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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | 5t4e20 12801 | 5 times 4 equals 20. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
⊢ (5 · 4) = ;20 | ||
Theorem | 5t5e25 12802 | 5 times 5 equals 25. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
⊢ (5 · 5) = ;25 | ||
Theorem | 6t2e12 12803 | 6 times 2 equals 12. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (6 · 2) = ;12 | ||
Theorem | 6t3e18 12804 | 6 times 3 equals 18. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (6 · 3) = ;18 | ||
Theorem | 6t4e24 12805 | 6 times 4 equals 24. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (6 · 4) = ;24 | ||
Theorem | 6t5e30 12806 | 6 times 5 equals 30. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
⊢ (6 · 5) = ;30 | ||
Theorem | 6t6e36 12807 | 6 times 6 equals 36. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
⊢ (6 · 6) = ;36 | ||
Theorem | 7t2e14 12808 | 7 times 2 equals 14. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (7 · 2) = ;14 | ||
Theorem | 7t3e21 12809 | 7 times 3 equals 21. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (7 · 3) = ;21 | ||
Theorem | 7t4e28 12810 | 7 times 4 equals 28. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (7 · 4) = ;28 | ||
Theorem | 7t5e35 12811 | 7 times 5 equals 35. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (7 · 5) = ;35 | ||
Theorem | 7t6e42 12812 | 7 times 6 equals 42. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (7 · 6) = ;42 | ||
Theorem | 7t7e49 12813 | 7 times 7 equals 49. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (7 · 7) = ;49 | ||
Theorem | 8t2e16 12814 | 8 times 2 equals 16. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (8 · 2) = ;16 | ||
Theorem | 8t3e24 12815 | 8 times 3 equals 24. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (8 · 3) = ;24 | ||
Theorem | 8t4e32 12816 | 8 times 4 equals 32. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (8 · 4) = ;32 | ||
Theorem | 8t5e40 12817 | 8 times 5 equals 40. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
⊢ (8 · 5) = ;40 | ||
Theorem | 8t6e48 12818 | 8 times 6 equals 48. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
⊢ (8 · 6) = ;48 | ||
Theorem | 8t7e56 12819 | 8 times 7 equals 56. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (8 · 7) = ;56 | ||
Theorem | 8t8e64 12820 | 8 times 8 equals 64. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (8 · 8) = ;64 | ||
Theorem | 9t2e18 12821 | 9 times 2 equals 18. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (9 · 2) = ;18 | ||
Theorem | 9t3e27 12822 | 9 times 3 equals 27. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (9 · 3) = ;27 | ||
Theorem | 9t4e36 12823 | 9 times 4 equals 36. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (9 · 4) = ;36 | ||
Theorem | 9t5e45 12824 | 9 times 5 equals 45. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (9 · 5) = ;45 | ||
Theorem | 9t6e54 12825 | 9 times 6 equals 54. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (9 · 6) = ;54 | ||
Theorem | 9t7e63 12826 | 9 times 7 equals 63. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (9 · 7) = ;63 | ||
Theorem | 9t8e72 12827 | 9 times 8 equals 72. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (9 · 8) = ;72 | ||
Theorem | 9t9e81 12828 | 9 times 9 equals 81. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (9 · 9) = ;81 | ||
Theorem | 9t11e99 12829 | 9 times 11 equals 99. (Contributed by AV, 14-Jun-2021.) (Revised by AV, 6-Sep-2021.) |
⊢ (9 · ;11) = ;99 | ||
Theorem | 9lt10 12830 | 9 is less than 10. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by AV, 8-Sep-2021.) |
⊢ 9 < ;10 | ||
Theorem | 8lt10 12831 | 8 is less than 10. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by AV, 8-Sep-2021.) |
⊢ 8 < ;10 | ||
Theorem | 7lt10 12832 | 7 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.) |
⊢ 7 < ;10 | ||
Theorem | 6lt10 12833 | 6 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.) |
⊢ 6 < ;10 | ||
Theorem | 5lt10 12834 | 5 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.) |
⊢ 5 < ;10 | ||
Theorem | 4lt10 12835 | 4 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.) |
⊢ 4 < ;10 | ||
Theorem | 3lt10 12836 | 3 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.) |
⊢ 3 < ;10 | ||
Theorem | 2lt10 12837 | 2 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.) |
⊢ 2 < ;10 | ||
Theorem | 1lt10 12838 | 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 12839 | Decompose base 4 into base 2. (Contributed by Mario Carneiro, 18-Feb-2014.) |
⊢ 𝐴 ∈ ℕ0 ⇒ ⊢ (4 · 𝐴) = (2 · (2 · 𝐴)) | ||
Theorem | decbin2 12840 | Decompose base 4 into base 2. (Contributed by Mario Carneiro, 18-Feb-2014.) |
⊢ 𝐴 ∈ ℕ0 ⇒ ⊢ ((4 · 𝐴) + 2) = (2 · ((2 · 𝐴) + 1)) | ||
Theorem | decbin3 12841 | 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 12842 | Half minus a third. (Contributed by Scott Fenton, 8-Jul-2015.) |
⊢ ((1 / 2) − (1 / 3)) = (1 / 6) | ||
Theorem | 5recm6rec 12843 | One fifth minus one sixth. (Contributed by Scott Fenton, 9-Jan-2017.) |
⊢ ((1 / 5) − (1 / 6)) = (1 / ;30) | ||
Syntax | cuz 12844 | 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 12845* | 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 12846 for its value, uzssz 12865 for its relationship to ℤ, nnuz 12887 and nn0uz 12886 for its relationships to ℕ and ℕ0, and eluz1 12848 and eluz2 12850 for its membership relations. (Contributed by NM, 5-Sep-2005.) |
⊢ ℤ≥ = (𝑗 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘}) | ||
Theorem | uzval 12846* | 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 12847 | The domain and codomain 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 12848 | 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 12849 | 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 12850 | 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 12851 | 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 12852 | Membership in an upper set of integers. (Contributed by NM, 5-Sep-2005.) |
⊢ 𝑀 ∈ ℤ ⇒ ⊢ (𝑁 ∈ (ℤ≥‘𝑀) ↔ (𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁)) | ||
Theorem | eluzuzle 12853 | 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 12854 | A member of an upper set of integers is an integer. (Contributed by NM, 6-Sep-2005.) |
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℤ) | ||
Theorem | eluzelre 12855 | 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 12856 | 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 12857 | Implication of membership in an upper set of integers. (Contributed by NM, 6-Sep-2005.) |
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ≤ 𝑁) | ||
Theorem | eluz 12858 | Membership in an upper set of integers. (Contributed by NM, 2-Oct-2005.) |
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ≥‘𝑀) ↔ 𝑀 ≤ 𝑁)) | ||
Theorem | uzid 12859 | Membership of the least member in an upper set of integers. (Contributed by NM, 2-Sep-2005.) |
⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ≥‘𝑀)) | ||
Theorem | uzidd 12860 | 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 12861 | The upper integers are all nonempty. (Contributed by Mario Carneiro, 16-Jan-2014.) |
⊢ (𝑀 ∈ ran ℤ≥ → 𝑀 ≠ ∅) | ||
Theorem | uztrn 12862 | Transitive law for sets of upper integers. (Contributed by NM, 20-Sep-2005.) |
⊢ ((𝑀 ∈ (ℤ≥‘𝐾) ∧ 𝐾 ∈ (ℤ≥‘𝑁)) → 𝑀 ∈ (ℤ≥‘𝑁)) | ||
Theorem | uztrn2 12863 | 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 12864 | Contraposition law for upper integers. (Contributed by NM, 28-Nov-2005.) |
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → -𝑀 ∈ (ℤ≥‘-𝑁)) | ||
Theorem | uzssz 12865 | 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 | uzssre 12866 | An upper set of integers is a subset of the reals. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
⊢ (ℤ≥‘𝑀) ⊆ ℝ | ||
Theorem | uzss 12867 | Subset relationship for two sets of upper integers. (Contributed by NM, 5-Sep-2005.) |
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (ℤ≥‘𝑁) ⊆ (ℤ≥‘𝑀)) | ||
Theorem | uztric 12868 | 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 12869 | The upper integers function is one-to-one. (Contributed by NM, 12-Dec-2005.) |
⊢ (𝑀 ∈ ℤ → ((ℤ≥‘𝑀) = (ℤ≥‘𝑁) ↔ 𝑀 = 𝑁)) | ||
Theorem | eluzp1m1 12870 | 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 12871 | 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 12872 | Membership in the next upper set of integers. (Contributed by NM, 5-Oct-2005.) |
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑁 + 1) ∈ (ℤ≥‘(𝑀 + 1))) | ||
Theorem | eluzadd 12873 | Membership in a later upper set of integers. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by SN, 7-Feb-2025.) |
⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝐾 ∈ ℤ) → (𝑁 + 𝐾) ∈ (ℤ≥‘(𝑀 + 𝐾))) | ||
Theorem | eluzsub 12874 | Membership in an earlier upper set of integers. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by SN, 7-Feb-2025.) |
⊢ ((𝑀 ∈ ℤ ∧ 𝐾 ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 𝐾))) → (𝑁 − 𝐾) ∈ (ℤ≥‘𝑀)) | ||
Theorem | eluzaddi 12875 | Membership in a later upper set of integers. (Contributed by Paul Chapman, 22-Nov-2007.) Shorten and remove 𝑀 ∈ ℤ hypothesis. (Revised by SN, 7-Feb-2025.) |
⊢ 𝐾 ∈ ℤ ⇒ ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑁 + 𝐾) ∈ (ℤ≥‘(𝑀 + 𝐾))) | ||
Theorem | eluzaddiOLD 12876 | Obsolete version of eluzaddi 12875 as of 7-Feb-2025. (Contributed by Paul Chapman, 22-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝑀 ∈ ℤ & ⊢ 𝐾 ∈ ℤ ⇒ ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑁 + 𝐾) ∈ (ℤ≥‘(𝑀 + 𝐾))) | ||
Theorem | eluzsubi 12877 | Membership in an earlier upper set of integers. (Contributed by Paul Chapman, 22-Nov-2007.) (Proof shortened by SN, 7-Feb-2025.) |
⊢ 𝑀 ∈ ℤ & ⊢ 𝐾 ∈ ℤ ⇒ ⊢ (𝑁 ∈ (ℤ≥‘(𝑀 + 𝐾)) → (𝑁 − 𝐾) ∈ (ℤ≥‘𝑀)) | ||
Theorem | eluzsubiOLD 12878 | Obsolete version of eluzsubi 12877 as of 7-Feb-2025. (Contributed by Paul Chapman, 22-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝑀 ∈ ℤ & ⊢ 𝐾 ∈ ℤ ⇒ ⊢ (𝑁 ∈ (ℤ≥‘(𝑀 + 𝐾)) → (𝑁 − 𝐾) ∈ (ℤ≥‘𝑀)) | ||
Theorem | eluzaddOLD 12879 | Obsolete version of eluzadd 12873 as of 7-Feb-2025. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝐾 ∈ ℤ) → (𝑁 + 𝐾) ∈ (ℤ≥‘(𝑀 + 𝐾))) | ||
Theorem | eluzsubOLD 12880 | Obsolete version of eluzsub 12874 as of 7-Feb-2025. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝑀 ∈ ℤ ∧ 𝐾 ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 𝐾))) → (𝑁 − 𝐾) ∈ (ℤ≥‘𝑀)) | ||
Theorem | subeluzsub 12881 | 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 12882 | 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 12883 | The nonnegative difference of integers is a nonnegative integer. (Contributed by NM, 4-Sep-2005.) |
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑁 − 𝑀) ∈ ℕ0) | ||
Theorem | uzin 12884 | 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 12885 | 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 12886 | Nonnegative integers expressed as an upper set of integers. (Contributed by NM, 2-Sep-2005.) |
⊢ ℕ0 = (ℤ≥‘0) | ||
Theorem | nnuz 12887 | Positive integers expressed as an upper set of integers. (Contributed by NM, 2-Sep-2005.) |
⊢ ℕ = (ℤ≥‘1) | ||
Theorem | elnnuz 12888 | 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 12889 | 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 12890 | 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 12891 | 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 12892 | 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 12893 | 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 12894 | An integer greater than or equal to 2 is not 0. (Contributed by AV, 25-May-2020.) |
⊢ (𝑁 ∈ (ℤ≥‘2) → 𝑁 ≠ 0) | ||
Theorem | uzuzle23 12895 | 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 12896 | 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 12897 | An integer greater than or equal to 3 decreased by 2 is a positive integer, analogous to uz2m1nn 12929. (Contributed by Alexander van der Vekens, 17-Sep-2018.) |
⊢ (𝑁 ∈ (ℤ≥‘3) → (𝑁 − 2) ∈ ℕ) | ||
Theorem | 1eluzge0 12898 | 1 is an integer greater than or equal to 0. (Contributed by Alexander van der Vekens, 8-Jun-2018.) |
⊢ 1 ∈ (ℤ≥‘0) | ||
Theorem | 2eluzge0 12899 | 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 12900 | 2 is an integer greater than or equal to 1. (Contributed by Alexander van der Vekens, 8-Jun-2018.) |
⊢ 2 ∈ (ℤ≥‘1) |
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