P. 140 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 13901-14000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	fldiv4lem1div2uz2 13901	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 13902	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	ceilval 13903	The value of the ceiling function. (Contributed by David A. Wheeler, 19-May-2015.)
⊢ (𝐴 ∈ ℝ → (⌈‘𝐴) = -(⌊‘-𝐴))
Theorem	dfceil2 13904*	Alternative definition of the ceiling function using restricted iota. (Contributed by AV, 1-Dec-2018.)
⊢ ⌈ = (𝑥 ∈ ℝ ↦ (℩𝑦 ∈ ℤ (𝑥 ≤ 𝑦 ∧ 𝑦 < (𝑥 + 1))))
Theorem	ceilval2 13905*	The value of the ceiling function using restricted iota. (Contributed by AV, 1-Dec-2018.)
⊢ (𝐴 ∈ ℝ → (⌈‘𝐴) = (℩𝑦 ∈ ℤ (𝐴 ≤ 𝑦 ∧ 𝑦 < (𝐴 + 1))))
Theorem	ceicl 13906	The ceiling function returns an integer (closure law). (Contributed by Jeff Hankins, 10-Jun-2007.)
⊢ (𝐴 ∈ ℝ → -(⌊‘-𝐴) ∈ ℤ)
Theorem	ceilcl 13907	Closure of the ceiling function. (Contributed by David A. Wheeler, 19-May-2015.)
⊢ (𝐴 ∈ ℝ → (⌈‘𝐴) ∈ ℤ)
Theorem	ceilcld 13908	Closure of the ceiling function. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    ⇒   ⊢ (𝜑 → (⌈‘𝐴) ∈ ℤ)
Theorem	ceige 13909	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))
Theorem	ceilge 13910	The ceiling of a real number is greater than or equal to that number. (Contributed by AV, 30-Nov-2018.)
⊢ (𝐴 ∈ ℝ → 𝐴 ≤ (⌈‘𝐴))
Theorem	ceilged 13911	The ceiling of a real number is greater than or equal to that number. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    ⇒   ⊢ (𝜑 → 𝐴 ≤ (⌈‘𝐴))
Theorem	ceim1l 13912	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)
Theorem	ceilm1lt 13913	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)
Theorem	ceile 13914	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 ∧ 𝐴 ≤ 𝐵) → -(⌊‘-𝐴) ≤ 𝐵)
Theorem	ceille 13915	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 ∧ 𝐴 ≤ 𝐵) → (⌈‘𝐴) ≤ 𝐵)
Theorem	ceilid 13916	An integer is its own ceiling. (Contributed by AV, 30-Nov-2018.)
⊢ (𝐴 ∈ ℤ → (⌈‘𝐴) = 𝐴)
Theorem	ceilidz 13917	A real number equals its ceiling iff it is an integer. (Contributed by AV, 30-Nov-2018.)
⊢ (𝐴 ∈ ℝ → (𝐴 ∈ ℤ ↔ (⌈‘𝐴) = 𝐴))
Theorem	flleceil 13918	The floor of a real number is less than or equal to its ceiling. (Contributed by AV, 30-Nov-2018.)
⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ≤ (⌈‘𝐴))
Theorem	fleqceilz 13919	A real number is an integer iff its floor equals its ceiling. (Contributed by AV, 30-Nov-2018.)
⊢ (𝐴 ∈ ℝ → (𝐴 ∈ ℤ ↔ (⌊‘𝐴) = (⌈‘𝐴)))
Theorem	quoremz 13920	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 16499. (Contributed by NM, 14-Aug-2008.)
⊢ 𝑄 = (⌊‘(𝐴 / 𝐵))    &   ⊢ 𝑅 = (𝐴 − (𝐵 · 𝑄))    ⇒   ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → ((𝑄 ∈ ℤ ∧ 𝑅 ∈ ℕ0) ∧ (𝑅 < 𝐵 ∧ 𝐴 = ((𝐵 · 𝑄) + 𝑅))))
Theorem	quoremnn0 13921	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 ∈ ℕ0 ∧ 𝐵 ∈ ℕ) → ((𝑄 ∈ ℕ0 ∧ 𝑅 ∈ ℕ0) ∧ (𝑅 < 𝐵 ∧ 𝐴 = ((𝐵 · 𝑄) + 𝑅))))
Theorem	quoremnn0ALT 13922	Alternate proof of quoremnn0 13921 not using quoremz 13920. TODO - Keep either quoremnn0ALT 13922 (if we don't keep quoremz 13920) or quoremnn0 13921? (Contributed by NM, 14-Aug-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝑄 = (⌊‘(𝐴 / 𝐵))    &   ⊢ 𝑅 = (𝐴 − (𝐵 · 𝑄))    ⇒   ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ) → ((𝑄 ∈ ℕ0 ∧ 𝑅 ∈ ℕ0) ∧ (𝑅 < 𝐵 ∧ 𝐴 = ((𝐵 · 𝑄) + 𝑅))))
Theorem	intfrac2 13923	Decompose a real into integer and fractional parts. TODO - should we replace this with intfrac 13951? (Contributed by NM, 16-Aug-2008.)
⊢ 𝑍 = (⌊‘𝐴)    &   ⊢ 𝐹 = (𝐴 − 𝑍)    ⇒   ⊢ (𝐴 ∈ ℝ → (0 ≤ 𝐹 ∧ 𝐹 < 1 ∧ 𝐴 = (𝑍 + 𝐹)))
Theorem	intfracq 13924	Decompose a rational number, expressed as a ratio, into integer and fractional parts. The fractional part has a tighter bound than that of intfrac2 13923. (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	fldiv 13925	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)))
Theorem	fldiv2 13926	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))))
Theorem	fznnfl 13927	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 ∧ 𝐾 ≤ 𝑁)))
Theorem	uzsup 13928	An upper set of integers is unbounded above. (Contributed by Mario Carneiro, 7-May-2016.)
⊢ 𝑍 = (ℤ≥‘𝑀)    ⇒   ⊢ (𝑀 ∈ ℤ → sup(𝑍, ℝ*, < ) = +∞)
Theorem	ioopnfsup 13929	An upper set of reals is unbounded above. (Contributed by Mario Carneiro, 7-May-2016.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ +∞) → sup((𝐴(,)+∞), ℝ*, < ) = +∞)
Theorem	icopnfsup 13930	An upper set of reals is unbounded above. (Contributed by Mario Carneiro, 7-May-2016.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ +∞) → sup((𝐴[,)+∞), ℝ*, < ) = +∞)
Theorem	rpsup 13931	The positive reals are unbounded above. (Contributed by Mario Carneiro, 7-May-2016.)
⊢ sup(ℝ+, ℝ*, < ) = +∞
Theorem	resup 13932	The real numbers are unbounded above. (Contributed by Mario Carneiro, 7-May-2016.)
⊢ sup(ℝ, ℝ*, < ) = +∞
Theorem	xrsup 13933	The extended real numbers are unbounded above. (Contributed by Mario Carneiro, 7-May-2016.)
⊢ sup(ℝ*, ℝ*, < ) = +∞
5.6.2  The modulo (remainder) operation
Syntax	cmo 13934	Extend class notation with the modulo operation.
class mod
Definition	df-mod 13935*	Define the modulo (remainder) operation. See modval 13936 for its value. For example, (5 mod 3) = 2 and (-7 mod 2) = 1 (ex-mod 30937). (Contributed by NM, 10-Nov-2008.)
⊢ mod = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ+ ↦ (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦)))))
Theorem	modval 13936	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 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)))))
Theorem	modvalr 13937	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)))
Theorem	modcl 13938	Closure law for the modulo operation. (Contributed by NM, 10-Nov-2008.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (𝐴 mod 𝐵) ∈ ℝ)
Theorem	flpmodeq 13939	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)
Theorem	modcld 13940	Closure law for the modulo operation. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ+)    ⇒   ⊢ (𝜑 → (𝐴 mod 𝐵) ∈ ℝ)
Theorem	mod0 13941	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))
Theorem	mulmod0 13942	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)
Theorem	negmod0 13943	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))
Theorem	modge0 13944	The modulo operation is nonnegative. (Contributed by NM, 10-Nov-2008.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → 0 ≤ (𝐴 mod 𝐵))
Theorem	modlt 13945	The modulo operation is less than its second argument. (Contributed by NM, 10-Nov-2008.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (𝐴 mod 𝐵) < 𝐵)
Theorem	modelico 13946	Modular reduction produces a half-open interval. (Contributed by Stefan O'Rear, 12-Sep-2014.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (𝐴 mod 𝐵) ∈ (0[,)𝐵))
Theorem	moddiffl 13947	Value of the modulo operation rewritten to give two ways of expressing the quotient when "𝐴 is divided by 𝐵 using Euclidean division." Multiplying both sides by 𝐵, this implies that 𝐴 mod 𝐵 differs from 𝐴 by an integer multiple of 𝐵. (Contributed by Jeff Madsen, 17-Jun-2010.) (Revised by Mario Carneiro, 6-Sep-2016.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → ((𝐴 − (𝐴 mod 𝐵)) / 𝐵) = (⌊‘(𝐴 / 𝐵)))
Theorem	moddifz 13948	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)
Theorem	modfrac 13949	The fractional part of a number is the number modulo 1. (Contributed by NM, 11-Nov-2008.)
⊢ (𝐴 ∈ ℝ → (𝐴 mod 1) = (𝐴 − (⌊‘𝐴)))
Theorem	flmod 13950	The floor function expressed in terms of the modulo operation. (Contributed by NM, 11-Nov-2008.)
⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) = (𝐴 − (𝐴 mod 1)))
Theorem	intfrac 13951	Break a number into its integer part and its fractional part. (Contributed by NM, 31-Dec-2008.)
⊢ (𝐴 ∈ ℝ → 𝐴 = ((⌊‘𝐴) + (𝐴 mod 1)))
Theorem	zmod10 13952	An integer modulo 1 is 0. (Contributed by Paul Chapman, 22-Jun-2011.)
⊢ (𝑁 ∈ ℤ → (𝑁 mod 1) = 0)
Theorem	zmod1congr 13953	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	modmulnn 13954	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)))
Theorem	modvalp1 13955	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))
Theorem	zmodcl 13956	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 13957	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 13958	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 13959	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 13960	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	modid 13961	Identity law for modulo. (Contributed by NM, 29-Dec-2008.)
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) ∧ (0 ≤ 𝐴 ∧ 𝐴 < 𝐵)) → (𝐴 mod 𝐵) = 𝐴)
Theorem	modid0 13962	A positive real number modulo itself is 0. (Contributed by Alexander van der Vekens, 15-May-2018.)
⊢ (𝑁 ∈ ℝ+ → (𝑁 mod 𝑁) = 0)
Theorem	modid2 13963	Identity law for modulo. (Contributed by NM, 29-Dec-2008.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → ((𝐴 mod 𝐵) = 𝐴 ↔ (0 ≤ 𝐴 ∧ 𝐴 < 𝐵)))
Theorem	zmodid2 13964	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 13965	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 13966	Identity law for modulo restricted to integers. (Contributed by AV, 27-Oct-2018.)
⊢ (𝑀 ∈ (0..^𝑁) → (𝑀 mod 𝑁) = 𝑀)
Theorem	0mod 13967	Special case: 0 modulo a positive real number is 0. (Contributed by Mario Carneiro, 22-Feb-2014.)
⊢ (𝑁 ∈ ℝ+ → (0 mod 𝑁) = 0)
Theorem	1mod 13968	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)
Theorem	modabs 13969	Absorption law for modulo. (Contributed by NM, 29-Dec-2008.)
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ∧ 𝐶 ∈ ℝ+) ∧ 𝐵 ≤ 𝐶) → ((𝐴 mod 𝐵) mod 𝐶) = (𝐴 mod 𝐵))
Theorem	modabs2 13970	Absorption law for modulo. (Contributed by NM, 29-Dec-2008.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → ((𝐴 mod 𝐵) mod 𝐵) = (𝐴 mod 𝐵))
Theorem	modcyc 13971	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))
Theorem	modcyc2 13972	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))
Theorem	modadd1 13973	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))
Theorem	modaddb 13974	Addition property of the modulo operation. Biconditional version of modadd1 13973 by applying modadd1 13973 twice. (Contributed by AV, 14-Nov-2025.)
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ+)) → ((𝐴 mod 𝐷) = (𝐵 mod 𝐷) ↔ ((𝐴 + 𝐶) mod 𝐷) = ((𝐵 + 𝐶) mod 𝐷)))
Theorem	modaddid 13975	The sums of two nonnegative integers less than the modulus and an integer are equal iff the two nonnegative integers are equal. (Contributed by AV, 14-Nov-2025.)
⊢ 𝐼 = (0..^𝑁)    ⇒   ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼) ∧ 𝐾 ∈ ℤ) → (((𝑋 + 𝐾) mod 𝑁) = ((𝑌 + 𝐾) mod 𝑁) ↔ 𝑋 = 𝑌))
Theorem	modaddabs 13976	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))
Theorem	modaddmod 13977	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))
Theorem	muladdmodid 13978	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)
Theorem	mulp1mod1 13979	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	muladdmod 13980	A real number is the sum of the number and a multiple of a positive real number modulo the positive real number. (Contributed by AV, 7-Sep-2025.)
⊢ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℝ+ ∧ 𝑁 ∈ ℤ) → (((𝑁 · 𝑀) + 𝐴) mod 𝑀) = (𝐴 mod 𝑀))
Theorem	modmuladd 13981*	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)))
Theorem	modmuladdim 13982*	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)))
Theorem	modmuladdnn0 13983*	Implication of a decomposition of a nonnegative integer into a multiple of a modulus and a remainder. (Contributed by AV, 14-Jul-2021.)
⊢ ((𝐴 ∈ ℕ0 ∧ 𝑀 ∈ ℝ+) → ((𝐴 mod 𝑀) = 𝐵 → ∃𝑘 ∈ ℕ0 𝐴 = ((𝑘 · 𝑀) + 𝐵)))
Theorem	negmod 13984	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) = ((𝑁 − 𝐴) mod 𝑁))
Theorem	m1modnnsub1 13985	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 13986	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 13987	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 13988	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	modadd2mod 13989	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))
Theorem	modm1p1mod0 13990	If a 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))
Theorem	modltm1p1mod 13991	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))
Theorem	modmul1 13992	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))
Theorem	modmul12d 13993	Multiplication property of the modulo operation, see theorem 5.2(b) in [ApostolNT] p. 107. (Contributed by Mario Carneiro, 5-Feb-2015.)
⊢ (𝜑 → 𝐴 ∈ ℤ)    &   ⊢ (𝜑 → 𝐵 ∈ ℤ)    &   ⊢ (𝜑 → 𝐶 ∈ ℤ)    &   ⊢ (𝜑 → 𝐷 ∈ ℤ)    &   ⊢ (𝜑 → 𝐸 ∈ ℝ+)    &   ⊢ (𝜑 → (𝐴 mod 𝐸) = (𝐵 mod 𝐸))    &   ⊢ (𝜑 → (𝐶 mod 𝐸) = (𝐷 mod 𝐸))    ⇒   ⊢ (𝜑 → ((𝐴 · 𝐶) mod 𝐸) = ((𝐵 · 𝐷) mod 𝐸))
Theorem	modnegd 13994	Negation property of the modulo operation. (Contributed by Mario Carneiro, 9-Sep-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ+)    &   ⊢ (𝜑 → (𝐴 mod 𝐶) = (𝐵 mod 𝐶))    ⇒   ⊢ (𝜑 → (-𝐴 mod 𝐶) = (-𝐵 mod 𝐶))
Theorem	modadd12d 13995	Additive property of the modulo operation. (Contributed by Mario Carneiro, 9-Sep-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐷 ∈ ℝ)    &   ⊢ (𝜑 → 𝐸 ∈ ℝ+)    &   ⊢ (𝜑 → (𝐴 mod 𝐸) = (𝐵 mod 𝐸))    &   ⊢ (𝜑 → (𝐶 mod 𝐸) = (𝐷 mod 𝐸))    ⇒   ⊢ (𝜑 → ((𝐴 + 𝐶) mod 𝐸) = ((𝐵 + 𝐷) mod 𝐸))
Theorem	modsub12d 13996	Subtraction property of the modulo operation. (Contributed by Mario Carneiro, 9-Sep-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐷 ∈ ℝ)    &   ⊢ (𝜑 → 𝐸 ∈ ℝ+)    &   ⊢ (𝜑 → (𝐴 mod 𝐸) = (𝐵 mod 𝐸))    &   ⊢ (𝜑 → (𝐶 mod 𝐸) = (𝐷 mod 𝐸))    ⇒   ⊢ (𝜑 → ((𝐴 − 𝐶) mod 𝐸) = ((𝐵 − 𝐷) mod 𝐸))
Theorem	modsubmod 13997	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) = ((𝐴 − 𝐵) mod 𝑀))
Theorem	modsubmodmod 13998	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) = ((𝐴 − 𝐵) mod 𝑀))
Theorem	2txmodxeq0 13999	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)
Theorem	2submod 14000	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+) ∧ (𝐵 ≤ 𝐴 ∧ 𝐴 < (2 · 𝐵))) → (𝐴 mod 𝐵) = (𝐴 − 𝐵))