Theorem List for Intuitionistic Logic Explorer - 9201-9300 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
|
Theorem | nnz 9201 |
A positive integer is an integer. (Contributed by NM, 9-May-2004.)
|
⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ) |
|
Theorem | nn0z 9202 |
A nonnegative integer is an integer. (Contributed by NM, 9-May-2004.)
|
⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈
ℤ) |
|
Theorem | nnzi 9203 |
A positive integer is an integer. (Contributed by Mario Carneiro,
18-Feb-2014.)
|
⊢ 𝑁 ∈ ℕ
⇒ ⊢ 𝑁 ∈ ℤ |
|
Theorem | nn0zi 9204 |
A nonnegative integer is an integer. (Contributed by Mario Carneiro,
18-Feb-2014.)
|
⊢ 𝑁 ∈
ℕ0 ⇒ ⊢ 𝑁 ∈ ℤ |
|
Theorem | elnnz1 9205 |
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 9206 |
Positive integers expressed as a subset of integers. (Contributed by NM,
3-Oct-2004.)
|
⊢ ℕ = {𝑥 ∈ ℤ ∣ 1 ≤ 𝑥} |
|
Theorem | nn0zrab 9207 |
Nonnegative integers expressed as a subset of integers. (Contributed by
NM, 3-Oct-2004.)
|
⊢ ℕ0 = {𝑥 ∈ ℤ ∣ 0 ≤ 𝑥} |
|
Theorem | 1z 9208 |
One is an integer. (Contributed by NM, 10-May-2004.)
|
⊢ 1 ∈ ℤ |
|
Theorem | 1zzd 9209 |
1 is an integer, deductive form (common case). (Contributed by David A.
Wheeler, 6-Dec-2018.)
|
⊢ (𝜑 → 1 ∈ ℤ) |
|
Theorem | 2z 9210 |
Two is an integer. (Contributed by NM, 10-May-2004.)
|
⊢ 2 ∈ ℤ |
|
Theorem | 3z 9211 |
3 is an integer. (Contributed by David A. Wheeler, 8-Dec-2018.)
|
⊢ 3 ∈ ℤ |
|
Theorem | 4z 9212 |
4 is an integer. (Contributed by BJ, 26-Mar-2020.)
|
⊢ 4 ∈ ℤ |
|
Theorem | znegcl 9213 |
Closure law for negative integers. (Contributed by NM, 9-May-2004.)
|
⊢ (𝑁 ∈ ℤ → -𝑁 ∈ ℤ) |
|
Theorem | neg1z 9214 |
-1 is an integer (common case). (Contributed by David A. Wheeler,
5-Dec-2018.)
|
⊢ -1 ∈ ℤ |
|
Theorem | znegclb 9215 |
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 9216 |
The negative of a nonnegative integer is an integer. (Contributed by NM,
9-May-2004.)
|
⊢ (𝑁 ∈ ℕ0 → -𝑁 ∈
ℤ) |
|
Theorem | nn0negzi 9217 |
The negative of a nonnegative integer is an integer. (Contributed by
Mario Carneiro, 18-Feb-2014.)
|
⊢ 𝑁 ∈
ℕ0 ⇒ ⊢ -𝑁 ∈ ℤ |
|
Theorem | peano2z 9218 |
Second Peano postulate generalized to integers. (Contributed by NM,
13-Feb-2005.)
|
⊢ (𝑁 ∈ ℤ → (𝑁 + 1) ∈ ℤ) |
|
Theorem | zaddcllempos 9219 |
Lemma for zaddcl 9222. 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 9220 |
"Reverse" second Peano postulate for integers. (Contributed by NM,
12-Sep-2005.)
|
⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈
ℤ) |
|
Theorem | zaddcllemneg 9221 |
Lemma for zaddcl 9222. 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 9222 |
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 9223 |
Closure of subtraction of integers. (Contributed by NM, 11-May-2004.)
|
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 − 𝑁) ∈ ℤ) |
|
Theorem | ztri3or0 9224 |
Integer trichotomy (with zero). (Contributed by Jim Kingdon,
14-Mar-2020.)
|
⊢ (𝑁 ∈ ℤ → (𝑁 < 0 ∨ 𝑁 = 0 ∨ 0 < 𝑁)) |
|
Theorem | ztri3or 9225 |
Integer trichotomy. (Contributed by Jim Kingdon, 14-Mar-2020.)
|
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 < 𝑁 ∨ 𝑀 = 𝑁 ∨ 𝑁 < 𝑀)) |
|
Theorem | zletric 9226 |
Trichotomy law. (Contributed by Jim Kingdon, 27-Mar-2020.)
|
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 ≤ 𝐵 ∨ 𝐵 ≤ 𝐴)) |
|
Theorem | zlelttric 9227 |
Trichotomy law. (Contributed by Jim Kingdon, 17-Apr-2020.)
|
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 ≤ 𝐵 ∨ 𝐵 < 𝐴)) |
|
Theorem | zltnle 9228 |
'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 9229 |
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 9230 |
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 9231 |
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 9232 |
Reverse closure law for addition of integers. (Contributed by NM,
11-May-2004.)
|
⊢ (𝑁 ∈ ℤ → ((𝑀 ∈ ℂ ∧ (𝑀 + 𝑁) ∈ ℤ) ↔ 𝑀 ∈ ℤ)) |
|
Theorem | znnsub 9233 |
The positive difference of unequal integers is a positive integer.
(Generalization of nnsub 8887.) (Contributed by NM, 11-May-2004.)
|
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 < 𝑁 ↔ (𝑁 − 𝑀) ∈ ℕ)) |
|
Theorem | nzadd 9234 |
The sum of a real number not being an integer and an integer is not an
integer. Note that "not being an integer" in this case means
"the
negation of is an integer" rather than "is apart from any
integer" (given
excluded middle, those two would be equivalent). (Contributed by AV,
19-Jul-2021.)
|
⊢ ((𝐴 ∈ (ℝ ∖ ℤ) ∧
𝐵 ∈ ℤ) →
(𝐴 + 𝐵) ∈ (ℝ ∖
ℤ)) |
|
Theorem | zmulcl 9235 |
Closure of multiplication of integers. (Contributed by NM,
30-Jul-2004.)
|
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 · 𝑁) ∈ ℤ) |
|
Theorem | zltp1le 9236 |
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 9237 |
Integer ordering relation. (Contributed by NM, 10-May-2004.)
|
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ≤ 𝑁 ↔ 𝑀 < (𝑁 + 1))) |
|
Theorem | zlem1lt 9238 |
Integer ordering relation. (Contributed by NM, 13-Nov-2004.)
|
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ≤ 𝑁 ↔ (𝑀 − 1) < 𝑁)) |
|
Theorem | zltlem1 9239 |
Integer ordering relation. (Contributed by NM, 13-Nov-2004.)
|
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 < 𝑁 ↔ 𝑀 ≤ (𝑁 − 1))) |
|
Theorem | zgt0ge1 9240 |
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 9241 |
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 9242 |
Positive integer ordering relation. (Contributed by NM, 19-Aug-2001.)
|
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 < 𝐵 ↔ (𝐴 + 1) ≤ 𝐵)) |
|
Theorem | nnaddm1cl 9243 |
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 9244 |
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 9245 |
Nonnegative integer ordering relation. (Contributed by Raph Levien,
10-Apr-2004.)
|
⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0)
→ (𝑀 ≤ 𝑁 ↔ 𝑀 < (𝑁 + 1))) |
|
Theorem | nn0ltlem1 9246 |
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 9247 |
The nonnegative difference of integers is a nonnegative integer.
(Generalization of nn0sub 9248.) (Contributed by NM, 14-Jul-2005.)
|
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ≤ 𝑁 ↔ (𝑁 − 𝑀) ∈
ℕ0)) |
|
Theorem | nn0sub 9248 |
Subtraction of nonnegative integers. (Contributed by NM, 9-May-2004.)
|
⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0)
→ (𝑀 ≤ 𝑁 ↔ (𝑁 − 𝑀) ∈
ℕ0)) |
|
Theorem | ltsubnn0 9249 |
Subtracting a nonnegative integer from a nonnegative integer which is
greater than the first one results in a nonnegative integer. (Contributed
by Alexander van der Vekens, 6-Apr-2018.)
|
⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0)
→ (𝐵 < 𝐴 → (𝐴 − 𝐵) ∈
ℕ0)) |
|
Theorem | nn0negleid 9250 |
A nonnegative integer is greater than or equal to its negative.
(Contributed by AV, 13-Aug-2021.)
|
⊢ (𝐴 ∈ ℕ0 → -𝐴 ≤ 𝐴) |
|
Theorem | difgtsumgt 9251 |
If the difference of a real number and a nonnegative integer is greater
than another real number, the sum of the real number and the nonnegative
integer is also greater than the other real number. (Contributed by AV,
13-Aug-2021.)
|
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℕ0 ∧ 𝐶 ∈ ℝ) → (𝐶 < (𝐴 − 𝐵) → 𝐶 < (𝐴 + 𝐵))) |
|
Theorem | nn0n0n1ge2 9252 |
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 9253* |
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 9254 |
Alternate definition of the integers, based on elz2 9253.
(Contributed by
Mario Carneiro, 16-May-2014.)
|
⊢ ℤ = ( − “ (ℕ ×
ℕ)) |
|
Theorem | nn0sub2 9255 |
Subtraction of nonnegative integers. (Contributed by NM, 4-Sep-2005.)
|
⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0
∧ 𝑀 ≤ 𝑁) → (𝑁 − 𝑀) ∈
ℕ0) |
|
Theorem | zapne 9256 |
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 9257 |
Equality of integers is decidable. (Contributed by Jim Kingdon,
14-Mar-2020.)
|
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) →
DECID 𝐴 =
𝐵) |
|
Theorem | zdcle 9258 |
Integer ≤ is decidable. (Contributed by Jim
Kingdon, 7-Apr-2020.)
|
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) →
DECID 𝐴
≤ 𝐵) |
|
Theorem | zdclt 9259 |
Integer < is decidable. (Contributed by Jim
Kingdon, 1-Jun-2020.)
|
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) →
DECID 𝐴
< 𝐵) |
|
Theorem | zltlen 9260 |
Integer 'Less than' expressed in terms of 'less than or equal to'. Also
see ltleap 8521 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 9261 |
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 9262 |
A nonnegative integer less than 1 is 0. (Contributed by Paul
Chapman, 22-Jun-2011.)
|
⊢ (𝑁 ∈ ℕ0 → (𝑁 < 1 ↔ 𝑁 = 0)) |
|
Theorem | nn0lt2 9263 |
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 9264 |
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 9265 |
Nonnegative integer ordering relation. (Contributed by NM,
21-Jun-2005.)
|
⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0)
→ (𝑀 ≤ 𝑁 ↔ (𝑀 − 1) < 𝑁)) |
|
Theorem | nnlem1lt 9266 |
Positive integer ordering relation. (Contributed by NM, 21-Jun-2005.)
|
⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 ≤ 𝑁 ↔ (𝑀 − 1) < 𝑁)) |
|
Theorem | nnltlem1 9267 |
Positive integer ordering relation. (Contributed by NM, 21-Jun-2005.)
|
⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 < 𝑁 ↔ 𝑀 ≤ (𝑁 − 1))) |
|
Theorem | nnm1ge0 9268 |
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 9269 |
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 9270* |
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 9271 |
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 9272 |
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 9273* |
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 9274* |
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 9275 |
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 9276 |
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 9277 |
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 9278 |
One-half is not an integer. (Contributed by NM, 31-Jul-2004.)
|
⊢ ¬ (1 / 2) ∈
ℤ |
|
Theorem | 3halfnz 9279 |
Three halves is not an integer. (Contributed by AV, 2-Jun-2020.)
|
⊢ ¬ (3 / 2) ∈
ℤ |
|
Theorem | suprzclex 9280* |
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 9281* |
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 9282 |
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 9283 |
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 9284 |
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 9285 |
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 9286 |
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 9287 |
An integer is even or odd. (Contributed by NM, 1-Jan-2006.)
|
⊢ (𝑁 ∈ ℤ → ((𝑁 / 2) ∈ ℤ ∨ ((𝑁 + 1) / 2) ∈
ℤ)) |
|
Theorem | zeo2 9288 |
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 9289* |
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 9290* |
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 9291* |
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 9292* |
An expression for the upper integers that start at 𝑁 that is
analogous to dfnn2 8850 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 9293* |
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 9294* |
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 9295* |
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 9296* |
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 9297* |
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 9298* |
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 9299* |
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 9300* |
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)) |