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Theorem List for Intuitionistic Logic Explorer - 9201-9300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem8t3e24 9201 8 times 3 equals 24. (Contributed by Mario Carneiro, 19-Apr-2015.)
(8 · 3) = 24
 
Theorem8t4e32 9202 8 times 4 equals 32. (Contributed by Mario Carneiro, 19-Apr-2015.)
(8 · 4) = 32
 
Theorem8t5e40 9203 8 times 5 equals 40. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
(8 · 5) = 40
 
Theorem8t6e48 9204 8 times 6 equals 48. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
(8 · 6) = 48
 
Theorem8t7e56 9205 8 times 7 equals 56. (Contributed by Mario Carneiro, 19-Apr-2015.)
(8 · 7) = 56
 
Theorem8t8e64 9206 8 times 8 equals 64. (Contributed by Mario Carneiro, 19-Apr-2015.)
(8 · 8) = 64
 
Theorem9t2e18 9207 9 times 2 equals 18. (Contributed by Mario Carneiro, 19-Apr-2015.)
(9 · 2) = 18
 
Theorem9t3e27 9208 9 times 3 equals 27. (Contributed by Mario Carneiro, 19-Apr-2015.)
(9 · 3) = 27
 
Theorem9t4e36 9209 9 times 4 equals 36. (Contributed by Mario Carneiro, 19-Apr-2015.)
(9 · 4) = 36
 
Theorem9t5e45 9210 9 times 5 equals 45. (Contributed by Mario Carneiro, 19-Apr-2015.)
(9 · 5) = 45
 
Theorem9t6e54 9211 9 times 6 equals 54. (Contributed by Mario Carneiro, 19-Apr-2015.)
(9 · 6) = 54
 
Theorem9t7e63 9212 9 times 7 equals 63. (Contributed by Mario Carneiro, 19-Apr-2015.)
(9 · 7) = 63
 
Theorem9t8e72 9213 9 times 8 equals 72. (Contributed by Mario Carneiro, 19-Apr-2015.)
(9 · 8) = 72
 
Theorem9t9e81 9214 9 times 9 equals 81. (Contributed by Mario Carneiro, 19-Apr-2015.)
(9 · 9) = 81
 
Theorem9t11e99 9215 9 times 11 equals 99. (Contributed by AV, 14-Jun-2021.) (Revised by AV, 6-Sep-2021.)
(9 · 11) = 99
 
Theorem9lt10 9216 9 is less than 10. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by AV, 8-Sep-2021.)
9 < 10
 
Theorem8lt10 9217 8 is less than 10. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by AV, 8-Sep-2021.)
8 < 10
 
Theorem7lt10 9218 7 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.)
7 < 10
 
Theorem6lt10 9219 6 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.)
6 < 10
 
Theorem5lt10 9220 5 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.)
5 < 10
 
Theorem4lt10 9221 4 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.)
4 < 10
 
Theorem3lt10 9222 3 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.)
3 < 10
 
Theorem2lt10 9223 2 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.)
2 < 10
 
Theorem1lt10 9224 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
 
Theoremdecbin0 9225 Decompose base 4 into base 2. (Contributed by Mario Carneiro, 18-Feb-2014.)
𝐴 ∈ ℕ0       (4 · 𝐴) = (2 · (2 · 𝐴))
 
Theoremdecbin2 9226 Decompose base 4 into base 2. (Contributed by Mario Carneiro, 18-Feb-2014.)
𝐴 ∈ ℕ0       ((4 · 𝐴) + 2) = (2 · ((2 · 𝐴) + 1))
 
Theoremdecbin3 9227 Decompose base 4 into base 2. (Contributed by Mario Carneiro, 18-Feb-2014.)
𝐴 ∈ ℕ0       ((4 · 𝐴) + 3) = ((2 · ((2 · 𝐴) + 1)) + 1)
 
4.4.11  Upper sets of integers
 
Syntaxcuz 9228 Extend class notation with the upper integer function. Read "𝑀 " as "the set of integers greater than or equal to 𝑀."
class
 
Definitiondf-uz 9229* 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 "𝑀 " as "the set of integers greater than or equal to 𝑀." See uzval 9230 for its value, uzssz 9247 for its relationship to , nnuz 9263 and nn0uz 9262 for its relationships to and 0, and eluz1 9232 and eluz2 9234 for its membership relations. (Contributed by NM, 5-Sep-2005.)
= (𝑗 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ 𝑗𝑘})
 
Theoremuzval 9230* The value of the upper integers function. (Contributed by NM, 5-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)
(𝑁 ∈ ℤ → (ℤ𝑁) = {𝑘 ∈ ℤ ∣ 𝑁𝑘})
 
Theoremuzf 9231 The domain and range of the upper integers function. (Contributed by Scott Fenton, 8-Aug-2013.) (Revised by Mario Carneiro, 3-Nov-2013.)
:ℤ⟶𝒫 ℤ
 
Theoremeluz1 9232 Membership in the upper set of integers starting at 𝑀. (Contributed by NM, 5-Sep-2005.)
(𝑀 ∈ ℤ → (𝑁 ∈ (ℤ𝑀) ↔ (𝑁 ∈ ℤ ∧ 𝑀𝑁)))
 
Theoremeluzel2 9233 Implication of membership in an upper set of integers. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)
(𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ ℤ)
 
Theoremeluz2 9234 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 ∈ (ℤ𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀𝑁))
 
Theoremeluz1i 9235 Membership in an upper set of integers. (Contributed by NM, 5-Sep-2005.)
𝑀 ∈ ℤ       (𝑁 ∈ (ℤ𝑀) ↔ (𝑁 ∈ ℤ ∧ 𝑀𝑁))
 
Theoremeluzuzle 9236 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 ∧ 𝐵𝐴) → (𝐶 ∈ (ℤ𝐴) → 𝐶 ∈ (ℤ𝐵)))
 
Theoremeluzelz 9237 A member of an upper set of integers is an integer. (Contributed by NM, 6-Sep-2005.)
(𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ ℤ)
 
Theoremeluzelre 9238 A member of an upper set of integers is a real. (Contributed by Mario Carneiro, 31-Aug-2013.)
(𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ ℝ)
 
Theoremeluzelcn 9239 A member of an upper set of integers is a complex number. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
(𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ ℂ)
 
Theoremeluzle 9240 Implication of membership in an upper set of integers. (Contributed by NM, 6-Sep-2005.)
(𝑁 ∈ (ℤ𝑀) → 𝑀𝑁)
 
Theoremeluz 9241 Membership in an upper set of integers. (Contributed by NM, 2-Oct-2005.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ𝑀) ↔ 𝑀𝑁))
 
Theoremuzid 9242 Membership of the least member in an upper set of integers. (Contributed by NM, 2-Sep-2005.)
(𝑀 ∈ ℤ → 𝑀 ∈ (ℤ𝑀))
 
Theoremuzn0 9243 The upper integers are all nonempty. (Contributed by Mario Carneiro, 16-Jan-2014.)
(𝑀 ∈ ran ℤ𝑀 ≠ ∅)
 
Theoremuztrn 9244 Transitive law for sets of upper integers. (Contributed by NM, 20-Sep-2005.)
((𝑀 ∈ (ℤ𝐾) ∧ 𝐾 ∈ (ℤ𝑁)) → 𝑀 ∈ (ℤ𝑁))
 
Theoremuztrn2 9245 Transitive law for sets of upper integers. (Contributed by Mario Carneiro, 26-Dec-2013.)
𝑍 = (ℤ𝐾)       ((𝑁𝑍𝑀 ∈ (ℤ𝑁)) → 𝑀𝑍)
 
Theoremuzneg 9246 Contraposition law for upper integers. (Contributed by NM, 28-Nov-2005.)
(𝑁 ∈ (ℤ𝑀) → -𝑀 ∈ (ℤ‘-𝑁))
 
Theoremuzssz 9247 An upper set of integers is a subset of all integers. (Contributed by NM, 2-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)
(ℤ𝑀) ⊆ ℤ
 
Theoremuzss 9248 Subset relationship for two sets of upper integers. (Contributed by NM, 5-Sep-2005.)
(𝑁 ∈ (ℤ𝑀) → (ℤ𝑁) ⊆ (ℤ𝑀))
 
Theoremuztric 9249 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 ∈ (ℤ𝑀) ∨ 𝑀 ∈ (ℤ𝑁)))
 
Theoremuz11 9250 The upper integers function is one-to-one. (Contributed by NM, 12-Dec-2005.)
(𝑀 ∈ ℤ → ((ℤ𝑀) = (ℤ𝑁) ↔ 𝑀 = 𝑁))
 
Theoremeluzp1m1 9251 Membership in the next upper set of integers. (Contributed by NM, 12-Sep-2005.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ (ℤ‘(𝑀 + 1))) → (𝑁 − 1) ∈ (ℤ𝑀))
 
Theoremeluzp1l 9252 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)
 
Theoremeluzp1p1 9253 Membership in the next upper set of integers. (Contributed by NM, 5-Oct-2005.)
(𝑁 ∈ (ℤ𝑀) → (𝑁 + 1) ∈ (ℤ‘(𝑀 + 1)))
 
Theoremeluzaddi 9254 Membership in a later upper set of integers. (Contributed by Paul Chapman, 22-Nov-2007.)
𝑀 ∈ ℤ    &   𝐾 ∈ ℤ       (𝑁 ∈ (ℤ𝑀) → (𝑁 + 𝐾) ∈ (ℤ‘(𝑀 + 𝐾)))
 
Theoremeluzsubi 9255 Membership in an earlier upper set of integers. (Contributed by Paul Chapman, 22-Nov-2007.)
𝑀 ∈ ℤ    &   𝐾 ∈ ℤ       (𝑁 ∈ (ℤ‘(𝑀 + 𝐾)) → (𝑁𝐾) ∈ (ℤ𝑀))
 
Theoremeluzadd 9256 Membership in a later upper set of integers. (Contributed by Jeff Madsen, 2-Sep-2009.)
((𝑁 ∈ (ℤ𝑀) ∧ 𝐾 ∈ ℤ) → (𝑁 + 𝐾) ∈ (ℤ‘(𝑀 + 𝐾)))
 
Theoremeluzsub 9257 Membership in an earlier upper set of integers. (Contributed by Jeff Madsen, 2-Sep-2009.)
((𝑀 ∈ ℤ ∧ 𝐾 ∈ ℤ ∧ 𝑁 ∈ (ℤ‘(𝑀 + 𝐾))) → (𝑁𝐾) ∈ (ℤ𝑀))
 
Theoremuzm1 9258 Choices for an element of an upper interval of integers. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝑁 ∈ (ℤ𝑀) → (𝑁 = 𝑀 ∨ (𝑁 − 1) ∈ (ℤ𝑀)))
 
Theoremuznn0sub 9259 The nonnegative difference of integers is a nonnegative integer. (Contributed by NM, 4-Sep-2005.)
(𝑁 ∈ (ℤ𝑀) → (𝑁𝑀) ∈ ℕ0)
 
Theoremuzin 9260 Intersection of two upper intervals of integers. (Contributed by Mario Carneiro, 24-Dec-2013.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((ℤ𝑀) ∩ (ℤ𝑁)) = (ℤ‘if(𝑀𝑁, 𝑁, 𝑀)))
 
Theoremuzp1 9261 Choices for an element of an upper interval of integers. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝑁 ∈ (ℤ𝑀) → (𝑁 = 𝑀𝑁 ∈ (ℤ‘(𝑀 + 1))))
 
Theoremnn0uz 9262 Nonnegative integers expressed as an upper set of integers. (Contributed by NM, 2-Sep-2005.)
0 = (ℤ‘0)
 
Theoremnnuz 9263 Positive integers expressed as an upper set of integers. (Contributed by NM, 2-Sep-2005.)
ℕ = (ℤ‘1)
 
Theoremelnnuz 9264 A positive integer expressed as a member of an upper set of integers. (Contributed by NM, 6-Jun-2006.)
(𝑁 ∈ ℕ ↔ 𝑁 ∈ (ℤ‘1))
 
Theoremelnn0uz 9265 A nonnegative integer expressed as a member an upper set of integers. (Contributed by NM, 6-Jun-2006.)
(𝑁 ∈ ℕ0𝑁 ∈ (ℤ‘0))
 
Theoremeluz2nn 9266 An integer is greater than or equal to 2 is a positive integer. (Contributed by AV, 3-Nov-2018.)
(𝐴 ∈ (ℤ‘2) → 𝐴 ∈ ℕ)
 
Theoremeluzge2nn0 9267 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)
 
Theoremuzuzle23 9268 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))
 
Theoremeluzge3nn 9269 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)
 
Theoremuz3m2nn 9270 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)
 
Theorem1eluzge0 9271 1 is an integer greater than or equal to 0. (Contributed by Alexander van der Vekens, 8-Jun-2018.)
1 ∈ (ℤ‘0)
 
Theorem2eluzge0 9272 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)
 
Theorem2eluzge1 9273 2 is an integer greater than or equal to 1. (Contributed by Alexander van der Vekens, 8-Jun-2018.)
2 ∈ (ℤ‘1)
 
Theoremuznnssnn 9274 The upper integers starting from a natural are a subset of the naturals. (Contributed by Scott Fenton, 29-Jun-2013.)
(𝑁 ∈ ℕ → (ℤ𝑁) ⊆ ℕ)
 
Theoremraluz 9275* Restricted universal quantification in an upper set of integers. (Contributed by NM, 9-Sep-2005.)
(𝑀 ∈ ℤ → (∀𝑛 ∈ (ℤ𝑀)𝜑 ↔ ∀𝑛 ∈ ℤ (𝑀𝑛𝜑)))
 
Theoremraluz2 9276* Restricted universal quantification in an upper set of integers. (Contributed by NM, 9-Sep-2005.)
(∀𝑛 ∈ (ℤ𝑀)𝜑 ↔ (𝑀 ∈ ℤ → ∀𝑛 ∈ ℤ (𝑀𝑛𝜑)))
 
Theoremrexuz 9277* Restricted existential quantification in an upper set of integers. (Contributed by NM, 9-Sep-2005.)
(𝑀 ∈ ℤ → (∃𝑛 ∈ (ℤ𝑀)𝜑 ↔ ∃𝑛 ∈ ℤ (𝑀𝑛𝜑)))
 
Theoremrexuz2 9278* Restricted existential quantification in an upper set of integers. (Contributed by NM, 9-Sep-2005.)
(∃𝑛 ∈ (ℤ𝑀)𝜑 ↔ (𝑀 ∈ ℤ ∧ ∃𝑛 ∈ ℤ (𝑀𝑛𝜑)))
 
Theorem2rexuz 9279* Double existential quantification in an upper set of integers. (Contributed by NM, 3-Nov-2005.)
(∃𝑚𝑛 ∈ (ℤ𝑚)𝜑 ↔ ∃𝑚 ∈ ℤ ∃𝑛 ∈ ℤ (𝑚𝑛𝜑))
 
Theorempeano2uz 9280 Second Peano postulate for an upper set of integers. (Contributed by NM, 7-Sep-2005.)
(𝑁 ∈ (ℤ𝑀) → (𝑁 + 1) ∈ (ℤ𝑀))
 
Theorempeano2uzs 9281 Second Peano postulate for an upper set of integers. (Contributed by Mario Carneiro, 26-Dec-2013.)
𝑍 = (ℤ𝑀)       (𝑁𝑍 → (𝑁 + 1) ∈ 𝑍)
 
Theorempeano2uzr 9282 Reversed second Peano axiom for upper integers. (Contributed by NM, 2-Jan-2006.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ (ℤ‘(𝑀 + 1))) → 𝑁 ∈ (ℤ𝑀))
 
Theoremuzaddcl 9283 Addition closure law for an upper set of integers. (Contributed by NM, 4-Jun-2006.)
((𝑁 ∈ (ℤ𝑀) ∧ 𝐾 ∈ ℕ0) → (𝑁 + 𝐾) ∈ (ℤ𝑀))
 
Theoremnn0pzuz 9284 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 ∈ ℕ0𝑍 ∈ ℤ) → (𝑁 + 𝑍) ∈ (ℤ𝑍))
 
Theoremuzind4 9285* 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 → (𝜑𝜓))    &   (𝑗 = 𝑘 → (𝜑𝜒))    &   (𝑗 = (𝑘 + 1) → (𝜑𝜃))    &   (𝑗 = 𝑁 → (𝜑𝜏))    &   (𝑀 ∈ ℤ → 𝜓)    &   (𝑘 ∈ (ℤ𝑀) → (𝜒𝜃))       (𝑁 ∈ (ℤ𝑀) → 𝜏)
 
Theoremuzind4ALT 9286* 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 9285 or uzind4ALT 9286 may be used; see comment for nnind 8646. (Contributed by NM, 7-Sep-2005.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝑀 ∈ ℤ → 𝜓)    &   (𝑘 ∈ (ℤ𝑀) → (𝜒𝜃))    &   (𝑗 = 𝑀 → (𝜑𝜓))    &   (𝑗 = 𝑘 → (𝜑𝜒))    &   (𝑗 = (𝑘 + 1) → (𝜑𝜃))    &   (𝑗 = 𝑁 → (𝜑𝜏))       (𝑁 ∈ (ℤ𝑀) → 𝜏)
 
Theoremuzind4s 9287* 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 ∈ (ℤ𝑀) → (𝜑[(𝑘 + 1) / 𝑘]𝜑))       (𝑁 ∈ (ℤ𝑀) → [𝑁 / 𝑘]𝜑)
 
Theoremuzind4s2 9288* 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 9287 when 𝑗 and 𝑘 must be distinct in [(𝑘 + 1) / 𝑗]𝜑. (Contributed by NM, 16-Nov-2005.)
(𝑀 ∈ ℤ → [𝑀 / 𝑗]𝜑)    &   (𝑘 ∈ (ℤ𝑀) → ([𝑘 / 𝑗]𝜑[(𝑘 + 1) / 𝑗]𝜑))       (𝑁 ∈ (ℤ𝑀) → [𝑁 / 𝑗]𝜑)
 
Theoremuzind4i 9289* 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 → (𝜑𝜓))    &   (𝑗 = 𝑘 → (𝜑𝜒))    &   (𝑗 = (𝑘 + 1) → (𝜑𝜃))    &   (𝑗 = 𝑁 → (𝜑𝜏))    &   𝜓    &   (𝑘 ∈ (ℤ𝑀) → (𝜒𝜃))       (𝑁 ∈ (ℤ𝑀) → 𝜏)
 
Theoremindstr 9290* Strong Mathematical Induction for positive integers (inference schema). (Contributed by NM, 17-Aug-2001.)
(𝑥 = 𝑦 → (𝜑𝜓))    &   (𝑥 ∈ ℕ → (∀𝑦 ∈ ℕ (𝑦 < 𝑥𝜓) → 𝜑))       (𝑥 ∈ ℕ → 𝜑)
 
Theoreminfrenegsupex 9291* 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 (∀𝑦𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)))    &   (𝜑𝐴 ⊆ ℝ)       (𝜑 → inf(𝐴, ℝ, < ) = -sup({𝑧 ∈ ℝ ∣ -𝑧𝐴}, ℝ, < ))
 
Theoremsupinfneg 9292* 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 9307. (Contributed by Jim Kingdon, 15-Jan-2022.)
(𝜑 → ∃𝑥 ∈ ℝ (∀𝑦𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧)))    &   (𝜑𝐴 ⊆ ℝ)       (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴}𝑧 < 𝑦)))
 
Theoreminfsupneg 9293* 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 9292. (Contributed by Jim Kingdon, 15-Jan-2022.)
(𝜑 → ∃𝑥 ∈ ℝ (∀𝑦𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)))    &   (𝜑𝐴 ⊆ ℝ)       (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴}𝑦 < 𝑧)))
 
Theoremsupminfex 9294* 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 (∀𝑦𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧)))    &   (𝜑𝐴 ⊆ ℝ)       (𝜑 → sup(𝐴, ℝ, < ) = -inf({𝑤 ∈ ℝ ∣ -𝑤𝐴}, ℝ, < ))
 
Theoremeluznn0 9295 Membership in a nonnegative upper set of integers implies membership in 0. (Contributed by Paul Chapman, 22-Jun-2011.)
((𝑁 ∈ ℕ0𝑀 ∈ (ℤ𝑁)) → 𝑀 ∈ ℕ0)
 
Theoremeluznn 9296 Membership in a positive upper set of integers implies membership in . (Contributed by JJ, 1-Oct-2018.)
((𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ𝑁)) → 𝑀 ∈ ℕ)
 
Theoremeluz2b1 9297 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))
 
Theoremeluz2gt1 9298 An integer greater than or equal to 2 is greater than 1. (Contributed by AV, 24-May-2020.)
(𝑁 ∈ (ℤ‘2) → 1 < 𝑁)
 
Theoremeluz2b2 9299 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))
 
Theoremeluz2b3 9300 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))
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