P. 473 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 47201-47300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	nnennex 47201*	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 47202	A positive integer is even or odd. (Contributed by AV, 30-May-2020.)
⊢ (𝑁 ∈ ℕ → ((𝑁 / 2) ∈ ℕ ∨ ((𝑁 + 1) / 2) ∈ ℕ))
Theorem	nneom 47203	A positive integer is even or odd. (Contributed by AV, 30-May-2020.)
⊢ (𝑁 ∈ ℕ → ((𝑁 / 2) ∈ ℕ ∨ ((𝑁 − 1) / 2) ∈ ℕ0))
Theorem	nn0eo 47204	A nonnegative integer is even or odd. (Contributed by AV, 27-May-2020.)
⊢ (𝑁 ∈ ℕ0 → ((𝑁 / 2) ∈ ℕ0 ∨ ((𝑁 + 1) / 2) ∈ ℕ0))
Theorem	nnpw2even 47205	2 to the power of a positive integer is even. (Contributed by AV, 2-Jun-2020.)
⊢ (𝑁 ∈ ℕ → ((2↑𝑁) / 2) ∈ ℕ)
Theorem	zefldiv2 47206	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 47207	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 47208	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 47209	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 47210	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)))
21.45.22.4  The natural logarithm on complex numbers (extension)
Theorem	logcxp0 47211	Logarithm of a complex power. Generalization of logcxp 26176. (Contributed by AV, 22-May-2020.)
⊢ ((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℂ ∧ (𝐵 · (log‘𝐴)) ∈ ran log) → (log‘(𝐴↑𝑐𝐵)) = (𝐵 · (log‘𝐴)))
Theorem	regt1loggt0 47212	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))
21.45.22.5  Division of functions
Syntax	cfdiv 47213	Extend class notation with the division operator of two functions.
class /f
Definition	df-fdiv 47214*	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 47215	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 47216*	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 47217	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 47218	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 47219	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 47220	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 47221	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 47222	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)))
21.45.22.6  Upper bounds
Syntax	cbigo 47223	Extend class notation with the class of the "big-O" function.
class Ο
Definition	df-bigo 47224*	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 15433 and df-lo1 15434. As explained in the comment of df-o1 , any big-O can be represented in terms of 𝑂(1) and division, see elbigolo1 47233. (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 47225*	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 47226	Reverse closure of the "big-O" function. (Contributed by AV, 16-May-2020.)
⊢ (𝐹 ∈ (Ο‘𝐺) → 𝐺 ∈ (ℝ ↑pm ℝ))
Theorem	elbigo 47227*	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 47228*	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 47229*	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 47230	A function of order G(x) is a function. (Contributed by AV, 18-May-2020.)
⊢ (𝐹 ∈ (Ο‘𝐺) → 𝐹:dom 𝐹⟶ℝ)
Theorem	elbigodm 47231	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 47232*	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 47233	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)))
21.45.22.7  Logarithm to an arbitrary base (extension)
Theorem	rege1logbrege0 47234	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 47235	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 47236	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 47237	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 47238	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 47239	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 47240	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))
21.45.22.8  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 47241	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 47242	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 47243	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 47244	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)
21.45.22.9  Binary length
Syntax	cblen 47245	Extend class notation with the class of the binary length function.
class #b
Definition	df-blen 47246	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 47247	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 47248	The binary length of 0. (Contributed by AV, 20-May-2020.)
⊢ (#b‘0) = 1
Theorem	blenn0 47249	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 47250	The binary length of a positive real number. (Contributed by AV, 20-May-2020.)
⊢ (𝑁 ∈ ℝ+ → (#b‘𝑁) = ((⌊‘(2 logb 𝑁)) + 1))
Theorem	blennn 47251	The binary length of a positive integer. (Contributed by AV, 21-May-2020.)
⊢ (𝑁 ∈ ℕ → (#b‘𝑁) = ((⌊‘(2 logb 𝑁)) + 1))
Theorem	blennnelnn 47252	The binary length of a positive integer is a positive integer. (Contributed by AV, 25-May-2020.)
⊢ (𝑁 ∈ ℕ → (#b‘𝑁) ∈ ℕ)
Theorem	blennn0elnn 47253	The binary length of a nonnegative integer is a positive integer. (Contributed by AV, 28-May-2020.)
⊢ (𝑁 ∈ ℕ0 → (#b‘𝑁) ∈ ℕ)
Theorem	blenpw2 47254	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 47255	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 47256	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 47257	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 47258	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 47259	Every positive integer 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 → 𝑁 = ((2↑((#b‘𝑁) − 1)) + (𝑁 mod (2↑((#b‘𝑁) − 1)))))
Theorem	blen1 47260	The binary length of 1. (Contributed by AV, 21-May-2020.)
⊢ (#b‘1) = 1
Theorem	blen2 47261	The binary length of 2. (Contributed by AV, 21-May-2020.)
⊢ (#b‘2) = 2
Theorem	nnpw2p 47262*	Every positive integer 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	nnpw2pb 47263*	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 47264	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 47265	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 47266	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 47267	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 47268	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 47269	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 47270	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))
21.45.22.10  Digits Generalization of df-bits 16362. In contrast to digit, bits are defined for integers only. The equivalence of both definitions for integers is shown in dig2bits 47290: if 𝐾 and 𝑁 are nonnegative integers, then ((𝐾(digit‘2)𝑁) = 1 ↔ 𝐾 ∈ (bits‘𝑁)).
Syntax	cdig 47271	Extend class notation with the class of the digit extraction operation.
class digit
Definition	df-dig 47272*	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 47273*	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 47274	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 47275	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 47276	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 47277	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 47278	Lemma for dignnld 47279. (Contributed by AV, 25-May-2020.)
⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ≥‘((⌊‘(𝐵 logb 𝑁)) + 1))) → 𝑁 < (𝐵↑𝐾))
Theorem	dignnld 47279	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 47280	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 47281	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 47282	All digits of 0 are 0. (Contributed by AV, 24-May-2020.)
⊢ ((𝐵 ∈ ℕ ∧ 𝐾 ∈ ℤ) → (𝐾(digit‘𝐵)0) = 0)
Theorem	digexp 47283	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 47284	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 47285	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 47286	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 47287	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 47288	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 47289	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 47290	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)))
21.45.22.11  Nonnegative integer as sum of its shifted digits
Theorem	dignn0flhalflem1 47291	Lemma 1 for dignn0flhalf 47294. (Contributed by AV, 7-Jun-2012.)
⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ) → (⌊‘((𝐴 / (2↑𝑁)) − 1)) < (⌊‘((𝐴 − 1) / (2↑𝑁))))
Theorem	dignn0flhalflem2 47292	Lemma 2 for dignn0flhalf 47294. (Contributed by AV, 7-Jun-2012.)
⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ0) → (⌊‘(𝐴 / (2↑(𝑁 + 1)))) = (⌊‘((⌊‘(𝐴 / 2)) / (2↑𝑁))))
Theorem	dignn0ehalf 47293	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 47294	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 47295*	Lemma for nn0sumshdig 47299 (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 47296*	Lemma for nn0sumshdig 47299 (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 47297*	Lemma 1 for nn0sumshdig 47299 (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 47298*	Lemma 2 for nn0sumshdig 47299. (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 47299*	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)))
21.45.22.12  Algorithms for the multiplication of nonnegative integers
Theorem	nn0mulfsum 47300*	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)