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Theorem List for Metamath Proof Explorer - 12601-12700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremflbi2 12601 A condition equivalent to floor. (Contributed by NM, 15-Aug-2008.)
((𝑁 ∈ ℤ ∧ 𝐹 ∈ ℝ) → ((⌊‘(𝑁 + 𝐹)) = 𝑁 ↔ (0 ≤ 𝐹𝐹 < 1)))

Theoremadddivflid 12602 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))

Theoremico01fl0 12603 The floor of a real number in [0, 1) is 0. Remark: may shorten the proof of modid 12678 or a version of it where the antecedent is membership in an interval. (Contributed by BJ, 29-Jun-2019.)
(𝐴 ∈ (0[,)1) → (⌊‘𝐴) = 0)

Theoremflge0nn0 12604 The floor of a number greater than or equal to 0 is a nonnegative integer. (Contributed by NM, 26-Apr-2005.)
((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (⌊‘𝐴) ∈ ℕ0)

Theoremflge1nn 12605 The floor of a number greater than or equal to 1 is a positive integer. (Contributed by NM, 26-Apr-2005.)
((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (⌊‘𝐴) ∈ ℕ)

Theoremfldivnn0 12606 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)

Theoremrefldivcl 12607 The floor function of a division of a real number by a positive real number is a real number. (Contributed by Alexander van der Vekens, 14-Apr-2018.)
((𝐾 ∈ ℝ ∧ 𝐿 ∈ ℝ+) → (⌊‘(𝐾 / 𝐿)) ∈ ℝ)

Theoremdivfl0 12608 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))

Theoremfladdz 12609 An integer can be moved in and out of the floor of a sum. (Contributed by NM, 27-Apr-2005.) (Proof shortened by Fan Zheng, 16-Jun-2016.)
((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) → (⌊‘(𝐴 + 𝑁)) = ((⌊‘𝐴) + 𝑁))

Theoremflzadd 12610 An integer can be moved in and out of the floor of a sum. (Contributed by NM, 2-Jan-2009.)
((𝑁 ∈ ℤ ∧ 𝐴 ∈ ℝ) → (⌊‘(𝑁 + 𝐴)) = (𝑁 + (⌊‘𝐴)))

Theoremflmulnn0 12611 Move a nonnegative integer in and out of a floor. (Contributed by NM, 2-Jan-2009.) (Proof shortened by Fan Zheng, 7-Jun-2016.)
((𝑁 ∈ ℕ0𝐴 ∈ ℝ) → (𝑁 · (⌊‘𝐴)) ≤ (⌊‘(𝑁 · 𝐴)))

Theorembtwnzge0 12612 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. (For the first half see rebtwnz 11772.) (Contributed by NM, 12-Mar-2005.)
(((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) ∧ (𝑁𝐴𝐴 < (𝑁 + 1))) → (0 ≤ 𝐴 ↔ 0 ≤ 𝑁))

Theorem2tnp1ge0ge0 12613 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 12614 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))))

Theoremfldivle 12615 The floor function of a division of a real number by a positive real number is less than or equal to the division. (Contributed by Alexander van der Vekens, 14-Apr-2018.)
((𝐾 ∈ ℝ ∧ 𝐿 ∈ ℝ+) → (⌊‘(𝐾 / 𝐿)) ≤ (𝐾 / 𝐿))

Theoremfldivnn0le 12616 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 12617 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)))

Theoremltdifltdiv 12618 If the dividend of a division is less than the difference between a real number and the divisor, the floor function of the division plus 1 is less than the division of the real number by the divisor. (Contributed by Alexander van der Vekens, 14-Apr-2018.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+𝐶 ∈ ℝ) → (𝐴 < (𝐶𝐵) → ((⌊‘(𝐴 / 𝐵)) + 1) < (𝐶 / 𝐵)))

Theoremfldiv4p1lem1div2 12619 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))

Theoremfldiv4lem1div2uz2 12620 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.)
(𝑁 ∈ (ℤ‘2) → (⌊‘(𝑁 / 4)) ≤ ((𝑁 − 1) / 2))

Theoremfldiv4lem1div2 12621 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))

Theoremceilval 12622 The value of the ceiling function. (Contributed by David A. Wheeler, 19-May-2015.)
(𝐴 ∈ ℝ → (⌈‘𝐴) = -(⌊‘-𝐴))

Theoremdfceil2 12623* Alternative definition of the ceiling function using restricted iota. (Contributed by AV, 1-Dec-2018.)
⌈ = (𝑥 ∈ ℝ ↦ (𝑦 ∈ ℤ (𝑥𝑦𝑦 < (𝑥 + 1))))

Theoremceilval2 12624* The value of the ceiling function using restricted iota. (Contributed by AV, 1-Dec-2018.)
(𝐴 ∈ ℝ → (⌈‘𝐴) = (𝑦 ∈ ℤ (𝐴𝑦𝑦 < (𝐴 + 1))))

Theoremceicl 12625 The ceiling function returns an integer (closure law). (Contributed by Jeff Hankins, 10-Jun-2007.)
(𝐴 ∈ ℝ → -(⌊‘-𝐴) ∈ ℤ)

Theoremceilcl 12626 Closure of the ceiling function. (Contributed by David A. Wheeler, 19-May-2015.)
(𝐴 ∈ ℝ → (⌈‘𝐴) ∈ ℤ)

Theoremceige 12627 The ceiling of a real number is greater than or equal to that number. (Contributed by Jeff Hankins, 10-Jun-2007.)
(𝐴 ∈ ℝ → 𝐴 ≤ -(⌊‘-𝐴))

Theoremceilge 12628 The ceiling of a real number is greater than or equal to that number. (Contributed by AV, 30-Nov-2018.)
(𝐴 ∈ ℝ → 𝐴 ≤ (⌈‘𝐴))

Theoremceim1l 12629 One less than the ceiling of a real number is strictly less than that number. (Contributed by Jeff Hankins, 10-Jun-2007.)
(𝐴 ∈ ℝ → (-(⌊‘-𝐴) − 1) < 𝐴)

Theoremceilm1lt 12630 One less than the ceiling of a real number is strictly less than that number. (Contributed by AV, 30-Nov-2018.)
(𝐴 ∈ ℝ → ((⌈‘𝐴) − 1) < 𝐴)

Theoremceile 12631 The ceiling of a real number is the smallest integer greater than or equal to it. (Contributed by Jeff Hankins, 10-Jun-2007.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℤ ∧ 𝐴𝐵) → -(⌊‘-𝐴) ≤ 𝐵)

Theoremceille 12632 The ceiling of a real number is the smallest integer greater than or equal to it. (Contributed by AV, 30-Nov-2018.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℤ ∧ 𝐴𝐵) → (⌈‘𝐴) ≤ 𝐵)

Theoremceilid 12633 An integer is its own ceiling. (Contributed by AV, 30-Nov-2018.)
(𝐴 ∈ ℤ → (⌈‘𝐴) = 𝐴)

Theoremceilidz 12634 A real number equals its ceiling iff it is an integer. (Contributed by AV, 30-Nov-2018.)
(𝐴 ∈ ℝ → (𝐴 ∈ ℤ ↔ (⌈‘𝐴) = 𝐴))

Theoremflleceil 12635 The floor of a real number is less than or equal to its ceiling. (Contributed by AV, 30-Nov-2018.)
(𝐴 ∈ ℝ → (⌊‘𝐴) ≤ (⌈‘𝐴))

Theoremfleqceilz 12636 A real number is an integer iff its floor equals its ceiling. (Contributed by AV, 30-Nov-2018.)
(𝐴 ∈ ℝ → (𝐴 ∈ ℤ ↔ (⌊‘𝐴) = (⌈‘𝐴)))

Theoremquoremz 12637 Quotient and remainder of an integer divided by a positive integer. TODO - is this really needed for anything? Should we use mod to simplify it? Remark (AV): This is a special case of divalg 15107. (Contributed by NM, 14-Aug-2008.)
𝑄 = (⌊‘(𝐴 / 𝐵))    &   𝑅 = (𝐴 − (𝐵 · 𝑄))       ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → ((𝑄 ∈ ℤ ∧ 𝑅 ∈ ℕ0) ∧ (𝑅 < 𝐵𝐴 = ((𝐵 · 𝑄) + 𝑅))))

Theoremquoremnn0 12638 Quotient and remainder of a nonnegative integer divided by a positive integer. (Contributed by NM, 14-Aug-2008.)
𝑄 = (⌊‘(𝐴 / 𝐵))    &   𝑅 = (𝐴 − (𝐵 · 𝑄))       ((𝐴 ∈ ℕ0𝐵 ∈ ℕ) → ((𝑄 ∈ ℕ0𝑅 ∈ ℕ0) ∧ (𝑅 < 𝐵𝐴 = ((𝐵 · 𝑄) + 𝑅))))

Theoremquoremnn0ALT 12639 Alternate proof of quoremnn0 12638 not using quoremz 12637. TODO - Keep either quoremnn0ALT 12639 (if we don't keep quoremz 12637) or quoremnn0 12638. (Contributed by NM, 14-Aug-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑄 = (⌊‘(𝐴 / 𝐵))    &   𝑅 = (𝐴 − (𝐵 · 𝑄))       ((𝐴 ∈ ℕ0𝐵 ∈ ℕ) → ((𝑄 ∈ ℕ0𝑅 ∈ ℕ0) ∧ (𝑅 < 𝐵𝐴 = ((𝐵 · 𝑄) + 𝑅))))

Theoremintfrac2 12640 Decompose a real into integer and fractional parts. TODO - should we replace this with intfrac 12668? (Contributed by NM, 16-Aug-2008.)
𝑍 = (⌊‘𝐴)    &   𝐹 = (𝐴𝑍)       (𝐴 ∈ ℝ → (0 ≤ 𝐹𝐹 < 1 ∧ 𝐴 = (𝑍 + 𝐹)))

Theoremintfracq 12641 Decompose a rational number, expressed as a ratio, into integer and fractional parts. The fractional part has a tighter bound than that of intfrac2 12640. (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)))

Theoremfldiv 12642 Cancellation of the embedded floor of a real divided by an integer. (Contributed by NM, 16-Aug-2008.)
((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ) → (⌊‘((⌊‘𝐴) / 𝑁)) = (⌊‘(𝐴 / 𝑁)))

Theoremfldiv2 12643 Cancellation of an embedded floor of a ratio. Generalization of Equation 2.4 in [CormenLeisersonRivest] p. 33 (where 𝐴 must be an integer). (Contributed by NM, 9-Nov-2008.)
((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (⌊‘((⌊‘(𝐴 / 𝑀)) / 𝑁)) = (⌊‘(𝐴 / (𝑀 · 𝑁))))

Theoremfznnfl 12644 Finite set of sequential integers starting at 1 and ending at a real number. (Contributed by Mario Carneiro, 3-May-2016.)
(𝑁 ∈ ℝ → (𝐾 ∈ (1...(⌊‘𝑁)) ↔ (𝐾 ∈ ℕ ∧ 𝐾𝑁)))

Theoremuzsup 12645 An upper set of integers is unbounded above. (Contributed by Mario Carneiro, 7-May-2016.)
𝑍 = (ℤ𝑀)       (𝑀 ∈ ℤ → sup(𝑍, ℝ*, < ) = +∞)

Theoremioopnfsup 12646 An upper set of reals is unbounded above. (Contributed by Mario Carneiro, 7-May-2016.)
((𝐴 ∈ ℝ*𝐴 ≠ +∞) → sup((𝐴(,)+∞), ℝ*, < ) = +∞)

Theoremicopnfsup 12647 An upper set of reals is unbounded above. (Contributed by Mario Carneiro, 7-May-2016.)
((𝐴 ∈ ℝ*𝐴 ≠ +∞) → sup((𝐴[,)+∞), ℝ*, < ) = +∞)

Theoremrpsup 12648 The positive reals are unbounded above. (Contributed by Mario Carneiro, 7-May-2016.)
sup(ℝ+, ℝ*, < ) = +∞

Theoremresup 12649 The real numbers are unbounded above. (Contributed by Mario Carneiro, 7-May-2016.)
sup(ℝ, ℝ*, < ) = +∞

Theoremxrsup 12650 The extended real numbers are unbounded above. (Contributed by Mario Carneiro, 7-May-2016.)
sup(ℝ*, ℝ*, < ) = +∞

5.6.2  The modulo (remainder) operation

Syntaxcmo 12651 Extend class notation with the modulo operation.
class mod

Definitiondf-mod 12652* Define the modulo (remainder) operation. See modval 12653 for its value. For example, (5 mod 3) = 2 and (-7 mod 2) = 1 (ex-mod 27276). (Contributed by NM, 10-Nov-2008.)
mod = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ+ ↦ (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦)))))

Theoremmodval 12653 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 reals 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.) (Contributed by NM, 10-Nov-2008.) (Revised by Mario Carneiro, 3-Nov-2013.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (𝐴 mod 𝐵) = (𝐴 − (𝐵 · (⌊‘(𝐴 / 𝐵)))))

Theoremmodvalr 12654 The value of the modulo operation (multiplication in reversed order). (Contributed by Alexander van der Vekens, 14-Apr-2018.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (𝐴 mod 𝐵) = (𝐴 − ((⌊‘(𝐴 / 𝐵)) · 𝐵)))

Theoremmodcl 12655 Closure law for the modulo operation. (Contributed by NM, 10-Nov-2008.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (𝐴 mod 𝐵) ∈ ℝ)

Theoremflpmodeq 12656 Partition of a division into its integer part and the remainder. (Contributed by Alexander van der Vekens, 14-Apr-2018.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (((⌊‘(𝐴 / 𝐵)) · 𝐵) + (𝐴 mod 𝐵)) = 𝐴)

Theoremmodcld 12657 Closure law for the modulo operation. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ+)       (𝜑 → (𝐴 mod 𝐵) ∈ ℝ)

Theoremmod0 12658 𝐴 mod 𝐵 is zero iff 𝐴 is evenly divisible by 𝐵. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Fan Zheng, 7-Jun-2016.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → ((𝐴 mod 𝐵) = 0 ↔ (𝐴 / 𝐵) ∈ ℤ))

Theoremmulmod0 12659 The product of an integer and a positive real number is 0 modulo the positive real number. (Contributed by Alexander van der Vekens, 17-May-2018.) (Revised by AV, 5-Jul-2020.)
((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℝ+) → ((𝐴 · 𝑀) mod 𝑀) = 0)

Theoremnegmod0 12660 𝐴 is divisible by 𝐵 iff its negative is. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Fan Zheng, 7-Jun-2016.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → ((𝐴 mod 𝐵) = 0 ↔ (-𝐴 mod 𝐵) = 0))

Theoremmodge0 12661 The modulo operation is nonnegative. (Contributed by NM, 10-Nov-2008.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → 0 ≤ (𝐴 mod 𝐵))

Theoremmodlt 12662 The modulo operation is less than its second argument. (Contributed by NM, 10-Nov-2008.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (𝐴 mod 𝐵) < 𝐵)

Theoremmodelico 12663 Modular reduction produces a half-open interval. (Contributed by Stefan O'Rear, 12-Sep-2014.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (𝐴 mod 𝐵) ∈ (0[,)𝐵))

Theoremmoddiffl 12664 The modulo operation differs from 𝐴 by an integer multiple of 𝐵. (Contributed by Mario Carneiro, 6-Sep-2016.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → ((𝐴 − (𝐴 mod 𝐵)) / 𝐵) = (⌊‘(𝐴 / 𝐵)))

Theoremmoddifz 12665 The modulo operation differs from 𝐴 by an integer multiple of 𝐵. (Contributed by Mario Carneiro, 15-Jul-2014.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → ((𝐴 − (𝐴 mod 𝐵)) / 𝐵) ∈ ℤ)

Theoremmodfrac 12666 The fractional part of a number is the number modulo 1. (Contributed by NM, 11-Nov-2008.)
(𝐴 ∈ ℝ → (𝐴 mod 1) = (𝐴 − (⌊‘𝐴)))

Theoremflmod 12667 The floor function expressed in terms of the modulo operation. (Contributed by NM, 11-Nov-2008.)
(𝐴 ∈ ℝ → (⌊‘𝐴) = (𝐴 − (𝐴 mod 1)))

Theoremintfrac 12668 Break a number into its integer part and its fractional part. (Contributed by NM, 31-Dec-2008.)
(𝐴 ∈ ℝ → 𝐴 = ((⌊‘𝐴) + (𝐴 mod 1)))

Theoremzmod10 12669 An integer modulo 1 is 0. (Contributed by Paul Chapman, 22-Jun-2011.)
(𝑁 ∈ ℤ → (𝑁 mod 1) = 0)

Theoremzmod1congr 12670 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))

Theoremmodmulnn 12671 Move a positive integer in and out of a floor in the first argument of a modulo operation. (Contributed by NM, 2-Jan-2009.)
((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ) → ((𝑁 · (⌊‘𝐴)) mod (𝑁 · 𝑀)) ≤ ((⌊‘(𝑁 · 𝐴)) mod (𝑁 · 𝑀)))

Theoremmodvalp1 12672 The value of the modulo operation (expressed with sum of denominator and nominator). (Contributed by Alexander van der Vekens, 14-Apr-2018.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → ((𝐴 + 𝐵) − (((⌊‘(𝐴 / 𝐵)) + 1) · 𝐵)) = (𝐴 mod 𝐵))

Theoremzmodcl 12673 Closure law for the modulo operation restricted to integers. (Contributed by NM, 27-Nov-2008.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → (𝐴 mod 𝐵) ∈ ℕ0)

Theoremzmodcld 12674 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 12675 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 12676 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 12677 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))

Theoremmodid 12678 Identity law for modulo. (Contributed by NM, 29-Dec-2008.)
(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) ∧ (0 ≤ 𝐴𝐴 < 𝐵)) → (𝐴 mod 𝐵) = 𝐴)

Theoremmodid0 12679 A positive real number modulo itself is 0. (Contributed by Alexander van der Vekens, 15-May-2018.)
(𝑁 ∈ ℝ+ → (𝑁 mod 𝑁) = 0)

Theoremmodid2 12680 Identity law for modulo. (Contributed by NM, 29-Dec-2008.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → ((𝐴 mod 𝐵) = 𝐴 ↔ (0 ≤ 𝐴𝐴 < 𝐵)))

Theoremzmodid2 12681 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 12682 Identity law for modulo restricted to integers. (Contributed by AV, 27-Oct-2018.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ) → ((𝑀 mod 𝑁) = 𝑀𝑀 ∈ (0..^𝑁)))

Theoremzmodidfzoimp 12683 Identity law for modulo restricted to integers. (Contributed by AV, 27-Oct-2018.)
(𝑀 ∈ (0..^𝑁) → (𝑀 mod 𝑁) = 𝑀)

Theorem0mod 12684 Special case: 0 modulo a positive real number is 0. (Contributed by Mario Carneiro, 22-Feb-2014.)
(𝑁 ∈ ℝ+ → (0 mod 𝑁) = 0)

Theorem1mod 12685 Special case: 1 modulo a real number greater than 1 is 1. (Contributed by Mario Carneiro, 18-Feb-2014.)
((𝑁 ∈ ℝ ∧ 1 < 𝑁) → (1 mod 𝑁) = 1)

Theoremmodabs 12686 Absorption law for modulo. (Contributed by NM, 29-Dec-2008.)
(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+𝐶 ∈ ℝ+) ∧ 𝐵𝐶) → ((𝐴 mod 𝐵) mod 𝐶) = (𝐴 mod 𝐵))

Theoremmodabs2 12687 Absorption law for modulo. (Contributed by NM, 29-Dec-2008.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → ((𝐴 mod 𝐵) mod 𝐵) = (𝐴 mod 𝐵))

Theoremmodcyc 12688 The modulo operation is periodic. (Contributed by NM, 10-Nov-2008.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+𝑁 ∈ ℤ) → ((𝐴 + (𝑁 · 𝐵)) mod 𝐵) = (𝐴 mod 𝐵))

Theoremmodcyc2 12689 The modulo operation is periodic. (Contributed by NM, 12-Nov-2008.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+𝑁 ∈ ℤ) → ((𝐴 − (𝐵 · 𝑁)) mod 𝐵) = (𝐴 mod 𝐵))

Theoremmodadd1 12690 Addition property of the modulo operation. (Contributed by NM, 12-Nov-2008.)
(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ+) ∧ (𝐴 mod 𝐷) = (𝐵 mod 𝐷)) → ((𝐴 + 𝐶) mod 𝐷) = ((𝐵 + 𝐶) mod 𝐷))

Theoremmodaddabs 12691 Absorption law for modulo. (Contributed by Paul Chapman, 22-Jun-2011.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ+) → (((𝐴 mod 𝐶) + (𝐵 mod 𝐶)) mod 𝐶) = ((𝐴 + 𝐵) mod 𝐶))

Theoremmodaddmod 12692 The sum of a real number modulo a positive real number and another real number equals the sum of the two real numbers modulo the positive real number. (Contributed by Alexander van der Vekens, 13-May-2018.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑀 ∈ ℝ+) → (((𝐴 mod 𝑀) + 𝐵) mod 𝑀) = ((𝐴 + 𝐵) mod 𝑀))

Theoremmuladdmodid 12693 The sum of a positive real number less than an upper bound and the product of an integer and the upper bound is the positive real number modulo the upper bound. (Contributed by AV, 5-Jul-2020.)
((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℝ+𝐴 ∈ (0[,)𝑀)) → (((𝑁 · 𝑀) + 𝐴) mod 𝑀) = 𝐴)

Theoremmulp1mod1 12694 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)

Theoremmodmuladd 12695* Decomposition of an integer into a multiple of a modulus and a remainder. (Contributed by AV, 14-Jul-2021.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ (0[,)𝑀) ∧ 𝑀 ∈ ℝ+) → ((𝐴 mod 𝑀) = 𝐵 ↔ ∃𝑘 ∈ ℤ 𝐴 = ((𝑘 · 𝑀) + 𝐵)))

Theoremmodmuladdim 12696* Implication of a decomposition of an integer into a multiple of a modulus and a remainder. (Contributed by AV, 14-Jul-2021.)
((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℝ+) → ((𝐴 mod 𝑀) = 𝐵 → ∃𝑘 ∈ ℤ 𝐴 = ((𝑘 · 𝑀) + 𝐵)))

Theoremmodmuladdnn0 12697* Implication of a decomposition of a nonnegative integer into a multiple of a modulus and a remainder. (Contributed by AV, 14-Jul-2021.)
((𝐴 ∈ ℕ0𝑀 ∈ ℝ+) → ((𝐴 mod 𝑀) = 𝐵 → ∃𝑘 ∈ ℕ0 𝐴 = ((𝑘 · 𝑀) + 𝐵)))

Theoremnegmod 12698 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 AV, 5-Jul-2020.)
((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℝ+) → (-𝐴 mod 𝑁) = ((𝑁𝐴) mod 𝑁))

Theoremm1modnnsub1 12699 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 12700 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))

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