HomeHome Intuitionistic Logic Explorer
Theorem List (p. 137 of 142)
< Previous  Next >
Bad symbols? Try the
GIF version.

Mirrors  >  Metamath Home Page  >  ILE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Theorem List for Intuitionistic Logic Explorer - 13601-13700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremlogled 13601 Natural logarithm preserves . (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ+)       (𝜑 → (𝐴𝐵 ↔ (log‘𝐴) ≤ (log‘𝐵)))
 
Theoremrelogefd 13602 Relationship between the natural logarithm function and the exponential function. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑 → (log‘(exp‘𝐴)) = 𝐴)
 
Theoremrplogcld 13603 Closure of the logarithm function in the positive reals. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑 → 1 < 𝐴)       (𝜑 → (log‘𝐴) ∈ ℝ+)
 
Theoremlogge0d 13604 The logarithm of a number greater than 1 is nonnegative. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑 → 1 ≤ 𝐴)       (𝜑 → 0 ≤ (log‘𝐴))
 
Theoremlogge0b 13605 The logarithm of a number is nonnegative iff the number is greater than or equal to 1. (Contributed by AV, 30-May-2020.)
(𝐴 ∈ ℝ+ → (0 ≤ (log‘𝐴) ↔ 1 ≤ 𝐴))
 
Theoremloggt0b 13606 The logarithm of a number is positive iff the number is greater than 1. (Contributed by AV, 30-May-2020.)
(𝐴 ∈ ℝ+ → (0 < (log‘𝐴) ↔ 1 < 𝐴))
 
Theoremlogle1b 13607 The logarithm of a number is less than or equal to 1 iff the number is less than or equal to Euler's constant. (Contributed by AV, 30-May-2020.)
(𝐴 ∈ ℝ+ → ((log‘𝐴) ≤ 1 ↔ 𝐴 ≤ e))
 
Theoremloglt1b 13608 The logarithm of a number is less than 1 iff the number is less than Euler's constant. (Contributed by AV, 30-May-2020.)
(𝐴 ∈ ℝ+ → ((log‘𝐴) < 1 ↔ 𝐴 < e))
 
Theoremrpcxpef 13609 Value of the complex power function. (Contributed by Mario Carneiro, 2-Aug-2014.) (Revised by Jim Kingdon, 12-Jun-2024.)
((𝐴 ∈ ℝ+𝐵 ∈ ℂ) → (𝐴𝑐𝐵) = (exp‘(𝐵 · (log‘𝐴))))
 
Theoremcxpexprp 13610 Relate the complex power function to the integer power function. (Contributed by Mario Carneiro, 2-Aug-2014.) (Revised by Jim Kingdon, 12-Jun-2024.)
((𝐴 ∈ ℝ+𝐵 ∈ ℤ) → (𝐴𝑐𝐵) = (𝐴𝐵))
 
Theoremcxpexpnn 13611 Relate the complex power function to the integer power function. (Contributed by Mario Carneiro, 2-Aug-2014.) (Revised by Jim Kingdon, 12-Jun-2024.)
((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℤ) → (𝐴𝑐𝐵) = (𝐴𝐵))
 
Theoremlogcxp 13612 Logarithm of a complex power. (Contributed by Mario Carneiro, 2-Aug-2014.)
((𝐴 ∈ ℝ+𝐵 ∈ ℝ) → (log‘(𝐴𝑐𝐵)) = (𝐵 · (log‘𝐴)))
 
Theoremrpcxp0 13613 Value of the complex power function when the second argument is zero. (Contributed by Mario Carneiro, 2-Aug-2014.) (Revised by Jim Kingdon, 12-Jun-2024.)
(𝐴 ∈ ℝ+ → (𝐴𝑐0) = 1)
 
Theoremrpcxp1 13614 Value of the complex power function at one. (Contributed by Mario Carneiro, 2-Aug-2014.)
(𝐴 ∈ ℝ+ → (𝐴𝑐1) = 𝐴)
 
Theorem1cxp 13615 Value of the complex power function at one. (Contributed by Mario Carneiro, 2-Aug-2014.)
(𝐴 ∈ ℂ → (1↑𝑐𝐴) = 1)
 
Theoremecxp 13616 Write the exponential function as an exponent to the power e. (Contributed by Mario Carneiro, 2-Aug-2014.)
(𝐴 ∈ ℂ → (e↑𝑐𝐴) = (exp‘𝐴))
 
Theoremrpcncxpcl 13617 Closure of the complex power function. (Contributed by Jim Kingdon, 12-Jun-2024.)
((𝐴 ∈ ℝ+𝐵 ∈ ℂ) → (𝐴𝑐𝐵) ∈ ℂ)
 
Theoremrpcxpcl 13618 Positive real closure of the complex power function. (Contributed by Mario Carneiro, 2-Aug-2014.)
((𝐴 ∈ ℝ+𝐵 ∈ ℝ) → (𝐴𝑐𝐵) ∈ ℝ+)
 
Theoremcxpap0 13619 Complex exponentiation is apart from zero. (Contributed by Mario Carneiro, 2-Aug-2014.) (Revised by Jim Kingdon, 12-Jun-2024.)
((𝐴 ∈ ℝ+𝐵 ∈ ℂ) → (𝐴𝑐𝐵) # 0)
 
Theoremrpcxpadd 13620 Sum of exponents law for complex exponentiation. (Contributed by Mario Carneiro, 2-Aug-2014.) (Revised by Jim Kingdon, 13-Jun-2024.)
((𝐴 ∈ ℝ+𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴𝑐(𝐵 + 𝐶)) = ((𝐴𝑐𝐵) · (𝐴𝑐𝐶)))
 
Theoremrpcxpp1 13621 Value of a nonzero complex number raised to a complex power plus one. (Contributed by Mario Carneiro, 2-Aug-2014.)
((𝐴 ∈ ℝ+𝐵 ∈ ℂ) → (𝐴𝑐(𝐵 + 1)) = ((𝐴𝑐𝐵) · 𝐴))
 
Theoremrpcxpneg 13622 Value of a complex number raised to a negative power. (Contributed by Mario Carneiro, 2-Aug-2014.)
((𝐴 ∈ ℝ+𝐵 ∈ ℂ) → (𝐴𝑐-𝐵) = (1 / (𝐴𝑐𝐵)))
 
Theoremrpcxpsub 13623 Exponent subtraction law for complex exponentiation. (Contributed by Mario Carneiro, 22-Sep-2014.)
((𝐴 ∈ ℝ+𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴𝑐(𝐵𝐶)) = ((𝐴𝑐𝐵) / (𝐴𝑐𝐶)))
 
Theoremrpmulcxp 13624 Complex exponentiation of a product. Proposition 10-4.2(c) of [Gleason] p. 135. (Contributed by Mario Carneiro, 2-Aug-2014.)
((𝐴 ∈ ℝ+𝐵 ∈ ℝ+𝐶 ∈ ℂ) → ((𝐴 · 𝐵)↑𝑐𝐶) = ((𝐴𝑐𝐶) · (𝐵𝑐𝐶)))
 
Theoremcxprec 13625 Complex exponentiation of a reciprocal. (Contributed by Mario Carneiro, 2-Aug-2014.)
((𝐴 ∈ ℝ+𝐵 ∈ ℂ) → ((1 / 𝐴)↑𝑐𝐵) = (1 / (𝐴𝑐𝐵)))
 
Theoremrpdivcxp 13626 Complex exponentiation of a quotient. (Contributed by Mario Carneiro, 8-Sep-2014.)
((𝐴 ∈ ℝ+𝐵 ∈ ℝ+𝐶 ∈ ℂ) → ((𝐴 / 𝐵)↑𝑐𝐶) = ((𝐴𝑐𝐶) / (𝐵𝑐𝐶)))
 
Theoremcxpmul 13627 Product of exponents law for complex exponentiation. Proposition 10-4.2(b) of [Gleason] p. 135. (Contributed by Mario Carneiro, 2-Aug-2014.)
((𝐴 ∈ ℝ+𝐵 ∈ ℝ ∧ 𝐶 ∈ ℂ) → (𝐴𝑐(𝐵 · 𝐶)) = ((𝐴𝑐𝐵)↑𝑐𝐶))
 
Theoremrpcxproot 13628 The complex power function allows us to write n-th roots via the idiom 𝐴𝑐(1 / 𝑁). (Contributed by Mario Carneiro, 6-May-2015.)
((𝐴 ∈ ℝ+𝑁 ∈ ℕ) → ((𝐴𝑐(1 / 𝑁))↑𝑁) = 𝐴)
 
Theoremabscxp 13629 Absolute value of a power, when the base is real. (Contributed by Mario Carneiro, 15-Sep-2014.)
((𝐴 ∈ ℝ+𝐵 ∈ ℂ) → (abs‘(𝐴𝑐𝐵)) = (𝐴𝑐(ℜ‘𝐵)))
 
Theoremcxplt 13630 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 2-Aug-2014.)
(((𝐴 ∈ ℝ ∧ 1 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → (𝐵 < 𝐶 ↔ (𝐴𝑐𝐵) < (𝐴𝑐𝐶)))
 
Theoremcxple 13631 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 2-Aug-2014.)
(((𝐴 ∈ ℝ ∧ 1 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → (𝐵𝐶 ↔ (𝐴𝑐𝐵) ≤ (𝐴𝑐𝐶)))
 
Theoremrpcxple2 13632 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 8-Sep-2014.)
((𝐴 ∈ ℝ+𝐵 ∈ ℝ+𝐶 ∈ ℝ+) → (𝐴𝐵 ↔ (𝐴𝑐𝐶) ≤ (𝐵𝑐𝐶)))
 
Theoremrpcxplt2 13633 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 15-Sep-2014.)
((𝐴 ∈ ℝ+𝐵 ∈ ℝ+𝐶 ∈ ℝ+) → (𝐴 < 𝐵 ↔ (𝐴𝑐𝐶) < (𝐵𝑐𝐶)))
 
Theoremcxplt3 13634 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 2-May-2016.)
(((𝐴 ∈ ℝ+𝐴 < 1) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → (𝐵 < 𝐶 ↔ (𝐴𝑐𝐶) < (𝐴𝑐𝐵)))
 
Theoremcxple3 13635 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 2-May-2016.)
(((𝐴 ∈ ℝ+𝐴 < 1) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → (𝐵𝐶 ↔ (𝐴𝑐𝐶) ≤ (𝐴𝑐𝐵)))
 
Theoremrpcxpsqrt 13636 The exponential function with exponent 1 / 2 exactly matches the square root function, and thus serves as a suitable generalization to other 𝑛-th roots and irrational roots. (Contributed by Mario Carneiro, 2-Aug-2014.) (Revised by Jim Kingdon, 16-Jun-2024.)
(𝐴 ∈ ℝ+ → (𝐴𝑐(1 / 2)) = (√‘𝐴))
 
Theoremlogsqrt 13637 Logarithm of a square root. (Contributed by Mario Carneiro, 5-May-2016.)
(𝐴 ∈ ℝ+ → (log‘(√‘𝐴)) = ((log‘𝐴) / 2))
 
Theoremrpcxp0d 13638 Value of the complex power function when the second argument is zero. (Contributed by Mario Carneiro, 30-May-2016.)
(𝜑𝐴 ∈ ℝ+)       (𝜑 → (𝐴𝑐0) = 1)
 
Theoremrpcxp1d 13639 Value of the complex power function at one. (Contributed by Mario Carneiro, 30-May-2016.)
(𝜑𝐴 ∈ ℝ+)       (𝜑 → (𝐴𝑐1) = 𝐴)
 
Theorem1cxpd 13640 Value of the complex power function at one. (Contributed by Mario Carneiro, 30-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (1↑𝑐𝐴) = 1)
 
Theoremrpcncxpcld 13641 Closure of the complex power function. (Contributed by Mario Carneiro, 30-May-2016.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (𝐴𝑐𝐵) ∈ ℂ)
 
Theoremcxpltd 13642 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑 → 1 < 𝐴)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → (𝐵 < 𝐶 ↔ (𝐴𝑐𝐵) < (𝐴𝑐𝐶)))
 
Theoremcxpled 13643 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑 → 1 < 𝐴)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → (𝐵𝐶 ↔ (𝐴𝑐𝐵) ≤ (𝐴𝑐𝐶)))
 
Theoremrpcxpsqrtth 13644 Square root theorem over the complex numbers for the complex power function. Compare with resqrtth 10995. (Contributed by AV, 23-Dec-2022.)
(𝐴 ∈ ℝ+ → ((√‘𝐴)↑𝑐2) = 𝐴)
 
Theoremcxprecd 13645 Complex exponentiation of a reciprocal. (Contributed by Mario Carneiro, 30-May-2016.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → ((1 / 𝐴)↑𝑐𝐵) = (1 / (𝐴𝑐𝐵)))
 
Theoremrpcxpcld 13646 Positive real closure of the complex power function. (Contributed by Mario Carneiro, 30-May-2016.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (𝐴𝑐𝐵) ∈ ℝ+)
 
Theoremlogcxpd 13647 Logarithm of a complex power. (Contributed by Mario Carneiro, 30-May-2016.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (log‘(𝐴𝑐𝐵)) = (𝐵 · (log‘𝐴)))
 
Theoremcxplt3d 13648 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < 1)    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → (𝐵 < 𝐶 ↔ (𝐴𝑐𝐶) < (𝐴𝑐𝐵)))
 
Theoremcxple3d 13649 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < 1)    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → (𝐵𝐶 ↔ (𝐴𝑐𝐶) ≤ (𝐴𝑐𝐵)))
 
Theoremcxpmuld 13650 Product of exponents law for complex exponentiation. Proposition 10-4.2(b) of [Gleason] p. 135. (Contributed by Mario Carneiro, 30-May-2016.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → (𝐴𝑐(𝐵 · 𝐶)) = ((𝐴𝑐𝐵)↑𝑐𝐶))
 
Theoremcxpcom 13651 Commutative law for real exponentiation. (Contributed by AV, 29-Dec-2022.)
((𝐴 ∈ ℝ+𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴𝑐𝐵)↑𝑐𝐶) = ((𝐴𝑐𝐶)↑𝑐𝐵))
 
Theoremapcxp2 13652 Apartness and real exponentiation. (Contributed by Jim Kingdon, 10-Jul-2024.)
(((𝐴 ∈ ℝ+𝐴 # 1) ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → (𝐵 # 𝐶 ↔ (𝐴𝑐𝐵) # (𝐴𝑐𝐶)))
 
Theoremrpabscxpbnd 13653 Bound on the absolute value of a complex power. (Contributed by Mario Carneiro, 15-Sep-2014.) (Revised by Jim Kingdon, 19-Jun-2024.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑 → 0 < (ℜ‘𝐵))    &   (𝜑𝑀 ∈ ℝ)    &   (𝜑 → (abs‘𝐴) ≤ 𝑀)       (𝜑 → (abs‘(𝐴𝑐𝐵)) ≤ ((𝑀𝑐(ℜ‘𝐵)) · (exp‘((abs‘𝐵) · π))))
 
Theoremltexp2 13654 Ordering law for exponentiation. (Contributed by NM, 2-Aug-2006.) (Revised by Mario Carneiro, 5-Jun-2014.)
(((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 1 < 𝐴) → (𝑀 < 𝑁 ↔ (𝐴𝑀) < (𝐴𝑁)))
 
10.1.4  Logarithms to an arbitrary base

Define "log using an arbitrary base" function and then prove some of its properties. As with df-relog 13573 this is for real logarithms rather than complex logarithms.

Metamath doesn't care what letters are used to represent classes. Usually classes begin with the letter "A", but here we use "B" and "X" to more clearly distinguish between "base" and "other parameter of log".

There are different ways this could be defined in Metamath. The approach used here is intentionally similar to existing 2-parameter Metamath functions (operations): (𝐵 logb 𝑋) where 𝐵 is the base and 𝑋 is the argument of the logarithm function. An alternative would be to support the notational form (( logb𝐵)‘𝑋); that looks a little more like traditional notation.

 
Syntaxclogb 13655 Extend class notation to include the logarithm generalized to an arbitrary base.
class logb
 
Definitiondf-logb 13656* Define the logb operator. This is the logarithm generalized to an arbitrary base. It can be used as (𝐵 logb 𝑋) for "log base B of X". In the most common traditional notation, base B is a subscript of "log". The definition will only be useful where 𝑥 is a positive real apart from one and where 𝑦 is a positive real, so the choice of (ℂ ∖ {0, 1}) and (ℂ ∖ {0}) is somewhat arbitrary (we adopt the definition used in set.mm). (Contributed by David A. Wheeler, 21-Jan-2017.)
logb = (𝑥 ∈ (ℂ ∖ {0, 1}), 𝑦 ∈ (ℂ ∖ {0}) ↦ ((log‘𝑦) / (log‘𝑥)))
 
Theoremrplogbval 13657 Define the value of the logb function, the logarithm generalized to an arbitrary base, when used as infix. Most Metamath statements select variables in order of their use, but to make the order clearer we use "B" for base and "X" for the argument of the logarithm function here. (Contributed by David A. Wheeler, 21-Jan-2017.) (Revised by Jim Kingdon, 3-Jul-2024.)
((𝐵 ∈ ℝ+𝐵 # 1 ∧ 𝑋 ∈ ℝ+) → (𝐵 logb 𝑋) = ((log‘𝑋) / (log‘𝐵)))
 
Theoremrplogbcl 13658 General logarithm closure. (Contributed by David A. Wheeler, 17-Jul-2017.)
((𝐵 ∈ ℝ+𝐵 # 1 ∧ 𝑋 ∈ ℝ+) → (𝐵 logb 𝑋) ∈ ℝ)
 
Theoremrplogbid1 13659 General logarithm is 1 when base and arg match. Property 1(a) of [Cohen4] p. 361. (Contributed by Stefan O'Rear, 19-Sep-2014.) (Revised by David A. Wheeler, 22-Jul-2017.)
((𝐴 ∈ ℝ+𝐴 # 1) → (𝐴 logb 𝐴) = 1)
 
Theoremrplogb1 13660 The logarithm of 1 to an arbitrary base 𝐵 is 0. Property 1(b) of [Cohen4] p. 361. See log1 13581. (Contributed by Stefan O'Rear, 19-Sep-2014.) (Revised by Thierry Arnoux, 27-Sep-2017.)
((𝐵 ∈ ℝ+𝐵 # 1) → (𝐵 logb 1) = 0)
 
Theoremrpelogb 13661 The general logarithm of a number to the base being Euler's constant is the natural logarithm of the number. Put another way, using e as the base in logb is the same as log. Definition in [Cohen4] p. 352. (Contributed by David A. Wheeler, 17-Oct-2017.) (Revised by David A. Wheeler and AV, 16-Jun-2020.)
(𝐴 ∈ ℝ+ → (e logb 𝐴) = (log‘𝐴))
 
Theoremrplogbchbase 13662 Change of base for logarithms. Property in [Cohen4] p. 367. (Contributed by AV, 11-Jun-2020.)
(((𝐴 ∈ ℝ+𝐴 # 1) ∧ (𝐵 ∈ ℝ+𝐵 # 1) ∧ 𝑋 ∈ ℝ+) → (𝐴 logb 𝑋) = ((𝐵 logb 𝑋) / (𝐵 logb 𝐴)))
 
Theoremrelogbval 13663 Value of the general logarithm with integer base. (Contributed by Thierry Arnoux, 27-Sep-2017.)
((𝐵 ∈ (ℤ‘2) ∧ 𝑋 ∈ ℝ+) → (𝐵 logb 𝑋) = ((log‘𝑋) / (log‘𝐵)))
 
Theoremrelogbzcl 13664 Closure of the general logarithm with integer base on positive reals. (Contributed by Thierry Arnoux, 27-Sep-2017.) (Proof shortened by AV, 9-Jun-2020.)
((𝐵 ∈ (ℤ‘2) ∧ 𝑋 ∈ ℝ+) → (𝐵 logb 𝑋) ∈ ℝ)
 
Theoremrplogbreexp 13665 Power law for the general logarithm for real powers: The logarithm of a positive real number to the power of a real number is equal to the product of the exponent and the logarithm of the base of the power. Property 4 of [Cohen4] p. 361. (Contributed by AV, 9-Jun-2020.)
(((𝐵 ∈ ℝ+𝐵 # 1) ∧ 𝐶 ∈ ℝ+𝐸 ∈ ℝ) → (𝐵 logb (𝐶𝑐𝐸)) = (𝐸 · (𝐵 logb 𝐶)))
 
Theoremrplogbzexp 13666 Power law for the general logarithm for integer powers: The logarithm of a positive real number to the power of an integer is equal to the product of the exponent and the logarithm of the base of the power. (Contributed by Stefan O'Rear, 19-Sep-2014.) (Revised by AV, 9-Jun-2020.)
(((𝐵 ∈ ℝ+𝐵 # 1) ∧ 𝐶 ∈ ℝ+𝑁 ∈ ℤ) → (𝐵 logb (𝐶𝑁)) = (𝑁 · (𝐵 logb 𝐶)))
 
Theoremrprelogbmul 13667 The logarithm of the product of two positive real numbers is the sum of logarithms. Property 2 of [Cohen4] p. 361. (Contributed by Stefan O'Rear, 19-Sep-2014.) (Revised by AV, 29-May-2020.)
(((𝐵 ∈ ℝ+𝐵 # 1) ∧ (𝐴 ∈ ℝ+𝐶 ∈ ℝ+)) → (𝐵 logb (𝐴 · 𝐶)) = ((𝐵 logb 𝐴) + (𝐵 logb 𝐶)))
 
Theoremrprelogbmulexp 13668 The logarithm of the product of a positive real and a positive real number to the power of a real number is the sum of the logarithm of the first real number and the scaled logarithm of the second real number. (Contributed by AV, 29-May-2020.)
(((𝐵 ∈ ℝ+𝐵 # 1) ∧ (𝐴 ∈ ℝ+𝐶 ∈ ℝ+𝐸 ∈ ℝ)) → (𝐵 logb (𝐴 · (𝐶𝑐𝐸))) = ((𝐵 logb 𝐴) + (𝐸 · (𝐵 logb 𝐶))))
 
Theoremrprelogbdiv 13669 The logarithm of the quotient of two positive real numbers is the difference of logarithms. Property 3 of [Cohen4] p. 361. (Contributed by AV, 29-May-2020.)
(((𝐵 ∈ ℝ+𝐵 # 1) ∧ (𝐴 ∈ ℝ+𝐶 ∈ ℝ+)) → (𝐵 logb (𝐴 / 𝐶)) = ((𝐵 logb 𝐴) − (𝐵 logb 𝐶)))
 
Theoremrelogbexpap 13670 Identity law for general logarithm: the logarithm of a power to the base is the exponent. Property 6 of [Cohen4] p. 361. (Contributed by Stefan O'Rear, 19-Sep-2014.) (Revised by AV, 9-Jun-2020.)
((𝐵 ∈ ℝ+𝐵 # 1 ∧ 𝑀 ∈ ℤ) → (𝐵 logb (𝐵𝑀)) = 𝑀)
 
Theoremnnlogbexp 13671 Identity law for general logarithm with integer base. (Contributed by Stefan O'Rear, 19-Sep-2014.) (Revised by Thierry Arnoux, 27-Sep-2017.)
((𝐵 ∈ (ℤ‘2) ∧ 𝑀 ∈ ℤ) → (𝐵 logb (𝐵𝑀)) = 𝑀)
 
Theoremlogbrec 13672 Logarithm of a reciprocal changes sign. Particular case of Property 3 of [Cohen4] p. 361. (Contributed by Thierry Arnoux, 27-Sep-2017.)
((𝐵 ∈ (ℤ‘2) ∧ 𝐴 ∈ ℝ+) → (𝐵 logb (1 / 𝐴)) = -(𝐵 logb 𝐴))
 
Theoremlogbleb 13673 The general logarithm function is monotone/increasing. See logleb 13590. (Contributed by Stefan O'Rear, 19-Oct-2014.) (Revised by AV, 31-May-2020.)
((𝐵 ∈ (ℤ‘2) ∧ 𝑋 ∈ ℝ+𝑌 ∈ ℝ+) → (𝑋𝑌 ↔ (𝐵 logb 𝑋) ≤ (𝐵 logb 𝑌)))
 
Theoremlogblt 13674 The general logarithm function is strictly monotone/increasing. Property 2 of [Cohen4] p. 377. See logltb 13589. (Contributed by Stefan O'Rear, 19-Oct-2014.) (Revised by Thierry Arnoux, 27-Sep-2017.)
((𝐵 ∈ (ℤ‘2) ∧ 𝑋 ∈ ℝ+𝑌 ∈ ℝ+) → (𝑋 < 𝑌 ↔ (𝐵 logb 𝑋) < (𝐵 logb 𝑌)))
 
Theoremrplogbcxp 13675 Identity law for the general logarithm for real numbers. (Contributed by AV, 22-May-2020.)
((𝐵 ∈ ℝ+𝐵 # 1 ∧ 𝑋 ∈ ℝ) → (𝐵 logb (𝐵𝑐𝑋)) = 𝑋)
 
Theoremrpcxplogb 13676 Identity law for the general logarithm. (Contributed by AV, 22-May-2020.)
((𝐵 ∈ ℝ+𝐵 # 1 ∧ 𝑋 ∈ ℝ+) → (𝐵𝑐(𝐵 logb 𝑋)) = 𝑋)
 
Theoremrelogbcxpbap 13677 The logarithm is the inverse of the exponentiation. Observation in [Cohen4] p. 348. (Contributed by AV, 11-Jun-2020.)
(((𝐵 ∈ ℝ+𝐵 # 1) ∧ 𝑋 ∈ ℝ+𝑌 ∈ ℝ) → ((𝐵 logb 𝑋) = 𝑌 ↔ (𝐵𝑐𝑌) = 𝑋))
 
Theoremlogbgt0b 13678 The logarithm of a positive real number to a real base greater than 1 is positive iff the number is greater than 1. (Contributed by AV, 29-Dec-2022.)
((𝐴 ∈ ℝ+ ∧ (𝐵 ∈ ℝ+ ∧ 1 < 𝐵)) → (0 < (𝐵 logb 𝐴) ↔ 1 < 𝐴))
 
Theoremlogbgcd1irr 13679 The logarithm of an integer greater than 1 to an integer base greater than 1 is not rational if the argument and the base are relatively prime. For example, (2 logb 9) ∈ (ℝ ∖ ℚ). (Contributed by AV, 29-Dec-2022.)
((𝑋 ∈ (ℤ‘2) ∧ 𝐵 ∈ (ℤ‘2) ∧ (𝑋 gcd 𝐵) = 1) → (𝐵 logb 𝑋) ∈ (ℝ ∖ ℚ))
 
Theoremlogbgcd1irraplemexp 13680 Lemma for logbgcd1irrap 13682. Apartness of 𝑋𝑁 and 𝐵𝑀. (Contributed by Jim Kingdon, 11-Jul-2024.)
(𝜑𝑋 ∈ (ℤ‘2))    &   (𝜑𝐵 ∈ (ℤ‘2))    &   (𝜑 → (𝑋 gcd 𝐵) = 1)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ ℕ)       (𝜑 → (𝑋𝑁) # (𝐵𝑀))
 
Theoremlogbgcd1irraplemap 13681 Lemma for logbgcd1irrap 13682. The result, with the rational number expressed as numerator and denominator. (Contributed by Jim Kingdon, 9-Jul-2024.)
(𝜑𝑋 ∈ (ℤ‘2))    &   (𝜑𝐵 ∈ (ℤ‘2))    &   (𝜑 → (𝑋 gcd 𝐵) = 1)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ ℕ)       (𝜑 → (𝐵 logb 𝑋) # (𝑀 / 𝑁))
 
Theoremlogbgcd1irrap 13682 The logarithm of an integer greater than 1 to an integer base greater than 1 is irrational (in the sense of being apart from any rational number) if the argument and the base are relatively prime. For example, (2 logb 9) # 𝑄 where 𝑄 is rational. (Contributed by AV, 29-Dec-2022.)
(((𝑋 ∈ (ℤ‘2) ∧ 𝐵 ∈ (ℤ‘2)) ∧ ((𝑋 gcd 𝐵) = 1 ∧ 𝑄 ∈ ℚ)) → (𝐵 logb 𝑋) # 𝑄)
 
Theorem2logb9irr 13683 Example for logbgcd1irr 13679. The logarithm of nine to base two is not rational. Also see 2logb9irrap 13689 which says that it is irrational (in the sense of being apart from any rational number). (Contributed by AV, 29-Dec-2022.)
(2 logb 9) ∈ (ℝ ∖ ℚ)
 
Theoremlogbprmirr 13684 The logarithm of a prime to a different prime base is not rational. For example, (2 logb 3) ∈ (ℝ ∖ ℚ) (see 2logb3irr 13685). (Contributed by AV, 31-Dec-2022.)
((𝑋 ∈ ℙ ∧ 𝐵 ∈ ℙ ∧ 𝑋𝐵) → (𝐵 logb 𝑋) ∈ (ℝ ∖ ℚ))
 
Theorem2logb3irr 13685 Example for logbprmirr 13684. The logarithm of three to base two is not rational. (Contributed by AV, 31-Dec-2022.)
(2 logb 3) ∈ (ℝ ∖ ℚ)
 
Theorem2logb9irrALT 13686 Alternate proof of 2logb9irr 13683: The logarithm of nine to base two is not rational. (Contributed by AV, 31-Dec-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
(2 logb 9) ∈ (ℝ ∖ ℚ)
 
Theoremsqrt2cxp2logb9e3 13687 The square root of two to the power of the logarithm of nine to base two is three. (√‘2) and (2 logb 9) are not rational (see sqrt2irr0 12118 resp. 2logb9irr 13683), satisfying the statement in 2irrexpq 13688. (Contributed by AV, 29-Dec-2022.)
((√‘2)↑𝑐(2 logb 9)) = 3
 
Theorem2irrexpq 13688* There exist real numbers 𝑎 and 𝑏 which are not rational such that (𝑎𝑏) is rational. Statement in the Metamath book, section 1.1.5, footnote 27 on page 17, and the "constructive proof" for theorem 1.2 of [Bauer], p. 483. This is a constructive proof because it is based on two explicitly named non-rational numbers (√‘2) and (2 logb 9), see sqrt2irr0 12118, 2logb9irr 13683 and sqrt2cxp2logb9e3 13687. Therefore, this proof is acceptable/usable in intuitionistic logic.

For a theorem which is the same but proves that 𝑎 and 𝑏 are irrational (in the sense of being apart from any rational number), see 2irrexpqap 13690. (Contributed by AV, 23-Dec-2022.)

𝑎 ∈ (ℝ ∖ ℚ)∃𝑏 ∈ (ℝ ∖ ℚ)(𝑎𝑐𝑏) ∈ ℚ
 
Theorem2logb9irrap 13689 Example for logbgcd1irrap 13682. The logarithm of nine to base two is irrational (in the sense of being apart from any rational number). (Contributed by Jim Kingdon, 12-Jul-2024.)
(𝑄 ∈ ℚ → (2 logb 9) # 𝑄)
 
Theorem2irrexpqap 13690* There exist real numbers 𝑎 and 𝑏 which are irrational (in the sense of being apart from any rational number) such that (𝑎𝑏) is rational. Statement in the Metamath book, section 1.1.5, footnote 27 on page 17, and the "constructive proof" for theorem 1.2 of [Bauer], p. 483. This is a constructive proof because it is based on two explicitly named irrational numbers (√‘2) and (2 logb 9), see sqrt2irrap 12134, 2logb9irrap 13689 and sqrt2cxp2logb9e3 13687. Therefore, this proof is acceptable/usable in intuitionistic logic. (Contributed by Jim Kingdon, 12-Jul-2024.)
𝑎 ∈ ℝ ∃𝑏 ∈ ℝ (∀𝑝 ∈ ℚ 𝑎 # 𝑝 ∧ ∀𝑞 ∈ ℚ 𝑏 # 𝑞 ∧ (𝑎𝑐𝑏) ∈ ℚ)
 
10.1.5  Quartic binomial expansion
 
Theorembinom4 13691 Work out a quartic binomial. (You would think that by this point it would be faster to use binom 11447, but it turns out to be just as much work to put it into this form after clearing all the sums and calculating binomial coefficients.) (Contributed by Mario Carneiro, 6-May-2015.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵)↑4) = (((𝐴↑4) + (4 · ((𝐴↑3) · 𝐵))) + ((6 · ((𝐴↑2) · (𝐵↑2))) + ((4 · (𝐴 · (𝐵↑3))) + (𝐵↑4)))))
 
10.2  Basic number theory
 
10.2.1  Quadratic residues and the Legendre symbol

If the congruence ((𝑥↑2) mod 𝑝) = (𝑛 mod 𝑝) has a solution we say that 𝑛 is a quadratic residue mod 𝑝. If the congruence has no solution we say that 𝑛 is a quadratic nonresidue mod 𝑝, see definition in [ApostolNT] p. 178. The Legendre symbol (𝑛 /L 𝑝) is defined in a way that its value is 1 if 𝑛 is a quadratic residue mod 𝑝 and -1 if 𝑛 is a quadratic nonresidue mod 𝑝 (and 0 if 𝑝 divides 𝑛).

Originally, the Legendre symbol (𝑁 /L 𝑃) was defined for odd primes 𝑃 only (and arbitrary integers 𝑁) by Adrien-Marie Legendre in 1798, see definition in [ApostolNT] p. 179. It was generalized to be defined for any positive odd integer by Carl Gustav Jacob Jacobi in 1837 (therefore called "Jacobi symbol" since then), see definition in [ApostolNT] p. 188. Finally, it was generalized to be defined for any integer by Leopold Kronecker in 1885 (therefore called "Kronecker symbol" since then). The definition df-lgs 13693 for the "Legendre symbol" /L is actually the definition of the "Kronecker symbol". Since only one definition (and one class symbol) are provided in set.mm, the names "Legendre symbol", "Jacobi symbol" and "Kronecker symbol" are used synonymously for /L, but mostly it is called "Legendre symbol", even if it is used in the context of a "Jacobi symbol" or "Kronecker symbol".

 
Syntaxclgs 13692 Extend class notation with the Legendre symbol function.
class /L
 
Definitiondf-lgs 13693* Define the Legendre symbol (actually the Kronecker symbol, which extends the Legendre symbol to all integers, and also the Jacobi symbol, which restricts the Kronecker symbol to positive odd integers). See definition in [ApostolNT] p. 179 resp. definition in [ApostolNT] p. 188. (Contributed by Mario Carneiro, 4-Feb-2015.)
/L = (𝑎 ∈ ℤ, 𝑛 ∈ ℤ ↦ if(𝑛 = 0, if((𝑎↑2) = 1, 1, 0), (if((𝑛 < 0 ∧ 𝑎 < 0), -1, 1) · (seq1( · , (𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, (if(𝑚 = 2, if(2 ∥ 𝑎, 0, if((𝑎 mod 8) ∈ {1, 7}, 1, -1)), ((((𝑎↑((𝑚 − 1) / 2)) + 1) mod 𝑚) − 1))↑(𝑚 pCnt 𝑛)), 1)))‘(abs‘𝑛)))))
 
Theoremzabsle1 13694 {-1, 0, 1} is the set of all integers with absolute value at most 1. (Contributed by AV, 13-Jul-2021.)
(𝑍 ∈ ℤ → (𝑍 ∈ {-1, 0, 1} ↔ (abs‘𝑍) ≤ 1))
 
Theoremlgslem1 13695 When 𝑎 is coprime to the prime 𝑝, 𝑎↑((𝑝 − 1) / 2) is equivalent mod 𝑝 to 1 or -1, and so adding 1 makes it equivalent to 0 or 2. (Contributed by Mario Carneiro, 4-Feb-2015.)
((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃𝐴) → (((𝐴↑((𝑃 − 1) / 2)) + 1) mod 𝑃) ∈ {0, 2})
 
Theoremlgslem2 13696 The set 𝑍 of all integers with absolute value at most 1 contains {-1, 0, 1}. (Contributed by Mario Carneiro, 4-Feb-2015.)
𝑍 = {𝑥 ∈ ℤ ∣ (abs‘𝑥) ≤ 1}       (-1 ∈ 𝑍 ∧ 0 ∈ 𝑍 ∧ 1 ∈ 𝑍)
 
Theoremlgslem3 13697* The set 𝑍 of all integers with absolute value at most 1 is closed under multiplication. (Contributed by Mario Carneiro, 4-Feb-2015.)
𝑍 = {𝑥 ∈ ℤ ∣ (abs‘𝑥) ≤ 1}       ((𝐴𝑍𝐵𝑍) → (𝐴 · 𝐵) ∈ 𝑍)
 
Theoremlgslem4 13698* Lemma for lgsfcl2 13701. (Contributed by Mario Carneiro, 4-Feb-2015.) (Proof shortened by AV, 19-Mar-2022.)
𝑍 = {𝑥 ∈ ℤ ∣ (abs‘𝑥) ≤ 1}       ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) → ((((𝐴↑((𝑃 − 1) / 2)) + 1) mod 𝑃) − 1) ∈ 𝑍)
 
Theoremlgsval 13699* Value of the Legendre symbol at an arbitrary integer. (Contributed by Mario Carneiro, 4-Feb-2015.)
𝐹 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (if(𝑛 = 2, if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)), ((((𝐴↑((𝑛 − 1) / 2)) + 1) mod 𝑛) − 1))↑(𝑛 pCnt 𝑁)), 1))       ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐴 /L 𝑁) = if(𝑁 = 0, if((𝐴↑2) = 1, 1, 0), (if((𝑁 < 0 ∧ 𝐴 < 0), -1, 1) · (seq1( · , 𝐹)‘(abs‘𝑁)))))
 
Theoremlgsfvalg 13700* Value of the function 𝐹 which defines the Legendre symbol at the primes. (Contributed by Mario Carneiro, 4-Feb-2015.) (Revised by Jim Kingdon, 4-Nov-2024.)
𝐹 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (if(𝑛 = 2, if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)), ((((𝐴↑((𝑛 − 1) / 2)) + 1) mod 𝑛) − 1))↑(𝑛 pCnt 𝑁)), 1))       ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ) → (𝐹𝑀) = if(𝑀 ∈ ℙ, (if(𝑀 = 2, if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)), ((((𝐴↑((𝑀 − 1) / 2)) + 1) mod 𝑀) − 1))↑(𝑀 pCnt 𝑁)), 1))
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14113
  Copyright terms: Public domain < Previous  Next >