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Theorem List for Intuitionistic Logic Explorer - 9001-9100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfzodisj 9001 Abutting half-open integer ranges are disjoint. (Contributed by Stefan O'Rear, 14-Aug-2015.)
((𝐴..^𝐵) ∩ (𝐵..^𝐶)) = ∅
 
Theoremfzouzsplit 9002 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 9003 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 9004 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 9005 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 9006 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 9007 Membership in a half-open range of nonnegative integers, generalization of elfzo0 9005 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 9008 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 9009 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 9010 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 9011 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 9012 Half-open integer ranges starting with 1 are subsets of NN. (Contributed by Thierry Arnoux, 28-Dec-2016.)
(1..^𝑁) ⊆ ℕ
 
Theoremelfzo1 9013 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 9014* 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 9015 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 9016 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 9017 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 9018 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 9019 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 9020 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 9021 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 9022 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 9023 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 9024 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 9025 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 9026 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 9027 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 9028 Expressing a singleton as a half-open range. (Contributed by Stefan O'Rear, 23-Aug-2015.)
(𝐴 ∈ ℤ → (𝐴..^(𝐴 + 1)) = {𝐴})
 
Theoremelfzomin 9029 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 9030 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 9031 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 9032 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 9033 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 9034 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 9035 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 9036 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 9037 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 9038 Half-open integer ranges starting with 0 are subsets of NN0. (Contributed by Thierry Arnoux, 8-Oct-2018.)
(0..^𝑁) ⊆ ℕ0
 
Theoremfzo01 9039 Expressing the singleton of 0 as a half-open integer range. (Contributed by Stefan O'Rear, 15-Aug-2015.)
(0..^1) = {0}
 
Theoremfzo12sn 9040 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 9041 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 9042 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 9043 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 9044 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 9045 The endpoint of a half-open integer range. (Contributed by Mario Carneiro, 29-Sep-2015.)
(𝐴 ∈ (𝐴..^𝐵) → (𝐵 − 1) ∈ (𝐴..^𝐵))
 
Theoremfzo0end 9046 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 9047 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 9048 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 9049 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 9050 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 9051 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 9052 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 9053 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 9054 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 9055 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 9056 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 9057 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 9058 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 9059 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 9060 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 9061* Shift the scanning order inside of a quantification over a half-open integer range, analogous to fzshftral 8937. (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 9062* 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 8325 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)
 
3.5.7  Miscellaneous theorems about integers
 
Theoremfrec2uz0d 9063* The mapping 𝐺 is a one-to-one mapping from ω onto upper integers that will be used to construct a recursive definition generator. Ordinal natural number 0 maps to complex number 𝐶 (normally 0 for the upper integers 0 or 1 for the upper integers ), 1 maps to 𝐶 + 1, etc. This theorem shows the value of 𝐺 at ordinal natural number zero. (Contributed by Jim Kingdon, 16-May-2020.)
(𝜑𝐶 ∈ ℤ)    &   𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)       (𝜑 → (𝐺‘∅) = 𝐶)
 
Theoremfrec2uzzd 9064* The value of 𝐺 (see frec2uz0d 9063) is an integer. (Contributed by Jim Kingdon, 16-May-2020.)
(𝜑𝐶 ∈ ℤ)    &   𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)    &   (𝜑𝐴 ∈ ω)       (𝜑 → (𝐺𝐴) ∈ ℤ)
 
Theoremfrec2uzsucd 9065* The value of 𝐺 (see frec2uz0d 9063) at a successor. (Contributed by Jim Kingdon, 16-May-2020.)
(𝜑𝐶 ∈ ℤ)    &   𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)    &   (𝜑𝐴 ∈ ω)       (𝜑 → (𝐺‘suc 𝐴) = ((𝐺𝐴) + 1))
 
Theoremfrec2uzuzd 9066* The value 𝐺 (see frec2uz0d 9063) at an ordinal natural number is in the upper integers. (Contributed by Jim Kingdon, 16-May-2020.)
(𝜑𝐶 ∈ ℤ)    &   𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)    &   (𝜑𝐴 ∈ ω)       (𝜑 → (𝐺𝐴) ∈ (ℤ𝐶))
 
Theoremfrec2uzltd 9067* Less-than relation for 𝐺 (see frec2uz0d 9063). (Contributed by Jim Kingdon, 16-May-2020.)
(𝜑𝐶 ∈ ℤ)    &   𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)    &   (𝜑𝐴 ∈ ω)    &   (𝜑𝐵 ∈ ω)       (𝜑 → (𝐴𝐵 → (𝐺𝐴) < (𝐺𝐵)))
 
Theoremfrec2uzlt2d 9068* The mapping 𝐺 (see frec2uz0d 9063) preserves order. (Contributed by Jim Kingdon, 16-May-2020.)
(𝜑𝐶 ∈ ℤ)    &   𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)    &   (𝜑𝐴 ∈ ω)    &   (𝜑𝐵 ∈ ω)       (𝜑 → (𝐴𝐵 ↔ (𝐺𝐴) < (𝐺𝐵)))
 
Theoremfrec2uzrand 9069* Range of 𝐺 (see frec2uz0d 9063). (Contributed by Jim Kingdon, 17-May-2020.)
(𝜑𝐶 ∈ ℤ)    &   𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)       (𝜑 → ran 𝐺 = (ℤ𝐶))
 
Theoremfrec2uzf1od 9070* 𝐺 (see frec2uz0d 9063) is a one-to-one onto mapping. (Contributed by Jim Kingdon, 17-May-2020.)
(𝜑𝐶 ∈ ℤ)    &   𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)       (𝜑𝐺:ω–1-1-onto→(ℤ𝐶))
 
Theoremfrec2uzisod 9071* 𝐺 (see frec2uz0d 9063) is an isomorphism from natural ordinals to upper integers. (Contributed by Jim Kingdon, 17-May-2020.)
(𝜑𝐶 ∈ ℤ)    &   𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)       (𝜑𝐺 Isom E , < (ω, (ℤ𝐶)))
 
Theoremfrecuzrdgrrn 9072* The function 𝑅 (used in the definition of the recursive definition generator on upper integers) yields ordered pairs of integers and elements of 𝑆. (Contributed by Jim Kingdon, 27-May-2020.)
(𝜑𝐶 ∈ ℤ)    &   𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)    &   (𝜑𝑆𝑉)    &   (𝜑𝐴𝑆)    &   ((𝜑 ∧ (𝑥 ∈ (ℤ𝐶) ∧ 𝑦𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)    &   𝑅 = frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)       ((𝜑𝐷 ∈ ω) → (𝑅𝐷) ∈ ((ℤ𝐶) × 𝑆))
 
Theoremfrec2uzrdg 9073* A helper lemma for the value of a recursive definition generator on upper integers (typically either or 0) with characteristic function 𝐹(𝑥, 𝑦) and initial value 𝐴. This lemma shows that evaluating 𝑅 at an element of ω gives an ordered pair whose first element is the index (translated from ω to (ℤ𝐶)). See comment in frec2uz0d 9063 which describes 𝐺 and the index translation. (Contributed by Jim Kingdon, 24-May-2020.)
(𝜑𝐶 ∈ ℤ)    &   𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)    &   (𝜑𝑆𝑉)    &   (𝜑𝐴𝑆)    &   ((𝜑 ∧ (𝑥 ∈ (ℤ𝐶) ∧ 𝑦𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)    &   𝑅 = frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)    &   (𝜑𝐵 ∈ ω)       (𝜑 → (𝑅𝐵) = ⟨(𝐺𝐵), (2nd ‘(𝑅𝐵))⟩)
 
Theoremfrecuzrdgrom 9074* The function 𝑅 (used in the definition of the recursive definition generator on upper integers) is a function defined for all natural numbers. (Contributed by Jim Kingdon, 26-May-2020.)
(𝜑𝐶 ∈ ℤ)    &   𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)    &   (𝜑𝑆𝑉)    &   (𝜑𝐴𝑆)    &   ((𝜑 ∧ (𝑥 ∈ (ℤ𝐶) ∧ 𝑦𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)    &   𝑅 = frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)       (𝜑𝑅 Fn ω)
 
Theoremfrecuzrdglem 9075* A helper lemma for the value of a recursive definition generator on upper integers. (Contributed by Jim Kingdon, 26-May-2020.)
(𝜑𝐶 ∈ ℤ)    &   𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)    &   (𝜑𝑆𝑉)    &   (𝜑𝐴𝑆)    &   ((𝜑 ∧ (𝑥 ∈ (ℤ𝐶) ∧ 𝑦𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)    &   𝑅 = frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)    &   (𝜑𝐵 ∈ (ℤ𝐶))       (𝜑 → ⟨𝐵, (2nd ‘(𝑅‘(𝐺𝐵)))⟩ ∈ ran 𝑅)
 
Theoremfrecuzrdgfn 9076* The recursive definition generator on upper integers is a function. See comment in frec2uz0d 9063 for the description of 𝐺 as the mapping from ω to (ℤ𝐶). (Contributed by Jim Kingdon, 26-May-2020.)
(𝜑𝐶 ∈ ℤ)    &   𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)    &   (𝜑𝑆𝑉)    &   (𝜑𝐴𝑆)    &   ((𝜑 ∧ (𝑥 ∈ (ℤ𝐶) ∧ 𝑦𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)    &   𝑅 = frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)    &   (𝜑𝑇 = ran 𝑅)       (𝜑𝑇 Fn (ℤ𝐶))
 
Theoremfrecuzrdgcl 9077* Closure law for the recursive definition generator on upper integers. See comment in frec2uz0d 9063 for the description of 𝐺 as the mapping from ω to (ℤ𝐶). (Contributed by Jim Kingdon, 31-May-2020.)
(𝜑𝐶 ∈ ℤ)    &   𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)    &   (𝜑𝑆𝑉)    &   (𝜑𝐴𝑆)    &   ((𝜑 ∧ (𝑥 ∈ (ℤ𝐶) ∧ 𝑦𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)    &   𝑅 = frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)    &   (𝜑𝑇 = ran 𝑅)       ((𝜑𝐵 ∈ (ℤ𝐶)) → (𝑇𝐵) ∈ 𝑆)
 
Theoremfrecuzrdg0 9078* Initial value of a recursive definition generator on upper integers. See comment in frec2uz0d 9063 for the description of 𝐺 as the mapping from ω to (ℤ𝐶). (Contributed by Jim Kingdon, 27-May-2020.)
(𝜑𝐶 ∈ ℤ)    &   𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)    &   (𝜑𝑆𝑉)    &   (𝜑𝐴𝑆)    &   ((𝜑 ∧ (𝑥 ∈ (ℤ𝐶) ∧ 𝑦𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)    &   𝑅 = frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)    &   (𝜑𝑇 = ran 𝑅)       (𝜑 → (𝑇𝐶) = 𝐴)
 
Theoremfrecuzrdgsuc 9079* Successor value of a recursive definition generator on upper integers. See comment in frec2uz0d 9063 for the description of 𝐺 as the mapping from ω to (ℤ𝐶). (Contributed by Jim Kingdon, 28-May-2020.)
(𝜑𝐶 ∈ ℤ)    &   𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)    &   (𝜑𝑆𝑉)    &   (𝜑𝐴𝑆)    &   ((𝜑 ∧ (𝑥 ∈ (ℤ𝐶) ∧ 𝑦𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)    &   𝑅 = frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)    &   (𝜑𝑇 = ran 𝑅)       ((𝜑𝐵 ∈ (ℤ𝐶)) → (𝑇‘(𝐵 + 1)) = (𝐵𝐹(𝑇𝐵)))
 
Theoremuzenom 9080 An upper integer set is denumerable. (Contributed by Mario Carneiro, 15-Oct-2015.)
𝑍 = (ℤ𝑀)       (𝑀 ∈ ℤ → 𝑍 ≈ ω)
 
Theoremfrecfzennn 9081 The cardinality of a finite set of sequential integers. (See frec2uz0d 9063 for a description of the hypothesis.) (Contributed by Jim Kingdon, 18-May-2020.)
𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)       (𝑁 ∈ ℕ0 → (1...𝑁) ≈ (𝐺𝑁))
 
Theoremfrecfzen2 9082 The cardinality of a finite set of sequential integers with arbitrary endpoints. (Contributed by Jim Kingdon, 18-May-2020.)
𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)       (𝑁 ∈ (ℤ𝑀) → (𝑀...𝑁) ≈ (𝐺‘((𝑁 + 1) − 𝑀)))
 
Theoremfrechashgf1o 9083 𝐺 maps ω one-to-one onto 0. (Contributed by Jim Kingdon, 19-May-2020.)
𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)       𝐺:ω–1-1-onto→ℕ0
 
Theoremfzfig 9084 A finite interval of integers is finite. (Contributed by Jim Kingdon, 19-May-2020.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀...𝑁) ∈ Fin)
 
Theoremfzfigd 9085 Deduction form of fzfig 9084. (Contributed by Jim Kingdon, 21-May-2020.)
(𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ ℤ)       (𝜑 → (𝑀...𝑁) ∈ Fin)
 
Theoremfzofig 9086 Half-open integer sets are finite. (Contributed by Jim Kingdon, 21-May-2020.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀..^𝑁) ∈ Fin)
 
Theoremnn0ennn 9087 The nonnegative integers are equinumerous to the positive integers. (Contributed by NM, 19-Jul-2004.)
0 ≈ ℕ
 
Theoremnnenom 9088 The set of positive integers (as a subset of complex numbers) is equinumerous to omega (the set of finite ordinal numbers). (Contributed by NM, 31-Jul-2004.) (Revised by Mario Carneiro, 15-Sep-2013.)
ℕ ≈ ω
 
3.5.8  The infinite sequence builder "seq"
 
Syntaxcseq 9089 Extend class notation with recursive sequence builder.
class seq𝑀( + , 𝐹, 𝑆)
 
Definitiondf-iseq 9090* Define a general-purpose operation that builds a recursive sequence (i.e. a function on the positive integers or some other upper integer set) whose value at an index is a function of its previous value and the value of an input sequence at that index. This definition is complicated, but fortunately it is not intended to be used directly. Instead, the only purpose of this definition is to provide us with an object that has the properties expressed by iseq1 9100 and iseqp1 9103. Typically, those are the main theorems that would be used in practice.

The first operand in the parentheses is the operation that is applied to the previous value and the value of the input sequence (second operand). The operand to the left of the parenthesis is the integer to start from. For example, for the operation +, an input sequence 𝐹 with values 1, 1/2, 1/4, 1/8,... would be transformed into the output sequence seq1( + , 𝐹, ℚ) with values 1, 3/2, 7/4, 15/8,.., so that (seq1( + , 𝐹, ℚ)‘1) = 1, (seq1( + , 𝐹, ℚ)‘2) = 3/2, etc. In other words, seq𝑀( + , 𝐹, ℚ) transforms a sequence 𝐹 into an infinite series.

Internally, the frec function generates as its values a set of ordered pairs starting at 𝑀, (𝐹𝑀)⟩, with the first member of each pair incremented by one in each successive value. So, the range of frec is exactly the sequence we want, and we just extract the range and throw away the domain.

(Contributed by Jim Kingdon, 29-May-2020.)

seq𝑀( + , 𝐹, 𝑆) = ran frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)
 
Theoremiseqex 9091 Existence of the sequence builder operation. (Contributed by Jim Kingdon, 20-Aug-2021.)
seq𝑀( + , 𝐹, 𝑆) ∈ V
 
Theoremiseqeq1 9092 Equality theorem for the sequence builder operation. (Contributed by Jim Kingdon, 30-May-2020.)
(𝑀 = 𝑁 → seq𝑀( + , 𝐹, 𝑆) = seq𝑁( + , 𝐹, 𝑆))
 
Theoremiseqeq2 9093 Equality theorem for the sequence builder operation. (Contributed by Jim Kingdon, 30-May-2020.)
( + = 𝑄 → seq𝑀( + , 𝐹, 𝑆) = seq𝑀(𝑄, 𝐹, 𝑆))
 
Theoremiseqeq3 9094 Equality theorem for the sequence builder operation. (Contributed by Jim Kingdon, 30-May-2020.)
(𝐹 = 𝐺 → seq𝑀( + , 𝐹, 𝑆) = seq𝑀( + , 𝐺, 𝑆))
 
Theoremiseqeq4 9095 Equality theorem for the sequence builder operation. (Contributed by Jim Kingdon, 30-May-2020.)
(𝑆 = 𝑇 → seq𝑀( + , 𝐹, 𝑆) = seq𝑀( + , 𝐹, 𝑇))
 
Theoremnfiseq 9096 Hypothesis builder for the sequence builder operation. (Contributed by Jim Kingdon, 30-May-2020.)
𝑥𝑀    &   𝑥 +    &   𝑥𝐹    &   𝑥𝑆       𝑥seq𝑀( + , 𝐹, 𝑆)
 
Theoremiseqovex 9097* Closure of a function used in proving sequence builder theorems. This can be thought of as a lemma for the small number of sequence builder theorems which need it. (Contributed by Jim Kingdon, 31-May-2020.)
((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)       ((𝜑 ∧ (𝑥 ∈ (ℤ𝑀) ∧ 𝑦𝑆)) → (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦) ∈ 𝑆)
 
Theoremiseqval 9098* Value of the sequence builder function. (Contributed by Jim Kingdon, 30-May-2020.)
𝑅 = frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩), ⟨𝑀, (𝐹𝑀)⟩)    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)       (𝜑 → seq𝑀( + , 𝐹, 𝑆) = ran 𝑅)
 
Theoremiseqfn 9099* The sequence builder function is a function. (Contributed by Jim Kingdon, 30-May-2020.)
(𝜑𝑀 ∈ ℤ)    &   (𝜑𝑆𝑉)    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)       (𝜑 → seq𝑀( + , 𝐹, 𝑆) Fn (ℤ𝑀))
 
Theoremiseq1 9100* Value of the sequence builder function at its initial value. (Contributed by Jim Kingdon, 31-May-2020.)
(𝜑𝑀 ∈ ℤ)    &   (𝜑𝑆𝑉)    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)       (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑀) = (𝐹𝑀))
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