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Theorem List for Metamath Proof Explorer - 47201-47300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem0dig2nn0o 47201 The last bit of an odd integer is 1. (Contributed by AV, 3-Jun-2010.)
((𝑁 ∈ ℕ0 ∧ ((𝑁 + 1) / 2) ∈ ℕ0) → (0(digit‘2)𝑁) = 1)
 
Theoremdig2bits 47202 The 𝐾 th digit of a nonnegative integer 𝑁 in a binary system is its 𝐾 th bit. (Contributed by AV, 24-May-2020.)
((𝑁 ∈ ℕ0𝐾 ∈ ℕ0) → ((𝐾(digit‘2)𝑁) = 1 ↔ 𝐾 ∈ (bits‘𝑁)))
 
21.45.22.11  Nonnegative integer as sum of its shifted digits
 
Theoremdignn0flhalflem1 47203 Lemma 1 for dignn0flhalf 47206. (Contributed by AV, 7-Jun-2012.)
((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ) → (⌊‘((𝐴 / (2↑𝑁)) − 1)) < (⌊‘((𝐴 − 1) / (2↑𝑁))))
 
Theoremdignn0flhalflem2 47204 Lemma 2 for dignn0flhalf 47206. (Contributed by AV, 7-Jun-2012.)
((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ0) → (⌊‘(𝐴 / (2↑(𝑁 + 1)))) = (⌊‘((⌊‘(𝐴 / 2)) / (2↑𝑁))))
 
Theoremdignn0ehalf 47205 The digits of the half of an even nonnegative integer are the digits of the integer shifted by 1. (Contributed by AV, 3-Jun-2010.)
(((𝐴 / 2) ∈ ℕ0𝐴 ∈ ℕ0𝐼 ∈ ℕ0) → ((𝐼 + 1)(digit‘2)𝐴) = (𝐼(digit‘2)(𝐴 / 2)))
 
Theoremdignn0flhalf 47206 The digits of the rounded half of a nonnegative integer are the digits of the integer shifted by 1. (Contributed by AV, 7-Jun-2010.)
((𝐴 ∈ (ℤ‘2) ∧ 𝐼 ∈ ℕ0) → ((𝐼 + 1)(digit‘2)𝐴) = (𝐼(digit‘2)(⌊‘(𝐴 / 2))))
 
Theoremnn0sumshdiglemA 47207* Lemma for nn0sumshdig 47211 (induction step, even multiplier). (Contributed by AV, 3-Jun-2020.)
(((𝑎 ∈ ℕ ∧ (𝑎 / 2) ∈ ℕ) ∧ 𝑦 ∈ ℕ) → (∀𝑥 ∈ ℕ0 ((#b𝑥) = 𝑦𝑥 = Σ𝑘 ∈ (0..^𝑦)((𝑘(digit‘2)𝑥) · (2↑𝑘))) → ((#b𝑎) = (𝑦 + 1) → 𝑎 = Σ𝑘 ∈ (0..^(𝑦 + 1))((𝑘(digit‘2)𝑎) · (2↑𝑘)))))
 
Theoremnn0sumshdiglemB 47208* Lemma for nn0sumshdig 47211 (induction step, odd multiplier). (Contributed by AV, 7-Jun-2020.)
(((𝑎 ∈ ℕ ∧ ((𝑎 − 1) / 2) ∈ ℕ0) ∧ 𝑦 ∈ ℕ) → (∀𝑥 ∈ ℕ0 ((#b𝑥) = 𝑦𝑥 = Σ𝑘 ∈ (0..^𝑦)((𝑘(digit‘2)𝑥) · (2↑𝑘))) → ((#b𝑎) = (𝑦 + 1) → 𝑎 = Σ𝑘 ∈ (0..^(𝑦 + 1))((𝑘(digit‘2)𝑎) · (2↑𝑘)))))
 
Theoremnn0sumshdiglem1 47209* Lemma 1 for nn0sumshdig 47211 (induction step). (Contributed by AV, 7-Jun-2020.)
(𝑦 ∈ ℕ → (∀𝑎 ∈ ℕ0 ((#b𝑎) = 𝑦𝑎 = Σ𝑘 ∈ (0..^𝑦)((𝑘(digit‘2)𝑎) · (2↑𝑘))) → ∀𝑎 ∈ ℕ0 ((#b𝑎) = (𝑦 + 1) → 𝑎 = Σ𝑘 ∈ (0..^(𝑦 + 1))((𝑘(digit‘2)𝑎) · (2↑𝑘)))))
 
Theoremnn0sumshdiglem2 47210* Lemma 2 for nn0sumshdig 47211. (Contributed by AV, 7-Jun-2020.)
(𝐿 ∈ ℕ → ∀𝑎 ∈ ℕ0 ((#b𝑎) = 𝐿𝑎 = Σ𝑘 ∈ (0..^𝐿)((𝑘(digit‘2)𝑎) · (2↑𝑘))))
 
Theoremnn0sumshdig 47211* A nonnegative integer can be represented as sum of its shifted bits. (Contributed by AV, 7-Jun-2020.)
(𝐴 ∈ ℕ0𝐴 = Σ𝑘 ∈ (0..^(#b𝐴))((𝑘(digit‘2)𝐴) · (2↑𝑘)))
 
21.45.22.12  Algorithms for the multiplication of nonnegative integers
 
Theoremnn0mulfsum 47212* Trivial algorithm to calculate the product of two nonnegative integers 𝑎 and 𝑏 by adding 𝑏 to itself 𝑎 times. (Contributed by AV, 17-May-2020.)
((𝐴 ∈ ℕ0𝐵 ∈ ℕ0) → (𝐴 · 𝐵) = Σ𝑘 ∈ (1...𝐴)𝐵)
 
Theoremnn0mullong 47213* Standard algorithm (also known as "long multiplication" or "grade-school multiplication") to calculate the product of two nonnegative integers 𝑎 and 𝑏 by multiplying the multiplicand 𝑏 by each digit of the multiplier 𝑎 and then add up all the properly shifted results. Here, the binary representation of the multiplier 𝑎 is used, i.e., the above mentioned "digits" are 0 or 1. This is a similar result as provided by smumul 16430. (Contributed by AV, 7-Jun-2020.)
((𝐴 ∈ ℕ0𝐵 ∈ ℕ0) → (𝐴 · 𝐵) = Σ𝑘 ∈ (0..^(#b𝐴))(((𝑘(digit‘2)𝐴) · (2↑𝑘)) · 𝐵))
 
21.45.22.13  N-ary functions

According to Wikipedia ("Arity", https://en.wikipedia.org/wiki/Arity, 19-May-2024): "In logic, mathematics, and computer science, arity is the number of arguments or operands taken by a function, operation or relation."

N-ary functions are often also called multivariate functions, without indicating the actual number of argumens. See also Wikipedia ("Multivariate functions", 19-May-2024, https://en.wikipedia.org/wiki/Function_(mathematics)#Multivariate_functions ): "A multivariate function, multivariable function, or function of several variables is a function that depends on several arguments. ... Formally, a function of n variables is a function whose domain is a set of n-tuples. For example, multiplication of integers is a function of two variables, or bivariate function, whose domain is the set of all ordered pairs (2-tuples) of integers, and whose codomain is the set of integers. The same is true for every binary operation. Commonly, an n-tuple is denoted enclosed between parentheses, such as in ( 1 , 2 , ... , n ). When using functional notation, one usually omits the parentheses surrounding tuples, writing f ( x1 , ... , xn ) instead of f ( ( x1 , ... , xn ) ). Given n sets X1 , ... , Xn , the set of all n-tuples ( x1 , ... , xn ) such that x1 is element of X1 , ... , xn is element of Xn is called the Cartesian product of X1 , ... , Xn , and denoted X1 X ... X Xn . Therefore, a multivariate function is a function that has a Cartesian product or a proper subset of a Cartesian product as a domain: 𝑓:𝑈𝑌 where where the domain 𝑈 has the form 𝑈 ⊆ ((...((𝑋‘1) × (𝑋‘2)) × ...) × (𝑋𝑛))."

In the following, n-ary functions are defined as mappings (see df-map 8818) from a finite sequence of arguments, which themselves are defined as mappings from the half-open range of nonnegative integers to the domain of each argument. Furthermore, the definition is restricted to endofunctions, meaning that the domain(s) of the argument(s) is identical with its codomain. This means that the domains of all arguments are identical (in contrast to the definition in Wikipedia, see above: here, we have X1 = X2 = ... = Xn = X).

For small n, n-ary functions correspond to "usual" functions with a different number of arguments:

- n = 0 (nullary functions): These correspond actually to constants, see 0aryfvalelfv 47223 and mapsn 8878: (𝑋m {∅})

- n = 1 (unary functions): These correspond actually to usual endofunctions, see 1aryenef 47233 and efmndbas 18748: (𝑋m 𝑋)

- n = 2 (binary functions): These correspond to usual operations on two elements of the same set, also called "binary operation" (according to Wikipedia ("Binary operation", 19-May-2024, https://en.wikipedia.org/wiki/Binary_operation 18748): "In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two. More specifically, a binary operation on a set is a binary operation whose two domains and the codomain are the same set." Sometimes also called "closed internal binary operation"), see 2aryenef 47244 and compare with df-clintop 46545: (𝑋m (𝑋 × 𝑋)).

Instead of using indexed arguments (represented by a mapping as described above), elements of Cartesian exponentiations (𝑈↑↑𝑁) (see df-finxp 36203) could have been used to represent multiple arguments. However, this concept is not fully developed yet (it is within a mathbox), and it is currently based on ordinal numbers, e.g., (𝑈↑↑2o), instead of integers, e.g., (𝑈↑↑2), which is not very practical.

The definition df-ixp of infinite Cartesian product could also have been used to represent multiple arguments, but this would have been more cumbersome without any additional advantage. naryfvalixp 47217 shows that both definitions are equivalent.

 
Syntaxcnaryf 47214 Extend the definition of a class to include the n-ary functions.
class -aryF
 
Definitiondf-naryf 47215* Define the n-ary (endo)functions. (Contributed by AV, 11-May-2024.) (Revised by TA and SN, 7-Jun-2024.)
-aryF = (𝑛 ∈ ℕ0, 𝑥 ∈ V ↦ (𝑥m (𝑥m (0..^𝑛))))
 
Theoremnaryfval 47216 The set of the n-ary (endo)functions on a class 𝑋. (Contributed by AV, 13-May-2024.)
𝐼 = (0..^𝑁)       (𝑁 ∈ ℕ0 → (𝑁-aryF 𝑋) = (𝑋m (𝑋m 𝐼)))
 
Theoremnaryfvalixp 47217* The set of the n-ary (endo)functions on a class 𝑋 expressed with the notation of infinite Cartesian products. (Contributed by AV, 19-May-2024.)
𝐼 = (0..^𝑁)       (𝑁 ∈ ℕ0 → (𝑁-aryF 𝑋) = (𝑋m X𝑥𝐼 𝑋))
 
Theoremnaryfvalel 47218 An n-ary (endo)function on a set 𝑋. (Contributed by AV, 14-May-2024.)
𝐼 = (0..^𝑁)       ((𝑁 ∈ ℕ0𝑋𝑉) → (𝐹 ∈ (𝑁-aryF 𝑋) ↔ 𝐹:(𝑋m 𝐼)⟶𝑋))
 
Theoremnaryrcl 47219 Reverse closure for n-ary (endo)functions. (Contributed by AV, 14-May-2024.)
𝐼 = (0..^𝑁)       (𝐹 ∈ (𝑁-aryF 𝑋) → (𝑁 ∈ ℕ0𝑋 ∈ V))
 
Theoremnaryfvalelfv 47220 The value of an n-ary (endo)function on a set 𝑋 is an element of 𝑋. (Contributed by AV, 14-May-2024.)
𝐼 = (0..^𝑁)       ((𝐹 ∈ (𝑁-aryF 𝑋) ∧ 𝐴:𝐼𝑋) → (𝐹𝐴) ∈ 𝑋)
 
Theoremnaryfvalelwrdf 47221* An n-ary (endo)function on a set 𝑋 expressed as a function over the set of words on 𝑋 of length 𝑛. (Contributed by AV, 4-Jun-2024.)
((𝑁 ∈ ℕ0𝑋𝑉) → (𝐹 ∈ (𝑁-aryF 𝑋) ↔ 𝐹:{𝑤 ∈ Word 𝑋 ∣ (♯‘𝑤) = 𝑁}⟶𝑋))
 
Theorem0aryfvalel 47222* A nullary (endo)function on a set 𝑋 is a singleton of an ordered pair with the empty set as first component. A nullary function represents a constant: (𝐹‘∅) = 𝐶 with 𝐶𝑋, see also 0aryfvalelfv 47223. Instead of (𝐹‘∅), nullary functions are usually written as 𝐹() in literature. (Contributed by AV, 15-May-2024.)
(𝑋𝑉 → (𝐹 ∈ (0-aryF 𝑋) ↔ ∃𝑥𝑋 𝐹 = {⟨∅, 𝑥⟩}))
 
Theorem0aryfvalelfv 47223* The value of a nullary (endo)function on a set 𝑋. (Contributed by AV, 19-May-2024.)
(𝐹 ∈ (0-aryF 𝑋) → ∃𝑥𝑋 (𝐹‘∅) = 𝑥)
 
Theorem1aryfvalel 47224 A unary (endo)function on a set 𝑋. (Contributed by AV, 15-May-2024.)
(𝑋𝑉 → (𝐹 ∈ (1-aryF 𝑋) ↔ 𝐹:(𝑋m {0})⟶𝑋))
 
Theoremfv1arycl 47225 Closure of a unary (endo)function. (Contributed by AV, 18-May-2024.)
((𝐺 ∈ (1-aryF 𝑋) ∧ 𝐴𝑋) → (𝐺‘{⟨0, 𝐴⟩}) ∈ 𝑋)
 
Theorem1arympt1 47226* A unary (endo)function in maps-to notation. (Contributed by AV, 16-May-2024.)
𝐹 = (𝑥 ∈ (𝑋m {0}) ↦ (𝐴‘(𝑥‘0)))       ((𝑋𝑉𝐴:𝑋𝑋) → 𝐹 ∈ (1-aryF 𝑋))
 
Theorem1arympt1fv 47227* The value of a unary (endo)function in maps-to notation. (Contributed by AV, 16-May-2024.)
𝐹 = (𝑥 ∈ (𝑋m {0}) ↦ (𝐴‘(𝑥‘0)))       ((𝑋𝑉𝐵𝑋) → (𝐹‘{⟨0, 𝐵⟩}) = (𝐴𝐵))
 
Theorem1arymaptfv 47228* The value of the mapping of unary (endo)functions. (Contributed by AV, 18-May-2024.)
𝐻 = ( ∈ (1-aryF 𝑋) ↦ (𝑥𝑋 ↦ (‘{⟨0, 𝑥⟩})))       (𝐹 ∈ (1-aryF 𝑋) → (𝐻𝐹) = (𝑥𝑋 ↦ (𝐹‘{⟨0, 𝑥⟩})))
 
Theorem1arymaptf 47229* The mapping of unary (endo)functions is a function into the set of endofunctions. (Contributed by AV, 18-May-2024.)
𝐻 = ( ∈ (1-aryF 𝑋) ↦ (𝑥𝑋 ↦ (‘{⟨0, 𝑥⟩})))       (𝑋𝑉𝐻:(1-aryF 𝑋)⟶(𝑋m 𝑋))
 
Theorem1arymaptf1 47230* The mapping of unary (endo)functions is a one-to-one function into the set of endofunctions. (Contributed by AV, 19-May-2024.)
𝐻 = ( ∈ (1-aryF 𝑋) ↦ (𝑥𝑋 ↦ (‘{⟨0, 𝑥⟩})))       (𝑋𝑉𝐻:(1-aryF 𝑋)–1-1→(𝑋m 𝑋))
 
Theorem1arymaptfo 47231* The mapping of unary (endo)functions is a function onto the set of endofunctions. (Contributed by AV, 18-May-2024.)
𝐻 = ( ∈ (1-aryF 𝑋) ↦ (𝑥𝑋 ↦ (‘{⟨0, 𝑥⟩})))       (𝑋𝑉𝐻:(1-aryF 𝑋)–onto→(𝑋m 𝑋))
 
Theorem1arymaptf1o 47232* The mapping of unary (endo)functions is a one-to-one function onto the set of endofunctions. (Contributed by AV, 19-May-2024.)
𝐻 = ( ∈ (1-aryF 𝑋) ↦ (𝑥𝑋 ↦ (‘{⟨0, 𝑥⟩})))       (𝑋𝑉𝐻:(1-aryF 𝑋)–1-1-onto→(𝑋m 𝑋))
 
Theorem1aryenef 47233 The set of unary (endo)functions and the set of endofunctions are equinumerous. (Contributed by AV, 19-May-2024.)
(1-aryF 𝑋) ≈ (𝑋m 𝑋)
 
Theorem1aryenefmnd 47234 The set of unary (endo)functions and the base set of the monoid of endofunctions are equinumerous. (Contributed by AV, 19-May-2024.)
(1-aryF 𝑋) ≈ (Base‘(EndoFMnd‘𝑋))
 
Theorem2aryfvalel 47235 A binary (endo)function on a set 𝑋. (Contributed by AV, 20-May-2024.)
(𝑋𝑉 → (𝐹 ∈ (2-aryF 𝑋) ↔ 𝐹:(𝑋m {0, 1})⟶𝑋))
 
Theoremfv2arycl 47236 Closure of a binary (endo)function. (Contributed by AV, 20-May-2024.)
((𝐺 ∈ (2-aryF 𝑋) ∧ 𝐴𝑋𝐵𝑋) → (𝐺‘{⟨0, 𝐴⟩, ⟨1, 𝐵⟩}) ∈ 𝑋)
 
Theorem2arympt 47237* A binary (endo)function in maps-to notation. (Contributed by AV, 20-May-2024.)
𝐹 = (𝑥 ∈ (𝑋m {0, 1}) ↦ ((𝑥‘0)𝑂(𝑥‘1)))       ((𝑋𝑉𝑂:(𝑋 × 𝑋)⟶𝑋) → 𝐹 ∈ (2-aryF 𝑋))
 
Theorem2arymptfv 47238* The value of a binary (endo)function in maps-to notation. (Contributed by AV, 20-May-2024.)
𝐹 = (𝑥 ∈ (𝑋m {0, 1}) ↦ ((𝑥‘0)𝑂(𝑥‘1)))       ((𝑋𝑉𝐴𝑋𝐵𝑋) → (𝐹‘{⟨0, 𝐴⟩, ⟨1, 𝐵⟩}) = (𝐴𝑂𝐵))
 
Theorem2arymaptfv 47239* The value of the mapping of binary (endo)functions. (Contributed by AV, 21-May-2024.)
𝐻 = ( ∈ (2-aryF 𝑋) ↦ (𝑥𝑋, 𝑦𝑋 ↦ (‘{⟨0, 𝑥⟩, ⟨1, 𝑦⟩})))       (𝐹 ∈ (2-aryF 𝑋) → (𝐻𝐹) = (𝑥𝑋, 𝑦𝑋 ↦ (𝐹‘{⟨0, 𝑥⟩, ⟨1, 𝑦⟩})))
 
Theorem2arymaptf 47240* The mapping of binary (endo)functions is a function into the set of binary operations. (Contributed by AV, 21-May-2024.)
𝐻 = ( ∈ (2-aryF 𝑋) ↦ (𝑥𝑋, 𝑦𝑋 ↦ (‘{⟨0, 𝑥⟩, ⟨1, 𝑦⟩})))       (𝑋𝑉𝐻:(2-aryF 𝑋)⟶(𝑋m (𝑋 × 𝑋)))
 
Theorem2arymaptf1 47241* The mapping of binary (endo)functions is a one-to-one function into the set of binary operations. (Contributed by AV, 22-May-2024.)
𝐻 = ( ∈ (2-aryF 𝑋) ↦ (𝑥𝑋, 𝑦𝑋 ↦ (‘{⟨0, 𝑥⟩, ⟨1, 𝑦⟩})))       (𝑋𝑉𝐻:(2-aryF 𝑋)–1-1→(𝑋m (𝑋 × 𝑋)))
 
Theorem2arymaptfo 47242* The mapping of binary (endo)functions is a function onto the set of binary operations. (Contributed by AV, 23-May-2024.)
𝐻 = ( ∈ (2-aryF 𝑋) ↦ (𝑥𝑋, 𝑦𝑋 ↦ (‘{⟨0, 𝑥⟩, ⟨1, 𝑦⟩})))       (𝑋𝑉𝐻:(2-aryF 𝑋)–onto→(𝑋m (𝑋 × 𝑋)))
 
Theorem2arymaptf1o 47243* The mapping of binary (endo)functions is a one-to-one function onto the set of binary operations. (Contributed by AV, 23-May-2024.)
𝐻 = ( ∈ (2-aryF 𝑋) ↦ (𝑥𝑋, 𝑦𝑋 ↦ (‘{⟨0, 𝑥⟩, ⟨1, 𝑦⟩})))       (𝑋𝑉𝐻:(2-aryF 𝑋)–1-1-onto→(𝑋m (𝑋 × 𝑋)))
 
Theorem2aryenef 47244 The set of binary (endo)functions and the set of binary operations are equinumerous. (Contributed by AV, 19-May-2024.)
(2-aryF 𝑋) ≈ (𝑋m (𝑋 × 𝑋))
 
21.45.22.14  The Ackermann function

According to Wikipedia ("Ackermann function", 8-May-2024, https://en.wikipedia.org/wiki/Ackermann_function): "In computability theory, the Ackermann function, named after Wilhelm Ackermann, is one of the simplest and earliest-discovered examples of a total computable function that is not primitive recursive. ... One common version is the two-argument Ackermann-Péter function developed by Rózsa Péter and Raphael Robinson. Its value grows very rapidly; for example, A(4,2) results in 2^65536-3 [see ackval42 47284)], an integer of 19,729 decimal digits."

In the following, the Ackermann function is defined as iterated 1-ary function (also mentioned in Wikipedia), see df-ack 47248, based on a definition IterComp of "the n-th iterate of (a class/function) f", see df-itco 47247. As an illustration, we have ((IterComp‘𝐹)‘3) = (𝐹 ∘ (𝐹𝐹))) (see itcoval3 47253).

The following recursive definition of the Ackermann function follows immediately from Definition df-ack 47248: ((Ack‘(𝑀 + 1))‘𝑁) = (((IterComp‘(Ack‘𝑀))‘(𝑁 + 1))‘1)).

That Definition df-ack 47248 is equivalent to Péter's definition is proven by the following three theorems:

ackval0val 47274: ((Ack‘0)‘𝑀) = (𝑀 + 1); ackvalsuc0val 47275: ((Ack‘(𝑀 + 1))‘0) = ((Ack‘𝑀)‘1); ackvalsucsucval 47276: ((Ack‘(𝑀 + 1))‘(𝑁 + 1)) = ((Ack‘𝑀)‘((Ack‘(𝑀 + 1))‘𝑁)).

The initial values of the Ackermann function are calculated in the following four theorems:

ackval0012 47277: 𝐴(0, 0) = 1, 𝐴(0, 1) = 2, 𝐴(0, 2) = 3; ackval1012 47278: 𝐴(1, 0) = 2, 𝐴(1, 1) = 3, 𝐴(1, 3) = 4; ackval2012 47279: 𝐴(2, 0) = 3, 𝐴(2, 1) = 5, 𝐴(2, 3) = 7; ackval3012 47280: 𝐴(3, 0) = 5, 𝐴(3, 1) = 13, 𝐴(3, 3) = 29.

 
Syntaxcitco 47245 Extend the definition of a class to include iterated functions.
class IterComp
 
Syntaxcack 47246 Extend the definition of a class to include the Ackermann function operator.
class Ack
 
Definitiondf-itco 47247* Define a function (recursively) that returns the n-th iterate of a class (usually a function) with regard to composition. (Contributed by Thierry Arnoux, 28-Apr-2024.) (Revised by AV, 2-May-2024.)
IterComp = (𝑓 ∈ V ↦ seq0((𝑔 ∈ V, 𝑗 ∈ V ↦ (𝑓𝑔)), (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, ( I ↾ dom 𝑓), 𝑓))))
 
Definitiondf-ack 47248* Define the Ackermann function (recursively). (Contributed by Thierry Arnoux, 28-Apr-2024.) (Revised by AV, 2-May-2024.)
Ack = seq0((𝑓 ∈ V, 𝑗 ∈ V ↦ (𝑛 ∈ ℕ0 ↦ (((IterComp‘𝑓)‘(𝑛 + 1))‘1))), (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, (𝑛 ∈ ℕ0 ↦ (𝑛 + 1)), 𝑖)))
 
Theoremitcoval 47249* The value of the function that returns the n-th iterate of a class (usually a function) with regard to composition. (Contributed by AV, 2-May-2024.)
(𝐹𝑉 → (IterComp‘𝐹) = seq0((𝑔 ∈ V, 𝑗 ∈ V ↦ (𝐹𝑔)), (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, ( I ↾ dom 𝐹), 𝐹))))
 
Theoremitcoval0 47250 A function iterated zero times (defined as identity function). (Contributed by AV, 2-May-2024.)
(𝐹𝑉 → ((IterComp‘𝐹)‘0) = ( I ↾ dom 𝐹))
 
Theoremitcoval1 47251 A function iterated once. (Contributed by AV, 2-May-2024.)
((Rel 𝐹𝐹𝑉) → ((IterComp‘𝐹)‘1) = 𝐹)
 
Theoremitcoval2 47252 A function iterated twice. (Contributed by AV, 2-May-2024.)
((Rel 𝐹𝐹𝑉) → ((IterComp‘𝐹)‘2) = (𝐹𝐹))
 
Theoremitcoval3 47253 A function iterated three times. (Contributed by AV, 2-May-2024.)
((Rel 𝐹𝐹𝑉) → ((IterComp‘𝐹)‘3) = (𝐹 ∘ (𝐹𝐹)))
 
Theoremitcoval0mpt 47254* A mapping iterated zero times (defined as identity function). (Contributed by AV, 4-May-2024.)
𝐹 = (𝑛𝐴𝐵)       ((𝐴𝑉 ∧ ∀𝑛𝐴 𝐵𝑊) → ((IterComp‘𝐹)‘0) = (𝑛𝐴𝑛))
 
Theoremitcovalsuc 47255* The value of the function that returns the n-th iterate of a function with regard to composition at a successor. (Contributed by AV, 4-May-2024.)
((𝐹𝑉𝑌 ∈ ℕ0 ∧ ((IterComp‘𝐹)‘𝑌) = 𝐺) → ((IterComp‘𝐹)‘(𝑌 + 1)) = (𝐺(𝑔 ∈ V, 𝑗 ∈ V ↦ (𝐹𝑔))𝐹))
 
Theoremitcovalsucov 47256 The value of the function that returns the n-th iterate of a function with regard to composition at a successor. (Contributed by AV, 4-May-2024.)
((𝐹𝑉𝑌 ∈ ℕ0 ∧ ((IterComp‘𝐹)‘𝑌) = 𝐺) → ((IterComp‘𝐹)‘(𝑌 + 1)) = (𝐹𝐺))
 
Theoremitcovalendof 47257 The n-th iterate of an endofunction is an endofunction. (Contributed by AV, 7-May-2024.)
(𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝐴)    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → ((IterComp‘𝐹)‘𝑁):𝐴𝐴)
 
Theoremitcovalpclem1 47258* Lemma 1 for itcovalpc 47260: induction basis. (Contributed by AV, 4-May-2024.)
𝐹 = (𝑛 ∈ ℕ0 ↦ (𝑛 + 𝐶))       (𝐶 ∈ ℕ0 → ((IterComp‘𝐹)‘0) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 0))))
 
Theoremitcovalpclem2 47259* Lemma 2 for itcovalpc 47260: induction step. (Contributed by AV, 4-May-2024.)
𝐹 = (𝑛 ∈ ℕ0 ↦ (𝑛 + 𝐶))       ((𝑦 ∈ ℕ0𝐶 ∈ ℕ0) → (((IterComp‘𝐹)‘𝑦) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑦))) → ((IterComp‘𝐹)‘(𝑦 + 1)) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · (𝑦 + 1))))))
 
Theoremitcovalpc 47260* The value of the function that returns the n-th iterate of the "plus a constant" function with regard to composition. (Contributed by AV, 4-May-2024.)
𝐹 = (𝑛 ∈ ℕ0 ↦ (𝑛 + 𝐶))       ((𝐼 ∈ ℕ0𝐶 ∈ ℕ0) → ((IterComp‘𝐹)‘𝐼) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝐼))))
 
Theoremitcovalt2lem2lem1 47261 Lemma 1 for itcovalt2lem2 47264. (Contributed by AV, 6-May-2024.)
(((𝑌 ∈ ℕ ∧ 𝐶 ∈ ℕ0) ∧ 𝑁 ∈ ℕ0) → (((𝑁 + 𝐶) · 𝑌) − 𝐶) ∈ ℕ0)
 
Theoremitcovalt2lem2lem2 47262 Lemma 2 for itcovalt2lem2 47264. (Contributed by AV, 7-May-2024.)
(((𝑌 ∈ ℕ0𝐶 ∈ ℕ0) ∧ 𝑁 ∈ ℕ0) → ((2 · (((𝑁 + 𝐶) · (2↑𝑌)) − 𝐶)) + 𝐶) = (((𝑁 + 𝐶) · (2↑(𝑌 + 1))) − 𝐶))
 
Theoremitcovalt2lem1 47263* Lemma 1 for itcovalt2 47265: induction basis. (Contributed by AV, 5-May-2024.)
𝐹 = (𝑛 ∈ ℕ0 ↦ ((2 · 𝑛) + 𝐶))       (𝐶 ∈ ℕ0 → ((IterComp‘𝐹)‘0) = (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑0)) − 𝐶)))
 
Theoremitcovalt2lem2 47264* Lemma 2 for itcovalt2 47265: induction step. (Contributed by AV, 7-May-2024.)
𝐹 = (𝑛 ∈ ℕ0 ↦ ((2 · 𝑛) + 𝐶))       ((𝑦 ∈ ℕ0𝐶 ∈ ℕ0) → (((IterComp‘𝐹)‘𝑦) = (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑𝑦)) − 𝐶)) → ((IterComp‘𝐹)‘(𝑦 + 1)) = (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑(𝑦 + 1))) − 𝐶))))
 
Theoremitcovalt2 47265* The value of the function that returns the n-th iterate of the "times 2 plus a constant" function with regard to composition. (Contributed by AV, 7-May-2024.)
𝐹 = (𝑛 ∈ ℕ0 ↦ ((2 · 𝑛) + 𝐶))       ((𝐼 ∈ ℕ0𝐶 ∈ ℕ0) → ((IterComp‘𝐹)‘𝐼) = (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑𝐼)) − 𝐶)))
 
Theoremackvalsuc1mpt 47266* The Ackermann function at a successor of the first argument as a mapping of the second argument. (Contributed by Thierry Arnoux, 28-Apr-2024.) (Revised by AV, 4-May-2024.)
(𝑀 ∈ ℕ0 → (Ack‘(𝑀 + 1)) = (𝑛 ∈ ℕ0 ↦ (((IterComp‘(Ack‘𝑀))‘(𝑛 + 1))‘1)))
 
Theoremackvalsuc1 47267 The Ackermann function at a successor of the first argument and an arbitrary second argument. (Contributed by Thierry Arnoux, 28-Apr-2024.) (Revised by AV, 4-May-2024.)
((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → ((Ack‘(𝑀 + 1))‘𝑁) = (((IterComp‘(Ack‘𝑀))‘(𝑁 + 1))‘1))
 
Theoremackval0 47268 The Ackermann function at 0. (Contributed by AV, 2-May-2024.)
(Ack‘0) = (𝑛 ∈ ℕ0 ↦ (𝑛 + 1))
 
Theoremackval1 47269 The Ackermann function at 1. (Contributed by AV, 4-May-2024.)
(Ack‘1) = (𝑛 ∈ ℕ0 ↦ (𝑛 + 2))
 
Theoremackval2 47270 The Ackermann function at 2. (Contributed by AV, 4-May-2024.)
(Ack‘2) = (𝑛 ∈ ℕ0 ↦ ((2 · 𝑛) + 3))
 
Theoremackval3 47271 The Ackermann function at 3. (Contributed by AV, 7-May-2024.)
(Ack‘3) = (𝑛 ∈ ℕ0 ↦ ((2↑(𝑛 + 3)) − 3))
 
Theoremackendofnn0 47272 The Ackermann function at any nonnegative integer is an endofunction on the nonnegative integers. (Contributed by AV, 8-May-2024.)
(𝑀 ∈ ℕ0 → (Ack‘𝑀):ℕ0⟶ℕ0)
 
Theoremackfnnn0 47273 The Ackermann function at any nonnegative integer is a function on the nonnegative integers. (Contributed by AV, 4-May-2024.) (Proof shortened by AV, 8-May-2024.)
(𝑀 ∈ ℕ0 → (Ack‘𝑀) Fn ℕ0)
 
Theoremackval0val 47274 The Ackermann function at 0 (for the first argument). This is the first equation of Péter's definition of the Ackermann function. (Contributed by AV, 4-May-2024.)
(𝑀 ∈ ℕ0 → ((Ack‘0)‘𝑀) = (𝑀 + 1))
 
Theoremackvalsuc0val 47275 The Ackermann function at a successor (of the first argument). This is the second equation of Péter's definition of the Ackermann function. (Contributed by AV, 4-May-2024.)
(𝑀 ∈ ℕ0 → ((Ack‘(𝑀 + 1))‘0) = ((Ack‘𝑀)‘1))
 
Theoremackvalsucsucval 47276 The Ackermann function at the successors. This is the third equation of Péter's definition of the Ackermann function. (Contributed by AV, 8-May-2024.)
((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → ((Ack‘(𝑀 + 1))‘(𝑁 + 1)) = ((Ack‘𝑀)‘((Ack‘(𝑀 + 1))‘𝑁)))
 
Theoremackval0012 47277 The Ackermann function at (0,0), (0,1), (0,2). (Contributed by AV, 2-May-2024.)
⟨((Ack‘0)‘0), ((Ack‘0)‘1), ((Ack‘0)‘2)⟩ = ⟨1, 2, 3⟩
 
Theoremackval1012 47278 The Ackermann function at (1,0), (1,1), (1,2). (Contributed by AV, 4-May-2024.)
⟨((Ack‘1)‘0), ((Ack‘1)‘1), ((Ack‘1)‘2)⟩ = ⟨2, 3, 4⟩
 
Theoremackval2012 47279 The Ackermann function at (2,0), (2,1), (2,2). (Contributed by AV, 4-May-2024.)
⟨((Ack‘2)‘0), ((Ack‘2)‘1), ((Ack‘2)‘2)⟩ = ⟨3, 5, 7⟩
 
Theoremackval3012 47280 The Ackermann function at (3,0), (3,1), (3,2). (Contributed by AV, 7-May-2024.)
⟨((Ack‘3)‘0), ((Ack‘3)‘1), ((Ack‘3)‘2)⟩ = ⟨5, 13, 29⟩
 
Theoremackval40 47281 The Ackermann function at (4,0). (Contributed by AV, 9-May-2024.)
((Ack‘4)‘0) = 13
 
Theoremackval41a 47282 The Ackermann function at (4,1). (Contributed by AV, 9-May-2024.)
((Ack‘4)‘1) = ((2↑16) − 3)
 
Theoremackval41 47283 The Ackermann function at (4,1). (Contributed by AV, 9-May-2024.)
((Ack‘4)‘1) = 65533
 
Theoremackval42 47284 The Ackermann function at (4,2). (Contributed by AV, 9-May-2024.)
((Ack‘4)‘2) = ((2↑65536) − 3)
 
Theoremackval42a 47285 The Ackermann function at (4,2), expressed with powers of 2. (Contributed by AV, 9-May-2024.)
((Ack‘4)‘2) = ((2↑(2↑(2↑(2↑2)))) − 3)
 
Theoremackval50 47286 The Ackermann function at (5,0). (Contributed by AV, 9-May-2024.)
((Ack‘5)‘0) = 65533
 
21.45.23  Elementary geometry (extension)
 
21.45.23.1  Auxiliary theorems
 
Theoremfv1prop 47287 The function value of unordered pair of ordered pairs with first components 1 and 2 at 1. (Contributed by AV, 4-Feb-2023.)
(𝐴𝑉 → ({⟨1, 𝐴⟩, ⟨2, 𝐵⟩}‘1) = 𝐴)
 
Theoremfv2prop 47288 The function value of unordered pair of ordered pairs with first components 1 and 2 at 1. (Contributed by AV, 4-Feb-2023.)
(𝐵𝑉 → ({⟨1, 𝐴⟩, ⟨2, 𝐵⟩}‘2) = 𝐵)
 
Theoremsubmuladdmuld 47289 Transformation of a sum of a product of a difference and a product with the subtrahend of the difference. (Contributed by AV, 2-Feb-2023.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐷 ∈ ℂ)       (𝜑 → (((𝐴𝐵) · 𝐶) + (𝐵 · 𝐷)) = ((𝐴 · 𝐶) + (𝐵 · (𝐷𝐶))))
 
Theoremaffinecomb1 47290* Combination of two real affine combinations, one class variable resolved. (Contributed by AV, 22-Jan-2023.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐵𝐶)    &   (𝜑𝐸 ∈ ℝ)    &   (𝜑𝐹 ∈ ℝ)    &   (𝜑𝐺 ∈ ℝ)    &   𝑆 = ((𝐺𝐹) / (𝐶𝐵))       (𝜑 → (∃𝑡 ∈ ℝ (𝐴 = (((1 − 𝑡) · 𝐵) + (𝑡 · 𝐶)) ∧ 𝐸 = (((1 − 𝑡) · 𝐹) + (𝑡 · 𝐺))) ↔ 𝐸 = ((𝑆 · (𝐴𝐵)) + 𝐹)))
 
Theoremaffinecomb2 47291* Combination of two real affine combinations, presented without fraction. (Contributed by AV, 22-Jan-2023.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐵𝐶)    &   (𝜑𝐸 ∈ ℝ)    &   (𝜑𝐹 ∈ ℝ)    &   (𝜑𝐺 ∈ ℝ)       (𝜑 → (∃𝑡 ∈ ℝ (𝐴 = (((1 − 𝑡) · 𝐵) + (𝑡 · 𝐶)) ∧ 𝐸 = (((1 − 𝑡) · 𝐹) + (𝑡 · 𝐺))) ↔ ((𝐶𝐵) · 𝐸) = (((𝐺𝐹) · 𝐴) + ((𝐹 · 𝐶) − (𝐵 · 𝐺)))))
 
Theoremaffineid 47292 Identity of an affine combination. (Contributed by AV, 2-Feb-2023.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝑇 ∈ ℂ)       (𝜑 → (((1 − 𝑇) · 𝐴) + (𝑇 · 𝐴)) = 𝐴)
 
Theorem1subrec1sub 47293 Subtract the reciprocal of 1 minus a number from 1 results in the number divided by the number minus 1. (Contributed by AV, 15-Feb-2023.)
((𝐴 ∈ ℂ ∧ 𝐴 ≠ 1) → (1 − (1 / (1 − 𝐴))) = (𝐴 / (𝐴 − 1)))
 
Theoremresum2sqcl 47294 The sum of two squares of real numbers is a real number. (Contributed by AV, 7-Feb-2023.)
𝑄 = ((𝐴↑2) + (𝐵↑2))       ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝑄 ∈ ℝ)
 
Theoremresum2sqgt0 47295 The sum of the square of a nonzero real number and the square of another real number is greater than zero. (Contributed by AV, 7-Feb-2023.)
𝑄 = ((𝐴↑2) + (𝐵↑2))       (((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℝ) → 0 < 𝑄)
 
Theoremresum2sqrp 47296 The sum of the square of a nonzero real number and the square of another real number is a positive real number. (Contributed by AV, 2-May-2023.)
𝑄 = ((𝐴↑2) + (𝐵↑2))       (((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℝ) → 𝑄 ∈ ℝ+)
 
Theoremresum2sqorgt0 47297 The sum of the square of two real numbers is greater than zero if at least one of the real numbers is nonzero. (Contributed by AV, 26-Feb-2023.)
𝑄 = ((𝐴↑2) + (𝐵↑2))       ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐴 ≠ 0 ∨ 𝐵 ≠ 0)) → 0 < 𝑄)
 
Theoremreorelicc 47298 Membership in and outside of a closed real interval. (Contributed by AV, 15-Feb-2023.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐶 < 𝐴𝐶 ∈ (𝐴[,]𝐵) ∨ 𝐵 < 𝐶))
 
21.45.23.2  Real euclidean space of dimension 2
 
Theoremrrx2pxel 47299 The x-coordinate of a point in a real Euclidean space of dimension 2 is a real number. (Contributed by AV, 2-Feb-2023.)
𝐼 = {1, 2}    &   𝑃 = (ℝ ↑m 𝐼)       (𝑋𝑃 → (𝑋‘1) ∈ ℝ)
 
Theoremrrx2pyel 47300 The y-coordinate of a point in a real Euclidean space of dimension 2 is a real number. (Contributed by AV, 2-Feb-2023.)
𝐼 = {1, 2}    &   𝑃 = (ℝ ↑m 𝐼)       (𝑋𝑃 → (𝑋‘2) ∈ ℝ)
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144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 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454 45301-45400 455 45401-45500 456 45501-45600 457 45601-45700 458 45701-45800 459 45801-45900 460 45901-46000 461 46001-46100 462 46101-46200 463 46201-46300 464 46301-46400 465 46401-46500 466 46501-46600 467 46601-46700 468 46701-46800 469 46801-46900 470 46901-47000 471 47001-47100 472 47101-47200 473 47201-47300 474 47301-47400 475 47401-47500 476 47501-47600 477 47601-47700 478 47701-47754
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