P. 498 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 49701-49800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	nnolog2flm1 49701	The floor of the binary logarithm of an odd integer greater than 1 is the floor of the binary logarithm of the integer decreased by 1. (Contributed by AV, 2-Jun-2020.)
⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ ((𝑁 + 1) / 2) ∈ ℕ) → (⌊‘(2 logb 𝑁)) = (⌊‘(2 logb (𝑁 − 1))))
Theorem	blennn0em1 49702	The binary length of the half of an even positive integer is the binary length of the integer minus 1. (Contributed by AV, 30-May-2010.)
⊢ ((𝑁 ∈ ℕ ∧ (𝑁 / 2) ∈ ℕ0) → (#b‘(𝑁 / 2)) = ((#b‘𝑁) − 1))
Theorem	blennngt2o2 49703	The binary length of an odd integer greater than 1 is the binary length of the half of the integer decreased by 1, increased by 1. (Contributed by AV, 3-Jun-2020.)
⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ ((𝑁 + 1) / 2) ∈ ℕ0) → (#b‘𝑁) = ((#b‘((𝑁 − 1) / 2)) + 1))
Theorem	blengt1fldiv2p1 49704	The binary length of an integer greater than 1 is the binary length of the integer divided by 2, increased by one. (Contributed by AV, 3-Jun-2020.)
⊢ (𝑁 ∈ (ℤ≥‘2) → (#b‘𝑁) = ((#b‘(⌊‘(𝑁 / 2))) + 1))
Theorem	blennn0e2 49705	The binary length of an even positive integer is the binary length of the half of the integer, increased by 1. (Contributed by AV, 29-May-2020.)
⊢ ((𝑁 ∈ ℕ ∧ (𝑁 / 2) ∈ ℕ0) → (#b‘𝑁) = ((#b‘(𝑁 / 2)) + 1))
21.50.22.9  Digits Generalization of df-bits 16592. In contrast to digit, bits are defined for integers only. The equivalence of both definitions for integers is shown in dig2bits 49725: if 𝐾 and 𝑁 are nonnegative integers, then ((𝐾(digit‘2)𝑁) = 1 ↔ 𝐾 ∈ (bits‘𝑁)).
Syntax	cdig 49706	Extend class notation with the class of the digit extraction operation.
class digit
Definition	df-dig 49707*	Definition of an operation to obtain the 𝑘 th digit of a nonnegative real number 𝑟 in the positional system with base 𝑏. 𝑘 = − 1 corresponds to the first digit of the fractional part (for 𝑏 = 10 the first digit after the decimal point), 𝑘 = 0 corresponds to the last digit of the integer part (for 𝑏 = 10 the first digit before the decimal point). See also digit1 14381. Examples (not formal): ( 234.567 ( digit ` 10 ) 0 ) = 4; ( 2.567 ( digit ` 10 ) -2 ) = 6; ( 2345.67 ( digit ` 10 ) 2 ) = 3. (Contributed by AV, 16-May-2020.)
⊢ digit = (𝑏 ∈ ℕ ↦ (𝑘 ∈ ℤ, 𝑟 ∈ (0[,)+∞) ↦ ((⌊‘((𝑏↑-𝑘) · 𝑟)) mod 𝑏)))
Theorem	digfval 49708*	Operation to obtain 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 → (digit‘𝐵) = (𝑘 ∈ ℤ, 𝑟 ∈ (0[,)+∞) ↦ ((⌊‘((𝐵↑-𝑘) · 𝑟)) mod 𝐵)))
Theorem	digval 49709	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 ∈ ℤ ∧ 𝑅 ∈ (0[,)+∞)) → (𝐾(digit‘𝐵)𝑅) = ((⌊‘((𝐵↑-𝐾) · 𝑅)) mod 𝐵))
Theorem	digvalnn0 49710	The 𝐾 th digit of a nonnegative real number 𝑅 in the positional system with base 𝐵 is a nonnegative integer. (Contributed by AV, 28-May-2020.)
⊢ ((𝐵 ∈ ℕ ∧ 𝐾 ∈ ℤ ∧ 𝑅 ∈ (0[,)+∞)) → (𝐾(digit‘𝐵)𝑅) ∈ ℕ0)
Theorem	nn0digval 49711	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 ∈ ℕ0 ∧ 𝑅 ∈ (0[,)+∞)) → (𝐾(digit‘𝐵)𝑅) = ((⌊‘(𝑅 / (𝐵↑𝐾))) mod 𝐵))
Theorem	dignn0fr 49712	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)
Theorem	dignn0ldlem 49713	Lemma for dignnld 49714. (Contributed by AV, 25-May-2020.)
⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ≥‘((⌊‘(𝐵 logb 𝑁)) + 1))) → 𝑁 < (𝐵↑𝐾))
Theorem	dignnld 49714	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)
Theorem	dig2nn0ld 49715	The leading digits of a positive integer in a binary system are 0. (Contributed by AV, 25-May-2020.)
⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ≥‘(#b‘𝑁))) → (𝐾(digit‘2)𝑁) = 0)
Theorem	dig2nn1st 49716	The first (relevant) digit of a positive integer in a binary system is 1. (Contributed by AV, 26-May-2020.)
⊢ (𝑁 ∈ ℕ → (((#b‘𝑁) − 1)(digit‘2)𝑁) = 1)
Theorem	dig0 49717	All digits of 0 are 0. (Contributed by AV, 24-May-2020.)
⊢ ((𝐵 ∈ ℕ ∧ 𝐾 ∈ ℤ) → (𝐾(digit‘𝐵)0) = 0)
Theorem	digexp 49718	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 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝐾(digit‘𝐵)(𝐵↑𝑁)) = if(𝐾 = 𝑁, 1, 0))
Theorem	dig1 49719	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))
Theorem	0dig1 49720	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)
Theorem	0dig2pr01 49721	The integers 0 and 1 correspond to their last bit. (Contributed by AV, 28-May-2010.)
⊢ (𝑁 ∈ {0, 1} → (0(digit‘2)𝑁) = 𝑁)
Theorem	dig2nn0 49722	A digit of a nonnegative integer 𝑁 in a binary system is either 0 or 1. (Contributed by AV, 24-May-2020.)
⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ ℤ) → (𝐾(digit‘2)𝑁) ∈ {0, 1})
Theorem	0dig2nn0e 49723	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)
Theorem	0dig2nn0o 49724	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)
Theorem	dig2bits 49725	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 ∈ ℕ0 ∧ 𝐾 ∈ ℕ0) → ((𝐾(digit‘2)𝑁) = 1 ↔ 𝐾 ∈ (bits‘𝑁)))
21.50.22.10  Nonnegative integer as sum of its shifted digits
Theorem	dignn0flhalflem1 49726	Lemma 1 for dignn0flhalf 49729. (Contributed by AV, 7-Jun-2012.)
⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ) → (⌊‘((𝐴 / (2↑𝑁)) − 1)) < (⌊‘((𝐴 − 1) / (2↑𝑁))))
Theorem	dignn0flhalflem2 49727	Lemma 2 for dignn0flhalf 49729. (Contributed by AV, 7-Jun-2012.)
⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ0) → (⌊‘(𝐴 / (2↑(𝑁 + 1)))) = (⌊‘((⌊‘(𝐴 / 2)) / (2↑𝑁))))
Theorem	dignn0ehalf 49728	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) ∈ ℕ0 ∧ 𝐴 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0) → ((𝐼 + 1)(digit‘2)𝐴) = (𝐼(digit‘2)(𝐴 / 2)))
Theorem	dignn0flhalf 49729	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))))
Theorem	nn0sumshdiglemA 49730*	Lemma for nn0sumshdig 49734 (induction step, even multiplier). (Contributed by AV, 3-Jun-2020.)
⊢ (((𝑎 ∈ ℕ ∧ (𝑎 / 2) ∈ ℕ) ∧ 𝑦 ∈ ℕ) → (∀𝑥 ∈ ℕ0 ((#b‘𝑥) = 𝑦 → 𝑥 = Σ𝑘 ∈ (0..^𝑦)((𝑘(digit‘2)𝑥) · (2↑𝑘))) → ((#b‘𝑎) = (𝑦 + 1) → 𝑎 = Σ𝑘 ∈ (0..^(𝑦 + 1))((𝑘(digit‘2)𝑎) · (2↑𝑘)))))
Theorem	nn0sumshdiglemB 49731*	Lemma for nn0sumshdig 49734 (induction step, odd multiplier). (Contributed by AV, 7-Jun-2020.)
⊢ (((𝑎 ∈ ℕ ∧ ((𝑎 − 1) / 2) ∈ ℕ0) ∧ 𝑦 ∈ ℕ) → (∀𝑥 ∈ ℕ0 ((#b‘𝑥) = 𝑦 → 𝑥 = Σ𝑘 ∈ (0..^𝑦)((𝑘(digit‘2)𝑥) · (2↑𝑘))) → ((#b‘𝑎) = (𝑦 + 1) → 𝑎 = Σ𝑘 ∈ (0..^(𝑦 + 1))((𝑘(digit‘2)𝑎) · (2↑𝑘)))))
Theorem	nn0sumshdiglem1 49732*	Lemma 1 for nn0sumshdig 49734 (induction step). (Contributed by AV, 7-Jun-2020.)
⊢ (𝑦 ∈ ℕ → (∀𝑎 ∈ ℕ0 ((#b‘𝑎) = 𝑦 → 𝑎 = Σ𝑘 ∈ (0..^𝑦)((𝑘(digit‘2)𝑎) · (2↑𝑘))) → ∀𝑎 ∈ ℕ0 ((#b‘𝑎) = (𝑦 + 1) → 𝑎 = Σ𝑘 ∈ (0..^(𝑦 + 1))((𝑘(digit‘2)𝑎) · (2↑𝑘)))))
Theorem	nn0sumshdiglem2 49733*	Lemma 2 for nn0sumshdig 49734. (Contributed by AV, 7-Jun-2020.)
⊢ (𝐿 ∈ ℕ → ∀𝑎 ∈ ℕ0 ((#b‘𝑎) = 𝐿 → 𝑎 = Σ𝑘 ∈ (0..^𝐿)((𝑘(digit‘2)𝑎) · (2↑𝑘))))
Theorem	nn0sumshdig 49734*	A nonnegative integer can be represented as sum of its shifted bits. (Contributed by AV, 7-Jun-2020.)
⊢ (𝐴 ∈ ℕ0 → 𝐴 = Σ𝑘 ∈ (0..^(#b‘𝐴))((𝑘(digit‘2)𝐴) · (2↑𝑘)))
21.50.22.11  Algorithms for the multiplication of nonnegative integers
Theorem	nn0mulfsum 49735*	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 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → (𝐴 · 𝐵) = Σ𝑘 ∈ (1...𝐴)𝐵)
Theorem	nn0mullong 49736*	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 16663. (Contributed by AV, 7-Jun-2020.)
⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → (𝐴 · 𝐵) = Σ𝑘 ∈ (0..^(#b‘𝐴))(((𝑘(digit‘2)𝐴) · (2↑𝑘)) · 𝐵))
21.50.22.12  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 arguments. 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 an element of X1 , ... , xn is an 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:𝑈⟶𝑌 where where the domain 𝑈 has the form 𝑈 ⊆ ((...((𝑋‘1) × (𝑋‘2)) × ...) × (𝑋‘𝑛))." In the following, n-ary functions are defined as mappings (see df-map 8849) 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 49746 and mapsn 8916: (𝑋 ↑m {∅}) - n = 1 (unary functions): These correspond actually to usual endofunctions, see 1aryenef 49756 and efmndbas 19067: (𝑋 ↑m 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 19067): "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 49767 and compare with df-clintop 49296: (𝑋 ↑m (𝑋 × 𝑋)). Instead of using indexed arguments (represented by a mapping as described above), elements of Cartesian exponentiations (𝑈↑↑𝑁) (see df-finxp 38307) 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 49740 shows that both definitions are equivalent.
Syntax	cnaryf 49737	Extend the definition of a class to include the n-ary functions.
class -aryF
Definition	df-naryf 49738*	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 ↦ (𝑥 ↑m (𝑥 ↑m (0..^𝑛))))
Theorem	naryfval 49739	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) = (𝑋 ↑m (𝑋 ↑m 𝐼)))
Theorem	naryfvalixp 49740*	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) = (𝑋 ↑m X𝑥 ∈ 𝐼 𝑋))
Theorem	naryfvalel 49741	An n-ary (endo)function on a set 𝑋. (Contributed by AV, 14-May-2024.)
⊢ 𝐼 = (0..^𝑁)    ⇒   ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑉) → (𝐹 ∈ (𝑁-aryF 𝑋) ↔ 𝐹:(𝑋 ↑m 𝐼)⟶𝑋))
Theorem	naryrcl 49742	Reverse closure for n-ary (endo)functions. (Contributed by AV, 14-May-2024.)
⊢ 𝐼 = (0..^𝑁)    ⇒   ⊢ (𝐹 ∈ (𝑁-aryF 𝑋) → (𝑁 ∈ ℕ0 ∧ 𝑋 ∈ V))
Theorem	naryfvalelfv 49743	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:𝐼⟶𝑋) → (𝐹‘𝐴) ∈ 𝑋)
Theorem	naryfvalelwrdf 49744*	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 ∈ ℕ0 ∧ 𝑋 ∈ 𝑉) → (𝐹 ∈ (𝑁-aryF 𝑋) ↔ 𝐹:{𝑤 ∈ Word 𝑋 ∣ (♯‘𝑤) = 𝑁}⟶𝑋))
Theorem	0aryfvalel 49745*	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 𝐶 ∈ 𝑋, see also 0aryfvalelfv 49746. Instead of (𝐹‘∅), nullary functions are usually written as 𝐹() in literature. (Contributed by AV, 15-May-2024.)
⊢ (𝑋 ∈ 𝑉 → (𝐹 ∈ (0-aryF 𝑋) ↔ ∃𝑥 ∈ 𝑋 𝐹 = {⟨∅, 𝑥⟩}))
Theorem	0aryfvalelfv 49746*	The value of a nullary (endo)function on a set 𝑋. (Contributed by AV, 19-May-2024.)
⊢ (𝐹 ∈ (0-aryF 𝑋) → ∃𝑥 ∈ 𝑋 (𝐹‘∅) = 𝑥)
Theorem	1aryfvalel 49747	A unary (endo)function on a set 𝑋. (Contributed by AV, 15-May-2024.)
⊢ (𝑋 ∈ 𝑉 → (𝐹 ∈ (1-aryF 𝑋) ↔ 𝐹:(𝑋 ↑m {0})⟶𝑋))
Theorem	fv1arycl 49748	Closure of a unary (endo)function. (Contributed by AV, 18-May-2024.)
⊢ ((𝐺 ∈ (1-aryF 𝑋) ∧ 𝐴 ∈ 𝑋) → (𝐺‘{⟨0, 𝐴⟩}) ∈ 𝑋)
Theorem	1arympt1 49749*	A unary (endo)function in maps-to notation. (Contributed by AV, 16-May-2024.)
⊢ 𝐹 = (𝑥 ∈ (𝑋 ↑m {0}) ↦ (𝐴‘(𝑥‘0)))    ⇒   ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴:𝑋⟶𝑋) → 𝐹 ∈ (1-aryF 𝑋))
Theorem	1arympt1fv 49750*	The value of a unary (endo)function in maps-to notation. (Contributed by AV, 16-May-2024.)
⊢ 𝐹 = (𝑥 ∈ (𝑋 ↑m {0}) ↦ (𝐴‘(𝑥‘0)))    ⇒   ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐵 ∈ 𝑋) → (𝐹‘{⟨0, 𝐵⟩}) = (𝐴‘𝐵))
Theorem	1arymaptfv 49751*	The value of the mapping of unary (endo)functions. (Contributed by AV, 18-May-2024.)
⊢ 𝐻 = (ℎ ∈ (1-aryF 𝑋) ↦ (𝑥 ∈ 𝑋 ↦ (ℎ‘{⟨0, 𝑥⟩})))    ⇒   ⊢ (𝐹 ∈ (1-aryF 𝑋) → (𝐻‘𝐹) = (𝑥 ∈ 𝑋 ↦ (𝐹‘{⟨0, 𝑥⟩})))
Theorem	1arymaptf 49752*	The mapping of unary (endo)functions is a function into the set of endofunctions. (Contributed by AV, 18-May-2024.)
⊢ 𝐻 = (ℎ ∈ (1-aryF 𝑋) ↦ (𝑥 ∈ 𝑋 ↦ (ℎ‘{⟨0, 𝑥⟩})))    ⇒   ⊢ (𝑋 ∈ 𝑉 → 𝐻:(1-aryF 𝑋)⟶(𝑋 ↑m 𝑋))
Theorem	1arymaptf1 49753*	The mapping of unary (endo)functions is a one-to-one function into the set of endofunctions. (Contributed by AV, 19-May-2024.)
⊢ 𝐻 = (ℎ ∈ (1-aryF 𝑋) ↦ (𝑥 ∈ 𝑋 ↦ (ℎ‘{⟨0, 𝑥⟩})))    ⇒   ⊢ (𝑋 ∈ 𝑉 → 𝐻:(1-aryF 𝑋)–1-1→(𝑋 ↑m 𝑋))
Theorem	1arymaptfo 49754*	The mapping of unary (endo)functions is a function onto the set of endofunctions. (Contributed by AV, 18-May-2024.)
⊢ 𝐻 = (ℎ ∈ (1-aryF 𝑋) ↦ (𝑥 ∈ 𝑋 ↦ (ℎ‘{⟨0, 𝑥⟩})))    ⇒   ⊢ (𝑋 ∈ 𝑉 → 𝐻:(1-aryF 𝑋)–onto→(𝑋 ↑m 𝑋))
Theorem	1arymaptf1o 49755*	The mapping of unary (endo)functions is a one-to-one function onto the set of endofunctions. (Contributed by AV, 19-May-2024.)
⊢ 𝐻 = (ℎ ∈ (1-aryF 𝑋) ↦ (𝑥 ∈ 𝑋 ↦ (ℎ‘{⟨0, 𝑥⟩})))    ⇒   ⊢ (𝑋 ∈ 𝑉 → 𝐻:(1-aryF 𝑋)–1-1-onto→(𝑋 ↑m 𝑋))
Theorem	1aryenef 49756	The set of unary (endo)functions and the set of endofunctions are equinumerous. (Contributed by AV, 19-May-2024.)
⊢ (1-aryF 𝑋) ≈ (𝑋 ↑m 𝑋)
Theorem	1aryenefmnd 49757	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))
Theorem	2aryfvalel 49758	A binary (endo)function on a set 𝑋. (Contributed by AV, 20-May-2024.)
⊢ (𝑋 ∈ 𝑉 → (𝐹 ∈ (2-aryF 𝑋) ↔ 𝐹:(𝑋 ↑m {0, 1})⟶𝑋))
Theorem	fv2arycl 49759	Closure of a binary (endo)function. (Contributed by AV, 20-May-2024.)
⊢ ((𝐺 ∈ (2-aryF 𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐺‘{⟨0, 𝐴⟩, ⟨1, 𝐵⟩}) ∈ 𝑋)
Theorem	2arympt 49760*	A binary (endo)function in maps-to notation. (Contributed by AV, 20-May-2024.)
⊢ 𝐹 = (𝑥 ∈ (𝑋 ↑m {0, 1}) ↦ ((𝑥‘0)𝑂(𝑥‘1)))    ⇒   ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑂:(𝑋 × 𝑋)⟶𝑋) → 𝐹 ∈ (2-aryF 𝑋))
Theorem	2arymptfv 49761*	The value of a binary (endo)function in maps-to notation. (Contributed by AV, 20-May-2024.)
⊢ 𝐹 = (𝑥 ∈ (𝑋 ↑m {0, 1}) ↦ ((𝑥‘0)𝑂(𝑥‘1)))    ⇒   ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐹‘{⟨0, 𝐴⟩, ⟨1, 𝐵⟩}) = (𝐴𝑂𝐵))
Theorem	2arymaptfv 49762*	The value of the mapping of binary (endo)functions. (Contributed by AV, 21-May-2024.)
⊢ 𝐻 = (ℎ ∈ (2-aryF 𝑋) ↦ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (ℎ‘{⟨0, 𝑥⟩, ⟨1, 𝑦⟩})))    ⇒   ⊢ (𝐹 ∈ (2-aryF 𝑋) → (𝐻‘𝐹) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝐹‘{⟨0, 𝑥⟩, ⟨1, 𝑦⟩})))
Theorem	2arymaptf 49763*	The mapping of binary (endo)functions is a function into the set of binary operations. (Contributed by AV, 21-May-2024.)
⊢ 𝐻 = (ℎ ∈ (2-aryF 𝑋) ↦ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (ℎ‘{⟨0, 𝑥⟩, ⟨1, 𝑦⟩})))    ⇒   ⊢ (𝑋 ∈ 𝑉 → 𝐻:(2-aryF 𝑋)⟶(𝑋 ↑m (𝑋 × 𝑋)))
Theorem	2arymaptf1 49764*	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 = (ℎ ∈ (2-aryF 𝑋) ↦ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (ℎ‘{⟨0, 𝑥⟩, ⟨1, 𝑦⟩})))    ⇒   ⊢ (𝑋 ∈ 𝑉 → 𝐻:(2-aryF 𝑋)–1-1→(𝑋 ↑m (𝑋 × 𝑋)))
Theorem	2arymaptfo 49765*	The mapping of binary (endo)functions is a function onto the set of binary operations. (Contributed by AV, 23-May-2024.)
⊢ 𝐻 = (ℎ ∈ (2-aryF 𝑋) ↦ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (ℎ‘{⟨0, 𝑥⟩, ⟨1, 𝑦⟩})))    ⇒   ⊢ (𝑋 ∈ 𝑉 → 𝐻:(2-aryF 𝑋)–onto→(𝑋 ↑m (𝑋 × 𝑋)))
Theorem	2arymaptf1o 49766*	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 = (ℎ ∈ (2-aryF 𝑋) ↦ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (ℎ‘{⟨0, 𝑥⟩, ⟨1, 𝑦⟩})))    ⇒   ⊢ (𝑋 ∈ 𝑉 → 𝐻:(2-aryF 𝑋)–1-1-onto→(𝑋 ↑m (𝑋 × 𝑋)))
Theorem	2aryenef 49767	The set of binary (endo)functions and the set of binary operations are equinumerous. (Contributed by AV, 19-May-2024.)
⊢ (2-aryF 𝑋) ≈ (𝑋 ↑m (𝑋 × 𝑋))
21.50.22.13  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 49807)], 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 49771, based on a definition IterComp of "the n-th iterate of (a class/function) f", see df-itco 49770. As an illustration, we have ((IterComp‘𝐹)‘3) = (𝐹 ∘ (𝐹 ∘ 𝐹))) (see itcoval3 49776). The following recursive definition of the Ackermann function follows immediately from Definition df-ack 49771: ((Ack‘(𝑀 + 1))‘𝑁) = (((IterComp‘(Ack‘𝑀))‘(𝑁 + 1))‘1)). That Definition df-ack 49771 is equivalent to Péter's definition is proven by the following three theorems: ackval0val 49797: ((Ack‘0)‘𝑀) = (𝑀 + 1); ackvalsuc0val 49798: ((Ack‘(𝑀 + 1))‘0) = ((Ack‘𝑀)‘1); ackvalsucsucval 49799: ((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 49800: 𝐴(0, 0) = 1, 𝐴(0, 1) = 2, 𝐴(0, 2) = 3; ackval1012 49801: 𝐴(1, 0) = 2, 𝐴(1, 1) = 3, 𝐴(1, 3) = 4; ackval2012 49802: 𝐴(2, 0) = 3, 𝐴(2, 1) = 5, 𝐴(2, 3) = 7; ackval3012 49803: 𝐴(3, 0) = 5, 𝐴(3, 1) = ;13, 𝐴(3, 3) = ;29.
Syntax	citco 49768	Extend the definition of a class to include iterated functions.
class IterComp
Syntax	cack 49769	Extend the definition of a class to include the Ackermann function operator.
class Ack
Definition	df-itco 49770*	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 ↦ (𝑓 ∘ 𝑔)), (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, ( I ↾ dom 𝑓), 𝑓))))
Definition	df-ack 49771*	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)))
Theorem	itcoval 49772*	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.)
⊢ (𝐹 ∈ 𝑉 → (IterComp‘𝐹) = seq0((𝑔 ∈ V, 𝑗 ∈ V ↦ (𝐹 ∘ 𝑔)), (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, ( I ↾ dom 𝐹), 𝐹))))
Theorem	itcoval0 49773	A function iterated zero times (defined as identity function). (Contributed by AV, 2-May-2024.)
⊢ (𝐹 ∈ 𝑉 → ((IterComp‘𝐹)‘0) = ( I ↾ dom 𝐹))
Theorem	itcoval1 49774	A function iterated once. (Contributed by AV, 2-May-2024.)
⊢ ((Rel 𝐹 ∧ 𝐹 ∈ 𝑉) → ((IterComp‘𝐹)‘1) = 𝐹)
Theorem	itcoval2 49775	A function iterated twice. (Contributed by AV, 2-May-2024.)
⊢ ((Rel 𝐹 ∧ 𝐹 ∈ 𝑉) → ((IterComp‘𝐹)‘2) = (𝐹 ∘ 𝐹))
Theorem	itcoval3 49776	A function iterated three times. (Contributed by AV, 2-May-2024.)
⊢ ((Rel 𝐹 ∧ 𝐹 ∈ 𝑉) → ((IterComp‘𝐹)‘3) = (𝐹 ∘ (𝐹 ∘ 𝐹)))
Theorem	itcoval0mpt 49777*	A mapping iterated zero times (defined as identity function). (Contributed by AV, 4-May-2024.)
⊢ 𝐹 = (𝑛 ∈ 𝐴 ↦ 𝐵)    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑛 ∈ 𝐴 𝐵 ∈ 𝑊) → ((IterComp‘𝐹)‘0) = (𝑛 ∈ 𝐴 ↦ 𝑛))
Theorem	itcovalsuc 49778*	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.)
⊢ ((𝐹 ∈ 𝑉 ∧ 𝑌 ∈ ℕ0 ∧ ((IterComp‘𝐹)‘𝑌) = 𝐺) → ((IterComp‘𝐹)‘(𝑌 + 1)) = (𝐺(𝑔 ∈ V, 𝑗 ∈ V ↦ (𝐹 ∘ 𝑔))𝐹))
Theorem	itcovalsucov 49779	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.)
⊢ ((𝐹 ∈ 𝑉 ∧ 𝑌 ∈ ℕ0 ∧ ((IterComp‘𝐹)‘𝑌) = 𝐺) → ((IterComp‘𝐹)‘(𝑌 + 1)) = (𝐹 ∘ 𝐺))
Theorem	itcovalendof 49780	The n-th iterate of an endofunction is an endofunction. (Contributed by AV, 7-May-2024.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐴)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    ⇒   ⊢ (𝜑 → ((IterComp‘𝐹)‘𝑁):𝐴⟶𝐴)
Theorem	itcovalpclem1 49781*	Lemma 1 for itcovalpc 49783: induction basis. (Contributed by AV, 4-May-2024.)
⊢ 𝐹 = (𝑛 ∈ ℕ0 ↦ (𝑛 + 𝐶))    ⇒   ⊢ (𝐶 ∈ ℕ0 → ((IterComp‘𝐹)‘0) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 0))))
Theorem	itcovalpclem2 49782*	Lemma 2 for itcovalpc 49783: induction step. (Contributed by AV, 4-May-2024.)
⊢ 𝐹 = (𝑛 ∈ ℕ0 ↦ (𝑛 + 𝐶))    ⇒   ⊢ ((𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) → (((IterComp‘𝐹)‘𝑦) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑦))) → ((IterComp‘𝐹)‘(𝑦 + 1)) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · (𝑦 + 1))))))
Theorem	itcovalpc 49783*	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 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) → ((IterComp‘𝐹)‘𝐼) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝐼))))
Theorem	itcovalt2lem2lem1 49784	Lemma 1 for itcovalt2lem2 49787. (Contributed by AV, 6-May-2024.)
⊢ (((𝑌 ∈ ℕ ∧ 𝐶 ∈ ℕ0) ∧ 𝑁 ∈ ℕ0) → (((𝑁 + 𝐶) · 𝑌) − 𝐶) ∈ ℕ0)
Theorem	itcovalt2lem2lem2 49785	Lemma 2 for itcovalt2lem2 49787. (Contributed by AV, 7-May-2024.)
⊢ (((𝑌 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) ∧ 𝑁 ∈ ℕ0) → ((2 · (((𝑁 + 𝐶) · (2↑𝑌)) − 𝐶)) + 𝐶) = (((𝑁 + 𝐶) · (2↑(𝑌 + 1))) − 𝐶))
Theorem	itcovalt2lem1 49786*	Lemma 1 for itcovalt2 49788: 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)))
Theorem	itcovalt2lem2 49787*	Lemma 2 for itcovalt2 49788: induction step. (Contributed by AV, 7-May-2024.)
⊢ 𝐹 = (𝑛 ∈ ℕ0 ↦ ((2 · 𝑛) + 𝐶))    ⇒   ⊢ ((𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) → (((IterComp‘𝐹)‘𝑦) = (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑𝑦)) − 𝐶)) → ((IterComp‘𝐹)‘(𝑦 + 1)) = (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑(𝑦 + 1))) − 𝐶))))
Theorem	itcovalt2 49788*	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 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) → ((IterComp‘𝐹)‘𝐼) = (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑𝐼)) − 𝐶)))
Theorem	ackvalsuc1mpt 49789*	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)))
Theorem	ackvalsuc1 49790	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 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ((Ack‘(𝑀 + 1))‘𝑁) = (((IterComp‘(Ack‘𝑀))‘(𝑁 + 1))‘1))
Theorem	ackval0 49791	The Ackermann function at 0. (Contributed by AV, 2-May-2024.)
⊢ (Ack‘0) = (𝑛 ∈ ℕ0 ↦ (𝑛 + 1))
Theorem	ackval1 49792	The Ackermann function at 1. (Contributed by AV, 4-May-2024.)
⊢ (Ack‘1) = (𝑛 ∈ ℕ0 ↦ (𝑛 + 2))
Theorem	ackval2 49793	The Ackermann function at 2. (Contributed by AV, 4-May-2024.)
⊢ (Ack‘2) = (𝑛 ∈ ℕ0 ↦ ((2 · 𝑛) + 3))
Theorem	ackval3 49794	The Ackermann function at 3. (Contributed by AV, 7-May-2024.)
⊢ (Ack‘3) = (𝑛 ∈ ℕ0 ↦ ((2↑(𝑛 + 3)) − 3))
Theorem	ackendofnn0 49795	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)
Theorem	ackfnnn0 49796	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)
Theorem	ackval0val 49797	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))
Theorem	ackvalsuc0val 49798	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))
Theorem	ackvalsucsucval 49799	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 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ((Ack‘(𝑀 + 1))‘(𝑁 + 1)) = ((Ack‘𝑀)‘((Ack‘(𝑀 + 1))‘𝑁)))
Theorem	ackval0012 49800	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⟩