Theorem List for Intuitionistic Logic Explorer - 9601-9700 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
|
Theorem | eluzge2nn0 9601 |
If an integer is greater than or equal to 2, then it is a nonnegative
integer. (Contributed by AV, 27-Aug-2018.) (Proof shortened by AV,
3-Nov-2018.)
|
⊢ (𝑁 ∈ (ℤ≥‘2)
→ 𝑁 ∈
ℕ0) |
|
Theorem | eluz2n0 9602 |
An integer greater than or equal to 2 is not 0. (Contributed by AV,
25-May-2020.)
|
⊢ (𝑁 ∈ (ℤ≥‘2)
→ 𝑁 ≠
0) |
|
Theorem | uzuzle23 9603 |
An integer in the upper set of integers starting at 3 is element of the
upper set of integers starting at 2. (Contributed by Alexander van der
Vekens, 17-Sep-2018.)
|
⊢ (𝐴 ∈ (ℤ≥‘3)
→ 𝐴 ∈
(ℤ≥‘2)) |
|
Theorem | eluzge3nn 9604 |
If an integer is greater than 3, then it is a positive integer.
(Contributed by Alexander van der Vekens, 17-Sep-2018.)
|
⊢ (𝑁 ∈ (ℤ≥‘3)
→ 𝑁 ∈
ℕ) |
|
Theorem | uz3m2nn 9605 |
An integer greater than or equal to 3 decreased by 2 is a positive
integer. (Contributed by Alexander van der Vekens, 17-Sep-2018.)
|
⊢ (𝑁 ∈ (ℤ≥‘3)
→ (𝑁 − 2)
∈ ℕ) |
|
Theorem | 1eluzge0 9606 |
1 is an integer greater than or equal to 0. (Contributed by Alexander van
der Vekens, 8-Jun-2018.)
|
⊢ 1 ∈
(ℤ≥‘0) |
|
Theorem | 2eluzge0 9607 |
2 is an integer greater than or equal to 0. (Contributed by Alexander van
der Vekens, 8-Jun-2018.) (Proof shortened by OpenAI, 25-Mar-2020.)
|
⊢ 2 ∈
(ℤ≥‘0) |
|
Theorem | 2eluzge1 9608 |
2 is an integer greater than or equal to 1. (Contributed by Alexander van
der Vekens, 8-Jun-2018.)
|
⊢ 2 ∈
(ℤ≥‘1) |
|
Theorem | uznnssnn 9609 |
The upper integers starting from a natural are a subset of the naturals.
(Contributed by Scott Fenton, 29-Jun-2013.)
|
⊢ (𝑁 ∈ ℕ →
(ℤ≥‘𝑁) ⊆ ℕ) |
|
Theorem | raluz 9610* |
Restricted universal quantification in an upper set of integers.
(Contributed by NM, 9-Sep-2005.)
|
⊢ (𝑀 ∈ ℤ → (∀𝑛 ∈
(ℤ≥‘𝑀)𝜑 ↔ ∀𝑛 ∈ ℤ (𝑀 ≤ 𝑛 → 𝜑))) |
|
Theorem | raluz2 9611* |
Restricted universal quantification in an upper set of integers.
(Contributed by NM, 9-Sep-2005.)
|
⊢ (∀𝑛 ∈ (ℤ≥‘𝑀)𝜑 ↔ (𝑀 ∈ ℤ → ∀𝑛 ∈ ℤ (𝑀 ≤ 𝑛 → 𝜑))) |
|
Theorem | rexuz 9612* |
Restricted existential quantification in an upper set of integers.
(Contributed by NM, 9-Sep-2005.)
|
⊢ (𝑀 ∈ ℤ → (∃𝑛 ∈
(ℤ≥‘𝑀)𝜑 ↔ ∃𝑛 ∈ ℤ (𝑀 ≤ 𝑛 ∧ 𝜑))) |
|
Theorem | rexuz2 9613* |
Restricted existential quantification in an upper set of integers.
(Contributed by NM, 9-Sep-2005.)
|
⊢ (∃𝑛 ∈ (ℤ≥‘𝑀)𝜑 ↔ (𝑀 ∈ ℤ ∧ ∃𝑛 ∈ ℤ (𝑀 ≤ 𝑛 ∧ 𝜑))) |
|
Theorem | 2rexuz 9614* |
Double existential quantification in an upper set of integers.
(Contributed by NM, 3-Nov-2005.)
|
⊢ (∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)𝜑 ↔ ∃𝑚 ∈ ℤ ∃𝑛 ∈ ℤ (𝑚 ≤ 𝑛 ∧ 𝜑)) |
|
Theorem | peano2uz 9615 |
Second Peano postulate for an upper set of integers. (Contributed by NM,
7-Sep-2005.)
|
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑁 + 1) ∈
(ℤ≥‘𝑀)) |
|
Theorem | peano2uzs 9616 |
Second Peano postulate for an upper set of integers. (Contributed by
Mario Carneiro, 26-Dec-2013.)
|
⊢ 𝑍 = (ℤ≥‘𝑀)
⇒ ⊢ (𝑁 ∈ 𝑍 → (𝑁 + 1) ∈ 𝑍) |
|
Theorem | peano2uzr 9617 |
Reversed second Peano axiom for upper integers. (Contributed by NM,
2-Jan-2006.)
|
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈
(ℤ≥‘(𝑀 + 1))) → 𝑁 ∈ (ℤ≥‘𝑀)) |
|
Theorem | uzaddcl 9618 |
Addition closure law for an upper set of integers. (Contributed by NM,
4-Jun-2006.)
|
⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝐾 ∈ ℕ0) → (𝑁 + 𝐾) ∈
(ℤ≥‘𝑀)) |
|
Theorem | nn0pzuz 9619 |
The sum of a nonnegative integer and an integer is an integer greater than
or equal to that integer. (Contributed by Alexander van der Vekens,
3-Oct-2018.)
|
⊢ ((𝑁 ∈ ℕ0 ∧ 𝑍 ∈ ℤ) → (𝑁 + 𝑍) ∈
(ℤ≥‘𝑍)) |
|
Theorem | uzind4 9620* |
Induction on the upper set of integers that starts 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, 7-Sep-2005.)
|
⊢ (𝑗 = 𝑀 → (𝜑 ↔ 𝜓)) & ⊢ (𝑗 = 𝑘 → (𝜑 ↔ 𝜒)) & ⊢ (𝑗 = (𝑘 + 1) → (𝜑 ↔ 𝜃)) & ⊢ (𝑗 = 𝑁 → (𝜑 ↔ 𝜏)) & ⊢ (𝑀 ∈ ℤ → 𝜓) & ⊢ (𝑘 ∈
(ℤ≥‘𝑀) → (𝜒 → 𝜃)) ⇒ ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝜏) |
|
Theorem | uzind4ALT 9621* |
Induction on the upper set of integers that starts at an integer 𝑀.
The last four hypotheses give us the substitution instances we need; the
first two are the basis and the induction step. Either uzind4 9620 or
uzind4ALT 9621 may be used; see comment for nnind 8966. (Contributed by NM,
7-Sep-2005.) (New usage is discouraged.)
(Proof modification is discouraged.)
|
⊢ (𝑀 ∈ ℤ → 𝜓)
& ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → (𝜒 → 𝜃)) & ⊢ (𝑗 = 𝑀 → (𝜑 ↔ 𝜓)) & ⊢ (𝑗 = 𝑘 → (𝜑 ↔ 𝜒)) & ⊢ (𝑗 = (𝑘 + 1) → (𝜑 ↔ 𝜃)) & ⊢ (𝑗 = 𝑁 → (𝜑 ↔ 𝜏)) ⇒ ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝜏) |
|
Theorem | uzind4s 9622* |
Induction on the upper set of integers that starts at an integer 𝑀,
using explicit substitution. The hypotheses are the basis and the
induction step. (Contributed by NM, 4-Nov-2005.)
|
⊢ (𝑀 ∈ ℤ → [𝑀 / 𝑘]𝜑)
& ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → (𝜑 → [(𝑘 + 1) / 𝑘]𝜑)) ⇒ ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → [𝑁 / 𝑘]𝜑) |
|
Theorem | uzind4s2 9623* |
Induction on the upper set of integers that starts at an integer 𝑀,
using explicit substitution. The hypotheses are the basis and the
induction step. Use this instead of uzind4s 9622 when 𝑗 and 𝑘 must
be distinct in [(𝑘 + 1) / 𝑗]𝜑. (Contributed by NM,
16-Nov-2005.)
|
⊢ (𝑀 ∈ ℤ → [𝑀 / 𝑗]𝜑)
& ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → ([𝑘 / 𝑗]𝜑 → [(𝑘 + 1) / 𝑗]𝜑)) ⇒ ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → [𝑁 / 𝑗]𝜑) |
|
Theorem | uzind4i 9624* |
Induction on the upper integers that start at 𝑀. The first four
give us the substitution instances we need, and the last two are the
basis and the induction step. This is a stronger version of uzind4 9620
assuming that 𝜓 holds unconditionally. Notice that
𝑁
∈ (ℤ≥‘𝑀) implies that the lower bound 𝑀 is an
integer
(𝑀
∈ ℤ, see eluzel2 9564). (Contributed by NM, 4-Sep-2005.)
(Revised by AV, 13-Jul-2022.)
|
⊢ (𝑗 = 𝑀 → (𝜑 ↔ 𝜓)) & ⊢ (𝑗 = 𝑘 → (𝜑 ↔ 𝜒)) & ⊢ (𝑗 = (𝑘 + 1) → (𝜑 ↔ 𝜃)) & ⊢ (𝑗 = 𝑁 → (𝜑 ↔ 𝜏)) & ⊢ 𝜓 & ⊢ (𝑘 ∈
(ℤ≥‘𝑀) → (𝜒 → 𝜃)) ⇒ ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝜏) |
|
Theorem | indstr 9625* |
Strong Mathematical Induction for positive integers (inference schema).
(Contributed by NM, 17-Aug-2001.)
|
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) & ⊢ (𝑥 ∈ ℕ →
(∀𝑦 ∈ ℕ
(𝑦 < 𝑥 → 𝜓) → 𝜑)) ⇒ ⊢ (𝑥 ∈ ℕ → 𝜑) |
|
Theorem | infrenegsupex 9626* |
The infimum of a set of reals 𝐴 is the negative of the supremum of
the negatives of its elements. (Contributed by Jim Kingdon,
14-Jan-2022.)
|
⊢ (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))) & ⊢ (𝜑 → 𝐴 ⊆ ℝ)
⇒ ⊢ (𝜑 → inf(𝐴, ℝ, < ) = -sup({𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴}, ℝ, < )) |
|
Theorem | supinfneg 9627* |
If a set of real numbers has a least upper bound, the set of the
negation of those numbers has a greatest lower bound. For a theorem
which is similar but only for the boundedness part, see ublbneg 9645.
(Contributed by Jim Kingdon, 15-Jan-2022.)
|
⊢ (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) & ⊢ (𝜑 → 𝐴 ⊆ ℝ)
⇒ ⊢ (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴} ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴}𝑧 < 𝑦))) |
|
Theorem | infsupneg 9628* |
If a set of real numbers has a greatest lower bound, the set of the
negation of those numbers has a least upper bound. To go in the other
direction see supinfneg 9627. (Contributed by Jim Kingdon,
15-Jan-2022.)
|
⊢ (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))) & ⊢ (𝜑 → 𝐴 ⊆ ℝ)
⇒ ⊢ (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴}𝑦 < 𝑧))) |
|
Theorem | supminfex 9629* |
A supremum is the negation of the infimum of that set's image under
negation. (Contributed by Jim Kingdon, 14-Jan-2022.)
|
⊢ (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) & ⊢ (𝜑 → 𝐴 ⊆ ℝ)
⇒ ⊢ (𝜑 → sup(𝐴, ℝ, < ) = -inf({𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴}, ℝ, < )) |
|
Theorem | infregelbex 9630* |
Any lower bound of a set of real numbers with an infimum is less than or
equal to the infimum. (Contributed by Jim Kingdon, 27-Sep-2024.)
|
⊢ (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))) & ⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ)
⇒ ⊢ (𝜑 → (𝐵 ≤ inf(𝐴, ℝ, < ) ↔ ∀𝑧 ∈ 𝐴 𝐵 ≤ 𝑧)) |
|
Theorem | eluznn0 9631 |
Membership in a nonnegative upper set of integers implies membership in
ℕ0. (Contributed by Paul
Chapman, 22-Jun-2011.)
|
⊢ ((𝑁 ∈ ℕ0 ∧ 𝑀 ∈
(ℤ≥‘𝑁)) → 𝑀 ∈
ℕ0) |
|
Theorem | eluznn 9632 |
Membership in a positive upper set of integers implies membership in
ℕ. (Contributed by JJ, 1-Oct-2018.)
|
⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ≥‘𝑁)) → 𝑀 ∈ ℕ) |
|
Theorem | eluz2b1 9633 |
Two ways to say "an integer greater than or equal to 2".
(Contributed by
Paul Chapman, 23-Nov-2012.)
|
⊢ (𝑁 ∈ (ℤ≥‘2)
↔ (𝑁 ∈ ℤ
∧ 1 < 𝑁)) |
|
Theorem | eluz2gt1 9634 |
An integer greater than or equal to 2 is greater than 1. (Contributed by
AV, 24-May-2020.)
|
⊢ (𝑁 ∈ (ℤ≥‘2)
→ 1 < 𝑁) |
|
Theorem | eluz2b2 9635 |
Two ways to say "an integer greater than or equal to 2".
(Contributed by
Paul Chapman, 23-Nov-2012.)
|
⊢ (𝑁 ∈ (ℤ≥‘2)
↔ (𝑁 ∈ ℕ
∧ 1 < 𝑁)) |
|
Theorem | eluz2b3 9636 |
Two ways to say "an integer greater than or equal to 2".
(Contributed by
Paul Chapman, 23-Nov-2012.)
|
⊢ (𝑁 ∈ (ℤ≥‘2)
↔ (𝑁 ∈ ℕ
∧ 𝑁 ≠
1)) |
|
Theorem | uz2m1nn 9637 |
One less than an integer greater than or equal to 2 is a positive integer.
(Contributed by Paul Chapman, 17-Nov-2012.)
|
⊢ (𝑁 ∈ (ℤ≥‘2)
→ (𝑁 − 1)
∈ ℕ) |
|
Theorem | 1nuz2 9638 |
1 is not in (ℤ≥‘2).
(Contributed by Paul Chapman,
21-Nov-2012.)
|
⊢ ¬ 1 ∈
(ℤ≥‘2) |
|
Theorem | elnn1uz2 9639 |
A positive integer is either 1 or greater than or equal to 2.
(Contributed by Paul Chapman, 17-Nov-2012.)
|
⊢ (𝑁 ∈ ℕ ↔ (𝑁 = 1 ∨ 𝑁 ∈
(ℤ≥‘2))) |
|
Theorem | uz2mulcl 9640 |
Closure of multiplication of integers greater than or equal to 2.
(Contributed by Paul Chapman, 26-Oct-2012.)
|
⊢ ((𝑀 ∈ (ℤ≥‘2)
∧ 𝑁 ∈
(ℤ≥‘2)) → (𝑀 · 𝑁) ∈
(ℤ≥‘2)) |
|
Theorem | indstr2 9641* |
Strong Mathematical Induction for positive integers (inference schema).
The first two hypotheses give us the substitution instances we need; the
last two are the basis and the induction step. (Contributed by Paul
Chapman, 21-Nov-2012.)
|
⊢ (𝑥 = 1 → (𝜑 ↔ 𝜒)) & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) & ⊢ 𝜒 & ⊢ (𝑥 ∈
(ℤ≥‘2) → (∀𝑦 ∈ ℕ (𝑦 < 𝑥 → 𝜓) → 𝜑)) ⇒ ⊢ (𝑥 ∈ ℕ → 𝜑) |
|
Theorem | eluzdc 9642 |
Membership of an integer in an upper set of integers is decidable.
(Contributed by Jim Kingdon, 18-Apr-2020.)
|
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) →
DECID 𝑁
∈ (ℤ≥‘𝑀)) |
|
Theorem | elnn0dc 9643 |
Membership of an integer in ℕ0 is
decidable. (Contributed by Jim
Kingdon, 8-Oct-2024.)
|
⊢ (𝑁 ∈ ℤ → DECID
𝑁 ∈
ℕ0) |
|
Theorem | elnndc 9644 |
Membership of an integer in ℕ is decidable.
(Contributed by Jim
Kingdon, 17-Oct-2024.)
|
⊢ (𝑁 ∈ ℤ → DECID
𝑁 ∈
ℕ) |
|
Theorem | ublbneg 9645* |
The image under negation of a bounded-above set of reals is bounded
below. For a theorem which is similar but also adds that the bounds
need to be the tightest possible, see supinfneg 9627. (Contributed by
Paul Chapman, 21-Mar-2011.)
|
⊢ (∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴}𝑥 ≤ 𝑦) |
|
Theorem | eqreznegel 9646* |
Two ways to express the image under negation of a set of integers.
(Contributed by Paul Chapman, 21-Mar-2011.)
|
⊢ (𝐴 ⊆ ℤ → {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴} = {𝑧 ∈ ℤ ∣ -𝑧 ∈ 𝐴}) |
|
Theorem | negm 9647* |
The image under negation of an inhabited set of reals is inhabited.
(Contributed by Jim Kingdon, 10-Apr-2020.)
|
⊢ ((𝐴 ⊆ ℝ ∧ ∃𝑥 𝑥 ∈ 𝐴) → ∃𝑦 𝑦 ∈ {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴}) |
|
Theorem | lbzbi 9648* |
If a set of reals is bounded below, it is bounded below by an integer.
(Contributed by Paul Chapman, 21-Mar-2011.)
|
⊢ (𝐴 ⊆ ℝ → (∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦 ↔ ∃𝑥 ∈ ℤ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦)) |
|
Theorem | nn01to3 9649 |
A (nonnegative) integer between 1 and 3 must be 1, 2 or 3. (Contributed
by Alexander van der Vekens, 13-Sep-2018.)
|
⊢ ((𝑁 ∈ ℕ0 ∧ 1 ≤
𝑁 ∧ 𝑁 ≤ 3) → (𝑁 = 1 ∨ 𝑁 = 2 ∨ 𝑁 = 3)) |
|
Theorem | nn0ge2m1nnALT 9650 |
Alternate proof of nn0ge2m1nn 9267: If a nonnegative integer is greater
than or equal to two, the integer decreased by 1 is a positive integer.
This version is proved using eluz2 9565, a theorem for upper sets of
integers, which are defined later than the positive and nonnegative
integers. This proof is, however, much shorter than the proof of
nn0ge2m1nn 9267. (Contributed by Alexander van der Vekens,
1-Aug-2018.)
(New usage is discouraged.) (Proof modification is discouraged.)
|
⊢ ((𝑁 ∈ ℕ0 ∧ 2 ≤
𝑁) → (𝑁 − 1) ∈
ℕ) |
|
4.4.12 Rational numbers (as a subset of complex
numbers)
|
|
Syntax | cq 9651 |
Extend class notation to include the class of rationals.
|
class ℚ |
|
Definition | df-q 9652 |
Define the set of rational numbers. Based on definition of rationals in
[Apostol] p. 22. See elq 9654
for the relation "is rational". (Contributed
by NM, 8-Jan-2002.)
|
⊢ ℚ = ( / “ (ℤ ×
ℕ)) |
|
Theorem | divfnzn 9653 |
Division restricted to ℤ × ℕ is a
function. Given excluded
middle, it would be easy to prove this for ℂ
× (ℂ ∖ {0}).
The key difference is that an element of ℕ
is apart from zero,
whereas being an element of ℂ ∖ {0}
implies being not equal to
zero. (Contributed by Jim Kingdon, 19-Mar-2020.)
|
⊢ ( / ↾ (ℤ × ℕ)) Fn
(ℤ × ℕ) |
|
Theorem | elq 9654* |
Membership in the set of rationals. (Contributed by NM, 8-Jan-2002.)
(Revised by Mario Carneiro, 28-Jan-2014.)
|
⊢ (𝐴 ∈ ℚ ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦)) |
|
Theorem | qmulz 9655* |
If 𝐴 is rational, then some integer
multiple of it is an integer.
(Contributed by NM, 7-Nov-2008.) (Revised by Mario Carneiro,
22-Jul-2014.)
|
⊢ (𝐴 ∈ ℚ → ∃𝑥 ∈ ℕ (𝐴 · 𝑥) ∈ ℤ) |
|
Theorem | znq 9656 |
The ratio of an integer and a positive integer is a rational number.
(Contributed by NM, 12-Jan-2002.)
|
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → (𝐴 / 𝐵) ∈ ℚ) |
|
Theorem | qre 9657 |
A rational number is a real number. (Contributed by NM,
14-Nov-2002.)
|
⊢ (𝐴 ∈ ℚ → 𝐴 ∈ ℝ) |
|
Theorem | zq 9658 |
An integer is a rational number. (Contributed by NM, 9-Jan-2002.)
|
⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℚ) |
|
Theorem | zssq 9659 |
The integers are a subset of the rationals. (Contributed by NM,
9-Jan-2002.)
|
⊢ ℤ ⊆ ℚ |
|
Theorem | nn0ssq 9660 |
The nonnegative integers are a subset of the rationals. (Contributed by
NM, 31-Jul-2004.)
|
⊢ ℕ0 ⊆
ℚ |
|
Theorem | nnssq 9661 |
The positive integers are a subset of the rationals. (Contributed by NM,
31-Jul-2004.)
|
⊢ ℕ ⊆ ℚ |
|
Theorem | qssre 9662 |
The rationals are a subset of the reals. (Contributed by NM,
9-Jan-2002.)
|
⊢ ℚ ⊆ ℝ |
|
Theorem | qsscn 9663 |
The rationals are a subset of the complex numbers. (Contributed by NM,
2-Aug-2004.)
|
⊢ ℚ ⊆ ℂ |
|
Theorem | qex 9664 |
The set of rational numbers exists. (Contributed by NM, 30-Jul-2004.)
(Revised by Mario Carneiro, 17-Nov-2014.)
|
⊢ ℚ ∈ V |
|
Theorem | nnq 9665 |
A positive integer is rational. (Contributed by NM, 17-Nov-2004.)
|
⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℚ) |
|
Theorem | qcn 9666 |
A rational number is a complex number. (Contributed by NM,
2-Aug-2004.)
|
⊢ (𝐴 ∈ ℚ → 𝐴 ∈ ℂ) |
|
Theorem | qaddcl 9667 |
Closure of addition of rationals. (Contributed by NM, 1-Aug-2004.)
|
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 + 𝐵) ∈ ℚ) |
|
Theorem | qnegcl 9668 |
Closure law for the negative of a rational. (Contributed by NM,
2-Aug-2004.) (Revised by Mario Carneiro, 15-Sep-2014.)
|
⊢ (𝐴 ∈ ℚ → -𝐴 ∈ ℚ) |
|
Theorem | qmulcl 9669 |
Closure of multiplication of rationals. (Contributed by NM,
1-Aug-2004.)
|
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 · 𝐵) ∈ ℚ) |
|
Theorem | qsubcl 9670 |
Closure of subtraction of rationals. (Contributed by NM, 2-Aug-2004.)
|
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 − 𝐵) ∈ ℚ) |
|
Theorem | qapne 9671 |
Apartness is equivalent to not equal for rationals. (Contributed by Jim
Kingdon, 20-Mar-2020.)
|
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 # 𝐵 ↔ 𝐴 ≠ 𝐵)) |
|
Theorem | qltlen 9672 |
Rational 'Less than' expressed in terms of 'less than or equal to'. Also
see ltleap 8620 which is a similar result for real numbers.
(Contributed by
Jim Kingdon, 11-Oct-2021.)
|
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 < 𝐵 ↔ (𝐴 ≤ 𝐵 ∧ 𝐵 ≠ 𝐴))) |
|
Theorem | qlttri2 9673 |
Apartness is equivalent to not equal for rationals. (Contributed by Jim
Kingdon, 9-Nov-2021.)
|
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 ≠ 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴))) |
|
Theorem | qreccl 9674 |
Closure of reciprocal of rationals. (Contributed by NM, 3-Aug-2004.)
|
⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) → (1 / 𝐴) ∈ ℚ) |
|
Theorem | qdivcl 9675 |
Closure of division of rationals. (Contributed by NM, 3-Aug-2004.)
|
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 𝐵 ≠ 0) → (𝐴 / 𝐵) ∈ ℚ) |
|
Theorem | qrevaddcl 9676 |
Reverse closure law for addition of rationals. (Contributed by NM,
2-Aug-2004.)
|
⊢ (𝐵 ∈ ℚ → ((𝐴 ∈ ℂ ∧ (𝐴 + 𝐵) ∈ ℚ) ↔ 𝐴 ∈ ℚ)) |
|
Theorem | nnrecq 9677 |
The reciprocal of a positive integer is rational. (Contributed by NM,
17-Nov-2004.)
|
⊢ (𝐴 ∈ ℕ → (1 / 𝐴) ∈
ℚ) |
|
Theorem | irradd 9678 |
The sum of an irrational number and a rational number is irrational.
(Contributed by NM, 7-Nov-2008.)
|
⊢ ((𝐴 ∈ (ℝ ∖ ℚ) ∧
𝐵 ∈ ℚ) →
(𝐴 + 𝐵) ∈ (ℝ ∖
ℚ)) |
|
Theorem | irrmul 9679 |
The product of a real which is not rational with a nonzero rational is not
rational. Note that by "not rational" we mean the negation of
"is
rational" (whereas "irrational" is often defined to mean
apart from any
rational number - given excluded middle these two definitions would be
equivalent). (Contributed by NM, 7-Nov-2008.)
|
⊢ ((𝐴 ∈ (ℝ ∖ ℚ) ∧
𝐵 ∈ ℚ ∧
𝐵 ≠ 0) → (𝐴 · 𝐵) ∈ (ℝ ∖
ℚ)) |
|
Theorem | elpq 9680* |
A positive rational is the quotient of two positive integers.
(Contributed by AV, 29-Dec-2022.)
|
⊢ ((𝐴 ∈ ℚ ∧ 0 < 𝐴) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦)) |
|
Theorem | elpqb 9681* |
A class is a positive rational iff it is the quotient of two positive
integers. (Contributed by AV, 30-Dec-2022.)
|
⊢ ((𝐴 ∈ ℚ ∧ 0 < 𝐴) ↔ ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦)) |
|
4.4.13 Complex numbers as pairs of
reals
|
|
Theorem | cnref1o 9682* |
There is a natural one-to-one mapping from (ℝ ×
ℝ) to ℂ,
where we map 〈𝑥, 𝑦〉 to (𝑥 + (i · 𝑦)). In our
construction of the complex numbers, this is in fact our
definition of
ℂ (see df-c 7848), but in the axiomatic treatment we can only
show
that there is the expected mapping between these two sets. (Contributed
by Mario Carneiro, 16-Jun-2013.) (Revised by Mario Carneiro,
17-Feb-2014.)
|
⊢ 𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + (i · 𝑦))) ⇒ ⊢ 𝐹:(ℝ × ℝ)–1-1-onto→ℂ |
|
Theorem | addex 9683 |
The addition operation is a set. (Contributed by NM, 19-Oct-2004.)
(Revised by Mario Carneiro, 17-Nov-2014.)
|
⊢ + ∈ V |
|
Theorem | mulex 9684 |
The multiplication operation is a set. (Contributed by NM, 19-Oct-2004.)
(Revised by Mario Carneiro, 17-Nov-2014.)
|
⊢ · ∈ V |
|
4.5 Order sets
|
|
4.5.1 Positive reals (as a subset of complex
numbers)
|
|
Syntax | crp 9685 |
Extend class notation to include the class of positive reals.
|
class ℝ+ |
|
Definition | df-rp 9686 |
Define the set of positive reals. Definition of positive numbers in
[Apostol] p. 20. (Contributed by NM,
27-Oct-2007.)
|
⊢ ℝ+ = {𝑥 ∈ ℝ ∣ 0 < 𝑥} |
|
Theorem | elrp 9687 |
Membership in the set of positive reals. (Contributed by NM,
27-Oct-2007.)
|
⊢ (𝐴 ∈ ℝ+ ↔ (𝐴 ∈ ℝ ∧ 0 <
𝐴)) |
|
Theorem | elrpii 9688 |
Membership in the set of positive reals. (Contributed by NM,
23-Feb-2008.)
|
⊢ 𝐴 ∈ ℝ & ⊢ 0 < 𝐴 ⇒ ⊢ 𝐴 ∈
ℝ+ |
|
Theorem | 1rp 9689 |
1 is a positive real. (Contributed by Jeff Hankins, 23-Nov-2008.)
|
⊢ 1 ∈
ℝ+ |
|
Theorem | 2rp 9690 |
2 is a positive real. (Contributed by Mario Carneiro, 28-May-2016.)
|
⊢ 2 ∈
ℝ+ |
|
Theorem | 3rp 9691 |
3 is a positive real. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|
⊢ 3 ∈
ℝ+ |
|
Theorem | rpre 9692 |
A positive real is a real. (Contributed by NM, 27-Oct-2007.)
|
⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈
ℝ) |
|
Theorem | rpxr 9693 |
A positive real is an extended real. (Contributed by Mario Carneiro,
21-Aug-2015.)
|
⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈
ℝ*) |
|
Theorem | rpcn 9694 |
A positive real is a complex number. (Contributed by NM, 11-Nov-2008.)
|
⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈
ℂ) |
|
Theorem | nnrp 9695 |
A positive integer is a positive real. (Contributed by NM,
28-Nov-2008.)
|
⊢ (𝐴 ∈ ℕ → 𝐴 ∈
ℝ+) |
|
Theorem | rpssre 9696 |
The positive reals are a subset of the reals. (Contributed by NM,
24-Feb-2008.)
|
⊢ ℝ+ ⊆
ℝ |
|
Theorem | rpgt0 9697 |
A positive real is greater than zero. (Contributed by FL,
27-Dec-2007.)
|
⊢ (𝐴 ∈ ℝ+ → 0 <
𝐴) |
|
Theorem | rpge0 9698 |
A positive real is greater than or equal to zero. (Contributed by NM,
22-Feb-2008.)
|
⊢ (𝐴 ∈ ℝ+ → 0 ≤
𝐴) |
|
Theorem | rpregt0 9699 |
A positive real is a positive real number. (Contributed by NM,
11-Nov-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
|
⊢ (𝐴 ∈ ℝ+ → (𝐴 ∈ ℝ ∧ 0 <
𝐴)) |
|
Theorem | rprege0 9700 |
A positive real is a nonnegative real number. (Contributed by Mario
Carneiro, 31-Jan-2014.)
|
⊢ (𝐴 ∈ ℝ+ → (𝐴 ∈ ℝ ∧ 0 ≤
𝐴)) |