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Theorem List for Metamath Proof Explorer - 44101-44200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfdivmpt 44101* 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 44102 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 44103 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 44104 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 44105 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 44106 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 44107 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)))
 
20.38.21.6  Upper bounds
 
Syntaxcbigo 44108 Extend class notation with the class of the "big-O" function.
class Ο
 
Definitiondf-bigo 44109* 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 14681 and df-lo1 14682. As explained in the comment of df-o1 , any big-O can be represented in terms of 𝑂(1) and division, see elbigolo1 44118. (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 44110* 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 44111 Reverse closure of the "big-O" function. (Contributed by AV, 16-May-2020.)
(𝐹 ∈ (Ο‘𝐺) → 𝐺 ∈ (ℝ ↑pm ℝ))
 
Theoremelbigo 44112* 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 44113* 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 44114* 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 44115 A function of order G(x) is a function. (Contributed by AV, 18-May-2020.)
(𝐹 ∈ (Ο‘𝐺) → 𝐹:dom 𝐹⟶ℝ)
 
Theoremelbigodm 44116 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 44117* 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 44118 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)))
 
20.38.21.7  Logarithm to an arbitrary base (extension)
 
Theoremrege1logbrege0 44119 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 44120 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 44121 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 44122 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 44123 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 44124 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 44125 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))
 
20.38.21.8  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 44126 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 44127 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 44128 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 44129 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)
 
20.38.21.9  Binary length
 
Syntaxcblen 44130 Extend class notation with the class of the binary length function.
class #b
 
Definitiondf-blen 44131 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 44132 The binary length of an integer. (Contributed by AV, 20-May-2020.)
(𝑁𝑉 → (#b𝑁) = if(𝑁 = 0, 1, ((⌊‘(2 logb (abs‘𝑁))) + 1)))
 
Theoremblen0 44133 The binary length of 0. (Contributed by AV, 20-May-2020.)
(#b‘0) = 1
 
Theoremblenn0 44134 The binary length of a "number" not being 0. (Contributed by AV, 20-May-2020.)
((𝑁𝑉𝑁 ≠ 0) → (#b𝑁) = ((⌊‘(2 logb (abs‘𝑁))) + 1))
 
Theoremblenre 44135 The binary length of a positive real number. (Contributed by AV, 20-May-2020.)
(𝑁 ∈ ℝ+ → (#b𝑁) = ((⌊‘(2 logb 𝑁)) + 1))
 
Theoremblennn 44136 The binary length of a positive integer. (Contributed by AV, 21-May-2020.)
(𝑁 ∈ ℕ → (#b𝑁) = ((⌊‘(2 logb 𝑁)) + 1))
 
Theoremblennnelnn 44137 The binary length of a positive integer is a positive integer. (Contributed by AV, 25-May-2020.)
(𝑁 ∈ ℕ → (#b𝑁) ∈ ℕ)
 
Theoremblennn0elnn 44138 The binary length of a nonnegative integer is a positive integer. (Contributed by AV, 28-May-2020.)
(𝑁 ∈ ℕ0 → (#b𝑁) ∈ ℕ)
 
Theoremblenpw2 44139 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 44140 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 44141 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 44142 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 44143 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 44144 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 44145 The binary length of 1. (Contributed by AV, 21-May-2020.)
(#b‘1) = 1
 
Theoremblen2 44146 The binary length of 2. (Contributed by AV, 21-May-2020.)
(#b‘2) = 2
 
Theoremnnpw2p 44147* 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 44148* 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 44149 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 44150 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 44151 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 44152 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 44153 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 44154 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 44155 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))
 
20.38.21.10  Digits

Generalization of df-bits 15604. In contrast to digit, bits are defined for integers only. The equivalence of both definitions for integers is shown in dig2bits 44175: ((𝐾(digit 2 ) N ) = 1 <-> K e. ( bits 𝑁)).

 
Syntaxcdig 44156 Extend class notation with the class of the digit extraction operation.
class digit
 
Definitiondf-dig 44157* 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 13448. 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 44158* 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 44159 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 44160 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 44161 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 44162 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 44163 Lemma for dignnld 44164. (Contributed by AV, 25-May-2020.)
((𝐵 ∈ (ℤ‘2) ∧ 𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ‘((⌊‘(𝐵 logb 𝑁)) + 1))) → 𝑁 < (𝐵𝐾))
 
Theoremdignnld 44164 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 44165 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 44166 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 44167 All digits of 0 are 0. (Contributed by AV, 24-May-2020.)
((𝐵 ∈ ℕ ∧ 𝐾 ∈ ℤ) → (𝐾(digit‘𝐵)0) = 0)
 
Theoremdigexp 44168 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 44169 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 44170 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 44171 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 44172 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 44173 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 44174 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 44175 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)))
 
20.38.21.11  Nonnegative integer as sum of its shifted digits
 
Theoremdignn0flhalflem1 44176 Lemma 1 for dignn0flhalf 44179. (Contributed by AV, 7-Jun-2012.)
((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ) → (⌊‘((𝐴 / (2↑𝑁)) − 1)) < (⌊‘((𝐴 − 1) / (2↑𝑁))))
 
Theoremdignn0flhalflem2 44177 Lemma 2 for dignn0flhalf 44179. (Contributed by AV, 7-Jun-2012.)
((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ0) → (⌊‘(𝐴 / (2↑(𝑁 + 1)))) = (⌊‘((⌊‘(𝐴 / 2)) / (2↑𝑁))))
 
Theoremdignn0ehalf 44178 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 44179 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 44180* Lemma for nn0sumshdig 44184 (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 44181* Lemma for nn0sumshdig 44184 (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 44182* Lemma 1 for nn0sumshdig 44184 (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 44183* Lemma 2 for nn0sumshdig 44184. (Contributed by AV, 7-Jun-2020.)
(𝐿 ∈ ℕ → ∀𝑎 ∈ ℕ0 ((#b𝑎) = 𝐿𝑎 = Σ𝑘 ∈ (0..^𝐿)((𝑘(digit‘2)𝑎) · (2↑𝑘))))
 
Theoremnn0sumshdig 44184* 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)))
 
20.38.21.12  Algorithms for the multiplication of nonnegative integers
 
Theoremnn0mulfsum 44185* 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 44186* 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 15675. (Contributed by AV, 7-Jun-2020.)
((𝐴 ∈ ℕ0𝐵 ∈ ℕ0) → (𝐴 · 𝐵) = Σ𝑘 ∈ (0..^(#b𝐴))(((𝑘(digit‘2)𝐴) · (2↑𝑘)) · 𝐵))
 
20.38.22  Elementary geometry (extension)
 
20.38.22.1  Auxiliary theorems
 
Theoremfv1prop 44187 The function value of unordered pair of ordered pairs with first components 1 and 2 at 1. (Contributed by AV, 4-Feb-2023.)
(𝐴𝑉 → ({⟨1, 𝐴⟩, ⟨2, 𝐵⟩}‘1) = 𝐴)
 
Theoremfv2prop 44188 The function value of unordered pair of ordered pairs with first components 1 and 2 at 1. (Contributed by AV, 4-Feb-2023.)
(𝐵𝑉 → ({⟨1, 𝐴⟩, ⟨2, 𝐵⟩}‘2) = 𝐵)
 
Theoremsubmuladdmuld 44189 Transformation of a sum of a product of a difference and a product with the subtrahend of the difference. (Contributed by AV, 2-Feb-2023.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐷 ∈ ℂ)       (𝜑 → (((𝐴𝐵) · 𝐶) + (𝐵 · 𝐷)) = ((𝐴 · 𝐶) + (𝐵 · (𝐷𝐶))))
 
Theoremaffinecomb1 44190* Combination of two real affine combinations, one class variable resolved. (Contributed by AV, 22-Jan-2023.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐵𝐶)    &   (𝜑𝐸 ∈ ℝ)    &   (𝜑𝐹 ∈ ℝ)    &   (𝜑𝐺 ∈ ℝ)    &   𝑆 = ((𝐺𝐹) / (𝐶𝐵))       (𝜑 → (∃𝑡 ∈ ℝ (𝐴 = (((1 − 𝑡) · 𝐵) + (𝑡 · 𝐶)) ∧ 𝐸 = (((1 − 𝑡) · 𝐹) + (𝑡 · 𝐺))) ↔ 𝐸 = ((𝑆 · (𝐴𝐵)) + 𝐹)))
 
Theoremaffinecomb2 44191* Combination of two real affine combinations, presented without fraction. (Contributed by AV, 22-Jan-2023.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐵𝐶)    &   (𝜑𝐸 ∈ ℝ)    &   (𝜑𝐹 ∈ ℝ)    &   (𝜑𝐺 ∈ ℝ)       (𝜑 → (∃𝑡 ∈ ℝ (𝐴 = (((1 − 𝑡) · 𝐵) + (𝑡 · 𝐶)) ∧ 𝐸 = (((1 − 𝑡) · 𝐹) + (𝑡 · 𝐺))) ↔ ((𝐶𝐵) · 𝐸) = (((𝐺𝐹) · 𝐴) + ((𝐹 · 𝐶) − (𝐵 · 𝐺)))))
 
Theoremaffineid 44192 Identity of an affine combination. (Contributed by AV, 2-Feb-2023.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝑇 ∈ ℂ)       (𝜑 → (((1 − 𝑇) · 𝐴) + (𝑇 · 𝐴)) = 𝐴)
 
Theorem1subrec1sub 44193 Subtract the reciprocal of 1 minus a number from 1 results in the number divided by the number minus 1. (Contributed by AV, 15-Feb-2023.)
((𝐴 ∈ ℂ ∧ 𝐴 ≠ 1) → (1 − (1 / (1 − 𝐴))) = (𝐴 / (𝐴 − 1)))
 
Theoremresum2sqcl 44194 The sum of two squares of real numbers is a real number. (Contributed by AV, 7-Feb-2023.)
𝑄 = ((𝐴↑2) + (𝐵↑2))       ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝑄 ∈ ℝ)
 
Theoremresum2sqgt0 44195 The sum of the square of a nonzero real number and the square of another real number is greater than zero. (Contributed by AV, 7-Feb-2023.)
𝑄 = ((𝐴↑2) + (𝐵↑2))       (((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℝ) → 0 < 𝑄)
 
Theoremresum2sqrp 44196 The sum of the square of a nonzero real number and the square of another real number is a positive real number. (Contributed by AV, 2-May-2023.)
𝑄 = ((𝐴↑2) + (𝐵↑2))       (((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℝ) → 𝑄 ∈ ℝ+)
 
Theoremresum2sqorgt0 44197 The sum of the square of two real numbers is greater than zero if at least one of the real numbers is nonzero. (Contributed by AV, 26-Feb-2023.)
𝑄 = ((𝐴↑2) + (𝐵↑2))       ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐴 ≠ 0 ∨ 𝐵 ≠ 0)) → 0 < 𝑄)
 
Theoremreorelicc 44198 Membership in and outside of a closed real interval. (Contributed by AV, 15-Feb-2023.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐶 < 𝐴𝐶 ∈ (𝐴[,]𝐵) ∨ 𝐵 < 𝐶))
 
20.38.22.2  Real euclidean space of dimension 2
 
Theoremrrx2pxel 44199 The x-coordinate of a point in a real Euclidean space of dimension 2 is a real number. (Contributed by AV, 2-Feb-2023.)
𝐼 = {1, 2}    &   𝑃 = (ℝ ↑𝑚 𝐼)       (𝑋𝑃 → (𝑋‘1) ∈ ℝ)
 
Theoremrrx2pyel 44200 The y-coordinate of a point in a real Euclidean space of dimension 2 is a real number. (Contributed by AV, 2-Feb-2023.)
𝐼 = {1, 2}    &   𝑃 = (ℝ ↑𝑚 𝐼)       (𝑋𝑃 → (𝑋‘2) ∈ ℝ)
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