P. 104 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 10301-10400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	zmodcld 10301	Closure law for the modulo operation restricted to integers. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℤ)    &   ⊢ (𝜑 → 𝐵 ∈ ℕ)    ⇒   ⊢ (𝜑 → (𝐴 mod 𝐵) ∈ ℕ0)
Theorem	zmodfz 10302	An integer mod 𝐵 lies in the first 𝐵 nonnegative integers. (Contributed by Jeff Madsen, 17-Jun-2010.)
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → (𝐴 mod 𝐵) ∈ (0...(𝐵 − 1)))
Theorem	zmodfzo 10303	An integer mod 𝐵 lies in the first 𝐵 nonnegative integers. (Contributed by Stefan O'Rear, 6-Sep-2015.)
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → (𝐴 mod 𝐵) ∈ (0..^𝐵))
Theorem	zmodfzp1 10304	An integer mod 𝐵 lies in the first 𝐵 + 1 nonnegative integers. (Contributed by AV, 27-Oct-2018.)
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → (𝐴 mod 𝐵) ∈ (0...𝐵))
Theorem	modqid 10305	Identity law for modulo. (Contributed by Jim Kingdon, 21-Oct-2021.)
⊢ (((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ (0 ≤ 𝐴 ∧ 𝐴 < 𝐵)) → (𝐴 mod 𝐵) = 𝐴)
Theorem	modqid0 10306	A positive real number modulo itself is 0. (Contributed by Jim Kingdon, 21-Oct-2021.)
⊢ ((𝑁 ∈ ℚ ∧ 0 < 𝑁) → (𝑁 mod 𝑁) = 0)
Theorem	modqid2 10307	Identity law for modulo. (Contributed by Jim Kingdon, 21-Oct-2021.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → ((𝐴 mod 𝐵) = 𝐴 ↔ (0 ≤ 𝐴 ∧ 𝐴 < 𝐵)))
Theorem	zmodid2 10308	Identity law for modulo restricted to integers. (Contributed by Paul Chapman, 22-Jun-2011.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ) → ((𝑀 mod 𝑁) = 𝑀 ↔ 𝑀 ∈ (0...(𝑁 − 1))))
Theorem	zmodidfzo 10309	Identity law for modulo restricted to integers. (Contributed by AV, 27-Oct-2018.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ) → ((𝑀 mod 𝑁) = 𝑀 ↔ 𝑀 ∈ (0..^𝑁)))
Theorem	zmodidfzoimp 10310	Identity law for modulo restricted to integers. (Contributed by AV, 27-Oct-2018.)
⊢ (𝑀 ∈ (0..^𝑁) → (𝑀 mod 𝑁) = 𝑀)
Theorem	q0mod 10311	Special case: 0 modulo a positive real number is 0. (Contributed by Jim Kingdon, 21-Oct-2021.)
⊢ ((𝑁 ∈ ℚ ∧ 0 < 𝑁) → (0 mod 𝑁) = 0)
Theorem	q1mod 10312	Special case: 1 modulo a real number greater than 1 is 1. (Contributed by Jim Kingdon, 21-Oct-2021.)
⊢ ((𝑁 ∈ ℚ ∧ 1 < 𝑁) → (1 mod 𝑁) = 1)
Theorem	modqabs 10313	Absorption law for modulo. (Contributed by Jim Kingdon, 21-Oct-2021.)
⊢ (𝜑 → 𝐴 ∈ ℚ)    &   ⊢ (𝜑 → 𝐵 ∈ ℚ)    &   ⊢ (𝜑 → 0 < 𝐵)    &   ⊢ (𝜑 → 𝐶 ∈ ℚ)    &   ⊢ (𝜑 → 𝐵 ≤ 𝐶)    ⇒   ⊢ (𝜑 → ((𝐴 mod 𝐵) mod 𝐶) = (𝐴 mod 𝐵))
Theorem	modqabs2 10314	Absorption law for modulo. (Contributed by Jim Kingdon, 21-Oct-2021.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → ((𝐴 mod 𝐵) mod 𝐵) = (𝐴 mod 𝐵))
Theorem	modqcyc 10315	The modulo operation is periodic. (Contributed by Jim Kingdon, 21-Oct-2021.)
⊢ (((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℤ) ∧ (𝐵 ∈ ℚ ∧ 0 < 𝐵)) → ((𝐴 + (𝑁 · 𝐵)) mod 𝐵) = (𝐴 mod 𝐵))
Theorem	modqcyc2 10316	The modulo operation is periodic. (Contributed by Jim Kingdon, 21-Oct-2021.)
⊢ (((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℤ) ∧ (𝐵 ∈ ℚ ∧ 0 < 𝐵)) → ((𝐴 − (𝐵 · 𝑁)) mod 𝐵) = (𝐴 mod 𝐵))
Theorem	modqadd1 10317	Addition property of the modulo operation. (Contributed by Jim Kingdon, 22-Oct-2021.)
⊢ (𝜑 → 𝐴 ∈ ℚ)    &   ⊢ (𝜑 → 𝐵 ∈ ℚ)    &   ⊢ (𝜑 → 𝐶 ∈ ℚ)    &   ⊢ (𝜑 → 𝐷 ∈ ℚ)    &   ⊢ (𝜑 → 0 < 𝐷)    &   ⊢ (𝜑 → (𝐴 mod 𝐷) = (𝐵 mod 𝐷))    ⇒   ⊢ (𝜑 → ((𝐴 + 𝐶) mod 𝐷) = ((𝐵 + 𝐶) mod 𝐷))
Theorem	modqaddabs 10318	Absorption law for modulo. (Contributed by Jim Kingdon, 22-Oct-2021.)
⊢ (((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ (𝐶 ∈ ℚ ∧ 0 < 𝐶)) → (((𝐴 mod 𝐶) + (𝐵 mod 𝐶)) mod 𝐶) = ((𝐴 + 𝐵) mod 𝐶))
Theorem	modqaddmod 10319	The sum of a number modulo a modulus and another number equals the sum of the two numbers modulo the same modulus. (Contributed by Jim Kingdon, 23-Oct-2021.)
⊢ (((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ (𝑀 ∈ ℚ ∧ 0 < 𝑀)) → (((𝐴 mod 𝑀) + 𝐵) mod 𝑀) = ((𝐴 + 𝐵) mod 𝑀))
Theorem	mulqaddmodid 10320	The sum of a positive rational number less than an upper bound and the product of an integer and the upper bound is the positive rational number modulo the upper bound. (Contributed by Jim Kingdon, 23-Oct-2021.)
⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℚ) ∧ (𝐴 ∈ ℚ ∧ 𝐴 ∈ (0[,)𝑀))) → (((𝑁 · 𝑀) + 𝐴) mod 𝑀) = 𝐴)
Theorem	mulp1mod1 10321	The product of an integer and an integer greater than 1 increased by 1 is 1 modulo the integer greater than 1. (Contributed by AV, 15-Jul-2021.)
⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘2)) → (((𝑁 · 𝐴) + 1) mod 𝑁) = 1)
Theorem	modqmuladd 10322*	Decomposition of an integer into a multiple of a modulus and a remainder. (Contributed by Jim Kingdon, 23-Oct-2021.)
⊢ (𝜑 → 𝐴 ∈ ℤ)    &   ⊢ (𝜑 → 𝐵 ∈ ℚ)    &   ⊢ (𝜑 → 𝐵 ∈ (0[,)𝑀))    &   ⊢ (𝜑 → 𝑀 ∈ ℚ)    &   ⊢ (𝜑 → 0 < 𝑀)    ⇒   ⊢ (𝜑 → ((𝐴 mod 𝑀) = 𝐵 ↔ ∃𝑘 ∈ ℤ 𝐴 = ((𝑘 · 𝑀) + 𝐵)))
Theorem	modqmuladdim 10323*	Implication of a decomposition of an integer into a multiple of a modulus and a remainder. (Contributed by Jim Kingdon, 23-Oct-2021.)
⊢ ((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℚ ∧ 0 < 𝑀) → ((𝐴 mod 𝑀) = 𝐵 → ∃𝑘 ∈ ℤ 𝐴 = ((𝑘 · 𝑀) + 𝐵)))
Theorem	modqmuladdnn0 10324*	Implication of a decomposition of a nonnegative integer into a multiple of a modulus and a remainder. (Contributed by Jim Kingdon, 23-Oct-2021.)
⊢ ((𝐴 ∈ ℕ0 ∧ 𝑀 ∈ ℚ ∧ 0 < 𝑀) → ((𝐴 mod 𝑀) = 𝐵 → ∃𝑘 ∈ ℕ0 𝐴 = ((𝑘 · 𝑀) + 𝐵)))
Theorem	qnegmod 10325	The negation of a number modulo a positive number is equal to the difference of the modulus and the number modulo the modulus. (Contributed by Jim Kingdon, 24-Oct-2021.)
⊢ ((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℚ ∧ 0 < 𝑁) → (-𝐴 mod 𝑁) = ((𝑁 − 𝐴) mod 𝑁))
Theorem	m1modnnsub1 10326	Minus one modulo a positive integer is equal to the integer minus one. (Contributed by AV, 14-Jul-2021.)
⊢ (𝑀 ∈ ℕ → (-1 mod 𝑀) = (𝑀 − 1))
Theorem	m1modge3gt1 10327	Minus one modulo an integer greater than two is greater than one. (Contributed by AV, 14-Jul-2021.)
⊢ (𝑀 ∈ (ℤ≥‘3) → 1 < (-1 mod 𝑀))
Theorem	addmodid 10328	The sum of a positive integer and a nonnegative integer less than the positive integer is equal to the nonnegative integer modulo the positive integer. (Contributed by Alexander van der Vekens, 30-Oct-2018.) (Proof shortened by AV, 5-Jul-2020.)
⊢ ((𝐴 ∈ ℕ0 ∧ 𝑀 ∈ ℕ ∧ 𝐴 < 𝑀) → ((𝑀 + 𝐴) mod 𝑀) = 𝐴)
Theorem	addmodidr 10329	The sum of a positive integer and a nonnegative integer less than the positive integer is equal to the nonnegative integer modulo the positive integer. (Contributed by AV, 19-Mar-2021.)
⊢ ((𝐴 ∈ ℕ0 ∧ 𝑀 ∈ ℕ ∧ 𝐴 < 𝑀) → ((𝐴 + 𝑀) mod 𝑀) = 𝐴)
Theorem	modqadd2mod 10330	The sum of a number modulo a modulus and another number equals the sum of the two numbers modulo the modulus. (Contributed by Jim Kingdon, 24-Oct-2021.)
⊢ (((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ (𝑀 ∈ ℚ ∧ 0 < 𝑀)) → ((𝐵 + (𝐴 mod 𝑀)) mod 𝑀) = ((𝐵 + 𝐴) mod 𝑀))
Theorem	modqm1p1mod0 10331	If a number modulo a modulus equals the modulus decreased by 1, the first number increased by 1 modulo the modulus equals 0. (Contributed by Jim Kingdon, 24-Oct-2021.)
⊢ ((𝐴 ∈ ℚ ∧ 𝑀 ∈ ℚ ∧ 0 < 𝑀) → ((𝐴 mod 𝑀) = (𝑀 − 1) → ((𝐴 + 1) mod 𝑀) = 0))
Theorem	modqltm1p1mod 10332	If a number modulo a modulus is less than the modulus decreased by 1, the first number increased by 1 modulo the modulus equals the first number modulo the modulus, increased by 1. (Contributed by Jim Kingdon, 24-Oct-2021.)
⊢ (((𝐴 ∈ ℚ ∧ (𝐴 mod 𝑀) < (𝑀 − 1)) ∧ (𝑀 ∈ ℚ ∧ 0 < 𝑀)) → ((𝐴 + 1) mod 𝑀) = ((𝐴 mod 𝑀) + 1))
Theorem	modqmul1 10333	Multiplication property of the modulo operation. Note that the multiplier 𝐶 must be an integer. (Contributed by Jim Kingdon, 24-Oct-2021.)
⊢ (𝜑 → 𝐴 ∈ ℚ)    &   ⊢ (𝜑 → 𝐵 ∈ ℚ)    &   ⊢ (𝜑 → 𝐶 ∈ ℤ)    &   ⊢ (𝜑 → 𝐷 ∈ ℚ)    &   ⊢ (𝜑 → 0 < 𝐷)    &   ⊢ (𝜑 → (𝐴 mod 𝐷) = (𝐵 mod 𝐷))    ⇒   ⊢ (𝜑 → ((𝐴 · 𝐶) mod 𝐷) = ((𝐵 · 𝐶) mod 𝐷))
Theorem	modqmul12d 10334	Multiplication property of the modulo operation, see theorem 5.2(b) in [ApostolNT] p. 107. (Contributed by Jim Kingdon, 24-Oct-2021.)
⊢ (𝜑 → 𝐴 ∈ ℤ)    &   ⊢ (𝜑 → 𝐵 ∈ ℤ)    &   ⊢ (𝜑 → 𝐶 ∈ ℤ)    &   ⊢ (𝜑 → 𝐷 ∈ ℤ)    &   ⊢ (𝜑 → 𝐸 ∈ ℚ)    &   ⊢ (𝜑 → 0 < 𝐸)    &   ⊢ (𝜑 → (𝐴 mod 𝐸) = (𝐵 mod 𝐸))    &   ⊢ (𝜑 → (𝐶 mod 𝐸) = (𝐷 mod 𝐸))    ⇒   ⊢ (𝜑 → ((𝐴 · 𝐶) mod 𝐸) = ((𝐵 · 𝐷) mod 𝐸))
Theorem	modqnegd 10335	Negation property of the modulo operation. (Contributed by Jim Kingdon, 24-Oct-2021.)
⊢ (𝜑 → 𝐴 ∈ ℚ)    &   ⊢ (𝜑 → 𝐵 ∈ ℚ)    &   ⊢ (𝜑 → 𝐶 ∈ ℚ)    &   ⊢ (𝜑 → 0 < 𝐶)    &   ⊢ (𝜑 → (𝐴 mod 𝐶) = (𝐵 mod 𝐶))    ⇒   ⊢ (𝜑 → (-𝐴 mod 𝐶) = (-𝐵 mod 𝐶))
Theorem	modqadd12d 10336	Additive property of the modulo operation. (Contributed by Jim Kingdon, 25-Oct-2021.)
⊢ (𝜑 → 𝐴 ∈ ℚ)    &   ⊢ (𝜑 → 𝐵 ∈ ℚ)    &   ⊢ (𝜑 → 𝐶 ∈ ℚ)    &   ⊢ (𝜑 → 𝐷 ∈ ℚ)    &   ⊢ (𝜑 → 𝐸 ∈ ℚ)    &   ⊢ (𝜑 → 0 < 𝐸)    &   ⊢ (𝜑 → (𝐴 mod 𝐸) = (𝐵 mod 𝐸))    &   ⊢ (𝜑 → (𝐶 mod 𝐸) = (𝐷 mod 𝐸))    ⇒   ⊢ (𝜑 → ((𝐴 + 𝐶) mod 𝐸) = ((𝐵 + 𝐷) mod 𝐸))
Theorem	modqsub12d 10337	Subtraction property of the modulo operation. (Contributed by Jim Kingdon, 25-Oct-2021.)
⊢ (𝜑 → 𝐴 ∈ ℚ)    &   ⊢ (𝜑 → 𝐵 ∈ ℚ)    &   ⊢ (𝜑 → 𝐶 ∈ ℚ)    &   ⊢ (𝜑 → 𝐷 ∈ ℚ)    &   ⊢ (𝜑 → 𝐸 ∈ ℚ)    &   ⊢ (𝜑 → 0 < 𝐸)    &   ⊢ (𝜑 → (𝐴 mod 𝐸) = (𝐵 mod 𝐸))    &   ⊢ (𝜑 → (𝐶 mod 𝐸) = (𝐷 mod 𝐸))    ⇒   ⊢ (𝜑 → ((𝐴 − 𝐶) mod 𝐸) = ((𝐵 − 𝐷) mod 𝐸))
Theorem	modqsubmod 10338	The difference of a number modulo a modulus and another number equals the difference of the two numbers modulo the modulus. (Contributed by Jim Kingdon, 25-Oct-2021.)
⊢ (((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ (𝑀 ∈ ℚ ∧ 0 < 𝑀)) → (((𝐴 mod 𝑀) − 𝐵) mod 𝑀) = ((𝐴 − 𝐵) mod 𝑀))
Theorem	modqsubmodmod 10339	The difference of a number modulo a modulus and another number modulo the same modulus equals the difference of the two numbers modulo the modulus. (Contributed by Jim Kingdon, 25-Oct-2021.)
⊢ (((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ (𝑀 ∈ ℚ ∧ 0 < 𝑀)) → (((𝐴 mod 𝑀) − (𝐵 mod 𝑀)) mod 𝑀) = ((𝐴 − 𝐵) mod 𝑀))
Theorem	q2txmodxeq0 10340	Two times a positive number modulo the number is zero. (Contributed by Jim Kingdon, 25-Oct-2021.)
⊢ ((𝑋 ∈ ℚ ∧ 0 < 𝑋) → ((2 · 𝑋) mod 𝑋) = 0)
Theorem	q2submod 10341	If a number is between a modulus and twice the modulus, the first number modulo the modulus equals the first number minus the modulus. (Contributed by Jim Kingdon, 25-Oct-2021.)
⊢ (((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) ∧ (𝐵 ≤ 𝐴 ∧ 𝐴 < (2 · 𝐵))) → (𝐴 mod 𝐵) = (𝐴 − 𝐵))
Theorem	modifeq2int 10342	If a nonnegative integer is less than twice a positive integer, the nonnegative integer modulo the positive integer equals the nonnegative integer or the nonnegative integer minus the positive integer. (Contributed by Alexander van der Vekens, 21-May-2018.)
⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ∧ 𝐴 < (2 · 𝐵)) → (𝐴 mod 𝐵) = if(𝐴 < 𝐵, 𝐴, (𝐴 − 𝐵)))
Theorem	modaddmodup 10343	The sum of an integer modulo a positive integer and another integer minus the positive integer equals the sum of the two integers modulo the positive integer if the other integer is in the upper part of the range between 0 and the positive integer. (Contributed by AV, 30-Oct-2018.)
⊢ ((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℕ) → (𝐵 ∈ ((𝑀 − (𝐴 mod 𝑀))..^𝑀) → ((𝐵 + (𝐴 mod 𝑀)) − 𝑀) = ((𝐵 + 𝐴) mod 𝑀)))
Theorem	modaddmodlo 10344	The sum of an integer modulo a positive integer and another integer equals the sum of the two integers modulo the positive integer if the other integer is in the lower part of the range between 0 and the positive integer. (Contributed by AV, 30-Oct-2018.)
⊢ ((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℕ) → (𝐵 ∈ (0..^(𝑀 − (𝐴 mod 𝑀))) → (𝐵 + (𝐴 mod 𝑀)) = ((𝐵 + 𝐴) mod 𝑀)))
Theorem	modqmulmod 10345	The product of a rational number modulo a modulus and an integer equals the product of the rational number and the integer modulo the modulus. (Contributed by Jim Kingdon, 25-Oct-2021.)
⊢ (((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ) ∧ (𝑀 ∈ ℚ ∧ 0 < 𝑀)) → (((𝐴 mod 𝑀) · 𝐵) mod 𝑀) = ((𝐴 · 𝐵) mod 𝑀))
Theorem	modqmulmodr 10346	The product of an integer and a rational number modulo a modulus equals the product of the integer and the rational number modulo the modulus. (Contributed by Jim Kingdon, 26-Oct-2021.)
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℚ) ∧ (𝑀 ∈ ℚ ∧ 0 < 𝑀)) → ((𝐴 · (𝐵 mod 𝑀)) mod 𝑀) = ((𝐴 · 𝐵) mod 𝑀))
Theorem	modqaddmulmod 10347	The sum of a rational number and the product of a second rational number modulo a modulus and an integer equals the sum of the rational number and the product of the other rational number and the integer modulo the modulus. (Contributed by Jim Kingdon, 26-Oct-2021.)
⊢ (((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 𝐶 ∈ ℤ) ∧ (𝑀 ∈ ℚ ∧ 0 < 𝑀)) → ((𝐴 + ((𝐵 mod 𝑀) · 𝐶)) mod 𝑀) = ((𝐴 + (𝐵 · 𝐶)) mod 𝑀))
Theorem	modqdi 10348	Distribute multiplication over a modulo operation. (Contributed by Jim Kingdon, 26-Oct-2021.)
⊢ (((𝐴 ∈ ℚ ∧ 0 < 𝐴) ∧ 𝐵 ∈ ℚ ∧ (𝐶 ∈ ℚ ∧ 0 < 𝐶)) → (𝐴 · (𝐵 mod 𝐶)) = ((𝐴 · 𝐵) mod (𝐴 · 𝐶)))
Theorem	modqsubdir 10349	Distribute the modulo operation over a subtraction. (Contributed by Jim Kingdon, 26-Oct-2021.)
⊢ (((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ (𝐶 ∈ ℚ ∧ 0 < 𝐶)) → ((𝐵 mod 𝐶) ≤ (𝐴 mod 𝐶) ↔ ((𝐴 − 𝐵) mod 𝐶) = ((𝐴 mod 𝐶) − (𝐵 mod 𝐶))))
Theorem	modqeqmodmin 10350	A rational number equals the difference of the rational number and a modulus modulo the modulus. (Contributed by Jim Kingdon, 26-Oct-2021.)
⊢ ((𝐴 ∈ ℚ ∧ 𝑀 ∈ ℚ ∧ 0 < 𝑀) → (𝐴 mod 𝑀) = ((𝐴 − 𝑀) mod 𝑀))
Theorem	modfzo0difsn 10351*	For a number within a half-open range of nonnegative integers with one excluded integer there is a positive integer so that the number is equal to the sum of the positive integer and the excluded integer modulo the upper bound of the range. (Contributed by AV, 19-Mar-2021.)
⊢ ((𝐽 ∈ (0..^𝑁) ∧ 𝐾 ∈ ((0..^𝑁) ∖ {𝐽})) → ∃𝑖 ∈ (1..^𝑁)𝐾 = ((𝑖 + 𝐽) mod 𝑁))
Theorem	modsumfzodifsn 10352	The sum of a number within a half-open range of positive integers is an element of the corresponding open range of nonnegative integers with one excluded integer modulo the excluded integer. (Contributed by AV, 19-Mar-2021.)
⊢ ((𝐽 ∈ (0..^𝑁) ∧ 𝐾 ∈ (1..^𝑁)) → ((𝐾 + 𝐽) mod 𝑁) ∈ ((0..^𝑁) ∖ {𝐽}))
Theorem	modlteq 10353	Two nonnegative integers less than the modulus are equal iff they are equal modulo the modulus. (Contributed by AV, 14-Mar-2021.)
⊢ ((𝐼 ∈ (0..^𝑁) ∧ 𝐽 ∈ (0..^𝑁)) → ((𝐼 mod 𝑁) = (𝐽 mod 𝑁) ↔ 𝐼 = 𝐽))
Theorem	addmodlteq 10354	Two nonnegative integers less than the modulus are equal iff the sums of these integer with another integer are equal modulo the modulus. (Contributed by AV, 20-Mar-2021.)
⊢ ((𝐼 ∈ (0..^𝑁) ∧ 𝐽 ∈ (0..^𝑁) ∧ 𝑆 ∈ ℤ) → (((𝐼 + 𝑆) mod 𝑁) = ((𝐽 + 𝑆) mod 𝑁) ↔ 𝐼 = 𝐽))
4.6.3  Miscellaneous theorems about integers
Theorem	frec2uz0d 10355*	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 10356*	The value of 𝐺 (see frec2uz0d 10355) 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 10357*	The value of 𝐺 (see frec2uz0d 10355) 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 10358*	The value 𝐺 (see frec2uz0d 10355) 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 10359*	Less-than relation for 𝐺 (see frec2uz0d 10355). (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 10360*	The mapping 𝐺 (see frec2uz0d 10355) 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 10361*	Range of 𝐺 (see frec2uz0d 10355). (Contributed by Jim Kingdon, 17-May-2020.)
⊢ (𝜑 → 𝐶 ∈ ℤ)    &   ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)    ⇒   ⊢ (𝜑 → ran 𝐺 = (ℤ≥‘𝐶))
Theorem	frec2uzf1od 10362*	𝐺 (see frec2uz0d 10355) 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 10363*	𝐺 (see frec2uz0d 10355) 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 10364*	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, 28-Mar-2022.)
⊢ (𝜑 → 𝐶 ∈ ℤ)    &   ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)    &   ⊢ (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 10365*	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 10355 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 → 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	frecuzrdgrcl 10366*	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, 1-Apr-2022.)
⊢ (𝜑 → 𝐶 ∈ ℤ)    &   ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)    &   ⊢ (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:ω⟶((ℤ≥‘𝐶) × 𝑆))
Theorem	frecuzrdglem 10367*	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 → 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	frecuzrdgtcl 10368*	The recursive definition generator on upper integers is a function. See comment in frec2uz0d 10355 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 → 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:(ℤ≥‘𝐶)⟶𝑆)
Theorem	frecuzrdg0 10369*	Initial value of a recursive definition generator on upper integers. See comment in frec2uz0d 10355 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 → 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 10370*	Successor value of a recursive definition generator on upper integers. See comment in frec2uz0d 10355 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 → 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	frecuzrdgrclt 10371*	The function 𝑅 (used in the definition of the recursive definition generator on upper integers) yields ordered pairs of integers and elements of 𝑆. Similar to frecuzrdgrcl 10366 except that 𝑆 and 𝑇 need not be the same. (Contributed by Jim Kingdon, 22-Apr-2022.)
⊢ (𝜑 → 𝐶 ∈ ℤ)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑆 ⊆ 𝑇)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)    &   ⊢ 𝑅 = frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)    ⇒   ⊢ (𝜑 → 𝑅:ω⟶((ℤ≥‘𝐶) × 𝑆))
Theorem	frecuzrdgg 10372*	Lemma for other theorems involving the the recursive definition generator on upper integers. Evaluating 𝑅 at a natural number gives an ordered pair whose first element is the mapping of that natural number via 𝐺. (Contributed by Jim Kingdon, 23-Apr-2022.)
⊢ (𝜑 → 𝐶 ∈ ℤ)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑆 ⊆ 𝑇)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)    &   ⊢ 𝑅 = frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)    &   ⊢ (𝜑 → 𝑁 ∈ ω)    &   ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)    ⇒   ⊢ (𝜑 → (1st ‘(𝑅‘𝑁)) = (𝐺‘𝑁))
Theorem	frecuzrdgdomlem 10373*	The domain of the result of the recursive definition generator on upper integers. (Contributed by Jim Kingdon, 24-Apr-2022.)
⊢ (𝜑 → 𝐶 ∈ ℤ)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑆 ⊆ 𝑇)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)    &   ⊢ 𝑅 = frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)    &   ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)    ⇒   ⊢ (𝜑 → dom ran 𝑅 = (ℤ≥‘𝐶))
Theorem	frecuzrdgdom 10374*	The domain of the result of the recursive definition generator on upper integers. (Contributed by Jim Kingdon, 24-Apr-2022.)
⊢ (𝜑 → 𝐶 ∈ ℤ)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑆 ⊆ 𝑇)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)    &   ⊢ 𝑅 = frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)    ⇒   ⊢ (𝜑 → dom ran 𝑅 = (ℤ≥‘𝐶))
Theorem	frecuzrdgfunlem 10375*	The recursive definition generator on upper integers produces a a function. (Contributed by Jim Kingdon, 24-Apr-2022.)
⊢ (𝜑 → 𝐶 ∈ ℤ)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑆 ⊆ 𝑇)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)    &   ⊢ 𝑅 = frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)    &   ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)    ⇒   ⊢ (𝜑 → Fun ran 𝑅)
Theorem	frecuzrdgfun 10376*	The recursive definition generator on upper integers produces a a function. (Contributed by Jim Kingdon, 24-Apr-2022.)
⊢ (𝜑 → 𝐶 ∈ ℤ)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑆 ⊆ 𝑇)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)    &   ⊢ 𝑅 = frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)    ⇒   ⊢ (𝜑 → Fun ran 𝑅)
Theorem	frecuzrdgtclt 10377*	The recursive definition generator on upper integers is a function. (Contributed by Jim Kingdon, 22-Apr-2022.)
⊢ (𝜑 → 𝐶 ∈ ℤ)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑆 ⊆ 𝑇)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)    &   ⊢ 𝑅 = frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)    &   ⊢ (𝜑 → 𝑃 = ran 𝑅)    ⇒   ⊢ (𝜑 → 𝑃:(ℤ≥‘𝐶)⟶𝑆)
Theorem	frecuzrdg0t 10378*	Initial value of a recursive definition generator on upper integers. (Contributed by Jim Kingdon, 28-Apr-2022.)
⊢ (𝜑 → 𝐶 ∈ ℤ)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑆 ⊆ 𝑇)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)    &   ⊢ 𝑅 = frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)    &   ⊢ (𝜑 → 𝑃 = ran 𝑅)    ⇒   ⊢ (𝜑 → (𝑃‘𝐶) = 𝐴)
Theorem	frecuzrdgsuctlem 10379*	Successor value of a recursive definition generator on upper integers. See comment in frec2uz0d 10355 for the description of 𝐺 as the mapping from ω to (ℤ≥‘𝐶). (Contributed by Jim Kingdon, 29-Apr-2022.)
⊢ (𝜑 → 𝐶 ∈ ℤ)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑆 ⊆ 𝑇)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)    &   ⊢ 𝑅 = frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)    &   ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)    &   ⊢ (𝜑 → 𝑃 = ran 𝑅)    ⇒   ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (𝑃‘(𝐵 + 1)) = (𝐵𝐹(𝑃‘𝐵)))
Theorem	frecuzrdgsuct 10380*	Successor value of a recursive definition generator on upper integers. (Contributed by Jim Kingdon, 29-Apr-2022.)
⊢ (𝜑 → 𝐶 ∈ ℤ)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑆 ⊆ 𝑇)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)    &   ⊢ 𝑅 = frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)    &   ⊢ (𝜑 → 𝑃 = ran 𝑅)    ⇒   ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (𝑃‘(𝐵 + 1)) = (𝐵𝐹(𝑃‘𝐵)))
Theorem	uzenom 10381	An upper integer set is denumerable. (Contributed by Mario Carneiro, 15-Oct-2015.)
⊢ 𝑍 = (ℤ≥‘𝑀)    ⇒   ⊢ (𝑀 ∈ ℤ → 𝑍 ≈ ω)
Theorem	frecfzennn 10382	The cardinality of a finite set of sequential integers. (See frec2uz0d 10355 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 10383	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 10384	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	frec2uzled 10385*	The mapping 𝐺 (see frec2uz0d 10355) preserves order. (Contributed by Jim Kingdon, 24-Feb-2022.)
⊢ (𝜑 → 𝐶 ∈ ℤ)    &   ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)    &   ⊢ (𝜑 → 𝐴 ∈ ω)    &   ⊢ (𝜑 → 𝐵 ∈ ω)    ⇒   ⊢ (𝜑 → (𝐴 ⊆ 𝐵 ↔ (𝐺‘𝐴) ≤ (𝐺‘𝐵)))
Theorem	fzfig 10386	A finite interval of integers is finite. (Contributed by Jim Kingdon, 19-May-2020.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀...𝑁) ∈ Fin)
Theorem	fzfigd 10387	Deduction form of fzfig 10386. (Contributed by Jim Kingdon, 21-May-2020.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝑁 ∈ ℤ)    ⇒   ⊢ (𝜑 → (𝑀...𝑁) ∈ Fin)
Theorem	fzofig 10388	Half-open integer sets are finite. (Contributed by Jim Kingdon, 21-May-2020.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀..^𝑁) ∈ Fin)
Theorem	nn0ennn 10389	The nonnegative integers are equinumerous to the positive integers. (Contributed by NM, 19-Jul-2004.)
⊢ ℕ0 ≈ ℕ
Theorem	nnenom 10390	The set of positive integers (as a subset of complex numbers) is equinumerous to omega (the set of natural numbers as ordinals). (Contributed by NM, 31-Jul-2004.) (Revised by Mario Carneiro, 15-Sep-2013.)
⊢ ℕ ≈ ω
Theorem	nnct 10391	ℕ is dominated by ω. (Contributed by Thierry Arnoux, 29-Dec-2016.)
⊢ ℕ ≼ ω
Theorem	uzennn 10392	An upper integer set is equinumerous to the set of natural numbers. (Contributed by Jim Kingdon, 30-Jul-2023.)
⊢ (𝑀 ∈ ℤ → (ℤ≥‘𝑀) ≈ ℕ)
Theorem	fnn0nninf 10393*	A function from ℕ0 into ℕ∞. (Contributed by Jim Kingdon, 16-Jul-2022.)
⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)    &   ⊢ 𝐹 = (𝑛 ∈ ω ↦ (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅)))    ⇒   ⊢ (𝐹 ∘ ◡𝐺):ℕ0⟶ℕ∞
Theorem	fxnn0nninf 10394*	A function from ℕ0* into ℕ∞. (Contributed by Jim Kingdon, 16-Jul-2022.) TODO: use infnninf 7100 instead of infnninfOLD 7101. More generally, this theorem and most theorems in this section could use an extended 𝐺 defined by 𝐺 = (frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ ⟨ω, +∞⟩) and 𝐹 = (𝑛 ∈ suc ω ↦ (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) as in nnnninf2 7103.
⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)    &   ⊢ 𝐹 = (𝑛 ∈ ω ↦ (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅)))    &   ⊢ 𝐼 = ((𝐹 ∘ ◡𝐺) ∪ {⟨+∞, (ω × {1o})⟩})    ⇒   ⊢ 𝐼:ℕ0*⟶ℕ∞
Theorem	0tonninf 10395*	The mapping of zero into ℕ∞ is the sequence of all zeroes. (Contributed by Jim Kingdon, 17-Jul-2022.)
⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)    &   ⊢ 𝐹 = (𝑛 ∈ ω ↦ (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅)))    &   ⊢ 𝐼 = ((𝐹 ∘ ◡𝐺) ∪ {⟨+∞, (ω × {1o})⟩})    ⇒   ⊢ (𝐼‘0) = (𝑥 ∈ ω ↦ ∅)
Theorem	1tonninf 10396*	The mapping of one into ℕ∞ is a sequence which is a one followed by zeroes. (Contributed by Jim Kingdon, 17-Jul-2022.)
⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)    &   ⊢ 𝐹 = (𝑛 ∈ ω ↦ (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅)))    &   ⊢ 𝐼 = ((𝐹 ∘ ◡𝐺) ∪ {⟨+∞, (ω × {1o})⟩})    ⇒   ⊢ (𝐼‘1) = (𝑥 ∈ ω ↦ if(𝑥 = ∅, 1o, ∅))
Theorem	inftonninf 10397*	The mapping of +∞ into ℕ∞ is the sequence of all ones. (Contributed by Jim Kingdon, 17-Jul-2022.)
⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)    &   ⊢ 𝐹 = (𝑛 ∈ ω ↦ (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅)))    &   ⊢ 𝐼 = ((𝐹 ∘ ◡𝐺) ∪ {⟨+∞, (ω × {1o})⟩})    ⇒   ⊢ (𝐼‘+∞) = (𝑥 ∈ ω ↦ 1o)
4.6.4  Strong induction over upper sets of integers
Theorem	uzsinds 10398*	Strong (or "total") induction principle over an upper set of integers. (Contributed by Scott Fenton, 16-May-2014.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝑁 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 ∈ (ℤ≥‘𝑀) → (∀𝑦 ∈ (𝑀...(𝑥 − 1))𝜓 → 𝜑))    ⇒   ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝜒)
Theorem	nnsinds 10399*	Strong (or "total") induction principle over the naturals. (Contributed by Scott Fenton, 16-May-2014.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝑁 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 ∈ ℕ → (∀𝑦 ∈ (1...(𝑥 − 1))𝜓 → 𝜑))    ⇒   ⊢ (𝑁 ∈ ℕ → 𝜒)
Theorem	nn0sinds 10400*	Strong (or "total") induction principle over the nonnegative integers. (Contributed by Scott Fenton, 16-May-2014.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝑁 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 ∈ ℕ0 → (∀𝑦 ∈ (0...(𝑥 − 1))𝜓 → 𝜑))    ⇒   ⊢ (𝑁 ∈ ℕ0 → 𝜒)