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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | qavgle 9601 | 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)) | ||
Theorem | ioo0 9602 | An empty open interval of extended reals. (Contributed by NM, 6-Feb-2007.) |
⊢ ((𝐴 ∈ ℝ^{*} ∧ 𝐵 ∈ ℝ^{*}) → ((𝐴(,)𝐵) = ∅ ↔ 𝐵 ≤ 𝐴)) | ||
Theorem | ioom 9603* | 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)) | ||
Theorem | ico0 9604 | An empty open interval of extended reals. (Contributed by FL, 30-May-2014.) |
⊢ ((𝐴 ∈ ℝ^{*} ∧ 𝐵 ∈ ℝ^{*}) → ((𝐴[,)𝐵) = ∅ ↔ 𝐵 ≤ 𝐴)) | ||
Theorem | ioc0 9605 | An empty open interval of extended reals. (Contributed by FL, 30-May-2014.) |
⊢ ((𝐴 ∈ ℝ^{*} ∧ 𝐵 ∈ ℝ^{*}) → ((𝐴(,]𝐵) = ∅ ↔ 𝐵 ≤ 𝐴)) | ||
Syntax | cfl 9606 | Extend class notation with floor (greatest integer) function. |
class ⌊ | ||
Syntax | cceil 9607 | Extend class notation to include the ceiling function. |
class ⌈ | ||
Definition | df-fl 9608* |
Define the floor (greatest integer less than or equal to) function. See
flval 9610 for its value, flqlelt 9614 for its basic property, and flqcl 9611 for
its closure. For example, (⌊‘(3 / 2)) =
1 while
(⌊‘-(3 / 2)) = -2 (ex-fl 11141).
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 (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1)))) | ||
Definition | df-ceil 9609 |
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 9644 for its value, ceilqge 9648 and ceilqm1lt 9650 for its basic
properties, and ceilqcl 9646 for its closure. For example,
(⌈‘(3 / 2)) = 2 while (⌈‘-(3 / 2)) = -1
(ex-ceil 11142).
As described in df-fl 9608 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)) | ||
Theorem | flval 9610* | 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 (𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)))) | ||
Theorem | flqcl 9611 | 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 9613. (Contributed by Jim Kingdon, 8-Oct-2021.) |
⊢ (𝐴 ∈ ℚ → (⌊‘𝐴) ∈ ℤ) | ||
Theorem | apbtwnz 9612* | 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 (𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1))) | ||
Theorem | flapcl 9613* | 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 11083) would satisfy this condition. (Contributed by Jim Kingdon, 11-May-2022.) |
⊢ ((𝐴 ∈ ℝ ∧ ∀𝑛 ∈ ℤ 𝐴 # 𝑛) → (⌊‘𝐴) ∈ ℤ) | ||
Theorem | flqlelt 9614 | A basic property of the floor (greatest integer) function. (Contributed by Jim Kingdon, 8-Oct-2021.) |
⊢ (𝐴 ∈ ℚ → ((⌊‘𝐴) ≤ 𝐴 ∧ 𝐴 < ((⌊‘𝐴) + 1))) | ||
Theorem | flqcld 9615 | The floor (greatest integer) function is an integer (closure law). (Contributed by Jim Kingdon, 8-Oct-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℚ) ⇒ ⊢ (𝜑 → (⌊‘𝐴) ∈ ℤ) | ||
Theorem | flqle 9616 | A basic property of the floor (greatest integer) function. (Contributed by Jim Kingdon, 8-Oct-2021.) |
⊢ (𝐴 ∈ ℚ → (⌊‘𝐴) ≤ 𝐴) | ||
Theorem | flqltp1 9617 | A basic property of the floor (greatest integer) function. (Contributed by Jim Kingdon, 8-Oct-2021.) |
⊢ (𝐴 ∈ ℚ → 𝐴 < ((⌊‘𝐴) + 1)) | ||
Theorem | qfraclt1 9618 | The fractional part of a rational number is less than one. (Contributed by Jim Kingdon, 8-Oct-2021.) |
⊢ (𝐴 ∈ ℚ → (𝐴 − (⌊‘𝐴)) < 1) | ||
Theorem | qfracge0 9619 | The fractional part of a rational number is nonnegative. (Contributed by Jim Kingdon, 8-Oct-2021.) |
⊢ (𝐴 ∈ ℚ → 0 ≤ (𝐴 − (⌊‘𝐴))) | ||
Theorem | flqge 9620 | 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) → (𝐵 ≤ 𝐴 ↔ 𝐵 ≤ (⌊‘𝐴))) | ||
Theorem | flqlt 9621 | 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)) | ||
Theorem | flid 9622 | An integer is its own floor. (Contributed by NM, 15-Nov-2004.) |
⊢ (𝐴 ∈ ℤ → (⌊‘𝐴) = 𝐴) | ||
Theorem | flqidm 9623 | The floor function is idempotent. (Contributed by Jim Kingdon, 8-Oct-2021.) |
⊢ (𝐴 ∈ ℚ → (⌊‘(⌊‘𝐴)) = (⌊‘𝐴)) | ||
Theorem | flqidz 9624 | A rational number equals its floor iff it is an integer. (Contributed by Jim Kingdon, 9-Oct-2021.) |
⊢ (𝐴 ∈ ℚ → ((⌊‘𝐴) = 𝐴 ↔ 𝐴 ∈ ℤ)) | ||
Theorem | flqltnz 9625 | 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) | ||
Theorem | flqwordi 9626 | Ordering relationship for the greatest integer function. (Contributed by Jim Kingdon, 9-Oct-2021.) |
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 𝐴 ≤ 𝐵) → (⌊‘𝐴) ≤ (⌊‘𝐵)) | ||
Theorem | flqword2 9627 | Ordering relationship for the greatest integer function. (Contributed by Jim Kingdon, 9-Oct-2021.) |
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 𝐴 ≤ 𝐵) → (⌊‘𝐵) ∈ (ℤ_{≥}‘(⌊‘𝐴))) | ||
Theorem | flqbi 9628 | A condition equivalent to floor. (Contributed by Jim Kingdon, 9-Oct-2021.) |
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ) → ((⌊‘𝐴) = 𝐵 ↔ (𝐵 ≤ 𝐴 ∧ 𝐴 < (𝐵 + 1)))) | ||
Theorem | flqbi2 9629 | A condition equivalent to floor. (Contributed by Jim Kingdon, 9-Oct-2021.) |
⊢ ((𝑁 ∈ ℤ ∧ 𝐹 ∈ ℚ) → ((⌊‘(𝑁 + 𝐹)) = 𝑁 ↔ (0 ≤ 𝐹 ∧ 𝐹 < 1))) | ||
Theorem | adddivflid 9630 | 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 ∈ ℕ_{0} ∧ 𝐶 ∈ ℕ) → (𝐵 < 𝐶 ↔ (⌊‘(𝐴 + (𝐵 / 𝐶))) = 𝐴)) | ||
Theorem | flqge0nn0 9631 | 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) ∈ ℕ_{0}) | ||
Theorem | flqge1nn 9632 | 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) | ||
Theorem | fldivnn0 9633 | 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 ∈ ℕ_{0} ∧ 𝐿 ∈ ℕ) → (⌊‘(𝐾 / 𝐿)) ∈ ℕ_{0}) | ||
Theorem | divfl0 9634 | The floor of a fraction is 0 iff the denominator is less than the numerator. (Contributed by AV, 8-Jul-2021.) |
⊢ ((𝐴 ∈ ℕ_{0} ∧ 𝐵 ∈ ℕ) → (𝐴 < 𝐵 ↔ (⌊‘(𝐴 / 𝐵)) = 0)) | ||
Theorem | flqaddz 9635 | 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)) | ||
Theorem | flqzadd 9636 | 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))) | ||
Theorem | flqmulnn0 9637 | Move a nonnegative integer in and out of a floor. (Contributed by Jim Kingdon, 10-Oct-2021.) |
⊢ ((𝑁 ∈ ℕ_{0} ∧ 𝐴 ∈ ℚ) → (𝑁 · (⌊‘𝐴)) ≤ (⌊‘(𝑁 · 𝐴))) | ||
Theorem | btwnzge0 9638 | 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) ∧ (𝑁 ≤ 𝐴 ∧ 𝐴 < (𝑁 + 1))) → (0 ≤ 𝐴 ↔ 0 ≤ 𝑁)) | ||
Theorem | 2tnp1ge0ge0 9639 | 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)) | ||
Theorem | flhalf 9640 | 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 | fldivnn0le 9641 | 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 ∈ ℕ_{0} ∧ 𝐿 ∈ ℕ) → (⌊‘(𝐾 / 𝐿)) ≤ (𝐾 / 𝐿)) | ||
Theorem | flltdivnn0lt 9642 | 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 ∈ ℕ_{0} ∧ 𝑁 ∈ ℕ_{0} ∧ 𝐿 ∈ ℕ) → (𝐾 < 𝑁 → (⌊‘(𝐾 / 𝐿)) < (𝑁 / 𝐿))) | ||
Theorem | fldiv4p1lem1div2 9643 | 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 | ceilqval 9644 | The value of the ceiling function. (Contributed by Jim Kingdon, 10-Oct-2021.) |
⊢ (𝐴 ∈ ℚ → (⌈‘𝐴) = -(⌊‘-𝐴)) | ||
Theorem | ceiqcl 9645 | The ceiling function returns an integer (closure law). (Contributed by Jim Kingdon, 11-Oct-2021.) |
⊢ (𝐴 ∈ ℚ → -(⌊‘-𝐴) ∈ ℤ) | ||
Theorem | ceilqcl 9646 | Closure of the ceiling function. (Contributed by Jim Kingdon, 11-Oct-2021.) |
⊢ (𝐴 ∈ ℚ → (⌈‘𝐴) ∈ ℤ) | ||
Theorem | ceiqge 9647 | 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)) | ||
Theorem | ceilqge 9648 | 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)) | ||
Theorem | ceiqm1l 9649 | 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) | ||
Theorem | ceilqm1lt 9650 | 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) | ||
Theorem | ceiqle 9651 | 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 ∧ 𝐴 ≤ 𝐵) → -(⌊‘-𝐴) ≤ 𝐵) | ||
Theorem | ceilqle 9652 | 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 ∧ 𝐴 ≤ 𝐵) → (⌈‘𝐴) ≤ 𝐵) | ||
Theorem | ceilid 9653 | An integer is its own ceiling. (Contributed by AV, 30-Nov-2018.) |
⊢ (𝐴 ∈ ℤ → (⌈‘𝐴) = 𝐴) | ||
Theorem | ceilqidz 9654 | A rational number equals its ceiling iff it is an integer. (Contributed by Jim Kingdon, 11-Oct-2021.) |
⊢ (𝐴 ∈ ℚ → (𝐴 ∈ ℤ ↔ (⌈‘𝐴) = 𝐴)) | ||
Theorem | flqleceil 9655 | 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)) | ||
Theorem | flqeqceilz 9656 | 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))) | ||
Theorem | intqfrac2 9657 | Decompose a real into integer and fractional parts. (Contributed by Jim Kingdon, 18-Oct-2021.) |
⊢ 𝑍 = (⌊‘𝐴) & ⊢ 𝐹 = (𝐴 − 𝑍) ⇒ ⊢ (𝐴 ∈ ℚ → (0 ≤ 𝐹 ∧ 𝐹 < 1 ∧ 𝐴 = (𝑍 + 𝐹))) | ||
Theorem | intfracq 9658 | Decompose a rational number, expressed as a ratio, into integer and fractional parts. The fractional part has a tighter bound than that of intqfrac2 9657. (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 | flqdiv 9659 | 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))) | ||
Syntax | cmo 9660 | Extend class notation with the modulo operation. |
class mod | ||
Definition | df-mod 9661* | Define the modulo (remainder) operation. See modqval 9662 for its value. For example, (5 mod 3) = 2 and (-7 mod 2) = 1. As with df-fl 9608 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))))) | ||
Theorem | modqval 9662 | 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 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 9611 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))))) | ||
Theorem | modqvalr 9663 | 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))) | ||
Theorem | modqcl 9664 | Closure law for the modulo operation. (Contributed by Jim Kingdon, 16-Oct-2021.) |
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → (𝐴 mod 𝐵) ∈ ℚ) | ||
Theorem | flqpmodeq 9665 | 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) | ||
Theorem | modqcld 9666 | Closure law for the modulo operation. (Contributed by Jim Kingdon, 16-Oct-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℚ) & ⊢ (𝜑 → 𝐵 ∈ ℚ) & ⊢ (𝜑 → 0 < 𝐵) ⇒ ⊢ (𝜑 → (𝐴 mod 𝐵) ∈ ℚ) | ||
Theorem | modq0 9667 | 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)) | ||
Theorem | mulqmod0 9668 | 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) | ||
Theorem | negqmod0 9669 | 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)) | ||
Theorem | modqge0 9670 | The modulo operation is nonnegative. (Contributed by Jim Kingdon, 18-Oct-2021.) |
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → 0 ≤ (𝐴 mod 𝐵)) | ||
Theorem | modqlt 9671 | 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) | ||
Theorem | modqelico 9672 | 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)) | ||
Theorem | modqdiffl 9673 | 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))) | ||
Theorem | modqdifz 9674 | 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) | ||
Theorem | modqfrac 9675 | 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))) | ||
Theorem | flqmod 9676 | The floor function expressed in terms of the modulo operation. (Contributed by Jim Kingdon, 18-Oct-2021.) |
⊢ (𝐴 ∈ ℚ → (⌊‘𝐴) = (𝐴 − (𝐴 mod 1))) | ||
Theorem | intqfrac 9677 | 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))) | ||
Theorem | zmod10 9678 | An integer modulo 1 is 0. (Contributed by Paul Chapman, 22-Jun-2011.) |
⊢ (𝑁 ∈ ℤ → (𝑁 mod 1) = 0) | ||
Theorem | zmod1congr 9679 | 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 | modqmulnn 9680 | 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))) | ||
Theorem | modqvalp1 9681 | 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)) | ||
Theorem | zmodcl 9682 | Closure law for the modulo operation restricted to integers. (Contributed by NM, 27-Nov-2008.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → (𝐴 mod 𝐵) ∈ ℕ_{0}) | ||
Theorem | zmodcld 9683 | 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) ∈ ℕ_{0}) | ||
Theorem | zmodfz 9684 | 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 9685 | 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 9686 | 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 | modqid 9687 | Identity law for modulo. (Contributed by Jim Kingdon, 21-Oct-2021.) |
⊢ (((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ (0 ≤ 𝐴 ∧ 𝐴 < 𝐵)) → (𝐴 mod 𝐵) = 𝐴) | ||
Theorem | modqid0 9688 | A positive real number modulo itself is 0. (Contributed by Jim Kingdon, 21-Oct-2021.) |
⊢ ((𝑁 ∈ ℚ ∧ 0 < 𝑁) → (𝑁 mod 𝑁) = 0) | ||
Theorem | modqid2 9689 | Identity law for modulo. (Contributed by Jim Kingdon, 21-Oct-2021.) |
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → ((𝐴 mod 𝐵) = 𝐴 ↔ (0 ≤ 𝐴 ∧ 𝐴 < 𝐵))) | ||
Theorem | zmodid2 9690 | 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 9691 | 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 9692 | Identity law for modulo restricted to integers. (Contributed by AV, 27-Oct-2018.) |
⊢ (𝑀 ∈ (0..^𝑁) → (𝑀 mod 𝑁) = 𝑀) | ||
Theorem | q0mod 9693 | 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) | ||
Theorem | q1mod 9694 | 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) | ||
Theorem | modqabs 9695 | Absorption law for modulo. (Contributed by Jim Kingdon, 21-Oct-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℚ) & ⊢ (𝜑 → 𝐵 ∈ ℚ) & ⊢ (𝜑 → 0 < 𝐵) & ⊢ (𝜑 → 𝐶 ∈ ℚ) & ⊢ (𝜑 → 𝐵 ≤ 𝐶) ⇒ ⊢ (𝜑 → ((𝐴 mod 𝐵) mod 𝐶) = (𝐴 mod 𝐵)) | ||
Theorem | modqabs2 9696 | 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)) | ||
Theorem | modqcyc 9697 | 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)) | ||
Theorem | modqcyc2 9698 | The modulo operation is periodic. (Contributed by Jim Kingdon, 21-Oct-2021.) |
⊢ (((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℤ) ∧ (𝐵 ∈ ℚ ∧ 0 < 𝐵)) → ((𝐴 − (𝐵 · 𝑁)) mod 𝐵) = (𝐴 mod 𝐵)) | ||
Theorem | modqadd1 9699 | Addition property of the modulo operation. (Contributed by Jim Kingdon, 22-Oct-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℚ) & ⊢ (𝜑 → 𝐵 ∈ ℚ) & ⊢ (𝜑 → 𝐶 ∈ ℚ) & ⊢ (𝜑 → 𝐷 ∈ ℚ) & ⊢ (𝜑 → 0 < 𝐷) & ⊢ (𝜑 → (𝐴 mod 𝐷) = (𝐵 mod 𝐷)) ⇒ ⊢ (𝜑 → ((𝐴 + 𝐶) mod 𝐷) = ((𝐵 + 𝐶) mod 𝐷)) | ||
Theorem | modqaddabs 9700 | Absorption law for modulo. (Contributed by Jim Kingdon, 22-Oct-2021.) |
⊢ (((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ (𝐶 ∈ ℚ ∧ 0 < 𝐶)) → (((𝐴 mod 𝐶) + (𝐵 mod 𝐶)) mod 𝐶) = ((𝐴 + 𝐵) mod 𝐶)) |
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