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Theorem List for Intuitionistic Logic Explorer - 9901-10000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremflqltnz 9901 If A is not an integer, then the floor of A is less than A. (Contributed by Jim Kingdon, 9-Oct-2021.)
((𝐴 ∈ ℚ ∧ ¬ 𝐴 ∈ ℤ) → (⌊‘𝐴) < 𝐴)
 
Theoremflqwordi 9902 Ordering relationship for the greatest integer function. (Contributed by Jim Kingdon, 9-Oct-2021.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 𝐴𝐵) → (⌊‘𝐴) ≤ (⌊‘𝐵))
 
Theoremflqword2 9903 Ordering relationship for the greatest integer function. (Contributed by Jim Kingdon, 9-Oct-2021.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 𝐴𝐵) → (⌊‘𝐵) ∈ (ℤ‘(⌊‘𝐴)))
 
Theoremflqbi 9904 A condition equivalent to floor. (Contributed by Jim Kingdon, 9-Oct-2021.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ) → ((⌊‘𝐴) = 𝐵 ↔ (𝐵𝐴𝐴 < (𝐵 + 1))))
 
Theoremflqbi2 9905 A condition equivalent to floor. (Contributed by Jim Kingdon, 9-Oct-2021.)
((𝑁 ∈ ℤ ∧ 𝐹 ∈ ℚ) → ((⌊‘(𝑁 + 𝐹)) = 𝑁 ↔ (0 ≤ 𝐹𝐹 < 1)))
 
Theoremadddivflid 9906 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 ∈ ℕ0𝐶 ∈ ℕ) → (𝐵 < 𝐶 ↔ (⌊‘(𝐴 + (𝐵 / 𝐶))) = 𝐴))
 
Theoremflqge0nn0 9907 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)
 
Theoremflqge1nn 9908 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)
 
Theoremfldivnn0 9909 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 ∈ ℕ0𝐿 ∈ ℕ) → (⌊‘(𝐾 / 𝐿)) ∈ ℕ0)
 
Theoremdivfl0 9910 The floor of a fraction is 0 iff the denominator is less than the numerator. (Contributed by AV, 8-Jul-2021.)
((𝐴 ∈ ℕ0𝐵 ∈ ℕ) → (𝐴 < 𝐵 ↔ (⌊‘(𝐴 / 𝐵)) = 0))
 
Theoremflqaddz 9911 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))
 
Theoremflqzadd 9912 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)))
 
Theoremflqmulnn0 9913 Move a nonnegative integer in and out of a floor. (Contributed by Jim Kingdon, 10-Oct-2021.)
((𝑁 ∈ ℕ0𝐴 ∈ ℚ) → (𝑁 · (⌊‘𝐴)) ≤ (⌊‘(𝑁 · 𝐴)))
 
Theorembtwnzge0 9914 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) ∧ (𝑁𝐴𝐴 < (𝑁 + 1))) → (0 ≤ 𝐴 ↔ 0 ≤ 𝑁))
 
Theorem2tnp1ge0ge0 9915 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))
 
Theoremflhalf 9916 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))))
 
Theoremfldivnn0le 9917 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 ∈ ℕ0𝐿 ∈ ℕ) → (⌊‘(𝐾 / 𝐿)) ≤ (𝐾 / 𝐿))
 
Theoremflltdivnn0lt 9918 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 ∈ ℕ0𝑁 ∈ ℕ0𝐿 ∈ ℕ) → (𝐾 < 𝑁 → (⌊‘(𝐾 / 𝐿)) < (𝑁 / 𝐿)))
 
Theoremfldiv4p1lem1div2 9919 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))
 
Theoremceilqval 9920 The value of the ceiling function. (Contributed by Jim Kingdon, 10-Oct-2021.)
(𝐴 ∈ ℚ → (⌈‘𝐴) = -(⌊‘-𝐴))
 
Theoremceiqcl 9921 The ceiling function returns an integer (closure law). (Contributed by Jim Kingdon, 11-Oct-2021.)
(𝐴 ∈ ℚ → -(⌊‘-𝐴) ∈ ℤ)
 
Theoremceilqcl 9922 Closure of the ceiling function. (Contributed by Jim Kingdon, 11-Oct-2021.)
(𝐴 ∈ ℚ → (⌈‘𝐴) ∈ ℤ)
 
Theoremceiqge 9923 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))
 
Theoremceilqge 9924 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))
 
Theoremceiqm1l 9925 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)
 
Theoremceilqm1lt 9926 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)
 
Theoremceiqle 9927 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 ∧ 𝐴𝐵) → -(⌊‘-𝐴) ≤ 𝐵)
 
Theoremceilqle 9928 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 ∧ 𝐴𝐵) → (⌈‘𝐴) ≤ 𝐵)
 
Theoremceilid 9929 An integer is its own ceiling. (Contributed by AV, 30-Nov-2018.)
(𝐴 ∈ ℤ → (⌈‘𝐴) = 𝐴)
 
Theoremceilqidz 9930 A rational number equals its ceiling iff it is an integer. (Contributed by Jim Kingdon, 11-Oct-2021.)
(𝐴 ∈ ℚ → (𝐴 ∈ ℤ ↔ (⌈‘𝐴) = 𝐴))
 
Theoremflqleceil 9931 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))
 
Theoremflqeqceilz 9932 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)))
 
Theoremintqfrac2 9933 Decompose a real into integer and fractional parts. (Contributed by Jim Kingdon, 18-Oct-2021.)
𝑍 = (⌊‘𝐴)    &   𝐹 = (𝐴𝑍)       (𝐴 ∈ ℚ → (0 ≤ 𝐹𝐹 < 1 ∧ 𝐴 = (𝑍 + 𝐹)))
 
Theoremintfracq 9934 Decompose a rational number, expressed as a ratio, into integer and fractional parts. The fractional part has a tighter bound than that of intqfrac2 9933. (Contributed by NM, 16-Aug-2008.)
𝑍 = (⌊‘(𝑀 / 𝑁))    &   𝐹 = ((𝑀 / 𝑁) − 𝑍)       ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (0 ≤ 𝐹𝐹 ≤ ((𝑁 − 1) / 𝑁) ∧ (𝑀 / 𝑁) = (𝑍 + 𝐹)))
 
Theoremflqdiv 9935 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)))
 
3.6.2  The modulo (remainder) operation
 
Syntaxcmo 9936 Extend class notation with the modulo operation.
class mod
 
Definitiondf-mod 9937* Define the modulo (remainder) operation. See modqval 9938 for its value. For example, (5 mod 3) = 2 and (-7 mod 2) = 1. As with df-fl 9884 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)))))
 
Theoremmodqval 9938 The value of the modulo operation. The modulo congruence notation of number theory, 𝐽𝐾 (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 9887 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)))))
 
Theoremmodqvalr 9939 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)))
 
Theoremmodqcl 9940 Closure law for the modulo operation. (Contributed by Jim Kingdon, 16-Oct-2021.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → (𝐴 mod 𝐵) ∈ ℚ)
 
Theoremflqpmodeq 9941 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)
 
Theoremmodqcld 9942 Closure law for the modulo operation. (Contributed by Jim Kingdon, 16-Oct-2021.)
(𝜑𝐴 ∈ ℚ)    &   (𝜑𝐵 ∈ ℚ)    &   (𝜑 → 0 < 𝐵)       (𝜑 → (𝐴 mod 𝐵) ∈ ℚ)
 
Theoremmodq0 9943 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))
 
Theoremmulqmod0 9944 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)
 
Theoremnegqmod0 9945 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))
 
Theoremmodqge0 9946 The modulo operation is nonnegative. (Contributed by Jim Kingdon, 18-Oct-2021.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → 0 ≤ (𝐴 mod 𝐵))
 
Theoremmodqlt 9947 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)
 
Theoremmodqelico 9948 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))
 
Theoremmodqdiffl 9949 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)))
 
Theoremmodqdifz 9950 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)
 
Theoremmodqfrac 9951 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)))
 
Theoremflqmod 9952 The floor function expressed in terms of the modulo operation. (Contributed by Jim Kingdon, 18-Oct-2021.)
(𝐴 ∈ ℚ → (⌊‘𝐴) = (𝐴 − (𝐴 mod 1)))
 
Theoremintqfrac 9953 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)))
 
Theoremzmod10 9954 An integer modulo 1 is 0. (Contributed by Paul Chapman, 22-Jun-2011.)
(𝑁 ∈ ℤ → (𝑁 mod 1) = 0)
 
Theoremzmod1congr 9955 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))
 
Theoremmodqmulnn 9956 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)))
 
Theoremmodqvalp1 9957 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))
 
Theoremzmodcl 9958 Closure law for the modulo operation restricted to integers. (Contributed by NM, 27-Nov-2008.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → (𝐴 mod 𝐵) ∈ ℕ0)
 
Theoremzmodcld 9959 Closure law for the modulo operation restricted to integers. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℤ)    &   (𝜑𝐵 ∈ ℕ)       (𝜑 → (𝐴 mod 𝐵) ∈ ℕ0)
 
Theoremzmodfz 9960 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)))
 
Theoremzmodfzo 9961 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))
 
Theoremzmodfzp1 9962 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))
 
Theoremmodqid 9963 Identity law for modulo. (Contributed by Jim Kingdon, 21-Oct-2021.)
(((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ (0 ≤ 𝐴𝐴 < 𝐵)) → (𝐴 mod 𝐵) = 𝐴)
 
Theoremmodqid0 9964 A positive real number modulo itself is 0. (Contributed by Jim Kingdon, 21-Oct-2021.)
((𝑁 ∈ ℚ ∧ 0 < 𝑁) → (𝑁 mod 𝑁) = 0)
 
Theoremmodqid2 9965 Identity law for modulo. (Contributed by Jim Kingdon, 21-Oct-2021.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → ((𝐴 mod 𝐵) = 𝐴 ↔ (0 ≤ 𝐴𝐴 < 𝐵)))
 
Theoremzmodid2 9966 Identity law for modulo restricted to integers. (Contributed by Paul Chapman, 22-Jun-2011.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ) → ((𝑀 mod 𝑁) = 𝑀𝑀 ∈ (0...(𝑁 − 1))))
 
Theoremzmodidfzo 9967 Identity law for modulo restricted to integers. (Contributed by AV, 27-Oct-2018.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ) → ((𝑀 mod 𝑁) = 𝑀𝑀 ∈ (0..^𝑁)))
 
Theoremzmodidfzoimp 9968 Identity law for modulo restricted to integers. (Contributed by AV, 27-Oct-2018.)
(𝑀 ∈ (0..^𝑁) → (𝑀 mod 𝑁) = 𝑀)
 
Theoremq0mod 9969 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)
 
Theoremq1mod 9970 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)
 
Theoremmodqabs 9971 Absorption law for modulo. (Contributed by Jim Kingdon, 21-Oct-2021.)
(𝜑𝐴 ∈ ℚ)    &   (𝜑𝐵 ∈ ℚ)    &   (𝜑 → 0 < 𝐵)    &   (𝜑𝐶 ∈ ℚ)    &   (𝜑𝐵𝐶)       (𝜑 → ((𝐴 mod 𝐵) mod 𝐶) = (𝐴 mod 𝐵))
 
Theoremmodqabs2 9972 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))
 
Theoremmodqcyc 9973 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))
 
Theoremmodqcyc2 9974 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))
 
Theoremmodqadd1 9975 Addition property of the modulo operation. (Contributed by Jim Kingdon, 22-Oct-2021.)
(𝜑𝐴 ∈ ℚ)    &   (𝜑𝐵 ∈ ℚ)    &   (𝜑𝐶 ∈ ℚ)    &   (𝜑𝐷 ∈ ℚ)    &   (𝜑 → 0 < 𝐷)    &   (𝜑 → (𝐴 mod 𝐷) = (𝐵 mod 𝐷))       (𝜑 → ((𝐴 + 𝐶) mod 𝐷) = ((𝐵 + 𝐶) mod 𝐷))
 
Theoremmodqaddabs 9976 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))
 
Theoremmodqaddmod 9977 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))
 
Theoremmulqaddmodid 9978 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)
 
Theoremmulp1mod1 9979 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)
 
Theoremmodqmuladd 9980* Decomposition of an integer into a multiple of a modulus and a remainder. (Contributed by Jim Kingdon, 23-Oct-2021.)
(𝜑𝐴 ∈ ℤ)    &   (𝜑𝐵 ∈ ℚ)    &   (𝜑𝐵 ∈ (0[,)𝑀))    &   (𝜑𝑀 ∈ ℚ)    &   (𝜑 → 0 < 𝑀)       (𝜑 → ((𝐴 mod 𝑀) = 𝐵 ↔ ∃𝑘 ∈ ℤ 𝐴 = ((𝑘 · 𝑀) + 𝐵)))
 
Theoremmodqmuladdim 9981* 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)))
 
Theoremmodqmuladdnn0 9982* 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 ∈ ℕ0𝑀 ∈ ℚ ∧ 0 < 𝑀) → ((𝐴 mod 𝑀) = 𝐵 → ∃𝑘 ∈ ℕ0 𝐴 = ((𝑘 · 𝑀) + 𝐵)))
 
Theoremqnegmod 9983 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) = ((𝑁𝐴) mod 𝑁))
 
Theoremm1modnnsub1 9984 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))
 
Theoremm1modge3gt1 9985 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))
 
Theoremaddmodid 9986 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 ∈ ℕ0𝑀 ∈ ℕ ∧ 𝐴 < 𝑀) → ((𝑀 + 𝐴) mod 𝑀) = 𝐴)
 
Theoremaddmodidr 9987 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 ∈ ℕ0𝑀 ∈ ℕ ∧ 𝐴 < 𝑀) → ((𝐴 + 𝑀) mod 𝑀) = 𝐴)
 
Theoremmodqadd2mod 9988 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))
 
Theoremmodqm1p1mod0 9989 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))
 
Theoremmodqltm1p1mod 9990 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))
 
Theoremmodqmul1 9991 Multiplication property of the modulo operation. Note that the multiplier 𝐶 must be an integer. (Contributed by Jim Kingdon, 24-Oct-2021.)
(𝜑𝐴 ∈ ℚ)    &   (𝜑𝐵 ∈ ℚ)    &   (𝜑𝐶 ∈ ℤ)    &   (𝜑𝐷 ∈ ℚ)    &   (𝜑 → 0 < 𝐷)    &   (𝜑 → (𝐴 mod 𝐷) = (𝐵 mod 𝐷))       (𝜑 → ((𝐴 · 𝐶) mod 𝐷) = ((𝐵 · 𝐶) mod 𝐷))
 
Theoremmodqmul12d 9992 Multiplication property of the modulo operation, see theorem 5.2(b) in [ApostolNT] p. 107. (Contributed by Jim Kingdon, 24-Oct-2021.)
(𝜑𝐴 ∈ ℤ)    &   (𝜑𝐵 ∈ ℤ)    &   (𝜑𝐶 ∈ ℤ)    &   (𝜑𝐷 ∈ ℤ)    &   (𝜑𝐸 ∈ 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))
 
Theoremmodqnegd 9993 Negation property of the modulo operation. (Contributed by Jim Kingdon, 24-Oct-2021.)
(𝜑𝐴 ∈ ℚ)    &   (𝜑𝐵 ∈ ℚ)    &   (𝜑𝐶 ∈ ℚ)    &   (𝜑 → 0 < 𝐶)    &   (𝜑 → (𝐴 mod 𝐶) = (𝐵 mod 𝐶))       (𝜑 → (-𝐴 mod 𝐶) = (-𝐵 mod 𝐶))
 
Theoremmodqadd12d 9994 Additive property of the modulo operation. (Contributed by Jim Kingdon, 25-Oct-2021.)
(𝜑𝐴 ∈ ℚ)    &   (𝜑𝐵 ∈ ℚ)    &   (𝜑𝐶 ∈ ℚ)    &   (𝜑𝐷 ∈ ℚ)    &   (𝜑𝐸 ∈ 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))
 
Theoremmodqsub12d 9995 Subtraction property of the modulo operation. (Contributed by Jim Kingdon, 25-Oct-2021.)
(𝜑𝐴 ∈ ℚ)    &   (𝜑𝐵 ∈ ℚ)    &   (𝜑𝐶 ∈ ℚ)    &   (𝜑𝐷 ∈ ℚ)    &   (𝜑𝐸 ∈ ℚ)    &   (𝜑 → 0 < 𝐸)    &   (𝜑 → (𝐴 mod 𝐸) = (𝐵 mod 𝐸))    &   (𝜑 → (𝐶 mod 𝐸) = (𝐷 mod 𝐸))       (𝜑 → ((𝐴𝐶) mod 𝐸) = ((𝐵𝐷) mod 𝐸))
 
Theoremmodqsubmod 9996 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) = ((𝐴𝐵) mod 𝑀))
 
Theoremmodqsubmodmod 9997 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) = ((𝐴𝐵) mod 𝑀))
 
Theoremq2txmodxeq0 9998 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)
 
Theoremq2submod 9999 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) ∧ (𝐵𝐴𝐴 < (2 · 𝐵))) → (𝐴 mod 𝐵) = (𝐴𝐵))
 
Theoremmodifeq2int 10000 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 ∈ ℕ0𝐵 ∈ ℕ ∧ 𝐴 < (2 · 𝐵)) → (𝐴 mod 𝐵) = if(𝐴 < 𝐵, 𝐴, (𝐴𝐵)))
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