Theorem List for Intuitionistic Logic Explorer - 10101-10200 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
|
Theorem | modqcld 10101 |
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 10102 |
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 10103 |
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 10104 |
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 10105 |
The modulo operation is nonnegative. (Contributed by Jim Kingdon,
18-Oct-2021.)
|
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → 0 ≤ (𝐴 mod 𝐵)) |
|
Theorem | modqlt 10106 |
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 10107 |
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 10108 |
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 10109 |
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 10110 |
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 10111 |
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 10112 |
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 10113 |
An integer modulo 1 is 0. (Contributed by Paul Chapman, 22-Jun-2011.)
|
⊢ (𝑁 ∈ ℤ → (𝑁 mod 1) = 0) |
|
Theorem | zmod1congr 10114 |
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 10115 |
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 10116 |
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 10117 |
Closure law for the modulo operation restricted to integers. (Contributed
by NM, 27-Nov-2008.)
|
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → (𝐴 mod 𝐵) ∈
ℕ0) |
|
Theorem | zmodcld 10118 |
Closure law for the modulo operation restricted to integers.
(Contributed by Mario Carneiro, 28-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℤ) & ⊢ (𝜑 → 𝐵 ∈ ℕ)
⇒ ⊢ (𝜑 → (𝐴 mod 𝐵) ∈
ℕ0) |
|
Theorem | zmodfz 10119 |
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 10120 |
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 10121 |
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 10122 |
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 10123 |
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 10124 |
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 10125 |
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 10126 |
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 10127 |
Identity law for modulo restricted to integers. (Contributed by AV,
27-Oct-2018.)
|
⊢ (𝑀 ∈ (0..^𝑁) → (𝑀 mod 𝑁) = 𝑀) |
|
Theorem | q0mod 10128 |
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 10129 |
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 10130 |
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 10131 |
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 10132 |
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 10133 |
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 10134 |
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 10135 |
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)) |
|
Theorem | modqaddmod 10136 |
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)) |
|
Theorem | mulqaddmodid 10137 |
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) |
|
Theorem | mulp1mod1 10138 |
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) |
|
Theorem | modqmuladd 10139* |
Decomposition of an integer into a multiple of a modulus and a
remainder. (Contributed by Jim Kingdon, 23-Oct-2021.)
|
⊢ (𝜑 → 𝐴 ∈ ℤ) & ⊢ (𝜑 → 𝐵 ∈ ℚ) & ⊢ (𝜑 → 𝐵 ∈ (0[,)𝑀)) & ⊢ (𝜑 → 𝑀 ∈ ℚ) & ⊢ (𝜑 → 0 < 𝑀) ⇒ ⊢ (𝜑 → ((𝐴 mod 𝑀) = 𝐵 ↔ ∃𝑘 ∈ ℤ 𝐴 = ((𝑘 · 𝑀) + 𝐵))) |
|
Theorem | modqmuladdim 10140* |
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))) |
|
Theorem | modqmuladdnn0 10141* |
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 ∈ ℕ0 ∧ 𝑀 ∈ ℚ ∧ 0 <
𝑀) → ((𝐴 mod 𝑀) = 𝐵 → ∃𝑘 ∈ ℕ0 𝐴 = ((𝑘 · 𝑀) + 𝐵))) |
|
Theorem | qnegmod 10142 |
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) = ((𝑁 − 𝐴) mod 𝑁)) |
|
Theorem | m1modnnsub1 10143 |
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)) |
|
Theorem | m1modge3gt1 10144 |
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)) |
|
Theorem | addmodid 10145 |
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 ∈ ℕ0 ∧ 𝑀 ∈ ℕ ∧ 𝐴 < 𝑀) → ((𝑀 + 𝐴) mod 𝑀) = 𝐴) |
|
Theorem | addmodidr 10146 |
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 ∈ ℕ0 ∧ 𝑀 ∈ ℕ ∧ 𝐴 < 𝑀) → ((𝐴 + 𝑀) mod 𝑀) = 𝐴) |
|
Theorem | modqadd2mod 10147 |
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)) |
|
Theorem | modqm1p1mod0 10148 |
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)) |
|
Theorem | modqltm1p1mod 10149 |
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)) |
|
Theorem | modqmul1 10150 |
Multiplication property of the modulo operation. Note that the
multiplier 𝐶 must be an integer. (Contributed by
Jim Kingdon,
24-Oct-2021.)
|
⊢ (𝜑 → 𝐴 ∈ ℚ) & ⊢ (𝜑 → 𝐵 ∈ ℚ) & ⊢ (𝜑 → 𝐶 ∈ ℤ) & ⊢ (𝜑 → 𝐷 ∈ ℚ) & ⊢ (𝜑 → 0 < 𝐷)
& ⊢ (𝜑 → (𝐴 mod 𝐷) = (𝐵 mod 𝐷)) ⇒ ⊢ (𝜑 → ((𝐴 · 𝐶) mod 𝐷) = ((𝐵 · 𝐶) mod 𝐷)) |
|
Theorem | modqmul12d 10151 |
Multiplication property of the modulo operation, see theorem 5.2(b) in
[ApostolNT] p. 107. (Contributed by
Jim Kingdon, 24-Oct-2021.)
|
⊢ (𝜑 → 𝐴 ∈ ℤ) & ⊢ (𝜑 → 𝐵 ∈ ℤ) & ⊢ (𝜑 → 𝐶 ∈ ℤ) & ⊢ (𝜑 → 𝐷 ∈ ℤ) & ⊢ (𝜑 → 𝐸 ∈ 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)) |
|
Theorem | modqnegd 10152 |
Negation property of the modulo operation. (Contributed by Jim Kingdon,
24-Oct-2021.)
|
⊢ (𝜑 → 𝐴 ∈ ℚ) & ⊢ (𝜑 → 𝐵 ∈ ℚ) & ⊢ (𝜑 → 𝐶 ∈ ℚ) & ⊢ (𝜑 → 0 < 𝐶)
& ⊢ (𝜑 → (𝐴 mod 𝐶) = (𝐵 mod 𝐶)) ⇒ ⊢ (𝜑 → (-𝐴 mod 𝐶) = (-𝐵 mod 𝐶)) |
|
Theorem | modqadd12d 10153 |
Additive property of the modulo operation. (Contributed by Jim Kingdon,
25-Oct-2021.)
|
⊢ (𝜑 → 𝐴 ∈ ℚ) & ⊢ (𝜑 → 𝐵 ∈ ℚ) & ⊢ (𝜑 → 𝐶 ∈ ℚ) & ⊢ (𝜑 → 𝐷 ∈ ℚ) & ⊢ (𝜑 → 𝐸 ∈ 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)) |
|
Theorem | modqsub12d 10154 |
Subtraction property of the modulo operation. (Contributed by Jim
Kingdon, 25-Oct-2021.)
|
⊢ (𝜑 → 𝐴 ∈ ℚ) & ⊢ (𝜑 → 𝐵 ∈ ℚ) & ⊢ (𝜑 → 𝐶 ∈ ℚ) & ⊢ (𝜑 → 𝐷 ∈ ℚ) & ⊢ (𝜑 → 𝐸 ∈ 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)) |
|
Theorem | modqsubmod 10155 |
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) = ((𝐴 − 𝐵) mod 𝑀)) |
|
Theorem | modqsubmodmod 10156 |
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) = ((𝐴 − 𝐵) mod 𝑀)) |
|
Theorem | q2txmodxeq0 10157 |
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) |
|
Theorem | q2submod 10158 |
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) ∧ (𝐵 ≤ 𝐴 ∧ 𝐴 < (2 · 𝐵))) → (𝐴 mod 𝐵) = (𝐴 − 𝐵)) |
|
Theorem | modifeq2int 10159 |
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 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ∧ 𝐴 < (2 · 𝐵)) → (𝐴 mod 𝐵) = if(𝐴 < 𝐵, 𝐴, (𝐴 − 𝐵))) |
|
Theorem | modaddmodup 10160 |
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))) |
|
Theorem | modaddmodlo 10161 |
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))) |
|
Theorem | modqmulmod 10162 |
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)) |
|
Theorem | modqmulmodr 10163 |
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)) |
|
Theorem | modqaddmulmod 10164 |
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)) |
|
Theorem | modqdi 10165 |
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))) |
|
Theorem | modqsubdir 10166 |
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) ↔ ((𝐴 − 𝐵) mod 𝐶) = ((𝐴 mod 𝐶) − (𝐵 mod 𝐶)))) |
|
Theorem | modqeqmodmin 10167 |
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) = ((𝐴 − 𝑀) mod 𝑀)) |
|
Theorem | modfzo0difsn 10168* |
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)) |
|
Theorem | modsumfzodifsn 10169 |
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})) |
|
Theorem | modlteq 10170 |
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)) |
|
Theorem | addmodlteq 10171 |
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
|
|
Theorem | frec2uz0d 10172* |
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 ℕ0 or 1 for the upper integers
ℕ), 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.)
|
⊢ (𝜑 → 𝐶 ∈ ℤ) & ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶) ⇒ ⊢ (𝜑 → (𝐺‘∅) = 𝐶) |
|
Theorem | frec2uzzd 10173* |
The value of 𝐺 (see frec2uz0d 10172) is an integer. (Contributed by
Jim Kingdon, 16-May-2020.)
|
⊢ (𝜑 → 𝐶 ∈ ℤ) & ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)
& ⊢ (𝜑 → 𝐴 ∈ ω)
⇒ ⊢ (𝜑 → (𝐺‘𝐴) ∈ ℤ) |
|
Theorem | frec2uzsucd 10174* |
The value of 𝐺 (see frec2uz0d 10172) at a successor. (Contributed by
Jim Kingdon, 16-May-2020.)
|
⊢ (𝜑 → 𝐶 ∈ ℤ) & ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)
& ⊢ (𝜑 → 𝐴 ∈ ω)
⇒ ⊢ (𝜑 → (𝐺‘suc 𝐴) = ((𝐺‘𝐴) + 1)) |
|
Theorem | frec2uzuzd 10175* |
The value 𝐺 (see frec2uz0d 10172) at an ordinal natural number is in
the upper integers. (Contributed by Jim Kingdon, 16-May-2020.)
|
⊢ (𝜑 → 𝐶 ∈ ℤ) & ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)
& ⊢ (𝜑 → 𝐴 ∈ ω)
⇒ ⊢ (𝜑 → (𝐺‘𝐴) ∈
(ℤ≥‘𝐶)) |
|
Theorem | frec2uzltd 10176* |
Less-than relation for 𝐺 (see frec2uz0d 10172). (Contributed by Jim
Kingdon, 16-May-2020.)
|
⊢ (𝜑 → 𝐶 ∈ ℤ) & ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)
& ⊢ (𝜑 → 𝐴 ∈ ω) & ⊢ (𝜑 → 𝐵 ∈ ω)
⇒ ⊢ (𝜑 → (𝐴 ∈ 𝐵 → (𝐺‘𝐴) < (𝐺‘𝐵))) |
|
Theorem | frec2uzlt2d 10177* |
The mapping 𝐺 (see frec2uz0d 10172) preserves order. (Contributed by
Jim Kingdon, 16-May-2020.)
|
⊢ (𝜑 → 𝐶 ∈ ℤ) & ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)
& ⊢ (𝜑 → 𝐴 ∈ ω) & ⊢ (𝜑 → 𝐵 ∈ ω)
⇒ ⊢ (𝜑 → (𝐴 ∈ 𝐵 ↔ (𝐺‘𝐴) < (𝐺‘𝐵))) |
|
Theorem | frec2uzrand 10178* |
Range of 𝐺 (see frec2uz0d 10172). (Contributed by Jim Kingdon,
17-May-2020.)
|
⊢ (𝜑 → 𝐶 ∈ ℤ) & ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶) ⇒ ⊢ (𝜑 → ran 𝐺 = (ℤ≥‘𝐶)) |
|
Theorem | frec2uzf1od 10179* |
𝐺
(see frec2uz0d 10172) is a one-to-one onto mapping. (Contributed
by Jim Kingdon, 17-May-2020.)
|
⊢ (𝜑 → 𝐶 ∈ ℤ) & ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶) ⇒ ⊢ (𝜑 → 𝐺:ω–1-1-onto→(ℤ≥‘𝐶)) |
|
Theorem | frec2uzisod 10180* |
𝐺
(see frec2uz0d 10172) is an isomorphism from natural ordinals to
upper integers. (Contributed by Jim Kingdon, 17-May-2020.)
|
⊢ (𝜑 → 𝐶 ∈ ℤ) & ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶) ⇒ ⊢ (𝜑 → 𝐺 Isom E , < (ω,
(ℤ≥‘𝐶))) |
|
Theorem | frecuzrdgrrn 10181* |
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.)
|
⊢ (𝜑 → 𝐶 ∈ ℤ) & ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)
& ⊢ (𝜑 → 𝐴 ∈ 𝑆)
& ⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
& ⊢ 𝑅 = frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉), 〈𝐶, 𝐴〉) ⇒ ⊢ ((𝜑 ∧ 𝐷 ∈ ω) → (𝑅‘𝐷) ∈
((ℤ≥‘𝐶) × 𝑆)) |
|
Theorem | frec2uzrdg 10182* |
A helper lemma for the value of a recursive definition generator on
upper integers (typically either ℕ or
ℕ0) 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 (ℤ≥‘𝐶)). See comment in frec2uz0d 10172
which describes 𝐺 and the index translation.
(Contributed by
Jim Kingdon, 24-May-2020.)
|
⊢ (𝜑 → 𝐶 ∈ ℤ) & ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)
& ⊢ (𝜑 → 𝐴 ∈ 𝑆)
& ⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
& ⊢ 𝑅 = frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉), 〈𝐶, 𝐴〉) & ⊢ (𝜑 → 𝐵 ∈ ω)
⇒ ⊢ (𝜑 → (𝑅‘𝐵) = 〈(𝐺‘𝐵), (2nd ‘(𝑅‘𝐵))〉) |
|
Theorem | frecuzrdgrcl 10183* |
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.)
|
⊢ (𝜑 → 𝐶 ∈ ℤ) & ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)
& ⊢ (𝜑 → 𝐴 ∈ 𝑆)
& ⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
& ⊢ 𝑅 = frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉), 〈𝐶, 𝐴〉) ⇒ ⊢ (𝜑 → 𝑅:ω⟶((ℤ≥‘𝐶) × 𝑆)) |
|
Theorem | frecuzrdglem 10184* |
A helper lemma for the value of a recursive definition generator on
upper integers. (Contributed by Jim Kingdon, 26-May-2020.)
|
⊢ (𝜑 → 𝐶 ∈ ℤ) & ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)
& ⊢ (𝜑 → 𝐴 ∈ 𝑆)
& ⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
& ⊢ 𝑅 = frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉), 〈𝐶, 𝐴〉) & ⊢ (𝜑 → 𝐵 ∈ (ℤ≥‘𝐶))
⇒ ⊢ (𝜑 → 〈𝐵, (2nd ‘(𝑅‘(◡𝐺‘𝐵)))〉 ∈ ran 𝑅) |
|
Theorem | frecuzrdgtcl 10185* |
The recursive definition generator on upper integers is a function.
See comment in frec2uz0d 10172 for the description of 𝐺 as the
mapping from ω to (ℤ≥‘𝐶). (Contributed by Jim
Kingdon, 26-May-2020.)
|
⊢ (𝜑 → 𝐶 ∈ ℤ) & ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)
& ⊢ (𝜑 → 𝐴 ∈ 𝑆)
& ⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
& ⊢ 𝑅 = frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉), 〈𝐶, 𝐴〉) & ⊢ (𝜑 → 𝑇 = ran 𝑅) ⇒ ⊢ (𝜑 → 𝑇:(ℤ≥‘𝐶)⟶𝑆) |
|
Theorem | frecuzrdg0 10186* |
Initial value of a recursive definition generator on upper integers.
See comment in frec2uz0d 10172 for the description of 𝐺 as the
mapping from ω to (ℤ≥‘𝐶). (Contributed by Jim
Kingdon, 27-May-2020.)
|
⊢ (𝜑 → 𝐶 ∈ ℤ) & ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)
& ⊢ (𝜑 → 𝐴 ∈ 𝑆)
& ⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
& ⊢ 𝑅 = frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉), 〈𝐶, 𝐴〉) & ⊢ (𝜑 → 𝑇 = ran 𝑅) ⇒ ⊢ (𝜑 → (𝑇‘𝐶) = 𝐴) |
|
Theorem | frecuzrdgsuc 10187* |
Successor value of a recursive definition generator on upper
integers. See comment in frec2uz0d 10172 for the description of 𝐺
as the mapping from ω to (ℤ≥‘𝐶). (Contributed
by Jim Kingdon, 28-May-2020.)
|
⊢ (𝜑 → 𝐶 ∈ ℤ) & ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)
& ⊢ (𝜑 → 𝐴 ∈ 𝑆)
& ⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
& ⊢ 𝑅 = frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉), 〈𝐶, 𝐴〉) & ⊢ (𝜑 → 𝑇 = ran 𝑅) ⇒ ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (𝑇‘(𝐵 + 1)) = (𝐵𝐹(𝑇‘𝐵))) |
|
Theorem | frecuzrdgrclt 10188* |
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 10183 except that 𝑆 and
𝑇 need not be the same. (Contributed
by Jim Kingdon,
22-Apr-2022.)
|
⊢ (𝜑 → 𝐶 ∈ ℤ) & ⊢ (𝜑 → 𝐴 ∈ 𝑆)
& ⊢ (𝜑 → 𝑆 ⊆ 𝑇)
& ⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
& ⊢ 𝑅 = frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉), 〈𝐶, 𝐴〉) ⇒ ⊢ (𝜑 → 𝑅:ω⟶((ℤ≥‘𝐶) × 𝑆)) |
|
Theorem | frecuzrdgg 10189* |
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.)
|
⊢ (𝜑 → 𝐶 ∈ ℤ) & ⊢ (𝜑 → 𝐴 ∈ 𝑆)
& ⊢ (𝜑 → 𝑆 ⊆ 𝑇)
& ⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
& ⊢ 𝑅 = frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉), 〈𝐶, 𝐴〉) & ⊢ (𝜑 → 𝑁 ∈ ω) & ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶) ⇒ ⊢ (𝜑 → (1st ‘(𝑅‘𝑁)) = (𝐺‘𝑁)) |
|
Theorem | frecuzrdgdomlem 10190* |
The domain of the result of the recursive definition generator on
upper integers. (Contributed by Jim Kingdon, 24-Apr-2022.)
|
⊢ (𝜑 → 𝐶 ∈ ℤ) & ⊢ (𝜑 → 𝐴 ∈ 𝑆)
& ⊢ (𝜑 → 𝑆 ⊆ 𝑇)
& ⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
& ⊢ 𝑅 = frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉), 〈𝐶, 𝐴〉) & ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶) ⇒ ⊢ (𝜑 → dom ran 𝑅 = (ℤ≥‘𝐶)) |
|
Theorem | frecuzrdgdom 10191* |
The domain of the result of the recursive definition generator on
upper integers. (Contributed by Jim Kingdon, 24-Apr-2022.)
|
⊢ (𝜑 → 𝐶 ∈ ℤ) & ⊢ (𝜑 → 𝐴 ∈ 𝑆)
& ⊢ (𝜑 → 𝑆 ⊆ 𝑇)
& ⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
& ⊢ 𝑅 = frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉), 〈𝐶, 𝐴〉) ⇒ ⊢ (𝜑 → dom ran 𝑅 = (ℤ≥‘𝐶)) |
|
Theorem | frecuzrdgfunlem 10192* |
The recursive definition generator on upper integers produces a a
function. (Contributed by Jim Kingdon, 24-Apr-2022.)
|
⊢ (𝜑 → 𝐶 ∈ ℤ) & ⊢ (𝜑 → 𝐴 ∈ 𝑆)
& ⊢ (𝜑 → 𝑆 ⊆ 𝑇)
& ⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
& ⊢ 𝑅 = frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉), 〈𝐶, 𝐴〉) & ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶) ⇒ ⊢ (𝜑 → Fun ran 𝑅) |
|
Theorem | frecuzrdgfun 10193* |
The recursive definition generator on upper integers produces a a
function. (Contributed by Jim Kingdon, 24-Apr-2022.)
|
⊢ (𝜑 → 𝐶 ∈ ℤ) & ⊢ (𝜑 → 𝐴 ∈ 𝑆)
& ⊢ (𝜑 → 𝑆 ⊆ 𝑇)
& ⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
& ⊢ 𝑅 = frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉), 〈𝐶, 𝐴〉) ⇒ ⊢ (𝜑 → Fun ran 𝑅) |
|
Theorem | frecuzrdgtclt 10194* |
The recursive definition generator on upper integers is a function.
(Contributed by Jim Kingdon, 22-Apr-2022.)
|
⊢ (𝜑 → 𝐶 ∈ ℤ) & ⊢ (𝜑 → 𝐴 ∈ 𝑆)
& ⊢ (𝜑 → 𝑆 ⊆ 𝑇)
& ⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
& ⊢ 𝑅 = frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉), 〈𝐶, 𝐴〉) & ⊢ (𝜑 → 𝑃 = ran 𝑅) ⇒ ⊢ (𝜑 → 𝑃:(ℤ≥‘𝐶)⟶𝑆) |
|
Theorem | frecuzrdg0t 10195* |
Initial value of a recursive definition generator on upper integers.
(Contributed by Jim Kingdon, 28-Apr-2022.)
|
⊢ (𝜑 → 𝐶 ∈ ℤ) & ⊢ (𝜑 → 𝐴 ∈ 𝑆)
& ⊢ (𝜑 → 𝑆 ⊆ 𝑇)
& ⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
& ⊢ 𝑅 = frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉), 〈𝐶, 𝐴〉) & ⊢ (𝜑 → 𝑃 = ran 𝑅) ⇒ ⊢ (𝜑 → (𝑃‘𝐶) = 𝐴) |
|
Theorem | frecuzrdgsuctlem 10196* |
Successor value of a recursive definition generator on upper integers.
See comment in frec2uz0d 10172 for the description of 𝐺 as the
mapping
from ω to (ℤ≥‘𝐶). (Contributed by Jim Kingdon,
29-Apr-2022.)
|
⊢ (𝜑 → 𝐶 ∈ ℤ) & ⊢ (𝜑 → 𝐴 ∈ 𝑆)
& ⊢ (𝜑 → 𝑆 ⊆ 𝑇)
& ⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
& ⊢ 𝑅 = frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉), 〈𝐶, 𝐴〉) & ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)
& ⊢ (𝜑 → 𝑃 = ran 𝑅) ⇒ ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (𝑃‘(𝐵 + 1)) = (𝐵𝐹(𝑃‘𝐵))) |
|
Theorem | frecuzrdgsuct 10197* |
Successor value of a recursive definition generator on upper integers.
(Contributed by Jim Kingdon, 29-Apr-2022.)
|
⊢ (𝜑 → 𝐶 ∈ ℤ) & ⊢ (𝜑 → 𝐴 ∈ 𝑆)
& ⊢ (𝜑 → 𝑆 ⊆ 𝑇)
& ⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
& ⊢ 𝑅 = frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉), 〈𝐶, 𝐴〉) & ⊢ (𝜑 → 𝑃 = ran 𝑅) ⇒ ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (𝑃‘(𝐵 + 1)) = (𝐵𝐹(𝑃‘𝐵))) |
|
Theorem | uzenom 10198 |
An upper integer set is denumerable. (Contributed by Mario Carneiro,
15-Oct-2015.)
|
⊢ 𝑍 = (ℤ≥‘𝑀)
⇒ ⊢ (𝑀 ∈ ℤ → 𝑍 ≈ ω) |
|
Theorem | frecfzennn 10199 |
The cardinality of a finite set of sequential integers. (See
frec2uz0d 10172 for a description of the hypothesis.)
(Contributed by Jim
Kingdon, 18-May-2020.)
|
⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ⇒ ⊢ (𝑁 ∈ ℕ0 →
(1...𝑁) ≈ (◡𝐺‘𝑁)) |
|
Theorem | frecfzen2 10200 |
The cardinality of a finite set of sequential integers with arbitrary
endpoints. (Contributed by Jim Kingdon, 18-May-2020.)
|
⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ⇒ ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑀...𝑁) ≈ (◡𝐺‘((𝑁 + 1) − 𝑀))) |