Theorem List for Intuitionistic Logic Explorer - 9301-9400 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
|
Theorem | 8t7e56 9301 |
8 times 7 equals 56. (Contributed by Mario Carneiro, 19-Apr-2015.)
|
⊢ (8 · 7) = ;56 |
|
Theorem | 8t8e64 9302 |
8 times 8 equals 64. (Contributed by Mario Carneiro, 19-Apr-2015.)
|
⊢ (8 · 8) = ;64 |
|
Theorem | 9t2e18 9303 |
9 times 2 equals 18. (Contributed by Mario Carneiro, 19-Apr-2015.)
|
⊢ (9 · 2) = ;18 |
|
Theorem | 9t3e27 9304 |
9 times 3 equals 27. (Contributed by Mario Carneiro, 19-Apr-2015.)
|
⊢ (9 · 3) = ;27 |
|
Theorem | 9t4e36 9305 |
9 times 4 equals 36. (Contributed by Mario Carneiro, 19-Apr-2015.)
|
⊢ (9 · 4) = ;36 |
|
Theorem | 9t5e45 9306 |
9 times 5 equals 45. (Contributed by Mario Carneiro, 19-Apr-2015.)
|
⊢ (9 · 5) = ;45 |
|
Theorem | 9t6e54 9307 |
9 times 6 equals 54. (Contributed by Mario Carneiro, 19-Apr-2015.)
|
⊢ (9 · 6) = ;54 |
|
Theorem | 9t7e63 9308 |
9 times 7 equals 63. (Contributed by Mario Carneiro, 19-Apr-2015.)
|
⊢ (9 · 7) = ;63 |
|
Theorem | 9t8e72 9309 |
9 times 8 equals 72. (Contributed by Mario Carneiro, 19-Apr-2015.)
|
⊢ (9 · 8) = ;72 |
|
Theorem | 9t9e81 9310 |
9 times 9 equals 81. (Contributed by Mario Carneiro, 19-Apr-2015.)
|
⊢ (9 · 9) = ;81 |
|
Theorem | 9t11e99 9311 |
9 times 11 equals 99. (Contributed by AV, 14-Jun-2021.) (Revised by AV,
6-Sep-2021.)
|
⊢ (9 · ;11) = ;99 |
|
Theorem | 9lt10 9312 |
9 is less than 10. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised
by AV, 8-Sep-2021.)
|
⊢ 9 < ;10 |
|
Theorem | 8lt10 9313 |
8 is less than 10. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised
by AV, 8-Sep-2021.)
|
⊢ 8 < ;10 |
|
Theorem | 7lt10 9314 |
7 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.)
(Revised by AV, 8-Sep-2021.)
|
⊢ 7 < ;10 |
|
Theorem | 6lt10 9315 |
6 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.)
(Revised by AV, 8-Sep-2021.)
|
⊢ 6 < ;10 |
|
Theorem | 5lt10 9316 |
5 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.)
(Revised by AV, 8-Sep-2021.)
|
⊢ 5 < ;10 |
|
Theorem | 4lt10 9317 |
4 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.)
(Revised by AV, 8-Sep-2021.)
|
⊢ 4 < ;10 |
|
Theorem | 3lt10 9318 |
3 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.)
(Revised by AV, 8-Sep-2021.)
|
⊢ 3 < ;10 |
|
Theorem | 2lt10 9319 |
2 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.)
(Revised by AV, 8-Sep-2021.)
|
⊢ 2 < ;10 |
|
Theorem | 1lt10 9320 |
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 9321 |
Decompose base 4 into base 2. (Contributed by Mario Carneiro,
18-Feb-2014.)
|
⊢ 𝐴 ∈
ℕ0 ⇒ ⊢ (4 · 𝐴) = (2 · (2 · 𝐴)) |
|
Theorem | decbin2 9322 |
Decompose base 4 into base 2. (Contributed by Mario Carneiro,
18-Feb-2014.)
|
⊢ 𝐴 ∈
ℕ0 ⇒ ⊢ ((4 · 𝐴) + 2) = (2 · ((2 · 𝐴) + 1)) |
|
Theorem | decbin3 9323 |
Decompose base 4 into base 2. (Contributed by Mario Carneiro,
18-Feb-2014.)
|
⊢ 𝐴 ∈
ℕ0 ⇒ ⊢ ((4 · 𝐴) + 3) = ((2 · ((2 · 𝐴) + 1)) + 1) |
|
Theorem | halfthird 9324 |
Half minus a third. (Contributed by Scott Fenton, 8-Jul-2015.)
|
⊢ ((1 / 2) − (1 / 3)) = (1 /
6) |
|
Theorem | 5recm6rec 9325 |
One fifth minus one sixth. (Contributed by Scott Fenton, 9-Jan-2017.)
|
⊢ ((1 / 5) − (1 / 6)) = (1 / ;30) |
|
4.4.11 Upper sets of integers
|
|
Syntax | cuz 9326 |
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 9327* |
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 9328 for its
value, uzssz 9345 for its relationship to ℤ, nnuz 9361 and nn0uz 9360 for
its relationships to ℕ and ℕ0, and eluz1 9330 and eluz2 9332 for
its membership relations. (Contributed by NM, 5-Sep-2005.)
|
⊢ ℤ≥ = (𝑗 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘}) |
|
Theorem | uzval 9328* |
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 9329 |
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 9330 |
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 9331 |
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 9332 |
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 9333 |
Membership in an upper set of integers. (Contributed by NM,
5-Sep-2005.)
|
⊢ 𝑀 ∈ ℤ
⇒ ⊢ (𝑁 ∈ (ℤ≥‘𝑀) ↔ (𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁)) |
|
Theorem | eluzuzle 9334 |
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 9335 |
A member of an upper set of integers is an integer. (Contributed by NM,
6-Sep-2005.)
|
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℤ) |
|
Theorem | eluzelre 9336 |
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 9337 |
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 9338 |
Implication of membership in an upper set of integers. (Contributed by
NM, 6-Sep-2005.)
|
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ≤ 𝑁) |
|
Theorem | eluz 9339 |
Membership in an upper set of integers. (Contributed by NM,
2-Oct-2005.)
|
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ≥‘𝑀) ↔ 𝑀 ≤ 𝑁)) |
|
Theorem | uzid 9340 |
Membership of the least member in an upper set of integers. (Contributed
by NM, 2-Sep-2005.)
|
⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ≥‘𝑀)) |
|
Theorem | uzn0 9341 |
The upper integers are all nonempty. (Contributed by Mario Carneiro,
16-Jan-2014.)
|
⊢ (𝑀 ∈ ran ℤ≥ →
𝑀 ≠
∅) |
|
Theorem | uztrn 9342 |
Transitive law for sets of upper integers. (Contributed by NM,
20-Sep-2005.)
|
⊢ ((𝑀 ∈ (ℤ≥‘𝐾) ∧ 𝐾 ∈ (ℤ≥‘𝑁)) → 𝑀 ∈ (ℤ≥‘𝑁)) |
|
Theorem | uztrn2 9343 |
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 9344 |
Contraposition law for upper integers. (Contributed by NM,
28-Nov-2005.)
|
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → -𝑀 ∈
(ℤ≥‘-𝑁)) |
|
Theorem | uzssz 9345 |
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 9346 |
Subset relationship for two sets of upper integers. (Contributed by NM,
5-Sep-2005.)
|
⊢ (𝑁 ∈ (ℤ≥‘𝑀) →
(ℤ≥‘𝑁) ⊆
(ℤ≥‘𝑀)) |
|
Theorem | uztric 9347 |
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 9348 |
The upper integers function is one-to-one. (Contributed by NM,
12-Dec-2005.)
|
⊢ (𝑀 ∈ ℤ →
((ℤ≥‘𝑀) = (ℤ≥‘𝑁) ↔ 𝑀 = 𝑁)) |
|
Theorem | eluzp1m1 9349 |
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 9350 |
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 9351 |
Membership in the next upper set of integers. (Contributed by NM,
5-Oct-2005.)
|
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑁 + 1) ∈
(ℤ≥‘(𝑀 + 1))) |
|
Theorem | eluzaddi 9352 |
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 9353 |
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 9354 |
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 9355 |
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 9356 |
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 9357 |
The nonnegative difference of integers is a nonnegative integer.
(Contributed by NM, 4-Sep-2005.)
|
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑁 − 𝑀) ∈
ℕ0) |
|
Theorem | uzin 9358 |
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 9359 |
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 9360 |
Nonnegative integers expressed as an upper set of integers. (Contributed
by NM, 2-Sep-2005.)
|
⊢ ℕ0 =
(ℤ≥‘0) |
|
Theorem | nnuz 9361 |
Positive integers expressed as an upper set of integers. (Contributed by
NM, 2-Sep-2005.)
|
⊢ ℕ =
(ℤ≥‘1) |
|
Theorem | elnnuz 9362 |
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 9363 |
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 9364 |
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 9365 |
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 9366 |
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 9367 |
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 9368 |
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 9369 |
1 is an integer greater than or equal to 0. (Contributed by Alexander van
der Vekens, 8-Jun-2018.)
|
⊢ 1 ∈
(ℤ≥‘0) |
|
Theorem | 2eluzge0 9370 |
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 9371 |
2 is an integer greater than or equal to 1. (Contributed by Alexander van
der Vekens, 8-Jun-2018.)
|
⊢ 2 ∈
(ℤ≥‘1) |
|
Theorem | uznnssnn 9372 |
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 9373* |
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 9374* |
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 9375* |
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 9376* |
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 9377* |
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 9378 |
Second Peano postulate for an upper set of integers. (Contributed by NM,
7-Sep-2005.)
|
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑁 + 1) ∈
(ℤ≥‘𝑀)) |
|
Theorem | peano2uzs 9379 |
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 9380 |
Reversed second Peano axiom for upper integers. (Contributed by NM,
2-Jan-2006.)
|
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈
(ℤ≥‘(𝑀 + 1))) → 𝑁 ∈ (ℤ≥‘𝑀)) |
|
Theorem | uzaddcl 9381 |
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 9382 |
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 9383* |
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 9384* |
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 9383 or
uzind4ALT 9384 may be used; see comment for nnind 8736. (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 9385* |
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 9386* |
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 9385 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 9387* |
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 9383
assuming that 𝜓 holds unconditionally. Notice that
𝑁
∈ (ℤ≥‘𝑀) implies that the lower bound 𝑀 is an
integer
(𝑀
∈ ℤ, see eluzel2 9331). (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 9388* |
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 9389* |
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 9390* |
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 9405.
(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 9391* |
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 9390. (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 9392* |
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 9393 |
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 9394 |
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 9395 |
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 9396 |
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 9397 |
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 9398 |
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 9399 |
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 9400 |
1 is not in (ℤ≥‘2).
(Contributed by Paul Chapman,
21-Nov-2012.)
|
⊢ ¬ 1 ∈
(ℤ≥‘2) |