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Theorem List for Intuitionistic Logic Explorer - 8201-8300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem8p5e13 8201 8 + 5 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.)
(8 + 5) = 13
 
Theorem8p6e14 8202 8 + 6 = 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
(8 + 6) = 14
 
Theorem8p7e15 8203 8 + 7 = 15. (Contributed by Mario Carneiro, 19-Apr-2015.)
(8 + 7) = 15
 
Theorem8p8e16 8204 8 + 8 = 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
(8 + 8) = 16
 
Theorem9p2e11 8205 9 + 2 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.)
(9 + 2) = 11
 
Theorem9p3e12 8206 9 + 3 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
(9 + 3) = 12
 
Theorem9p4e13 8207 9 + 4 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.)
(9 + 4) = 13
 
Theorem9p5e14 8208 9 + 5 = 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
(9 + 5) = 14
 
Theorem9p6e15 8209 9 + 6 = 15. (Contributed by Mario Carneiro, 19-Apr-2015.)
(9 + 6) = 15
 
Theorem9p7e16 8210 9 + 7 = 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
(9 + 7) = 16
 
Theorem9p8e17 8211 9 + 8 = 17. (Contributed by Mario Carneiro, 19-Apr-2015.)
(9 + 8) = 17
 
Theorem9p9e18 8212 9 + 9 = 18. (Contributed by Mario Carneiro, 19-Apr-2015.)
(9 + 9) = 18
 
Theorem10p10e20 8213 10 + 10 = 20. (Contributed by Mario Carneiro, 19-Apr-2015.)
(10 + 10) = 20
 
Theorem4t3lem 8214 Lemma for 4t3e12 8215 and related theorems. (Contributed by Mario Carneiro, 19-Apr-2015.)
A 0    &   B 0    &   𝐶 = (B + 1)    &   (A · B) = 𝐷    &   (𝐷 + A) = 𝐸       (A · 𝐶) = 𝐸
 
Theorem4t3e12 8215 4 times 3 equals 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
(4 · 3) = 12
 
Theorem4t4e16 8216 4 times 4 equals 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
(4 · 4) = 16
 
Theorem5t3e15 8217 5 times 3 equals 15. (Contributed by Mario Carneiro, 19-Apr-2015.)
(5 · 3) = 15
 
Theorem5t4e20 8218 5 times 4 equals 20. (Contributed by Mario Carneiro, 19-Apr-2015.)
(5 · 4) = 20
 
Theorem5t5e25 8219 5 times 5 equals 25. (Contributed by Mario Carneiro, 19-Apr-2015.)
(5 · 5) = 25
 
Theorem6t2e12 8220 6 times 2 equals 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
(6 · 2) = 12
 
Theorem6t3e18 8221 6 times 3 equals 18. (Contributed by Mario Carneiro, 19-Apr-2015.)
(6 · 3) = 18
 
Theorem6t4e24 8222 6 times 4 equals 24. (Contributed by Mario Carneiro, 19-Apr-2015.)
(6 · 4) = 24
 
Theorem6t5e30 8223 6 times 5 equals 30. (Contributed by Mario Carneiro, 19-Apr-2015.)
(6 · 5) = 30
 
Theorem6t6e36 8224 6 times 6 equals 36. (Contributed by Mario Carneiro, 19-Apr-2015.)
(6 · 6) = 36
 
Theorem7t2e14 8225 7 times 2 equals 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
(7 · 2) = 14
 
Theorem7t3e21 8226 7 times 3 equals 21. (Contributed by Mario Carneiro, 19-Apr-2015.)
(7 · 3) = 21
 
Theorem7t4e28 8227 7 times 4 equals 28. (Contributed by Mario Carneiro, 19-Apr-2015.)
(7 · 4) = 28
 
Theorem7t5e35 8228 7 times 5 equals 35. (Contributed by Mario Carneiro, 19-Apr-2015.)
(7 · 5) = 35
 
Theorem7t6e42 8229 7 times 6 equals 42. (Contributed by Mario Carneiro, 19-Apr-2015.)
(7 · 6) = 42
 
Theorem7t7e49 8230 7 times 7 equals 49. (Contributed by Mario Carneiro, 19-Apr-2015.)
(7 · 7) = 49
 
Theorem8t2e16 8231 8 times 2 equals 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
(8 · 2) = 16
 
Theorem8t3e24 8232 8 times 3 equals 24. (Contributed by Mario Carneiro, 19-Apr-2015.)
(8 · 3) = 24
 
Theorem8t4e32 8233 8 times 4 equals 32. (Contributed by Mario Carneiro, 19-Apr-2015.)
(8 · 4) = 32
 
Theorem8t5e40 8234 8 times 5 equals 40. (Contributed by Mario Carneiro, 19-Apr-2015.)
(8 · 5) = 40
 
Theorem8t6e48 8235 8 times 6 equals 48. (Contributed by Mario Carneiro, 19-Apr-2015.)
(8 · 6) = 48
 
Theorem8t7e56 8236 8 times 7 equals 56. (Contributed by Mario Carneiro, 19-Apr-2015.)
(8 · 7) = 56
 
Theorem8t8e64 8237 8 times 8 equals 64. (Contributed by Mario Carneiro, 19-Apr-2015.)
(8 · 8) = 64
 
Theorem9t2e18 8238 9 times 2 equals 18. (Contributed by Mario Carneiro, 19-Apr-2015.)
(9 · 2) = 18
 
Theorem9t3e27 8239 9 times 3 equals 27. (Contributed by Mario Carneiro, 19-Apr-2015.)
(9 · 3) = 27
 
Theorem9t4e36 8240 9 times 4 equals 36. (Contributed by Mario Carneiro, 19-Apr-2015.)
(9 · 4) = 36
 
Theorem9t5e45 8241 9 times 5 equals 45. (Contributed by Mario Carneiro, 19-Apr-2015.)
(9 · 5) = 45
 
Theorem9t6e54 8242 9 times 6 equals 54. (Contributed by Mario Carneiro, 19-Apr-2015.)
(9 · 6) = 54
 
Theorem9t7e63 8243 9 times 7 equals 63. (Contributed by Mario Carneiro, 19-Apr-2015.)
(9 · 7) = 63
 
Theorem9t8e72 8244 9 times 8 equals 72. (Contributed by Mario Carneiro, 19-Apr-2015.)
(9 · 8) = 72
 
Theorem9t9e81 8245 9 times 9 equals 81. (Contributed by Mario Carneiro, 19-Apr-2015.)
(9 · 9) = 81
 
Theoremdecbin0 8246 Decompose base 4 into base 2. (Contributed by Mario Carneiro, 18-Feb-2014.)
A 0       (4 · A) = (2 · (2 · A))
 
Theoremdecbin2 8247 Decompose base 4 into base 2. (Contributed by Mario Carneiro, 18-Feb-2014.)
A 0       ((4 · A) + 2) = (2 · ((2 · A) + 1))
 
Theoremdecbin3 8248 Decompose base 4 into base 2. (Contributed by Mario Carneiro, 18-Feb-2014.)
A 0       ((4 · A) + 3) = ((2 · ((2 · A) + 1)) + 1)
 
3.4.10  Upper sets of integers
 
Syntaxcuz 8249 Extend class notation with the upper integer function. Read "𝑀 " as "the set of integers greater than or equal to 𝑀."
class
 
Definitiondf-uz 8250* 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 8251 for its value, uzssz 8268 for its relationship to , nnuz 8284 and nn0uz 8283 for its relationships to and 0, and eluz1 8253 and eluz2 8255 for its membership relations. (Contributed by NM, 5-Sep-2005.)
= (𝑗 ℤ ↦ {𝑘 ℤ ∣ 𝑗𝑘})
 
Theoremuzval 8251* 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 8252 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 8253 Membership in the upper set of integers starting at 𝑀. (Contributed by NM, 5-Sep-2005.)
(𝑀 ℤ → (𝑁 (ℤ𝑀) ↔ (𝑁 𝑀𝑁)))
 
Theoremeluzel2 8254 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 8255 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 . (Contributed by NM, 5-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)
(𝑁 (ℤ𝑀) ↔ (𝑀 𝑁 𝑀𝑁))
 
Theoremeluz1i 8256 Membership in an upper set of integers. (Contributed by NM, 5-Sep-2005.)
𝑀        (𝑁 (ℤ𝑀) ↔ (𝑁 𝑀𝑁))
 
Theoremeluzuzle 8257 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 BA) → (𝐶 (ℤA) → 𝐶 (ℤB)))
 
Theoremeluzelz 8258 A member of an upper set of integers is an integer. (Contributed by NM, 6-Sep-2005.)
(𝑁 (ℤ𝑀) → 𝑁 ℤ)
 
Theoremeluzelre 8259 A member of an upper set of integers is a real. (Contributed by Mario Carneiro, 31-Aug-2013.)
(𝑁 (ℤ𝑀) → 𝑁 ℝ)
 
Theoremeluzelcn 8260 A member of an upper set of integers is a complex number. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
(𝑁 (ℤ𝑀) → 𝑁 ℂ)
 
Theoremeluzle 8261 Implication of membership in an upper set of integers. (Contributed by NM, 6-Sep-2005.)
(𝑁 (ℤ𝑀) → 𝑀𝑁)
 
Theoremeluz 8262 Membership in an upper set of integers. (Contributed by NM, 2-Oct-2005.)
((𝑀 𝑁 ℤ) → (𝑁 (ℤ𝑀) ↔ 𝑀𝑁))
 
Theoremuzid 8263 Membership of the least member in an upper set of integers. (Contributed by NM, 2-Sep-2005.)
(𝑀 ℤ → 𝑀 (ℤ𝑀))
 
Theoremuzn0 8264 The upper integers are all nonempty. (Contributed by Mario Carneiro, 16-Jan-2014.)
(𝑀 ran ℤ𝑀 ≠ ∅)
 
Theoremuztrn 8265 Transitive law for sets of upper integers. (Contributed by NM, 20-Sep-2005.)
((𝑀 (ℤ𝐾) 𝐾 (ℤ𝑁)) → 𝑀 (ℤ𝑁))
 
Theoremuztrn2 8266 Transitive law for sets of upper integers. (Contributed by Mario Carneiro, 26-Dec-2013.)
𝑍 = (ℤ𝐾)       ((𝑁 𝑍 𝑀 (ℤ𝑁)) → 𝑀 𝑍)
 
Theoremuzneg 8267 Contraposition law for upper integers. (Contributed by NM, 28-Nov-2005.)
(𝑁 (ℤ𝑀) → -𝑀 (ℤ‘-𝑁))
 
Theoremuzssz 8268 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 8269 Subset relationship for two sets of upper integers. (Contributed by NM, 5-Sep-2005.)
(𝑁 (ℤ𝑀) → (ℤ𝑁) ⊆ (ℤ𝑀))
 
Theoremuztric 8270 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 𝑁 ℤ) → (𝑁 (ℤ𝑀) 𝑀 (ℤ𝑁)))
 
Theoremuz11 8271 The upper integers function is one-to-one. (Contributed by NM, 12-Dec-2005.)
(𝑀 ℤ → ((ℤ𝑀) = (ℤ𝑁) ↔ 𝑀 = 𝑁))
 
Theoremeluzp1m1 8272 Membership in the next upper set of integers. (Contributed by NM, 12-Sep-2005.)
((𝑀 𝑁 (ℤ‘(𝑀 + 1))) → (𝑁 − 1) (ℤ𝑀))
 
Theoremeluzp1l 8273 Strict ordering implied by membership in the next upper set of integers. (Contributed by NM, 12-Sep-2005.)
((𝑀 𝑁 (ℤ‘(𝑀 + 1))) → 𝑀 < 𝑁)
 
Theoremeluzp1p1 8274 Membership in the next upper set of integers. (Contributed by NM, 5-Oct-2005.)
(𝑁 (ℤ𝑀) → (𝑁 + 1) (ℤ‘(𝑀 + 1)))
 
Theoremeluzaddi 8275 Membership in a later upper set of integers. (Contributed by Paul Chapman, 22-Nov-2007.)
𝑀     &   𝐾        (𝑁 (ℤ𝑀) → (𝑁 + 𝐾) (ℤ‘(𝑀 + 𝐾)))
 
Theoremeluzsubi 8276 Membership in an earlier upper set of integers. (Contributed by Paul Chapman, 22-Nov-2007.)
𝑀     &   𝐾        (𝑁 (ℤ‘(𝑀 + 𝐾)) → (𝑁𝐾) (ℤ𝑀))
 
Theoremeluzadd 8277 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 8278 Membership in an earlier upper set of integers. (Contributed by Jeff Madsen, 2-Sep-2009.)
((𝑀 𝐾 𝑁 (ℤ‘(𝑀 + 𝐾))) → (𝑁𝐾) (ℤ𝑀))
 
Theoremuzm1 8279 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 8280 The nonnegative difference of integers is a nonnegative integer. (Contributed by NM, 4-Sep-2005.)
(𝑁 (ℤ𝑀) → (𝑁𝑀) 0)
 
Theoremuzin 8281 Intersection of two upper intervals of integers. (Contributed by Mario Carneiro, 24-Dec-2013.)
((𝑀 𝑁 ℤ) → ((ℤ𝑀) ∩ (ℤ𝑁)) = (ℤ‘if(𝑀𝑁, 𝑁, 𝑀)))
 
Theoremuzp1 8282 Choices for an element of an upper interval of integers. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝑁 (ℤ𝑀) → (𝑁 = 𝑀 𝑁 (ℤ‘(𝑀 + 1))))
 
Theoremnn0uz 8283 Nonnegative integers expressed as an upper set of integers. (Contributed by NM, 2-Sep-2005.)
0 = (ℤ‘0)
 
Theoremnnuz 8284 Positive integers expressed as an upper set of integers. (Contributed by NM, 2-Sep-2005.)
ℕ = (ℤ‘1)
 
Theoremelnnuz 8285 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 8286 A nonnegative integer expressed as a member an upper set of integers. (Contributed by NM, 6-Jun-2006.)
(𝑁 0𝑁 (ℤ‘0))
 
Theoremeluz2nn 8287 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 8288 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 0)
 
Theoremuzuzle23 8289 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 8290 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 8291 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 8292 1 is an integer greater than or equal to 0. (Contributed by Alexander van der Vekens, 8-Jun-2018.)
1 (ℤ‘0)
 
Theorem2eluzge0 8293 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)
 
Theorem2eluzge0OLD 8294 2 is an integer greater than or equal to 0. (Contributed by Alexander van der Vekens, 8-Jun-2018.) Obsolete version of 2eluzge0 8293 as of 25-Mar-2020. (New usage is discouraged.) (Proof modification is discouraged.)
2 (ℤ‘0)
 
Theorem2eluzge1 8295 2 is an integer greater than or equal to 1. (Contributed by Alexander van der Vekens, 8-Jun-2018.)
2 (ℤ‘1)
 
Theoremuznnssnn 8296 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 8297* Restricted universal quantification in an upper set of integers. (Contributed by NM, 9-Sep-2005.)
(𝑀 ℤ → (𝑛 (ℤ𝑀)φ𝑛 ℤ (𝑀𝑛φ)))
 
Theoremraluz2 8298* Restricted universal quantification in an upper set of integers. (Contributed by NM, 9-Sep-2005.)
(𝑛 (ℤ𝑀)φ ↔ (𝑀 ℤ → 𝑛 ℤ (𝑀𝑛φ)))
 
Theoremrexuz 8299* Restricted existential quantification in an upper set of integers. (Contributed by NM, 9-Sep-2005.)
(𝑀 ℤ → (𝑛 (ℤ𝑀)φ𝑛 ℤ (𝑀𝑛 φ)))
 
Theoremrexuz2 8300* Restricted existential quantification in an upper set of integers. (Contributed by NM, 9-Sep-2005.)
(𝑛 (ℤ𝑀)φ ↔ (𝑀 𝑛 ℤ (𝑀𝑛 φ)))
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