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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | fdivmpt 44101* | 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 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 ∧ 𝐴 ∈ 𝑉) → (𝐹 /f 𝐺):(𝐺 supp 0)⟶ℂ) | ||
Theorem | refdivmptf 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 ∧ 𝐴 ∈ 𝑉) → (𝐹 /f 𝐺):(𝐺 supp 0)⟶ℝ) | ||
Theorem | fdivpm 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 ∧ 𝐴 ∈ 𝑉) → (𝐹 /f 𝐺) ∈ (ℂ ↑pm 𝐴)) | ||
Theorem | refdivpm 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 ∧ 𝐴 ∈ 𝑉) → (𝐹 /f 𝐺) ∈ (ℝ ↑pm 𝐴)) | ||
Theorem | fdivmptfv 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 ∧ 𝐴 ∈ 𝑉) ∧ 𝑋 ∈ (𝐺 supp 0)) → ((𝐹 /f 𝐺)‘𝑋) = ((𝐹‘𝑋) / (𝐺‘𝑋))) | ||
Theorem | refdivmptfv 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 ∧ 𝐴 ∈ 𝑉) ∧ 𝑋 ∈ (𝐺 supp 0)) → ((𝐹 /f 𝐺)‘𝑋) = ((𝐹‘𝑋) / (𝐺‘𝑋))) | ||
Syntax | cbigo 44108 | Extend class notation with the class of the "big-O" function. |
class Ο | ||
Definition | df-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[,)+∞))(𝑓‘𝑦) ≤ (𝑚 · (𝑔‘𝑦))}) | ||
Theorem | bigoval 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[,)+∞))(𝑓‘𝑦) ≤ (𝑚 · (𝐺‘𝑦))}) | ||
Theorem | elbigofrcl 44111 | Reverse closure of the "big-O" function. (Contributed by AV, 16-May-2020.) |
⊢ (𝐹 ∈ (Ο‘𝐺) → 𝐺 ∈ (ℝ ↑pm ℝ)) | ||
Theorem | elbigo 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[,)+∞))(𝐹‘𝑦) ≤ (𝑚 · (𝐺‘𝑦)))) | ||
Theorem | elbigo2 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 ∧ 𝐵 ⊆ 𝐴)) → (𝐹 ∈ (Ο‘𝐺) ↔ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 → (𝐹‘𝑦) ≤ (𝑚 · (𝐺‘𝑦))))) | ||
Theorem | elbigo2r 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 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ ℝ ∧ 𝑀 ∈ ℝ ∧ ∀𝑥 ∈ 𝐵 (𝐶 ≤ 𝑥 → (𝐹‘𝑥) ≤ (𝑀 · (𝐺‘𝑥))))) → 𝐹 ∈ (Ο‘𝐺)) | ||
Theorem | elbigof 44115 | A function of order G(x) is a function. (Contributed by AV, 18-May-2020.) |
⊢ (𝐹 ∈ (Ο‘𝐺) → 𝐹:dom 𝐹⟶ℝ) | ||
Theorem | elbigodm 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) | ||
Theorem | elbigoimp 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 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑦) ≤ (𝑚 · (𝐺‘𝑦)))) | ||
Theorem | elbigolo1 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))) | ||
Theorem | rege1logbrege0 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)) | ||
Theorem | rege1logbzge0 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)) | ||
Theorem | fllogbd 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.) |
⊢ (𝜑 → 𝐵 ∈ (ℤ≥‘2)) & ⊢ (𝜑 → 𝑋 ∈ ℝ+) & ⊢ 𝐸 = (⌊‘(𝐵 logb 𝑋)) ⇒ ⊢ (𝜑 → ((𝐵↑𝐸) ≤ 𝑋 ∧ 𝑋 < (𝐵↑(𝐸 + 1)))) | ||
Theorem | relogbmulbexp 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 · (𝐵↑𝑐𝐶))) = ((𝐵 logb 𝐴) + 𝐶)) | ||
Theorem | relogbdivb 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)) | ||
Theorem | logbge0b 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)) | ||
Theorem | logblt1b 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)) | ||
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 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) | ||
Theorem | nnlog2ge0lt1 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))) | ||
Theorem | logbpw2m1 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)) | ||
Theorem | fllog2 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 ∈ ℕ0 ∧ 𝑁 ∈ ((2↑𝐼)..^(2↑(𝐼 + 1)))) → (⌊‘(2 logb 𝑁)) = 𝐼) | ||
Syntax | cblen 44130 | Extend class notation with the class of the binary length function. |
class #b | ||
Definition | df-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))) | ||
Theorem | blenval 44132 | 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 44133 | The binary length of 0. (Contributed by AV, 20-May-2020.) |
⊢ (#b‘0) = 1 | ||
Theorem | blenn0 44134 | 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 44135 | The binary length of a positive real number. (Contributed by AV, 20-May-2020.) |
⊢ (𝑁 ∈ ℝ+ → (#b‘𝑁) = ((⌊‘(2 logb 𝑁)) + 1)) | ||
Theorem | blennn 44136 | The binary length of a positive integer. (Contributed by AV, 21-May-2020.) |
⊢ (𝑁 ∈ ℕ → (#b‘𝑁) = ((⌊‘(2 logb 𝑁)) + 1)) | ||
Theorem | blennnelnn 44137 | The binary length of a positive integer is a positive integer. (Contributed by AV, 25-May-2020.) |
⊢ (𝑁 ∈ ℕ → (#b‘𝑁) ∈ ℕ) | ||
Theorem | blennn0elnn 44138 | The binary length of a nonnegative integer is a positive integer. (Contributed by AV, 28-May-2020.) |
⊢ (𝑁 ∈ ℕ0 → (#b‘𝑁) ∈ ℕ) | ||
Theorem | blenpw2 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)) | ||
Theorem | blenpw2m1 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) | ||
Theorem | nnpw2blen 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↑((#b‘𝑁) − 1)) ≤ 𝑁 ∧ 𝑁 < (2↑(#b‘𝑁)))) | ||
Theorem | nnpw2blenfzo 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↑((#b‘𝑁) − 1))..^(2↑(#b‘𝑁)))) | ||
Theorem | nnpw2blenfzo2 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↑((#b‘𝑁) − 1)) ∨ 𝑁 ∈ (((2↑((#b‘𝑁) − 1)) + 1)..^(2↑(#b‘𝑁))))) | ||
Theorem | nnpw2pmod 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↑((#b‘𝑁) − 1)) + (𝑁 mod (2↑((#b‘𝑁) − 1))))) | ||
Theorem | blen1 44145 | The binary length of 1. (Contributed by AV, 21-May-2020.) |
⊢ (#b‘1) = 1 | ||
Theorem | blen2 44146 | The binary length of 2. (Contributed by AV, 21-May-2020.) |
⊢ (#b‘2) = 2 | ||
Theorem | nnpw2p 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 ∈ ℕ0 ∃𝑟 ∈ (0..^(2↑𝑖))𝑁 = ((2↑𝑖) + 𝑟)) | ||
Theorem | nnpw2pb 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 ∈ ℕ0 ∃𝑟 ∈ (0..^(2↑𝑖))𝑁 = ((2↑𝑖) + 𝑟)) | ||
Theorem | blen1b 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 → ((#b‘𝑁) = 1 ↔ (𝑁 = 0 ∨ 𝑁 = 1))) | ||
Theorem | blennnt2 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)) = ((#b‘𝑁) + 1)) | ||
Theorem | nnolog2flm1 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)))) | ||
Theorem | blennn0em1 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)) = ((#b‘𝑁) − 1)) | ||
Theorem | blennngt2o2 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) → (#b‘𝑁) = ((#b‘((𝑁 − 1) / 2)) + 1)) | ||
Theorem | blengt1fldiv2p1 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) → (#b‘𝑁) = ((#b‘(⌊‘(𝑁 / 2))) + 1)) | ||
Theorem | blennn0e2 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) → (#b‘𝑁) = ((#b‘(𝑁 / 2)) + 1)) | ||
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)). | ||
Syntax | cdig 44156 | Extend class notation with the class of the digit extraction operation. |
class digit | ||
Definition | df-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))) | ||
Theorem | digfval 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))) | ||
Theorem | digval 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)) | ||
Theorem | digvalnn0 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) | ||
Theorem | nn0digval 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 ∈ ℕ0 ∧ 𝑅 ∈ (0[,)+∞)) → (𝐾(digit‘𝐵)𝑅) = ((⌊‘(𝑅 / (𝐵↑𝐾))) mod 𝐵)) | ||
Theorem | dignn0fr 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) | ||
Theorem | dignn0ldlem 44163 | Lemma for dignnld 44164. (Contributed by AV, 25-May-2020.) |
⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ≥‘((⌊‘(𝐵 logb 𝑁)) + 1))) → 𝑁 < (𝐵↑𝐾)) | ||
Theorem | dignnld 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) | ||
Theorem | dig2nn0ld 44165 | 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 44166 | 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 44167 | All digits of 0 are 0. (Contributed by AV, 24-May-2020.) |
⊢ ((𝐵 ∈ ℕ ∧ 𝐾 ∈ ℤ) → (𝐾(digit‘𝐵)0) = 0) | ||
Theorem | digexp 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 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝐾(digit‘𝐵)(𝐵↑𝑁)) = if(𝐾 = 𝑁, 1, 0)) | ||
Theorem | dig1 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)) | ||
Theorem | 0dig1 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) | ||
Theorem | 0dig2pr01 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) | ||
Theorem | dig2nn0 44172 | 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 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) | ||
Theorem | 0dig2nn0o 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) | ||
Theorem | dig2bits 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 ∈ ℕ0 ∧ 𝐾 ∈ ℕ0) → ((𝐾(digit‘2)𝑁) = 1 ↔ 𝐾 ∈ (bits‘𝑁))) | ||
Theorem | dignn0flhalflem1 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)))) | ||
Theorem | dignn0flhalflem2 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)))) | ||
Theorem | dignn0ehalf 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) ∈ ℕ0 ∧ 𝐴 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0) → ((𝐼 + 1)(digit‘2)𝐴) = (𝐼(digit‘2)(𝐴 / 2))) | ||
Theorem | dignn0flhalf 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)))) | ||
Theorem | nn0sumshdiglemA 44180* | Lemma for nn0sumshdig 44184 (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 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 ((#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 44182* | Lemma 1 for nn0sumshdig 44184 (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 44183* | Lemma 2 for nn0sumshdig 44184. (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 44184* | 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))) | ||
Theorem | nn0mulfsum 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 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → (𝐴 · 𝐵) = Σ𝑘 ∈ (1...𝐴)𝐵) | ||
Theorem | nn0mullong 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 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → (𝐴 · 𝐵) = Σ𝑘 ∈ (0..^(#b‘𝐴))(((𝑘(digit‘2)𝐴) · (2↑𝑘)) · 𝐵)) | ||
Theorem | fv1prop 44187 | The function value of unordered pair of ordered pairs with first components 1 and 2 at 1. (Contributed by AV, 4-Feb-2023.) |
⊢ (𝐴 ∈ 𝑉 → ({〈1, 𝐴〉, 〈2, 𝐵〉}‘1) = 𝐴) | ||
Theorem | fv2prop 44188 | The function value of unordered pair of ordered pairs with first components 1 and 2 at 1. (Contributed by AV, 4-Feb-2023.) |
⊢ (𝐵 ∈ 𝑉 → ({〈1, 𝐴〉, 〈2, 𝐵〉}‘2) = 𝐵) | ||
Theorem | submuladdmuld 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.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐷 ∈ ℂ) ⇒ ⊢ (𝜑 → (((𝐴 − 𝐵) · 𝐶) + (𝐵 · 𝐷)) = ((𝐴 · 𝐶) + (𝐵 · (𝐷 − 𝐶)))) | ||
Theorem | affinecomb1 44190* | Combination of two real affine combinations, one class variable resolved. (Contributed by AV, 22-Jan-2023.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ≠ 𝐶) & ⊢ (𝜑 → 𝐸 ∈ ℝ) & ⊢ (𝜑 → 𝐹 ∈ ℝ) & ⊢ (𝜑 → 𝐺 ∈ ℝ) & ⊢ 𝑆 = ((𝐺 − 𝐹) / (𝐶 − 𝐵)) ⇒ ⊢ (𝜑 → (∃𝑡 ∈ ℝ (𝐴 = (((1 − 𝑡) · 𝐵) + (𝑡 · 𝐶)) ∧ 𝐸 = (((1 − 𝑡) · 𝐹) + (𝑡 · 𝐺))) ↔ 𝐸 = ((𝑆 · (𝐴 − 𝐵)) + 𝐹))) | ||
Theorem | affinecomb2 44191* | Combination of two real affine combinations, presented without fraction. (Contributed by AV, 22-Jan-2023.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ≠ 𝐶) & ⊢ (𝜑 → 𝐸 ∈ ℝ) & ⊢ (𝜑 → 𝐹 ∈ ℝ) & ⊢ (𝜑 → 𝐺 ∈ ℝ) ⇒ ⊢ (𝜑 → (∃𝑡 ∈ ℝ (𝐴 = (((1 − 𝑡) · 𝐵) + (𝑡 · 𝐶)) ∧ 𝐸 = (((1 − 𝑡) · 𝐹) + (𝑡 · 𝐺))) ↔ ((𝐶 − 𝐵) · 𝐸) = (((𝐺 − 𝐹) · 𝐴) + ((𝐹 · 𝐶) − (𝐵 · 𝐺))))) | ||
Theorem | affineid 44192 | Identity of an affine combination. (Contributed by AV, 2-Feb-2023.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝑇 ∈ ℂ) ⇒ ⊢ (𝜑 → (((1 − 𝑇) · 𝐴) + (𝑇 · 𝐴)) = 𝐴) | ||
Theorem | 1subrec1sub 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))) | ||
Theorem | resum2sqcl 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) | ||
Theorem | resum2sqgt0 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) | ||
Theorem | resum2sqrp 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+) | ||
Theorem | resum2sqorgt0 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) | ||
Theorem | reorelicc 44198 | Membership in and outside of a closed real interval. (Contributed by AV, 15-Feb-2023.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐶 < 𝐴 ∨ 𝐶 ∈ (𝐴[,]𝐵) ∨ 𝐵 < 𝐶)) | ||
Theorem | rrx2pxel 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) ⇒ ⊢ (𝑋 ∈ 𝑃 → (𝑋‘1) ∈ ℝ) | ||
Theorem | rrx2pyel 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) ⇒ ⊢ (𝑋 ∈ 𝑃 → (𝑋‘2) ∈ ℝ) |
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