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Theorem List for Intuitionistic Logic Explorer - 9401-9500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremintqfrac2 9401 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 9402 Decompose a rational number, expressed as a ratio, into integer and fractional parts. The fractional part has a tighter bound than that of intqfrac2 9401. (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 9403 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 9404 Extend class notation with the modulo operation.
class mod
 
Definitiondf-mod 9405* Define the modulo (remainder) operation. See modqval 9406 for its value. For example, (5 mod 3) = 2 and (-7 mod 2) = 1. As with df-fl 9352 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 9406 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 9355 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 9407 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 9408 Closure law for the modulo operation. (Contributed by Jim Kingdon, 16-Oct-2021.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → (𝐴 mod 𝐵) ∈ ℚ)
 
Theoremflqpmodeq 9409 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 9410 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 9411 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 9412 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 9413 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 9414 The modulo operation is nonnegative. (Contributed by Jim Kingdon, 18-Oct-2021.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → 0 ≤ (𝐴 mod 𝐵))
 
Theoremmodqlt 9415 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 9416 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 9417 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 9418 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 9419 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 9420 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 9421 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 9422 An integer modulo 1 is 0. (Contributed by Paul Chapman, 22-Jun-2011.)
(𝑁 ∈ ℤ → (𝑁 mod 1) = 0)
 
Theoremzmod1congr 9423 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 9424 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)))
 
Theoremmodqvalp1 9425 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))
 
Theoremzmodcl 9426 Closure law for the modulo operation restricted to integers. (Contributed by NM, 27-Nov-2008.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → (𝐴 mod 𝐵) ∈ ℕ0)
 
Theoremzmodcld 9427 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 9428 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 9429 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 9430 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))
 
Theoremmodqid 9431 Identity law for modulo. (Contributed by Jim Kingdon, 21-Oct-2021.)
(((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ (0 ≤ 𝐴𝐴 < 𝐵)) → (𝐴 mod 𝐵) = 𝐴)
 
Theoremmodqid0 9432 A positive real number modulo itself is 0. (Contributed by Jim Kingdon, 21-Oct-2021.)
((𝑁 ∈ ℚ ∧ 0 < 𝑁) → (𝑁 mod 𝑁) = 0)
 
Theoremmodqid2 9433 Identity law for modulo. (Contributed by Jim Kingdon, 21-Oct-2021.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → ((𝐴 mod 𝐵) = 𝐴 ↔ (0 ≤ 𝐴𝐴 < 𝐵)))
 
Theoremzmodid2 9434 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 9435 Identity law for modulo restricted to integers. (Contributed by AV, 27-Oct-2018.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ) → ((𝑀 mod 𝑁) = 𝑀𝑀 ∈ (0..^𝑁)))
 
Theoremzmodidfzoimp 9436 Identity law for modulo restricted to integers. (Contributed by AV, 27-Oct-2018.)
(𝑀 ∈ (0..^𝑁) → (𝑀 mod 𝑁) = 𝑀)
 
Theoremq0mod 9437 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)
 
Theoremq1mod 9438 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)
 
Theoremmodqabs 9439 Absorption law for modulo. (Contributed by Jim Kingdon, 21-Oct-2021.)
(𝜑𝐴 ∈ ℚ)    &   (𝜑𝐵 ∈ ℚ)    &   (𝜑 → 0 < 𝐵)    &   (𝜑𝐶 ∈ ℚ)    &   (𝜑𝐵𝐶)       (𝜑 → ((𝐴 mod 𝐵) mod 𝐶) = (𝐴 mod 𝐵))
 
Theoremmodqabs2 9440 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))
 
Theoremmodqcyc 9441 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))
 
Theoremmodqcyc2 9442 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))
 
Theoremmodqadd1 9443 Addition property of the modulo operation. (Contributed by Jim Kingdon, 22-Oct-2021.)
(𝜑𝐴 ∈ ℚ)    &   (𝜑𝐵 ∈ ℚ)    &   (𝜑𝐶 ∈ ℚ)    &   (𝜑𝐷 ∈ ℚ)    &   (𝜑 → 0 < 𝐷)    &   (𝜑 → (𝐴 mod 𝐷) = (𝐵 mod 𝐷))       (𝜑 → ((𝐴 + 𝐶) mod 𝐷) = ((𝐵 + 𝐶) mod 𝐷))
 
Theoremmodqaddabs 9444 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))
 
Theoremmodqaddmod 9445 The sum of a number modulo a modulus and another number equals the sum of the two numbers modulo the same modulus. (Contributed by Jim Kingdon, 23-Oct-2021.)
(((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ (𝑀 ∈ ℚ ∧ 0 < 𝑀)) → (((𝐴 mod 𝑀) + 𝐵) mod 𝑀) = ((𝐴 + 𝐵) mod 𝑀))
 
Theoremmulqaddmodid 9446 The sum of a positive rational number less than an upper bound and the product of an integer and the upper bound is the positive rational number modulo the upper bound. (Contributed by Jim Kingdon, 23-Oct-2021.)
(((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℚ) ∧ (𝐴 ∈ ℚ ∧ 𝐴 ∈ (0[,)𝑀))) → (((𝑁 · 𝑀) + 𝐴) mod 𝑀) = 𝐴)
 
Theoremmulp1mod1 9447 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)
 
Theoremmodqmuladd 9448* Decomposition of an integer into a multiple of a modulus and a remainder. (Contributed by Jim Kingdon, 23-Oct-2021.)
(𝜑𝐴 ∈ ℤ)    &   (𝜑𝐵 ∈ ℚ)    &   (𝜑𝐵 ∈ (0[,)𝑀))    &   (𝜑𝑀 ∈ ℚ)    &   (𝜑 → 0 < 𝑀)       (𝜑 → ((𝐴 mod 𝑀) = 𝐵 ↔ ∃𝑘 ∈ ℤ 𝐴 = ((𝑘 · 𝑀) + 𝐵)))
 
Theoremmodqmuladdim 9449* Implication of a decomposition of an integer into a multiple of a modulus and a remainder. (Contributed by Jim Kingdon, 23-Oct-2021.)
((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℚ ∧ 0 < 𝑀) → ((𝐴 mod 𝑀) = 𝐵 → ∃𝑘 ∈ ℤ 𝐴 = ((𝑘 · 𝑀) + 𝐵)))
 
Theoremmodqmuladdnn0 9450* Implication of a decomposition of a nonnegative integer into a multiple of a modulus and a remainder. (Contributed by Jim Kingdon, 23-Oct-2021.)
((𝐴 ∈ ℕ0𝑀 ∈ ℚ ∧ 0 < 𝑀) → ((𝐴 mod 𝑀) = 𝐵 → ∃𝑘 ∈ ℕ0 𝐴 = ((𝑘 · 𝑀) + 𝐵)))
 
Theoremqnegmod 9451 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 Jim Kingdon, 24-Oct-2021.)
((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℚ ∧ 0 < 𝑁) → (-𝐴 mod 𝑁) = ((𝑁𝐴) mod 𝑁))
 
Theoremm1modnnsub1 9452 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 9453 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 9454 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 9455 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)
 
Theoremmodqadd2mod 9456 The sum of a number modulo a modulus and another number equals the sum of the two numbers modulo the modulus. (Contributed by Jim Kingdon, 24-Oct-2021.)
(((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ (𝑀 ∈ ℚ ∧ 0 < 𝑀)) → ((𝐵 + (𝐴 mod 𝑀)) mod 𝑀) = ((𝐵 + 𝐴) mod 𝑀))
 
Theoremmodqm1p1mod0 9457 If a number modulo a modulus equals the modulus decreased by 1, the first number increased by 1 modulo the modulus equals 0. (Contributed by Jim Kingdon, 24-Oct-2021.)
((𝐴 ∈ ℚ ∧ 𝑀 ∈ ℚ ∧ 0 < 𝑀) → ((𝐴 mod 𝑀) = (𝑀 − 1) → ((𝐴 + 1) mod 𝑀) = 0))
 
Theoremmodqltm1p1mod 9458 If a number modulo a modulus is less than the modulus decreased by 1, the first number increased by 1 modulo the modulus equals the first number modulo the modulus, increased by 1. (Contributed by Jim Kingdon, 24-Oct-2021.)
(((𝐴 ∈ ℚ ∧ (𝐴 mod 𝑀) < (𝑀 − 1)) ∧ (𝑀 ∈ ℚ ∧ 0 < 𝑀)) → ((𝐴 + 1) mod 𝑀) = ((𝐴 mod 𝑀) + 1))
 
Theoremmodqmul1 9459 Multiplication property of the modulo operation. Note that the multiplier 𝐶 must be an integer. (Contributed by Jim Kingdon, 24-Oct-2021.)
(𝜑𝐴 ∈ ℚ)    &   (𝜑𝐵 ∈ ℚ)    &   (𝜑𝐶 ∈ ℤ)    &   (𝜑𝐷 ∈ ℚ)    &   (𝜑 → 0 < 𝐷)    &   (𝜑 → (𝐴 mod 𝐷) = (𝐵 mod 𝐷))       (𝜑 → ((𝐴 · 𝐶) mod 𝐷) = ((𝐵 · 𝐶) mod 𝐷))
 
Theoremmodqmul12d 9460 Multiplication property of the modulo operation, see theorem 5.2(b) in [ApostolNT] p. 107. (Contributed by Jim Kingdon, 24-Oct-2021.)
(𝜑𝐴 ∈ ℤ)    &   (𝜑𝐵 ∈ ℤ)    &   (𝜑𝐶 ∈ ℤ)    &   (𝜑𝐷 ∈ ℤ)    &   (𝜑𝐸 ∈ ℚ)    &   (𝜑 → 0 < 𝐸)    &   (𝜑 → (𝐴 mod 𝐸) = (𝐵 mod 𝐸))    &   (𝜑 → (𝐶 mod 𝐸) = (𝐷 mod 𝐸))       (𝜑 → ((𝐴 · 𝐶) mod 𝐸) = ((𝐵 · 𝐷) mod 𝐸))
 
Theoremmodqnegd 9461 Negation property of the modulo operation. (Contributed by Jim Kingdon, 24-Oct-2021.)
(𝜑𝐴 ∈ ℚ)    &   (𝜑𝐵 ∈ ℚ)    &   (𝜑𝐶 ∈ ℚ)    &   (𝜑 → 0 < 𝐶)    &   (𝜑 → (𝐴 mod 𝐶) = (𝐵 mod 𝐶))       (𝜑 → (-𝐴 mod 𝐶) = (-𝐵 mod 𝐶))
 
Theoremmodqadd12d 9462 Additive property of the modulo operation. (Contributed by Jim Kingdon, 25-Oct-2021.)
(𝜑𝐴 ∈ ℚ)    &   (𝜑𝐵 ∈ ℚ)    &   (𝜑𝐶 ∈ ℚ)    &   (𝜑𝐷 ∈ ℚ)    &   (𝜑𝐸 ∈ ℚ)    &   (𝜑 → 0 < 𝐸)    &   (𝜑 → (𝐴 mod 𝐸) = (𝐵 mod 𝐸))    &   (𝜑 → (𝐶 mod 𝐸) = (𝐷 mod 𝐸))       (𝜑 → ((𝐴 + 𝐶) mod 𝐸) = ((𝐵 + 𝐷) mod 𝐸))
 
Theoremmodqsub12d 9463 Subtraction property of the modulo operation. (Contributed by Jim Kingdon, 25-Oct-2021.)
(𝜑𝐴 ∈ ℚ)    &   (𝜑𝐵 ∈ ℚ)    &   (𝜑𝐶 ∈ ℚ)    &   (𝜑𝐷 ∈ ℚ)    &   (𝜑𝐸 ∈ ℚ)    &   (𝜑 → 0 < 𝐸)    &   (𝜑 → (𝐴 mod 𝐸) = (𝐵 mod 𝐸))    &   (𝜑 → (𝐶 mod 𝐸) = (𝐷 mod 𝐸))       (𝜑 → ((𝐴𝐶) mod 𝐸) = ((𝐵𝐷) mod 𝐸))
 
Theoremmodqsubmod 9464 The difference of a number modulo a modulus and another number equals the difference of the two numbers modulo the modulus. (Contributed by Jim Kingdon, 25-Oct-2021.)
(((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ (𝑀 ∈ ℚ ∧ 0 < 𝑀)) → (((𝐴 mod 𝑀) − 𝐵) mod 𝑀) = ((𝐴𝐵) mod 𝑀))
 
Theoremmodqsubmodmod 9465 The difference of a number modulo a modulus and another number modulo the same modulus equals the difference of the two numbers modulo the modulus. (Contributed by Jim Kingdon, 25-Oct-2021.)
(((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ (𝑀 ∈ ℚ ∧ 0 < 𝑀)) → (((𝐴 mod 𝑀) − (𝐵 mod 𝑀)) mod 𝑀) = ((𝐴𝐵) mod 𝑀))
 
Theoremq2txmodxeq0 9466 Two times a positive number modulo the number is zero. (Contributed by Jim Kingdon, 25-Oct-2021.)
((𝑋 ∈ ℚ ∧ 0 < 𝑋) → ((2 · 𝑋) mod 𝑋) = 0)
 
Theoremq2submod 9467 If a number is between a modulus and twice the modulus, the first number modulo the modulus equals the first number minus the modulus. (Contributed by Jim Kingdon, 25-Oct-2021.)
(((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) ∧ (𝐵𝐴𝐴 < (2 · 𝐵))) → (𝐴 mod 𝐵) = (𝐴𝐵))
 
Theoremmodifeq2int 9468 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 9469 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 9470 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)))
 
Theoremmodqmulmod 9471 The product of a rational number modulo a modulus and an integer equals the product of the rational number and the integer modulo the modulus. (Contributed by Jim Kingdon, 25-Oct-2021.)
(((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ) ∧ (𝑀 ∈ ℚ ∧ 0 < 𝑀)) → (((𝐴 mod 𝑀) · 𝐵) mod 𝑀) = ((𝐴 · 𝐵) mod 𝑀))
 
Theoremmodqmulmodr 9472 The product of an integer and a rational number modulo a modulus equals the product of the integer and the rational number modulo the modulus. (Contributed by Jim Kingdon, 26-Oct-2021.)
(((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℚ) ∧ (𝑀 ∈ ℚ ∧ 0 < 𝑀)) → ((𝐴 · (𝐵 mod 𝑀)) mod 𝑀) = ((𝐴 · 𝐵) mod 𝑀))
 
Theoremmodqaddmulmod 9473 The sum of a rational number and the product of a second rational number modulo a modulus and an integer equals the sum of the rational number and the product of the other rational number and the integer modulo the modulus. (Contributed by Jim Kingdon, 26-Oct-2021.)
(((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 𝐶 ∈ ℤ) ∧ (𝑀 ∈ ℚ ∧ 0 < 𝑀)) → ((𝐴 + ((𝐵 mod 𝑀) · 𝐶)) mod 𝑀) = ((𝐴 + (𝐵 · 𝐶)) mod 𝑀))
 
Theoremmodqdi 9474 Distribute multiplication over a modulo operation. (Contributed by Jim Kingdon, 26-Oct-2021.)
(((𝐴 ∈ ℚ ∧ 0 < 𝐴) ∧ 𝐵 ∈ ℚ ∧ (𝐶 ∈ ℚ ∧ 0 < 𝐶)) → (𝐴 · (𝐵 mod 𝐶)) = ((𝐴 · 𝐵) mod (𝐴 · 𝐶)))
 
Theoremmodqsubdir 9475 Distribute the modulo operation over a subtraction. (Contributed by Jim Kingdon, 26-Oct-2021.)
(((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ (𝐶 ∈ ℚ ∧ 0 < 𝐶)) → ((𝐵 mod 𝐶) ≤ (𝐴 mod 𝐶) ↔ ((𝐴𝐵) mod 𝐶) = ((𝐴 mod 𝐶) − (𝐵 mod 𝐶))))
 
Theoremmodqeqmodmin 9476 A rational number equals the difference of the rational number and a modulus modulo the modulus. (Contributed by Jim Kingdon, 26-Oct-2021.)
((𝐴 ∈ ℚ ∧ 𝑀 ∈ ℚ ∧ 0 < 𝑀) → (𝐴 mod 𝑀) = ((𝐴𝑀) mod 𝑀))
 
Theoremmodfzo0difsn 9477* For a number within a half-open range of nonnegative integers with one excluded integer there is a positive integer so that the number is equal to the sum of the positive integer and the excluded integer modulo the upper bound of the range. (Contributed by AV, 19-Mar-2021.)
((𝐽 ∈ (0..^𝑁) ∧ 𝐾 ∈ ((0..^𝑁) ∖ {𝐽})) → ∃𝑖 ∈ (1..^𝑁)𝐾 = ((𝑖 + 𝐽) mod 𝑁))
 
Theoremmodsumfzodifsn 9478 The sum of a number within a half-open range of positive integers is an element of the corresponding open range of nonnegative integers with one excluded integer modulo the excluded integer. (Contributed by AV, 19-Mar-2021.)
((𝐽 ∈ (0..^𝑁) ∧ 𝐾 ∈ (1..^𝑁)) → ((𝐾 + 𝐽) mod 𝑁) ∈ ((0..^𝑁) ∖ {𝐽}))
 
Theoremmodlteq 9479 Two nonnegative integers less than the modulus are equal iff they are equal modulo the modulus. (Contributed by AV, 14-Mar-2021.)
((𝐼 ∈ (0..^𝑁) ∧ 𝐽 ∈ (0..^𝑁)) → ((𝐼 mod 𝑁) = (𝐽 mod 𝑁) ↔ 𝐼 = 𝐽))
 
Theoremaddmodlteq 9480 Two nonnegative integers less than the modulus are equal iff the sums of these integer with another integer are equal modulo the modulus. (Contributed by AV, 20-Mar-2021.)
((𝐼 ∈ (0..^𝑁) ∧ 𝐽 ∈ (0..^𝑁) ∧ 𝑆 ∈ ℤ) → (((𝐼 + 𝑆) mod 𝑁) = ((𝐽 + 𝑆) mod 𝑁) ↔ 𝐼 = 𝐽))
 
3.6.3  Miscellaneous theorems about integers
 
Theoremfrec2uz0d 9481* 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 9482* The value of 𝐺 (see frec2uz0d 9481) 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 9483* The value of 𝐺 (see frec2uz0d 9481) 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 9484* The value 𝐺 (see frec2uz0d 9481) 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 9485* Less-than relation for 𝐺 (see frec2uz0d 9481). (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 9486* The mapping 𝐺 (see frec2uz0d 9481) 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 9487* Range of 𝐺 (see frec2uz0d 9481). (Contributed by Jim Kingdon, 17-May-2020.)
(𝜑𝐶 ∈ ℤ)    &   𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)       (𝜑 → ran 𝐺 = (ℤ𝐶))
 
Theoremfrec2uzf1od 9488* 𝐺 (see frec2uz0d 9481) 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 9489* 𝐺 (see frec2uz0d 9481) 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 9490* 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, 28-Mar-2022.)
(𝜑𝐶 ∈ ℤ)    &   𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)    &   (𝜑𝐴𝑆)    &   ((𝜑 ∧ (𝑥 ∈ (ℤ𝐶) ∧ 𝑦𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)    &   𝑅 = frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)       ((𝜑𝐷 ∈ ω) → (𝑅𝐷) ∈ ((ℤ𝐶) × 𝑆))
 
Theoremfrec2uzrdg 9491* 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 9481 which describes 𝐺 and the index translation. (Contributed by Jim Kingdon, 24-May-2020.)
(𝜑𝐶 ∈ ℤ)    &   𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)    &   (𝜑𝐴𝑆)    &   ((𝜑 ∧ (𝑥 ∈ (ℤ𝐶) ∧ 𝑦𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)    &   𝑅 = frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)    &   (𝜑𝐵 ∈ ω)       (𝜑 → (𝑅𝐵) = ⟨(𝐺𝐵), (2nd ‘(𝑅𝐵))⟩)
 
Theoremfrecuzrdgrcl 9492* 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, 1-Apr-2022.)
(𝜑𝐶 ∈ ℤ)    &   𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)    &   (𝜑𝐴𝑆)    &   ((𝜑 ∧ (𝑥 ∈ (ℤ𝐶) ∧ 𝑦𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)    &   𝑅 = frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)       (𝜑𝑅:ω⟶((ℤ𝐶) × 𝑆))
 
Theoremfrecuzrdglem 9493* 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)    &   (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)
 
Theoremfrecuzrdgtcl 9494* The recursive definition generator on upper integers is a function. See comment in frec2uz0d 9481 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)    &   (𝜑𝐴𝑆)    &   ((𝜑 ∧ (𝑥 ∈ (ℤ𝐶) ∧ 𝑦𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)    &   𝑅 = frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)    &   (𝜑𝑇 = ran 𝑅)       (𝜑𝑇:(ℤ𝐶)⟶𝑆)
 
Theoremfrecuzrdg0 9495* Initial value of a recursive definition generator on upper integers. See comment in frec2uz0d 9481 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)    &   (𝜑𝐴𝑆)    &   ((𝜑 ∧ (𝑥 ∈ (ℤ𝐶) ∧ 𝑦𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)    &   𝑅 = frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)    &   (𝜑𝑇 = ran 𝑅)       (𝜑 → (𝑇𝐶) = 𝐴)
 
Theoremfrecuzrdgsuc 9496* Successor value of a recursive definition generator on upper integers. See comment in frec2uz0d 9481 for the description of 𝐺 as the mapping from ω to (ℤ𝐶). (Contributed by Jim Kingdon, 28-May-2020.)
(𝜑𝐶 ∈ ℤ)    &   𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)    &   (𝜑𝐴𝑆)    &   ((𝜑 ∧ (𝑥 ∈ (ℤ𝐶) ∧ 𝑦𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)    &   𝑅 = frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)    &   (𝜑𝑇 = ran 𝑅)       ((𝜑𝐵 ∈ (ℤ𝐶)) → (𝑇‘(𝐵 + 1)) = (𝐵𝐹(𝑇𝐵)))
 
Theoremfrecuzrdgrclt 9497* The function 𝑅 (used in the definition of the recursive definition generator on upper integers) yields ordered pairs of integers and elements of 𝑆. Similar to frecuzrdgrcl 9492 except that 𝑆 and 𝑇 need not be the same. (Contributed by Jim Kingdon, 22-Apr-2022.)
(𝜑𝐶 ∈ ℤ)    &   (𝜑𝐴𝑆)    &   (𝜑𝑆𝑇)    &   ((𝜑 ∧ (𝑥 ∈ (ℤ𝐶) ∧ 𝑦𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)    &   𝑅 = frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)       (𝜑𝑅:ω⟶((ℤ𝐶) × 𝑆))
 
Theoremfrecuzrdgg 9498* Lemma for other theorems involving the the recursive definition generator on upper integers. Evaluating 𝑅 at a natural number gives an ordered pair whose first element is the mapping of that natural number via 𝐺. (Contributed by Jim Kingdon, 23-Apr-2022.)
(𝜑𝐶 ∈ ℤ)    &   (𝜑𝐴𝑆)    &   (𝜑𝑆𝑇)    &   ((𝜑 ∧ (𝑥 ∈ (ℤ𝐶) ∧ 𝑦𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)    &   𝑅 = frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)    &   (𝜑𝑁 ∈ ω)    &   𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)       (𝜑 → (1st ‘(𝑅𝑁)) = (𝐺𝑁))
 
Theoremfrecuzrdgdomlem 9499* The domain of the result of the recursive definition generator on upper integers. (Contributed by Jim Kingdon, 24-Apr-2022.)
(𝜑𝐶 ∈ ℤ)    &   (𝜑𝐴𝑆)    &   (𝜑𝑆𝑇)    &   ((𝜑 ∧ (𝑥 ∈ (ℤ𝐶) ∧ 𝑦𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)    &   𝑅 = frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)    &   𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)       (𝜑 → dom ran 𝑅 = (ℤ𝐶))
 
Theoremfrecuzrdgdom 9500* The domain of the result of the recursive definition generator on upper integers. (Contributed by Jim Kingdon, 24-Apr-2022.)
(𝜑𝐶 ∈ ℤ)    &   (𝜑𝐴𝑆)    &   (𝜑𝑆𝑇)    &   ((𝜑 ∧ (𝑥 ∈ (ℤ𝐶) ∧ 𝑦𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)    &   𝑅 = frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)       (𝜑 → dom ran 𝑅 = (ℤ𝐶))
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