P. 162 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 16101-16200   *Has distinct variable group(s)

Type	Label	Description
Statement
11.2.4  Logarithms to an arbitrary base Define "log using an arbitrary base" function and then prove some of its properties. As with df-relog 16012 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 (( logb ‘𝐵)‘𝑋); that looks a little more like traditional notation.
Syntax	clogb 16101	Extend class notation to include the logarithm generalized to an arbitrary base.
class logb
Definition	df-logb 16102*	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)))
Theorem	rplogbval 16103	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)))
Theorem	rplogbcl 16104	General logarithm closure. (Contributed by David A. Wheeler, 17-Jul-2017.)
⊢ ((𝐵 ∈ ℝ+ ∧ 𝐵 # 1 ∧ 𝑋 ∈ ℝ+) → (𝐵 logb 𝑋) ∈ ℝ)
Theorem	rplogbid1 16105	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)
Theorem	rplogb1 16106	The logarithm of 1 to an arbitrary base 𝐵 is 0. Property 1(b) of [Cohen4] p. 361. See log1 16020. (Contributed by Stefan O'Rear, 19-Sep-2014.) (Revised by Thierry Arnoux, 27-Sep-2017.)
⊢ ((𝐵 ∈ ℝ+ ∧ 𝐵 # 1) → (𝐵 logb 1) = 0)
Theorem	rpelogb 16107	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))
Theorem	rplogbchbase 16108	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)))
Theorem	relogbval 16109	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)))
Theorem	relogbzcl 16110	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)
Theorem	rplogbreexp 16111	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 (𝐶↑𝑐𝐸)) = (𝐸 · (𝐵 logb 𝐶)))
Theorem	rplogbzexp 16112	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 (𝐶↑𝑁)) = (𝑁 · (𝐵 logb 𝐶)))
Theorem	rprelogbmul 16113	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)))
Theorem	rprelogbmulexp 16114	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 · (𝐶↑𝑐𝐸))) = ((𝐵 logb 𝐴) + (𝐸 · (𝐵 logb 𝐶))))
Theorem	rprelogbdiv 16115	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)))
Theorem	relogbexpap 16116	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 (𝐵↑𝑀)) = 𝑀)
Theorem	nnlogbexp 16117	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 (𝐵↑𝑀)) = 𝑀)
Theorem	logbrec 16118	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))
Theorem	logbleb 16119	The general logarithm function is monotone/increasing. See logleb 16030. (Contributed by Stefan O'Rear, 19-Oct-2014.) (Revised by AV, 31-May-2020.)
⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ 𝑋 ∈ ℝ+ ∧ 𝑌 ∈ ℝ+) → (𝑋 ≤ 𝑌 ↔ (𝐵 logb 𝑋) ≤ (𝐵 logb 𝑌)))
Theorem	logblt 16120	The general logarithm function is strictly monotone/increasing. Property 2 of [Cohen4] p. 377. See logltb 16029. (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)))
Theorem	rplogbcxp 16121	Identity law for the general logarithm for real numbers. (Contributed by AV, 22-May-2020.)
⊢ ((𝐵 ∈ ℝ+ ∧ 𝐵 # 1 ∧ 𝑋 ∈ ℝ) → (𝐵 logb (𝐵↑𝑐𝑋)) = 𝑋)
Theorem	rpcxplogb 16122	Identity law for the general logarithm. (Contributed by AV, 22-May-2020.)
⊢ ((𝐵 ∈ ℝ+ ∧ 𝐵 # 1 ∧ 𝑋 ∈ ℝ+) → (𝐵↑𝑐(𝐵 logb 𝑋)) = 𝑋)
Theorem	relogbcxpbap 16123	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 ↔ (𝐵↑𝑐𝑌) = 𝑋))
Theorem	logbgt0b 16124	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))
Theorem	logbgcd1irr 16125	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))
Theorem	logbgcd1irraplemexp 16126	Lemma for logbgcd1irrap 16128. Apartness of 𝑋↑𝑁 and 𝐵↑𝑀. (Contributed by Jim Kingdon, 11-Jul-2024.)
⊢ (𝜑 → 𝑋 ∈ (ℤ≥‘2))    &   ⊢ (𝜑 → 𝐵 ∈ (ℤ≥‘2))    &   ⊢ (𝜑 → (𝑋 gcd 𝐵) = 1)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    ⇒   ⊢ (𝜑 → (𝑋↑𝑁) # (𝐵↑𝑀))
Theorem	logbgcd1irraplemap 16127	Lemma for logbgcd1irrap 16128. The result, with the rational number expressed as numerator and denominator. (Contributed by Jim Kingdon, 9-Jul-2024.)
⊢ (𝜑 → 𝑋 ∈ (ℤ≥‘2))    &   ⊢ (𝜑 → 𝐵 ∈ (ℤ≥‘2))    &   ⊢ (𝜑 → (𝑋 gcd 𝐵) = 1)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    ⇒   ⊢ (𝜑 → (𝐵 logb 𝑋) # (𝑀 / 𝑁))
Theorem	logbgcd1irrap 16128	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)
Theorem	2logb9irr 16129	Example for logbgcd1irr 16125. The logarithm of nine to base two is not rational. Also see 2logb9irrap 16135 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)
Theorem	logbprmirr 16130	The logarithm of a prime to a different prime base is not rational. For example, (2 logb 3) ∈ (ℝ ∖ ℚ) (see 2logb3irr 16131). (Contributed by AV, 31-Dec-2022.)
⊢ ((𝑋 ∈ ℙ ∧ 𝐵 ∈ ℙ ∧ 𝑋 ≠ 𝐵) → (𝐵 logb 𝑋) ∈ (ℝ ∖ ℚ))
Theorem	2logb3irr 16131	Example for logbprmirr 16130. The logarithm of three to base two is not rational. (Contributed by AV, 31-Dec-2022.)
⊢ (2 logb 3) ∈ (ℝ ∖ ℚ)
Theorem	2logb9irrALT 16132	Alternate proof of 2logb9irr 16129: 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)
Theorem	sqrt2cxp2logb9e3 16133	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 12961 resp. 2logb9irr 16129), satisfying the statement in 2irrexpq 16134. (Contributed by AV, 29-Dec-2022.)
⊢ ((√‘2)↑𝑐(2 logb 9)) = 3
Theorem	2irrexpq 16134*	There exist real numbers 𝑎 and 𝑏 which are not rational such that (𝑎↑𝑏) 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 12961, 2logb9irr 16129 and sqrt2cxp2logb9e3 16133. 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 16136. (Contributed by AV, 23-Dec-2022.)
⊢ ∃𝑎 ∈ (ℝ ∖ ℚ)∃𝑏 ∈ (ℝ ∖ ℚ)(𝑎↑𝑐𝑏) ∈ ℚ
Theorem	2logb9irrap 16135	Example for logbgcd1irrap 16128. 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)
Theorem	2irrexpqap 16136*	There exist real numbers 𝑎 and 𝑏 which are irrational (in the sense of being apart from any rational number) such that (𝑎↑𝑏) 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 12978, 2logb9irrap 16135 and sqrt2cxp2logb9e3 16133. 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 ∧ (𝑎↑𝑐𝑏) ∈ ℚ)
Theorem	zprmlogbaplem1 16137	Lemma for zprmlogbap 16140. Rearranging an expression involving logarithms. (Contributed by Jim Kingdon, 20-Aug-2026.)
⊢ (𝜑 → 𝐵 ∈ ℙ)    &   ⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → ¬ 𝐵 ∥ 𝑀)    &   ⊢ (𝜑 → 𝐴 ∈ ℕ0)    ⇒   ⊢ (𝜑 → (𝐵 logb ((𝐵↑𝐴) · 𝑀)) = (𝐴 + (𝐵 logb 𝑀)))
Theorem	zprmlogbaplem2 16138*	Lemma for zprmlogbap 16140. The logarithm is either rational or irrational. (Contributed by Jim Kingdon, 20-Aug-2026.)
⊢ (𝜑 → 𝐵 ∈ ℙ)    &   ⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → ¬ 𝐵 ∥ 𝑀)    &   ⊢ (𝜑 → 𝐴 ∈ ℕ0)    &   ⊢ 𝑋 = ((𝐵↑𝐴) · 𝑀)    ⇒   ⊢ (𝜑 → ((𝐵 logb 𝑋) ∈ ℚ ∨ ((𝐵 logb 𝑋) ∈ ℝ ∧ ∀𝑞 ∈ ℚ (𝐵 logb 𝑋) # 𝑞)))
Theorem	zprmlogbaplem3 16139*	Lemma for zprmlogbap 16140. Decomposing a natural number into a power of a prime base and a factor not divisible by that prime. (Contributed by Jim Kingdon, 20-Aug-2026.)
⊢ 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 𝐵 ∥ 𝑧}    &   ⊢ 𝐹 = (𝑥 ∈ 𝐽, 𝑦 ∈ ℕ0 ↦ ((𝐵↑𝑦) · 𝑥))    ⇒   ⊢ ((𝑋 ∈ ℕ ∧ 𝐵 ∈ ℙ) → ∃𝑚 ∈ ℕ ∃𝑎 ∈ ℕ0 (¬ 𝐵 ∥ 𝑚 ∧ 𝑋 = ((𝐵↑𝑎) · 𝑚)))
Theorem	zprmlogbap 16140*	The logarithm of a natural number to a prime base is either rational or irrational. The proof decomposes 𝑋 into 𝑚 ∈ ℕ and 𝑎 ∈ ℕ0 such that 𝑋 = ((𝐵↑𝑎) · 𝑚) (using nnmaxpw 12971). If 𝑚 = 1 the logarithm is 𝑎, which is rational. If 1 < 𝑚 then we can apply logbgcd1irrap 16128 to show that the logarithm is irrational. (Contributed by Jim Kingdon and Taylor Barrella, 20-Aug-2026.)
⊢ ((𝑋 ∈ ℕ ∧ 𝐵 ∈ ℙ) → ((𝐵 logb 𝑋) ∈ ℚ ∨ ((𝐵 logb 𝑋) ∈ ℝ ∧ ∀𝑞 ∈ ℚ (𝐵 logb 𝑋) # 𝑞)))
11.2.5  Quartic binomial expansion
Theorem	binom4 16141	Work out a quartic binomial. (You would think that by this point it would be faster to use binom 12269, 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)))))
11.2.6  Logarithms (cont.)
Theorem	log2tlbndlog2 16142*	Bound the error term in the series of the hypothesis. The presence of the hypothesis here is a temporary measure until it can be proved as log2cnv . (Contributed by Mario Carneiro, 7-Apr-2015.)
⊢ seq0( + , (𝑘 ∈ ℕ0 ↦ (2 / ((3 · ((2 · 𝑘) + 1)) · (9↑𝑘))))) ⇝ (log‘2)    ⇒   ⊢ (𝑁 ∈ ℕ0 → ((log‘2) − Σ𝑛 ∈ (0...(𝑁 − 1))(2 / ((3 · ((2 · 𝑛) + 1)) · (9↑𝑛)))) ∈ (0[,](3 / ((4 · ((2 · 𝑁) + 1)) · (9↑𝑁)))))
11.2.7  The Birthday Problem
Theorem	log2ublem1 16143	Lemma for log2ublog2 16146. The proof of log2ublog2 16146, which is simply the evaluation of log2tlbndlog2 16142 for 𝑁 = 4, takes the form of the addition of five fractions and showing this is less than another fraction. We could just perform exact arithmetic on these fractions, get a large rational number, and just multiply everything to verify the claim, but as anyone who uses decimal numbers for this task knows, it is often better to pick a common denominator 𝑑 (usually a large power of 10) and work with the closest approximations of the form 𝑛 / 𝑑 for some integer 𝑛 instead. It turns out that for our purposes it is sufficient to take 𝑑 = (3↑7) · 5 · 7, which is also nice because it shares many factors in common with the fractions in question. (Contributed by Mario Carneiro, 17-Apr-2015.)
⊢ (((3↑7) · (5 · 7)) · 𝐴) ≤ 𝐵    &   ⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐷 ∈ ℕ0    &   ⊢ 𝐸 ∈ ℕ    &   ⊢ 𝐵 ∈ ℕ0    &   ⊢ 𝐹 ∈ ℕ0    &   ⊢ 𝐶 = (𝐴 + (𝐷 / 𝐸))    &   ⊢ (𝐵 + 𝐹) = 𝐺    &   ⊢ (((3↑7) · (5 · 7)) · 𝐷) ≤ (𝐸 · 𝐹)    ⇒   ⊢ (((3↑7) · (5 · 7)) · 𝐶) ≤ 𝐺
Theorem	log2ublem2 16144*	Lemma for log2ublog2 16146. (Contributed by Mario Carneiro, 17-Apr-2015.)
⊢ (((3↑7) · (5 · 7)) · Σ𝑛 ∈ (0...𝐾)(2 / ((3 · ((2 · 𝑛) + 1)) · (9↑𝑛)))) ≤ (2 · 𝐵)    &   ⊢ 𝐵 ∈ ℕ0    &   ⊢ 𝐹 ∈ ℕ0    &   ⊢ 𝑁 ∈ ℕ0    &   ⊢ (𝑁 − 1) = 𝐾    &   ⊢ (𝐵 + 𝐹) = 𝐺    &   ⊢ 𝑀 ∈ ℕ0    &   ⊢ (𝑀 + 𝑁) = 3    &   ⊢ ((5 · 7) · (9↑𝑀)) = (((2 · 𝑁) + 1) · 𝐹)    ⇒   ⊢ (((3↑7) · (5 · 7)) · Σ𝑛 ∈ (0...𝑁)(2 / ((3 · ((2 · 𝑛) + 1)) · (9↑𝑛)))) ≤ (2 · 𝐺)
Theorem	log2ublem3 16145	Lemma for log2ublog2 16146. In decimal, this is a proof that the first four terms of the series for log2 is less than 53056 / 76545. (Contributed by Mario Carneiro, 17-Apr-2015.) (Proof shortened by AV, 15-Sep-2021.)
⊢ (((3↑7) · (5 · 7)) · Σ𝑛 ∈ (0...3)(2 / ((3 · ((2 · 𝑛) + 1)) · (9↑𝑛)))) ≤ ;;;;53056
Theorem	log2ublog2 16146	log2 is less than 253 / 365. If written in decimal, this is because log2 = 0.693147... is less than 253/365 = 0.693151... , so this is a very tight bound, at five decimal places. The presence of the hypothesis here is a temporary measure until it can be proved as log2cnv . (Contributed by Mario Carneiro, 7-Apr-2015.) (Proof shortened by AV, 16-Sep-2021.)
⊢ seq0( + , (𝑘 ∈ ℕ0 ↦ (2 / ((3 · ((2 · 𝑘) + 1)) · (9↑𝑘))))) ⇝ (log‘2)    ⇒   ⊢ (log‘2) < (;;253 / ;;365)
Theorem	birthdaylem1g 16147*	Lemma for birthdaylog2 16150. (Contributed by Mario Carneiro, 17-Apr-2015.)
⊢ 𝑆 = {𝑓 ∣ 𝑓:(1...𝐾)⟶(1...𝑁)}    &   ⊢ 𝑇 = {𝑓 ∣ 𝑓:(1...𝐾)–1-1→(1...𝑁)}    ⇒   ⊢ ((𝐾 ∈ ℕ0 ∧ 𝑁 ∈ ℕ) → (𝑇 ⊆ 𝑆 ∧ 𝑆 ∈ Fin ∧ 𝑆 ≠ ∅))
Theorem	birthdaylem2 16148*	For general 𝑁 and 𝐾, count the fraction of injective functions from 1...𝐾 to 1...𝑁. (Contributed by Mario Carneiro, 7-May-2015.)
⊢ 𝑆 = {𝑓 ∣ 𝑓:(1...𝐾)⟶(1...𝑁)}    &   ⊢ 𝑇 = {𝑓 ∣ 𝑓:(1...𝐾)–1-1→(1...𝑁)}    ⇒   ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁)) → ((♯‘𝑇) / (♯‘𝑆)) = (exp‘Σ𝑘 ∈ (0...(𝐾 − 1))(log‘(1 − (𝑘 / 𝑁)))))
Theorem	birthdaylem3 16149*	For general 𝑁 and 𝐾, upper-bound the fraction of injective functions from 1...𝐾 to 1...𝑁. (Contributed by Mario Carneiro, 17-Apr-2015.)
⊢ 𝑆 = {𝑓 ∣ 𝑓:(1...𝐾)⟶(1...𝑁)}    &   ⊢ 𝑇 = {𝑓 ∣ 𝑓:(1...𝐾)–1-1→(1...𝑁)}    ⇒   ⊢ ((𝐾 ∈ ℕ0 ∧ 𝑁 ∈ ℕ) → ((♯‘𝑇) / (♯‘𝑆)) ≤ (exp‘-((((𝐾↑2) − 𝐾) / 2) / 𝑁)))
Theorem	birthdaylog2 16150*	The Birthday Problem. There is a more than even chance that out of 23 people in a room, at least two of them have the same birthday. Mathematically, this is asserting that for 𝐾 = 23 and 𝑁 = 365, fewer than half of the set of all functions from 1...𝐾 to 1...𝑁 are injective. The presence of the hypothesis giving a series which converges to (log‘2) is a temporary measure until it can be proved as log2cnv . Although this is Metamath 100 proof #93, we cannot consider it proved until we prove the missing log2cnv piece (or prove the theorem another way which does not require it). (Contributed by Mario Carneiro, 17-Apr-2015.)
⊢ 𝑆 = {𝑓 ∣ 𝑓:(1...𝐾)⟶(1...𝑁)}    &   ⊢ 𝑇 = {𝑓 ∣ 𝑓:(1...𝐾)–1-1→(1...𝑁)}    &   ⊢ 𝐾 = ;23    &   ⊢ 𝑁 = ;;365    &   ⊢ seq0( + , (𝑘 ∈ ℕ0 ↦ (2 / ((3 · ((2 · 𝑘) + 1)) · (9↑𝑘))))) ⇝ (log‘2)    ⇒   ⊢ ((♯‘𝑇) / (♯‘𝑆)) < (1 / 2)
11.3  Pell equations
11.3.1  Pell equations 1: A nontrivial solution always exists
Theorem	pellexlem1 16151	Lemma for pellex . Arithmetical core of pellexlem3, norm lower bound. This begins Dirichlet's proof of the Pell equation solution existence; the proof here follows theorem 62 of [vandenDries] p. 43. (Contributed by Stefan O'Rear, 14-Sep-2014.)
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ ¬ (√‘𝐷) ∈ ℚ) → ((𝐴↑2) − (𝐷 · (𝐵↑2))) ≠ 0)
Theorem	pellexlem2 16152	Lemma for pellex . Arithmetical core of pellexlem3, norm upper bound. (Contributed by Stefan O'Rear, 14-Sep-2014.)
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (abs‘((𝐴↑2) − (𝐷 · (𝐵↑2)))) < (1 + (2 · (√‘𝐷))))
Theorem	pellexlem3 16153*	Lemma for pellex . To each good rational approximation of (√‘𝐷), there exists a near-solution. (Contributed by Stefan O'Rear, 14-Sep-2014.)
⊢ ((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) → {𝑥 ∈ ℚ ∣ (0 < 𝑥 ∧ (abs‘(𝑥 − (√‘𝐷))) < ((denom‘𝑥)↑-2))} ≼ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷)))))})
11.4  Basic number theory
11.4.1  Wilson's theorem
Theorem	wilthlem1 16154	The only elements that are equal to their own inverses in the multiplicative group of nonzero elements in ℤ / 𝑃ℤ are 1 and -1≡𝑃 − 1. (Note that from prmdiveq 13036, (𝑁↑(𝑃 − 2)) mod 𝑃 is the modular inverse of 𝑁 in ℤ / 𝑃ℤ. (Contributed by Mario Carneiro, 24-Jan-2015.)
⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (𝑁 = ((𝑁↑(𝑃 − 2)) mod 𝑃) ↔ (𝑁 = 1 ∨ 𝑁 = (𝑃 − 1))))
11.4.2  Number-theoretical functions
Syntax	ccht 16155	Extend class notation with the first Chebyshev function.
class θ
Syntax	cppi 16156	Extend class notation with the prime-counting function pi.
class π
Syntax	csgm 16157	Extend class notation with the divisor function.
class σ
Definition	df-cht 16158*	Define the first Chebyshev function, which adds up the logarithms of all primes less than 𝑥, see definition in [ApostolNT] p. 75. The symbol used to represent this function is sometimes the variant greek letter theta shown here and sometimes the greek letter psi, ψ; however, this notation can also refer to the second Chebyshev function, which adds up the logarithms of prime powers instead. See https://en.wikipedia.org/wiki/Chebyshev_function for a discussion of the two functions. (Contributed by Mario Carneiro, 15-Sep-2014.)
⊢ θ = (𝑥 ∈ ℝ ↦ Σ𝑝 ∈ ((0[,]𝑥) ∩ ℙ)(log‘𝑝))
Definition	df-ppi 16159	Define the prime π function, which counts the number of primes less than or equal to 𝑥, see definition in [ApostolNT] p. 8. Most often 𝑥 will be an integer, but many of our theorems support rational numbers (for example at ppiqsval 16162), and the definition would also work for cases such as numbers known to be irrational. (Contributed by Mario Carneiro, 15-Sep-2014.)
⊢ π = (𝑥 ∈ ℝ ↦ (♯‘((0[,]𝑥) ∩ ℙ)))
Definition	df-sgm 16160*	Define the sum of positive divisors function (𝑥 σ 𝑛), which is the sum of the xth powers of the positive integer divisors of n, see definition in [ApostolNT] p. 38. For 𝑥 = 0, (𝑥 σ 𝑛) counts the number of divisors of 𝑛, i.e. (0 σ 𝑛) is the divisor function, see remark in [ApostolNT] p. 38. (Contributed by Mario Carneiro, 22-Sep-2014.)
⊢ σ = (𝑥 ∈ ℂ, 𝑛 ∈ ℕ ↦ Σ𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝑛} (𝑘↑𝑐𝑥))
Theorem	efnnfsumcl 16161*	Finite sum closure in the log-integers. (Contributed by Mario Carneiro, 7-Apr-2016.)
⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (exp‘𝐵) ∈ ℕ)    ⇒   ⊢ (𝜑 → (exp‘Σ𝑘 ∈ 𝐴 𝐵) ∈ ℕ)
Theorem	ppiqsval 16162	The set of primes less than 𝐴 expressed using a finite set of integers. (Contributed by Mario Carneiro, 22-Sep-2014.)
⊢ (𝐴 ∈ ℚ → ((0[,]𝐴) ∩ ℙ) = ((2...(⌊‘𝐴)) ∩ ℙ))
Theorem	ppiqsval2 16163	The set of primes less than 𝐴 expressed using a finite set of integers. (Contributed by Mario Carneiro, 22-Sep-2014.)
⊢ ((𝐴 ∈ ℚ ∧ 2 ∈ (ℤ≥‘𝑀)) → ((0[,]𝐴) ∩ ℙ) = ((𝑀...(⌊‘𝐴)) ∩ ℙ))
Theorem	ppiqfi 16164	The set of primes less than 𝐴 is a finite set. (Contributed by Mario Carneiro, 15-Sep-2014.)
⊢ (𝐴 ∈ ℚ → ((0[,]𝐴) ∩ ℙ) ∈ Fin)
Theorem	prmdvdsfi 16165*	The set of prime divisors of a number is a finite set. (Contributed by Mario Carneiro, 7-Apr-2016.)
⊢ (𝐴 ∈ ℕ → {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ∈ Fin)
Theorem	chtqcl 16166	Rational closure of the Chebyshev function. (Contributed by Mario Carneiro, 15-Sep-2014.)
⊢ (𝐴 ∈ ℚ → (θ‘𝐴) ∈ ℝ)
Theorem	chtqval 16167*	Value of the Chebyshev function. (Contributed by Mario Carneiro, 15-Sep-2014.)
⊢ (𝐴 ∈ ℚ → (θ‘𝐴) = Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)(log‘𝑝))
Theorem	efchtqcl 16168	The Chebyshev function is closed in the log-integers. (Contributed by Mario Carneiro, 22-Sep-2014.) (Revised by Mario Carneiro, 7-Apr-2016.)
⊢ (𝐴 ∈ ℚ → (exp‘(θ‘𝐴)) ∈ ℕ)
Theorem	chtqge0 16169	The Chebyshev function is always positive. (Contributed by Mario Carneiro, 15-Sep-2014.)
⊢ (𝐴 ∈ ℚ → 0 ≤ (θ‘𝐴))
Theorem	ppiqval 16170	Value of the prime-counting function pi. (Contributed by Mario Carneiro, 15-Sep-2014.)
⊢ (𝐴 ∈ ℚ → (π‘𝐴) = (♯‘((0[,]𝐴) ∩ ℙ)))
Theorem	ppival2 16171	Value of the prime-counting function pi. (Contributed by Mario Carneiro, 18-Sep-2014.)
⊢ (𝐴 ∈ ℤ → (π‘𝐴) = (♯‘((2...𝐴) ∩ ℙ)))
Theorem	ppival2g 16172	Value of the prime-counting function pi. (Contributed by Mario Carneiro, 22-Sep-2014.)
⊢ ((𝐴 ∈ ℤ ∧ 2 ∈ (ℤ≥‘𝑀)) → (π‘𝐴) = (♯‘((𝑀...𝐴) ∩ ℙ)))
Theorem	ppiqcl 16173	Rational closure of the prime-counting function pi. (Contributed by Mario Carneiro, 15-Sep-2014.)
⊢ (𝐴 ∈ ℚ → (π‘𝐴) ∈ ℕ0)
Theorem	sgmval 16174*	The value of the divisor function. (Contributed by Mario Carneiro, 22-Sep-2014.) (Revised by Mario Carneiro, 21-Jun-2015.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℕ) → (𝐴 σ 𝐵) = Σ𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐵} (𝑘↑𝑐𝐴))
Theorem	sgmval2 16175*	The value of the divisor function. (Contributed by Mario Carneiro, 21-Jun-2015.)
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → (𝐴 σ 𝐵) = Σ𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐵} (𝑘↑𝐴))
Theorem	0sgm 16176*	The value of the sum-of-divisors function, usually denoted σ<SUB>0</SUB>(<i>n</i>). (Contributed by Mario Carneiro, 21-Jun-2015.)
⊢ (𝐴 ∈ ℕ → (0 σ 𝐴) = (♯‘{𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴}))
Theorem	sgmf 16177	The divisor function is a function into the complex numbers. (Contributed by Mario Carneiro, 22-Sep-2014.) (Revised by Mario Carneiro, 21-Jun-2015.)
⊢ σ :(ℂ × ℕ)⟶ℂ
Theorem	sgmcl 16178	Closure of the divisor function. (Contributed by Mario Carneiro, 22-Sep-2014.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℕ) → (𝐴 σ 𝐵) ∈ ℂ)
Theorem	sgmnncl 16179	Closure of the divisor function. (Contributed by Mario Carneiro, 21-Jun-2015.)
⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ) → (𝐴 σ 𝐵) ∈ ℕ)
Theorem	chtqfl 16180	The Chebyshev function does not change off the integers. (Contributed by Mario Carneiro, 22-Sep-2014.)
⊢ (𝐴 ∈ ℚ → (θ‘(⌊‘𝐴)) = (θ‘𝐴))
Theorem	ppiprm 16181	The prime-counting function π at a prime. (Contributed by Mario Carneiro, 19-Sep-2014.)
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → (π‘(𝐴 + 1)) = ((π‘𝐴) + 1))
Theorem	ppinprm 16182	The prime-counting function π at a non-prime. (Contributed by Mario Carneiro, 19-Sep-2014.)
⊢ ((𝐴 ∈ ℤ ∧ ¬ (𝐴 + 1) ∈ ℙ) → (π‘(𝐴 + 1)) = (π‘𝐴))
Theorem	chtprm 16183	The Chebyshev function at a prime. (Contributed by Mario Carneiro, 22-Sep-2014.)
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → (θ‘(𝐴 + 1)) = ((θ‘𝐴) + (log‘(𝐴 + 1))))
Theorem	chtnprm 16184	The Chebyshev function at a non-prime. (Contributed by Mario Carneiro, 19-Sep-2014.)
⊢ ((𝐴 ∈ ℤ ∧ ¬ (𝐴 + 1) ∈ ℙ) → (θ‘(𝐴 + 1)) = (θ‘𝐴))
Theorem	chtqwordi 16185	The Chebyshev function is weakly increasing. (Contributed by Mario Carneiro, 22-Sep-2014.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 𝐴 ≤ 𝐵) → (θ‘𝐴) ≤ (θ‘𝐵))
Theorem	chtdif 16186*	The difference of the Chebyshev function at two points sums the logarithms of the primes in an interval. (Contributed by Mario Carneiro, 22-Sep-2014.)
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → ((θ‘𝑁) − (θ‘𝑀)) = Σ𝑝 ∈ (((𝑀 + 1)...𝑁) ∩ ℙ)(log‘𝑝))
Theorem	efchtqdvds 16187	The exponentiated Chebyshev function forms a divisibility chain between any two points. (Contributed by Mario Carneiro, 22-Sep-2014.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 𝐴 ≤ 𝐵) → (exp‘(θ‘𝐴)) ∥ (exp‘(θ‘𝐵)))
Theorem	ppiqfl 16188	The prime-counting function π does not change off the integers. (Contributed by Mario Carneiro, 18-Sep-2014.)
⊢ (𝐴 ∈ ℚ → (π‘(⌊‘𝐴)) = (π‘𝐴))
Theorem	ppiqp1le 16189	The prime-counting function π cannot locally increase faster than the identity function. (Contributed by Mario Carneiro, 21-Sep-2014.)
⊢ (𝐴 ∈ ℚ → (π‘(𝐴 + 1)) ≤ ((π‘𝐴) + 1))
Theorem	ppiqwordi 16190	The prime-counting function π is weakly increasing. (Contributed by Mario Carneiro, 19-Sep-2014.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 𝐴 ≤ 𝐵) → (π‘𝐴) ≤ (π‘𝐵))
Theorem	ppidif 16191	The difference of the prime-counting function π at two points counts the number of primes in an interval. (Contributed by Mario Carneiro, 21-Sep-2014.)
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → ((π‘𝑁) − (π‘𝑀)) = (♯‘(((𝑀 + 1)...𝑁) ∩ ℙ)))
Theorem	ppi1 16192	The prime-counting function π at 1. (Contributed by Mario Carneiro, 21-Sep-2014.)
⊢ (π‘1) = 0
Theorem	cht1 16193	The Chebyshev function at 1. (Contributed by Mario Carneiro, 22-Sep-2014.)
⊢ (θ‘1) = 0
Theorem	ppi1i 16194	Inference form of ppiprm 16181. (Contributed by Mario Carneiro, 21-Sep-2014.)
⊢ 𝑀 ∈ ℕ0    &   ⊢ 𝑁 = (𝑀 + 1)    &   ⊢ (π‘𝑀) = 𝐾    &   ⊢ 𝑁 ∈ ℙ    ⇒   ⊢ (π‘𝑁) = (𝐾 + 1)
Theorem	ppi2i 16195	Inference form of ppinprm 16182. (Contributed by Mario Carneiro, 21-Sep-2014.)
⊢ 𝑀 ∈ ℕ0    &   ⊢ 𝑁 = (𝑀 + 1)    &   ⊢ (π‘𝑀) = 𝐾    &   ⊢ ¬ 𝑁 ∈ ℙ    ⇒   ⊢ (π‘𝑁) = 𝐾
Theorem	ppi2 16196	The prime-counting function π at 2. (Contributed by Mario Carneiro, 21-Sep-2014.)
⊢ (π‘2) = 1
Theorem	ppi3 16197	The prime-counting function π at 3. (Contributed by Mario Carneiro, 21-Sep-2014.)
⊢ (π‘3) = 2
Theorem	cht2 16198	The Chebyshev function at 2. (Contributed by Mario Carneiro, 22-Sep-2014.)
⊢ (θ‘2) = (log‘2)
Theorem	cht3 16199	The Chebyshev function at 3. (Contributed by Mario Carneiro, 22-Sep-2014.)
⊢ (θ‘3) = (log‘6)
Theorem	ppiqnncl 16200	Closure of the prime-counting function π in the positive integers. (Contributed by Mario Carneiro, 21-Sep-2014.)
⊢ ((𝐴 ∈ ℚ ∧ 2 ≤ 𝐴) → (π‘𝐴) ∈ ℕ)