HomeHome Intuitionistic Logic Explorer
Theorem List (p. 102 of 142)
< Previous  Next >
Bad symbols? Try the
GIF version.

Mirrors  >  Metamath Home Page  >  ILE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Theorem List for Intuitionistic Logic Explorer - 10101-10200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremelfzoel1 10101 Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.)
(𝐴 ∈ (𝐵..^𝐶) → 𝐵 ∈ ℤ)
 
Theoremelfzoel2 10102 Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.)
(𝐴 ∈ (𝐵..^𝐶) → 𝐶 ∈ ℤ)
 
Theoremelfzoelz 10103 Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.)
(𝐴 ∈ (𝐵..^𝐶) → 𝐴 ∈ ℤ)
 
Theoremfzoval 10104 Value of the half-open integer set in terms of the closed integer set. (Contributed by Stefan O'Rear, 14-Aug-2015.)
(𝑁 ∈ ℤ → (𝑀..^𝑁) = (𝑀...(𝑁 − 1)))
 
Theoremelfzo 10105 Membership in a half-open finite set of integers. (Contributed by Stefan O'Rear, 15-Aug-2015.)
((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 ∈ (𝑀..^𝑁) ↔ (𝑀𝐾𝐾 < 𝑁)))
 
Theoremelfzo2 10106 Membership in a half-open integer interval. (Contributed by Mario Carneiro, 29-Sep-2015.)
(𝐾 ∈ (𝑀..^𝑁) ↔ (𝐾 ∈ (ℤ𝑀) ∧ 𝑁 ∈ ℤ ∧ 𝐾 < 𝑁))
 
Theoremelfzouz 10107 Membership in a half-open integer interval. (Contributed by Mario Carneiro, 29-Sep-2015.)
(𝐾 ∈ (𝑀..^𝑁) → 𝐾 ∈ (ℤ𝑀))
 
Theoremfzodcel 10108 Decidability of membership in a half-open integer interval. (Contributed by Jim Kingdon, 25-Aug-2022.)
((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID 𝐾 ∈ (𝑀..^𝑁))
 
Theoremfzolb 10109 The left endpoint of a half-open integer interval is in the set iff the two arguments are integers with 𝑀 < 𝑁. This provides an alternate notation for the "strict upper integer" predicate by analogy to the "weak upper integer" predicate 𝑀 ∈ (ℤ𝑁). (Contributed by Mario Carneiro, 29-Sep-2015.)
(𝑀 ∈ (𝑀..^𝑁) ↔ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 < 𝑁))
 
Theoremfzolb2 10110 The left endpoint of a half-open integer interval is in the set iff the two arguments are integers with 𝑀 < 𝑁. This provides an alternate notation for the "strict upper integer" predicate by analogy to the "weak upper integer" predicate 𝑀 ∈ (ℤ𝑁). (Contributed by Mario Carneiro, 29-Sep-2015.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∈ (𝑀..^𝑁) ↔ 𝑀 < 𝑁))
 
Theoremelfzole1 10111 A member in a half-open integer interval is greater than or equal to the lower bound. (Contributed by Stefan O'Rear, 15-Aug-2015.)
(𝐾 ∈ (𝑀..^𝑁) → 𝑀𝐾)
 
Theoremelfzolt2 10112 A member in a half-open integer interval is less than the upper bound. (Contributed by Stefan O'Rear, 15-Aug-2015.)
(𝐾 ∈ (𝑀..^𝑁) → 𝐾 < 𝑁)
 
Theoremelfzolt3 10113 Membership in a half-open integer interval implies that the bounds are unequal. (Contributed by Stefan O'Rear, 15-Aug-2015.)
(𝐾 ∈ (𝑀..^𝑁) → 𝑀 < 𝑁)
 
Theoremelfzolt2b 10114 A member in a half-open integer interval is less than the upper bound. (Contributed by Mario Carneiro, 29-Sep-2015.)
(𝐾 ∈ (𝑀..^𝑁) → 𝐾 ∈ (𝐾..^𝑁))
 
Theoremelfzolt3b 10115 Membership in a half-open integer interval implies that the bounds are unequal. (Contributed by Mario Carneiro, 29-Sep-2015.)
(𝐾 ∈ (𝑀..^𝑁) → 𝑀 ∈ (𝑀..^𝑁))
 
Theoremfzonel 10116 A half-open range does not contain its right endpoint. (Contributed by Stefan O'Rear, 25-Aug-2015.)
¬ 𝐵 ∈ (𝐴..^𝐵)
 
Theoremelfzouz2 10117 The upper bound of a half-open range is greater or equal to an element of the range. (Contributed by Mario Carneiro, 29-Sep-2015.)
(𝐾 ∈ (𝑀..^𝑁) → 𝑁 ∈ (ℤ𝐾))
 
Theoremelfzofz 10118 A half-open range is contained in the corresponding closed range. (Contributed by Stefan O'Rear, 23-Aug-2015.)
(𝐾 ∈ (𝑀..^𝑁) → 𝐾 ∈ (𝑀...𝑁))
 
Theoremelfzo3 10119 Express membership in a half-open integer interval in terms of the "less than or equal" and "less than" predicates on integers, resp. 𝐾 ∈ (ℤ𝑀) ↔ 𝑀𝐾, 𝐾 ∈ (𝐾..^𝑁) ↔ 𝐾 < 𝑁. (Contributed by Mario Carneiro, 29-Sep-2015.)
(𝐾 ∈ (𝑀..^𝑁) ↔ (𝐾 ∈ (ℤ𝑀) ∧ 𝐾 ∈ (𝐾..^𝑁)))
 
Theoremfzom 10120* A half-open integer interval is inhabited iff it contains its left endpoint. (Contributed by Jim Kingdon, 20-Apr-2020.)
(∃𝑥 𝑥 ∈ (𝑀..^𝑁) ↔ 𝑀 ∈ (𝑀..^𝑁))
 
Theoremfzossfz 10121 A half-open range is contained in the corresponding closed range. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Mario Carneiro, 29-Sep-2015.)
(𝐴..^𝐵) ⊆ (𝐴...𝐵)
 
Theoremfzon 10122 A half-open set of sequential integers is empty if the bounds are equal or reversed. (Contributed by Alexander van der Vekens, 30-Oct-2017.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁𝑀 ↔ (𝑀..^𝑁) = ∅))
 
Theoremfzonlt0 10123 A half-open integer range is empty if the bounds are equal or reversed. (Contributed by AV, 20-Oct-2018.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (¬ 𝑀 < 𝑁 ↔ (𝑀..^𝑁) = ∅))
 
Theoremfzo0 10124 Half-open sets with equal endpoints are empty. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 29-Sep-2015.)
(𝐴..^𝐴) = ∅
 
Theoremfzonnsub 10125 If 𝐾 < 𝑁 then 𝑁𝐾 is a positive integer. (Contributed by Mario Carneiro, 29-Sep-2015.) (Revised by Mario Carneiro, 1-Jan-2017.)
(𝐾 ∈ (𝑀..^𝑁) → (𝑁𝐾) ∈ ℕ)
 
Theoremfzonnsub2 10126 If 𝑀 < 𝑁 then 𝑁𝑀 is a positive integer. (Contributed by Mario Carneiro, 1-Jan-2017.)
(𝐾 ∈ (𝑀..^𝑁) → (𝑁𝑀) ∈ ℕ)
 
Theoremfzoss1 10127 Subset relationship for half-open sequences of integers. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 29-Sep-2015.)
(𝐾 ∈ (ℤ𝑀) → (𝐾..^𝑁) ⊆ (𝑀..^𝑁))
 
Theoremfzoss2 10128 Subset relationship for half-open sequences of integers. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 29-Sep-2015.)
(𝑁 ∈ (ℤ𝐾) → (𝑀..^𝐾) ⊆ (𝑀..^𝑁))
 
Theoremfzossrbm1 10129 Subset of a half open range. (Contributed by Alexander van der Vekens, 1-Nov-2017.)
(𝑁 ∈ ℤ → (0..^(𝑁 − 1)) ⊆ (0..^𝑁))
 
Theoremfzo0ss1 10130 Subset relationship for half-open integer ranges with lower bounds 0 and 1. (Contributed by Alexander van der Vekens, 18-Mar-2018.)
(1..^𝑁) ⊆ (0..^𝑁)
 
Theoremfzossnn0 10131 A half-open integer range starting at a nonnegative integer is a subset of the nonnegative integers. (Contributed by Alexander van der Vekens, 13-May-2018.)
(𝑀 ∈ ℕ0 → (𝑀..^𝑁) ⊆ ℕ0)
 
Theoremfzospliti 10132 One direction of splitting a half-open integer range in half. (Contributed by Stefan O'Rear, 14-Aug-2015.)
((𝐴 ∈ (𝐵..^𝐶) ∧ 𝐷 ∈ ℤ) → (𝐴 ∈ (𝐵..^𝐷) ∨ 𝐴 ∈ (𝐷..^𝐶)))
 
Theoremfzosplit 10133 Split a half-open integer range in half. (Contributed by Stefan O'Rear, 14-Aug-2015.)
(𝐷 ∈ (𝐵...𝐶) → (𝐵..^𝐶) = ((𝐵..^𝐷) ∪ (𝐷..^𝐶)))
 
Theoremfzodisj 10134 Abutting half-open integer ranges are disjoint. (Contributed by Stefan O'Rear, 14-Aug-2015.)
((𝐴..^𝐵) ∩ (𝐵..^𝐶)) = ∅
 
Theoremfzouzsplit 10135 Split an upper integer set into a half-open integer range and another upper integer set. (Contributed by Mario Carneiro, 21-Sep-2016.)
(𝐵 ∈ (ℤ𝐴) → (ℤ𝐴) = ((𝐴..^𝐵) ∪ (ℤ𝐵)))
 
Theoremfzouzdisj 10136 A half-open integer range does not overlap the upper integer range starting at the endpoint of the first range. (Contributed by Mario Carneiro, 21-Sep-2016.)
((𝐴..^𝐵) ∩ (ℤ𝐵)) = ∅
 
Theoremlbfzo0 10137 An integer is strictly greater than zero iff it is a member of . (Contributed by Mario Carneiro, 29-Sep-2015.)
(0 ∈ (0..^𝐴) ↔ 𝐴 ∈ ℕ)
 
Theoremelfzo0 10138 Membership in a half-open integer range based at 0. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 29-Sep-2015.)
(𝐴 ∈ (0..^𝐵) ↔ (𝐴 ∈ ℕ0𝐵 ∈ ℕ ∧ 𝐴 < 𝐵))
 
Theoremfzo1fzo0n0 10139 An integer between 1 and an upper bound of a half-open integer range is not 0 and between 0 and the upper bound of the half-open integer range. (Contributed by Alexander van der Vekens, 21-Mar-2018.)
(𝐾 ∈ (1..^𝑁) ↔ (𝐾 ∈ (0..^𝑁) ∧ 𝐾 ≠ 0))
 
Theoremelfzo0z 10140 Membership in a half-open range of nonnegative integers, generalization of elfzo0 10138 requiring the upper bound to be an integer only. (Contributed by Alexander van der Vekens, 23-Sep-2018.)
(𝐴 ∈ (0..^𝐵) ↔ (𝐴 ∈ ℕ0𝐵 ∈ ℤ ∧ 𝐴 < 𝐵))
 
Theoremelfzo0le 10141 A member in a half-open range of nonnegative integers is less than or equal to the upper bound of the range. (Contributed by Alexander van der Vekens, 23-Sep-2018.)
(𝐴 ∈ (0..^𝐵) → 𝐴𝐵)
 
Theoremelfzonn0 10142 A member of a half-open range of nonnegative integers is a nonnegative integer. (Contributed by Alexander van der Vekens, 21-May-2018.)
(𝐾 ∈ (0..^𝑁) → 𝐾 ∈ ℕ0)
 
Theoremfzonmapblen 10143 The result of subtracting a nonnegative integer from a positive integer and adding another nonnegative integer which is less than the first one is less then the positive integer. (Contributed by Alexander van der Vekens, 19-May-2018.)
((𝐴 ∈ (0..^𝑁) ∧ 𝐵 ∈ (0..^𝑁) ∧ 𝐵 < 𝐴) → (𝐵 + (𝑁𝐴)) < 𝑁)
 
Theoremfzofzim 10144 If a nonnegative integer in a finite interval of integers is not the upper bound of the interval, it is contained in the corresponding half-open integer range. (Contributed by Alexander van der Vekens, 15-Jun-2018.)
((𝐾𝑀𝐾 ∈ (0...𝑀)) → 𝐾 ∈ (0..^𝑀))
 
Theoremfzossnn 10145 Half-open integer ranges starting with 1 are subsets of . (Contributed by Thierry Arnoux, 28-Dec-2016.)
(1..^𝑁) ⊆ ℕ
 
Theoremelfzo1 10146 Membership in a half-open integer range based at 1. (Contributed by Thierry Arnoux, 14-Feb-2017.)
(𝑁 ∈ (1..^𝑀) ↔ (𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑁 < 𝑀))
 
Theoremfzo0m 10147* A half-open integer range based at 0 is inhabited precisely if the upper bound is a positive integer. (Contributed by Jim Kingdon, 20-Apr-2020.)
(∃𝑥 𝑥 ∈ (0..^𝐴) ↔ 𝐴 ∈ ℕ)
 
Theoremfzoaddel 10148 Translate membership in a half-open integer range. (Contributed by Stefan O'Rear, 15-Aug-2015.)
((𝐴 ∈ (𝐵..^𝐶) ∧ 𝐷 ∈ ℤ) → (𝐴 + 𝐷) ∈ ((𝐵 + 𝐷)..^(𝐶 + 𝐷)))
 
Theoremfzoaddel2 10149 Translate membership in a shifted-down half-open integer range. (Contributed by Stefan O'Rear, 15-Aug-2015.)
((𝐴 ∈ (0..^(𝐵𝐶)) ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (𝐴 + 𝐶) ∈ (𝐶..^𝐵))
 
Theoremfzosubel 10150 Translate membership in a half-open integer range. (Contributed by Stefan O'Rear, 15-Aug-2015.)
((𝐴 ∈ (𝐵..^𝐶) ∧ 𝐷 ∈ ℤ) → (𝐴𝐷) ∈ ((𝐵𝐷)..^(𝐶𝐷)))
 
Theoremfzosubel2 10151 Membership in a translated half-open integer range implies translated membership in the original range. (Contributed by Stefan O'Rear, 15-Aug-2015.)
((𝐴 ∈ ((𝐵 + 𝐶)..^(𝐵 + 𝐷)) ∧ (𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → (𝐴𝐵) ∈ (𝐶..^𝐷))
 
Theoremfzosubel3 10152 Membership in a translated half-open integer range when the original range is zero-based. (Contributed by Stefan O'Rear, 15-Aug-2015.)
((𝐴 ∈ (𝐵..^(𝐵 + 𝐷)) ∧ 𝐷 ∈ ℤ) → (𝐴𝐵) ∈ (0..^𝐷))
 
Theoremeluzgtdifelfzo 10153 Membership of the difference of integers in a half-open range of nonnegative integers. (Contributed by Alexander van der Vekens, 17-Sep-2018.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝑁 ∈ (ℤ𝐴) ∧ 𝐵 < 𝐴) → (𝑁𝐴) ∈ (0..^(𝑁𝐵))))
 
Theoremige2m2fzo 10154 Membership of an integer greater than 1 decreased by 2 in a half-open range of nonnegative integers. (Contributed by Alexander van der Vekens, 3-Oct-2018.)
(𝑁 ∈ (ℤ‘2) → (𝑁 − 2) ∈ (0..^(𝑁 − 1)))
 
Theoremfzocatel 10155 Translate membership in a half-open integer range. (Contributed by Thierry Arnoux, 28-Sep-2018.)
(((𝐴 ∈ (0..^(𝐵 + 𝐶)) ∧ ¬ 𝐴 ∈ (0..^𝐵)) ∧ (𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → (𝐴𝐵) ∈ (0..^𝐶))
 
Theoremubmelfzo 10156 If an integer in a 1 based finite set of sequential integers is subtracted from the upper bound of this finite set of sequential integers, the result is contained in a half-open range of nonnegative integers with the same upper bound. (Contributed by AV, 18-Mar-2018.) (Revised by AV, 30-Oct-2018.)
(𝐾 ∈ (1...𝑁) → (𝑁𝐾) ∈ (0..^𝑁))
 
Theoremelfzodifsumelfzo 10157 If an integer is in a half-open range of nonnegative integers with a difference as upper bound, the sum of the integer with the subtrahend of the difference is in the a half-open range of nonnegative integers containing the minuend of the difference. (Contributed by AV, 13-Nov-2018.)
((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...𝑃)) → (𝐼 ∈ (0..^(𝑁𝑀)) → (𝐼 + 𝑀) ∈ (0..^𝑃)))
 
Theoremelfzom1elp1fzo 10158 Membership of an integer incremented by one in a half-open range of nonnegative integers. (Contributed by Alexander van der Vekens, 24-Jun-2018.) (Proof shortened by AV, 5-Jan-2020.)
((𝑁 ∈ ℤ ∧ 𝐼 ∈ (0..^(𝑁 − 1))) → (𝐼 + 1) ∈ (0..^𝑁))
 
Theoremelfzom1elfzo 10159 Membership in a half-open range of nonnegative integers. (Contributed by Alexander van der Vekens, 18-Jun-2018.)
((𝑁 ∈ ℤ ∧ 𝐼 ∈ (0..^(𝑁 − 1))) → 𝐼 ∈ (0..^𝑁))
 
Theoremfzval3 10160 Expressing a closed integer range as a half-open integer range. (Contributed by Stefan O'Rear, 15-Aug-2015.)
(𝑁 ∈ ℤ → (𝑀...𝑁) = (𝑀..^(𝑁 + 1)))
 
Theoremfzosn 10161 Expressing a singleton as a half-open range. (Contributed by Stefan O'Rear, 23-Aug-2015.)
(𝐴 ∈ ℤ → (𝐴..^(𝐴 + 1)) = {𝐴})
 
Theoremelfzomin 10162 Membership of an integer in the smallest open range of integers. (Contributed by Alexander van der Vekens, 22-Sep-2018.)
(𝑍 ∈ ℤ → 𝑍 ∈ (𝑍..^(𝑍 + 1)))
 
Theoremzpnn0elfzo 10163 Membership of an integer increased by a nonnegative integer in a half- open integer range. (Contributed by Alexander van der Vekens, 22-Sep-2018.)
((𝑍 ∈ ℤ ∧ 𝑁 ∈ ℕ0) → (𝑍 + 𝑁) ∈ (𝑍..^((𝑍 + 𝑁) + 1)))
 
Theoremzpnn0elfzo1 10164 Membership of an integer increased by a nonnegative integer in a half- open integer range. (Contributed by Alexander van der Vekens, 22-Sep-2018.)
((𝑍 ∈ ℤ ∧ 𝑁 ∈ ℕ0) → (𝑍 + 𝑁) ∈ (𝑍..^(𝑍 + (𝑁 + 1))))
 
Theoremfzosplitsnm1 10165 Removing a singleton from a half-open integer range at the end. (Contributed by Alexander van der Vekens, 23-Mar-2018.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ (ℤ‘(𝐴 + 1))) → (𝐴..^𝐵) = ((𝐴..^(𝐵 − 1)) ∪ {(𝐵 − 1)}))
 
Theoremelfzonlteqm1 10166 If an element of a half-open integer range is not less than the upper bound of the range decreased by 1, it must be equal to the upper bound of the range decreased by 1. (Contributed by AV, 3-Nov-2018.)
((𝐴 ∈ (0..^𝐵) ∧ ¬ 𝐴 < (𝐵 − 1)) → 𝐴 = (𝐵 − 1))
 
Theoremfzonn0p1 10167 A nonnegative integer is element of the half-open range of nonnegative integers with the element increased by one as an upper bound. (Contributed by Alexander van der Vekens, 5-Aug-2018.)
(𝑁 ∈ ℕ0𝑁 ∈ (0..^(𝑁 + 1)))
 
Theoremfzossfzop1 10168 A half-open range of nonnegative integers is a subset of a half-open range of nonnegative integers with the upper bound increased by one. (Contributed by Alexander van der Vekens, 5-Aug-2018.)
(𝑁 ∈ ℕ0 → (0..^𝑁) ⊆ (0..^(𝑁 + 1)))
 
Theoremfzonn0p1p1 10169 If a nonnegative integer is element of a half-open range of nonnegative integers, increasing this integer by one results in an element of a half- open range of nonnegative integers with the upper bound increased by one. (Contributed by Alexander van der Vekens, 5-Aug-2018.)
(𝐼 ∈ (0..^𝑁) → (𝐼 + 1) ∈ (0..^(𝑁 + 1)))
 
Theoremelfzom1p1elfzo 10170 Increasing an element of a half-open range of nonnegative integers by 1 results in an element of the half-open range of nonnegative integers with an upper bound increased by 1. (Contributed by Alexander van der Vekens, 1-Aug-2018.)
((𝑁 ∈ ℕ ∧ 𝑋 ∈ (0..^(𝑁 − 1))) → (𝑋 + 1) ∈ (0..^𝑁))
 
Theoremfzo0ssnn0 10171 Half-open integer ranges starting with 0 are subsets of NN0. (Contributed by Thierry Arnoux, 8-Oct-2018.)
(0..^𝑁) ⊆ ℕ0
 
Theoremfzo01 10172 Expressing the singleton of 0 as a half-open integer range. (Contributed by Stefan O'Rear, 15-Aug-2015.)
(0..^1) = {0}
 
Theoremfzo12sn 10173 A 1-based half-open integer interval up to, but not including, 2 is a singleton. (Contributed by Alexander van der Vekens, 31-Jan-2018.)
(1..^2) = {1}
 
Theoremfzo0to2pr 10174 A half-open integer range from 0 to 2 is an unordered pair. (Contributed by Alexander van der Vekens, 4-Dec-2017.)
(0..^2) = {0, 1}
 
Theoremfzo0to3tp 10175 A half-open integer range from 0 to 3 is an unordered triple. (Contributed by Alexander van der Vekens, 9-Nov-2017.)
(0..^3) = {0, 1, 2}
 
Theoremfzo0to42pr 10176 A half-open integer range from 0 to 4 is a union of two unordered pairs. (Contributed by Alexander van der Vekens, 17-Nov-2017.)
(0..^4) = ({0, 1} ∪ {2, 3})
 
Theoremfzo0sn0fzo1 10177 A half-open range of nonnegative integers is the union of the singleton set containing 0 and a half-open range of positive integers. (Contributed by Alexander van der Vekens, 18-May-2018.)
(𝑁 ∈ ℕ → (0..^𝑁) = ({0} ∪ (1..^𝑁)))
 
Theoremfzoend 10178 The endpoint of a half-open integer range. (Contributed by Mario Carneiro, 29-Sep-2015.)
(𝐴 ∈ (𝐴..^𝐵) → (𝐵 − 1) ∈ (𝐴..^𝐵))
 
Theoremfzo0end 10179 The endpoint of a zero-based half-open range. (Contributed by Stefan O'Rear, 27-Aug-2015.) (Revised by Mario Carneiro, 29-Sep-2015.)
(𝐵 ∈ ℕ → (𝐵 − 1) ∈ (0..^𝐵))
 
Theoremssfzo12 10180 Subset relationship for half-open integer ranges. (Contributed by Alexander van der Vekens, 16-Mar-2018.)
((𝐾 ∈ ℤ ∧ 𝐿 ∈ ℤ ∧ 𝐾 < 𝐿) → ((𝐾..^𝐿) ⊆ (𝑀..^𝑁) → (𝑀𝐾𝐿𝑁)))
 
Theoremssfzo12bi 10181 Subset relationship for half-open integer ranges. (Contributed by Alexander van der Vekens, 5-Nov-2018.)
(((𝐾 ∈ ℤ ∧ 𝐿 ∈ ℤ) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐾 < 𝐿) → ((𝐾..^𝐿) ⊆ (𝑀..^𝑁) ↔ (𝑀𝐾𝐿𝑁)))
 
Theoremubmelm1fzo 10182 The result of subtracting 1 and an integer of a half-open range of nonnegative integers from the upper bound of this range is contained in this range. (Contributed by AV, 23-Mar-2018.) (Revised by AV, 30-Oct-2018.)
(𝐾 ∈ (0..^𝑁) → ((𝑁𝐾) − 1) ∈ (0..^𝑁))
 
Theoremfzofzp1 10183 If a point is in a half-open range, the next point is in the closed range. (Contributed by Stefan O'Rear, 23-Aug-2015.)
(𝐶 ∈ (𝐴..^𝐵) → (𝐶 + 1) ∈ (𝐴...𝐵))
 
Theoremfzofzp1b 10184 If a point is in a half-open range, the next point is in the closed range. (Contributed by Mario Carneiro, 27-Sep-2015.)
(𝐶 ∈ (ℤ𝐴) → (𝐶 ∈ (𝐴..^𝐵) ↔ (𝐶 + 1) ∈ (𝐴...𝐵)))
 
Theoremelfzom1b 10185 An integer is a member of a 1-based finite set of sequential integers iff its predecessor is a member of the corresponding 0-based set. (Contributed by Mario Carneiro, 27-Sep-2015.)
((𝐾 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 ∈ (1..^𝑁) ↔ (𝐾 − 1) ∈ (0..^(𝑁 − 1))))
 
Theoremelfzonelfzo 10186 If an element of a half-open integer range is not contained in the lower subrange, it must be in the upper subrange. (Contributed by Alexander van der Vekens, 30-Mar-2018.)
(𝑁 ∈ ℤ → ((𝐾 ∈ (𝑀..^𝑅) ∧ ¬ 𝐾 ∈ (𝑀..^𝑁)) → 𝐾 ∈ (𝑁..^𝑅)))
 
Theoremelfzomelpfzo 10187 An integer increased by another integer is an element of a half-open integer range if and only if the integer is contained in the half-open integer range with bounds decreased by the other integer. (Contributed by Alexander van der Vekens, 30-Mar-2018.)
(((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐾 ∈ ℤ ∧ 𝐿 ∈ ℤ)) → (𝐾 ∈ ((𝑀𝐿)..^(𝑁𝐿)) ↔ (𝐾 + 𝐿) ∈ (𝑀..^𝑁)))
 
Theorempeano2fzor 10188 A Peano-postulate-like theorem for downward closure of a half-open integer range. (Contributed by Mario Carneiro, 1-Oct-2015.)
((𝐾 ∈ (ℤ𝑀) ∧ (𝐾 + 1) ∈ (𝑀..^𝑁)) → 𝐾 ∈ (𝑀..^𝑁))
 
Theoremfzosplitsn 10189 Extending a half-open range by a singleton on the end. (Contributed by Stefan O'Rear, 23-Aug-2015.)
(𝐵 ∈ (ℤ𝐴) → (𝐴..^(𝐵 + 1)) = ((𝐴..^𝐵) ∪ {𝐵}))
 
Theoremfzosplitprm1 10190 Extending a half-open integer range by an unordered pair at the end. (Contributed by Alexander van der Vekens, 22-Sep-2018.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐴 < 𝐵) → (𝐴..^(𝐵 + 1)) = ((𝐴..^(𝐵 − 1)) ∪ {(𝐵 − 1), 𝐵}))
 
Theoremfzosplitsni 10191 Membership in a half-open range extended by a singleton. (Contributed by Stefan O'Rear, 23-Aug-2015.)
(𝐵 ∈ (ℤ𝐴) → (𝐶 ∈ (𝐴..^(𝐵 + 1)) ↔ (𝐶 ∈ (𝐴..^𝐵) ∨ 𝐶 = 𝐵)))
 
Theoremfzisfzounsn 10192 A finite interval of integers as union of a half-open integer range and a singleton. (Contributed by Alexander van der Vekens, 15-Jun-2018.)
(𝐵 ∈ (ℤ𝐴) → (𝐴...𝐵) = ((𝐴..^𝐵) ∪ {𝐵}))
 
Theoremfzostep1 10193 Two possibilities for a number one greater than a number in a half-open range. (Contributed by Stefan O'Rear, 23-Aug-2015.)
(𝐴 ∈ (𝐵..^𝐶) → ((𝐴 + 1) ∈ (𝐵..^𝐶) ∨ (𝐴 + 1) = 𝐶))
 
Theoremfzoshftral 10194* Shift the scanning order inside of a quantification over a half-open integer range, analogous to fzshftral 10064. (Contributed by Alexander van der Vekens, 23-Sep-2018.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (∀𝑗 ∈ (𝑀..^𝑁)𝜑 ↔ ∀𝑘 ∈ ((𝑀 + 𝐾)..^(𝑁 + 𝐾))[(𝑘𝐾) / 𝑗]𝜑))
 
Theoremfzind2 10195* Induction on the integers from 𝑀 to 𝑁 inclusive. The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction step. Version of fzind 9327 using integer range definitions. (Contributed by Mario Carneiro, 6-Feb-2016.)
(𝑥 = 𝑀 → (𝜑𝜓))    &   (𝑥 = 𝑦 → (𝜑𝜒))    &   (𝑥 = (𝑦 + 1) → (𝜑𝜃))    &   (𝑥 = 𝐾 → (𝜑𝜏))    &   (𝑁 ∈ (ℤ𝑀) → 𝜓)    &   (𝑦 ∈ (𝑀..^𝑁) → (𝜒𝜃))       (𝐾 ∈ (𝑀...𝑁) → 𝜏)
 
Theoremexfzdc 10196* Decidability of the existence of an integer defined by a decidable proposition. (Contributed by Jim Kingdon, 28-Jan-2022.)
(𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ ℤ)    &   ((𝜑𝑛 ∈ (𝑀...𝑁)) → DECID 𝜓)       (𝜑DECID𝑛 ∈ (𝑀...𝑁)𝜓)
 
Theoremfvinim0ffz 10197 The function values for the borders of a finite interval of integers, which is the domain of the function, are not in the image of the interior of the interval iff the intersection of the images of the interior and the borders is empty. (Contributed by Alexander van der Vekens, 31-Oct-2017.) (Revised by AV, 5-Feb-2021.)
((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ ↔ ((𝐹‘0) ∉ (𝐹 “ (1..^𝐾)) ∧ (𝐹𝐾) ∉ (𝐹 “ (1..^𝐾)))))
 
Theoremsubfzo0 10198 The difference between two elements in a half-open range of nonnegative integers is greater than the negation of the upper bound and less than the upper bound of the range. (Contributed by AV, 20-Mar-2021.)
((𝐼 ∈ (0..^𝑁) ∧ 𝐽 ∈ (0..^𝑁)) → (-𝑁 < (𝐼𝐽) ∧ (𝐼𝐽) < 𝑁))
 
4.5.7  Rational numbers (cont.)
 
Theoremqtri3or 10199 Rational trichotomy. (Contributed by Jim Kingdon, 6-Oct-2021.)
((𝑀 ∈ ℚ ∧ 𝑁 ∈ ℚ) → (𝑀 < 𝑁𝑀 = 𝑁𝑁 < 𝑀))
 
Theoremqletric 10200 Rational trichotomy. (Contributed by Jim Kingdon, 6-Oct-2021.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴𝐵𝐵𝐴))
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14113
  Copyright terms: Public domain < Previous  Next >