Theorem List for Intuitionistic Logic Explorer - 9101-9200 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
|
Theorem | zltp1le 9101 |
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 9102 |
Integer ordering relation. (Contributed by NM, 10-May-2004.)
|
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ≤ 𝑁 ↔ 𝑀 < (𝑁 + 1))) |
|
Theorem | zlem1lt 9103 |
Integer ordering relation. (Contributed by NM, 13-Nov-2004.)
|
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ≤ 𝑁 ↔ (𝑀 − 1) < 𝑁)) |
|
Theorem | zltlem1 9104 |
Integer ordering relation. (Contributed by NM, 13-Nov-2004.)
|
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 < 𝑁 ↔ 𝑀 ≤ (𝑁 − 1))) |
|
Theorem | zgt0ge1 9105 |
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 9106 |
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 9107 |
Positive integer ordering relation. (Contributed by NM, 19-Aug-2001.)
|
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 < 𝐵 ↔ (𝐴 + 1) ≤ 𝐵)) |
|
Theorem | nnaddm1cl 9108 |
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 9109 |
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 9110 |
Nonnegative integer ordering relation. (Contributed by Raph Levien,
10-Apr-2004.)
|
⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0)
→ (𝑀 ≤ 𝑁 ↔ 𝑀 < (𝑁 + 1))) |
|
Theorem | nn0ltlem1 9111 |
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 9112 |
The nonnegative difference of integers is a nonnegative integer.
(Generalization of nn0sub 9113.) (Contributed by NM, 14-Jul-2005.)
|
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ≤ 𝑁 ↔ (𝑁 − 𝑀) ∈
ℕ0)) |
|
Theorem | nn0sub 9113 |
Subtraction of nonnegative integers. (Contributed by NM, 9-May-2004.)
|
⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0)
→ (𝑀 ≤ 𝑁 ↔ (𝑁 − 𝑀) ∈
ℕ0)) |
|
Theorem | nn0n0n1ge2 9114 |
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 9115* |
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 9116 |
Alternate definition of the integers, based on elz2 9115.
(Contributed by
Mario Carneiro, 16-May-2014.)
|
⊢ ℤ = ( − “ (ℕ ×
ℕ)) |
|
Theorem | nn0sub2 9117 |
Subtraction of nonnegative integers. (Contributed by NM, 4-Sep-2005.)
|
⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0
∧ 𝑀 ≤ 𝑁) → (𝑁 − 𝑀) ∈
ℕ0) |
|
Theorem | zapne 9118 |
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 9119 |
Equality of integers is decidable. (Contributed by Jim Kingdon,
14-Mar-2020.)
|
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) →
DECID 𝐴 =
𝐵) |
|
Theorem | zdcle 9120 |
Integer ≤ is decidable. (Contributed by Jim
Kingdon, 7-Apr-2020.)
|
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) →
DECID 𝐴
≤ 𝐵) |
|
Theorem | zdclt 9121 |
Integer < is decidable. (Contributed by Jim
Kingdon, 1-Jun-2020.)
|
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) →
DECID 𝐴
< 𝐵) |
|
Theorem | zltlen 9122 |
Integer 'Less than' expressed in terms of 'less than or equal to'. Also
see ltleap 8387 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 9123 |
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 9124 |
A nonnegative integer less than 1 is 0. (Contributed by Paul
Chapman, 22-Jun-2011.)
|
⊢ (𝑁 ∈ ℕ0 → (𝑁 < 1 ↔ 𝑁 = 0)) |
|
Theorem | nn0lt2 9125 |
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 9126 |
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 9127 |
Nonnegative integer ordering relation. (Contributed by NM,
21-Jun-2005.)
|
⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0)
→ (𝑀 ≤ 𝑁 ↔ (𝑀 − 1) < 𝑁)) |
|
Theorem | nnlem1lt 9128 |
Positive integer ordering relation. (Contributed by NM, 21-Jun-2005.)
|
⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 ≤ 𝑁 ↔ (𝑀 − 1) < 𝑁)) |
|
Theorem | nnltlem1 9129 |
Positive integer ordering relation. (Contributed by NM, 21-Jun-2005.)
|
⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 < 𝑁 ↔ 𝑀 ≤ (𝑁 − 1))) |
|
Theorem | nnm1ge0 9130 |
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 9131 |
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 9132* |
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 9133 |
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 9134 |
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 9135* |
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 9136* |
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 9137 |
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 9138 |
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 9139 |
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 9140 |
One-half is not an integer. (Contributed by NM, 31-Jul-2004.)
|
⊢ ¬ (1 / 2) ∈
ℤ |
|
Theorem | 3halfnz 9141 |
Three halves is not an integer. (Contributed by AV, 2-Jun-2020.)
|
⊢ ¬ (3 / 2) ∈
ℤ |
|
Theorem | suprzclex 9142* |
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 9143* |
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 9144 |
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 9145 |
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 9146 |
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 9147 |
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 9148 |
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 9149 |
An integer is even or odd. (Contributed by NM, 1-Jan-2006.)
|
⊢ (𝑁 ∈ ℤ → ((𝑁 / 2) ∈ ℤ ∨ ((𝑁 + 1) / 2) ∈
ℤ)) |
|
Theorem | zeo2 9150 |
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 9151* |
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 9152* |
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 9153* |
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 9154* |
An expression for the upper integers that start at 𝑁 that is
analogous to dfnn2 8715 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 9155* |
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 9156* |
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 9157* |
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 9158* |
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 9159* |
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 9160* |
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 9161* |
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 9162* |
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 9163* |
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 9164 |
A positive integer is an integer. (Contributed by Mario Carneiro,
28-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈
ℕ0) ⇒ ⊢ (𝜑 → 𝐴 ∈ ℤ) |
|
Theorem | nnzd 9165 |
A nonnegative integer is an integer. (Contributed by Mario Carneiro,
28-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℕ)
⇒ ⊢ (𝜑 → 𝐴 ∈ ℤ) |
|
Theorem | zred 9166 |
An integer is a real number. (Contributed by Mario Carneiro,
28-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℤ)
⇒ ⊢ (𝜑 → 𝐴 ∈ ℝ) |
|
Theorem | zcnd 9167 |
An integer is a complex number. (Contributed by Mario Carneiro,
28-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℤ)
⇒ ⊢ (𝜑 → 𝐴 ∈ ℂ) |
|
Theorem | znegcld 9168 |
Closure law for negative integers. (Contributed by Mario Carneiro,
28-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℤ)
⇒ ⊢ (𝜑 → -𝐴 ∈ ℤ) |
|
Theorem | peano2zd 9169 |
Deduction from second Peano postulate generalized to integers.
(Contributed by Mario Carneiro, 28-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℤ)
⇒ ⊢ (𝜑 → (𝐴 + 1) ∈ ℤ) |
|
Theorem | zaddcld 9170 |
Closure of addition of integers. (Contributed by Mario Carneiro,
28-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℤ) & ⊢ (𝜑 → 𝐵 ∈ ℤ)
⇒ ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℤ) |
|
Theorem | zsubcld 9171 |
Closure of subtraction of integers. (Contributed by Mario Carneiro,
28-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℤ) & ⊢ (𝜑 → 𝐵 ∈ ℤ)
⇒ ⊢ (𝜑 → (𝐴 − 𝐵) ∈ ℤ) |
|
Theorem | zmulcld 9172 |
Closure of multiplication of integers. (Contributed by Mario Carneiro,
28-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℤ) & ⊢ (𝜑 → 𝐵 ∈ ℤ)
⇒ ⊢ (𝜑 → (𝐴 · 𝐵) ∈ ℤ) |
|
Theorem | zadd2cl 9173 |
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 9174 |
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 9175 |
Constant used for decimal constructor.
|
class ;𝐴𝐵 |
|
Definition | df-dec 9176 |
Define the "decimal constructor", which is used to build up
"decimal
integers" or "numeric terms" in base 10. For example,
(;;;1000 + ;;;2000) = ;;;3000 1kp2ke3k 12925.
(Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV,
1-Aug-2021.)
|
⊢ ;𝐴𝐵 = (((9 + 1) · 𝐴) + 𝐵) |
|
Theorem | 9p1e10 9177 |
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 9178 |
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 9179 |
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 9180 |
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 9181 |
Equality theorem for the decimal constructor. (Contributed by Mario
Carneiro, 17-Apr-2015.)
|
⊢ 𝐴 = 𝐵 ⇒ ⊢ ;𝐴𝐶 = ;𝐵𝐶 |
|
Theorem | deceq2i 9182 |
Equality theorem for the decimal constructor. (Contributed by Mario
Carneiro, 17-Apr-2015.)
|
⊢ 𝐴 = 𝐵 ⇒ ⊢ ;𝐶𝐴 = ;𝐶𝐵 |
|
Theorem | deceq12i 9183 |
Equality theorem for the decimal constructor. (Contributed by Mario
Carneiro, 17-Apr-2015.)
|
⊢ 𝐴 = 𝐵
& ⊢ 𝐶 = 𝐷 ⇒ ⊢ ;𝐴𝐶 = ;𝐵𝐷 |
|
Theorem | numnncl 9184 |
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 9185 |
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 9186 |
Add a zero in the higher places. (Contributed by Mario Carneiro,
18-Feb-2014.)
|
⊢ 𝑇 ∈ ℕ0 & ⊢ 𝐴 ∈
ℕ0 ⇒ ⊢ 𝐴 = ((𝑇 · 0) + 𝐴) |
|
Theorem | numcl 9187 |
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 9188 |
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 9189 |
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 9190 |
10 is a positive integer. (Contributed by NM, 8-Nov-2012.) (Revised by
AV, 6-Sep-2021.)
|
⊢ ;10 ∈ ℕ |
|
Theorem | 10pos 9191 |
The number 10 is positive. (Contributed by NM, 5-Feb-2007.) (Revised by
AV, 8-Sep-2021.)
|
⊢ 0 < ;10 |
|
Theorem | 10nn0 9192 |
10 is a nonnegative integer. (Contributed by Mario Carneiro,
19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
|
⊢ ;10 ∈ ℕ0 |
|
Theorem | 10re 9193 |
The number 10 is real. (Contributed by NM, 5-Feb-2007.) (Revised by AV,
8-Sep-2021.)
|
⊢ ;10 ∈ ℝ |
|
Theorem | decnncl 9194 |
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 9195 |
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 9196 |
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 9197 |
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 9198 |
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 9199 |
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 9200 |
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) |