P. 91 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 9001-9100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	fzodisj 9001	Abutting half-open integer ranges are disjoint. (Contributed by Stefan O'Rear, 14-Aug-2015.)
⊢ ((𝐴..^𝐵) ∩ (𝐵..^𝐶)) = ∅
Theorem	fzouzsplit 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 ∈ (ℤ≥‘𝐴) → (ℤ≥‘𝐴) = ((𝐴..^𝐵) ∪ (ℤ≥‘𝐵)))
Theorem	fzouzdisj 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) ∩ (ℤ≥‘𝐵)) = ∅
Theorem	lbfzo0 9004	An integer is strictly greater than zero iff it is a member of ℕ. (Contributed by Mario Carneiro, 29-Sep-2015.)
⊢ (0 ∈ (0..^𝐴) ↔ 𝐴 ∈ ℕ)
Theorem	elfzo0 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 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ∧ 𝐴 < 𝐵))
Theorem	fzo1fzo0n0 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))
Theorem	elfzo0z 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 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐴 < 𝐵))
Theorem	elfzo0le 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) → 𝐴 ≤ 𝐵)
Theorem	elfzonn0 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)
Theorem	fzonmapblen 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 + (𝑁 − 𝐴)) < 𝑁)
Theorem	fzofzim 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.)
⊢ ((𝐾 ≠ 𝑀 ∧ 𝐾 ∈ (0...𝑀)) → 𝐾 ∈ (0..^𝑀))
Theorem	fzossnn 9012	Half-open integer ranges starting with 1 are subsets of NN. (Contributed by Thierry Arnoux, 28-Dec-2016.)
⊢ (1..^𝑁) ⊆ ℕ
Theorem	elfzo1 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))
Theorem	fzo0m 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)
Theorem	fzoaddel 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)))
Theorem	fzoaddel2 9016	Translate membership in a shifted-down half-open integer range. (Contributed by Stefan O'Rear, 15-Aug-2015.)
⊢ ((𝐴 ∈ (0..^(𝐵 − 𝐶)) ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (𝐴 + 𝐶) ∈ (𝐶..^𝐵))
Theorem	fzosubel 9017	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)))
Theorem	fzosubel2 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)) → (𝐴 − 𝐵) ∈ (𝐶..^𝐷))
Theorem	fzosubel3 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) → (𝐴 − 𝐵) ∈ (0..^𝐷))
Theorem	eluzgtdifelfzo 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 ∈ (ℤ≥‘𝐴) ∧ 𝐵 < 𝐴) → (𝑁 − 𝐴) ∈ (0..^(𝑁 − 𝐵))))
Theorem	ige2m2fzo 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)))
Theorem	fzocatel 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)) → (𝐴 − 𝐵) ∈ (0..^𝐶))
Theorem	ubmelfzo 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) → (𝑁 − 𝐾) ∈ (0..^𝑁))
Theorem	elfzodifsumelfzo 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..^(𝑁 − 𝑀)) → (𝐼 + 𝑀) ∈ (0..^𝑃)))
Theorem	elfzom1elp1fzo 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))
Theorem	elfzom1elfzo 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))
Theorem	fzval3 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)))
Theorem	fzosn 9028	Expressing a singleton as a half-open range. (Contributed by Stefan O'Rear, 23-Aug-2015.)
⊢ (𝐴 ∈ ℤ → (𝐴..^(𝐴 + 1)) = {𝐴})
Theorem	elfzomin 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)))
Theorem	zpnn0elfzo 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)))
Theorem	zpnn0elfzo1 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))))
Theorem	fzosplitsnm1 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)}))
Theorem	elfzonlteqm1 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))
Theorem	fzonn0p1 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 ∈ ℕ0 → 𝑁 ∈ (0..^(𝑁 + 1)))
Theorem	fzossfzop1 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)))
Theorem	fzonn0p1p1 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)))
Theorem	elfzom1p1elfzo 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))
Theorem	fzo0ssnn0 9038	Half-open integer ranges starting with 0 are subsets of NN0. (Contributed by Thierry Arnoux, 8-Oct-2018.)
⊢ (0..^𝑁) ⊆ ℕ0
Theorem	fzo01 9039	Expressing the singleton of 0 as a half-open integer range. (Contributed by Stefan O'Rear, 15-Aug-2015.)
⊢ (0..^1) = {0}
Theorem	fzo12sn 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}
Theorem	fzo0to2pr 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}
Theorem	fzo0to3tp 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}
Theorem	fzo0to42pr 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})
Theorem	fzo0sn0fzo1 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)))
Theorem	fzoend 9045	The endpoint of a half-open integer range. (Contributed by Mario Carneiro, 29-Sep-2015.)
⊢ (𝐴 ∈ (𝐴..^𝐵) → (𝐵 − 1) ∈ (𝐴..^𝐵))
Theorem	fzo0end 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))
Theorem	ssfzo12 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) → (𝑀 ≤ 𝐾 ∧ 𝐿 ≤ 𝑁)))
Theorem	ssfzo12bi 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) ↔ (𝑀 ≤ 𝐾 ∧ 𝐿 ≤ 𝑁)))
Theorem	ubmelm1fzo 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) → ((𝑁 − 𝐾) − 1) ∈ (0..^𝑁))
Theorem	fzofzp1 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))
Theorem	fzofzp1b 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 ∈ (ℤ≥‘𝐴) → (𝐶 ∈ (𝐴..^𝐵) ↔ (𝐶 + 1) ∈ (𝐴...𝐵)))
Theorem	elfzom1b 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))))
Theorem	elfzonelfzo 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)))
Theorem	elfzomelpfzo 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 ∈ ((𝑀 − 𝐿)..^(𝑁 − 𝐿)) ↔ (𝐾 + 𝐿) ∈ (𝑀..^𝑁)))
Theorem	peano2fzor 9055	A Peano-postulate-like theorem for downward closure of a half-open integer range. (Contributed by Mario Carneiro, 1-Oct-2015.)
⊢ ((𝐾 ∈ (ℤ≥‘𝑀) ∧ (𝐾 + 1) ∈ (𝑀..^𝑁)) → 𝐾 ∈ (𝑀..^𝑁))
Theorem	fzosplitsn 9056	Extending a half-open range by a singleton on the end. (Contributed by Stefan O'Rear, 23-Aug-2015.)
⊢ (𝐵 ∈ (ℤ≥‘𝐴) → (𝐴..^(𝐵 + 1)) = ((𝐴..^𝐵) ∪ {𝐵}))
Theorem	fzosplitprm1 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}))
Theorem	fzosplitsni 9058	Membership in a half-open range extended by a singleton. (Contributed by Stefan O'Rear, 23-Aug-2015.)
⊢ (𝐵 ∈ (ℤ≥‘𝐴) → (𝐶 ∈ (𝐴..^(𝐵 + 1)) ↔ (𝐶 ∈ (𝐴..^𝐵) ∨ 𝐶 = 𝐵)))
Theorem	fzisfzounsn 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 ∈ (ℤ≥‘𝐴) → (𝐴...𝐵) = ((𝐴..^𝐵) ∪ {𝐵}))
Theorem	fzostep1 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))
Theorem	fzoshftral 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))[(𝑘 − 𝐾) / 𝑗]𝜑))
Theorem	fzind2 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 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 = (𝑦 + 1) → (𝜑 ↔ 𝜃))    &   ⊢ (𝑥 = 𝐾 → (𝜑 ↔ 𝜏))    &   ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝜓)    &   ⊢ (𝑦 ∈ (𝑀..^𝑁) → (𝜒 → 𝜃))    ⇒   ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝜏)
3.5.7  Miscellaneous theorems about integers
Theorem	frec2uz0d 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 ℕ0 or 1 for the upper integers ℕ), 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.)
⊢ (𝜑 → 𝐶 ∈ ℤ)    &   ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)    ⇒   ⊢ (𝜑 → (𝐺‘∅) = 𝐶)
Theorem	frec2uzzd 9064*	The value of 𝐺 (see frec2uz0d 9063) is an integer. (Contributed by Jim Kingdon, 16-May-2020.)
⊢ (𝜑 → 𝐶 ∈ ℤ)    &   ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)    &   ⊢ (𝜑 → 𝐴 ∈ ω)    ⇒   ⊢ (𝜑 → (𝐺‘𝐴) ∈ ℤ)
Theorem	frec2uzsucd 9065*	The value of 𝐺 (see frec2uz0d 9063) at a successor. (Contributed by Jim Kingdon, 16-May-2020.)
⊢ (𝜑 → 𝐶 ∈ ℤ)    &   ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)    &   ⊢ (𝜑 → 𝐴 ∈ ω)    ⇒   ⊢ (𝜑 → (𝐺‘suc 𝐴) = ((𝐺‘𝐴) + 1))
Theorem	frec2uzuzd 9066*	The value 𝐺 (see frec2uz0d 9063) at an ordinal natural number is in the upper integers. (Contributed by Jim Kingdon, 16-May-2020.)
⊢ (𝜑 → 𝐶 ∈ ℤ)    &   ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)    &   ⊢ (𝜑 → 𝐴 ∈ ω)    ⇒   ⊢ (𝜑 → (𝐺‘𝐴) ∈ (ℤ≥‘𝐶))
Theorem	frec2uzltd 9067*	Less-than relation for 𝐺 (see frec2uz0d 9063). (Contributed by Jim Kingdon, 16-May-2020.)
⊢ (𝜑 → 𝐶 ∈ ℤ)    &   ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)    &   ⊢ (𝜑 → 𝐴 ∈ ω)    &   ⊢ (𝜑 → 𝐵 ∈ ω)    ⇒   ⊢ (𝜑 → (𝐴 ∈ 𝐵 → (𝐺‘𝐴) < (𝐺‘𝐵)))
Theorem	frec2uzlt2d 9068*	The mapping 𝐺 (see frec2uz0d 9063) preserves order. (Contributed by Jim Kingdon, 16-May-2020.)
⊢ (𝜑 → 𝐶 ∈ ℤ)    &   ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)    &   ⊢ (𝜑 → 𝐴 ∈ ω)    &   ⊢ (𝜑 → 𝐵 ∈ ω)    ⇒   ⊢ (𝜑 → (𝐴 ∈ 𝐵 ↔ (𝐺‘𝐴) < (𝐺‘𝐵)))
Theorem	frec2uzrand 9069*	Range of 𝐺 (see frec2uz0d 9063). (Contributed by Jim Kingdon, 17-May-2020.)
⊢ (𝜑 → 𝐶 ∈ ℤ)    &   ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)    ⇒   ⊢ (𝜑 → ran 𝐺 = (ℤ≥‘𝐶))
Theorem	frec2uzf1od 9070*	𝐺 (see frec2uz0d 9063) is a one-to-one onto mapping. (Contributed by Jim Kingdon, 17-May-2020.)
⊢ (𝜑 → 𝐶 ∈ ℤ)    &   ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)    ⇒   ⊢ (𝜑 → 𝐺:ω–1-1-onto→(ℤ≥‘𝐶))
Theorem	frec2uzisod 9071*	𝐺 (see frec2uz0d 9063) is an isomorphism from natural ordinals to upper integers. (Contributed by Jim Kingdon, 17-May-2020.)
⊢ (𝜑 → 𝐶 ∈ ℤ)    &   ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)    ⇒   ⊢ (𝜑 → 𝐺 Isom E , < (ω, (ℤ≥‘𝐶)))
Theorem	frecuzrdgrrn 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.)
⊢ (𝜑 → 𝐶 ∈ ℤ)    &   ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑆)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)    &   ⊢ 𝑅 = frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)    ⇒   ⊢ ((𝜑 ∧ 𝐷 ∈ ω) → (𝑅‘𝐷) ∈ ((ℤ≥‘𝐶) × 𝑆))
Theorem	frec2uzrdg 9073*	A helper lemma for the value of a recursive definition generator on upper integers (typically either ℕ or ℕ0) 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 (ℤ≥‘𝐶)). See comment in frec2uz0d 9063 which describes 𝐺 and the index translation. (Contributed by Jim Kingdon, 24-May-2020.)
⊢ (𝜑 → 𝐶 ∈ ℤ)    &   ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑆)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)    &   ⊢ 𝑅 = frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)    &   ⊢ (𝜑 → 𝐵 ∈ ω)    ⇒   ⊢ (𝜑 → (𝑅‘𝐵) = ⟨(𝐺‘𝐵), (2nd ‘(𝑅‘𝐵))⟩)
Theorem	frecuzrdgrom 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.)
⊢ (𝜑 → 𝐶 ∈ ℤ)    &   ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑆)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)    &   ⊢ 𝑅 = frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)    ⇒   ⊢ (𝜑 → 𝑅 Fn ω)
Theorem	frecuzrdglem 9075*	A helper lemma for the value of a recursive definition generator on upper integers. (Contributed by Jim Kingdon, 26-May-2020.)
⊢ (𝜑 → 𝐶 ∈ ℤ)    &   ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑆)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)    &   ⊢ 𝑅 = frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)    &   ⊢ (𝜑 → 𝐵 ∈ (ℤ≥‘𝐶))    ⇒   ⊢ (𝜑 → ⟨𝐵, (2nd ‘(𝑅‘(◡𝐺‘𝐵)))⟩ ∈ ran 𝑅)
Theorem	frecuzrdgfn 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 (ℤ≥‘𝐶). (Contributed by Jim Kingdon, 26-May-2020.)
⊢ (𝜑 → 𝐶 ∈ ℤ)    &   ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑆)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)    &   ⊢ 𝑅 = frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)    &   ⊢ (𝜑 → 𝑇 = ran 𝑅)    ⇒   ⊢ (𝜑 → 𝑇 Fn (ℤ≥‘𝐶))
Theorem	frecuzrdgcl 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 (ℤ≥‘𝐶). (Contributed by Jim Kingdon, 31-May-2020.)
⊢ (𝜑 → 𝐶 ∈ ℤ)    &   ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑆)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)    &   ⊢ 𝑅 = frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)    &   ⊢ (𝜑 → 𝑇 = ran 𝑅)    ⇒   ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (𝑇‘𝐵) ∈ 𝑆)
Theorem	frecuzrdg0 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 (ℤ≥‘𝐶). (Contributed by Jim Kingdon, 27-May-2020.)
⊢ (𝜑 → 𝐶 ∈ ℤ)    &   ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑆)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)    &   ⊢ 𝑅 = frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)    &   ⊢ (𝜑 → 𝑇 = ran 𝑅)    ⇒   ⊢ (𝜑 → (𝑇‘𝐶) = 𝐴)
Theorem	frecuzrdgsuc 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 (ℤ≥‘𝐶). (Contributed by Jim Kingdon, 28-May-2020.)
⊢ (𝜑 → 𝐶 ∈ ℤ)    &   ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑆)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)    &   ⊢ 𝑅 = frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)    &   ⊢ (𝜑 → 𝑇 = ran 𝑅)    ⇒   ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (𝑇‘(𝐵 + 1)) = (𝐵𝐹(𝑇‘𝐵)))
Theorem	uzenom 9080	An upper integer set is denumerable. (Contributed by Mario Carneiro, 15-Oct-2015.)
⊢ 𝑍 = (ℤ≥‘𝑀)    ⇒   ⊢ (𝑀 ∈ ℤ → 𝑍 ≈ ω)
Theorem	frecfzennn 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) ≈ (◡𝐺‘𝑁))
Theorem	frecfzen2 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 ∈ (ℤ≥‘𝑀) → (𝑀...𝑁) ≈ (◡𝐺‘((𝑁 + 1) − 𝑀)))
Theorem	frechashgf1o 9083	𝐺 maps ω one-to-one onto ℕ0. (Contributed by Jim Kingdon, 19-May-2020.)
⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)    ⇒   ⊢ 𝐺:ω–1-1-onto→ℕ0
Theorem	fzfig 9084	A finite interval of integers is finite. (Contributed by Jim Kingdon, 19-May-2020.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀...𝑁) ∈ Fin)
Theorem	fzfigd 9085	Deduction form of fzfig 9084. (Contributed by Jim Kingdon, 21-May-2020.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝑁 ∈ ℤ)    ⇒   ⊢ (𝜑 → (𝑀...𝑁) ∈ Fin)
Theorem	fzofig 9086	Half-open integer sets are finite. (Contributed by Jim Kingdon, 21-May-2020.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀..^𝑁) ∈ Fin)
Theorem	nn0ennn 9087	The nonnegative integers are equinumerous to the positive integers. (Contributed by NM, 19-Jul-2004.)
⊢ ℕ0 ≈ ℕ
Theorem	nnenom 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"
Syntax	cseq 9089	Extend class notation with recursive sequence builder.
class seq𝑀( + , 𝐹, 𝑆)
Definition	df-iseq 9090*	Define a general-purpose operation that builds a recursive sequence (i.e. a function on the positive integers ℕ 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, (𝐹‘𝑀)⟩, 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 ∈ (ℤ≥‘𝑀), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)
Theorem	iseqex 9091	Existence of the sequence builder operation. (Contributed by Jim Kingdon, 20-Aug-2021.)
⊢ seq𝑀( + , 𝐹, 𝑆) ∈ V
Theorem	iseqeq1 9092	Equality theorem for the sequence builder operation. (Contributed by Jim Kingdon, 30-May-2020.)
⊢ (𝑀 = 𝑁 → seq𝑀( + , 𝐹, 𝑆) = seq𝑁( + , 𝐹, 𝑆))
Theorem	iseqeq2 9093	Equality theorem for the sequence builder operation. (Contributed by Jim Kingdon, 30-May-2020.)
⊢ ( + = 𝑄 → seq𝑀( + , 𝐹, 𝑆) = seq𝑀(𝑄, 𝐹, 𝑆))
Theorem	iseqeq3 9094	Equality theorem for the sequence builder operation. (Contributed by Jim Kingdon, 30-May-2020.)
⊢ (𝐹 = 𝐺 → seq𝑀( + , 𝐹, 𝑆) = seq𝑀( + , 𝐺, 𝑆))
Theorem	iseqeq4 9095	Equality theorem for the sequence builder operation. (Contributed by Jim Kingdon, 30-May-2020.)
⊢ (𝑆 = 𝑇 → seq𝑀( + , 𝐹, 𝑆) = seq𝑀( + , 𝐹, 𝑇))
Theorem	nfiseq 9096	Hypothesis builder for the sequence builder operation. (Contributed by Jim Kingdon, 30-May-2020.)
⊢ Ⅎ𝑥𝑀    &   ⊢ Ⅎ𝑥 +    &   ⊢ Ⅎ𝑥𝐹    &   ⊢ Ⅎ𝑥𝑆    ⇒   ⊢ Ⅎ𝑥seq𝑀( + , 𝐹, 𝑆)
Theorem	iseqovex 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.)
⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    ⇒   ⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝑀) ∧ 𝑦 ∈ 𝑆)) → (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦) ∈ 𝑆)
Theorem	iseqval 9098*	Value of the sequence builder function. (Contributed by Jim Kingdon, 30-May-2020.)
⊢ 𝑅 = frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩), ⟨𝑀, (𝐹‘𝑀)⟩)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    ⇒   ⊢ (𝜑 → seq𝑀( + , 𝐹, 𝑆) = ran 𝑅)
Theorem	iseqfn 9099*	The sequence builder function is a function. (Contributed by Jim Kingdon, 30-May-2020.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    ⇒   ⊢ (𝜑 → seq𝑀( + , 𝐹, 𝑆) Fn (ℤ≥‘𝑀))
Theorem	iseq1 9100*	Value of the sequence builder function at its initial value. (Contributed by Jim Kingdon, 31-May-2020.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    ⇒   ⊢ (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑀) = (𝐹‘𝑀))