P. 95 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 9401-9500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	nn0n0n1ge2 9401	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 9402*	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 9403	Alternate definition of the integers, based on elz2 9402. (Contributed by Mario Carneiro, 16-May-2014.)
⊢ ℤ = ( − “ (ℕ × ℕ))
Theorem	nn0sub2 9404	Subtraction of nonnegative integers. (Contributed by NM, 4-Sep-2005.)
⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑀 ≤ 𝑁) → (𝑁 − 𝑀) ∈ ℕ0)
Theorem	zapne 9405	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 9406	Equality of integers is decidable. (Contributed by Jim Kingdon, 14-Mar-2020.)
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → DECID 𝐴 = 𝐵)
Theorem	zdcle 9407	Integer ≤ is decidable. (Contributed by Jim Kingdon, 7-Apr-2020.)
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → DECID 𝐴 ≤ 𝐵)
Theorem	zdclt 9408	Integer < is decidable. (Contributed by Jim Kingdon, 1-Jun-2020.)
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → DECID 𝐴 < 𝐵)
Theorem	zltlen 9409	Integer 'Less than' expressed in terms of 'less than or equal to'. Also see ltleap 8664 which is a similar result for real numbers. (Contributed by Jim Kingdon, 14-Mar-2020.)
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 < 𝐵 ↔ (𝐴 ≤ 𝐵 ∧ 𝐵 ≠ 𝐴)))
Theorem	nn0n0n1ge2b 9410	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 9411	A nonnegative integer less than 1 is 0. (Contributed by Paul Chapman, 22-Jun-2011.)
⊢ (𝑁 ∈ ℕ0 → (𝑁 < 1 ↔ 𝑁 = 0))
Theorem	nn0lt2 9412	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	nn0le2is012 9413	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 ∈ ℕ0 ∧ 𝑁 ≤ 2) → (𝑁 = 0 ∨ 𝑁 = 1 ∨ 𝑁 = 2))
Theorem	nn0lem1lt 9414	Nonnegative integer ordering relation. (Contributed by NM, 21-Jun-2005.)
⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑀 ≤ 𝑁 ↔ (𝑀 − 1) < 𝑁))
Theorem	nnlem1lt 9415	Positive integer ordering relation. (Contributed by NM, 21-Jun-2005.)
⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 ≤ 𝑁 ↔ (𝑀 − 1) < 𝑁))
Theorem	nnltlem1 9416	Positive integer ordering relation. (Contributed by NM, 21-Jun-2005.)
⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 < 𝑁 ↔ 𝑀 ≤ (𝑁 − 1)))
Theorem	nnm1ge0 9417	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 9418	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 9419*	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 9420	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 9421	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 9422*	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 9423*	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 9424	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 9425	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 9426	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 9427	One-half is not an integer. (Contributed by NM, 31-Jul-2004.)
⊢ ¬ (1 / 2) ∈ ℤ
Theorem	3halfnz 9428	Three halves is not an integer. (Contributed by AV, 2-Jun-2020.)
⊢ ¬ (3 / 2) ∈ ℤ
Theorem	suprzclex 9429*	The supremum of a set of integers is an element of the set. (Contributed by Jim Kingdon, 20-Dec-2021.)
⊢ (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))    &   ⊢ (𝜑 → 𝐴 ⊆ ℤ)    ⇒   ⊢ (𝜑 → sup(𝐴, ℝ, < ) ∈ 𝐴)
Theorem	prime 9430*	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 9431	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 9432	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 9433	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 9434	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 9435	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 9436	An integer is even or odd. (Contributed by NM, 1-Jan-2006.)
⊢ (𝑁 ∈ ℤ → ((𝑁 / 2) ∈ ℤ ∨ ((𝑁 + 1) / 2) ∈ ℤ))
Theorem	zeo2 9437	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 9438*	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 9439*	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 9440*	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 9441*	An expression for the upper integers that start at 𝑁 that is analogous to dfnn2 8997 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 9442*	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 9443*	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 9444*	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 9445*	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 9446*	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 9447*	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 9448*	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 9449*	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 9450*	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 9451	A positive integer is an integer. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℕ0)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ℤ)
Theorem	nnzd 9452	A nonnegative integer is an integer. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℕ)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ℤ)
Theorem	zred 9453	An integer is a real number. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℤ)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ℝ)
Theorem	zcnd 9454	An integer is a complex number. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℤ)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ℂ)
Theorem	znegcld 9455	Closure law for negative integers. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℤ)    ⇒   ⊢ (𝜑 → -𝐴 ∈ ℤ)
Theorem	peano2zd 9456	Deduction from second Peano postulate generalized to integers. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℤ)    ⇒   ⊢ (𝜑 → (𝐴 + 1) ∈ ℤ)
Theorem	zaddcld 9457	Closure of addition of integers. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℤ)    &   ⊢ (𝜑 → 𝐵 ∈ ℤ)    ⇒   ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℤ)
Theorem	zsubcld 9458	Closure of subtraction of integers. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℤ)    &   ⊢ (𝜑 → 𝐵 ∈ ℤ)    ⇒   ⊢ (𝜑 → (𝐴 − 𝐵) ∈ ℤ)
Theorem	zmulcld 9459	Closure of multiplication of integers. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℤ)    &   ⊢ (𝜑 → 𝐵 ∈ ℤ)    ⇒   ⊢ (𝜑 → (𝐴 · 𝐵) ∈ ℤ)
Theorem	zadd2cl 9460	Increasing an integer by 2 results in an integer. (Contributed by Alexander van der Vekens, 16-Sep-2018.)
⊢ (𝑁 ∈ ℤ → (𝑁 + 2) ∈ ℤ)
Theorem	btwnapz 9461	A number between an integer and its successor is apart from any integer. (Contributed by Jim Kingdon, 6-Jan-2023.)
⊢ (𝜑 → 𝐴 ∈ ℤ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℤ)    &   ⊢ (𝜑 → 𝐴 < 𝐵)    &   ⊢ (𝜑 → 𝐵 < (𝐴 + 1))    ⇒   ⊢ (𝜑 → 𝐵 # 𝐶)
4.4.10  Decimal arithmetic
Syntax	cdc 9462	Constant used for decimal constructor.
class ;𝐴𝐵
Definition	df-dec 9463	Define the "decimal constructor", which is used to build up "decimal integers" or "numeric terms" in base 10. For example, (;;;1000 + ;;;2000) = ;;;3000 1kp2ke3k 15417. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 1-Aug-2021.)
⊢ ;𝐴𝐵 = (((9 + 1) · 𝐴) + 𝐵)
Theorem	9p1e10 9464	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) = ;10
Theorem	dfdec10 9465	Version of the definition of the "decimal constructor" using ;10 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.)
⊢ ;𝐴𝐵 = ((;10 · 𝐴) + 𝐵)
Theorem	deceq1 9466	Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ (𝐴 = 𝐵 → ;𝐴𝐶 = ;𝐵𝐶)
Theorem	deceq2 9467	Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ (𝐴 = 𝐵 → ;𝐶𝐴 = ;𝐶𝐵)
Theorem	deceq1i 9468	Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ ;𝐴𝐶 = ;𝐵𝐶
Theorem	deceq2i 9469	Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ ;𝐶𝐴 = ;𝐶𝐵
Theorem	deceq12i 9470	Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐶 = 𝐷    ⇒   ⊢ ;𝐴𝐶 = ;𝐵𝐷
Theorem	numnncl 9471	Closure for a numeral (with units place). (Contributed by Mario Carneiro, 18-Feb-2014.)
⊢ 𝑇 ∈ ℕ0    &   ⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℕ    ⇒   ⊢ ((𝑇 · 𝐴) + 𝐵) ∈ ℕ
Theorem	num0u 9472	Add a zero in the units place. (Contributed by Mario Carneiro, 18-Feb-2014.)
⊢ 𝑇 ∈ ℕ0    &   ⊢ 𝐴 ∈ ℕ0    ⇒   ⊢ (𝑇 · 𝐴) = ((𝑇 · 𝐴) + 0)
Theorem	num0h 9473	Add a zero in the higher places. (Contributed by Mario Carneiro, 18-Feb-2014.)
⊢ 𝑇 ∈ ℕ0    &   ⊢ 𝐴 ∈ ℕ0    ⇒   ⊢ 𝐴 = ((𝑇 · 0) + 𝐴)
Theorem	numcl 9474	Closure for a decimal integer (with units place). (Contributed by Mario Carneiro, 18-Feb-2014.)
⊢ 𝑇 ∈ ℕ0    &   ⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℕ0    ⇒   ⊢ ((𝑇 · 𝐴) + 𝐵) ∈ ℕ0
Theorem	numsuc 9475	The successor of a decimal integer (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
⊢ 𝑇 ∈ ℕ0    &   ⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℕ0    &   ⊢ (𝐵 + 1) = 𝐶    &   ⊢ 𝑁 = ((𝑇 · 𝐴) + 𝐵)    ⇒   ⊢ (𝑁 + 1) = ((𝑇 · 𝐴) + 𝐶)
Theorem	deccl 9476	Closure for a numeral. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℕ0    ⇒   ⊢ ;𝐴𝐵 ∈ ℕ0
Theorem	10nn 9477	10 is a positive integer. (Contributed by NM, 8-Nov-2012.) (Revised by AV, 6-Sep-2021.)
⊢ ;10 ∈ ℕ
Theorem	10pos 9478	The number 10 is positive. (Contributed by NM, 5-Feb-2007.) (Revised by AV, 8-Sep-2021.)
⊢ 0 < ;10
Theorem	10nn0 9479	10 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ ;10 ∈ ℕ0
Theorem	10re 9480	The number 10 is real. (Contributed by NM, 5-Feb-2007.) (Revised by AV, 8-Sep-2021.)
⊢ ;10 ∈ ℝ
Theorem	decnncl 9481	Closure for a numeral. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℕ    ⇒   ⊢ ;𝐴𝐵 ∈ ℕ
Theorem	dec0u 9482	Add a zero in the units place. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ 𝐴 ∈ ℕ0    ⇒   ⊢ (;10 · 𝐴) = ;𝐴0
Theorem	dec0h 9483	Add a zero in the higher places. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ 𝐴 ∈ ℕ0    ⇒   ⊢ 𝐴 = ;0𝐴
Theorem	numnncl2 9484	Closure for a decimal integer (zero units place). (Contributed by Mario Carneiro, 9-Mar-2015.)
⊢ 𝑇 ∈ ℕ    &   ⊢ 𝐴 ∈ ℕ    ⇒   ⊢ ((𝑇 · 𝐴) + 0) ∈ ℕ
Theorem	decnncl2 9485	Closure for a decimal integer (zero units place). (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ 𝐴 ∈ ℕ    ⇒   ⊢ ;𝐴0 ∈ ℕ
Theorem	numlt 9486	Comparing two decimal integers (equal higher places). (Contributed by Mario Carneiro, 18-Feb-2014.)
⊢ 𝑇 ∈ ℕ    &   ⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℕ0    &   ⊢ 𝐶 ∈ ℕ    &   ⊢ 𝐵 < 𝐶    ⇒   ⊢ ((𝑇 · 𝐴) + 𝐵) < ((𝑇 · 𝐴) + 𝐶)
Theorem	numltc 9487	Comparing two decimal integers (unequal higher places). (Contributed by Mario Carneiro, 18-Feb-2014.)
⊢ 𝑇 ∈ ℕ    &   ⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℕ0    &   ⊢ 𝐶 ∈ ℕ0    &   ⊢ 𝐷 ∈ ℕ0    &   ⊢ 𝐶 < 𝑇    &   ⊢ 𝐴 < 𝐵    ⇒   ⊢ ((𝑇 · 𝐴) + 𝐶) < ((𝑇 · 𝐵) + 𝐷)
Theorem	le9lt10 9488	A "decimal digit" (i.e. a nonnegative integer less than or equal to 9) is less then 10. (Contributed by AV, 8-Sep-2021.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐴 ≤ 9    ⇒   ⊢ 𝐴 < ;10
Theorem	declt 9489	Comparing two decimal integers (equal higher places). (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℕ0    &   ⊢ 𝐶 ∈ ℕ    &   ⊢ 𝐵 < 𝐶    ⇒   ⊢ ;𝐴𝐵 < ;𝐴𝐶
Theorem	decltc 9490	Comparing two decimal integers (unequal higher places). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℕ0    &   ⊢ 𝐶 ∈ ℕ0    &   ⊢ 𝐷 ∈ ℕ0    &   ⊢ 𝐶 < ;10    &   ⊢ 𝐴 < 𝐵    ⇒   ⊢ ;𝐴𝐶 < ;𝐵𝐷
Theorem	declth 9491	Comparing two decimal integers (unequal higher places). (Contributed by AV, 8-Sep-2021.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℕ0    &   ⊢ 𝐶 ∈ ℕ0    &   ⊢ 𝐷 ∈ ℕ0    &   ⊢ 𝐶 ≤ 9    &   ⊢ 𝐴 < 𝐵    ⇒   ⊢ ;𝐴𝐶 < ;𝐵𝐷
Theorem	decsuc 9492	The successor of a decimal integer (no carry). (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℕ0    &   ⊢ (𝐵 + 1) = 𝐶    &   ⊢ 𝑁 = ;𝐴𝐵    ⇒   ⊢ (𝑁 + 1) = ;𝐴𝐶
Theorem	3declth 9493	Comparing two decimal integers with three "digits" (unequal higher places). (Contributed by AV, 8-Sep-2021.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℕ0    &   ⊢ 𝐶 ∈ ℕ0    &   ⊢ 𝐷 ∈ ℕ0    &   ⊢ 𝐸 ∈ ℕ0    &   ⊢ 𝐹 ∈ ℕ0    &   ⊢ 𝐴 < 𝐵    &   ⊢ 𝐶 ≤ 9    &   ⊢ 𝐸 ≤ 9    ⇒   ⊢ ;;𝐴𝐶𝐸 < ;;𝐵𝐷𝐹
Theorem	3decltc 9494	Comparing two decimal integers with three "digits" (unequal higher places). (Contributed by AV, 15-Jun-2021.) (Revised by AV, 6-Sep-2021.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℕ0    &   ⊢ 𝐶 ∈ ℕ0    &   ⊢ 𝐷 ∈ ℕ0    &   ⊢ 𝐸 ∈ ℕ0    &   ⊢ 𝐹 ∈ ℕ0    &   ⊢ 𝐴 < 𝐵    &   ⊢ 𝐶 < ;10    &   ⊢ 𝐸 < ;10    ⇒   ⊢ ;;𝐴𝐶𝐸 < ;;𝐵𝐷𝐹
Theorem	decle 9495	Comparing two decimal integers (equal higher places). (Contributed by AV, 17-Aug-2021.) (Revised by AV, 8-Sep-2021.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℕ0    &   ⊢ 𝐶 ∈ ℕ0    &   ⊢ 𝐵 ≤ 𝐶    ⇒   ⊢ ;𝐴𝐵 ≤ ;𝐴𝐶
Theorem	decleh 9496	Comparing two decimal integers (unequal higher places). (Contributed by AV, 17-Aug-2021.) (Revised by AV, 8-Sep-2021.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℕ0    &   ⊢ 𝐶 ∈ ℕ0    &   ⊢ 𝐷 ∈ ℕ0    &   ⊢ 𝐶 ≤ 9    &   ⊢ 𝐴 < 𝐵    ⇒   ⊢ ;𝐴𝐶 ≤ ;𝐵𝐷
Theorem	declei 9497	Comparing a digit to a decimal integer. (Contributed by AV, 17-Aug-2021.)
⊢ 𝐴 ∈ ℕ    &   ⊢ 𝐵 ∈ ℕ0    &   ⊢ 𝐶 ∈ ℕ0    &   ⊢ 𝐶 ≤ 9    ⇒   ⊢ 𝐶 ≤ ;𝐴𝐵
Theorem	numlti 9498	Comparing a digit to a decimal integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
⊢ 𝑇 ∈ ℕ    &   ⊢ 𝐴 ∈ ℕ    &   ⊢ 𝐵 ∈ ℕ0    &   ⊢ 𝐶 ∈ ℕ0    &   ⊢ 𝐶 < 𝑇    ⇒   ⊢ 𝐶 < ((𝑇 · 𝐴) + 𝐵)
Theorem	declti 9499	Comparing a digit to a decimal integer. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
⊢ 𝐴 ∈ ℕ    &   ⊢ 𝐵 ∈ ℕ0    &   ⊢ 𝐶 ∈ ℕ0    &   ⊢ 𝐶 < ;10    ⇒   ⊢ 𝐶 < ;𝐴𝐵
Theorem	decltdi 9500	Comparing a digit to a decimal integer. (Contributed by AV, 8-Sep-2021.)
⊢ 𝐴 ∈ ℕ    &   ⊢ 𝐵 ∈ ℕ0    &   ⊢ 𝐶 ∈ ℕ0    &   ⊢ 𝐶 ≤ 9    ⇒   ⊢ 𝐶 < ;𝐴𝐵