P. 138 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 13701-13800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	flcl 13701	The floor (greatest integer) function is an integer (closure law). (Contributed by NM, 15-Nov-2004.) (Revised by Mario Carneiro, 2-Nov-2013.)
⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ∈ ℤ)
Theorem	reflcl 13702	The floor (greatest integer) function is real. (Contributed by NM, 15-Jul-2008.)
⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ∈ ℝ)
Theorem	fllelt 13703	A basic property of the floor (greatest integer) function. (Contributed by NM, 15-Nov-2004.) (Revised by Mario Carneiro, 2-Nov-2013.)
⊢ (𝐴 ∈ ℝ → ((⌊‘𝐴) ≤ 𝐴 ∧ 𝐴 < ((⌊‘𝐴) + 1)))
Theorem	flcld 13704	The floor (greatest integer) function is an integer (closure law). (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    ⇒   ⊢ (𝜑 → (⌊‘𝐴) ∈ ℤ)
Theorem	flle 13705	A basic property of the floor (greatest integer) function. (Contributed by NM, 24-Feb-2005.)
⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ≤ 𝐴)
Theorem	flltp1 13706	A basic property of the floor (greatest integer) function. (Contributed by NM, 24-Feb-2005.)
⊢ (𝐴 ∈ ℝ → 𝐴 < ((⌊‘𝐴) + 1))
Theorem	fllep1 13707	A basic property of the floor (greatest integer) function. (Contributed by Mario Carneiro, 21-May-2016.)
⊢ (𝐴 ∈ ℝ → 𝐴 ≤ ((⌊‘𝐴) + 1))
Theorem	fraclt1 13708	The fractional part of a real number is less than one. (Contributed by NM, 15-Jul-2008.)
⊢ (𝐴 ∈ ℝ → (𝐴 − (⌊‘𝐴)) < 1)
Theorem	fracle1 13709	The fractional part of a real number is less than or equal to one. (Contributed by Mario Carneiro, 21-May-2016.)
⊢ (𝐴 ∈ ℝ → (𝐴 − (⌊‘𝐴)) ≤ 1)
Theorem	fracge0 13710	The fractional part of a real number is nonnegative. (Contributed by NM, 17-Jul-2008.)
⊢ (𝐴 ∈ ℝ → 0 ≤ (𝐴 − (⌊‘𝐴)))
Theorem	flge 13711	The floor function value is the greatest integer less than or equal to its argument. (Contributed by NM, 15-Nov-2004.) (Proof shortened by Fan Zheng, 14-Jul-2016.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℤ) → (𝐵 ≤ 𝐴 ↔ 𝐵 ≤ (⌊‘𝐴)))
Theorem	fllt 13712	The floor function value is less than the next integer. (Contributed by NM, 24-Feb-2005.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℤ) → (𝐴 < 𝐵 ↔ (⌊‘𝐴) < 𝐵))
Theorem	flflp1 13713	Move floor function between strict and non-strict inequality. (Contributed by Brendan Leahy, 25-Oct-2017.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((⌊‘𝐴) ≤ 𝐵 ↔ 𝐴 < ((⌊‘𝐵) + 1)))
Theorem	flid 13714	An integer is its own floor. (Contributed by NM, 15-Nov-2004.)
⊢ (𝐴 ∈ ℤ → (⌊‘𝐴) = 𝐴)
Theorem	flidm 13715	The floor function is idempotent. (Contributed by NM, 17-Aug-2008.)
⊢ (𝐴 ∈ ℝ → (⌊‘(⌊‘𝐴)) = (⌊‘𝐴))
Theorem	flidz 13716	A real number equals its floor iff it is an integer. (Contributed by NM, 11-Nov-2008.)
⊢ (𝐴 ∈ ℝ → ((⌊‘𝐴) = 𝐴 ↔ 𝐴 ∈ ℤ))
Theorem	flltnz 13717	The floor of a non-integer real is less than it. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ((𝐴 ∈ ℝ ∧ ¬ 𝐴 ∈ ℤ) → (⌊‘𝐴) < 𝐴)
Theorem	flwordi 13718	Ordering relation for the floor function. (Contributed by NM, 31-Dec-2005.) (Proof shortened by Fan Zheng, 14-Jul-2016.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → (⌊‘𝐴) ≤ (⌊‘𝐵))
Theorem	flword2 13719	Ordering relation for the floor function. (Contributed by Mario Carneiro, 7-Jun-2016.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → (⌊‘𝐵) ∈ (ℤ≥‘(⌊‘𝐴)))
Theorem	flval2 13720*	An alternate way to define the floor function. (Contributed by NM, 16-Nov-2004.)
⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) = (℩𝑥 ∈ ℤ (𝑥 ≤ 𝐴 ∧ ∀𝑦 ∈ ℤ (𝑦 ≤ 𝐴 → 𝑦 ≤ 𝑥))))
Theorem	flval3 13721*	An alternate way to define the floor function, as the supremum of all integers less than or equal to its argument. (Contributed by NM, 15-Nov-2004.) (Proof shortened by Mario Carneiro, 6-Sep-2014.)
⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) = sup({𝑥 ∈ ℤ ∣ 𝑥 ≤ 𝐴}, ℝ, < ))
Theorem	flbi 13722	A condition equivalent to floor. (Contributed by NM, 11-Mar-2005.) (Revised by Mario Carneiro, 2-Nov-2013.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℤ) → ((⌊‘𝐴) = 𝐵 ↔ (𝐵 ≤ 𝐴 ∧ 𝐴 < (𝐵 + 1))))
Theorem	flbi2 13723	A condition equivalent to floor. (Contributed by NM, 15-Aug-2008.)
⊢ ((𝑁 ∈ ℤ ∧ 𝐹 ∈ ℝ) → ((⌊‘(𝑁 + 𝐹)) = 𝑁 ↔ (0 ≤ 𝐹 ∧ 𝐹 < 1)))
Theorem	adddivflid 13724	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 ∈ ℕ0 ∧ 𝐶 ∈ ℕ) → (𝐵 < 𝐶 ↔ (⌊‘(𝐴 + (𝐵 / 𝐶))) = 𝐴))
Theorem	ico01fl0 13725	The floor of a real number in [0, 1) is 0. Remark: may shorten the proof of modid 13802 or a version of it where the antecedent is membership in an interval. (Contributed by BJ, 29-Jun-2019.)
⊢ (𝐴 ∈ (0[,)1) → (⌊‘𝐴) = 0)
Theorem	flge0nn0 13726	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)
Theorem	flge1nn 13727	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)
Theorem	fldivnn0 13728	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 ∈ ℕ0 ∧ 𝐿 ∈ ℕ) → (⌊‘(𝐾 / 𝐿)) ∈ ℕ0)
Theorem	refldivcl 13729	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)
Theorem	divfl0 13730	The floor of a fraction is 0 iff the denominator is less than the numerator. (Contributed by AV, 8-Jul-2021.)
⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ) → (𝐴 < 𝐵 ↔ (⌊‘(𝐴 / 𝐵)) = 0))
Theorem	fladdz 13731	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))
Theorem	flzadd 13732	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)))
Theorem	flmulnn0 13733	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 ∈ ℕ0 ∧ 𝐴 ∈ ℝ) → (𝑁 · (⌊‘𝐴)) ≤ (⌊‘(𝑁 · 𝐴)))
Theorem	btwnzge0 13734	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 12847.) (Contributed by NM, 12-Mar-2005.)
⊢ (((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) ∧ (𝑁 ≤ 𝐴 ∧ 𝐴 < (𝑁 + 1))) → (0 ≤ 𝐴 ↔ 0 ≤ 𝑁))
Theorem	2tnp1ge0ge0 13735	Two times an integer plus one is not negative iff the integer is not negative. (Contributed by AV, 19-Jun-2021.) (Proof shortened by AV, 10-Jul-2022.)
⊢ (𝑁 ∈ ℤ → (0 ≤ ((2 · 𝑁) + 1) ↔ 0 ≤ 𝑁))
Theorem	flhalf 13736	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))))
Theorem	fldivle 13737	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))
Theorem	fldivnn0le 13738	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 ∈ ℕ0 ∧ 𝐿 ∈ ℕ) → (⌊‘(𝐾 / 𝐿)) ≤ (𝐾 / 𝐿))
Theorem	flltdivnn0lt 13739	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 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝐿 ∈ ℕ) → (𝐾 < 𝑁 → (⌊‘(𝐾 / 𝐿)) < (𝑁 / 𝐿)))
Theorem	ltdifltdiv 13740	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 < (𝐶 − 𝐵) → ((⌊‘(𝐴 / 𝐵)) + 1) < (𝐶 / 𝐵)))
Theorem	fldiv4p1lem1div2 13741	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))
Theorem	fldiv4lem1div2uz2 13742	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 13743	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 13744	The value of the ceiling function. (Contributed by David A. Wheeler, 19-May-2015.)
⊢ (𝐴 ∈ ℝ → (⌈‘𝐴) = -(⌊‘-𝐴))
Theorem	dfceil2 13745*	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 13746*	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 13747	The ceiling function returns an integer (closure law). (Contributed by Jeff Hankins, 10-Jun-2007.)
⊢ (𝐴 ∈ ℝ → -(⌊‘-𝐴) ∈ ℤ)
Theorem	ceilcl 13748	Closure of the ceiling function. (Contributed by David A. Wheeler, 19-May-2015.)
⊢ (𝐴 ∈ ℝ → (⌈‘𝐴) ∈ ℤ)
Theorem	ceilcld 13749	Closure of the ceiling function. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    ⇒   ⊢ (𝜑 → (⌈‘𝐴) ∈ ℤ)
Theorem	ceige 13750	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 13751	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 13752	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 13753	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 13754	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 13755	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 13756	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 13757	An integer is its own ceiling. (Contributed by AV, 30-Nov-2018.)
⊢ (𝐴 ∈ ℤ → (⌈‘𝐴) = 𝐴)
Theorem	ceilidz 13758	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 13759	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 13760	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 13761	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 16316. (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 13762	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 13763	Alternate proof of quoremnn0 13762 not using quoremz 13761. TODO - Keep either quoremnn0ALT 13763 (if we don't keep quoremz 13761) or quoremnn0 13762? (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 13764	Decompose a real into integer and fractional parts. TODO - should we replace this with intfrac 13792? (Contributed by NM, 16-Aug-2008.)
⊢ 𝑍 = (⌊‘𝐴)    &   ⊢ 𝐹 = (𝐴 − 𝑍)    ⇒   ⊢ (𝐴 ∈ ℝ → (0 ≤ 𝐹 ∧ 𝐹 < 1 ∧ 𝐴 = (𝑍 + 𝐹)))
Theorem	intfracq 13765	Decompose a rational number, expressed as a ratio, into integer and fractional parts. The fractional part has a tighter bound than that of intfrac2 13764. (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 13766	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 13767	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 13768	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 13769	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 13770	An upper set of reals is unbounded above. (Contributed by Mario Carneiro, 7-May-2016.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ +∞) → sup((𝐴(,)+∞), ℝ*, < ) = +∞)
Theorem	icopnfsup 13771	An upper set of reals is unbounded above. (Contributed by Mario Carneiro, 7-May-2016.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ +∞) → sup((𝐴[,)+∞), ℝ*, < ) = +∞)
Theorem	rpsup 13772	The positive reals are unbounded above. (Contributed by Mario Carneiro, 7-May-2016.)
⊢ sup(ℝ+, ℝ*, < ) = +∞
Theorem	resup 13773	The real numbers are unbounded above. (Contributed by Mario Carneiro, 7-May-2016.)
⊢ sup(ℝ, ℝ*, < ) = +∞
Theorem	xrsup 13774	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 13775	Extend class notation with the modulo operation.
class mod
Definition	df-mod 13776*	Define the modulo (remainder) operation. See modval 13777 for its value. For example, (5 mod 3) = 2 and (-7 mod 2) = 1 (ex-mod 30431). (Contributed by NM, 10-Nov-2008.)
⊢ mod = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ+ ↦ (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦)))))
Theorem	modval 13777	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 13778	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 13779	Closure law for the modulo operation. (Contributed by NM, 10-Nov-2008.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (𝐴 mod 𝐵) ∈ ℝ)
Theorem	flpmodeq 13780	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 13781	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 13782	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 13783	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 13784	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 13785	The modulo operation is nonnegative. (Contributed by NM, 10-Nov-2008.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → 0 ≤ (𝐴 mod 𝐵))
Theorem	modlt 13786	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 13787	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 13788	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 13789	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 13790	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 13791	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 13792	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 13793	An integer modulo 1 is 0. (Contributed by Paul Chapman, 22-Jun-2011.)
⊢ (𝑁 ∈ ℤ → (𝑁 mod 1) = 0)
Theorem	zmod1congr 13794	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 13795	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 13796	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 13797	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 13798	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 13799	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 13800	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))