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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | 8t2e16 12201 | 8 times 2 equals 16. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (8 · 2) = ;16 | ||
Theorem | 8t3e24 12202 | 8 times 3 equals 24. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (8 · 3) = ;24 | ||
Theorem | 8t4e32 12203 | 8 times 4 equals 32. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (8 · 4) = ;32 | ||
Theorem | 8t5e40 12204 | 8 times 5 equals 40. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
⊢ (8 · 5) = ;40 | ||
Theorem | 8t6e48 12205 | 8 times 6 equals 48. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
⊢ (8 · 6) = ;48 | ||
Theorem | 8t7e56 12206 | 8 times 7 equals 56. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (8 · 7) = ;56 | ||
Theorem | 8t8e64 12207 | 8 times 8 equals 64. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (8 · 8) = ;64 | ||
Theorem | 9t2e18 12208 | 9 times 2 equals 18. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (9 · 2) = ;18 | ||
Theorem | 9t3e27 12209 | 9 times 3 equals 27. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (9 · 3) = ;27 | ||
Theorem | 9t4e36 12210 | 9 times 4 equals 36. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (9 · 4) = ;36 | ||
Theorem | 9t5e45 12211 | 9 times 5 equals 45. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (9 · 5) = ;45 | ||
Theorem | 9t6e54 12212 | 9 times 6 equals 54. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (9 · 6) = ;54 | ||
Theorem | 9t7e63 12213 | 9 times 7 equals 63. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (9 · 7) = ;63 | ||
Theorem | 9t8e72 12214 | 9 times 8 equals 72. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (9 · 8) = ;72 | ||
Theorem | 9t9e81 12215 | 9 times 9 equals 81. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (9 · 9) = ;81 | ||
Theorem | 9t11e99 12216 | 9 times 11 equals 99. (Contributed by AV, 14-Jun-2021.) (Revised by AV, 6-Sep-2021.) |
⊢ (9 · ;11) = ;99 | ||
Theorem | 9lt10 12217 | 9 is less than 10. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by AV, 8-Sep-2021.) |
⊢ 9 < ;10 | ||
Theorem | 8lt10 12218 | 8 is less than 10. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by AV, 8-Sep-2021.) |
⊢ 8 < ;10 | ||
Theorem | 7lt10 12219 | 7 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.) |
⊢ 7 < ;10 | ||
Theorem | 6lt10 12220 | 6 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.) |
⊢ 6 < ;10 | ||
Theorem | 5lt10 12221 | 5 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.) |
⊢ 5 < ;10 | ||
Theorem | 4lt10 12222 | 4 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.) |
⊢ 4 < ;10 | ||
Theorem | 3lt10 12223 | 3 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.) |
⊢ 3 < ;10 | ||
Theorem | 2lt10 12224 | 2 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.) |
⊢ 2 < ;10 | ||
Theorem | 1lt10 12225 | 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 12226 | Decompose base 4 into base 2. (Contributed by Mario Carneiro, 18-Feb-2014.) |
⊢ 𝐴 ∈ ℕ0 ⇒ ⊢ (4 · 𝐴) = (2 · (2 · 𝐴)) | ||
Theorem | decbin2 12227 | Decompose base 4 into base 2. (Contributed by Mario Carneiro, 18-Feb-2014.) |
⊢ 𝐴 ∈ ℕ0 ⇒ ⊢ ((4 · 𝐴) + 2) = (2 · ((2 · 𝐴) + 1)) | ||
Theorem | decbin3 12228 | 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 12229 | Half minus a third. (Contributed by Scott Fenton, 8-Jul-2015.) |
⊢ ((1 / 2) − (1 / 3)) = (1 / 6) | ||
Theorem | 5recm6rec 12230 | One fifth minus one sixth. (Contributed by Scott Fenton, 9-Jan-2017.) |
⊢ ((1 / 5) − (1 / 6)) = (1 / ;30) | ||
Syntax | cuz 12231 | 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 12232* | 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 12233 for its value, uzssz 12252 for its relationship to ℤ, nnuz 12269 and nn0uz 12268 for its relationships to ℕ and ℕ0, and eluz1 12235 and eluz2 12237 for its membership relations. (Contributed by NM, 5-Sep-2005.) |
⊢ ℤ≥ = (𝑗 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘}) | ||
Theorem | uzval 12233* | 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 12234 | 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 12235 | 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 12236 | 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 12237 | 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 12238 | 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 12239 | Membership in an upper set of integers. (Contributed by NM, 5-Sep-2005.) |
⊢ 𝑀 ∈ ℤ ⇒ ⊢ (𝑁 ∈ (ℤ≥‘𝑀) ↔ (𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁)) | ||
Theorem | eluzuzle 12240 | 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 12241 | A member of an upper set of integers is an integer. (Contributed by NM, 6-Sep-2005.) |
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℤ) | ||
Theorem | eluzelre 12242 | 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 12243 | 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 12244 | Implication of membership in an upper set of integers. (Contributed by NM, 6-Sep-2005.) |
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ≤ 𝑁) | ||
Theorem | eluz 12245 | Membership in an upper set of integers. (Contributed by NM, 2-Oct-2005.) |
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ≥‘𝑀) ↔ 𝑀 ≤ 𝑁)) | ||
Theorem | uzid 12246 | Membership of the least member in an upper set of integers. (Contributed by NM, 2-Sep-2005.) |
⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ≥‘𝑀)) | ||
Theorem | uzidd 12247 | 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 12248 | The upper integers are all nonempty. (Contributed by Mario Carneiro, 16-Jan-2014.) |
⊢ (𝑀 ∈ ran ℤ≥ → 𝑀 ≠ ∅) | ||
Theorem | uztrn 12249 | Transitive law for sets of upper integers. (Contributed by NM, 20-Sep-2005.) |
⊢ ((𝑀 ∈ (ℤ≥‘𝐾) ∧ 𝐾 ∈ (ℤ≥‘𝑁)) → 𝑀 ∈ (ℤ≥‘𝑁)) | ||
Theorem | uztrn2 12250 | 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 12251 | Contraposition law for upper integers. (Contributed by NM, 28-Nov-2005.) |
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → -𝑀 ∈ (ℤ≥‘-𝑁)) | ||
Theorem | uzssz 12252 | 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 12253 | Subset relationship for two sets of upper integers. (Contributed by NM, 5-Sep-2005.) |
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (ℤ≥‘𝑁) ⊆ (ℤ≥‘𝑀)) | ||
Theorem | uztric 12254 | 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 12255 | The upper integers function is one-to-one. (Contributed by NM, 12-Dec-2005.) |
⊢ (𝑀 ∈ ℤ → ((ℤ≥‘𝑀) = (ℤ≥‘𝑁) ↔ 𝑀 = 𝑁)) | ||
Theorem | eluzp1m1 12256 | 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 12257 | 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 12258 | Membership in the next upper set of integers. (Contributed by NM, 5-Oct-2005.) |
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑁 + 1) ∈ (ℤ≥‘(𝑀 + 1))) | ||
Theorem | eluzaddi 12259 | 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 12260 | 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 12261 | 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 12262 | 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 12263 | 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 12264 | 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 12265 | The nonnegative difference of integers is a nonnegative integer. (Contributed by NM, 4-Sep-2005.) |
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑁 − 𝑀) ∈ ℕ0) | ||
Theorem | uzin 12266 | 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 12267 | 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 12268 | Nonnegative integers expressed as an upper set of integers. (Contributed by NM, 2-Sep-2005.) |
⊢ ℕ0 = (ℤ≥‘0) | ||
Theorem | nnuz 12269 | Positive integers expressed as an upper set of integers. (Contributed by NM, 2-Sep-2005.) |
⊢ ℕ = (ℤ≥‘1) | ||
Theorem | elnnuz 12270 | 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 12271 | 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 12272 | 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 12273 | 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 12274 | 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 12275 | 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 12276 | An integer greater than or equal to 2 is not 0. (Contributed by AV, 25-May-2020.) |
⊢ (𝑁 ∈ (ℤ≥‘2) → 𝑁 ≠ 0) | ||
Theorem | uzuzle23 12277 | 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 12278 | 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 12279 | An integer greater than or equal to 3 decreased by 2 is a positive integer, analogous to uz2m1nn 12311. (Contributed by Alexander van der Vekens, 17-Sep-2018.) |
⊢ (𝑁 ∈ (ℤ≥‘3) → (𝑁 − 2) ∈ ℕ) | ||
Theorem | 1eluzge0 12280 | 1 is an integer greater than or equal to 0. (Contributed by Alexander van der Vekens, 8-Jun-2018.) |
⊢ 1 ∈ (ℤ≥‘0) | ||
Theorem | 2eluzge0 12281 | 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 12282 | 2 is an integer greater than or equal to 1. (Contributed by Alexander van der Vekens, 8-Jun-2018.) |
⊢ 2 ∈ (ℤ≥‘1) | ||
Theorem | uznnssnn 12283 | 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 12284* | 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 12285* | 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 12286* | 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 12287* | 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 12288* | 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 12289 | Second Peano postulate for an upper set of integers. (Contributed by NM, 7-Sep-2005.) |
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑁 + 1) ∈ (ℤ≥‘𝑀)) | ||
Theorem | peano2uzs 12290 | 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 12291 | Reversed second Peano axiom for upper integers. (Contributed by NM, 2-Jan-2006.) |
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 1))) → 𝑁 ∈ (ℤ≥‘𝑀)) | ||
Theorem | uzaddcl 12292 | 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 12293 | 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 12294* | 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 12295* | 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 12294 or uzind4ALT 12295 may be used; see comment for nnind 11643. (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 12296* | 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 12297* | 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 12296 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 12298* | 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 12294 assuming that 𝜓 holds unconditionally. Notice that 𝑁 ∈ (ℤ≥‘𝑀) implies that the lower bound 𝑀 is an integer (𝑀 ∈ ℤ, see eluzel2 12236). (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 | uzwo 12299* | Well-ordering principle: any nonempty subset of an upper set of integers has a least element. (Contributed by NM, 8-Oct-2005.) |
⊢ ((𝑆 ⊆ (ℤ≥‘𝑀) ∧ 𝑆 ≠ ∅) → ∃𝑗 ∈ 𝑆 ∀𝑘 ∈ 𝑆 𝑗 ≤ 𝑘) | ||
Theorem | uzwo2 12300* | Well-ordering principle: any nonempty subset of an upper set of integers has a unique least element. (Contributed by NM, 8-Oct-2005.) |
⊢ ((𝑆 ⊆ (ℤ≥‘𝑀) ∧ 𝑆 ≠ ∅) → ∃!𝑗 ∈ 𝑆 ∀𝑘 ∈ 𝑆 𝑗 ≤ 𝑘) |
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