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Theorem List for Metamath Proof Explorer - 45401-45500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremnn0digval 45401 The 𝐾 th digit of a nonnegative real number 𝑅 in the positional system with base 𝐵. (Contributed by AV, 23-May-2020.)
((𝐵 ∈ ℕ ∧ 𝐾 ∈ ℕ0𝑅 ∈ (0[,)+∞)) → (𝐾(digit‘𝐵)𝑅) = ((⌊‘(𝑅 / (𝐵𝐾))) mod 𝐵))

Theoremdignn0fr 45402 The digits of the fractional part of a nonnegative integer are 0. (Contributed by AV, 23-May-2020.)
((𝐵 ∈ ℕ ∧ 𝐾 ∈ (ℤ ∖ ℕ0) ∧ 𝑁 ∈ ℕ0) → (𝐾(digit‘𝐵)𝑁) = 0)

Theoremdignn0ldlem 45403 Lemma for dignnld 45404. (Contributed by AV, 25-May-2020.)
((𝐵 ∈ (ℤ‘2) ∧ 𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ‘((⌊‘(𝐵 logb 𝑁)) + 1))) → 𝑁 < (𝐵𝐾))

Theoremdignnld 45404 The leading digits of a positive integer are 0. (Contributed by AV, 25-May-2020.)
((𝐵 ∈ (ℤ‘2) ∧ 𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ‘((⌊‘(𝐵 logb 𝑁)) + 1))) → (𝐾(digit‘𝐵)𝑁) = 0)

Theoremdig2nn0ld 45405 The leading digits of a positive integer in a binary system are 0. (Contributed by AV, 25-May-2020.)
((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ‘(#b𝑁))) → (𝐾(digit‘2)𝑁) = 0)

Theoremdig2nn1st 45406 The first (relevant) digit of a positive integer in a binary system is 1. (Contributed by AV, 26-May-2020.)
(𝑁 ∈ ℕ → (((#b𝑁) − 1)(digit‘2)𝑁) = 1)

Theoremdig0 45407 All digits of 0 are 0. (Contributed by AV, 24-May-2020.)
((𝐵 ∈ ℕ ∧ 𝐾 ∈ ℤ) → (𝐾(digit‘𝐵)0) = 0)

Theoremdigexp 45408 The 𝐾 th digit of a power to the base is either 1 or 0. (Contributed by AV, 24-May-2020.)
((𝐵 ∈ (ℤ‘2) ∧ 𝐾 ∈ ℕ0𝑁 ∈ ℕ0) → (𝐾(digit‘𝐵)(𝐵𝑁)) = if(𝐾 = 𝑁, 1, 0))

Theoremdig1 45409 All but one digits of 1 are 0. (Contributed by AV, 24-May-2020.)
((𝐵 ∈ (ℤ‘2) ∧ 𝐾 ∈ ℤ) → (𝐾(digit‘𝐵)1) = if(𝐾 = 0, 1, 0))

Theorem0dig1 45410 The 0 th digit of 1 is 1 in any positional system. (Contributed by AV, 28-May-2020.)
(𝐵 ∈ (ℤ‘2) → (0(digit‘𝐵)1) = 1)

Theorem0dig2pr01 45411 The integers 0 and 1 correspond to their last bit. (Contributed by AV, 28-May-2010.)
(𝑁 ∈ {0, 1} → (0(digit‘2)𝑁) = 𝑁)

Theoremdig2nn0 45412 A digit of a nonnegative integer 𝑁 in a binary system is either 0 or 1. (Contributed by AV, 24-May-2020.)
((𝑁 ∈ ℕ0𝐾 ∈ ℤ) → (𝐾(digit‘2)𝑁) ∈ {0, 1})

Theorem0dig2nn0e 45413 The last bit of an even integer is 0. (Contributed by AV, 3-Jun-2010.)
((𝑁 ∈ ℕ0 ∧ (𝑁 / 2) ∈ ℕ0) → (0(digit‘2)𝑁) = 0)

Theorem0dig2nn0o 45414 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 45415 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)))

20.41.22.11  Nonnegative integer as sum of its shifted digits

Theoremdignn0flhalflem1 45416 Lemma 1 for dignn0flhalf 45419. (Contributed by AV, 7-Jun-2012.)
((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ) → (⌊‘((𝐴 / (2↑𝑁)) − 1)) < (⌊‘((𝐴 − 1) / (2↑𝑁))))

Theoremdignn0flhalflem2 45417 Lemma 2 for dignn0flhalf 45419. (Contributed by AV, 7-Jun-2012.)
((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ0) → (⌊‘(𝐴 / (2↑(𝑁 + 1)))) = (⌊‘((⌊‘(𝐴 / 2)) / (2↑𝑁))))

Theoremdignn0ehalf 45418 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 45419 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 45420* Lemma for nn0sumshdig 45424 (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 45421* Lemma for nn0sumshdig 45424 (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 45422* Lemma 1 for nn0sumshdig 45424 (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 45423* Lemma 2 for nn0sumshdig 45424. (Contributed by AV, 7-Jun-2020.)
(𝐿 ∈ ℕ → ∀𝑎 ∈ ℕ0 ((#b𝑎) = 𝐿𝑎 = Σ𝑘 ∈ (0..^𝐿)((𝑘(digit‘2)𝑎) · (2↑𝑘))))

Theoremnn0sumshdig 45424* 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)))

20.41.22.12  Algorithms for the multiplication of nonnegative integers

Theoremnn0mulfsum 45425* 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 45426* 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 15892. (Contributed by AV, 7-Jun-2020.)
((𝐴 ∈ ℕ0𝐵 ∈ ℕ0) → (𝐴 · 𝐵) = Σ𝑘 ∈ (0..^(#b𝐴))(((𝑘(digit‘2)𝐴) · (2↑𝑘)) · 𝐵))

20.41.22.13  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 8418) 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 45436 and mapsn 8470: (𝑋m {∅})

- n = 1 (unary functions): These correspond actually to usual endofunctions, see 1aryenef 45446 and efmndbas 18102: (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 18102): "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 45457 and compare with df-clintop 44849: (𝑋m (𝑋 × 𝑋)).

Instead of using indexed arguments (represented by a mapping as described above), elements of Cartesian exponentiations (𝑈↑↑𝑁) (see df-finxp 35103) 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 45430 shows that both definitions are equivalent.

Syntaxcnaryf 45427 Extend the definition of a class to include the n-ary functions.
class -aryF

Definitiondf-naryf 45428* 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 45429 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 45430* 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 45431 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 45432 Reverse closure for n-ary (endo)functions. (Contributed by AV, 14-May-2024.)
𝐼 = (0..^𝑁)       (𝐹 ∈ (𝑁-aryF 𝑋) → (𝑁 ∈ ℕ0𝑋 ∈ V))

Theoremnaryfvalelfv 45433 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 45434* 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 45435* 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 45436. 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 45436* 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 45437 A unary (endo)function on a set 𝑋. (Contributed by AV, 15-May-2024.)
(𝑋𝑉 → (𝐹 ∈ (1-aryF 𝑋) ↔ 𝐹:(𝑋m {0})⟶𝑋))

Theoremfv1arycl 45438 Closure of a unary (endo)function. (Contributed by AV, 18-May-2024.)
((𝐺 ∈ (1-aryF 𝑋) ∧ 𝐴𝑋) → (𝐺‘{⟨0, 𝐴⟩}) ∈ 𝑋)

Theorem1arympt1 45439* 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 45440* 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 45441* 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 45442* 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 45443* 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 45444* 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 45445* 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 45446 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 45447 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 45448 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 45449 Closure of a binary (endo)function. (Contributed by AV, 20-May-2024.)
((𝐺 ∈ (2-aryF 𝑋) ∧ 𝐴𝑋𝐵𝑋) → (𝐺‘{⟨0, 𝐴⟩, ⟨1, 𝐵⟩}) ∈ 𝑋)

Theorem2arympt 45450* 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 45451* 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 45452* 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 45453* 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 45454* 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 45455* 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 45456* 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 45457 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))

20.41.22.14  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 45497)], 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 45461, based on a definition IterComp of "the n-th iterate of (a class/function) f", see df-itco 45460. As an illustration, we have ((IterComp‘𝐹)‘3) = (𝐹 ∘ (𝐹𝐹))) (see itcoval3 45466).

The following recursive definition of the Ackermann function follows immediately from Definition df-ack 45461: ((Ack‘(𝑀 + 1))‘𝑁) = (((IterComp‘(Ack‘𝑀))‘(𝑁 + 1))‘1)).

That Definition df-ack 45461 is equivalent to Péter's definition is proven by the following three theorems:

ackval0val 45487: ((Ack‘0)‘𝑀) = (𝑀 + 1); ackvalsuc0val 45488: ((Ack‘(𝑀 + 1))‘0) = ((Ack‘𝑀)‘1); ackvalsucsucval 45489: ((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 45490: 𝐴(0, 0) = 1, 𝐴(0, 1) = 2, 𝐴(0, 2) = 3; ackval1012 45491: 𝐴(1, 0) = 2, 𝐴(1, 1) = 3, 𝐴(1, 3) = 4; ackval2012 45492: 𝐴(2, 0) = 3, 𝐴(2, 1) = 5, 𝐴(2, 3) = 7; ackval3012 45493: 𝐴(3, 0) = 5, 𝐴(3, 1) = 13, 𝐴(3, 3) = 29.

Syntaxcitco 45458 Extend the definition of a class to include iterated functions.
class IterComp

Syntaxcack 45459 Extend the definition of a class to include the Ackermann function operator.
class Ack

Definitiondf-itco 45460* 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 45461* 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 45462* 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 45463 A function iterated zero times (defined as identity function). (Contributed by AV, 2-May-2024.)
(𝐹𝑉 → ((IterComp‘𝐹)‘0) = ( I ↾ dom 𝐹))

Theoremitcoval1 45464 A function iterated once. (Contributed by AV, 2-May-2024.)
((Rel 𝐹𝐹𝑉) → ((IterComp‘𝐹)‘1) = 𝐹)

Theoremitcoval2 45465 A function iterated twice. (Contributed by AV, 2-May-2024.)
((Rel 𝐹𝐹𝑉) → ((IterComp‘𝐹)‘2) = (𝐹𝐹))

Theoremitcoval3 45466 A function iterated three times. (Contributed by AV, 2-May-2024.)
((Rel 𝐹𝐹𝑉) → ((IterComp‘𝐹)‘3) = (𝐹 ∘ (𝐹𝐹)))

Theoremitcoval0mpt 45467* 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 45468* 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 45469 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 45470 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 45471* Lemma 1 for itcovalpc 45473: induction basis. (Contributed by AV, 4-May-2024.)
𝐹 = (𝑛 ∈ ℕ0 ↦ (𝑛 + 𝐶))       (𝐶 ∈ ℕ0 → ((IterComp‘𝐹)‘0) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 0))))

Theoremitcovalpclem2 45472* Lemma 2 for itcovalpc 45473: 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 45473* 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 45474 Lemma 1 for itcovalt2lem2 45477. (Contributed by AV, 6-May-2024.)
(((𝑌 ∈ ℕ ∧ 𝐶 ∈ ℕ0) ∧ 𝑁 ∈ ℕ0) → (((𝑁 + 𝐶) · 𝑌) − 𝐶) ∈ ℕ0)

Theoremitcovalt2lem2lem2 45475 Lemma 2 for itcovalt2lem2 45477. (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 45476* Lemma 1 for itcovalt2 45478: 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 45477* Lemma 2 for itcovalt2 45478: 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 45478* 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 45479* 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 45480 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 45481 The Ackermann function at 0. (Contributed by AV, 2-May-2024.)
(Ack‘0) = (𝑛 ∈ ℕ0 ↦ (𝑛 + 1))

Theoremackval1 45482 The Ackermann function at 1. (Contributed by AV, 4-May-2024.)
(Ack‘1) = (𝑛 ∈ ℕ0 ↦ (𝑛 + 2))

Theoremackval2 45483 The Ackermann function at 2. (Contributed by AV, 4-May-2024.)
(Ack‘2) = (𝑛 ∈ ℕ0 ↦ ((2 · 𝑛) + 3))

Theoremackval3 45484 The Ackermann function at 3. (Contributed by AV, 7-May-2024.)
(Ack‘3) = (𝑛 ∈ ℕ0 ↦ ((2↑(𝑛 + 3)) − 3))

Theoremackendofnn0 45485 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 45486 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 45487 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 45488 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 45489 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 45490 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 45491 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 45492 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 45493 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 45494 The Ackermann function at (4,0). (Contributed by AV, 9-May-2024.)
((Ack‘4)‘0) = 13

Theoremackval41a 45495 The Ackermann function at (4,1). (Contributed by AV, 9-May-2024.)
((Ack‘4)‘1) = ((2↑16) − 3)

Theoremackval41 45496 The Ackermann function at (4,1). (Contributed by AV, 9-May-2024.)
((Ack‘4)‘1) = 65533

Theoremackval42 45497 The Ackermann function at (4,2). (Contributed by AV, 9-May-2024.)
((Ack‘4)‘2) = ((2↑65536) − 3)

Theoremackval42a 45498 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 45499 The Ackermann function at (5,0). (Contributed by AV, 9-May-2024.)
((Ack‘5)‘0) = 65533

20.41.23  Elementary geometry (extension)

20.41.23.1  Auxiliary theorems

Theoremfv1prop 45500 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)

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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 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44900 450 44901-45000 451 45001-45100 452 45101-45200 453 45201-45300 454 45301-45400 455 45401-45500 456 45501-45600 457 45601-45700 458 45701-45746
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