Theorem List for Intuitionistic Logic Explorer - 8601-8700 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
|
Theorem | 9t5e45 8601 |
9 times 5 equals 45. (Contributed by Mario Carneiro, 19-Apr-2015.)
|
⊢ (9 · 5) = ;45 |
|
Theorem | 9t6e54 8602 |
9 times 6 equals 54. (Contributed by Mario Carneiro, 19-Apr-2015.)
|
⊢ (9 · 6) = ;54 |
|
Theorem | 9t7e63 8603 |
9 times 7 equals 63. (Contributed by Mario Carneiro, 19-Apr-2015.)
|
⊢ (9 · 7) = ;63 |
|
Theorem | 9t8e72 8604 |
9 times 8 equals 72. (Contributed by Mario Carneiro, 19-Apr-2015.)
|
⊢ (9 · 8) = ;72 |
|
Theorem | 9t9e81 8605 |
9 times 9 equals 81. (Contributed by Mario Carneiro, 19-Apr-2015.)
|
⊢ (9 · 9) = ;81 |
|
Theorem | 9t11e99 8606 |
9 times 11 equals 99. (Contributed by AV, 14-Jun-2021.) (Revised by AV,
6-Sep-2021.)
|
⊢ (9 · ;11) = ;99 |
|
Theorem | 9lt10 8607 |
9 is less than 10. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised
by AV, 8-Sep-2021.)
|
⊢ 9 < ;10 |
|
Theorem | 8lt10 8608 |
8 is less than 10. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised
by AV, 8-Sep-2021.)
|
⊢ 8 < ;10 |
|
Theorem | 7lt10 8609 |
7 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.)
(Revised by AV, 8-Sep-2021.)
|
⊢ 7 < ;10 |
|
Theorem | 6lt10 8610 |
6 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.)
(Revised by AV, 8-Sep-2021.)
|
⊢ 6 < ;10 |
|
Theorem | 5lt10 8611 |
5 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.)
(Revised by AV, 8-Sep-2021.)
|
⊢ 5 < ;10 |
|
Theorem | 4lt10 8612 |
4 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.)
(Revised by AV, 8-Sep-2021.)
|
⊢ 4 < ;10 |
|
Theorem | 3lt10 8613 |
3 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.)
(Revised by AV, 8-Sep-2021.)
|
⊢ 3 < ;10 |
|
Theorem | 2lt10 8614 |
2 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.)
(Revised by AV, 8-Sep-2021.)
|
⊢ 2 < ;10 |
|
Theorem | 1lt10 8615 |
1 is less than 10. (Contributed by NM, 7-Nov-2012.) (Revised by Mario
Carneiro, 9-Mar-2015.) (Revised by AV, 8-Sep-2021.)
|
⊢ 1 < ;10 |
|
Theorem | decbin0 8616 |
Decompose base 4 into base 2. (Contributed by Mario Carneiro,
18-Feb-2014.)
|
⊢ 𝐴 ∈
ℕ0 ⇒ ⊢ (4 · 𝐴) = (2 · (2 · 𝐴)) |
|
Theorem | decbin2 8617 |
Decompose base 4 into base 2. (Contributed by Mario Carneiro,
18-Feb-2014.)
|
⊢ 𝐴 ∈
ℕ0 ⇒ ⊢ ((4 · 𝐴) + 2) = (2 · ((2 · 𝐴) + 1)) |
|
Theorem | decbin3 8618 |
Decompose base 4 into base 2. (Contributed by Mario Carneiro,
18-Feb-2014.)
|
⊢ 𝐴 ∈
ℕ0 ⇒ ⊢ ((4 · 𝐴) + 3) = ((2 · ((2 · 𝐴) + 1)) + 1) |
|
3.4.10 Upper sets of integers
|
|
Syntax | cuz 8619 |
Extend class notation with the upper integer function.
Read "ℤ≥‘𝑀 " as "the set of integers
greater than or equal to
𝑀."
|
class ℤ≥ |
|
Definition | df-uz 8620* |
Define a function whose value at 𝑗 is the semi-infinite set of
contiguous integers starting at 𝑗, which we will also call the
upper integers starting at 𝑗. Read "ℤ≥‘𝑀 " as "the set
of integers greater than or equal to 𝑀." See uzval 8621 for its
value, uzssz 8638 for its relationship to ℤ, nnuz 8654 and nn0uz 8653 for
its relationships to ℕ and ℕ0, and eluz1 8623 and eluz2 8625 for
its membership relations. (Contributed by NM, 5-Sep-2005.)
|
⊢ ℤ≥ = (𝑗 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘}) |
|
Theorem | uzval 8621* |
The value of the upper integers function. (Contributed by NM,
5-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)
|
⊢ (𝑁 ∈ ℤ →
(ℤ≥‘𝑁) = {𝑘 ∈ ℤ ∣ 𝑁 ≤ 𝑘}) |
|
Theorem | uzf 8622 |
The domain and range of the upper integers function. (Contributed by
Scott Fenton, 8-Aug-2013.) (Revised by Mario Carneiro, 3-Nov-2013.)
|
⊢
ℤ≥:ℤ⟶𝒫
ℤ |
|
Theorem | eluz1 8623 |
Membership in the upper set of integers starting at 𝑀.
(Contributed by NM, 5-Sep-2005.)
|
⊢ (𝑀 ∈ ℤ → (𝑁 ∈ (ℤ≥‘𝑀) ↔ (𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁))) |
|
Theorem | eluzel2 8624 |
Implication of membership in an upper set of integers. (Contributed by
NM, 6-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)
|
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ) |
|
Theorem | eluz2 8625 |
Membership in an upper set of integers. We use the fact that a
function's value (under our function value definition) is empty outside
of its domain to show 𝑀 ∈ ℤ. (Contributed by NM,
5-Sep-2005.)
(Revised by Mario Carneiro, 3-Nov-2013.)
|
⊢ (𝑁 ∈ (ℤ≥‘𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁)) |
|
Theorem | eluz1i 8626 |
Membership in an upper set of integers. (Contributed by NM,
5-Sep-2005.)
|
⊢ 𝑀 ∈ ℤ
⇒ ⊢ (𝑁 ∈ (ℤ≥‘𝑀) ↔ (𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁)) |
|
Theorem | eluzuzle 8627 |
An integer in an upper set of integers is an element of an upper set of
integers with a smaller bound. (Contributed by Alexander van der Vekens,
17-Jun-2018.)
|
⊢ ((𝐵 ∈ ℤ ∧ 𝐵 ≤ 𝐴) → (𝐶 ∈ (ℤ≥‘𝐴) → 𝐶 ∈ (ℤ≥‘𝐵))) |
|
Theorem | eluzelz 8628 |
A member of an upper set of integers is an integer. (Contributed by NM,
6-Sep-2005.)
|
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℤ) |
|
Theorem | eluzelre 8629 |
A member of an upper set of integers is a real. (Contributed by Mario
Carneiro, 31-Aug-2013.)
|
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℝ) |
|
Theorem | eluzelcn 8630 |
A member of an upper set of integers is a complex number. (Contributed by
Glauco Siliprandi, 29-Jun-2017.)
|
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℂ) |
|
Theorem | eluzle 8631 |
Implication of membership in an upper set of integers. (Contributed by
NM, 6-Sep-2005.)
|
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ≤ 𝑁) |
|
Theorem | eluz 8632 |
Membership in an upper set of integers. (Contributed by NM,
2-Oct-2005.)
|
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ≥‘𝑀) ↔ 𝑀 ≤ 𝑁)) |
|
Theorem | uzid 8633 |
Membership of the least member in an upper set of integers. (Contributed
by NM, 2-Sep-2005.)
|
⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ≥‘𝑀)) |
|
Theorem | uzn0 8634 |
The upper integers are all nonempty. (Contributed by Mario Carneiro,
16-Jan-2014.)
|
⊢ (𝑀 ∈ ran ℤ≥ →
𝑀 ≠
∅) |
|
Theorem | uztrn 8635 |
Transitive law for sets of upper integers. (Contributed by NM,
20-Sep-2005.)
|
⊢ ((𝑀 ∈ (ℤ≥‘𝐾) ∧ 𝐾 ∈ (ℤ≥‘𝑁)) → 𝑀 ∈ (ℤ≥‘𝑁)) |
|
Theorem | uztrn2 8636 |
Transitive law for sets of upper integers. (Contributed by Mario
Carneiro, 26-Dec-2013.)
|
⊢ 𝑍 = (ℤ≥‘𝐾)
⇒ ⊢ ((𝑁 ∈ 𝑍 ∧ 𝑀 ∈ (ℤ≥‘𝑁)) → 𝑀 ∈ 𝑍) |
|
Theorem | uzneg 8637 |
Contraposition law for upper integers. (Contributed by NM,
28-Nov-2005.)
|
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → -𝑀 ∈
(ℤ≥‘-𝑁)) |
|
Theorem | uzssz 8638 |
An upper set of integers is a subset of all integers. (Contributed by
NM, 2-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)
|
⊢ (ℤ≥‘𝑀) ⊆
ℤ |
|
Theorem | uzss 8639 |
Subset relationship for two sets of upper integers. (Contributed by NM,
5-Sep-2005.)
|
⊢ (𝑁 ∈ (ℤ≥‘𝑀) →
(ℤ≥‘𝑁) ⊆
(ℤ≥‘𝑀)) |
|
Theorem | uztric 8640 |
Trichotomy of the ordering relation on integers, stated in terms of upper
integers. (Contributed by NM, 6-Jul-2005.) (Revised by Mario Carneiro,
25-Jun-2013.)
|
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ≥‘𝑀) ∨ 𝑀 ∈ (ℤ≥‘𝑁))) |
|
Theorem | uz11 8641 |
The upper integers function is one-to-one. (Contributed by NM,
12-Dec-2005.)
|
⊢ (𝑀 ∈ ℤ →
((ℤ≥‘𝑀) = (ℤ≥‘𝑁) ↔ 𝑀 = 𝑁)) |
|
Theorem | eluzp1m1 8642 |
Membership in the next upper set of integers. (Contributed by NM,
12-Sep-2005.)
|
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈
(ℤ≥‘(𝑀 + 1))) → (𝑁 − 1) ∈
(ℤ≥‘𝑀)) |
|
Theorem | eluzp1l 8643 |
Strict ordering implied by membership in the next upper set of integers.
(Contributed by NM, 12-Sep-2005.)
|
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈
(ℤ≥‘(𝑀 + 1))) → 𝑀 < 𝑁) |
|
Theorem | eluzp1p1 8644 |
Membership in the next upper set of integers. (Contributed by NM,
5-Oct-2005.)
|
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑁 + 1) ∈
(ℤ≥‘(𝑀 + 1))) |
|
Theorem | eluzaddi 8645 |
Membership in a later upper set of integers. (Contributed by Paul
Chapman, 22-Nov-2007.)
|
⊢ 𝑀 ∈ ℤ & ⊢ 𝐾 ∈
ℤ ⇒ ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑁 + 𝐾) ∈
(ℤ≥‘(𝑀 + 𝐾))) |
|
Theorem | eluzsubi 8646 |
Membership in an earlier upper set of integers. (Contributed by Paul
Chapman, 22-Nov-2007.)
|
⊢ 𝑀 ∈ ℤ & ⊢ 𝐾 ∈
ℤ ⇒ ⊢ (𝑁 ∈
(ℤ≥‘(𝑀 + 𝐾)) → (𝑁 − 𝐾) ∈
(ℤ≥‘𝑀)) |
|
Theorem | eluzadd 8647 |
Membership in a later upper set of integers. (Contributed by Jeff Madsen,
2-Sep-2009.)
|
⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝐾 ∈ ℤ) → (𝑁 + 𝐾) ∈
(ℤ≥‘(𝑀 + 𝐾))) |
|
Theorem | eluzsub 8648 |
Membership in an earlier upper set of integers. (Contributed by Jeff
Madsen, 2-Sep-2009.)
|
⊢ ((𝑀 ∈ ℤ ∧ 𝐾 ∈ ℤ ∧ 𝑁 ∈
(ℤ≥‘(𝑀 + 𝐾))) → (𝑁 − 𝐾) ∈
(ℤ≥‘𝑀)) |
|
Theorem | uzm1 8649 |
Choices for an element of an upper interval of integers. (Contributed by
Jeff Madsen, 2-Sep-2009.)
|
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑁 = 𝑀 ∨ (𝑁 − 1) ∈
(ℤ≥‘𝑀))) |
|
Theorem | uznn0sub 8650 |
The nonnegative difference of integers is a nonnegative integer.
(Contributed by NM, 4-Sep-2005.)
|
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑁 − 𝑀) ∈
ℕ0) |
|
Theorem | uzin 8651 |
Intersection of two upper intervals of integers. (Contributed by Mario
Carneiro, 24-Dec-2013.)
|
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) →
((ℤ≥‘𝑀) ∩ (ℤ≥‘𝑁)) =
(ℤ≥‘if(𝑀 ≤ 𝑁, 𝑁, 𝑀))) |
|
Theorem | uzp1 8652 |
Choices for an element of an upper interval of integers. (Contributed by
Jeff Madsen, 2-Sep-2009.)
|
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑁 = 𝑀 ∨ 𝑁 ∈
(ℤ≥‘(𝑀 + 1)))) |
|
Theorem | nn0uz 8653 |
Nonnegative integers expressed as an upper set of integers. (Contributed
by NM, 2-Sep-2005.)
|
⊢ ℕ0 =
(ℤ≥‘0) |
|
Theorem | nnuz 8654 |
Positive integers expressed as an upper set of integers. (Contributed by
NM, 2-Sep-2005.)
|
⊢ ℕ =
(ℤ≥‘1) |
|
Theorem | elnnuz 8655 |
A positive integer expressed as a member of an upper set of integers.
(Contributed by NM, 6-Jun-2006.)
|
⊢ (𝑁 ∈ ℕ ↔ 𝑁 ∈
(ℤ≥‘1)) |
|
Theorem | elnn0uz 8656 |
A nonnegative integer expressed as a member an upper set of integers.
(Contributed by NM, 6-Jun-2006.)
|
⊢ (𝑁 ∈ ℕ0 ↔ 𝑁 ∈
(ℤ≥‘0)) |
|
Theorem | eluz2nn 8657 |
An integer is greater than or equal to 2 is a positive integer.
(Contributed by AV, 3-Nov-2018.)
|
⊢ (𝐴 ∈ (ℤ≥‘2)
→ 𝐴 ∈
ℕ) |
|
Theorem | eluzge2nn0 8658 |
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 | uzuzle23 8659 |
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 8660 |
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 8661 |
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 8662 |
1 is an integer greater than or equal to 0. (Contributed by Alexander van
der Vekens, 8-Jun-2018.)
|
⊢ 1 ∈
(ℤ≥‘0) |
|
Theorem | 2eluzge0 8663 |
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 8664 |
2 is an integer greater than or equal to 1. (Contributed by Alexander van
der Vekens, 8-Jun-2018.)
|
⊢ 2 ∈
(ℤ≥‘1) |
|
Theorem | uznnssnn 8665 |
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 8666* |
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 8667* |
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 8668* |
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 8669* |
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 8670* |
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 8671 |
Second Peano postulate for an upper set of integers. (Contributed by NM,
7-Sep-2005.)
|
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑁 + 1) ∈
(ℤ≥‘𝑀)) |
|
Theorem | peano2uzs 8672 |
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 8673 |
Reversed second Peano axiom for upper integers. (Contributed by NM,
2-Jan-2006.)
|
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈
(ℤ≥‘(𝑀 + 1))) → 𝑁 ∈ (ℤ≥‘𝑀)) |
|
Theorem | uzaddcl 8674 |
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 8675 |
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 8676* |
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 8677* |
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 8676 or
uzind4ALT 8677 may be used; see comment for nnind 8055. (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 8678* |
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 8679* |
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 8678 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 8680* |
Induction on the upper integers that start at 𝑀. The first
hypothesis specifies the lower bound, the next four give us the
substitution instances we need, and the last two are the basis and the
induction step. (Contributed by NM, 4-Sep-2005.)
|
⊢ 𝑀 ∈ ℤ & ⊢ (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 8681* |
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 8682* |
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 8683* |
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 8698.
(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 8684* |
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 8683. (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 8685* |
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 | eluznn0 8686 |
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 8687 |
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 8688 |
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 8689 |
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 8690 |
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 8691 |
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 8692 |
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 8693 |
1 is not in (ℤ≥‘2).
(Contributed by Paul Chapman,
21-Nov-2012.)
|
⊢ ¬ 1 ∈
(ℤ≥‘2) |
|
Theorem | elnn1uz2 8694 |
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 8695 |
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 8696* |
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 8697 |
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 | ublbneg 8698* |
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 8683. (Contributed by
Paul Chapman, 21-Mar-2011.)
|
⊢ (∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴}𝑥 ≤ 𝑦) |
|
Theorem | eqreznegel 8699* |
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 8700* |
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}) |