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Theorem List for Metamath Proof Explorer - 12501-12600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremceilval 12501 The value of the ceiling function. (Contributed by David A. Wheeler, 19-May-2015.)
(𝐴 ∈ ℝ → (⌈‘𝐴) = -(⌊‘-𝐴))

Theoremdfceil2 12502* Alternative definition of the ceiling function using restricted iota. (Contributed by AV, 1-Dec-2018.)
⌈ = (𝑥 ∈ ℝ ↦ (𝑦 ∈ ℤ (𝑥𝑦𝑦 < (𝑥 + 1))))

Theoremceilval2 12503* The value of the ceiling function using restricted iota. (Contributed by AV, 1-Dec-2018.)
(𝐴 ∈ ℝ → (⌈‘𝐴) = (𝑦 ∈ ℤ (𝐴𝑦𝑦 < (𝐴 + 1))))

Theoremceicl 12504 The ceiling function returns an integer (closure law). (Contributed by Jeff Hankins, 10-Jun-2007.)
(𝐴 ∈ ℝ → -(⌊‘-𝐴) ∈ ℤ)

Theoremceilcl 12505 Closure of the ceiling function. (Contributed by David A. Wheeler, 19-May-2015.)
(𝐴 ∈ ℝ → (⌈‘𝐴) ∈ ℤ)

Theoremceige 12506 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 12507 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 12508 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 12509 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 12510 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 12511 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 12512 An integer is its own ceiling. (Contributed by AV, 30-Nov-2018.)
(𝐴 ∈ ℤ → (⌈‘𝐴) = 𝐴)

Theoremceilidz 12513 A real number equals its ceiling iff it is an integer. (Contributed by AV, 30-Nov-2018.)
(𝐴 ∈ ℝ → (𝐴 ∈ ℤ ↔ (⌈‘𝐴) = 𝐴))

Theoremflleceil 12514 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 12515 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 12516 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 14964. (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 12517 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 12518 Alternate proof of quoremnn0 12517 not using quoremz 12516. TODO - Keep either quoremnn0ALT 12518 (if we don't keep quoremz 12516) or quoremnn0 12517. (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 12519 Decompose a real into integer and fractional parts. TODO - should we replace this with intfrac 12547? (Contributed by NM, 16-Aug-2008.)
𝑍 = (⌊‘𝐴)    &   𝐹 = (𝐴𝑍)       (𝐴 ∈ ℝ → (0 ≤ 𝐹𝐹 < 1 ∧ 𝐴 = (𝑍 + 𝐹)))

Theoremintfracq 12520 Decompose a rational number, expressed as a ratio, into integer and fractional parts. The fractional part has a tighter bound than that of intfrac2 12519. (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 12521 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 12522 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 12523 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 12524 An upper set of integers is unbounded above. (Contributed by Mario Carneiro, 7-May-2016.)
𝑍 = (ℤ𝑀)       (𝑀 ∈ ℤ → sup(𝑍, ℝ*, < ) = +∞)

Theoremioopnfsup 12525 An upper set of reals is unbounded above. (Contributed by Mario Carneiro, 7-May-2016.)
((𝐴 ∈ ℝ*𝐴 ≠ +∞) → sup((𝐴(,)+∞), ℝ*, < ) = +∞)

Theoremicopnfsup 12526 An upper set of reals is unbounded above. (Contributed by Mario Carneiro, 7-May-2016.)
((𝐴 ∈ ℝ*𝐴 ≠ +∞) → sup((𝐴[,)+∞), ℝ*, < ) = +∞)

Theoremrpsup 12527 The positive reals are unbounded above. (Contributed by Mario Carneiro, 7-May-2016.)
sup(ℝ+, ℝ*, < ) = +∞

Theoremresup 12528 The real numbers are unbounded above. (Contributed by Mario Carneiro, 7-May-2016.)
sup(ℝ, ℝ*, < ) = +∞

Theoremxrsup 12529 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 12530 Extend class notation with the modulo operation.
class mod

Definitiondf-mod 12531* Define the modulo (remainder) operation. See modval 12532 for its value. For example, (5 mod 3) = 2 and (-7 mod 2) = 1 (ex-mod 26698). (Contributed by NM, 10-Nov-2008.)
mod = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ+ ↦ (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦)))))

Theoremmodval 12532 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 12533 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 12534 Closure law for the modulo operation. (Contributed by NM, 10-Nov-2008.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (𝐴 mod 𝐵) ∈ ℝ)

Theoremflpmodeq 12535 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 12536 Closure law for the modulo operation. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ+)       (𝜑 → (𝐴 mod 𝐵) ∈ ℝ)

Theoremmod0 12537 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 12538 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 12539 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 12540 The modulo operation is nonnegative. (Contributed by NM, 10-Nov-2008.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → 0 ≤ (𝐴 mod 𝐵))

Theoremmodlt 12541 The modulo operation is less than its second argument. (Contributed by NM, 10-Nov-2008.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (𝐴 mod 𝐵) < 𝐵)

Theoremmodelico 12542 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 12543 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 12544 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 12545 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 12546 The floor function expressed in terms of the modulo operation. (Contributed by NM, 11-Nov-2008.)
(𝐴 ∈ ℝ → (⌊‘𝐴) = (𝐴 − (𝐴 mod 1)))

Theoremintfrac 12547 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 12548 An integer modulo 1 is 0. (Contributed by Paul Chapman, 22-Jun-2011.)
(𝑁 ∈ ℤ → (𝑁 mod 1) = 0)

Theoremzmod1congr 12549 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 12550 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 12551 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 12552 Closure law for the modulo operation restricted to integers. (Contributed by NM, 27-Nov-2008.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → (𝐴 mod 𝐵) ∈ ℕ0)

Theoremzmodcld 12553 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 12554 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 12555 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 12556 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 12557 Identity law for modulo. (Contributed by NM, 29-Dec-2008.)
(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) ∧ (0 ≤ 𝐴𝐴 < 𝐵)) → (𝐴 mod 𝐵) = 𝐴)

Theoremmodid0 12558 A positive real number modulo itself is 0. (Contributed by Alexander van der Vekens, 15-May-2018.)
(𝑁 ∈ ℝ+ → (𝑁 mod 𝑁) = 0)

Theoremmodid2 12559 Identity law for modulo. (Contributed by NM, 29-Dec-2008.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → ((𝐴 mod 𝐵) = 𝐴 ↔ (0 ≤ 𝐴𝐴 < 𝐵)))

Theoremzmodid2 12560 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 12561 Identity law for modulo restricted to integers. (Contributed by AV, 27-Oct-2018.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ) → ((𝑀 mod 𝑁) = 𝑀𝑀 ∈ (0..^𝑁)))

Theoremzmodidfzoimp 12562 Identity law for modulo restricted to integers. (Contributed by AV, 27-Oct-2018.)
(𝑀 ∈ (0..^𝑁) → (𝑀 mod 𝑁) = 𝑀)

Theorem0mod 12563 Special case: 0 modulo a positive real number is 0. (Contributed by Mario Carneiro, 22-Feb-2014.)
(𝑁 ∈ ℝ+ → (0 mod 𝑁) = 0)

Theorem1mod 12564 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 12565 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 12566 Absorption law for modulo. (Contributed by NM, 29-Dec-2008.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → ((𝐴 mod 𝐵) mod 𝐵) = (𝐴 mod 𝐵))

Theoremmodcyc 12567 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 12568 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 12569 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 12570 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 12571 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 12572 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 12573 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 12574* 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 12575* 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 12576* 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 12577 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 12578 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 12579 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 12580 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 12581 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)

Theoremmodadd2mod 12582 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, 17-May-2018.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑀 ∈ ℝ+) → ((𝐵 + (𝐴 mod 𝑀)) mod 𝑀) = ((𝐵 + 𝐴) mod 𝑀))

Theoremmodm1p1mod0 12583 If an real number modulo a positive real number equals the positive real number decreased by 1, the real number increased by 1 modulo the positive real number equals 0. (Contributed by AV, 2-Nov-2018.)
((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℝ+) → ((𝐴 mod 𝑀) = (𝑀 − 1) → ((𝐴 + 1) mod 𝑀) = 0))

Theoremmodltm1p1mod 12584 If a real number modulo a positive real number is less than the positive real number decreased by 1, the real number increased by 1 modulo the positive real number equals the real number modulo the positive real number increased by 1. (Contributed by AV, 2-Nov-2018.)
((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℝ+ ∧ (𝐴 mod 𝑀) < (𝑀 − 1)) → ((𝐴 + 1) mod 𝑀) = ((𝐴 mod 𝑀) + 1))

Theoremmodmul1 12585 Multiplication property of the modulo operation. Note that the multiplier 𝐶 must be an integer. (Contributed by NM, 12-Nov-2008.)
(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℝ+) ∧ (𝐴 mod 𝐷) = (𝐵 mod 𝐷)) → ((𝐴 · 𝐶) mod 𝐷) = ((𝐵 · 𝐶) mod 𝐷))

Theoremmodmul12d 12586 Multiplication property of the modulo operation, see theorem 5.2(b) in [ApostolNT] p. 107. (Contributed by Mario Carneiro, 5-Feb-2015.)
(𝜑𝐴 ∈ ℤ)    &   (𝜑𝐵 ∈ ℤ)    &   (𝜑𝐶 ∈ ℤ)    &   (𝜑𝐷 ∈ ℤ)    &   (𝜑𝐸 ∈ ℝ+)    &   (𝜑 → (𝐴 mod 𝐸) = (𝐵 mod 𝐸))    &   (𝜑 → (𝐶 mod 𝐸) = (𝐷 mod 𝐸))       (𝜑 → ((𝐴 · 𝐶) mod 𝐸) = ((𝐵 · 𝐷) mod 𝐸))

Theoremmodnegd 12587 Negation property of the modulo operation. (Contributed by Mario Carneiro, 9-Sep-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ+)    &   (𝜑 → (𝐴 mod 𝐶) = (𝐵 mod 𝐶))       (𝜑 → (-𝐴 mod 𝐶) = (-𝐵 mod 𝐶))

Theoremmodadd12d 12588 Additive property of the modulo operation. (Contributed by Mario Carneiro, 9-Sep-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐷 ∈ ℝ)    &   (𝜑𝐸 ∈ ℝ+)    &   (𝜑 → (𝐴 mod 𝐸) = (𝐵 mod 𝐸))    &   (𝜑 → (𝐶 mod 𝐸) = (𝐷 mod 𝐸))       (𝜑 → ((𝐴 + 𝐶) mod 𝐸) = ((𝐵 + 𝐷) mod 𝐸))

Theoremmodsub12d 12589 Subtraction property of the modulo operation. (Contributed by Mario Carneiro, 9-Sep-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐷 ∈ ℝ)    &   (𝜑𝐸 ∈ ℝ+)    &   (𝜑 → (𝐴 mod 𝐸) = (𝐵 mod 𝐸))    &   (𝜑 → (𝐶 mod 𝐸) = (𝐷 mod 𝐸))       (𝜑 → ((𝐴𝐶) mod 𝐸) = ((𝐵𝐷) mod 𝐸))

Theoremmodsubmod 12590 The difference of a real number modulo a positive real number and another real number equals the difference of the two real numbers modulo the positive real number. (Contributed by Alexander van der Vekens, 17-May-2018.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑀 ∈ ℝ+) → (((𝐴 mod 𝑀) − 𝐵) mod 𝑀) = ((𝐴𝐵) mod 𝑀))

Theoremmodsubmodmod 12591 The difference of a real number modulo a positive real number and another real number modulo this positive real number equals the difference of the two real numbers modulo the positive real number. (Contributed by Alexander van der Vekens, 17-May-2018.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑀 ∈ ℝ+) → (((𝐴 mod 𝑀) − (𝐵 mod 𝑀)) mod 𝑀) = ((𝐴𝐵) mod 𝑀))

Theorem2txmodxeq0 12592 Two times a positive real number modulo the real number is zero. (Contributed by Alexander van der Vekens, 8-Jun-2018.)
(𝑋 ∈ ℝ+ → ((2 · 𝑋) mod 𝑋) = 0)

Theorem2submod 12593 If a real number is between a positive real number and twice the positive real number, the real number modulo the positive real number equals the real number minus the positive real number. (Contributed by Alexander van der Vekens, 13-May-2018.)
(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) ∧ (𝐵𝐴𝐴 < (2 · 𝐵))) → (𝐴 mod 𝐵) = (𝐴𝐵))

Theoremmodifeq2int 12594 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)))

Theoremmodaddmodup 12595 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)))

Theoremmodaddmodlo 12596 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)))

Theoremmodmulmod 12597 The product of a real number modulo a positive real number and an integer equals the product of the real number and the integer modulo the positive real number. (Contributed by Alexander van der Vekens, 17-May-2018.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℤ ∧ 𝑀 ∈ ℝ+) → (((𝐴 mod 𝑀) · 𝐵) mod 𝑀) = ((𝐴 · 𝐵) mod 𝑀))

Theoremmodmulmodr 12598 The product of an integer and a real number modulo a positive real number equals the product of the integer and the real number modulo the positive real number. (Contributed by Alexander van der Vekens, 9-Jul-2021.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℝ ∧ 𝑀 ∈ ℝ+) → ((𝐴 · (𝐵 mod 𝑀)) mod 𝑀) = ((𝐴 · 𝐵) mod 𝑀))

Theoremmodaddmulmod 12599 The sum of a real number and the product of a second real number modulo a positive real number and an integer equals the sum of the real number and the product of the other real number and the integer modulo the positive real number. (Contributed by Alexander van der Vekens, 17-May-2018.)
(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℤ) ∧ 𝑀 ∈ ℝ+) → ((𝐴 + ((𝐵 mod 𝑀) · 𝐶)) mod 𝑀) = ((𝐴 + (𝐵 · 𝐶)) mod 𝑀))

Theoremmoddi 12600 Distribute multiplication over a modulo operation. (Contributed by NM, 11-Nov-2008.)
((𝐴 ∈ ℝ+𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ+) → (𝐴 · (𝐵 mod 𝐶)) = ((𝐴 · 𝐵) mod (𝐴 · 𝐶)))

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