P. 451 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 45001-45100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	nnpw2pb 45001*	A number is a positive integer iff it can be represented as the sum of a power of 2 and a "remainder" smaller than the power. (Contributed by AV, 31-May-2020.)
⊢ (𝑁 ∈ ℕ ↔ ∃𝑖 ∈ ℕ0 ∃𝑟 ∈ (0..^(2↑𝑖))𝑁 = ((2↑𝑖) + 𝑟))
Theorem	blen1b 45002	The binary length of a nonnegative integer is 1 if the integer is 0 or 1. (Contributed by AV, 30-May-2020.)
⊢ (𝑁 ∈ ℕ0 → ((#b‘𝑁) = 1 ↔ (𝑁 = 0 ∨ 𝑁 = 1)))
Theorem	blennnt2 45003	The binary length of a positive integer, doubled and increased by 1, is the binary length of the integer plus 1. (Contributed by AV, 30-May-2010.)
⊢ (𝑁 ∈ ℕ → (#b‘(2 · 𝑁)) = ((#b‘𝑁) + 1))
Theorem	nnolog2flm1 45004	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 45005	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 45006	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 45007	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 45008	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))
20.41.22.10  Digits Generalization of df-bits 15761. In contrast to digit, bits are defined for integers only. The equivalence of both definitions for integers is shown in dig2bits 45028: ((𝐾(digit 2 ) N ) = 1 <-> K e. ( bits 𝑁)).
Syntax	cdig 45009	Extend class notation with the class of the digit extraction operation.
class digit
Definition	df-dig 45010*	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 13594. 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 45011*	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 45012	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 45013	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 45014	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 45015	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 45016	Lemma for dignnld 45017. (Contributed by AV, 25-May-2020.)
⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ≥‘((⌊‘(𝐵 logb 𝑁)) + 1))) → 𝑁 < (𝐵↑𝐾))
Theorem	dignnld 45017	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 45018	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 45019	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 45020	All digits of 0 are 0. (Contributed by AV, 24-May-2020.)
⊢ ((𝐵 ∈ ℕ ∧ 𝐾 ∈ ℤ) → (𝐾(digit‘𝐵)0) = 0)
Theorem	digexp 45021	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 45022	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 45023	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 45024	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 45025	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 45026	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 45027	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 45028	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)))
20.41.22.11  Nonnegative integer as sum of its shifted digits
Theorem	dignn0flhalflem1 45029	Lemma 1 for dignn0flhalf 45032. (Contributed by AV, 7-Jun-2012.)
⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ) → (⌊‘((𝐴 / (2↑𝑁)) − 1)) < (⌊‘((𝐴 − 1) / (2↑𝑁))))
Theorem	dignn0flhalflem2 45030	Lemma 2 for dignn0flhalf 45032. (Contributed by AV, 7-Jun-2012.)
⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ0) → (⌊‘(𝐴 / (2↑(𝑁 + 1)))) = (⌊‘((⌊‘(𝐴 / 2)) / (2↑𝑁))))
Theorem	dignn0ehalf 45031	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 45032	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 45033*	Lemma for nn0sumshdig 45037 (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 45034*	Lemma for nn0sumshdig 45037 (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 45035*	Lemma 1 for nn0sumshdig 45037 (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 45036*	Lemma 2 for nn0sumshdig 45037. (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 45037*	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)))
20.41.22.12  Algorithms for the multiplication of nonnegative integers
Theorem	nn0mulfsum 45038*	Trivial algorithm to calculate the product of two nonnegative integers 𝑎 and 𝑏 by adding up 𝑏 𝑎 times. (Contributed by AV, 17-May-2020.)
⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → (𝐴 · 𝐵) = Σ𝑘 ∈ (1...𝐴)𝐵)
Theorem	nn0mullong 45039*	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 15832. (Contributed by AV, 7-Jun-2020.)
⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → (𝐴 · 𝐵) = Σ𝑘 ∈ (0..^(#b‘𝐴))(((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: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 8391) 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 45049 and mapsn 8435: (𝑋 ↑m {∅}) - n = 1 (unary functions): These correspond actually to usual endofunctions, see 1aryenef 45059 and efmndbas 18028: (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 18028): "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 45070 and compare with df-clintop 44460: (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 34801) 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 45043 shows that both definitions are equivalent.
Syntax	cnaryf 45040	Extend the definition of a class to include the n-ary functions.
class -aryF
Definition	df-naryf 45041*	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 45042	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 45043*	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 45044	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 45045	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 45046	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 45047*	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 45048*	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 45049. 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 45049*	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 45050	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 45051	Closure of a unary (endo)function. (Contributed by AV, 18-May-2024.)
⊢ ((𝐺 ∈ (1-aryF 𝑋) ∧ 𝐴 ∈ 𝑋) → (𝐺‘{⟨0, 𝐴⟩}) ∈ 𝑋)
Theorem	1arympt1 45052*	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 45053*	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 45054*	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 45055*	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 45056*	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 45057*	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 45058*	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 45059	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 45060	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 45061	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 45062	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 45063*	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 45064*	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 45065*	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 45066*	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 45067*	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 45068*	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 45069*	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 45070	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))
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 45110)], 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 45074, based on a definition IterComp of "the n-th iterate of (a class/function) f", see df-itco 45073. As an illustration, we have ((IterComp‘𝐹)‘3) = (𝐹 ∘ (𝐹 ∘ 𝐹))) (see itcoval3 45079). The following recursive definition of the Ackermann function follows immediately from the definition df-ack 45074: ((Ack‘(𝑀 + 1))‘𝑁) = (((IterComp‘(Ack‘𝑀))‘(𝑁 + 1))‘1)). That the definition df-ack 45074 is equivalent to Péter's definition is proven by the following three theorems: ackval0val 45100: ((Ack‘0)‘𝑀) = (𝑀 + 1) ackvalsuc0val 45101: ((Ack‘(𝑀 + 1))‘0) = ((Ack‘𝑀)‘1) ackvalsucsucval 45102: ((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 45103: 𝐴(0, 0) = 1, 𝐴(0, 1) = 2, 𝐴(0, 2) = 3 ackval1012 45104: 𝐴(1, 0) = 2, 𝐴(1, 1) = 3, 𝐴(1, 3) = 4 ackval2012 45105: 𝐴(2, 0) = 3, 𝐴(2, 1) = 5, 𝐴(2, 3) = 7 ackval3012 45106: 𝐴(3, 0) = 5, 𝐴(3, 1) = ;13, 𝐴(3, 3) = ;29
Syntax	citco 45071	Extend the definition of a class to include iterated functions.
class IterComp
Syntax	cack 45072	Extend the definition of a class to include the Ackermann function operator.
class Ack
Definition	df-itco 45073*	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 45074*	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 45075*	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 45076	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 45077	A function iterated once. (Contributed by AV, 2-May-2024.)
⊢ ((Rel 𝐹 ∧ 𝐹 ∈ 𝑉) → ((IterComp‘𝐹)‘1) = 𝐹)
Theorem	itcoval2 45078	A function iterated twice. (Contributed by AV, 2-May-2024.)
⊢ ((Rel 𝐹 ∧ 𝐹 ∈ 𝑉) → ((IterComp‘𝐹)‘2) = (𝐹 ∘ 𝐹))
Theorem	itcoval3 45079	A function iterated three times. (Contributed by AV, 2-May-2024.)
⊢ ((Rel 𝐹 ∧ 𝐹 ∈ 𝑉) → ((IterComp‘𝐹)‘3) = (𝐹 ∘ (𝐹 ∘ 𝐹)))
Theorem	itcoval0mpt 45080*	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 45081*	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 45082	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 45083	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 45084*	Lemma 1 for itcovalpc 45086: 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 45085*	Lemma 2 for itcovalpc 45086: 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 45086*	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 45087	Lemma 1 for itcovalt2lem2 45090. (Contributed by AV, 6-May-2024.)
⊢ (((𝑌 ∈ ℕ ∧ 𝐶 ∈ ℕ0) ∧ 𝑁 ∈ ℕ0) → (((𝑁 + 𝐶) · 𝑌) − 𝐶) ∈ ℕ0)
Theorem	itcovalt2lem2lem2 45088	Lemma 2 for itcovalt2lem2 45090. (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 45089*	Lemma 1 for itcovalt2 45091: 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 45090*	Lemma 2 for itcovalt2 45091: 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 45091*	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 45092*	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 45093	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 45094	The Ackermann function at 0. (Contributed by AV, 2-May-2024.)
⊢ (Ack‘0) = (𝑛 ∈ ℕ0 ↦ (𝑛 + 1))
Theorem	ackval1 45095	The Ackermann function at 1. (Contributed by AV, 4-May-2024.)
⊢ (Ack‘1) = (𝑛 ∈ ℕ0 ↦ (𝑛 + 2))
Theorem	ackval2 45096	The Ackermann function at 2. (Contributed by AV, 4-May-2024.)
⊢ (Ack‘2) = (𝑛 ∈ ℕ0 ↦ ((2 · 𝑛) + 3))
Theorem	ackval3 45097	The Ackermann function at 3. (Contributed by AV, 7-May-2024.)
⊢ (Ack‘3) = (𝑛 ∈ ℕ0 ↦ ((2↑(𝑛 + 3)) − 3))
Theorem	ackendofnn0 45098	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 45099	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 45100	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))