon, 7-Apr-2020.)
((A B ℤ) → DECID AB)
 
Theoremzdclt 8054 Integer < is decidable. (Contributed by Jim Kingdon, 1-Jun-2020.)
((A B ℤ) → DECID A < B)
 
Theoremzltlen 8055 Integer 'Less than' expressed in terms of 'less than or equal to'. Also see ltleap 7373 which is a similar result for complex numbers. (Contributed by Jim Kingdon, 14-Mar-2020.)
((A B ℤ) → (A < B ↔ (AB BA)))
 
Theoremnn0n0n1ge2b 8056 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 0 → ((𝑁 ≠ 0 𝑁 ≠ 1) ↔ 2 ≤ 𝑁))
 
Theoremnn0lt10b 8057 A nonnegative integer less than 1 is 0. (Contributed by Paul Chapman, 22-Jun-2011.)
(𝑁 0 → (𝑁 < 1 ↔ 𝑁 = 0))
 
Theoremnn0lt2 8058 A nonnegative integer less than 2 must be 0 or 1. (Contributed by Alexander van der Vekens, 16-Sep-2018.)
((𝑁 0 𝑁 < 2) → (𝑁 = 0 𝑁 = 1))
 
Theoremnn0lem1lt 8059 Nonnegative integer ordering relation. (Contributed by NM, 21-Jun-2005.)
((𝑀 0 𝑁 0) → (𝑀𝑁 ↔ (𝑀 − 1) < 𝑁))
 
Theoremnnlem1lt 8060 Positive integer ordering relation. (Contributed by NM, 21-Jun-2005.)
((𝑀 𝑁 ℕ) → (𝑀𝑁 ↔ (𝑀 − 1) < 𝑁))
 
Theoremnnltlem1 8061 Positive integer ordering relation. (Contributed by NM, 21-Jun-2005.)
((𝑀 𝑁 ℕ) → (𝑀 < 𝑁𝑀 ≤ (𝑁 − 1)))
 
Theoremnnm1ge0 8062 A positive integer decreased by 1 is greater than or equal to 0. (Contributed by AV, 30-Oct-2018.)
(𝑁 ℕ → 0 ≤ (𝑁 − 1))
 
Theoremnn0ge0div 8063 Division of a nonnegative integer by a positive number is not negative. (Contributed by Alexander van der Vekens, 14-Apr-2018.)
((𝐾 0 𝐿 ℕ) → 0 ≤ (𝐾 / 𝐿))
 
Theoremzdiv 8064* Two ways to express "𝑀 divides 𝑁. (Contributed by NM, 3-Oct-2008.)
((𝑀 𝑁 ℤ) → (𝑘 ℤ (𝑀 · 𝑘) = 𝑁 ↔ (𝑁 / 𝑀) ℤ))
 
Theoremzdivadd 8065 Property of divisibility: if 𝐷 divides A and B then it divides A + B. (Contributed by NM, 3-Oct-2008.)
(((𝐷 A B ℤ) ((A / 𝐷) (B / 𝐷) ℤ)) → ((A + B) / 𝐷) ℤ)
 
Theoremzdivmul 8066 Property of divisibility: if 𝐷 divides A then it divides B · A. (Contributed by NM, 3-Oct-2008.)
(((𝐷 A B ℤ) (A / 𝐷) ℤ) → ((B · A) / 𝐷) ℤ)
 
Theoremzextle 8067* An extensionality-like property for integer ordering. (Contributed by NM, 29-Oct-2005.)
((𝑀 𝑁 𝑘 ℤ (𝑘𝑀𝑘𝑁)) → 𝑀 = 𝑁)
 
Theoremzextlt 8068* An extensionality-like property for integer ordering. (Contributed by NM, 29-Oct-2005.)
((𝑀 𝑁 𝑘 ℤ (𝑘 < 𝑀𝑘 < 𝑁)) → 𝑀 = 𝑁)
 
Theoremrecnz 8069 The reciprocal of a number greater than 1 is not an integer. (Contributed by NM, 3-May-2005.)
((A 1 < A) → ¬ (1 / A) ℤ)
 
Theorembtwnnz 8070 A number between an integer and its successor is not an integer. (Contributed by NM, 3-May-2005.)
((A A < B B < (A + 1)) → ¬ B ℤ)
 
Theoremgtndiv 8071 A larger number does not divide a smaller positive integer. (Contributed by NM, 3-May-2005.)
((A B B < A) → ¬ (B / A) ℤ)
 
Theoremhalfnz 8072 One-half is not an integer. (Contributed by NM, 31-Jul-2004.)
¬ (1 / 2)
 
Theoremprime 8073* Two ways to express "A is a prime number (or 1)." (Contributed by NM, 4-May-2005.)
(A ℕ → (x ℕ ((A / x) ℕ → (x = 1 x = A)) ↔ x ℕ ((1 < x xA (A / x) ℕ) → x = A)))
 
Theoremmsqznn 8074 The square of a nonzero integer is a positive integer. (Contributed by NM, 2-Aug-2004.)
((A A ≠ 0) → (A · A) ℕ)
 
Theoremzneo 8075 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 B ℤ) → (2 · A) ≠ ((2 · B) + 1))
 
Theoremnneoor 8076 A positive integer is even or odd. (Contributed by Jim Kingdon, 15-Mar-2020.)
(𝑁 ℕ → ((𝑁 / 2) ((𝑁 + 1) / 2) ℕ))
 
Theoremnneo 8077 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 ℕ → ((𝑁 / 2) ℕ ↔ ¬ ((𝑁 + 1) / 2) ℕ))
 
Theoremnneoi 8078 A positive integer is even or odd but not both. (Contributed by NM, 20-Aug-2001.)
𝑁        ((𝑁 / 2) ℕ ↔ ¬ ((𝑁 + 1) / 2) ℕ)
 
Theoremzeo 8079 An integer is even or odd. (Contributed by NM, 1-Jan-2006.)
(𝑁 ℤ → ((𝑁 / 2) ((𝑁 + 1) / 2) ℤ))
 
Theoremzeo2 8080 An integer is even or odd but not both. (Contributed by Mario Carneiro, 12-Sep-2015.)
(𝑁 ℤ → ((𝑁 / 2) ℤ ↔ ¬ ((𝑁 + 1) / 2) ℤ))
 
Theorempeano2uz2 8081* Second Peano postulate for upper integers. (Contributed by NM, 3-Oct-2004.)
((A B {x ℤ ∣ Ax}) → (B + 1) {x ℤ ∣ Ax})
 
Theorempeano5uzti 8082* Peano's inductive postulate for upper integers. (Contributed by NM, 6-Jul-2005.) (Revised by Mario Carneiro, 25-Jul-2013.)
(𝑁 ℤ → ((𝑁 A x A (x + 1) A) → {𝑘 ℤ ∣ 𝑁𝑘} ⊆ A))
 
Theorempeano5uzi 8083* Peano's inductive postulate for upper integers. (Contributed by NM, 6-Jul-2005.) (Revised by Mario Carneiro, 3-May-2014.)
𝑁        ((𝑁 A x A (x + 1) A) → {𝑘 ℤ ∣ 𝑁𝑘} ⊆ A)
 
Theoremdfuzi 8084* An expression for the upper integers that start at 𝑁 that is analogous to dfnn2 7657 for positive integers. (Contributed by NM, 6-Jul-2005.) (Proof shortened by Mario Carneiro, 3-May-2014.)
𝑁        {z ℤ ∣ 𝑁z} = {x ∣ (𝑁 x y x (y + 1) x)}
 
Theoremuzind 8085* 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 → (φψ))    &   (𝑗 = 𝑘 → (φχ))    &   (𝑗 = (𝑘 + 1) → (φθ))    &   (𝑗 = 𝑁 → (φτ))    &   (𝑀 ℤ → ψ)    &   ((𝑀 𝑘 𝑀𝑘) → (χθ))       ((𝑀 𝑁 𝑀𝑁) → τ)
 
Theoremuzind2 8086* 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) → (φψ))    &   (𝑗 = 𝑘 → (φχ))    &   (𝑗 = (𝑘 + 1) → (φθ))    &   (𝑗 = 𝑁 → (φτ))    &   (𝑀 ℤ → ψ)    &   ((𝑀 𝑘 𝑀 < 𝑘) → (χθ))       ((𝑀 𝑁 𝑀 < 𝑁) → τ)
 
Theoremuzind3 8087* 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 → (φψ))    &   (𝑗 = 𝑚 → (φχ))    &   (𝑗 = (𝑚 + 1) → (φθ))    &   (𝑗 = 𝑁 → (φτ))    &   (𝑀 ℤ → ψ)    &   ((𝑀 𝑚 {𝑘 ℤ ∣ 𝑀𝑘}) → (χθ))       ((𝑀 𝑁 {𝑘 ℤ ∣ 𝑀𝑘}) → τ)
 
Theoremnn0ind 8088* 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 → (φψ))    &   (x = y → (φχ))    &   (x = (y + 1) → (φθ))    &   (x = A → (φτ))    &   ψ    &   (y 0 → (χθ))       (A 0τ)
 
Theoremfzind 8089* 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 → (φψ))    &   (x = y → (φχ))    &   (x = (y + 1) → (φθ))    &   (x = 𝐾 → (φτ))    &   ((𝑀 𝑁 𝑀𝑁) → ψ)    &   (((𝑀 𝑁 ℤ) (y 𝑀y y < 𝑁)) → (χθ))       (((𝑀 𝑁 ℤ) (𝐾 𝑀𝐾 𝐾𝑁)) → τ)
 
Theoremfnn0ind 8090* 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 → (φψ))    &   (x = y → (φχ))    &   (x = (y + 1) → (φθ))    &   (x = 𝐾 → (φτ))    &   (𝑁 0ψ)    &   ((𝑁 0 y 0 y < 𝑁) → (χθ))       ((𝑁 0 𝐾 0 𝐾𝑁) → τ)
 
Theoremnn0ind-raph 8091* 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 → (φψ))    &   (x = y → (φχ))    &   (x = (y + 1) → (φθ))    &   (x = A → (φτ))    &   ψ    &   (y 0 → (χθ))       (A 0τ)
 
Theoremzindd 8092* 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 → (φψ))    &   (x = y → (φχ))    &   (x = (y + 1) → (φτ))    &   (x = -y → (φθ))    &   (x = A → (φη))    &   (ζψ)    &   (ζ → (y 0 → (χτ)))    &   (ζ → (y ℕ → (χθ)))       (ζ → (A ℤ → η))
 
Theorembtwnz 8093* Any real number can be sandwiched between two integers. Exercise 2 of [Apostol] p. 28. (Contributed by NM, 10-Nov-2004.)
(A ℝ → (x x < A y A < y))
 
Theoremnn0zd 8094 A positive integer is an integer. (Contributed by Mario Carneiro, 28-May-2016.)
(φA 0)       (φA ℤ)
 
Theoremnnzd 8095 A nonnegative integer is an integer. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℕ)       (φA ℤ)
 
Theoremzred 8096 An integer is a real number. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℤ)       (φA ℝ)
 
Theoremzcnd 8097 An integer is a complex number. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℤ)       (φA ℂ)
 
Theoremznegcld 8098 Closure law for negative integers. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℤ)       (φ → -A ℤ)
 
Theorempeano2zd 8099 Deduction from second Peano postulate generalized to integers. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℤ)       (φ → (A + 1) ℤ)
 
Theoremzaddcld 8100 Closure of addition of integers. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℤ)    &   (φB ℤ)       (φ → (A + B) ℤ)
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