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Theorem List for Intuitionistic Logic Explorer - 8901-9000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem5t2e10 8901 5 times 2 equals 10. (Contributed by NM, 5-Feb-2007.) (Revised by AV, 4-Sep-2021.)
(5 · 2) = 10
 
Theorem5t3e15 8902 5 times 3 equals 15. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
(5 · 3) = 15
 
Theorem5t4e20 8903 5 times 4 equals 20. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
(5 · 4) = 20
 
Theorem5t5e25 8904 5 times 5 equals 25. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
(5 · 5) = 25
 
Theorem6t2e12 8905 6 times 2 equals 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
(6 · 2) = 12
 
Theorem6t3e18 8906 6 times 3 equals 18. (Contributed by Mario Carneiro, 19-Apr-2015.)
(6 · 3) = 18
 
Theorem6t4e24 8907 6 times 4 equals 24. (Contributed by Mario Carneiro, 19-Apr-2015.)
(6 · 4) = 24
 
Theorem6t5e30 8908 6 times 5 equals 30. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
(6 · 5) = 30
 
Theorem6t6e36 8909 6 times 6 equals 36. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
(6 · 6) = 36
 
Theorem7t2e14 8910 7 times 2 equals 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
(7 · 2) = 14
 
Theorem7t3e21 8911 7 times 3 equals 21. (Contributed by Mario Carneiro, 19-Apr-2015.)
(7 · 3) = 21
 
Theorem7t4e28 8912 7 times 4 equals 28. (Contributed by Mario Carneiro, 19-Apr-2015.)
(7 · 4) = 28
 
Theorem7t5e35 8913 7 times 5 equals 35. (Contributed by Mario Carneiro, 19-Apr-2015.)
(7 · 5) = 35
 
Theorem7t6e42 8914 7 times 6 equals 42. (Contributed by Mario Carneiro, 19-Apr-2015.)
(7 · 6) = 42
 
Theorem7t7e49 8915 7 times 7 equals 49. (Contributed by Mario Carneiro, 19-Apr-2015.)
(7 · 7) = 49
 
Theorem8t2e16 8916 8 times 2 equals 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
(8 · 2) = 16
 
Theorem8t3e24 8917 8 times 3 equals 24. (Contributed by Mario Carneiro, 19-Apr-2015.)
(8 · 3) = 24
 
Theorem8t4e32 8918 8 times 4 equals 32. (Contributed by Mario Carneiro, 19-Apr-2015.)
(8 · 4) = 32
 
Theorem8t5e40 8919 8 times 5 equals 40. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
(8 · 5) = 40
 
Theorem8t6e48 8920 8 times 6 equals 48. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
(8 · 6) = 48
 
Theorem8t7e56 8921 8 times 7 equals 56. (Contributed by Mario Carneiro, 19-Apr-2015.)
(8 · 7) = 56
 
Theorem8t8e64 8922 8 times 8 equals 64. (Contributed by Mario Carneiro, 19-Apr-2015.)
(8 · 8) = 64
 
Theorem9t2e18 8923 9 times 2 equals 18. (Contributed by Mario Carneiro, 19-Apr-2015.)
(9 · 2) = 18
 
Theorem9t3e27 8924 9 times 3 equals 27. (Contributed by Mario Carneiro, 19-Apr-2015.)
(9 · 3) = 27
 
Theorem9t4e36 8925 9 times 4 equals 36. (Contributed by Mario Carneiro, 19-Apr-2015.)
(9 · 4) = 36
 
Theorem9t5e45 8926 9 times 5 equals 45. (Contributed by Mario Carneiro, 19-Apr-2015.)
(9 · 5) = 45
 
Theorem9t6e54 8927 9 times 6 equals 54. (Contributed by Mario Carneiro, 19-Apr-2015.)
(9 · 6) = 54
 
Theorem9t7e63 8928 9 times 7 equals 63. (Contributed by Mario Carneiro, 19-Apr-2015.)
(9 · 7) = 63
 
Theorem9t8e72 8929 9 times 8 equals 72. (Contributed by Mario Carneiro, 19-Apr-2015.)
(9 · 8) = 72
 
Theorem9t9e81 8930 9 times 9 equals 81. (Contributed by Mario Carneiro, 19-Apr-2015.)
(9 · 9) = 81
 
Theorem9t11e99 8931 9 times 11 equals 99. (Contributed by AV, 14-Jun-2021.) (Revised by AV, 6-Sep-2021.)
(9 · 11) = 99
 
Theorem9lt10 8932 9 is less than 10. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by AV, 8-Sep-2021.)
9 < 10
 
Theorem8lt10 8933 8 is less than 10. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by AV, 8-Sep-2021.)
8 < 10
 
Theorem7lt10 8934 7 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.)
7 < 10
 
Theorem6lt10 8935 6 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.)
6 < 10
 
Theorem5lt10 8936 5 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.)
5 < 10
 
Theorem4lt10 8937 4 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.)
4 < 10
 
Theorem3lt10 8938 3 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.)
3 < 10
 
Theorem2lt10 8939 2 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.)
2 < 10
 
Theorem1lt10 8940 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
 
Theoremdecbin0 8941 Decompose base 4 into base 2. (Contributed by Mario Carneiro, 18-Feb-2014.)
𝐴 ∈ ℕ0       (4 · 𝐴) = (2 · (2 · 𝐴))
 
Theoremdecbin2 8942 Decompose base 4 into base 2. (Contributed by Mario Carneiro, 18-Feb-2014.)
𝐴 ∈ ℕ0       ((4 · 𝐴) + 2) = (2 · ((2 · 𝐴) + 1))
 
Theoremdecbin3 8943 Decompose base 4 into base 2. (Contributed by Mario Carneiro, 18-Feb-2014.)
𝐴 ∈ ℕ0       ((4 · 𝐴) + 3) = ((2 · ((2 · 𝐴) + 1)) + 1)
 
3.4.11  Upper sets of integers
 
Syntaxcuz 8944 Extend class notation with the upper integer function. Read "𝑀 " as "the set of integers greater than or equal to 𝑀."
class
 
Definitiondf-uz 8945* 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 8946 for its value, uzssz 8963 for its relationship to , nnuz 8979 and nn0uz 8978 for its relationships to and 0, and eluz1 8948 and eluz2 8950 for its membership relations. (Contributed by NM, 5-Sep-2005.)
= (𝑗 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ 𝑗𝑘})
 
Theoremuzval 8946* 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 8947 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 8948 Membership in the upper set of integers starting at 𝑀. (Contributed by NM, 5-Sep-2005.)
(𝑀 ∈ ℤ → (𝑁 ∈ (ℤ𝑀) ↔ (𝑁 ∈ ℤ ∧ 𝑀𝑁)))
 
Theoremeluzel2 8949 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 8950 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 ∈ (ℤ𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀𝑁))
 
Theoremeluz1i 8951 Membership in an upper set of integers. (Contributed by NM, 5-Sep-2005.)
𝑀 ∈ ℤ       (𝑁 ∈ (ℤ𝑀) ↔ (𝑁 ∈ ℤ ∧ 𝑀𝑁))
 
Theoremeluzuzle 8952 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 ∧ 𝐵𝐴) → (𝐶 ∈ (ℤ𝐴) → 𝐶 ∈ (ℤ𝐵)))
 
Theoremeluzelz 8953 A member of an upper set of integers is an integer. (Contributed by NM, 6-Sep-2005.)
(𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ ℤ)
 
Theoremeluzelre 8954 A member of an upper set of integers is a real. (Contributed by Mario Carneiro, 31-Aug-2013.)
(𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ ℝ)
 
Theoremeluzelcn 8955 A member of an upper set of integers is a complex number. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
(𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ ℂ)
 
Theoremeluzle 8956 Implication of membership in an upper set of integers. (Contributed by NM, 6-Sep-2005.)
(𝑁 ∈ (ℤ𝑀) → 𝑀𝑁)
 
Theoremeluz 8957 Membership in an upper set of integers. (Contributed by NM, 2-Oct-2005.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ𝑀) ↔ 𝑀𝑁))
 
Theoremuzid 8958 Membership of the least member in an upper set of integers. (Contributed by NM, 2-Sep-2005.)
(𝑀 ∈ ℤ → 𝑀 ∈ (ℤ𝑀))
 
Theoremuzn0 8959 The upper integers are all nonempty. (Contributed by Mario Carneiro, 16-Jan-2014.)
(𝑀 ∈ ran ℤ𝑀 ≠ ∅)
 
Theoremuztrn 8960 Transitive law for sets of upper integers. (Contributed by NM, 20-Sep-2005.)
((𝑀 ∈ (ℤ𝐾) ∧ 𝐾 ∈ (ℤ𝑁)) → 𝑀 ∈ (ℤ𝑁))
 
Theoremuztrn2 8961 Transitive law for sets of upper integers. (Contributed by Mario Carneiro, 26-Dec-2013.)
𝑍 = (ℤ𝐾)       ((𝑁𝑍𝑀 ∈ (ℤ𝑁)) → 𝑀𝑍)
 
Theoremuzneg 8962 Contraposition law for upper integers. (Contributed by NM, 28-Nov-2005.)
(𝑁 ∈ (ℤ𝑀) → -𝑀 ∈ (ℤ‘-𝑁))
 
Theoremuzssz 8963 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 8964 Subset relationship for two sets of upper integers. (Contributed by NM, 5-Sep-2005.)
(𝑁 ∈ (ℤ𝑀) → (ℤ𝑁) ⊆ (ℤ𝑀))
 
Theoremuztric 8965 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 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ𝑀) ∨ 𝑀 ∈ (ℤ𝑁)))
 
Theoremuz11 8966 The upper integers function is one-to-one. (Contributed by NM, 12-Dec-2005.)
(𝑀 ∈ ℤ → ((ℤ𝑀) = (ℤ𝑁) ↔ 𝑀 = 𝑁))
 
Theoremeluzp1m1 8967 Membership in the next upper set of integers. (Contributed by NM, 12-Sep-2005.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ (ℤ‘(𝑀 + 1))) → (𝑁 − 1) ∈ (ℤ𝑀))
 
Theoremeluzp1l 8968 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)
 
Theoremeluzp1p1 8969 Membership in the next upper set of integers. (Contributed by NM, 5-Oct-2005.)
(𝑁 ∈ (ℤ𝑀) → (𝑁 + 1) ∈ (ℤ‘(𝑀 + 1)))
 
Theoremeluzaddi 8970 Membership in a later upper set of integers. (Contributed by Paul Chapman, 22-Nov-2007.)
𝑀 ∈ ℤ    &   𝐾 ∈ ℤ       (𝑁 ∈ (ℤ𝑀) → (𝑁 + 𝐾) ∈ (ℤ‘(𝑀 + 𝐾)))
 
Theoremeluzsubi 8971 Membership in an earlier upper set of integers. (Contributed by Paul Chapman, 22-Nov-2007.)
𝑀 ∈ ℤ    &   𝐾 ∈ ℤ       (𝑁 ∈ (ℤ‘(𝑀 + 𝐾)) → (𝑁𝐾) ∈ (ℤ𝑀))
 
Theoremeluzadd 8972 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 8973 Membership in an earlier upper set of integers. (Contributed by Jeff Madsen, 2-Sep-2009.)
((𝑀 ∈ ℤ ∧ 𝐾 ∈ ℤ ∧ 𝑁 ∈ (ℤ‘(𝑀 + 𝐾))) → (𝑁𝐾) ∈ (ℤ𝑀))
 
Theoremuzm1 8974 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 8975 The nonnegative difference of integers is a nonnegative integer. (Contributed by NM, 4-Sep-2005.)
(𝑁 ∈ (ℤ𝑀) → (𝑁𝑀) ∈ ℕ0)
 
Theoremuzin 8976 Intersection of two upper intervals of integers. (Contributed by Mario Carneiro, 24-Dec-2013.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((ℤ𝑀) ∩ (ℤ𝑁)) = (ℤ‘if(𝑀𝑁, 𝑁, 𝑀)))
 
Theoremuzp1 8977 Choices for an element of an upper interval of integers. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝑁 ∈ (ℤ𝑀) → (𝑁 = 𝑀𝑁 ∈ (ℤ‘(𝑀 + 1))))
 
Theoremnn0uz 8978 Nonnegative integers expressed as an upper set of integers. (Contributed by NM, 2-Sep-2005.)
0 = (ℤ‘0)
 
Theoremnnuz 8979 Positive integers expressed as an upper set of integers. (Contributed by NM, 2-Sep-2005.)
ℕ = (ℤ‘1)
 
Theoremelnnuz 8980 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 8981 A nonnegative integer expressed as a member an upper set of integers. (Contributed by NM, 6-Jun-2006.)
(𝑁 ∈ ℕ0𝑁 ∈ (ℤ‘0))
 
Theoremeluz2nn 8982 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 8983 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)
 
Theoremuzuzle23 8984 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 8985 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 8986 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 8987 1 is an integer greater than or equal to 0. (Contributed by Alexander van der Vekens, 8-Jun-2018.)
1 ∈ (ℤ‘0)
 
Theorem2eluzge0 8988 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)
 
Theorem2eluzge1 8989 2 is an integer greater than or equal to 1. (Contributed by Alexander van der Vekens, 8-Jun-2018.)
2 ∈ (ℤ‘1)
 
Theoremuznnssnn 8990 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 8991* Restricted universal quantification in an upper set of integers. (Contributed by NM, 9-Sep-2005.)
(𝑀 ∈ ℤ → (∀𝑛 ∈ (ℤ𝑀)𝜑 ↔ ∀𝑛 ∈ ℤ (𝑀𝑛𝜑)))
 
Theoremraluz2 8992* Restricted universal quantification in an upper set of integers. (Contributed by NM, 9-Sep-2005.)
(∀𝑛 ∈ (ℤ𝑀)𝜑 ↔ (𝑀 ∈ ℤ → ∀𝑛 ∈ ℤ (𝑀𝑛𝜑)))
 
Theoremrexuz 8993* Restricted existential quantification in an upper set of integers. (Contributed by NM, 9-Sep-2005.)
(𝑀 ∈ ℤ → (∃𝑛 ∈ (ℤ𝑀)𝜑 ↔ ∃𝑛 ∈ ℤ (𝑀𝑛𝜑)))
 
Theoremrexuz2 8994* Restricted existential quantification in an upper set of integers. (Contributed by NM, 9-Sep-2005.)
(∃𝑛 ∈ (ℤ𝑀)𝜑 ↔ (𝑀 ∈ ℤ ∧ ∃𝑛 ∈ ℤ (𝑀𝑛𝜑)))
 
Theorem2rexuz 8995* Double existential quantification in an upper set of integers. (Contributed by NM, 3-Nov-2005.)
(∃𝑚𝑛 ∈ (ℤ𝑚)𝜑 ↔ ∃𝑚 ∈ ℤ ∃𝑛 ∈ ℤ (𝑚𝑛𝜑))
 
Theorempeano2uz 8996 Second Peano postulate for an upper set of integers. (Contributed by NM, 7-Sep-2005.)
(𝑁 ∈ (ℤ𝑀) → (𝑁 + 1) ∈ (ℤ𝑀))
 
Theorempeano2uzs 8997 Second Peano postulate for an upper set of integers. (Contributed by Mario Carneiro, 26-Dec-2013.)
𝑍 = (ℤ𝑀)       (𝑁𝑍 → (𝑁 + 1) ∈ 𝑍)
 
Theorempeano2uzr 8998 Reversed second Peano axiom for upper integers. (Contributed by NM, 2-Jan-2006.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ (ℤ‘(𝑀 + 1))) → 𝑁 ∈ (ℤ𝑀))
 
Theoremuzaddcl 8999 Addition closure law for an upper set of integers. (Contributed by NM, 4-Jun-2006.)
((𝑁 ∈ (ℤ𝑀) ∧ 𝐾 ∈ ℕ0) → (𝑁 + 𝐾) ∈ (ℤ𝑀))
 
Theoremnn0pzuz 9000 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 ∈ ℕ0𝑍 ∈ ℤ) → (𝑁 + 𝑍) ∈ (ℤ𝑍))
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