P. 491 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 49001-49100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	nnennex 49001*	For each even positive integer there is a positive integer which, multiplied by 2, results in the even positive integer. (Contributed by AV, 5-Jun-2023.)
⊢ ((𝑁 ∈ ℕ ∧ (𝑁 / 2) ∈ ℕ) → ∃𝑚 ∈ ℕ 𝑁 = (2 · 𝑚))
Theorem	nneop 49002	A positive integer is even or odd. (Contributed by AV, 30-May-2020.)
⊢ (𝑁 ∈ ℕ → ((𝑁 / 2) ∈ ℕ ∨ ((𝑁 + 1) / 2) ∈ ℕ))
Theorem	nneom 49003	A positive integer is even or odd. (Contributed by AV, 30-May-2020.)
⊢ (𝑁 ∈ ℕ → ((𝑁 / 2) ∈ ℕ ∨ ((𝑁 − 1) / 2) ∈ ℕ0))
Theorem	nn0eo 49004	A nonnegative integer is even or odd. (Contributed by AV, 27-May-2020.)
⊢ (𝑁 ∈ ℕ0 → ((𝑁 / 2) ∈ ℕ0 ∨ ((𝑁 + 1) / 2) ∈ ℕ0))
Theorem	nnpw2even 49005	2 to the power of a positive integer is even. (Contributed by AV, 2-Jun-2020.)
⊢ (𝑁 ∈ ℕ → ((2↑𝑁) / 2) ∈ ℕ)
Theorem	zefldiv2 49006	The floor of an even integer divided by 2 is equal to the integer divided by 2. (Contributed by AV, 7-Jun-2020.)
⊢ ((𝑁 ∈ ℤ ∧ (𝑁 / 2) ∈ ℤ) → (⌊‘(𝑁 / 2)) = (𝑁 / 2))
Theorem	zofldiv2 49007	The floor of an odd integer divided by 2 is equal to the integer first decreased by 1 and then divided by 2. (Contributed by AV, 7-Jun-2020.)
⊢ ((𝑁 ∈ ℤ ∧ ((𝑁 + 1) / 2) ∈ ℤ) → (⌊‘(𝑁 / 2)) = ((𝑁 − 1) / 2))
Theorem	nn0ofldiv2 49008	The floor of an odd nonnegative integer divided by 2 is equal to the integer first decreased by 1 and then divided by 2. (Contributed by AV, 1-Jun-2020.) (Proof shortened by AV, 7-Jun-2020.)
⊢ ((𝑁 ∈ ℕ0 ∧ ((𝑁 + 1) / 2) ∈ ℕ0) → (⌊‘(𝑁 / 2)) = ((𝑁 − 1) / 2))
Theorem	flnn0div2ge 49009	The floor of a positive integer divided by 2 is greater than or equal to the integer decreased by 1 and then divided by 2. (Contributed by AV, 1-Jun-2020.)
⊢ (𝑁 ∈ ℕ0 → ((𝑁 − 1) / 2) ≤ (⌊‘(𝑁 / 2)))
Theorem	flnn0ohalf 49010	The floor of the half of an odd positive integer is equal to the floor of the half of the integer decreased by 1. (Contributed by AV, 5-Jun-2012.)
⊢ ((𝑁 ∈ ℕ0 ∧ ((𝑁 + 1) / 2) ∈ ℕ0) → (⌊‘(𝑁 / 2)) = (⌊‘((𝑁 − 1) / 2)))
21.48.21.3  The natural logarithm on complex numbers (extension)
Theorem	logcxp0 49011	Logarithm of a complex power. Generalization of logcxp 26633. (Contributed by AV, 22-May-2020.)
⊢ ((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℂ ∧ (𝐵 · (log‘𝐴)) ∈ ran log) → (log‘(𝐴↑𝑐𝐵)) = (𝐵 · (log‘𝐴)))
Theorem	regt1loggt0 49012	The natural logarithm for a real number greater than 1 is greater than 0. (Contributed by AV, 25-May-2020.)
⊢ (𝐵 ∈ (1(,)+∞) → 0 < (log‘𝐵))
21.48.21.4  Division of functions
Syntax	cfdiv 49013	Extend class notation with the division operator of two functions.
class /f
Definition	df-fdiv 49014*	Define the division of two functions into the complex numbers. (Contributed by AV, 15-May-2020.)
⊢ /f = (𝑓 ∈ V, 𝑔 ∈ V ↦ ((𝑓 ∘f / 𝑔) ↾ (𝑔 supp 0)))
Theorem	fdivval 49015	The quotient of two functions into the complex numbers. (Contributed by AV, 15-May-2020.)
⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → (𝐹 /f 𝐺) = ((𝐹 ∘f / 𝐺) ↾ (𝐺 supp 0)))
Theorem	fdivmpt 49016*	The quotient of two functions into the complex numbers as mapping. (Contributed by AV, 16-May-2020.)
⊢ ((𝐹:𝐴⟶ℂ ∧ 𝐺:𝐴⟶ℂ ∧ 𝐴 ∈ 𝑉) → (𝐹 /f 𝐺) = (𝑥 ∈ (𝐺 supp 0) ↦ ((𝐹‘𝑥) / (𝐺‘𝑥))))
Theorem	fdivmptf 49017	The quotient of two functions into the complex numbers is a function into the complex numbers. (Contributed by AV, 16-May-2020.)
⊢ ((𝐹:𝐴⟶ℂ ∧ 𝐺:𝐴⟶ℂ ∧ 𝐴 ∈ 𝑉) → (𝐹 /f 𝐺):(𝐺 supp 0)⟶ℂ)
Theorem	refdivmptf 49018	The quotient of two functions into the real numbers is a function into the real numbers. (Contributed by AV, 16-May-2020.)
⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐺:𝐴⟶ℝ ∧ 𝐴 ∈ 𝑉) → (𝐹 /f 𝐺):(𝐺 supp 0)⟶ℝ)
Theorem	fdivpm 49019	The quotient of two functions into the complex numbers is a partial function. (Contributed by AV, 16-May-2020.)
⊢ ((𝐹:𝐴⟶ℂ ∧ 𝐺:𝐴⟶ℂ ∧ 𝐴 ∈ 𝑉) → (𝐹 /f 𝐺) ∈ (ℂ ↑pm 𝐴))
Theorem	refdivpm 49020	The quotient of two functions into the real numbers is a partial function. (Contributed by AV, 16-May-2020.)
⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐺:𝐴⟶ℝ ∧ 𝐴 ∈ 𝑉) → (𝐹 /f 𝐺) ∈ (ℝ ↑pm 𝐴))
Theorem	fdivmptfv 49021	The function value of a quotient of two functions into the complex numbers. (Contributed by AV, 19-May-2020.)
⊢ (((𝐹:𝐴⟶ℂ ∧ 𝐺:𝐴⟶ℂ ∧ 𝐴 ∈ 𝑉) ∧ 𝑋 ∈ (𝐺 supp 0)) → ((𝐹 /f 𝐺)‘𝑋) = ((𝐹‘𝑋) / (𝐺‘𝑋)))
Theorem	refdivmptfv 49022	The function value of a quotient of two functions into the real numbers. (Contributed by AV, 19-May-2020.)
⊢ (((𝐹:𝐴⟶ℝ ∧ 𝐺:𝐴⟶ℝ ∧ 𝐴 ∈ 𝑉) ∧ 𝑋 ∈ (𝐺 supp 0)) → ((𝐹 /f 𝐺)‘𝑋) = ((𝐹‘𝑋) / (𝐺‘𝑋)))
21.48.21.5  Upper bounds
Syntax	cbigo 49023	Extend class notation with the class of the "big-O" function.
class Ο
Definition	df-bigo 49024*	Define the function "big-O", mapping a real function g to the set of real functions "of order g(x)". Definition in section 1.1 of [AhoHopUll] p. 2. This is a generalization of "big-O of one", see df-o1 15452 and df-lo1 15453. As explained in the comment of df-o1 , any big-O can be represented in terms of 𝑂(1) and division, see elbigolo1 49033. (Contributed by AV, 15-May-2020.)
⊢ Ο = (𝑔 ∈ (ℝ ↑pm ℝ) ↦ {𝑓 ∈ (ℝ ↑pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(𝑓‘𝑦) ≤ (𝑚 · (𝑔‘𝑦))})
Theorem	bigoval 49025*	Set of functions of order G(x). (Contributed by AV, 15-May-2020.)
⊢ (𝐺 ∈ (ℝ ↑pm ℝ) → (Ο‘𝐺) = {𝑓 ∈ (ℝ ↑pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(𝑓‘𝑦) ≤ (𝑚 · (𝐺‘𝑦))})
Theorem	elbigofrcl 49026	Reverse closure of the "big-O" function. (Contributed by AV, 16-May-2020.)
⊢ (𝐹 ∈ (Ο‘𝐺) → 𝐺 ∈ (ℝ ↑pm ℝ))
Theorem	elbigo 49027*	Properties of a function of order G(x). (Contributed by AV, 16-May-2020.)
⊢ (𝐹 ∈ (Ο‘𝐺) ↔ (𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐺 ∈ (ℝ ↑pm ℝ) ∧ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝐹 ∩ (𝑥[,)+∞))(𝐹‘𝑦) ≤ (𝑚 · (𝐺‘𝑦))))
Theorem	elbigo2 49028*	Properties of a function of order G(x) under certain assumptions. (Contributed by AV, 17-May-2020.)
⊢ (((𝐺:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) ∧ (𝐹:𝐵⟶ℝ ∧ 𝐵 ⊆ 𝐴)) → (𝐹 ∈ (Ο‘𝐺) ↔ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 → (𝐹‘𝑦) ≤ (𝑚 · (𝐺‘𝑦)))))
Theorem	elbigo2r 49029*	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 49030	A function of order G(x) is a function. (Contributed by AV, 18-May-2020.)
⊢ (𝐹 ∈ (Ο‘𝐺) → 𝐹:dom 𝐹⟶ℝ)
Theorem	elbigodm 49031	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 49032*	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 49033	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)))
21.48.21.6  Logarithm to an arbitrary base (extension)
Theorem	rege1logbrege0 49034	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 49035	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 49036	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 49037	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 49038	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 49039	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 49040	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))
21.48.21.7  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 49041	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 49042	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 49043	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 49044	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)
21.48.21.8  Binary length
Syntax	cblen 49045	Extend class notation with the class of the binary length function.
class #b
Definition	df-blen 49046	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 49047	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 49048	The binary length of 0. (Contributed by AV, 20-May-2020.)
⊢ (#b‘0) = 1
Theorem	blenn0 49049	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 49050	The binary length of a positive real number. (Contributed by AV, 20-May-2020.)
⊢ (𝑁 ∈ ℝ+ → (#b‘𝑁) = ((⌊‘(2 logb 𝑁)) + 1))
Theorem	blennn 49051	The binary length of a positive integer. (Contributed by AV, 21-May-2020.)
⊢ (𝑁 ∈ ℕ → (#b‘𝑁) = ((⌊‘(2 logb 𝑁)) + 1))
Theorem	blennnelnn 49052	The binary length of a positive integer is a positive integer. (Contributed by AV, 25-May-2020.)
⊢ (𝑁 ∈ ℕ → (#b‘𝑁) ∈ ℕ)
Theorem	blennn0elnn 49053	The binary length of a nonnegative integer is a positive integer. (Contributed by AV, 28-May-2020.)
⊢ (𝑁 ∈ ℕ0 → (#b‘𝑁) ∈ ℕ)
Theorem	blenpw2 49054	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 49055	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 49056	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 49057	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 49058	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 49059	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 49060	The binary length of 1. (Contributed by AV, 21-May-2020.)
⊢ (#b‘1) = 1
Theorem	blen2 49061	The binary length of 2. (Contributed by AV, 21-May-2020.)
⊢ (#b‘2) = 2
Theorem	nnpw2p 49062*	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 49063*	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 49064	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 49065	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 49066	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 49067	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 49068	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 49069	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 49070	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))
21.48.21.9  Digits Generalization of df-bits 16391. In contrast to digit, bits are defined for integers only. The equivalence of both definitions for integers is shown in dig2bits 49090: if 𝐾 and 𝑁 are nonnegative integers, then ((𝐾(digit‘2)𝑁) = 1 ↔ 𝐾 ∈ (bits‘𝑁)).
Syntax	cdig 49071	Extend class notation with the class of the digit extraction operation.
class digit
Definition	df-dig 49072*	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 14199. 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 49073*	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 49074	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 49075	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 49076	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 49077	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 49078	Lemma for dignnld 49079. (Contributed by AV, 25-May-2020.)
⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ≥‘((⌊‘(𝐵 logb 𝑁)) + 1))) → 𝑁 < (𝐵↑𝐾))
Theorem	dignnld 49079	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 49080	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 49081	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 49082	All digits of 0 are 0. (Contributed by AV, 24-May-2020.)
⊢ ((𝐵 ∈ ℕ ∧ 𝐾 ∈ ℤ) → (𝐾(digit‘𝐵)0) = 0)
Theorem	digexp 49083	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 49084	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 49085	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 49086	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 49087	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 49088	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 49089	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 49090	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)))
21.48.21.10  Nonnegative integer as sum of its shifted digits
Theorem	dignn0flhalflem1 49091	Lemma 1 for dignn0flhalf 49094. (Contributed by AV, 7-Jun-2012.)
⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ) → (⌊‘((𝐴 / (2↑𝑁)) − 1)) < (⌊‘((𝐴 − 1) / (2↑𝑁))))
Theorem	dignn0flhalflem2 49092	Lemma 2 for dignn0flhalf 49094. (Contributed by AV, 7-Jun-2012.)
⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ0) → (⌊‘(𝐴 / (2↑(𝑁 + 1)))) = (⌊‘((⌊‘(𝐴 / 2)) / (2↑𝑁))))
Theorem	dignn0ehalf 49093	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 49094	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 49095*	Lemma for nn0sumshdig 49099 (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 49096*	Lemma for nn0sumshdig 49099 (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 49097*	Lemma 1 for nn0sumshdig 49099 (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 49098*	Lemma 2 for nn0sumshdig 49099. (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 49099*	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)))
21.48.21.11  Algorithms for the multiplication of nonnegative integers
Theorem	nn0mulfsum 49100*	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)