P. 105 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 10401-10500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	adddivflid 10401	The floor of a sum of an integer and a fraction is equal to the integer iff the denominator of the fraction is less than the numerator. (Contributed by AV, 14-Jul-2021.)
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ0 ∧ 𝐶 ∈ ℕ) → (𝐵 < 𝐶 ↔ (⌊‘(𝐴 + (𝐵 / 𝐶))) = 𝐴))
Theorem	flqge0nn0 10402	The floor of a number greater than or equal to 0 is a nonnegative integer. (Contributed by Jim Kingdon, 10-Oct-2021.)
⊢ ((𝐴 ∈ ℚ ∧ 0 ≤ 𝐴) → (⌊‘𝐴) ∈ ℕ0)
Theorem	flqge1nn 10403	The floor of a number greater than or equal to 1 is a positive integer. (Contributed by Jim Kingdon, 10-Oct-2021.)
⊢ ((𝐴 ∈ ℚ ∧ 1 ≤ 𝐴) → (⌊‘𝐴) ∈ ℕ)
Theorem	fldivnn0 10404	The floor function of a division of a nonnegative integer by a positive integer is a nonnegative integer. (Contributed by Alexander van der Vekens, 14-Apr-2018.)
⊢ ((𝐾 ∈ ℕ0 ∧ 𝐿 ∈ ℕ) → (⌊‘(𝐾 / 𝐿)) ∈ ℕ0)
Theorem	divfl0 10405	The floor of a fraction is 0 iff the denominator is less than the numerator. (Contributed by AV, 8-Jul-2021.)
⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ) → (𝐴 < 𝐵 ↔ (⌊‘(𝐴 / 𝐵)) = 0))
Theorem	flqaddz 10406	An integer can be moved in and out of the floor of a sum. (Contributed by Jim Kingdon, 10-Oct-2021.)
⊢ ((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℤ) → (⌊‘(𝐴 + 𝑁)) = ((⌊‘𝐴) + 𝑁))
Theorem	flqzadd 10407	An integer can be moved in and out of the floor of a sum. (Contributed by Jim Kingdon, 10-Oct-2021.)
⊢ ((𝑁 ∈ ℤ ∧ 𝐴 ∈ ℚ) → (⌊‘(𝑁 + 𝐴)) = (𝑁 + (⌊‘𝐴)))
Theorem	flqmulnn0 10408	Move a nonnegative integer in and out of a floor. (Contributed by Jim Kingdon, 10-Oct-2021.)
⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℚ) → (𝑁 · (⌊‘𝐴)) ≤ (⌊‘(𝑁 · 𝐴)))
Theorem	btwnzge0 10409	A real bounded between an integer and its successor is nonnegative iff the integer is nonnegative. Second half of Lemma 13-4.1 of [Gleason] p. 217. (Contributed by NM, 12-Mar-2005.)
⊢ (((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) ∧ (𝑁 ≤ 𝐴 ∧ 𝐴 < (𝑁 + 1))) → (0 ≤ 𝐴 ↔ 0 ≤ 𝑁))
Theorem	2tnp1ge0ge0 10410	Two times an integer plus one is not negative iff the integer is not negative. (Contributed by AV, 19-Jun-2021.)
⊢ (𝑁 ∈ ℤ → (0 ≤ ((2 · 𝑁) + 1) ↔ 0 ≤ 𝑁))
Theorem	flhalf 10411	Ordering relation for the floor of half of an integer. (Contributed by NM, 1-Jan-2006.) (Proof shortened by Mario Carneiro, 7-Jun-2016.)
⊢ (𝑁 ∈ ℤ → 𝑁 ≤ (2 · (⌊‘((𝑁 + 1) / 2))))
Theorem	fldivnn0le 10412	The floor function of a division of a nonnegative integer by a positive integer is less than or equal to the division. (Contributed by Alexander van der Vekens, 14-Apr-2018.)
⊢ ((𝐾 ∈ ℕ0 ∧ 𝐿 ∈ ℕ) → (⌊‘(𝐾 / 𝐿)) ≤ (𝐾 / 𝐿))
Theorem	flltdivnn0lt 10413	The floor function of a division of a nonnegative integer by a positive integer is less than the division of a greater dividend by the same positive integer. (Contributed by Alexander van der Vekens, 14-Apr-2018.)
⊢ ((𝐾 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝐿 ∈ ℕ) → (𝐾 < 𝑁 → (⌊‘(𝐾 / 𝐿)) < (𝑁 / 𝐿)))
Theorem	fldiv4p1lem1div2 10414	The floor of an integer equal to 3 or greater than 4, increased by 1, is less than or equal to the half of the integer minus 1. (Contributed by AV, 8-Jul-2021.)
⊢ ((𝑁 = 3 ∨ 𝑁 ∈ (ℤ≥‘5)) → ((⌊‘(𝑁 / 4)) + 1) ≤ ((𝑁 − 1) / 2))
Theorem	fldiv4lem1div2uz2 10415	The floor of an integer greater than 1, divided by 4 is less than or equal to the half of the integer minus 1. (Contributed by AV, 5-Jul-2021.) (Proof shortened by AV, 9-Jul-2022.)
⊢ (𝑁 ∈ (ℤ≥‘2) → (⌊‘(𝑁 / 4)) ≤ ((𝑁 − 1) / 2))
Theorem	fldiv4lem1div2 10416	The floor of a positive integer divided by 4 is less than or equal to the half of the integer minus 1. (Contributed by AV, 9-Jul-2021.)
⊢ (𝑁 ∈ ℕ → (⌊‘(𝑁 / 4)) ≤ ((𝑁 − 1) / 2))
Theorem	ceilqval 10417	The value of the ceiling function. (Contributed by Jim Kingdon, 10-Oct-2021.)
⊢ (𝐴 ∈ ℚ → (⌈‘𝐴) = -(⌊‘-𝐴))
Theorem	ceiqcl 10418	The ceiling function returns an integer (closure law). (Contributed by Jim Kingdon, 11-Oct-2021.)
⊢ (𝐴 ∈ ℚ → -(⌊‘-𝐴) ∈ ℤ)
Theorem	ceilqcl 10419	Closure of the ceiling function. (Contributed by Jim Kingdon, 11-Oct-2021.)
⊢ (𝐴 ∈ ℚ → (⌈‘𝐴) ∈ ℤ)
Theorem	ceiqge 10420	The ceiling of a real number is greater than or equal to that number. (Contributed by Jim Kingdon, 11-Oct-2021.)
⊢ (𝐴 ∈ ℚ → 𝐴 ≤ -(⌊‘-𝐴))
Theorem	ceilqge 10421	The ceiling of a real number is greater than or equal to that number. (Contributed by Jim Kingdon, 11-Oct-2021.)
⊢ (𝐴 ∈ ℚ → 𝐴 ≤ (⌈‘𝐴))
Theorem	ceiqm1l 10422	One less than the ceiling of a real number is strictly less than that number. (Contributed by Jim Kingdon, 11-Oct-2021.)
⊢ (𝐴 ∈ ℚ → (-(⌊‘-𝐴) − 1) < 𝐴)
Theorem	ceilqm1lt 10423	One less than the ceiling of a real number is strictly less than that number. (Contributed by Jim Kingdon, 11-Oct-2021.)
⊢ (𝐴 ∈ ℚ → ((⌈‘𝐴) − 1) < 𝐴)
Theorem	ceiqle 10424	The ceiling of a real number is the smallest integer greater than or equal to it. (Contributed by Jim Kingdon, 11-Oct-2021.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐴 ≤ 𝐵) → -(⌊‘-𝐴) ≤ 𝐵)
Theorem	ceilqle 10425	The ceiling of a real number is the smallest integer greater than or equal to it. (Contributed by Jim Kingdon, 11-Oct-2021.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐴 ≤ 𝐵) → (⌈‘𝐴) ≤ 𝐵)
Theorem	ceilid 10426	An integer is its own ceiling. (Contributed by AV, 30-Nov-2018.)
⊢ (𝐴 ∈ ℤ → (⌈‘𝐴) = 𝐴)
Theorem	ceilqidz 10427	A rational number equals its ceiling iff it is an integer. (Contributed by Jim Kingdon, 11-Oct-2021.)
⊢ (𝐴 ∈ ℚ → (𝐴 ∈ ℤ ↔ (⌈‘𝐴) = 𝐴))
Theorem	flqleceil 10428	The floor of a rational number is less than or equal to its ceiling. (Contributed by Jim Kingdon, 11-Oct-2021.)
⊢ (𝐴 ∈ ℚ → (⌊‘𝐴) ≤ (⌈‘𝐴))
Theorem	flqeqceilz 10429	A rational number is an integer iff its floor equals its ceiling. (Contributed by Jim Kingdon, 11-Oct-2021.)
⊢ (𝐴 ∈ ℚ → (𝐴 ∈ ℤ ↔ (⌊‘𝐴) = (⌈‘𝐴)))
Theorem	intqfrac2 10430	Decompose a real into integer and fractional parts. (Contributed by Jim Kingdon, 18-Oct-2021.)
⊢ 𝑍 = (⌊‘𝐴)    &   ⊢ 𝐹 = (𝐴 − 𝑍)    ⇒   ⊢ (𝐴 ∈ ℚ → (0 ≤ 𝐹 ∧ 𝐹 < 1 ∧ 𝐴 = (𝑍 + 𝐹)))
Theorem	intfracq 10431	Decompose a rational number, expressed as a ratio, into integer and fractional parts. The fractional part has a tighter bound than that of intqfrac2 10430. (Contributed by NM, 16-Aug-2008.)
⊢ 𝑍 = (⌊‘(𝑀 / 𝑁))    &   ⊢ 𝐹 = ((𝑀 / 𝑁) − 𝑍)    ⇒   ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (0 ≤ 𝐹 ∧ 𝐹 ≤ ((𝑁 − 1) / 𝑁) ∧ (𝑀 / 𝑁) = (𝑍 + 𝐹)))
Theorem	flqdiv 10432	Cancellation of the embedded floor of a real divided by an integer. (Contributed by Jim Kingdon, 18-Oct-2021.)
⊢ ((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℕ) → (⌊‘((⌊‘𝐴) / 𝑁)) = (⌊‘(𝐴 / 𝑁)))
4.6.2  The modulo (remainder) operation
Syntax	cmo 10433	Extend class notation with the modulo operation.
class mod
Definition	df-mod 10434*	Define the modulo (remainder) operation. See modqval 10435 for its value. For example, (5 mod 3) = 2 and (-7 mod 2) = 1. As with df-fl 10379 we define this for first and second arguments which are real and positive real, respectively, even though many theorems will need to be more restricted (for example, specify rational arguments). (Contributed by NM, 10-Nov-2008.)
⊢ mod = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ+ ↦ (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦)))))
Theorem	modqval 10435	The value of the modulo operation. The modulo congruence notation of number theory, 𝐽≡𝐾 (modulo 𝑁), can be expressed in our notation as (𝐽 mod 𝑁) = (𝐾 mod 𝑁). Definition 1 in Knuth, The Art of Computer Programming, Vol. I (1972), p. 38. Knuth uses "mod" for the operation and "modulo" for the congruence. Unlike Knuth, we restrict the second argument to positive numbers to simplify certain theorems. (This also gives us future flexibility to extend it to any one of several different conventions for a zero or negative second argument, should there be an advantage in doing so.) As with flqcl 10382 we only prove this for rationals although other particular kinds of real numbers may be possible. (Contributed by Jim Kingdon, 16-Oct-2021.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → (𝐴 mod 𝐵) = (𝐴 − (𝐵 · (⌊‘(𝐴 / 𝐵)))))
Theorem	modqvalr 10436	The value of the modulo operation (multiplication in reversed order). (Contributed by Jim Kingdon, 16-Oct-2021.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → (𝐴 mod 𝐵) = (𝐴 − ((⌊‘(𝐴 / 𝐵)) · 𝐵)))
Theorem	modqcl 10437	Closure law for the modulo operation. (Contributed by Jim Kingdon, 16-Oct-2021.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → (𝐴 mod 𝐵) ∈ ℚ)
Theorem	flqpmodeq 10438	Partition of a division into its integer part and the remainder. (Contributed by Jim Kingdon, 16-Oct-2021.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → (((⌊‘(𝐴 / 𝐵)) · 𝐵) + (𝐴 mod 𝐵)) = 𝐴)
Theorem	modqcld 10439	Closure law for the modulo operation. (Contributed by Jim Kingdon, 16-Oct-2021.)
⊢ (𝜑 → 𝐴 ∈ ℚ)    &   ⊢ (𝜑 → 𝐵 ∈ ℚ)    &   ⊢ (𝜑 → 0 < 𝐵)    ⇒   ⊢ (𝜑 → (𝐴 mod 𝐵) ∈ ℚ)
Theorem	modq0 10440	𝐴 mod 𝐵 is zero iff 𝐴 is evenly divisible by 𝐵. (Contributed by Jim Kingdon, 17-Oct-2021.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → ((𝐴 mod 𝐵) = 0 ↔ (𝐴 / 𝐵) ∈ ℤ))
Theorem	mulqmod0 10441	The product of an integer and a positive rational number is 0 modulo the positive real number. (Contributed by Jim Kingdon, 18-Oct-2021.)
⊢ ((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℚ ∧ 0 < 𝑀) → ((𝐴 · 𝑀) mod 𝑀) = 0)
Theorem	negqmod0 10442	𝐴 is divisible by 𝐵 iff its negative is. (Contributed by Jim Kingdon, 18-Oct-2021.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → ((𝐴 mod 𝐵) = 0 ↔ (-𝐴 mod 𝐵) = 0))
Theorem	modqge0 10443	The modulo operation is nonnegative. (Contributed by Jim Kingdon, 18-Oct-2021.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → 0 ≤ (𝐴 mod 𝐵))
Theorem	modqlt 10444	The modulo operation is less than its second argument. (Contributed by Jim Kingdon, 18-Oct-2021.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → (𝐴 mod 𝐵) < 𝐵)
Theorem	modqelico 10445	Modular reduction produces a half-open interval. (Contributed by Jim Kingdon, 18-Oct-2021.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → (𝐴 mod 𝐵) ∈ (0[,)𝐵))
Theorem	modqdiffl 10446	The modulo operation differs from 𝐴 by an integer multiple of 𝐵. (Contributed by Jim Kingdon, 18-Oct-2021.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → ((𝐴 − (𝐴 mod 𝐵)) / 𝐵) = (⌊‘(𝐴 / 𝐵)))
Theorem	modqdifz 10447	The modulo operation differs from 𝐴 by an integer multiple of 𝐵. (Contributed by Jim Kingdon, 18-Oct-2021.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → ((𝐴 − (𝐴 mod 𝐵)) / 𝐵) ∈ ℤ)
Theorem	modqfrac 10448	The fractional part of a number is the number modulo 1. (Contributed by Jim Kingdon, 18-Oct-2021.)
⊢ (𝐴 ∈ ℚ → (𝐴 mod 1) = (𝐴 − (⌊‘𝐴)))
Theorem	flqmod 10449	The floor function expressed in terms of the modulo operation. (Contributed by Jim Kingdon, 18-Oct-2021.)
⊢ (𝐴 ∈ ℚ → (⌊‘𝐴) = (𝐴 − (𝐴 mod 1)))
Theorem	intqfrac 10450	Break a number into its integer part and its fractional part. (Contributed by Jim Kingdon, 18-Oct-2021.)
⊢ (𝐴 ∈ ℚ → 𝐴 = ((⌊‘𝐴) + (𝐴 mod 1)))
Theorem	zmod10 10451	An integer modulo 1 is 0. (Contributed by Paul Chapman, 22-Jun-2011.)
⊢ (𝑁 ∈ ℤ → (𝑁 mod 1) = 0)
Theorem	zmod1congr 10452	Two arbitrary integers are congruent modulo 1, see example 4 in [ApostolNT] p. 107. (Contributed by AV, 21-Jul-2021.)
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 mod 1) = (𝐵 mod 1))
Theorem	modqmulnn 10453	Move a positive integer in and out of a floor in the first argument of a modulo operation. (Contributed by Jim Kingdon, 18-Oct-2021.)
⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℚ ∧ 𝑀 ∈ ℕ) → ((𝑁 · (⌊‘𝐴)) mod (𝑁 · 𝑀)) ≤ ((⌊‘(𝑁 · 𝐴)) mod (𝑁 · 𝑀)))
Theorem	modqvalp1 10454	The value of the modulo operation (expressed with sum of denominator and nominator). (Contributed by Jim Kingdon, 20-Oct-2021.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → ((𝐴 + 𝐵) − (((⌊‘(𝐴 / 𝐵)) + 1) · 𝐵)) = (𝐴 mod 𝐵))
Theorem	zmodcl 10455	Closure law for the modulo operation restricted to integers. (Contributed by NM, 27-Nov-2008.)
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → (𝐴 mod 𝐵) ∈ ℕ0)
Theorem	zmodcld 10456	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 10457	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 10458	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 10459	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 10460	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 10461	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 10462	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 10463	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 10464	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 10465	Identity law for modulo restricted to integers. (Contributed by AV, 27-Oct-2018.)
⊢ (𝑀 ∈ (0..^𝑁) → (𝑀 mod 𝑁) = 𝑀)
Theorem	q0mod 10466	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 10467	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 10468	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 10469	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 10470	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 10471	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 10472	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 10473	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 10474	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 10475	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 10476	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 10477*	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 10478*	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 10479*	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 10480	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 10481	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 10482	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 10483	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 10484	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 10485	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 10486	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 10487	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 10488	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 10489	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 10490	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 10491	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 10492	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 10493	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 10494	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 10495	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 10496	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 10497	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 10498	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 10499	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 10500	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))