P. 98 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 9701-9800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	nn0sub2 9701	Subtraction of nonnegative integers. (Contributed by NM, 4-Sep-2005.)
⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑀 ≤ 𝑁) → (𝑁 − 𝑀) ∈ ℕ0)
Theorem	zapne 9702	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 9703	Equality of integers is decidable. (Contributed by Jim Kingdon, 14-Mar-2020.)
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → DECID 𝐴 = 𝐵)
Theorem	zdcle 9704	Integer ≤ is decidable. (Contributed by Jim Kingdon, 7-Apr-2020.)
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → DECID 𝐴 ≤ 𝐵)
Theorem	zdclt 9705	Integer < is decidable. (Contributed by Jim Kingdon, 1-Jun-2020.)
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → DECID 𝐴 < 𝐵)
Theorem	zfidc 9706	Whether an integer is an element of a finite set of integers is decidable. (Contributed by Jim Kingdon, 8-Jun-2026.)
⊢ ((𝑆 ⊆ ℤ ∧ 𝐴 ∈ ℤ ∧ 𝑆 ∈ Fin) → DECID 𝐴 ∈ 𝑆)
Theorem	zltlen 9707	Integer 'Less than' expressed in terms of 'less than or equal to'. Also see ltleap 8954 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 9708	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 9709	A nonnegative integer less than 1 is 0. (Contributed by Paul Chapman, 22-Jun-2011.)
⊢ (𝑁 ∈ ℕ0 → (𝑁 < 1 ↔ 𝑁 = 0))
Theorem	nn0lt2 9710	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 9711	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 9712	Nonnegative integer ordering relation. (Contributed by NM, 21-Jun-2005.)
⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑀 ≤ 𝑁 ↔ (𝑀 − 1) < 𝑁))
Theorem	nnlem1lt 9713	Positive integer ordering relation. (Contributed by NM, 21-Jun-2005.)
⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 ≤ 𝑁 ↔ (𝑀 − 1) < 𝑁))
Theorem	nnltlem1 9714	Positive integer ordering relation. (Contributed by NM, 21-Jun-2005.)
⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 < 𝑁 ↔ 𝑀 ≤ (𝑁 − 1)))
Theorem	nnm1ge0 9715	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 9716	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 9717*	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 9718	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 9719	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 9720*	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 9721*	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 9722	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 9723	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 9724	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 9725	One-half is not an integer. (Contributed by NM, 31-Jul-2004.)
⊢ ¬ (1 / 2) ∈ ℤ
Theorem	3halfnz 9726	Three halves is not an integer. (Contributed by AV, 2-Jun-2020.)
⊢ ¬ (3 / 2) ∈ ℤ
Theorem	suprzclex 9727*	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 9728*	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 9729	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 9730	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 9731	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 9732	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 9733	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 9734	An integer is even or odd. (Contributed by NM, 1-Jan-2006.)
⊢ (𝑁 ∈ ℤ → ((𝑁 / 2) ∈ ℤ ∨ ((𝑁 + 1) / 2) ∈ ℤ))
Theorem	zeo2 9735	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 9736*	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 9737*	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 9738*	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 9739*	An expression for the upper integers that start at 𝑁 that is analogous to dfnn2 9289 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 9740*	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 9741*	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 9742*	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 9743*	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 9744*	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 9745*	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 9746*	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 9747*	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 9748*	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 9749	A positive integer is an integer. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℕ0)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ℤ)
Theorem	nnzd 9750	A nonnegative integer is an integer. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℕ)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ℤ)
Theorem	zred 9751	An integer is a real number. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℤ)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ℝ)
Theorem	zcnd 9752	An integer is a complex number. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℤ)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ℂ)
Theorem	znegcld 9753	Closure law for negative integers. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℤ)    ⇒   ⊢ (𝜑 → -𝐴 ∈ ℤ)
Theorem	peano2zd 9754	Deduction from second Peano postulate generalized to integers. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℤ)    ⇒   ⊢ (𝜑 → (𝐴 + 1) ∈ ℤ)
Theorem	zaddcld 9755	Closure of addition of integers. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℤ)    &   ⊢ (𝜑 → 𝐵 ∈ ℤ)    ⇒   ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℤ)
Theorem	zsubcld 9756	Closure of subtraction of integers. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℤ)    &   ⊢ (𝜑 → 𝐵 ∈ ℤ)    ⇒   ⊢ (𝜑 → (𝐴 − 𝐵) ∈ ℤ)
Theorem	zmulcld 9757	Closure of multiplication of integers. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℤ)    &   ⊢ (𝜑 → 𝐵 ∈ ℤ)    ⇒   ⊢ (𝜑 → (𝐴 · 𝐵) ∈ ℤ)
Theorem	zadd2cl 9758	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 9759	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 9760	Constant used for decimal constructor.
class ;𝐴𝐵
Definition	df-dec 9761	Define the "decimal constructor", which is used to build up "decimal integers" or "numeric terms" in base 10. For example, (;;;1000 + ;;;2000) = ;;;3000 1kp2ke3k 16721. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 1-Aug-2021.)
⊢ ;𝐴𝐵 = (((9 + 1) · 𝐴) + 𝐵)
Theorem	9p1e10 9762	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 9763	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 9764	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 9765	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 9766	Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ ;𝐴𝐶 = ;𝐵𝐶
Theorem	deceq2i 9767	Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ ;𝐶𝐴 = ;𝐶𝐵
Theorem	deceq12i 9768	Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐶 = 𝐷    ⇒   ⊢ ;𝐴𝐶 = ;𝐵𝐷
Theorem	numnncl 9769	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 9770	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 9771	Add a zero in the higher places. (Contributed by Mario Carneiro, 18-Feb-2014.)
⊢ 𝑇 ∈ ℕ0    &   ⊢ 𝐴 ∈ ℕ0    ⇒   ⊢ 𝐴 = ((𝑇 · 0) + 𝐴)
Theorem	numcl 9772	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 9773	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 9774	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 9775	10 is a positive integer. (Contributed by NM, 8-Nov-2012.) (Revised by AV, 6-Sep-2021.)
⊢ ;10 ∈ ℕ
Theorem	10pos 9776	The number 10 is positive. (Contributed by NM, 5-Feb-2007.) (Revised by AV, 8-Sep-2021.)
⊢ 0 < ;10
Theorem	10nn0 9777	10 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
⊢ ;10 ∈ ℕ0
Theorem	10re 9778	The number 10 is real. (Contributed by NM, 5-Feb-2007.) (Revised by AV, 8-Sep-2021.)
⊢ ;10 ∈ ℝ
Theorem	decnncl 9779	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 9780	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 9781	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 9782	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 9783	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 9784	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 9785	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 9786	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 9787	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 9788	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 9789	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 9790	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 9791	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 9792	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 9793	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 9794	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 9795	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 9796	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 9797	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 9798	Comparing a digit to a decimal integer. (Contributed by AV, 8-Sep-2021.)
⊢ 𝐴 ∈ ℕ    &   ⊢ 𝐵 ∈ ℕ0    &   ⊢ 𝐶 ∈ ℕ0    &   ⊢ 𝐶 ≤ 9    ⇒   ⊢ 𝐶 < ;𝐴𝐵
Theorem	numsucc 9799	The successor of a decimal integer (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
⊢ 𝑌 ∈ ℕ0    &   ⊢ 𝑇 = (𝑌 + 1)    &   ⊢ 𝐴 ∈ ℕ0    &   ⊢ (𝐴 + 1) = 𝐵    &   ⊢ 𝑁 = ((𝑇 · 𝐴) + 𝑌)    ⇒   ⊢ (𝑁 + 1) = ((𝑇 · 𝐵) + 0)
Theorem	decsucc 9800	The successor of a decimal integer (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ (𝐴 + 1) = 𝐵    &   ⊢ 𝑁 = ;𝐴9    ⇒   ⊢ (𝑁 + 1) = ;𝐵0