P. 499 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 49801-49900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	elbigo2r 49801*	Sufficient condition for a function to be of order G(x). (Contributed by AV, 18-May-2020.)
⊢ (((𝐺:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) ∧ (𝐹:𝐵⟶ℝ ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ ℝ ∧ 𝑀 ∈ ℝ ∧ ∀𝑥 ∈ 𝐵 (𝐶 ≤ 𝑥 → (𝐹‘𝑥) ≤ (𝑀 · (𝐺‘𝑥))))) → 𝐹 ∈ (Ο‘𝐺))
Theorem	elbigof 49802	A function of order G(x) is a function. (Contributed by AV, 18-May-2020.)
⊢ (𝐹 ∈ (Ο‘𝐺) → 𝐹:dom 𝐹⟶ℝ)
Theorem	elbigodm 49803	The domain of a function of order G(x) is a subset of the reals. (Contributed by AV, 18-May-2020.)
⊢ (𝐹 ∈ (Ο‘𝐺) → dom 𝐹 ⊆ ℝ)
Theorem	elbigoimp 49804*	The defining property of a function of order G(x). (Contributed by AV, 18-May-2020.)
⊢ ((𝐹 ∈ (Ο‘𝐺) ∧ 𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ dom 𝐺) → ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑦) ≤ (𝑚 · (𝐺‘𝑦))))
Theorem	elbigolo1 49805	A function (into the positive reals) is of order G(x) iff the quotient of the function and G(x) (also a function into the positive reals) is an eventually upper bounded function. (Contributed by AV, 20-May-2020.) (Proof shortened by II, 16-Feb-2023.)
⊢ ((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ+ ∧ 𝐹:𝐴⟶ℝ+) → (𝐹 ∈ (Ο‘𝐺) ↔ (𝐹 /f 𝐺) ∈ ≤𝑂(1)))
21.52.22.6  Logarithm to an arbitrary base (extension)
Theorem	rege1logbrege0 49806	The general logarithm, with a real base greater than 1, for a real number greater than or equal to 1 is greater than or equal to 0. (Contributed by AV, 25-May-2020.)
⊢ ((𝐵 ∈ (1(,)+∞) ∧ 𝑋 ∈ (1[,)+∞)) → 0 ≤ (𝐵 logb 𝑋))
Theorem	rege1logbzge0 49807	The general logarithm, with an integer base greater than 1, for a real number greater than or equal to 1 is greater than or equal to 0. (Contributed by AV, 25-May-2020.)
⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ 𝑋 ∈ (1[,)+∞)) → 0 ≤ (𝐵 logb 𝑋))
Theorem	fllogbd 49808	A real number is between the base of a logarithm to the power of the floor of the logarithm of the number and the base of the logarithm to the power of the floor of the logarithm of the number plus one. (Contributed by AV, 23-May-2020.)
⊢ (𝜑 → 𝐵 ∈ (ℤ≥‘2))    &   ⊢ (𝜑 → 𝑋 ∈ ℝ+)    &   ⊢ 𝐸 = (⌊‘(𝐵 logb 𝑋))    ⇒   ⊢ (𝜑 → ((𝐵↑𝐸) ≤ 𝑋 ∧ 𝑋 < (𝐵↑(𝐸 + 1))))
Theorem	relogbmulbexp 49809	The logarithm of the product of a positive real number and the base to the power of a real number is the logarithm of the positive real number plus the real number. (Contributed by AV, 29-May-2020.)
⊢ ((𝐵 ∈ (ℝ+ ∖ {1}) ∧ (𝐴 ∈ ℝ+ ∧ 𝐶 ∈ ℝ)) → (𝐵 logb (𝐴 · (𝐵↑𝑐𝐶))) = ((𝐵 logb 𝐴) + 𝐶))
Theorem	relogbdivb 49810	The logarithm of the quotient of a positive real number and the base is the logarithm of the number minus 1. (Contributed by AV, 29-May-2020.)
⊢ ((𝐵 ∈ (ℝ+ ∖ {1}) ∧ 𝐴 ∈ ℝ+) → (𝐵 logb (𝐴 / 𝐵)) = ((𝐵 logb 𝐴) − 1))
Theorem	logbge0b 49811	The logarithm of a number is nonnegative iff the number is greater than or equal to 1. (Contributed by AV, 30-May-2020.)
⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ 𝑋 ∈ ℝ+) → (0 ≤ (𝐵 logb 𝑋) ↔ 1 ≤ 𝑋))
Theorem	logblt1b 49812	The logarithm of a number is less than 1 iff the number is less than the base of the logarithm. (Contributed by AV, 30-May-2020.)
⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ 𝑋 ∈ ℝ+) → ((𝐵 logb 𝑋) < 1 ↔ 𝑋 < 𝐵))
21.52.22.7  The binary logarithm If the binary logarithm is used more often, a separate symbol/definition could be provided for it, e.g., log2 = (𝑥 ∈ (ℂ ∖ {0}) ↦ (2 logb 𝑋)). Then we can write "( log2 ` x )" (analogous to (log𝑥) for the natural logarithm) instead of (2 logb 𝑥).
Theorem	fldivexpfllog2 49813	The floor of a positive real number divided by 2 to the power of the floor of the logarithm to base 2 of the number is 1. (Contributed by AV, 26-May-2020.)
⊢ (𝑋 ∈ ℝ+ → (⌊‘(𝑋 / (2↑(⌊‘(2 logb 𝑋))))) = 1)
Theorem	nnlog2ge0lt1 49814	A positive integer is 1 iff its binary logarithm is between 0 and 1. (Contributed by AV, 30-May-2020.)
⊢ (𝑁 ∈ ℕ → (𝑁 = 1 ↔ (0 ≤ (2 logb 𝑁) ∧ (2 logb 𝑁) < 1)))
Theorem	logbpw2m1 49815	The floor of the binary logarithm of 2 to the power of a positive integer minus 1 is equal to the integer minus 1. (Contributed by AV, 31-May-2020.)
⊢ (𝐼 ∈ ℕ → (⌊‘(2 logb ((2↑𝐼) − 1))) = (𝐼 − 1))
Theorem	fllog2 49816	The floor of the binary logarithm of 2 to the power of an element of a half-open integer interval bounded by powers of 2 is equal to the integer. (Contributed by AV, 31-May-2020.)
⊢ ((𝐼 ∈ ℕ0 ∧ 𝑁 ∈ ((2↑𝐼)..^(2↑(𝐼 + 1)))) → (⌊‘(2 logb 𝑁)) = 𝐼)
21.52.22.8  Binary length
Syntax	cblen 49817	Extend class notation with the class of the binary length function.
class #b
Definition	df-blen 49818	Define the binary length of an integer. Definition in section 1.3 of [AhoHopUll] p. 12. Although not restricted to integers, this definition is only meaningful for 𝑛 ∈ ℤ or even for 𝑛 ∈ ℂ. (Contributed by AV, 16-May-2020.)
⊢ #b = (𝑛 ∈ V ↦ if(𝑛 = 0, 1, ((⌊‘(2 logb (abs‘𝑛))) + 1)))
Theorem	blenval 49819	The binary length of an integer. (Contributed by AV, 20-May-2020.)
⊢ (𝑁 ∈ 𝑉 → (#b‘𝑁) = if(𝑁 = 0, 1, ((⌊‘(2 logb (abs‘𝑁))) + 1)))
Theorem	blen0 49820	The binary length of 0. (Contributed by AV, 20-May-2020.)
⊢ (#b‘0) = 1
Theorem	blenn0 49821	The binary length of a "number" not being 0. (Contributed by AV, 20-May-2020.)
⊢ ((𝑁 ∈ 𝑉 ∧ 𝑁 ≠ 0) → (#b‘𝑁) = ((⌊‘(2 logb (abs‘𝑁))) + 1))
Theorem	blenre 49822	The binary length of a positive real number. (Contributed by AV, 20-May-2020.)
⊢ (𝑁 ∈ ℝ+ → (#b‘𝑁) = ((⌊‘(2 logb 𝑁)) + 1))
Theorem	blennn 49823	The binary length of a positive integer. (Contributed by AV, 21-May-2020.)
⊢ (𝑁 ∈ ℕ → (#b‘𝑁) = ((⌊‘(2 logb 𝑁)) + 1))
Theorem	blennnelnn 49824	The binary length of a positive integer is a positive integer. (Contributed by AV, 25-May-2020.)
⊢ (𝑁 ∈ ℕ → (#b‘𝑁) ∈ ℕ)
Theorem	blennn0elnn 49825	The binary length of a nonnegative integer is a positive integer. (Contributed by AV, 28-May-2020.)
⊢ (𝑁 ∈ ℕ0 → (#b‘𝑁) ∈ ℕ)
Theorem	blenpw2 49826	The binary length of a power of 2 is the exponent plus 1. (Contributed by AV, 30-May-2020.)
⊢ (𝐼 ∈ ℕ0 → (#b‘(2↑𝐼)) = (𝐼 + 1))
Theorem	blenpw2m1 49827	The binary length of a power of 2 minus 1 is the exponent. (Contributed by AV, 31-May-2020.)
⊢ (𝐼 ∈ ℕ → (#b‘((2↑𝐼) − 1)) = 𝐼)
Theorem	nnpw2blen 49828	A positive integer is between 2 to the power of its binary length minus 1 and 2 to the power of its binary length. (Contributed by AV, 31-May-2020.)
⊢ (𝑁 ∈ ℕ → ((2↑((#b‘𝑁) − 1)) ≤ 𝑁 ∧ 𝑁 < (2↑(#b‘𝑁))))
Theorem	nnpw2blenfzo 49829	A positive integer is between 2 to the power of the binary length of the integer minus 1, and 2 to the power of the binary length of the integer. (Contributed by AV, 2-Jun-2020.)
⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ((2↑((#b‘𝑁) − 1))..^(2↑(#b‘𝑁))))
Theorem	nnpw2blenfzo2 49830	A positive integer is either 2 to the power of the binary length of the integer minus 1, or between 2 to the power of the binary length of the integer minus 1, increased by 1, and 2 to the power of the binary length of the integer. (Contributed by AV, 2-Jun-2020.)
⊢ (𝑁 ∈ ℕ → (𝑁 = (2↑((#b‘𝑁) − 1)) ∨ 𝑁 ∈ (((2↑((#b‘𝑁) − 1)) + 1)..^(2↑(#b‘𝑁)))))
Theorem	nnpw2pmod 49831	Every positive integer can be represented as the sum of a power of 2 and a "remainder" less than the power. (Contributed by AV, 31-May-2020.)
⊢ (𝑁 ∈ ℕ → 𝑁 = ((2↑((#b‘𝑁) − 1)) + (𝑁 mod (2↑((#b‘𝑁) − 1)))))
Theorem	blen1 49832	The binary length of 1. (Contributed by AV, 21-May-2020.)
⊢ (#b‘1) = 1
Theorem	blen2 49833	The binary length of 2. (Contributed by AV, 21-May-2020.)
⊢ (#b‘2) = 2
Theorem	nnpw2p 49834*	Every positive integer can be represented as the sum of a power of 2 and a "remainder" less than the power. (Contributed by AV, 31-May-2020.)
⊢ (𝑁 ∈ ℕ → ∃𝑖 ∈ ℕ0 ∃𝑟 ∈ (0..^(2↑𝑖))𝑁 = ((2↑𝑖) + 𝑟))
Theorem	nnpw2pb 49835*	A number is a positive integer iff it can be represented as the sum of a power of 2 and a "remainder" less than the power. (Contributed by AV, 31-May-2020.)
⊢ (𝑁 ∈ ℕ ↔ ∃𝑖 ∈ ℕ0 ∃𝑟 ∈ (0..^(2↑𝑖))𝑁 = ((2↑𝑖) + 𝑟))
Theorem	blen1b 49836	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 49837	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 49838	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 49839	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 49840	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 49841	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 49842	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))
21.52.22.9  Digits Generalization of df-bits 16601. In contrast to digit, bits are defined for integers only. The equivalence of both definitions for integers is shown in dig2bits 49862: if 𝐾 and 𝑁 are nonnegative integers, then ((𝐾(digit‘2)𝑁) = 1 ↔ 𝐾 ∈ (bits‘𝑁)).
Syntax	cdig 49843	Extend class notation with the class of the digit extraction operation.
class digit
Definition	df-dig 49844*	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 14388. 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 49845*	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 49846	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 49847	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 49848	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 49849	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 49850	Lemma for dignnld 49851. (Contributed by AV, 25-May-2020.)
⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ≥‘((⌊‘(𝐵 logb 𝑁)) + 1))) → 𝑁 < (𝐵↑𝐾))
Theorem	dignnld 49851	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 49852	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 49853	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 49854	All digits of 0 are 0. (Contributed by AV, 24-May-2020.)
⊢ ((𝐵 ∈ ℕ ∧ 𝐾 ∈ ℤ) → (𝐾(digit‘𝐵)0) = 0)
Theorem	digexp 49855	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 49856	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 49857	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 49858	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 49859	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 49860	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 49861	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 49862	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)))
21.52.22.10  Nonnegative integer as sum of its shifted digits
Theorem	dignn0flhalflem1 49863	Lemma 1 for dignn0flhalf 49866. (Contributed by AV, 7-Jun-2012.)
⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ) → (⌊‘((𝐴 / (2↑𝑁)) − 1)) < (⌊‘((𝐴 − 1) / (2↑𝑁))))
Theorem	dignn0flhalflem2 49864	Lemma 2 for dignn0flhalf 49866. (Contributed by AV, 7-Jun-2012.)
⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ0) → (⌊‘(𝐴 / (2↑(𝑁 + 1)))) = (⌊‘((⌊‘(𝐴 / 2)) / (2↑𝑁))))
Theorem	dignn0ehalf 49865	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 49866	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 49867*	Lemma for nn0sumshdig 49871 (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 49868*	Lemma for nn0sumshdig 49871 (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 49869*	Lemma 1 for nn0sumshdig 49871 (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 49870*	Lemma 2 for nn0sumshdig 49871. (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 49871*	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)))
21.52.22.11  Algorithms for the multiplication of nonnegative integers
Theorem	nn0mulfsum 49872*	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 49873*	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 16672. (Contributed by AV, 7-Jun-2020.)
⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → (𝐴 · 𝐵) = Σ𝑘 ∈ (0..^(#b‘𝐴))(((𝑘(digit‘2)𝐴) · (2↑𝑘)) · 𝐵))
21.52.22.12  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 8857) 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 49883 and mapsn 8924: (𝑋 ↑m {∅}) - n = 1 (unary functions): These correspond actually to usual endofunctions, see 1aryenef 49893 and efmndbas 19076: (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 19076): "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 49904 and compare with df-clintop 49433: (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 38444) 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 49877 shows that both definitions are equivalent.
Syntax	cnaryf 49874	Extend the definition of a class to include the n-ary functions.
class -aryF
Definition	df-naryf 49875*	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 49876	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 49877*	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 49878	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 49879	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 49880	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 49881*	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 49882*	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 49883. 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 49883*	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 49884	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 49885	Closure of a unary (endo)function. (Contributed by AV, 18-May-2024.)
⊢ ((𝐺 ∈ (1-aryF 𝑋) ∧ 𝐴 ∈ 𝑋) → (𝐺‘{⟨0, 𝐴⟩}) ∈ 𝑋)
Theorem	1arympt1 49886*	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 49887*	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 49888*	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 49889*	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 49890*	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 49891*	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 49892*	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 49893	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 49894	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 49895	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 49896	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 49897*	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 49898*	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 49899*	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 49900*	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)))