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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | 5t4e20 12201 | 5 times 4 equals 20. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
⊢ (5 · 4) = ;20 | ||
Theorem | 5t5e25 12202 | 5 times 5 equals 25. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
⊢ (5 · 5) = ;25 | ||
Theorem | 6t2e12 12203 | 6 times 2 equals 12. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (6 · 2) = ;12 | ||
Theorem | 6t3e18 12204 | 6 times 3 equals 18. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (6 · 3) = ;18 | ||
Theorem | 6t4e24 12205 | 6 times 4 equals 24. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (6 · 4) = ;24 | ||
Theorem | 6t5e30 12206 | 6 times 5 equals 30. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
⊢ (6 · 5) = ;30 | ||
Theorem | 6t6e36 12207 | 6 times 6 equals 36. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
⊢ (6 · 6) = ;36 | ||
Theorem | 7t2e14 12208 | 7 times 2 equals 14. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (7 · 2) = ;14 | ||
Theorem | 7t3e21 12209 | 7 times 3 equals 21. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (7 · 3) = ;21 | ||
Theorem | 7t4e28 12210 | 7 times 4 equals 28. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (7 · 4) = ;28 | ||
Theorem | 7t5e35 12211 | 7 times 5 equals 35. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (7 · 5) = ;35 | ||
Theorem | 7t6e42 12212 | 7 times 6 equals 42. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (7 · 6) = ;42 | ||
Theorem | 7t7e49 12213 | 7 times 7 equals 49. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (7 · 7) = ;49 | ||
Theorem | 8t2e16 12214 | 8 times 2 equals 16. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (8 · 2) = ;16 | ||
Theorem | 8t3e24 12215 | 8 times 3 equals 24. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (8 · 3) = ;24 | ||
Theorem | 8t4e32 12216 | 8 times 4 equals 32. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (8 · 4) = ;32 | ||
Theorem | 8t5e40 12217 | 8 times 5 equals 40. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
⊢ (8 · 5) = ;40 | ||
Theorem | 8t6e48 12218 | 8 times 6 equals 48. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
⊢ (8 · 6) = ;48 | ||
Theorem | 8t7e56 12219 | 8 times 7 equals 56. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (8 · 7) = ;56 | ||
Theorem | 8t8e64 12220 | 8 times 8 equals 64. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (8 · 8) = ;64 | ||
Theorem | 9t2e18 12221 | 9 times 2 equals 18. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (9 · 2) = ;18 | ||
Theorem | 9t3e27 12222 | 9 times 3 equals 27. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (9 · 3) = ;27 | ||
Theorem | 9t4e36 12223 | 9 times 4 equals 36. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (9 · 4) = ;36 | ||
Theorem | 9t5e45 12224 | 9 times 5 equals 45. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (9 · 5) = ;45 | ||
Theorem | 9t6e54 12225 | 9 times 6 equals 54. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (9 · 6) = ;54 | ||
Theorem | 9t7e63 12226 | 9 times 7 equals 63. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (9 · 7) = ;63 | ||
Theorem | 9t8e72 12227 | 9 times 8 equals 72. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (9 · 8) = ;72 | ||
Theorem | 9t9e81 12228 | 9 times 9 equals 81. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (9 · 9) = ;81 | ||
Theorem | 9t11e99 12229 | 9 times 11 equals 99. (Contributed by AV, 14-Jun-2021.) (Revised by AV, 6-Sep-2021.) |
⊢ (9 · ;11) = ;99 | ||
Theorem | 9lt10 12230 | 9 is less than 10. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by AV, 8-Sep-2021.) |
⊢ 9 < ;10 | ||
Theorem | 8lt10 12231 | 8 is less than 10. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by AV, 8-Sep-2021.) |
⊢ 8 < ;10 | ||
Theorem | 7lt10 12232 | 7 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.) |
⊢ 7 < ;10 | ||
Theorem | 6lt10 12233 | 6 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.) |
⊢ 6 < ;10 | ||
Theorem | 5lt10 12234 | 5 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.) |
⊢ 5 < ;10 | ||
Theorem | 4lt10 12235 | 4 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.) |
⊢ 4 < ;10 | ||
Theorem | 3lt10 12236 | 3 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.) |
⊢ 3 < ;10 | ||
Theorem | 2lt10 12237 | 2 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.) |
⊢ 2 < ;10 | ||
Theorem | 1lt10 12238 | 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 12239 | Decompose base 4 into base 2. (Contributed by Mario Carneiro, 18-Feb-2014.) |
⊢ 𝐴 ∈ ℕ0 ⇒ ⊢ (4 · 𝐴) = (2 · (2 · 𝐴)) | ||
Theorem | decbin2 12240 | Decompose base 4 into base 2. (Contributed by Mario Carneiro, 18-Feb-2014.) |
⊢ 𝐴 ∈ ℕ0 ⇒ ⊢ ((4 · 𝐴) + 2) = (2 · ((2 · 𝐴) + 1)) | ||
Theorem | decbin3 12241 | 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 12242 | Half minus a third. (Contributed by Scott Fenton, 8-Jul-2015.) |
⊢ ((1 / 2) − (1 / 3)) = (1 / 6) | ||
Theorem | 5recm6rec 12243 | One fifth minus one sixth. (Contributed by Scott Fenton, 9-Jan-2017.) |
⊢ ((1 / 5) − (1 / 6)) = (1 / ;30) | ||
Syntax | cuz 12244 | 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 12245* | 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 12246 for its value, uzssz 12265 for its relationship to ℤ, nnuz 12282 and nn0uz 12281 for its relationships to ℕ and ℕ0, and eluz1 12248 and eluz2 12250 for its membership relations. (Contributed by NM, 5-Sep-2005.) |
⊢ ℤ≥ = (𝑗 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘}) | ||
Theorem | uzval 12246* | 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 12247 | 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 12248 | 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 12249 | 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 12250 | 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 | eluzmn 12251 | Membership in an earlier upper set of integers. (Contributed by Thierry Arnoux, 8-Oct-2018.) |
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ0) → 𝑀 ∈ (ℤ≥‘(𝑀 − 𝑁))) | ||
Theorem | eluz1i 12252 | Membership in an upper set of integers. (Contributed by NM, 5-Sep-2005.) |
⊢ 𝑀 ∈ ℤ ⇒ ⊢ (𝑁 ∈ (ℤ≥‘𝑀) ↔ (𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁)) | ||
Theorem | eluzuzle 12253 | 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 12254 | A member of an upper set of integers is an integer. (Contributed by NM, 6-Sep-2005.) |
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℤ) | ||
Theorem | eluzelre 12255 | 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 12256 | 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 12257 | Implication of membership in an upper set of integers. (Contributed by NM, 6-Sep-2005.) |
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ≤ 𝑁) | ||
Theorem | eluz 12258 | Membership in an upper set of integers. (Contributed by NM, 2-Oct-2005.) |
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ≥‘𝑀) ↔ 𝑀 ≤ 𝑁)) | ||
Theorem | uzid 12259 | Membership of the least member in an upper set of integers. (Contributed by NM, 2-Sep-2005.) |
⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ≥‘𝑀)) | ||
Theorem | uzidd 12260 | Membership of the least member in an upper set of integers. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
⊢ (𝜑 → 𝑀 ∈ ℤ) ⇒ ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘𝑀)) | ||
Theorem | uzn0 12261 | The upper integers are all nonempty. (Contributed by Mario Carneiro, 16-Jan-2014.) |
⊢ (𝑀 ∈ ran ℤ≥ → 𝑀 ≠ ∅) | ||
Theorem | uztrn 12262 | Transitive law for sets of upper integers. (Contributed by NM, 20-Sep-2005.) |
⊢ ((𝑀 ∈ (ℤ≥‘𝐾) ∧ 𝐾 ∈ (ℤ≥‘𝑁)) → 𝑀 ∈ (ℤ≥‘𝑁)) | ||
Theorem | uztrn2 12263 | 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 12264 | Contraposition law for upper integers. (Contributed by NM, 28-Nov-2005.) |
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → -𝑀 ∈ (ℤ≥‘-𝑁)) | ||
Theorem | uzssz 12265 | 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 12266 | Subset relationship for two sets of upper integers. (Contributed by NM, 5-Sep-2005.) |
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (ℤ≥‘𝑁) ⊆ (ℤ≥‘𝑀)) | ||
Theorem | uztric 12267 | Totality 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 12268 | The upper integers function is one-to-one. (Contributed by NM, 12-Dec-2005.) |
⊢ (𝑀 ∈ ℤ → ((ℤ≥‘𝑀) = (ℤ≥‘𝑁) ↔ 𝑀 = 𝑁)) | ||
Theorem | eluzp1m1 12269 | 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 12270 | 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 12271 | Membership in the next upper set of integers. (Contributed by NM, 5-Oct-2005.) |
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑁 + 1) ∈ (ℤ≥‘(𝑀 + 1))) | ||
Theorem | eluzaddi 12272 | 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 12273 | 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 12274 | 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 12275 | 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 | subeluzsub 12276 | Membership of a difference in an earlier upper set of integers. (Contributed by AV, 10-May-2022.) |
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘𝐾)) → (𝑀 − 𝐾) ∈ (ℤ≥‘(𝑀 − 𝑁))) | ||
Theorem | uzm1 12277 | 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 12278 | The nonnegative difference of integers is a nonnegative integer. (Contributed by NM, 4-Sep-2005.) |
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑁 − 𝑀) ∈ ℕ0) | ||
Theorem | uzin 12279 | 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 12280 | 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 12281 | Nonnegative integers expressed as an upper set of integers. (Contributed by NM, 2-Sep-2005.) |
⊢ ℕ0 = (ℤ≥‘0) | ||
Theorem | nnuz 12282 | Positive integers expressed as an upper set of integers. (Contributed by NM, 2-Sep-2005.) |
⊢ ℕ = (ℤ≥‘1) | ||
Theorem | elnnuz 12283 | 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 12284 | 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 12285 | An integer greater than or equal to 2 is a positive integer. (Contributed by AV, 3-Nov-2018.) |
⊢ (𝐴 ∈ (ℤ≥‘2) → 𝐴 ∈ ℕ) | ||
Theorem | eluz4eluz2 12286 | An integer greater than or equal to 4 is an integer greater than or equal to 2. (Contributed by AV, 30-May-2023.) |
⊢ (𝑋 ∈ (ℤ≥‘4) → 𝑋 ∈ (ℤ≥‘2)) | ||
Theorem | eluz4nn 12287 | An integer greater than or equal to 4 is a positive integer. (Contributed by AV, 30-May-2023.) |
⊢ (𝑋 ∈ (ℤ≥‘4) → 𝑋 ∈ ℕ) | ||
Theorem | eluzge2nn0 12288 | 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 12289 | An integer greater than or equal to 2 is not 0. (Contributed by AV, 25-May-2020.) |
⊢ (𝑁 ∈ (ℤ≥‘2) → 𝑁 ≠ 0) | ||
Theorem | uzuzle23 12290 | 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 12291 | 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 12292 | An integer greater than or equal to 3 decreased by 2 is a positive integer, analogous to uz2m1nn 12324. (Contributed by Alexander van der Vekens, 17-Sep-2018.) |
⊢ (𝑁 ∈ (ℤ≥‘3) → (𝑁 − 2) ∈ ℕ) | ||
Theorem | 1eluzge0 12293 | 1 is an integer greater than or equal to 0. (Contributed by Alexander van der Vekens, 8-Jun-2018.) |
⊢ 1 ∈ (ℤ≥‘0) | ||
Theorem | 2eluzge0 12294 | 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 12295 | 2 is an integer greater than or equal to 1. (Contributed by Alexander van der Vekens, 8-Jun-2018.) |
⊢ 2 ∈ (ℤ≥‘1) | ||
Theorem | uznnssnn 12296 | 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 12297* | 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 12298* | 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 12299* | 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 12300* | 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))) |
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