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Theorem List for Metamath Proof Explorer - 42801-42900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremnn0eo 42801 A nonnegative integer is even or odd. (Contributed by AV, 27-May-2020.)
(𝑁 ∈ ℕ0 → ((𝑁 / 2) ∈ ℕ0 ∨ ((𝑁 + 1) / 2) ∈ ℕ0))

Theoremnnpw2even 42802 2 to the power of a positive integer is even. (Contributed by AV, 2-Jun-2020.)
(𝑁 ∈ ℕ → ((2↑𝑁) / 2) ∈ ℕ)

Theoremzefldiv2 42803 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))

Theoremzofldiv2 42804 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))

Theoremnn0ofldiv2 42805 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))

Theoremflnn0div2ge 42806 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)))

Theoremflnn0ohalf 42807 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)))

20.35.17.4  The natural logarithm on complex numbers (extension)

Theoremlogcxp0 42808 Logarithm of a complex power. Generalisation of logcxp 24585. (Contributed by AV, 22-May-2020.)
((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℂ ∧ (𝐵 · (log‘𝐴)) ∈ ran log) → (log‘(𝐴𝑐𝐵)) = (𝐵 · (log‘𝐴)))

Theoremregt1loggt0 42809 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))

20.35.17.5  Division of functions

Syntaxcfdiv 42810 Extend class notation with the division operator of two functions.
class /f

Definitiondf-fdiv 42811* Define the division of two functions into the complex numbers. (Contributed by AV, 15-May-2020.)
/f = (𝑓 ∈ V, 𝑔 ∈ V ↦ ((𝑓𝑓 / 𝑔) ↾ (𝑔 supp 0)))

Theoremfdivval 42812 The quotient of two functions into the complex numbers. (Contributed by AV, 15-May-2020.)
((𝐹𝑉𝐺𝑊) → (𝐹 /f 𝐺) = ((𝐹𝑓 / 𝐺) ↾ (𝐺 supp 0)))

Theoremfdivmpt 42813* The quotient of two functions into the complex numbers as mapping. (Contributed by AV, 16-May-2020.)
((𝐹:𝐴⟶ℂ ∧ 𝐺:𝐴⟶ℂ ∧ 𝐴𝑉) → (𝐹 /f 𝐺) = (𝑥 ∈ (𝐺 supp 0) ↦ ((𝐹𝑥) / (𝐺𝑥))))

Theoremfdivmptf 42814 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 ∧ 𝐴𝑉) → (𝐹 /f 𝐺):(𝐺 supp 0)⟶ℂ)

Theoremrefdivmptf 42815 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 ∧ 𝐴𝑉) → (𝐹 /f 𝐺):(𝐺 supp 0)⟶ℝ)

Theoremfdivpm 42816 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 ∧ 𝐴𝑉) → (𝐹 /f 𝐺) ∈ (ℂ ↑pm 𝐴))

Theoremrefdivpm 42817 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 ∧ 𝐴𝑉) → (𝐹 /f 𝐺) ∈ (ℝ ↑pm 𝐴))

Theoremfdivmptfv 42818 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 ∧ 𝐴𝑉) ∧ 𝑋 ∈ (𝐺 supp 0)) → ((𝐹 /f 𝐺)‘𝑋) = ((𝐹𝑋) / (𝐺𝑋)))

Theoremrefdivmptfv 42819 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 ∧ 𝐴𝑉) ∧ 𝑋 ∈ (𝐺 supp 0)) → ((𝐹 /f 𝐺)‘𝑋) = ((𝐹𝑋) / (𝐺𝑋)))

20.35.17.6  Upper bounds

Syntaxcbigo 42820 Extend class notation with the class of the "big-O" function.
class Ο

Definitiondf-bigo 42821* 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 generalisation of "big-O of one", see df-o1 14391 and df-lo1 14392. As explained in the comment of df-o1 , any big-O can be represented in terms of 𝑂(1) and division, see elbigolo1 42830. (Contributed by AV, 15-May-2020.)
Ο = (𝑔 ∈ (ℝ ↑pm ℝ) ↦ {𝑓 ∈ (ℝ ↑pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(𝑓𝑦) ≤ (𝑚 · (𝑔𝑦))})

Theorembigoval 42822* 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[,)+∞))(𝑓𝑦) ≤ (𝑚 · (𝐺𝑦))})

Theoremelbigofrcl 42823 Reverse closure of the "big-O" function. (Contributed by AV, 16-May-2020.)
(𝐹 ∈ (Ο‘𝐺) → 𝐺 ∈ (ℝ ↑pm ℝ))

Theoremelbigo 42824* 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[,)+∞))(𝐹𝑦) ≤ (𝑚 · (𝐺𝑦))))

Theoremelbigo2 42825* 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 ∧ 𝐵𝐴)) → (𝐹 ∈ (Ο‘𝐺) ↔ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦𝐵 (𝑥𝑦 → (𝐹𝑦) ≤ (𝑚 · (𝐺𝑦)))))

Theoremelbigo2r 42826* 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 ∧ 𝐵𝐴) ∧ (𝐶 ∈ ℝ ∧ 𝑀 ∈ ℝ ∧ ∀𝑥𝐵 (𝐶𝑥 → (𝐹𝑥) ≤ (𝑀 · (𝐺𝑥))))) → 𝐹 ∈ (Ο‘𝐺))

Theoremelbigof 42827 A function of order G(x) is a function. (Contributed by AV, 18-May-2020.)
(𝐹 ∈ (Ο‘𝐺) → 𝐹:dom 𝐹⟶ℝ)

Theoremelbigodm 42828 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)

Theoremelbigoimp 42829* 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 ∀𝑦𝐴 (𝑥𝑦 → (𝐹𝑦) ≤ (𝑚 · (𝐺𝑦))))

Theoremelbigolo1 42830 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.)
((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ+𝐹:𝐴⟶ℝ+) → (𝐹 ∈ (Ο‘𝐺) ↔ (𝐹 /f 𝐺) ∈ ≤𝑂(1)))

20.35.17.7  Logarithm to an arbitrary base (extension)

Theoremrege1logbrege0 42831 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))

Theoremrege1logbzge0 42832 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))

Theoremfllogbd 42833 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.)
(𝜑𝐵 ∈ (ℤ‘2))    &   (𝜑𝑋 ∈ ℝ+)    &   𝐸 = (⌊‘(𝐵 logb 𝑋))       (𝜑 → ((𝐵𝐸) ≤ 𝑋𝑋 < (𝐵↑(𝐸 + 1))))

Theoremrelogbmulbexp 42834 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 · (𝐵𝑐𝐶))) = ((𝐵 logb 𝐴) + 𝐶))

Theoremrelogbdivb 42835 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))

Theoremlogbge0b 42836 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))

Theoremlogblt1b 42837 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))

20.35.17.8  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).

Theoremfldivexpfllog2 42838 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)

Theoremnnlog2ge0lt1 42839 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)))

Theoremlogbpw2m1 42840 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))

Theoremfllog2 42841 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 ∈ ℕ0𝑁 ∈ ((2↑𝐼)..^(2↑(𝐼 + 1)))) → (⌊‘(2 logb 𝑁)) = 𝐼)

20.35.17.9  Binary length

Syntaxcblen 42842 Extend class notation with the class of the binary length function.
class #b

Definitiondf-blen 42843 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)))

Theoremblenval 42844 The binary length of an integer. (Contributed by AV, 20-May-2020.)
(𝑁𝑉 → (#b𝑁) = if(𝑁 = 0, 1, ((⌊‘(2 logb (abs‘𝑁))) + 1)))

Theoremblen0 42845 The binary length of 0. (Contributed by AV, 20-May-2020.)
(#b‘0) = 1

Theoremblenn0 42846 The binary length of a "number" not being 0. (Contributed by AV, 20-May-2020.)
((𝑁𝑉𝑁 ≠ 0) → (#b𝑁) = ((⌊‘(2 logb (abs‘𝑁))) + 1))

Theoremblenre 42847 The binary length of a positive real number. (Contributed by AV, 20-May-2020.)
(𝑁 ∈ ℝ+ → (#b𝑁) = ((⌊‘(2 logb 𝑁)) + 1))

Theoremblennn 42848 The binary length of a positive integer. (Contributed by AV, 21-May-2020.)
(𝑁 ∈ ℕ → (#b𝑁) = ((⌊‘(2 logb 𝑁)) + 1))

Theoremblennnelnn 42849 The binary length of a positive integer is a positive integer. (Contributed by AV, 25-May-2020.)
(𝑁 ∈ ℕ → (#b𝑁) ∈ ℕ)

Theoremblennn0elnn 42850 The binary length of a nonnegative integer is a positive integer. (Contributed by AV, 28-May-2020.)
(𝑁 ∈ ℕ0 → (#b𝑁) ∈ ℕ)

Theoremblenpw2 42851 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))

Theoremblenpw2m1 42852 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)

Theoremnnpw2blen 42853 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↑((#b𝑁) − 1)) ≤ 𝑁𝑁 < (2↑(#b𝑁))))

Theoremnnpw2blenfzo 42854 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↑((#b𝑁) − 1))..^(2↑(#b𝑁))))

Theoremnnpw2blenfzo2 42855 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↑((#b𝑁) − 1)) ∨ 𝑁 ∈ (((2↑((#b𝑁) − 1)) + 1)..^(2↑(#b𝑁)))))

Theoremnnpw2pmod 42856 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↑((#b𝑁) − 1)) + (𝑁 mod (2↑((#b𝑁) − 1)))))

Theoremblen1 42857 The binary length of 1. (Contributed by AV, 21-May-2020.)
(#b‘1) = 1

Theoremblen2 42858 The binary length of 2. (Contributed by AV, 21-May-2020.)
(#b‘2) = 2

Theoremnnpw2p 42859* 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 ∈ ℕ0𝑟 ∈ (0..^(2↑𝑖))𝑁 = ((2↑𝑖) + 𝑟))

Theoremnnpw2pb 42860* 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 ∈ ℕ0𝑟 ∈ (0..^(2↑𝑖))𝑁 = ((2↑𝑖) + 𝑟))

Theoremblen1b 42861 The binary length of a nonnegative integer is 1 if the integer is 0 or 1. (Contributed by AV, 30-May-2020.)
(𝑁 ∈ ℕ0 → ((#b𝑁) = 1 ↔ (𝑁 = 0 ∨ 𝑁 = 1)))

Theoremblennnt2 42862 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)) = ((#b𝑁) + 1))

Theoremnnolog2flm1 42863 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))))

Theoremblennn0em1 42864 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)) = ((#b𝑁) − 1))

Theoremblennngt2o2 42865 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) → (#b𝑁) = ((#b‘((𝑁 − 1) / 2)) + 1))

Theoremblengt1fldiv2p1 42866 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) → (#b𝑁) = ((#b‘(⌊‘(𝑁 / 2))) + 1))

Theoremblennn0e2 42867 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) → (#b𝑁) = ((#b‘(𝑁 / 2)) + 1))

20.35.17.10  Digits

Generalisation of df-bits 15317. In contrast to digit, bits are defined for integers only. The equivalence of both definitions for integers is shown in dig2bits 42887: ((𝐾(digit 2 ) N ) = 1 <-> K e. ( bits 𝑁)).

Syntaxcdig 42868 Extend class notation with the class of the digit extraction operation.
class digit

Definitiondf-dig 42869* 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 13163. 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)))

Theoremdigfval 42870* 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)))

Theoremdigval 42871 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))

Theoremdigvalnn0 42872 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)

Theoremnn0digval 42873 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 ∈ ℕ0𝑅 ∈ (0[,)+∞)) → (𝐾(digit‘𝐵)𝑅) = ((⌊‘(𝑅 / (𝐵𝐾))) mod 𝐵))

Theoremdignn0fr 42874 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)

Theoremdignn0ldlem 42875 Lemma for dignnld 42876. (Contributed by AV, 25-May-2020.)
((𝐵 ∈ (ℤ‘2) ∧ 𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ‘((⌊‘(𝐵 logb 𝑁)) + 1))) → 𝑁 < (𝐵𝐾))

Theoremdignnld 42876 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)

Theoremdig2nn0ld 42877 The leading digits of a positive integer in a binary system are 0. (Contributed by AV, 25-May-2020.)
((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ‘(#b𝑁))) → (𝐾(digit‘2)𝑁) = 0)

Theoremdig2nn1st 42878 The first (relevant) digit of a positive integer in a binary system is 1. (Contributed by AV, 26-May-2020.)
(𝑁 ∈ ℕ → (((#b𝑁) − 1)(digit‘2)𝑁) = 1)

Theoremdig0 42879 All digits of 0 are 0. (Contributed by AV, 24-May-2020.)
((𝐵 ∈ ℕ ∧ 𝐾 ∈ ℤ) → (𝐾(digit‘𝐵)0) = 0)

Theoremdigexp 42880 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 ∈ ℕ0𝑁 ∈ ℕ0) → (𝐾(digit‘𝐵)(𝐵𝑁)) = if(𝐾 = 𝑁, 1, 0))

Theoremdig1 42881 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))

Theorem0dig1 42882 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)

Theorem0dig2pr01 42883 The integers 0 and 1 correspond to their last bit. (Contributed by AV, 28-May-2010.)
(𝑁 ∈ {0, 1} → (0(digit‘2)𝑁) = 𝑁)

Theoremdig2nn0 42884 A digit of a nonnegative integer 𝑁 in a binary system is either 0 or 1. (Contributed by AV, 24-May-2020.)
((𝑁 ∈ ℕ0𝐾 ∈ ℤ) → (𝐾(digit‘2)𝑁) ∈ {0, 1})

Theorem0dig2nn0e 42885 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)

Theorem0dig2nn0o 42886 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)

Theoremdig2bits 42887 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 ∈ ℕ0𝐾 ∈ ℕ0) → ((𝐾(digit‘2)𝑁) = 1 ↔ 𝐾 ∈ (bits‘𝑁)))

20.35.17.11  Nonnegative integer as sum of its shifted digits

Theoremdignn0flhalflem1 42888 Lemma 1 for dignn0flhalf 42891. (Contributed by AV, 7-Jun-2012.)
((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ) → (⌊‘((𝐴 / (2↑𝑁)) − 1)) < (⌊‘((𝐴 − 1) / (2↑𝑁))))

Theoremdignn0flhalflem2 42889 Lemma 2 for dignn0flhalf 42891. (Contributed by AV, 7-Jun-2012.)
((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ0) → (⌊‘(𝐴 / (2↑(𝑁 + 1)))) = (⌊‘((⌊‘(𝐴 / 2)) / (2↑𝑁))))

Theoremdignn0ehalf 42890 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) ∈ ℕ0𝐴 ∈ ℕ0𝐼 ∈ ℕ0) → ((𝐼 + 1)(digit‘2)𝐴) = (𝐼(digit‘2)(𝐴 / 2)))

Theoremdignn0flhalf 42891 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))))

Theoremnn0sumshdiglemA 42892* Lemma for nn0sumshdig 42896 (induction step, even multiplier). (Contributed by AV, 3-Jun-2020.)
(((𝑎 ∈ ℕ ∧ (𝑎 / 2) ∈ ℕ) ∧ 𝑦 ∈ ℕ) → (∀𝑥 ∈ ℕ0 ((#b𝑥) = 𝑦𝑥 = Σ𝑘 ∈ (0..^𝑦)((𝑘(digit‘2)𝑥) · (2↑𝑘))) → ((#b𝑎) = (𝑦 + 1) → 𝑎 = Σ𝑘 ∈ (0..^(𝑦 + 1))((𝑘(digit‘2)𝑎) · (2↑𝑘)))))

Theoremnn0sumshdiglemB 42893* Lemma for nn0sumshdig 42896 (induction step, odd multiplier). (Contributed by AV, 7-Jun-2020.)
(((𝑎 ∈ ℕ ∧ ((𝑎 − 1) / 2) ∈ ℕ0) ∧ 𝑦 ∈ ℕ) → (∀𝑥 ∈ ℕ0 ((#b𝑥) = 𝑦𝑥 = Σ𝑘 ∈ (0..^𝑦)((𝑘(digit‘2)𝑥) · (2↑𝑘))) → ((#b𝑎) = (𝑦 + 1) → 𝑎 = Σ𝑘 ∈ (0..^(𝑦 + 1))((𝑘(digit‘2)𝑎) · (2↑𝑘)))))

Theoremnn0sumshdiglem1 42894* Lemma 1 for nn0sumshdig 42896 (induction step). (Contributed by AV, 7-Jun-2020.)
(𝑦 ∈ ℕ → (∀𝑎 ∈ ℕ0 ((#b𝑎) = 𝑦𝑎 = Σ𝑘 ∈ (0..^𝑦)((𝑘(digit‘2)𝑎) · (2↑𝑘))) → ∀𝑎 ∈ ℕ0 ((#b𝑎) = (𝑦 + 1) → 𝑎 = Σ𝑘 ∈ (0..^(𝑦 + 1))((𝑘(digit‘2)𝑎) · (2↑𝑘)))))

Theoremnn0sumshdiglem2 42895* Lemma 2 for nn0sumshdig 42896. (Contributed by AV, 7-Jun-2020.)
(𝐿 ∈ ℕ → ∀𝑎 ∈ ℕ0 ((#b𝑎) = 𝐿𝑎 = Σ𝑘 ∈ (0..^𝐿)((𝑘(digit‘2)𝑎) · (2↑𝑘))))

Theoremnn0sumshdig 42896* A nonnegative integer can be represented as sum of its shifted bits. (Contributed by AV, 7-Jun-2020.)
(𝐴 ∈ ℕ0𝐴 = Σ𝑘 ∈ (0..^(#b𝐴))((𝑘(digit‘2)𝐴) · (2↑𝑘)))

20.35.17.12  Algorithms for the multiplication of nonnegative integers

Theoremnn0mulfsum 42897* 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 ∈ ℕ0𝐵 ∈ ℕ0) → (𝐴 · 𝐵) = Σ𝑘 ∈ (1...𝐴)𝐵)

Theoremnn0mullong 42898* 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 15388. (Contributed by AV, 7-Jun-2020.)
((𝐴 ∈ ℕ0𝐵 ∈ ℕ0) → (𝐴 · 𝐵) = Σ𝑘 ∈ (0..^(#b𝐴))(((𝑘(digit‘2)𝐴) · (2↑𝑘)) · 𝐵))

20.36  Mathbox for Emmett Weisz

20.36.1  Miscellaneous Theorems

Some of these theorems are used in the series of lemmas and theorems proving the defining properties of setrecs.

Theoremnfintd 42899 Bound-variable hypothesis builder for intersection. (Contributed by Emmett Weisz, 16-Jan-2020.)
(𝜑𝑥𝐴)       (𝜑𝑥 𝐴)

Theoremnfiund 42900 Bound-variable hypothesis builder for indexed union. (Contributed by Emmett Weisz, 6-Dec-2019.)
𝑥𝜑    &   (𝜑𝑦𝐴)    &   (𝜑𝑦𝐵)       (𝜑𝑦 𝑥𝐴 𝐵)

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