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Type | Label | Description |
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Statement | ||
Theorem | 5p5e10 12801 | 5 + 5 = 10. (Contributed by NM, 5-Feb-2007.) (Revised by Stanislas Polu, 7-Apr-2020.) (Revised by AV, 6-Sep-2021.) |
⊢ (5 + 5) = ;10 | ||
Theorem | 6p4e10 12802 | 6 + 4 = 10. (Contributed by NM, 5-Feb-2007.) (Revised by Stanislas Polu, 7-Apr-2020.) (Revised by AV, 6-Sep-2021.) |
⊢ (6 + 4) = ;10 | ||
Theorem | 6p5e11 12803 | 6 + 5 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
⊢ (6 + 5) = ;11 | ||
Theorem | 6p6e12 12804 | 6 + 6 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (6 + 6) = ;12 | ||
Theorem | 7p3e10 12805 | 7 + 3 = 10. (Contributed by NM, 5-Feb-2007.) (Revised by Stanislas Polu, 7-Apr-2020.) (Revised by AV, 6-Sep-2021.) |
⊢ (7 + 3) = ;10 | ||
Theorem | 7p4e11 12806 | 7 + 4 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
⊢ (7 + 4) = ;11 | ||
Theorem | 7p5e12 12807 | 7 + 5 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (7 + 5) = ;12 | ||
Theorem | 7p6e13 12808 | 7 + 6 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (7 + 6) = ;13 | ||
Theorem | 7p7e14 12809 | 7 + 7 = 14. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (7 + 7) = ;14 | ||
Theorem | 8p2e10 12810 | 8 + 2 = 10. (Contributed by NM, 5-Feb-2007.) (Revised by Stanislas Polu, 7-Apr-2020.) (Revised by AV, 6-Sep-2021.) |
⊢ (8 + 2) = ;10 | ||
Theorem | 8p3e11 12811 | 8 + 3 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
⊢ (8 + 3) = ;11 | ||
Theorem | 8p4e12 12812 | 8 + 4 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (8 + 4) = ;12 | ||
Theorem | 8p5e13 12813 | 8 + 5 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (8 + 5) = ;13 | ||
Theorem | 8p6e14 12814 | 8 + 6 = 14. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (8 + 6) = ;14 | ||
Theorem | 8p7e15 12815 | 8 + 7 = 15. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (8 + 7) = ;15 | ||
Theorem | 8p8e16 12816 | 8 + 8 = 16. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (8 + 8) = ;16 | ||
Theorem | 9p2e11 12817 | 9 + 2 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
⊢ (9 + 2) = ;11 | ||
Theorem | 9p3e12 12818 | 9 + 3 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (9 + 3) = ;12 | ||
Theorem | 9p4e13 12819 | 9 + 4 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (9 + 4) = ;13 | ||
Theorem | 9p5e14 12820 | 9 + 5 = 14. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (9 + 5) = ;14 | ||
Theorem | 9p6e15 12821 | 9 + 6 = 15. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (9 + 6) = ;15 | ||
Theorem | 9p7e16 12822 | 9 + 7 = 16. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (9 + 7) = ;16 | ||
Theorem | 9p8e17 12823 | 9 + 8 = 17. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (9 + 8) = ;17 | ||
Theorem | 9p9e18 12824 | 9 + 9 = 18. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (9 + 9) = ;18 | ||
Theorem | 10p10e20 12825 | 10 + 10 = 20. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
⊢ (;10 + ;10) = ;20 | ||
Theorem | 10m1e9 12826 | 10 - 1 = 9. (Contributed by AV, 6-Sep-2021.) |
⊢ (;10 − 1) = 9 | ||
Theorem | 4t3lem 12827 | Lemma for 4t3e12 12828 and related theorems. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 & ⊢ 𝐶 = (𝐵 + 1) & ⊢ (𝐴 · 𝐵) = 𝐷 & ⊢ (𝐷 + 𝐴) = 𝐸 ⇒ ⊢ (𝐴 · 𝐶) = 𝐸 | ||
Theorem | 4t3e12 12828 | 4 times 3 equals 12. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (4 · 3) = ;12 | ||
Theorem | 4t4e16 12829 | 4 times 4 equals 16. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (4 · 4) = ;16 | ||
Theorem | 5t2e10 12830 | 5 times 2 equals 10. (Contributed by NM, 5-Feb-2007.) (Revised by AV, 4-Sep-2021.) |
⊢ (5 · 2) = ;10 | ||
Theorem | 5t3e15 12831 | 5 times 3 equals 15. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
⊢ (5 · 3) = ;15 | ||
Theorem | 5t4e20 12832 | 5 times 4 equals 20. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
⊢ (5 · 4) = ;20 | ||
Theorem | 5t5e25 12833 | 5 times 5 equals 25. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
⊢ (5 · 5) = ;25 | ||
Theorem | 6t2e12 12834 | 6 times 2 equals 12. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (6 · 2) = ;12 | ||
Theorem | 6t3e18 12835 | 6 times 3 equals 18. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (6 · 3) = ;18 | ||
Theorem | 6t4e24 12836 | 6 times 4 equals 24. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (6 · 4) = ;24 | ||
Theorem | 6t5e30 12837 | 6 times 5 equals 30. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
⊢ (6 · 5) = ;30 | ||
Theorem | 6t6e36 12838 | 6 times 6 equals 36. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
⊢ (6 · 6) = ;36 | ||
Theorem | 7t2e14 12839 | 7 times 2 equals 14. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (7 · 2) = ;14 | ||
Theorem | 7t3e21 12840 | 7 times 3 equals 21. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (7 · 3) = ;21 | ||
Theorem | 7t4e28 12841 | 7 times 4 equals 28. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (7 · 4) = ;28 | ||
Theorem | 7t5e35 12842 | 7 times 5 equals 35. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (7 · 5) = ;35 | ||
Theorem | 7t6e42 12843 | 7 times 6 equals 42. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (7 · 6) = ;42 | ||
Theorem | 7t7e49 12844 | 7 times 7 equals 49. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (7 · 7) = ;49 | ||
Theorem | 8t2e16 12845 | 8 times 2 equals 16. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (8 · 2) = ;16 | ||
Theorem | 8t3e24 12846 | 8 times 3 equals 24. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (8 · 3) = ;24 | ||
Theorem | 8t4e32 12847 | 8 times 4 equals 32. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (8 · 4) = ;32 | ||
Theorem | 8t5e40 12848 | 8 times 5 equals 40. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
⊢ (8 · 5) = ;40 | ||
Theorem | 8t6e48 12849 | 8 times 6 equals 48. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
⊢ (8 · 6) = ;48 | ||
Theorem | 8t7e56 12850 | 8 times 7 equals 56. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (8 · 7) = ;56 | ||
Theorem | 8t8e64 12851 | 8 times 8 equals 64. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (8 · 8) = ;64 | ||
Theorem | 9t2e18 12852 | 9 times 2 equals 18. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (9 · 2) = ;18 | ||
Theorem | 9t3e27 12853 | 9 times 3 equals 27. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (9 · 3) = ;27 | ||
Theorem | 9t4e36 12854 | 9 times 4 equals 36. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (9 · 4) = ;36 | ||
Theorem | 9t5e45 12855 | 9 times 5 equals 45. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (9 · 5) = ;45 | ||
Theorem | 9t6e54 12856 | 9 times 6 equals 54. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (9 · 6) = ;54 | ||
Theorem | 9t7e63 12857 | 9 times 7 equals 63. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (9 · 7) = ;63 | ||
Theorem | 9t8e72 12858 | 9 times 8 equals 72. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (9 · 8) = ;72 | ||
Theorem | 9t9e81 12859 | 9 times 9 equals 81. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ (9 · 9) = ;81 | ||
Theorem | 9t11e99 12860 | 9 times 11 equals 99. (Contributed by AV, 14-Jun-2021.) (Revised by AV, 6-Sep-2021.) |
⊢ (9 · ;11) = ;99 | ||
Theorem | 9lt10 12861 | 9 is less than 10. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by AV, 8-Sep-2021.) |
⊢ 9 < ;10 | ||
Theorem | 8lt10 12862 | 8 is less than 10. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by AV, 8-Sep-2021.) |
⊢ 8 < ;10 | ||
Theorem | 7lt10 12863 | 7 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.) |
⊢ 7 < ;10 | ||
Theorem | 6lt10 12864 | 6 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.) |
⊢ 6 < ;10 | ||
Theorem | 5lt10 12865 | 5 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.) |
⊢ 5 < ;10 | ||
Theorem | 4lt10 12866 | 4 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.) |
⊢ 4 < ;10 | ||
Theorem | 3lt10 12867 | 3 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.) |
⊢ 3 < ;10 | ||
Theorem | 2lt10 12868 | 2 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.) |
⊢ 2 < ;10 | ||
Theorem | 1lt10 12869 | 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 12870 | Decompose base 4 into base 2. (Contributed by Mario Carneiro, 18-Feb-2014.) |
⊢ 𝐴 ∈ ℕ0 ⇒ ⊢ (4 · 𝐴) = (2 · (2 · 𝐴)) | ||
Theorem | decbin2 12871 | Decompose base 4 into base 2. (Contributed by Mario Carneiro, 18-Feb-2014.) |
⊢ 𝐴 ∈ ℕ0 ⇒ ⊢ ((4 · 𝐴) + 2) = (2 · ((2 · 𝐴) + 1)) | ||
Theorem | decbin3 12872 | 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 12873 | Half minus a third. (Contributed by Scott Fenton, 8-Jul-2015.) |
⊢ ((1 / 2) − (1 / 3)) = (1 / 6) | ||
Theorem | 5recm6rec 12874 | One fifth minus one sixth. (Contributed by Scott Fenton, 9-Jan-2017.) |
⊢ ((1 / 5) − (1 / 6)) = (1 / ;30) | ||
Syntax | cuz 12875 | 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 12876* | 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 12877 for its value, uzssz 12896 for its relationship to ℤ, nnuz 12918 and nn0uz 12917 for its relationships to ℕ and ℕ0, and eluz1 12879 and eluz2 12881 for its membership relations. (Contributed by NM, 5-Sep-2005.) |
⊢ ℤ≥ = (𝑗 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘}) | ||
Theorem | uzval 12877* | 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 12878 | 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 12879 | 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 12880 | 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 12881 | 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 12882 | 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 12883 | Membership in an upper set of integers. (Contributed by NM, 5-Sep-2005.) |
⊢ 𝑀 ∈ ℤ ⇒ ⊢ (𝑁 ∈ (ℤ≥‘𝑀) ↔ (𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁)) | ||
Theorem | eluzuzle 12884 | 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 12885 | A member of an upper set of integers is an integer. (Contributed by NM, 6-Sep-2005.) |
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℤ) | ||
Theorem | eluzelre 12886 | 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 12887 | 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 12888 | Implication of membership in an upper set of integers. (Contributed by NM, 6-Sep-2005.) |
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ≤ 𝑁) | ||
Theorem | eluz 12889 | Membership in an upper set of integers. (Contributed by NM, 2-Oct-2005.) |
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ≥‘𝑀) ↔ 𝑀 ≤ 𝑁)) | ||
Theorem | uzid 12890 | Membership of the least member in an upper set of integers. (Contributed by NM, 2-Sep-2005.) |
⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ≥‘𝑀)) | ||
Theorem | uzidd 12891 | 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 12892 | The upper integers are all nonempty. (Contributed by Mario Carneiro, 16-Jan-2014.) |
⊢ (𝑀 ∈ ran ℤ≥ → 𝑀 ≠ ∅) | ||
Theorem | uztrn 12893 | Transitive law for sets of upper integers. (Contributed by NM, 20-Sep-2005.) |
⊢ ((𝑀 ∈ (ℤ≥‘𝐾) ∧ 𝐾 ∈ (ℤ≥‘𝑁)) → 𝑀 ∈ (ℤ≥‘𝑁)) | ||
Theorem | uztrn2 12894 | 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 12895 | Contraposition law for upper integers. (Contributed by NM, 28-Nov-2005.) |
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → -𝑀 ∈ (ℤ≥‘-𝑁)) | ||
Theorem | uzssz 12896 | 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 12897 | An upper set of integers is a subset of the reals. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
⊢ (ℤ≥‘𝑀) ⊆ ℝ | ||
Theorem | uzss 12898 | Subset relationship for two sets of upper integers. (Contributed by NM, 5-Sep-2005.) |
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (ℤ≥‘𝑁) ⊆ (ℤ≥‘𝑀)) | ||
Theorem | uztric 12899 | 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 12900 | The upper integers function is one-to-one. (Contributed by NM, 12-Dec-2005.) |
⊢ (𝑀 ∈ ℤ → ((ℤ≥‘𝑀) = (ℤ≥‘𝑁) ↔ 𝑀 = 𝑁)) |
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