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Theorem List for Intuitionistic Logic Explorer - 10101-10200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremceilqidz 10101 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 10102 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 10103 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 10104 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 10105 Decompose a rational number, expressed as a ratio, into integer and fractional parts. The fractional part has a tighter bound than that of intqfrac2 10104. (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 10106 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)))

4.6.2  The modulo (remainder) operation

Syntaxcmo 10107 Extend class notation with the modulo operation.
class mod

Definitiondf-mod 10108* Define the modulo (remainder) operation. See modqval 10109 for its value. For example, (5 mod 3) = 2 and (-7 mod 2) = 1. As with df-fl 10055 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 10109 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 10058 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 10110 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 10111 Closure law for the modulo operation. (Contributed by Jim Kingdon, 16-Oct-2021.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → (𝐴 mod 𝐵) ∈ ℚ)

Theoremflqpmodeq 10112 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 10113 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 10114 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 10115 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 10116 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 10117 The modulo operation is nonnegative. (Contributed by Jim Kingdon, 18-Oct-2021.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → 0 ≤ (𝐴 mod 𝐵))

Theoremmodqlt 10118 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 10119 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 10120 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 10121 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 10122 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 10123 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 10124 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 10125 An integer modulo 1 is 0. (Contributed by Paul Chapman, 22-Jun-2011.)
(𝑁 ∈ ℤ → (𝑁 mod 1) = 0)

Theoremzmod1congr 10126 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 10127 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 10128 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 10129 Closure law for the modulo operation restricted to integers. (Contributed by NM, 27-Nov-2008.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → (𝐴 mod 𝐵) ∈ ℕ0)

Theoremzmodcld 10130 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 10131 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 10132 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 10133 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 10134 Identity law for modulo. (Contributed by Jim Kingdon, 21-Oct-2021.)
(((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ (0 ≤ 𝐴𝐴 < 𝐵)) → (𝐴 mod 𝐵) = 𝐴)

Theoremmodqid0 10135 A positive real number modulo itself is 0. (Contributed by Jim Kingdon, 21-Oct-2021.)
((𝑁 ∈ ℚ ∧ 0 < 𝑁) → (𝑁 mod 𝑁) = 0)

Theoremmodqid2 10136 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 10137 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 10138 Identity law for modulo restricted to integers. (Contributed by AV, 27-Oct-2018.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ) → ((𝑀 mod 𝑁) = 𝑀𝑀 ∈ (0..^𝑁)))

Theoremzmodidfzoimp 10139 Identity law for modulo restricted to integers. (Contributed by AV, 27-Oct-2018.)
(𝑀 ∈ (0..^𝑁) → (𝑀 mod 𝑁) = 𝑀)

Theoremq0mod 10140 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 10141 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 10142 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 10143 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 10144 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 10145 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 10146 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 10147 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 10148 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 10149 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 10150 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 10151* 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 10152* 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 10153* 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 10154 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 10155 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 10156 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 10157 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 10158 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 10159 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 10160 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 10161 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 10162 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 10163 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 10164 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 10165 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 10166 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 10167 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 10168 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 10169 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 10170 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 10171 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 10172 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 10173 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 10174 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 10175 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 10176 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 10177 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 10178 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 10179 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 10180* 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 10181 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 10182 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 10183 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))

4.6.3  Miscellaneous theorems about integers

Theoremfrec2uz0d 10184* 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 10185* The value of 𝐺 (see frec2uz0d 10184) 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 10186* The value of 𝐺 (see frec2uz0d 10184) 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 10187* The value 𝐺 (see frec2uz0d 10184) 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 10188* Less-than relation for 𝐺 (see frec2uz0d 10184). (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 10189* The mapping 𝐺 (see frec2uz0d 10184) 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 10190* Range of 𝐺 (see frec2uz0d 10184). (Contributed by Jim Kingdon, 17-May-2020.)
(𝜑𝐶 ∈ ℤ)    &   𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)       (𝜑 → ran 𝐺 = (ℤ𝐶))

Theoremfrec2uzf1od 10191* 𝐺 (see frec2uz0d 10184) 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 10192* 𝐺 (see frec2uz0d 10184) 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 10193* 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 10194* 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 10184 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 10195* 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 10196* 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 10197* The recursive definition generator on upper integers is a function. See comment in frec2uz0d 10184 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 10198* Initial value of a recursive definition generator on upper integers. See comment in frec2uz0d 10184 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 10199* Successor value of a recursive definition generator on upper integers. See comment in frec2uz0d 10184 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 10200* 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 10195 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))

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