P. 89 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 8801-8900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	fzosubel2 8801	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 8802	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 8803	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 8804	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 8805	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 8806	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 8807	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 8808	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 8809	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 8810	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 8811	Expressing a singleton as a half-open range. (Contributed by Stefan O'Rear, 23-Aug-2015.)
⊢ (A ∈ ℤ → (A..^(A + 1)) = {A})
Theorem	elfzomin 8812	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 8813	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 8814	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 8815	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 8816	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 8817	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 8818	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 8819	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 8820	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 8821	Half-open integer ranges starting with 0 are subsets of NN0. (Contributed by Thierry Arnoux, 8-Oct-2018.)
⊢ (0..^𝑁) ⊆ ℕ0
Theorem	fzo01 8822	Expressing the singleton of 0 as a half-open integer range. (Contributed by Stefan O'Rear, 15-Aug-2015.)
⊢ (0..^1) = {0}
Theorem	fzo12sn 8823	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 8824	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 8825	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 8826	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 8827	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 8828	The endpoint of a half-open integer range. (Contributed by Mario Carneiro, 29-Sep-2015.)
⊢ (A ∈ (A..^B) → (B − 1) ∈ (A..^B))
Theorem	fzo0end 8829	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 8830	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 8831	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 8832	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 8833	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 8834	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 8835	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 8836	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 8837	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 8838	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 8839	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 8840	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 8841	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 8842	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 8843	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 8844*	Shift the scanning order inside of a quantification over a half-open integer range, analogous to fzshftral 8720. (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 8845*	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 8109 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 8846*	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 8847*	The value of 𝐺 (see frec2uz0d 8846) 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 8848*	The value of 𝐺 (see frec2uz0d 8846) 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 8849*	The value 𝐺 (see frec2uz0d 8846) 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 8850*	Less-than relation for 𝐺 (see frec2uz0d 8846). (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 8851*	The mapping 𝐺 (see frec2uz0d 8846) 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 8852*	Range of 𝐺 (see frec2uz0d 8846). (Contributed by Jim Kingdon, 17-May-2020.)
⊢ (φ → 𝐶 ∈ ℤ)    &   ⊢ 𝐺 = frec((x ∈ ℤ ↦ (x + 1)), 𝐶)    ⇒   ⊢ (φ → ran 𝐺 = (ℤ≥‘𝐶))
Theorem	frec2uzf1od 8853*	𝐺 (see frec2uz0d 8846) 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 8854*	𝐺 (see frec2uz0d 8846) 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 8855*	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 8856*	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 8846 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 8857*	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 8858*	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 8859*	The recursive definition generator on upper integers is a function. See comment in frec2uz0d 8846 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 8860*	Closure law for the recursive definition generator on upper integers. See comment in frec2uz0d 8846 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 8861*	Initial value of a recursive definition generator on upper integers. See comment in frec2uz0d 8846 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 8862*	Successor value of a recursive definition generator on upper integers. See comment in frec2uz0d 8846 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 8863	An upper integer set is denumerable. (Contributed by Mario Carneiro, 15-Oct-2015.)
⊢ 𝑍 = (ℤ≥‘𝑀)    ⇒   ⊢ (𝑀 ∈ ℤ → 𝑍 ≈ 𝜔)
Theorem	frecfzennn 8864	The cardinality of a finite set of sequential integers. (See frec2uz0d 8846 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 8865	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 8866	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 8867	A finite interval of integers is finite. (Contributed by Jim Kingdon, 19-May-2020.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀...𝑁) ∈ Fin)
Theorem	fzfigd 8868	Deduction form of fzfig 8867. (Contributed by Jim Kingdon, 21-May-2020.)
⊢ (φ → 𝑀 ∈ ℤ)    &   ⊢ (φ → 𝑁 ∈ ℤ)    ⇒   ⊢ (φ → (𝑀...𝑁) ∈ Fin)
Theorem	fzofig 8869	Half-open integer sets are finite. (Contributed by Jim Kingdon, 21-May-2020.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀..^𝑁) ∈ Fin)
Theorem	nn0ennn 8870	The nonnegative integers are equinumerous to the positive integers. (Contributed by NM, 19-Jul-2004.)
⊢ ℕ0 ≈ ℕ
Theorem	nnenom 8871	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 8872	Extend class notation with recursive sequence builder.
class seq𝑀( + , 𝐹, 𝑆)
Definition	df-iseq 8873*	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 8882 and at successors. 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) with values 1, 3/2, 7/4, 15/8,.., so that (seq1( + , 𝐹)‘1) = 1, (seq1( + , 𝐹)‘2) = 3/2, etc. In other words, seq𝑀( + , 𝐹) 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	iseqeq1 8874	Equality theorem for the sequence builder operation. (Contributed by Jim Kingdon, 30-May-2020.)
⊢ (𝑀 = 𝑁 → seq𝑀( + , 𝐹, 𝑆) = seq𝑁( + , 𝐹, 𝑆))
Theorem	iseqeq2 8875	Equality theorem for the sequence builder operation. (Contributed by Jim Kingdon, 30-May-2020.)
⊢ ( + = 𝑄 → seq𝑀( + , 𝐹, 𝑆) = seq𝑀(𝑄, 𝐹, 𝑆))
Theorem	iseqeq3 8876	Equality theorem for the sequence builder operation. (Contributed by Jim Kingdon, 30-May-2020.)
⊢ (𝐹 = 𝐺 → seq𝑀( + , 𝐹, 𝑆) = seq𝑀( + , 𝐺, 𝑆))
Theorem	iseqeq4 8877	Equality theorem for the sequence builder operation. (Contributed by Jim Kingdon, 30-May-2020.)
⊢ (𝑆 = 𝑇 → seq𝑀( + , 𝐹, 𝑆) = seq𝑀( + , 𝐹, 𝑇))
Theorem	nfiseq 8878	Hypothesis builder for the sequence builder operation. (Contributed by Jim Kingdon, 30-May-2020.)
⊢ Ⅎx𝑀    &   ⊢ Ⅎx +    &   ⊢ Ⅎx𝐹    &   ⊢ Ⅎx𝑆    ⇒   ⊢ Ⅎxseq𝑀( + , 𝐹, 𝑆)
Theorem	iseqovex 8879*	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 8880*	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 8881*	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 8882*	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))
Theorem	iseqcl 8883*	Closure properties of the recursive sequence builder. (Contributed by Jim Kingdon, 1-Jun-2020.)
⊢ (φ → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ (φ → 𝑆 ∈ 𝑉)    &   ⊢ ((φ ∧ x ∈ (ℤ≥‘𝑀)) → (𝐹‘x) ∈ 𝑆)    &   ⊢ ((φ ∧ (x ∈ 𝑆 ∧ y ∈ 𝑆)) → (x + y) ∈ 𝑆)    ⇒   ⊢ (φ → (seq𝑀( + , 𝐹, 𝑆)‘𝑁) ∈ 𝑆)
Theorem	iseqp1 8884*	Value of the sequence builder function at a successor. (Contributed by Jim Kingdon, 31-May-2020.)
⊢ (φ → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ (φ → 𝑆 ∈ 𝑉)    &   ⊢ ((φ ∧ x ∈ (ℤ≥‘𝑀)) → (𝐹‘x) ∈ 𝑆)    &   ⊢ ((φ ∧ (x ∈ 𝑆 ∧ y ∈ 𝑆)) → (x + y) ∈ 𝑆)    ⇒   ⊢ (φ → (seq𝑀( + , 𝐹, 𝑆)‘(𝑁 + 1)) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑁) + (𝐹‘(𝑁 + 1))))
Theorem	iseqfveq2 8885*	Equality of sequences. (Contributed by Jim Kingdon, 3-Jun-2020.)
⊢ (φ → 𝐾 ∈ (ℤ≥‘𝑀))    &   ⊢ (φ → (seq𝑀( + , 𝐹, 𝑆)‘𝐾) = (𝐺‘𝐾))    &   ⊢ (φ → 𝑆 ∈ 𝑉)    &   ⊢ ((φ ∧ x ∈ (ℤ≥‘𝑀)) → (𝐹‘x) ∈ 𝑆)    &   ⊢ ((φ ∧ x ∈ (ℤ≥‘𝐾)) → (𝐺‘x) ∈ 𝑆)    &   ⊢ ((φ ∧ (x ∈ 𝑆 ∧ y ∈ 𝑆)) → (x + y) ∈ 𝑆)    &   ⊢ (φ → 𝑁 ∈ (ℤ≥‘𝐾))    &   ⊢ ((φ ∧ 𝑘 ∈ ((𝐾 + 1)...𝑁)) → (𝐹‘𝑘) = (𝐺‘𝑘))    ⇒   ⊢ (φ → (seq𝑀( + , 𝐹, 𝑆)‘𝑁) = (seq𝐾( + , 𝐺, 𝑆)‘𝑁))
Theorem	iseqfeq2 8886*	Equality of sequences. (Contributed by Jim Kingdon, 3-Jun-2020.)
⊢ (φ → 𝐾 ∈ (ℤ≥‘𝑀))    &   ⊢ (φ → (seq𝑀( + , 𝐹, 𝑆)‘𝐾) = (𝐺‘𝐾))    &   ⊢ (φ → 𝑆 ∈ 𝑉)    &   ⊢ ((φ ∧ x ∈ (ℤ≥‘𝑀)) → (𝐹‘x) ∈ 𝑆)    &   ⊢ ((φ ∧ x ∈ (ℤ≥‘𝐾)) → (𝐺‘x) ∈ 𝑆)    &   ⊢ ((φ ∧ (x ∈ 𝑆 ∧ y ∈ 𝑆)) → (x + y) ∈ 𝑆)    &   ⊢ ((φ ∧ 𝑘 ∈ (ℤ≥‘(𝐾 + 1))) → (𝐹‘𝑘) = (𝐺‘𝑘))    ⇒   ⊢ (φ → (seq𝑀( + , 𝐹, 𝑆) ↾ (ℤ≥‘𝐾)) = seq𝐾( + , 𝐺, 𝑆))
Theorem	iseqfveq 8887*	Equality of sequences. (Contributed by Jim Kingdon, 4-Jun-2020.)
⊢ (φ → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ ((φ ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) = (𝐺‘𝑘))    &   ⊢ (φ → 𝑆 ∈ 𝑉)    &   ⊢ ((φ ∧ x ∈ (ℤ≥‘𝑀)) → (𝐹‘x) ∈ 𝑆)    &   ⊢ ((φ ∧ x ∈ (ℤ≥‘𝑀)) → (𝐺‘x) ∈ 𝑆)    &   ⊢ ((φ ∧ (x ∈ 𝑆 ∧ y ∈ 𝑆)) → (x + y) ∈ 𝑆)    ⇒   ⊢ (φ → (seq𝑀( + , 𝐹, 𝑆)‘𝑁) = (seq𝑀( + , 𝐺, 𝑆)‘𝑁))
3.5.9  Integer powers
Syntax	cexp 8888	Extend class notation to include exponentiation of a complex number to an integer power.
class ↑
Definition	df-iexp 8889*	Define exponentiation to nonnegative integer powers. This definition is not meant to be used directly; instead, exp0 8893 and expp1 8896 provide the standard recursive definition. The up-arrow notation is used by Donald Knuth for iterated exponentiation (Science 194, 1235-1242, 1976) and is convenient for us since we don't have superscripts. 10-Jun-2005: The definition was extended to include zero exponents, so that 0↑0 = 1 per the convention of Definition 10-4.1 of [Gleason] p. 134. 4-Jun-2014: The definition was extended to include negative integer exponents. The case x = 0, y < 0 gives the value (1 / 0), so we will avoid this case in our theorems. (Contributed by Jim Kingodon, 7-Jun-2020.)
⊢ ↑ = (x ∈ ℂ, y ∈ ℤ ↦ if(y = 0, 1, if(0 < y, (seq1( · , (ℕ × {x}), ℂ)‘y), (1 / (seq1( · , (ℕ × {x}), ℂ)‘-y)))))
Theorem	expivallem 8890	Lemma for expival 8891. If we take a complex number apart from zero and raise it to a positive integer power, the result is apart from zero. (Contributed by Jim Kingodon, 7-Jun-2020.)
⊢ ((A ∈ ℂ ∧ A # 0 ∧ 𝑁 ∈ ℕ) → (seq1( · , (ℕ × {A}), ℂ)‘𝑁) # 0)
Theorem	expival 8891	Value of exponentiation to integer powers. (Contributed by Jim Kingdon, 7-Jun-2020.)
⊢ ((A ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (A # 0 ∨ 0 ≤ 𝑁)) → (A↑𝑁) = if(𝑁 = 0, 1, if(0 < 𝑁, (seq1( · , (ℕ × {A}), ℂ)‘𝑁), (1 / (seq1( · , (ℕ × {A}), ℂ)‘-𝑁)))))
Theorem	expinnval 8892	Value of exponentiation to positive integer powers. (Contributed by Jim Kingdon, 8-Jun-2020.)
⊢ ((A ∈ ℂ ∧ 𝑁 ∈ ℕ) → (A↑𝑁) = (seq1( · , (ℕ × {A}), ℂ)‘𝑁))
Theorem	exp0 8893	Value of a complex number raised to the 0th power. Note that under our definition, 0↑0 = 1, following the convention used by Gleason. Part of Definition 10-4.1 of [Gleason] p. 134. (Contributed by NM, 20-May-2004.) (Revised by Mario Carneiro, 4-Jun-2014.)
⊢ (A ∈ ℂ → (A↑0) = 1)
Theorem	0exp0e1 8894	0↑0 = 1 (common case). This is our convention. It follows the convention used by Gleason; see Part of Definition 10-4.1 of [Gleason] p. 134. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (0↑0) = 1
Theorem	exp1 8895	Value of a complex number raised to the first power. (Contributed by NM, 20-Oct-2004.) (Revised by Mario Carneiro, 2-Jul-2013.)
⊢ (A ∈ ℂ → (A↑1) = A)
Theorem	expp1 8896	Value of a complex number raised to a nonnegative integer power plus one. Part of Definition 10-4.1 of [Gleason] p. 134. (Contributed by NM, 20-May-2005.) (Revised by Mario Carneiro, 2-Jul-2013.)
⊢ ((A ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (A↑(𝑁 + 1)) = ((A↑𝑁) · A))
Theorem	expnegap0 8897	Value of a complex number raised to a negative integer power. (Contributed by Jim Kingdon, 8-Jun-2020.)
⊢ ((A ∈ ℂ ∧ A # 0 ∧ 𝑁 ∈ ℕ0) → (A↑-𝑁) = (1 / (A↑𝑁)))
Theorem	expineg2 8898	Value of a complex number raised to a negative integer power. (Contributed by Jim Kingdon, 8-Jun-2020.)
⊢ (((A ∈ ℂ ∧ A # 0) ∧ (𝑁 ∈ ℂ ∧ -𝑁 ∈ ℕ0)) → (A↑𝑁) = (1 / (A↑-𝑁)))
Theorem	expn1ap0 8899	A number to the negative one power is the reciprocal. (Contributed by Jim Kingdon, 8-Jun-2020.)
⊢ ((A ∈ ℂ ∧ A # 0) → (A↑-1) = (1 / A))
Theorem	expcllem 8900*	Lemma for proving nonnegative integer exponentiation closure laws. (Contributed by NM, 14-Dec-2005.)
⊢ 𝐹 ⊆ ℂ    &   ⊢ ((x ∈ 𝐹 ∧ y ∈ 𝐹) → (x · y) ∈ 𝐹)    &   ⊢ 1 ∈ 𝐹    ⇒   ⊢ ((A ∈ 𝐹 ∧ B ∈ ℕ0) → (A↑B) ∈ 𝐹)