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Theorem List for Intuitionistic Logic Explorer - 9101-9200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremqltnle 9101 'Less than' expressed in terms of 'less than or equal to'. (Contributed by Jim Kingdon, 8-Oct-2021.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 < 𝐵 ↔ ¬ 𝐵𝐴))

Theoremqdceq 9102 Equality of rationals is decidable. (Contributed by Jim Kingdon, 11-Oct-2021.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → DECID 𝐴 = 𝐵)

Theoremqbtwnzlemstep 9103* Lemma for qbtwnz 9106. Induction step. (Contributed by Jim Kingdon, 8-Oct-2021.)
((𝐾 ∈ ℕ ∧ 𝐴 ∈ ℚ ∧ ∃𝑚 ∈ ℤ (𝑚𝐴𝐴 < (𝑚 + (𝐾 + 1)))) → ∃𝑚 ∈ ℤ (𝑚𝐴𝐴 < (𝑚 + 𝐾)))

Theoremqbtwnzlemshrink 9104* Lemma for qbtwnz 9106. Shrinking the range around the given rational number. (Contributed by Jim Kingdon, 8-Oct-2021.)
((𝐴 ∈ ℚ ∧ 𝐽 ∈ ℕ ∧ ∃𝑚 ∈ ℤ (𝑚𝐴𝐴 < (𝑚 + 𝐽))) → ∃𝑥 ∈ ℤ (𝑥𝐴𝐴 < (𝑥 + 1)))

Theoremqbtwnzlemex 9105* Lemma for qbtwnz 9106. Existence of the integer.

The proof starts by finding two integers which are less than and greater than the given rational number. Then this range can be shrunk by choosing an integer in between the endpoints of the range and then deciding which half of the range to keep based on rational number trichotomy, and iterating until the range consists of two consecutive integers. (Contributed by Jim Kingdon, 8-Oct-2021.)

(𝐴 ∈ ℚ → ∃𝑥 ∈ ℤ (𝑥𝐴𝐴 < (𝑥 + 1)))

Theoremqbtwnz 9106* There is a unique greatest integer less than or equal to a rational number. (Contributed by Jim Kingdon, 8-Oct-2021.)
(𝐴 ∈ ℚ → ∃!𝑥 ∈ ℤ (𝑥𝐴𝐴 < (𝑥 + 1)))

Theoremrebtwn2zlemstep 9107* Lemma for rebtwn2z 9109. Induction step. (Contributed by Jim Kingdon, 13-Oct-2021.)
((𝐾 ∈ (ℤ‘2) ∧ 𝐴 ∈ ℝ ∧ ∃𝑚 ∈ ℤ (𝑚 < 𝐴𝐴 < (𝑚 + (𝐾 + 1)))) → ∃𝑚 ∈ ℤ (𝑚 < 𝐴𝐴 < (𝑚 + 𝐾)))

Theoremrebtwn2zlemshrink 9108* Lemma for rebtwn2z 9109. Shrinking the range around the given real number. (Contributed by Jim Kingdon, 13-Oct-2021.)
((𝐴 ∈ ℝ ∧ 𝐽 ∈ (ℤ‘2) ∧ ∃𝑚 ∈ ℤ (𝑚 < 𝐴𝐴 < (𝑚 + 𝐽))) → ∃𝑥 ∈ ℤ (𝑥 < 𝐴𝐴 < (𝑥 + 2)))

Theoremrebtwn2z 9109* A real number can be bounded by integers above and below which are two apart.

The proof starts by finding two integers which are less than and greater than the given real number. Then this range can be shrunk by choosing an integer in between the endpoints of the range and then deciding which half of the range to keep based on weak linearity, and iterating until the range consists of integers which are two apart. (Contributed by Jim Kingdon, 13-Oct-2021.)

(𝐴 ∈ ℝ → ∃𝑥 ∈ ℤ (𝑥 < 𝐴𝐴 < (𝑥 + 2)))

Theoremqbtwnrelemcalc 9110 Lemma for qbtwnre 9111. Calculations involved in showing the constructed rational number is less than 𝐵. (Contributed by Jim Kingdon, 14-Oct-2021.)
(𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝑀 < (𝐴 · (2 · 𝑁)))    &   (𝜑 → (1 / 𝑁) < (𝐵𝐴))       (𝜑 → ((𝑀 + 2) / (2 · 𝑁)) < 𝐵)

Theoremqbtwnre 9111* The rational numbers are dense in : any two real numbers have a rational between them. Exercise 6 of [Apostol] p. 28. (Contributed by NM, 18-Nov-2004.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → ∃𝑥 ∈ ℚ (𝐴 < 𝑥𝑥 < 𝐵))

3.6  Elementary integer functions

3.6.1  The floor and ceiling functions

Syntaxcfl 9112 Extend class notation with floor (greatest integer) function.
class

Syntaxcceil 9113 Extend class notation to include the ceiling function.
class

Definitiondf-fl 9114* Define the floor (greatest integer less than or equal to) function. See flval 9116 for its value, flqlelt 9118 for its basic property, and flqcl 9117 for its closure. For example, (⌊‘(3 / 2)) = 1 while (⌊‘-(3 / 2)) = -2 (ex-fl 9895).

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 beyond the rationals. 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 (𝑦𝑥𝑥 < (𝑦 + 1))))

Definitiondf-ceil 9115 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 9148 for its value, ceilqge 9152 and ceilqm1lt 9154 for its basic properties, and ceilqcl 9150 for its closure. For example, (⌈‘(3 / 2)) = 2 while (⌈‘-(3 / 2)) = -1 (ex-ceil 9896).

As described in df-fl 9114 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))

Theoremflval 9116* 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 (𝑥𝐴𝐴 < (𝑥 + 1))))

Theoremflqcl 9117 The floor (greatest integer) function yields an integer when applied to a rational (closure law). It would presumably be possible to prove a similar result for some real numbers (for example, those apart from any integer), but not real numbers in general. (Contributed by Jim Kingdon, 8-Oct-2021.)
(𝐴 ∈ ℚ → (⌊‘𝐴) ∈ ℤ)

Theoremflqlelt 9118 A basic property of the floor (greatest integer) function. (Contributed by Jim Kingdon, 8-Oct-2021.)
(𝐴 ∈ ℚ → ((⌊‘𝐴) ≤ 𝐴𝐴 < ((⌊‘𝐴) + 1)))

Theoremflqcld 9119 The floor (greatest integer) function is an integer (closure law). (Contributed by Jim Kingdon, 8-Oct-2021.)
(𝜑𝐴 ∈ ℚ)       (𝜑 → (⌊‘𝐴) ∈ ℤ)

Theoremflqle 9120 A basic property of the floor (greatest integer) function. (Contributed by Jim Kingdon, 8-Oct-2021.)
(𝐴 ∈ ℚ → (⌊‘𝐴) ≤ 𝐴)

Theoremflqltp1 9121 A basic property of the floor (greatest integer) function. (Contributed by Jim Kingdon, 8-Oct-2021.)
(𝐴 ∈ ℚ → 𝐴 < ((⌊‘𝐴) + 1))

Theoremqfraclt1 9122 The fractional part of a rational number is less than one. (Contributed by Jim Kingdon, 8-Oct-2021.)
(𝐴 ∈ ℚ → (𝐴 − (⌊‘𝐴)) < 1)

Theoremqfracge0 9123 The fractional part of a rational number is nonnegative. (Contributed by Jim Kingdon, 8-Oct-2021.)
(𝐴 ∈ ℚ → 0 ≤ (𝐴 − (⌊‘𝐴)))

Theoremflqge 9124 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) → (𝐵𝐴𝐵 ≤ (⌊‘𝐴)))

Theoremflqlt 9125 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))

Theoremflid 9126 An integer is its own floor. (Contributed by NM, 15-Nov-2004.)
(𝐴 ∈ ℤ → (⌊‘𝐴) = 𝐴)

Theoremflqidm 9127 The floor function is idempotent. (Contributed by Jim Kingdon, 8-Oct-2021.)
(𝐴 ∈ ℚ → (⌊‘(⌊‘𝐴)) = (⌊‘𝐴))

Theoremflqidz 9128 A rational number equals its floor iff it is an integer. (Contributed by Jim Kingdon, 9-Oct-2021.)
(𝐴 ∈ ℚ → ((⌊‘𝐴) = 𝐴𝐴 ∈ ℤ))

Theoremflqltnz 9129 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)

Theoremflqwordi 9130 Ordering relationship for the greatest integer function. (Contributed by Jim Kingdon, 9-Oct-2021.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 𝐴𝐵) → (⌊‘𝐴) ≤ (⌊‘𝐵))

Theoremflqword2 9131 Ordering relationship for the greatest integer function. (Contributed by Jim Kingdon, 9-Oct-2021.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 𝐴𝐵) → (⌊‘𝐵) ∈ (ℤ‘(⌊‘𝐴)))

Theoremflqbi 9132 A condition equivalent to floor. (Contributed by Jim Kingdon, 9-Oct-2021.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ) → ((⌊‘𝐴) = 𝐵 ↔ (𝐵𝐴𝐴 < (𝐵 + 1))))

Theoremflqbi2 9133 A condition equivalent to floor. (Contributed by Jim Kingdon, 9-Oct-2021.)
((𝑁 ∈ ℤ ∧ 𝐹 ∈ ℚ) → ((⌊‘(𝑁 + 𝐹)) = 𝑁 ↔ (0 ≤ 𝐹𝐹 < 1)))

Theoremadddivflid 9134 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))

Theoremflqge0nn0 9135 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) ∈ ℕ0)

Theoremflqge1nn 9136 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)

Theoremfldivnn0 9137 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)

Theoremdivfl0 9138 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))

Theoremflqaddz 9139 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))

Theoremflqzadd 9140 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)))

Theoremflqmulnn0 9141 Move a nonnegative integer in and out of a floor. (Contributed by Jim Kingdon, 10-Oct-2021.)
((𝑁 ∈ ℕ0𝐴 ∈ ℚ) → (𝑁 · (⌊‘𝐴)) ≤ (⌊‘(𝑁 · 𝐴)))

Theorembtwnzge0 9142 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) ∧ (𝑁𝐴𝐴 < (𝑁 + 1))) → (0 ≤ 𝐴 ↔ 0 ≤ 𝑁))

Theorem2tnp1ge0ge0 9143 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 9144 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))))

Theoremfldivnn0le 9145 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 9146 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)))

Theoremfldiv4p1lem1div2 9147 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))

Theoremceilqval 9148 The value of the ceiling function. (Contributed by Jim Kingdon, 10-Oct-2021.)
(𝐴 ∈ ℚ → (⌈‘𝐴) = -(⌊‘-𝐴))

Theoremceiqcl 9149 The ceiling function returns an integer (closure law). (Contributed by Jim Kingdon, 11-Oct-2021.)
(𝐴 ∈ ℚ → -(⌊‘-𝐴) ∈ ℤ)

Theoremceilqcl 9150 Closure of the ceiling function. (Contributed by Jim Kingdon, 11-Oct-2021.)
(𝐴 ∈ ℚ → (⌈‘𝐴) ∈ ℤ)

Theoremceiqge 9151 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))

Theoremceilqge 9152 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))

Theoremceiqm1l 9153 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)

Theoremceilqm1lt 9154 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)

Theoremceiqle 9155 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 ∧ 𝐴𝐵) → -(⌊‘-𝐴) ≤ 𝐵)

Theoremceilqle 9156 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 ∧ 𝐴𝐵) → (⌈‘𝐴) ≤ 𝐵)

Theoremceilid 9157 An integer is its own ceiling. (Contributed by AV, 30-Nov-2018.)
(𝐴 ∈ ℤ → (⌈‘𝐴) = 𝐴)

Theoremceilqidz 9158 A rational number equals its ceiling iff it is an integer. (Contributed by Jim Kingdon, 11-Oct-2021.)
(𝐴 ∈ ℚ → (𝐴 ∈ ℤ ↔ (⌈‘𝐴) = 𝐴))

Theoremflqleceil 9159 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))

Theoremflqeqceilz 9160 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)))

Theoremintqfrac2 9161 Decompose a real into integer and fractional parts. (Contributed by Jim Kingdon, 18-Oct-2021.)
𝑍 = (⌊‘𝐴)    &   𝐹 = (𝐴𝑍)       (𝐴 ∈ ℚ → (0 ≤ 𝐹𝐹 < 1 ∧ 𝐴 = (𝑍 + 𝐹)))

Theoremintfracq 9162 Decompose a rational number, expressed as a ratio, into integer and fractional parts. The fractional part has a tighter bound than that of intqfrac2 9161. (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)))

Theoremflqdiv 9163 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)))

3.6.2  The modulo (remainder) operation

Syntaxcmo 9164 Extend class notation with the modulo operation.
class mod

Definitiondf-mod 9165* Define the modulo (remainder) operation. See modqval 9166 for its value. For example, (5 mod 3) = 2 and (-7 mod 2) = 1. As with df-fl 9114 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)))))

Theoremmodqval 9166 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 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 9117 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)))))

Theoremmodqvalr 9167 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)))

Theoremmodqcl 9168 Closure law for the modulo operation. (Contributed by Jim Kingdon, 16-Oct-2021.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → (𝐴 mod 𝐵) ∈ ℚ)

Theoremflqpmodeq 9169 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)

Theoremmodqcld 9170 Closure law for the modulo operation. (Contributed by Jim Kingdon, 16-Oct-2021.)
(𝜑𝐴 ∈ ℚ)    &   (𝜑𝐵 ∈ ℚ)    &   (𝜑 → 0 < 𝐵)       (𝜑 → (𝐴 mod 𝐵) ∈ ℚ)

Theoremmodq0 9171 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))

Theoremmulqmod0 9172 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)

Theoremnegqmod0 9173 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))

Theoremmodqge0 9174 The modulo operation is nonnegative. (Contributed by Jim Kingdon, 18-Oct-2021.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → 0 ≤ (𝐴 mod 𝐵))

Theoremmodqlt 9175 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)

Theoremmodqelico 9176 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))

Theoremmodqdiffl 9177 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)))

Theoremmodqdifz 9178 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)

Theoremmodqfrac 9179 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)))

Theoremflqmod 9180 The floor function expressed in terms of the modulo operation. (Contributed by Jim Kingdon, 18-Oct-2021.)
(𝐴 ∈ ℚ → (⌊‘𝐴) = (𝐴 − (𝐴 mod 1)))

Theoremintqfrac 9181 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)))

Theoremzmod10 9182 An integer modulo 1 is 0. (Contributed by Paul Chapman, 22-Jun-2011.)
(𝑁 ∈ ℤ → (𝑁 mod 1) = 0)

Theoremzmod1congr 9183 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))

Theoremmodqmulnn 9184 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)))

Theoremfrec2uz0d 9185* The mapping 𝐺 is a one-to-one mapping from ω onto upper integers that will be used to construct a recursive definition generator. Ordinal natural number 0 maps to complex number 𝐶 (normally 0 for the upper integers 0 or 1 for the upper integers ), 1 maps to 𝐶 + 1, etc. This theorem shows the value of 𝐺 at ordinal natural number zero. (Contributed by Jim Kingdon, 16-May-2020.)
(𝜑𝐶 ∈ ℤ)    &   𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)       (𝜑 → (𝐺‘∅) = 𝐶)

Theoremfrec2uzzd 9186* The value of 𝐺 (see frec2uz0d 9185) is an integer. (Contributed by Jim Kingdon, 16-May-2020.)
(𝜑𝐶 ∈ ℤ)    &   𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)    &   (𝜑𝐴 ∈ ω)       (𝜑 → (𝐺𝐴) ∈ ℤ)

Theoremfrec2uzsucd 9187* The value of 𝐺 (see frec2uz0d 9185) at a successor. (Contributed by Jim Kingdon, 16-May-2020.)
(𝜑𝐶 ∈ ℤ)    &   𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)    &   (𝜑𝐴 ∈ ω)       (𝜑 → (𝐺‘suc 𝐴) = ((𝐺𝐴) + 1))

Theoremfrec2uzuzd 9188* The value 𝐺 (see frec2uz0d 9185) at an ordinal natural number is in the upper integers. (Contributed by Jim Kingdon, 16-May-2020.)
(𝜑𝐶 ∈ ℤ)    &   𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)    &   (𝜑𝐴 ∈ ω)       (𝜑 → (𝐺𝐴) ∈ (ℤ𝐶))

Theoremfrec2uzltd 9189* Less-than relation for 𝐺 (see frec2uz0d 9185). (Contributed by Jim Kingdon, 16-May-2020.)
(𝜑𝐶 ∈ ℤ)    &   𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)    &   (𝜑𝐴 ∈ ω)    &   (𝜑𝐵 ∈ ω)       (𝜑 → (𝐴𝐵 → (𝐺𝐴) < (𝐺𝐵)))

Theoremfrec2uzlt2d 9190* The mapping 𝐺 (see frec2uz0d 9185) preserves order. (Contributed by Jim Kingdon, 16-May-2020.)
(𝜑𝐶 ∈ ℤ)    &   𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)    &   (𝜑𝐴 ∈ ω)    &   (𝜑𝐵 ∈ ω)       (𝜑 → (𝐴𝐵 ↔ (𝐺𝐴) < (𝐺𝐵)))

Theoremfrec2uzrand 9191* Range of 𝐺 (see frec2uz0d 9185). (Contributed by Jim Kingdon, 17-May-2020.)
(𝜑𝐶 ∈ ℤ)    &   𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)       (𝜑 → ran 𝐺 = (ℤ𝐶))

Theoremfrec2uzf1od 9192* 𝐺 (see frec2uz0d 9185) is a one-to-one onto mapping. (Contributed by Jim Kingdon, 17-May-2020.)
(𝜑𝐶 ∈ ℤ)    &   𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)       (𝜑𝐺:ω–1-1-onto→(ℤ𝐶))

Theoremfrec2uzisod 9193* 𝐺 (see frec2uz0d 9185) is an isomorphism from natural ordinals to upper integers. (Contributed by Jim Kingdon, 17-May-2020.)
(𝜑𝐶 ∈ ℤ)    &   𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)       (𝜑𝐺 Isom E , < (ω, (ℤ𝐶)))

Theoremfrecuzrdgrrn 9194* The function 𝑅 (used in the definition of the recursive definition generator on upper integers) yields ordered pairs of integers and elements of 𝑆. (Contributed by Jim Kingdon, 27-May-2020.)
(𝜑𝐶 ∈ ℤ)    &   𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)    &   (𝜑𝑆𝑉)    &   (𝜑𝐴𝑆)    &   ((𝜑 ∧ (𝑥 ∈ (ℤ𝐶) ∧ 𝑦𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)    &   𝑅 = frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)       ((𝜑𝐷 ∈ ω) → (𝑅𝐷) ∈ ((ℤ𝐶) × 𝑆))

Theoremfrec2uzrdg 9195* A helper lemma for the value of a recursive definition generator on upper integers (typically either or 0) with characteristic function 𝐹(𝑥, 𝑦) and initial value 𝐴. This lemma shows that evaluating 𝑅 at an element of ω gives an ordered pair whose first element is the index (translated from ω to (ℤ𝐶)). See comment in frec2uz0d 9185 which describes 𝐺 and the index translation. (Contributed by Jim Kingdon, 24-May-2020.)
(𝜑𝐶 ∈ ℤ)    &   𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)    &   (𝜑𝑆𝑉)    &   (𝜑𝐴𝑆)    &   ((𝜑 ∧ (𝑥 ∈ (ℤ𝐶) ∧ 𝑦𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)    &   𝑅 = frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)    &   (𝜑𝐵 ∈ ω)       (𝜑 → (𝑅𝐵) = ⟨(𝐺𝐵), (2nd ‘(𝑅𝐵))⟩)

Theoremfrecuzrdgrom 9196* The function 𝑅 (used in the definition of the recursive definition generator on upper integers) is a function defined for all natural numbers. (Contributed by Jim Kingdon, 26-May-2020.)
(𝜑𝐶 ∈ ℤ)    &   𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)    &   (𝜑𝑆𝑉)    &   (𝜑𝐴𝑆)    &   ((𝜑 ∧ (𝑥 ∈ (ℤ𝐶) ∧ 𝑦𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)    &   𝑅 = frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)       (𝜑𝑅 Fn ω)

Theoremfrecuzrdglem 9197* A helper lemma for the value of a recursive definition generator on upper integers. (Contributed by Jim Kingdon, 26-May-2020.)
(𝜑𝐶 ∈ ℤ)    &   𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)    &   (𝜑𝑆𝑉)    &   (𝜑𝐴𝑆)    &   ((𝜑 ∧ (𝑥 ∈ (ℤ𝐶) ∧ 𝑦𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)    &   𝑅 = frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)    &   (𝜑𝐵 ∈ (ℤ𝐶))       (𝜑 → ⟨𝐵, (2nd ‘(𝑅‘(𝐺𝐵)))⟩ ∈ ran 𝑅)

Theoremfrecuzrdgfn 9198* The recursive definition generator on upper integers is a function. See comment in frec2uz0d 9185 for the description of 𝐺 as the mapping from ω to (ℤ𝐶). (Contributed by Jim Kingdon, 26-May-2020.)
(𝜑𝐶 ∈ ℤ)    &   𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)    &   (𝜑𝑆𝑉)    &   (𝜑𝐴𝑆)    &   ((𝜑 ∧ (𝑥 ∈ (ℤ𝐶) ∧ 𝑦𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)    &   𝑅 = frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)    &   (𝜑𝑇 = ran 𝑅)       (𝜑𝑇 Fn (ℤ𝐶))

Theoremfrecuzrdgcl 9199* Closure law for the recursive definition generator on upper integers. See comment in frec2uz0d 9185 for the description of 𝐺 as the mapping from ω to (ℤ𝐶). (Contributed by Jim Kingdon, 31-May-2020.)
(𝜑𝐶 ∈ ℤ)    &   𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)    &   (𝜑𝑆𝑉)    &   (𝜑𝐴𝑆)    &   ((𝜑 ∧ (𝑥 ∈ (ℤ𝐶) ∧ 𝑦𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)    &   𝑅 = frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)    &   (𝜑𝑇 = ran 𝑅)       ((𝜑𝐵 ∈ (ℤ𝐶)) → (𝑇𝐵) ∈ 𝑆)

Theoremfrecuzrdg0 9200* Initial value of a recursive definition generator on upper integers. See comment in frec2uz0d 9185 for the description of 𝐺 as the mapping from ω to (ℤ𝐶). (Contributed by Jim Kingdon, 27-May-2020.)
(𝜑𝐶 ∈ ℤ)    &   𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)    &   (𝜑𝑆𝑉)    &   (𝜑𝐴𝑆)    &   ((𝜑 ∧ (𝑥 ∈ (ℤ𝐶) ∧ 𝑦𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)    &   𝑅 = frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)    &   (𝜑𝑇 = ran 𝑅)       (𝜑 → (𝑇𝐶) = 𝐴)

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