P. 81 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 8001-8100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	nn0z 8001	A nonnegative integer is an integer. (Contributed by NM, 9-May-2004.)
⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ)
Theorem	nnzi 8002	A positive integer is an integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
⊢ 𝑁 ∈ ℕ    ⇒   ⊢ 𝑁 ∈ ℤ
Theorem	nn0zi 8003	A nonnegative integer is an integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
⊢ 𝑁 ∈ ℕ0    ⇒   ⊢ 𝑁 ∈ ℤ
Theorem	elnnz1 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 ∈ ℤ ∧ 1 ≤ 𝑁))
Theorem	nnzrab 8005	Positive integers expressed as a subset of integers. (Contributed by NM, 3-Oct-2004.)
⊢ ℕ = {x ∈ ℤ ∣ 1 ≤ x}
Theorem	nn0zrab 8006	Nonnegative integers expressed as a subset of integers. (Contributed by NM, 3-Oct-2004.)
⊢ ℕ0 = {x ∈ ℤ ∣ 0 ≤ x}
Theorem	1z 8007	One is an integer. (Contributed by NM, 10-May-2004.)
⊢ 1 ∈ ℤ
Theorem	1zzd 8008	1 is an integer, deductive form (common case). (Contributed by David A. Wheeler, 6-Dec-2018.)
⊢ (φ → 1 ∈ ℤ)
Theorem	2z 8009	Two is an integer. (Contributed by NM, 10-May-2004.)
⊢ 2 ∈ ℤ
Theorem	3z 8010	3 is an integer. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ 3 ∈ ℤ
Theorem	4z 8011	4 is an integer. (Contributed by BJ, 26-Mar-2020.)
⊢ 4 ∈ ℤ
Theorem	znegcl 8012	Closure law for negative integers. (Contributed by NM, 9-May-2004.)
⊢ (𝑁 ∈ ℤ → -𝑁 ∈ ℤ)
Theorem	neg1z 8013	-1 is an integer (common case). (Contributed by David A. Wheeler, 5-Dec-2018.)
⊢ -1 ∈ ℤ
Theorem	znegclb 8014	A number is an integer iff its negative is. (Contributed by Stefan O'Rear, 13-Sep-2014.)
⊢ (A ∈ ℂ → (A ∈ ℤ ↔ -A ∈ ℤ))
Theorem	nn0negz 8015	The negative of a nonnegative integer is an integer. (Contributed by NM, 9-May-2004.)
⊢ (𝑁 ∈ ℕ0 → -𝑁 ∈ ℤ)
Theorem	nn0negzi 8016	The negative of a nonnegative integer is an integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
⊢ 𝑁 ∈ ℕ0    ⇒   ⊢ -𝑁 ∈ ℤ
Theorem	peano2z 8017	Second Peano postulate generalized to integers. (Contributed by NM, 13-Feb-2005.)
⊢ (𝑁 ∈ ℤ → (𝑁 + 1) ∈ ℤ)
Theorem	zaddcllempos 8018	Lemma for zaddcl 8021. Special case in which 𝑁 is a positive integer. (Contributed by Jim Kingdon, 14-Mar-2020.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝑀 + 𝑁) ∈ ℤ)
Theorem	peano2zm 8019	"Reverse" second Peano postulate for integers. (Contributed by NM, 12-Sep-2005.)
⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ)
Theorem	zaddcllemneg 8020	Lemma for zaddcl 8021. Special case in which -𝑁 is a positive integer. (Contributed by Jim Kingdon, 14-Mar-2020.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) → (𝑀 + 𝑁) ∈ ℤ)
Theorem	zaddcl 8021	Closure of addition of integers. (Contributed by NM, 9-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 + 𝑁) ∈ ℤ)
Theorem	zsubcl 8022	Closure of subtraction of integers. (Contributed by NM, 11-May-2004.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 − 𝑁) ∈ ℤ)
Theorem	ztri3or0 8023	Integer trichotomy (with zero). (Contributed by Jim Kingdon, 14-Mar-2020.)
⊢ (𝑁 ∈ ℤ → (𝑁 < 0 ∨ 𝑁 = 0 ∨ 0 < 𝑁))
Theorem	ztri3or 8024	Integer trichotomy. (Contributed by Jim Kingdon, 14-Mar-2020.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 < 𝑁 ∨ 𝑀 = 𝑁 ∨ 𝑁 < 𝑀))
Theorem	zletric 8025	Trichotomy law. (Contributed by Jim Kingdon, 27-Mar-2020.)
⊢ ((A ∈ ℤ ∧ B ∈ ℤ) → (A ≤ B ∨ B ≤ A))
Theorem	zlelttric 8026	Trichotomy law. (Contributed by Jim Kingdon, 17-Apr-2020.)
⊢ ((A ∈ ℤ ∧ B ∈ ℤ) → (A ≤ B ∨ B < A))
Theorem	zltnle 8027	'Less than' expressed in terms of 'less than or equal to'. (Contributed by Jim Kingdon, 14-Mar-2020.)
⊢ ((A ∈ ℤ ∧ B ∈ ℤ) → (A < B ↔ ¬ B ≤ A))
Theorem	zleloe 8028	Integer 'Less than or equal to' expressed in terms of 'less than' or 'equals'. (Contributed by Jim Kingdon, 8-Apr-2020.)
⊢ ((A ∈ ℤ ∧ B ∈ ℤ) → (A ≤ B ↔ (A < B ∨ A = B)))
Theorem	znnnlt1 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))
Theorem	zletr 8030	Transitive law of ordering for integers. (Contributed by Alexander van der Vekens, 3-Apr-2018.)
⊢ ((𝐽 ∈ ℤ ∧ 𝐾 ∈ ℤ ∧ 𝐿 ∈ ℤ) → ((𝐽 ≤ 𝐾 ∧ 𝐾 ≤ 𝐿) → 𝐽 ≤ 𝐿))
Theorem	zrevaddcl 8031	Reverse closure law for addition of integers. (Contributed by NM, 11-May-2004.)
⊢ (𝑁 ∈ ℤ → ((𝑀 ∈ ℂ ∧ (𝑀 + 𝑁) ∈ ℤ) ↔ 𝑀 ∈ ℤ))
Theorem	znnsub 8032	The positive difference of unequal integers is a positive integer. (Generalization of nnsub 7693.) (Contributed by NM, 11-May-2004.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 < 𝑁 ↔ (𝑁 − 𝑀) ∈ ℕ))
Theorem	zmulcl 8033	Closure of multiplication of integers. (Contributed by NM, 30-Jul-2004.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 · 𝑁) ∈ ℤ)
Theorem	zltp1le 8034	Integer ordering relation. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 < 𝑁 ↔ (𝑀 + 1) ≤ 𝑁))
Theorem	zleltp1 8035	Integer ordering relation. (Contributed by NM, 10-May-2004.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ≤ 𝑁 ↔ 𝑀 < (𝑁 + 1)))
Theorem	zlem1lt 8036	Integer ordering relation. (Contributed by NM, 13-Nov-2004.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ≤ 𝑁 ↔ (𝑀 − 1) < 𝑁))
Theorem	zltlem1 8037	Integer ordering relation. (Contributed by NM, 13-Nov-2004.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 < 𝑁 ↔ 𝑀 ≤ (𝑁 − 1)))
Theorem	zgt0ge1 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))
Theorem	nnleltp1 8039	Positive integer ordering relation. (Contributed by NM, 13-Aug-2001.) (Proof shortened by Mario Carneiro, 16-May-2014.)
⊢ ((A ∈ ℕ ∧ B ∈ ℕ) → (A ≤ B ↔ A < (B + 1)))
Theorem	nnltp1le 8040	Positive integer ordering relation. (Contributed by NM, 19-Aug-2001.)
⊢ ((A ∈ ℕ ∧ B ∈ ℕ) → (A < B ↔ (A + 1) ≤ B))
Theorem	nnaddm1cl 8041	Closure of addition of positive integers minus one. (Contributed by NM, 6-Aug-2003.) (Proof shortened by Mario Carneiro, 16-May-2014.)
⊢ ((A ∈ ℕ ∧ B ∈ ℕ) → ((A + B) − 1) ∈ ℕ)
Theorem	nn0ltp1le 8042	Nonnegative integer ordering relation. (Contributed by Raph Levien, 10-Dec-2002.) (Proof shortened by Mario Carneiro, 16-May-2014.)
⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑀 < 𝑁 ↔ (𝑀 + 1) ≤ 𝑁))
Theorem	nn0leltp1 8043	Nonnegative integer ordering relation. (Contributed by Raph Levien, 10-Apr-2004.)
⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑀 ≤ 𝑁 ↔ 𝑀 < (𝑁 + 1)))
Theorem	nn0ltlem1 8044	Nonnegative integer ordering relation. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)
⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑀 < 𝑁 ↔ 𝑀 ≤ (𝑁 − 1)))
Theorem	znn0sub 8045	The nonnegative difference of integers is a nonnegative integer. (Generalization of nn0sub 8046.) (Contributed by NM, 14-Jul-2005.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ≤ 𝑁 ↔ (𝑁 − 𝑀) ∈ ℕ0))
Theorem	nn0sub 8046	Subtraction of nonnegative integers. (Contributed by NM, 9-May-2004.)
⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑀 ≤ 𝑁 ↔ (𝑁 − 𝑀) ∈ ℕ0))
Theorem	nn0n0n1ge2 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 ∈ ℕ0 ∧ 𝑁 ≠ 0 ∧ 𝑁 ≠ 1) → 2 ≤ 𝑁)
Theorem	elz2 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 ∈ ℕ ∃y ∈ ℕ 𝑁 = (x − y))
Theorem	dfz2 8049	Alternative definition of the integers, based on elz2 8048. (Contributed by Mario Carneiro, 16-May-2014.)
⊢ ℤ = ( − “ (ℕ × ℕ))
Theorem	nn0sub2 8050	Subtraction of nonnegative integers. (Contributed by NM, 4-Sep-2005.)
⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑀 ≤ 𝑁) → (𝑁 − 𝑀) ∈ ℕ0)
Theorem	zapne 8051	Apartness is equivalent to not equal for integers. (Contributed by Jim Kingdon, 14-Mar-2020.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 # 𝑁 ↔ 𝑀 ≠ 𝑁))
Theorem	zdceq 8052	Equality of integers is decidable. (Contributed by Jim Kingdon, 14-Mar-2020.)
⊢ ((A ∈ ℤ ∧ B ∈ ℤ) → DECID A = B)
Theorem	zdcle 8053	Integer ≤ is decidable. (Contributed by Jim Kingdon, 7-Apr-2020.)
⊢ ((A ∈ ℤ ∧ B ∈ ℤ) → DECID A ≤ B)
Theorem	zdclt 8054	Integer < is decidable. (Contributed by Jim Kingdon, 1-Jun-2020.)
⊢ ((A ∈ ℤ ∧ B ∈ ℤ) → DECID A < B)
Theorem	zltlen 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 ∈ ℤ ∧ B ∈ ℤ) → (A < B ↔ (A ≤ B ∧ B ≠ A)))
Theorem	nn0n0n1ge2b 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 ∈ ℕ0 → ((𝑁 ≠ 0 ∧ 𝑁 ≠ 1) ↔ 2 ≤ 𝑁))
Theorem	nn0lt10b 8057	A nonnegative integer less than 1 is 0. (Contributed by Paul Chapman, 22-Jun-2011.)
⊢ (𝑁 ∈ ℕ0 → (𝑁 < 1 ↔ 𝑁 = 0))
Theorem	nn0lt2 8058	A nonnegative integer less than 2 must be 0 or 1. (Contributed by Alexander van der Vekens, 16-Sep-2018.)
⊢ ((𝑁 ∈ ℕ0 ∧ 𝑁 < 2) → (𝑁 = 0 ∨ 𝑁 = 1))
Theorem	nn0lem1lt 8059	Nonnegative integer ordering relation. (Contributed by NM, 21-Jun-2005.)
⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑀 ≤ 𝑁 ↔ (𝑀 − 1) < 𝑁))
Theorem	nnlem1lt 8060	Positive integer ordering relation. (Contributed by NM, 21-Jun-2005.)
⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 ≤ 𝑁 ↔ (𝑀 − 1) < 𝑁))
Theorem	nnltlem1 8061	Positive integer ordering relation. (Contributed by NM, 21-Jun-2005.)
⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 < 𝑁 ↔ 𝑀 ≤ (𝑁 − 1)))
Theorem	nnm1ge0 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))
Theorem	nn0ge0div 8063	Division of a nonnegative integer by a positive number is not negative. (Contributed by Alexander van der Vekens, 14-Apr-2018.)
⊢ ((𝐾 ∈ ℕ0 ∧ 𝐿 ∈ ℕ) → 0 ≤ (𝐾 / 𝐿))
Theorem	zdiv 8064*	Two ways to express "𝑀 divides 𝑁. (Contributed by NM, 3-Oct-2008.)
⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℤ) → (∃𝑘 ∈ ℤ (𝑀 · 𝑘) = 𝑁 ↔ (𝑁 / 𝑀) ∈ ℤ))
Theorem	zdivadd 8065	Property of divisibility: if 𝐷 divides A and B then it divides A + B. (Contributed by NM, 3-Oct-2008.)
⊢ (((𝐷 ∈ ℕ ∧ A ∈ ℤ ∧ B ∈ ℤ) ∧ ((A / 𝐷) ∈ ℤ ∧ (B / 𝐷) ∈ ℤ)) → ((A + B) / 𝐷) ∈ ℤ)
Theorem	zdivmul 8066	Property of divisibility: if 𝐷 divides A then it divides B · A. (Contributed by NM, 3-Oct-2008.)
⊢ (((𝐷 ∈ ℕ ∧ A ∈ ℤ ∧ B ∈ ℤ) ∧ (A / 𝐷) ∈ ℤ) → ((B · A) / 𝐷) ∈ ℤ)
Theorem	zextle 8067*	An extensionality-like property for integer ordering. (Contributed by NM, 29-Oct-2005.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ ∀𝑘 ∈ ℤ (𝑘 ≤ 𝑀 ↔ 𝑘 ≤ 𝑁)) → 𝑀 = 𝑁)
Theorem	zextlt 8068*	An extensionality-like property for integer ordering. (Contributed by NM, 29-Oct-2005.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ ∀𝑘 ∈ ℤ (𝑘 < 𝑀 ↔ 𝑘 < 𝑁)) → 𝑀 = 𝑁)
Theorem	recnz 8069	The reciprocal of a number greater than 1 is not an integer. (Contributed by NM, 3-May-2005.)
⊢ ((A ∈ ℝ ∧ 1 < A) → ¬ (1 / A) ∈ ℤ)
Theorem	btwnnz 8070	A number between an integer and its successor is not an integer. (Contributed by NM, 3-May-2005.)
⊢ ((A ∈ ℤ ∧ A < B ∧ B < (A + 1)) → ¬ B ∈ ℤ)
Theorem	gtndiv 8071	A larger number does not divide a smaller positive integer. (Contributed by NM, 3-May-2005.)
⊢ ((A ∈ ℝ ∧ B ∈ ℕ ∧ B < A) → ¬ (B / A) ∈ ℤ)
Theorem	halfnz 8072	One-half is not an integer. (Contributed by NM, 31-Jul-2004.)
⊢ ¬ (1 / 2) ∈ ℤ
Theorem	prime 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 ∧ x ≤ A ∧ (A / x) ∈ ℕ) → x = A)))
Theorem	msqznn 8074	The square of a nonzero integer is a positive integer. (Contributed by NM, 2-Aug-2004.)
⊢ ((A ∈ ℤ ∧ A ≠ 0) → (A · A) ∈ ℕ)
Theorem	zneo 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 ∈ ℤ ∧ B ∈ ℤ) → (2 · A) ≠ ((2 · B) + 1))
Theorem	nneoor 8076	A positive integer is even or odd. (Contributed by Jim Kingdon, 15-Mar-2020.)
⊢ (𝑁 ∈ ℕ → ((𝑁 / 2) ∈ ℕ ∨ ((𝑁 + 1) / 2) ∈ ℕ))
Theorem	nneo 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))
Theorem	nneoi 8078	A positive integer is even or odd but not both. (Contributed by NM, 20-Aug-2001.)
⊢ 𝑁 ∈ ℕ    ⇒   ⊢ ((𝑁 / 2) ∈ ℕ ↔ ¬ ((𝑁 + 1) / 2) ∈ ℕ)
Theorem	zeo 8079	An integer is even or odd. (Contributed by NM, 1-Jan-2006.)
⊢ (𝑁 ∈ ℤ → ((𝑁 / 2) ∈ ℤ ∨ ((𝑁 + 1) / 2) ∈ ℤ))
Theorem	zeo2 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))
Theorem	peano2uz2 8081*	Second Peano postulate for upper integers. (Contributed by NM, 3-Oct-2004.)
⊢ ((A ∈ ℤ ∧ B ∈ {x ∈ ℤ ∣ A ≤ x}) → (B + 1) ∈ {x ∈ ℤ ∣ A ≤ x})
Theorem	peano5uzti 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 ∣ 𝑁 ≤ 𝑘} ⊆ A))
Theorem	peano5uzi 8083*	Peano's inductive postulate for upper integers. (Contributed by NM, 6-Jul-2005.) (Revised by Mario Carneiro, 3-May-2014.)
⊢ 𝑁 ∈ ℤ    ⇒   ⊢ ((𝑁 ∈ A ∧ ∀x ∈ A (x + 1) ∈ A) → {𝑘 ∈ ℤ ∣ 𝑁 ≤ 𝑘} ⊆ A)
Theorem	dfuzi 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 ∈ ℤ ∣ 𝑁 ≤ z} = ∩ {x ∣ (𝑁 ∈ x ∧ ∀y ∈ x (y + 1) ∈ x)}
Theorem	uzind 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 → (φ ↔ ψ))    &   ⊢ (𝑗 = 𝑘 → (φ ↔ χ))    &   ⊢ (𝑗 = (𝑘 + 1) → (φ ↔ θ))    &   ⊢ (𝑗 = 𝑁 → (φ ↔ τ))    &   ⊢ (𝑀 ∈ ℤ → ψ)    &   ⊢ ((𝑀 ∈ ℤ ∧ 𝑘 ∈ ℤ ∧ 𝑀 ≤ 𝑘) → (χ → θ))    ⇒   ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁) → τ)
Theorem	uzind2 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) → (φ ↔ ψ))    &   ⊢ (𝑗 = 𝑘 → (φ ↔ χ))    &   ⊢ (𝑗 = (𝑘 + 1) → (φ ↔ θ))    &   ⊢ (𝑗 = 𝑁 → (φ ↔ τ))    &   ⊢ (𝑀 ∈ ℤ → ψ)    &   ⊢ ((𝑀 ∈ ℤ ∧ 𝑘 ∈ ℤ ∧ 𝑀 < 𝑘) → (χ → θ))    ⇒   ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 < 𝑁) → τ)
Theorem	uzind3 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 → (φ ↔ ψ))    &   ⊢ (𝑗 = 𝑚 → (φ ↔ χ))    &   ⊢ (𝑗 = (𝑚 + 1) → (φ ↔ θ))    &   ⊢ (𝑗 = 𝑁 → (φ ↔ τ))    &   ⊢ (𝑀 ∈ ℤ → ψ)    &   ⊢ ((𝑀 ∈ ℤ ∧ 𝑚 ∈ {𝑘 ∈ ℤ ∣ 𝑀 ≤ 𝑘}) → (χ → θ))    ⇒   ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ {𝑘 ∈ ℤ ∣ 𝑀 ≤ 𝑘}) → τ)
Theorem	nn0ind 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 → (φ ↔ ψ))    &   ⊢ (x = y → (φ ↔ χ))    &   ⊢ (x = (y + 1) → (φ ↔ θ))    &   ⊢ (x = A → (φ ↔ τ))    &   ⊢ ψ    &   ⊢ (y ∈ ℕ0 → (χ → θ))    ⇒   ⊢ (A ∈ ℕ0 → τ)
Theorem	fzind 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 → (φ ↔ ψ))    &   ⊢ (x = y → (φ ↔ χ))    &   ⊢ (x = (y + 1) → (φ ↔ θ))    &   ⊢ (x = 𝐾 → (φ ↔ τ))    &   ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁) → ψ)    &   ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (y ∈ ℤ ∧ 𝑀 ≤ y ∧ y < 𝑁)) → (χ → θ))    ⇒   ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐾 ∈ ℤ ∧ 𝑀 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁)) → τ)
Theorem	fnn0ind 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 → (φ ↔ ψ))    &   ⊢ (x = y → (φ ↔ χ))    &   ⊢ (x = (y + 1) → (φ ↔ θ))    &   ⊢ (x = 𝐾 → (φ ↔ τ))    &   ⊢ (𝑁 ∈ ℕ0 → ψ)    &   ⊢ ((𝑁 ∈ ℕ0 ∧ y ∈ ℕ0 ∧ y < 𝑁) → (χ → θ))    ⇒   ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ ℕ0 ∧ 𝐾 ≤ 𝑁) → τ)
Theorem	nn0ind-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 → (φ ↔ ψ))    &   ⊢ (x = y → (φ ↔ χ))    &   ⊢ (x = (y + 1) → (φ ↔ θ))    &   ⊢ (x = A → (φ ↔ τ))    &   ⊢ ψ    &   ⊢ (y ∈ ℕ0 → (χ → θ))    ⇒   ⊢ (A ∈ ℕ0 → τ)
Theorem	zindd 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 → (φ ↔ ψ))    &   ⊢ (x = y → (φ ↔ χ))    &   ⊢ (x = (y + 1) → (φ ↔ τ))    &   ⊢ (x = -y → (φ ↔ θ))    &   ⊢ (x = A → (φ ↔ η))    &   ⊢ (ζ → ψ)    &   ⊢ (ζ → (y ∈ ℕ0 → (χ → τ)))    &   ⊢ (ζ → (y ∈ ℕ → (χ → θ)))    ⇒   ⊢ (ζ → (A ∈ ℤ → η))
Theorem	btwnz 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 ∈ ℤ x < A ∧ ∃y ∈ ℤ A < y))
Theorem	nn0zd 8094	A positive integer is an integer. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (φ → A ∈ ℕ0)    ⇒   ⊢ (φ → A ∈ ℤ)
Theorem	nnzd 8095	A nonnegative integer is an integer. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (φ → A ∈ ℕ)    ⇒   ⊢ (φ → A ∈ ℤ)
Theorem	zred 8096	An integer is a real number. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (φ → A ∈ ℤ)    ⇒   ⊢ (φ → A ∈ ℝ)
Theorem	zcnd 8097	An integer is a complex number. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (φ → A ∈ ℤ)    ⇒   ⊢ (φ → A ∈ ℂ)
Theorem	znegcld 8098	Closure law for negative integers. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (φ → A ∈ ℤ)    ⇒   ⊢ (φ → -A ∈ ℤ)
Theorem	peano2zd 8099	Deduction from second Peano postulate generalized to integers. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (φ → A ∈ ℤ)    ⇒   ⊢ (φ → (A + 1) ∈ ℤ)
Theorem	zaddcld 8100	Closure of addition of integers. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (φ → A ∈ ℤ)    &   ⊢ (φ → B ∈ ℤ)    ⇒   ⊢ (φ → (A + B) ∈ ℤ)