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Theorem List for Intuitionistic Logic Explorer - 10001-10100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremqbtwnrelemcalc 10001 Lemma for qbtwnre 10002. Calculations involved in showing the constructed rational number is less than 𝐵. (Contributed by Jim Kingdon, 14-Oct-2021.)
(𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝑀 < (𝐴 · (2 · 𝑁)))    &   (𝜑 → (1 / 𝑁) < (𝐵𝐴))       (𝜑 → ((𝑀 + 2) / (2 · 𝑁)) < 𝐵)
 
Theoremqbtwnre 10002* The rational numbers are dense in : any two real numbers have a rational between them. Exercise 6 of [Apostol] p. 28. (Contributed by NM, 18-Nov-2004.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → ∃𝑥 ∈ ℚ (𝐴 < 𝑥𝑥 < 𝐵))
 
Theoremqbtwnxr 10003* The rational numbers are dense in *: any two extended real numbers have a rational between them. (Contributed by NM, 6-Feb-2007.) (Proof shortened by Mario Carneiro, 23-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴 < 𝐵) → ∃𝑥 ∈ ℚ (𝐴 < 𝑥𝑥 < 𝐵))
 
Theoremqavgle 10004 The average of two rational numbers is less than or equal to at least one of them. (Contributed by Jim Kingdon, 3-Nov-2021.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (((𝐴 + 𝐵) / 2) ≤ 𝐴 ∨ ((𝐴 + 𝐵) / 2) ≤ 𝐵))
 
Theoremioo0 10005 An empty open interval of extended reals. (Contributed by NM, 6-Feb-2007.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((𝐴(,)𝐵) = ∅ ↔ 𝐵𝐴))
 
Theoremioom 10006* An open interval of extended reals is inhabited iff the lower argument is less than the upper argument. (Contributed by Jim Kingdon, 27-Nov-2021.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (∃𝑥 𝑥 ∈ (𝐴(,)𝐵) ↔ 𝐴 < 𝐵))
 
Theoremico0 10007 An empty open interval of extended reals. (Contributed by FL, 30-May-2014.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((𝐴[,)𝐵) = ∅ ↔ 𝐵𝐴))
 
Theoremioc0 10008 An empty open interval of extended reals. (Contributed by FL, 30-May-2014.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((𝐴(,]𝐵) = ∅ ↔ 𝐵𝐴))
 
4.6  Elementary integer functions
 
4.6.1  The floor and ceiling functions
 
Syntaxcfl 10009 Extend class notation with floor (greatest integer) function.
class
 
Syntaxcceil 10010 Extend class notation to include the ceiling function.
class
 
Definitiondf-fl 10011* Define the floor (greatest integer less than or equal to) function. See flval 10013 for its value, flqlelt 10017 for its basic property, and flqcl 10014 for its closure. For example, (⌊‘(3 / 2)) = 1 while (⌊‘-(3 / 2)) = -2 (ex-fl 12864).

Although we define this on real numbers so that notations are similar to the Metamath Proof Explorer, in the absence of excluded middle few theorems will be possible for all real numbers. Imagine a real number which is around 2.99995 or 3.00001 . In order to determine whether its floor is 2 or 3, it would be necessary to compute the number to arbitrary precision.

The term "floor" was coined by Ken Iverson. He also invented a mathematical notation for floor, consisting of an L-shaped left bracket and its reflection as a right bracket. In APL, the left-bracket alone is used, and we borrow this idea. (Thanks to Paul Chapman for this information.) (Contributed by NM, 14-Nov-2004.)

⌊ = (𝑥 ∈ ℝ ↦ (𝑦 ∈ ℤ (𝑦𝑥𝑥 < (𝑦 + 1))))
 
Definitiondf-ceil 10012 The ceiling (least integer greater than or equal to) function. Defined in ISO 80000-2:2009(E) operation 2-9.18 and the "NIST Digital Library of Mathematical Functions" , front introduction, "Common Notations and Definitions" section at http://dlmf.nist.gov/front/introduction#Sx4. See ceilqval 10047 for its value, ceilqge 10051 and ceilqm1lt 10053 for its basic properties, and ceilqcl 10049 for its closure. For example, (⌈‘(3 / 2)) = 2 while (⌈‘-(3 / 2)) = -1 (ex-ceil 12865).

As described in df-fl 10011 most theorems are only for rationals, not reals.

The symbol is inspired by the gamma shaped left bracket of the usual notation. (Contributed by David A. Wheeler, 19-May-2015.)

⌈ = (𝑥 ∈ ℝ ↦ -(⌊‘-𝑥))
 
Theoremflval 10013* Value of the floor (greatest integer) function. The floor of 𝐴 is the (unique) integer less than or equal to 𝐴 whose successor is strictly greater than 𝐴. (Contributed by NM, 14-Nov-2004.) (Revised by Mario Carneiro, 2-Nov-2013.)
(𝐴 ∈ ℝ → (⌊‘𝐴) = (𝑥 ∈ ℤ (𝑥𝐴𝐴 < (𝑥 + 1))))
 
Theoremflqcl 10014 The floor (greatest integer) function yields an integer when applied to a rational (closure law). For a similar closure law for real numbers apart from any integer, see flapcl 10016. (Contributed by Jim Kingdon, 8-Oct-2021.)
(𝐴 ∈ ℚ → (⌊‘𝐴) ∈ ℤ)
 
Theoremapbtwnz 10015* There is a unique greatest integer less than or equal to a real number which is apart from all integers. (Contributed by Jim Kingdon, 11-May-2022.)
((𝐴 ∈ ℝ ∧ ∀𝑛 ∈ ℤ 𝐴 # 𝑛) → ∃!𝑥 ∈ ℤ (𝑥𝐴𝐴 < (𝑥 + 1)))
 
Theoremflapcl 10016* The floor (greatest integer) function yields an integer when applied to a real number apart from any integer. For example, an irrational number (see for example sqrt2irrap 11785) would satisfy this condition. (Contributed by Jim Kingdon, 11-May-2022.)
((𝐴 ∈ ℝ ∧ ∀𝑛 ∈ ℤ 𝐴 # 𝑛) → (⌊‘𝐴) ∈ ℤ)
 
Theoremflqlelt 10017 A basic property of the floor (greatest integer) function. (Contributed by Jim Kingdon, 8-Oct-2021.)
(𝐴 ∈ ℚ → ((⌊‘𝐴) ≤ 𝐴𝐴 < ((⌊‘𝐴) + 1)))
 
Theoremflqcld 10018 The floor (greatest integer) function is an integer (closure law). (Contributed by Jim Kingdon, 8-Oct-2021.)
(𝜑𝐴 ∈ ℚ)       (𝜑 → (⌊‘𝐴) ∈ ℤ)
 
Theoremflqle 10019 A basic property of the floor (greatest integer) function. (Contributed by Jim Kingdon, 8-Oct-2021.)
(𝐴 ∈ ℚ → (⌊‘𝐴) ≤ 𝐴)
 
Theoremflqltp1 10020 A basic property of the floor (greatest integer) function. (Contributed by Jim Kingdon, 8-Oct-2021.)
(𝐴 ∈ ℚ → 𝐴 < ((⌊‘𝐴) + 1))
 
Theoremqfraclt1 10021 The fractional part of a rational number is less than one. (Contributed by Jim Kingdon, 8-Oct-2021.)
(𝐴 ∈ ℚ → (𝐴 − (⌊‘𝐴)) < 1)
 
Theoremqfracge0 10022 The fractional part of a rational number is nonnegative. (Contributed by Jim Kingdon, 8-Oct-2021.)
(𝐴 ∈ ℚ → 0 ≤ (𝐴 − (⌊‘𝐴)))
 
Theoremflqge 10023 The floor function value is the greatest integer less than or equal to its argument. (Contributed by Jim Kingdon, 8-Oct-2021.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ) → (𝐵𝐴𝐵 ≤ (⌊‘𝐴)))
 
Theoremflqlt 10024 The floor function value is less than the next integer. (Contributed by Jim Kingdon, 8-Oct-2021.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ) → (𝐴 < 𝐵 ↔ (⌊‘𝐴) < 𝐵))
 
Theoremflid 10025 An integer is its own floor. (Contributed by NM, 15-Nov-2004.)
(𝐴 ∈ ℤ → (⌊‘𝐴) = 𝐴)
 
Theoremflqidm 10026 The floor function is idempotent. (Contributed by Jim Kingdon, 8-Oct-2021.)
(𝐴 ∈ ℚ → (⌊‘(⌊‘𝐴)) = (⌊‘𝐴))
 
Theoremflqidz 10027 A rational number equals its floor iff it is an integer. (Contributed by Jim Kingdon, 9-Oct-2021.)
(𝐴 ∈ ℚ → ((⌊‘𝐴) = 𝐴𝐴 ∈ ℤ))
 
Theoremflqltnz 10028 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 10029 Ordering relationship for the greatest integer function. (Contributed by Jim Kingdon, 9-Oct-2021.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 𝐴𝐵) → (⌊‘𝐴) ≤ (⌊‘𝐵))
 
Theoremflqword2 10030 Ordering relationship for the greatest integer function. (Contributed by Jim Kingdon, 9-Oct-2021.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 𝐴𝐵) → (⌊‘𝐵) ∈ (ℤ‘(⌊‘𝐴)))
 
Theoremflqbi 10031 A condition equivalent to floor. (Contributed by Jim Kingdon, 9-Oct-2021.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ) → ((⌊‘𝐴) = 𝐵 ↔ (𝐵𝐴𝐴 < (𝐵 + 1))))
 
Theoremflqbi2 10032 A condition equivalent to floor. (Contributed by Jim Kingdon, 9-Oct-2021.)
((𝑁 ∈ ℤ ∧ 𝐹 ∈ ℚ) → ((⌊‘(𝑁 + 𝐹)) = 𝑁 ↔ (0 ≤ 𝐹𝐹 < 1)))
 
Theoremadddivflid 10033 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 10034 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 10035 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 10036 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 10037 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 10038 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 10039 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 10040 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 10041 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 10042 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 10043 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 10044 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 10045 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 10046 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 10047 The value of the ceiling function. (Contributed by Jim Kingdon, 10-Oct-2021.)
(𝐴 ∈ ℚ → (⌈‘𝐴) = -(⌊‘-𝐴))
 
Theoremceiqcl 10048 The ceiling function returns an integer (closure law). (Contributed by Jim Kingdon, 11-Oct-2021.)
(𝐴 ∈ ℚ → -(⌊‘-𝐴) ∈ ℤ)
 
Theoremceilqcl 10049 Closure of the ceiling function. (Contributed by Jim Kingdon, 11-Oct-2021.)
(𝐴 ∈ ℚ → (⌈‘𝐴) ∈ ℤ)
 
Theoremceiqge 10050 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 10051 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 10052 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 10053 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 10054 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 10055 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 10056 An integer is its own ceiling. (Contributed by AV, 30-Nov-2018.)
(𝐴 ∈ ℤ → (⌈‘𝐴) = 𝐴)
 
Theoremceilqidz 10057 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 10058 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 10059 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 10060 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 10061 Decompose a rational number, expressed as a ratio, into integer and fractional parts. The fractional part has a tighter bound than that of intqfrac2 10060. (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 10062 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
 
Syntaxcmo 10063 Extend class notation with the modulo operation.
class mod
 
Definitiondf-mod 10064* Define the modulo (remainder) operation. See modqval 10065 for its value. For example, (5 mod 3) = 2 and (-7 mod 2) = 1. As with df-fl 10011 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 10065 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 10014 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 10066 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 10067 Closure law for the modulo operation. (Contributed by Jim Kingdon, 16-Oct-2021.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → (𝐴 mod 𝐵) ∈ ℚ)
 
Theoremflqpmodeq 10068 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 10069 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 10070 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 10071 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 10072 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 10073 The modulo operation is nonnegative. (Contributed by Jim Kingdon, 18-Oct-2021.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → 0 ≤ (𝐴 mod 𝐵))
 
Theoremmodqlt 10074 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 10075 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 10076 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 10077 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 10078 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 10079 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 10080 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 10081 An integer modulo 1 is 0. (Contributed by Paul Chapman, 22-Jun-2011.)
(𝑁 ∈ ℤ → (𝑁 mod 1) = 0)
 
Theoremzmod1congr 10082 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 10083 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 10084 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 10085 Closure law for the modulo operation restricted to integers. (Contributed by NM, 27-Nov-2008.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → (𝐴 mod 𝐵) ∈ ℕ0)
 
Theoremzmodcld 10086 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 10087 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 10088 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 10089 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 10090 Identity law for modulo. (Contributed by Jim Kingdon, 21-Oct-2021.)
(((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ (0 ≤ 𝐴𝐴 < 𝐵)) → (𝐴 mod 𝐵) = 𝐴)
 
Theoremmodqid0 10091 A positive real number modulo itself is 0. (Contributed by Jim Kingdon, 21-Oct-2021.)
((𝑁 ∈ ℚ ∧ 0 < 𝑁) → (𝑁 mod 𝑁) = 0)
 
Theoremmodqid2 10092 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 10093 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 10094 Identity law for modulo restricted to integers. (Contributed by AV, 27-Oct-2018.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ) → ((𝑀 mod 𝑁) = 𝑀𝑀 ∈ (0..^𝑁)))
 
Theoremzmodidfzoimp 10095 Identity law for modulo restricted to integers. (Contributed by AV, 27-Oct-2018.)
(𝑀 ∈ (0..^𝑁) → (𝑀 mod 𝑁) = 𝑀)
 
Theoremq0mod 10096 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 10097 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 10098 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 10099 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 10100 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))
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