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Theorem List for Metamath Proof Explorer - 25701-25800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremppifi 25701 The set of primes less than 𝐴 is a finite set. (Contributed by Mario Carneiro, 15-Sep-2014.)
(𝐴 ∈ ℝ → ((0[,]𝐴) ∩ ℙ) ∈ Fin)

Theoremprmdvdsfi 25702* The set of prime divisors of a number is a finite set. (Contributed by Mario Carneiro, 7-Apr-2016.)
(𝐴 ∈ ℕ → {𝑝 ∈ ℙ ∣ 𝑝𝐴} ∈ Fin)

Theoremchtf 25703 Domain and range of the Chebyshev function. (Contributed by Mario Carneiro, 15-Sep-2014.)
θ:ℝ⟶ℝ

Theoremchtcl 25704 Real closure of the Chebyshev function. (Contributed by Mario Carneiro, 15-Sep-2014.)
(𝐴 ∈ ℝ → (θ‘𝐴) ∈ ℝ)

Theoremchtval 25705* Value of the Chebyshev function. (Contributed by Mario Carneiro, 15-Sep-2014.)
(𝐴 ∈ ℝ → (θ‘𝐴) = Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)(log‘𝑝))

Theoremefchtcl 25706 The Chebyshev function is closed in the log-integers. (Contributed by Mario Carneiro, 22-Sep-2014.) (Revised by Mario Carneiro, 7-Apr-2016.)
(𝐴 ∈ ℝ → (exp‘(θ‘𝐴)) ∈ ℕ)

Theoremchtge0 25707 The Chebyshev function is always positive. (Contributed by Mario Carneiro, 15-Sep-2014.)
(𝐴 ∈ ℝ → 0 ≤ (θ‘𝐴))

Theoremvmaval 25708* Value of the von Mangoldt function. (Contributed by Mario Carneiro, 7-Apr-2016.)
𝑆 = {𝑝 ∈ ℙ ∣ 𝑝𝐴}       (𝐴 ∈ ℕ → (Λ‘𝐴) = if((♯‘𝑆) = 1, (log‘ 𝑆), 0))

Theoremisppw 25709* Two ways to say that 𝐴 is a prime power. (Contributed by Mario Carneiro, 7-Apr-2016.)
(𝐴 ∈ ℕ → ((Λ‘𝐴) ≠ 0 ↔ ∃!𝑝 ∈ ℙ 𝑝𝐴))

Theoremisppw2 25710* Two ways to say that 𝐴 is a prime power. (Contributed by Mario Carneiro, 7-Apr-2016.)
(𝐴 ∈ ℕ → ((Λ‘𝐴) ≠ 0 ↔ ∃𝑝 ∈ ℙ ∃𝑘 ∈ ℕ 𝐴 = (𝑝𝑘)))

Theoremvmappw 25711 Value of the von Mangoldt function at a prime power. (Contributed by Mario Carneiro, 7-Apr-2016.)
((𝑃 ∈ ℙ ∧ 𝐾 ∈ ℕ) → (Λ‘(𝑃𝐾)) = (log‘𝑃))

Theoremvmaprm 25712 Value of the von Mangoldt function at a prime. (Contributed by Mario Carneiro, 7-Apr-2016.)
(𝑃 ∈ ℙ → (Λ‘𝑃) = (log‘𝑃))

Theoremvmacl 25713 Closure for the von Mangoldt function. (Contributed by Mario Carneiro, 7-Apr-2016.)
(𝐴 ∈ ℕ → (Λ‘𝐴) ∈ ℝ)

Theoremvmaf 25714 Functionality of the von Mangoldt function. (Contributed by Mario Carneiro, 7-Apr-2016.)
Λ:ℕ⟶ℝ

Theoremefvmacl 25715 The von Mangoldt is closed in the log-integers. (Contributed by Mario Carneiro, 7-Apr-2016.)
(𝐴 ∈ ℕ → (exp‘(Λ‘𝐴)) ∈ ℕ)

Theoremvmage0 25716 The von Mangoldt function is nonnegative. (Contributed by Mario Carneiro, 7-Apr-2016.)
(𝐴 ∈ ℕ → 0 ≤ (Λ‘𝐴))

Theoremchpval 25717* Value of the second Chebyshev function, or summary von Mangoldt function. (Contributed by Mario Carneiro, 7-Apr-2016.)
(𝐴 ∈ ℝ → (ψ‘𝐴) = Σ𝑛 ∈ (1...(⌊‘𝐴))(Λ‘𝑛))

Theoremchpf 25718 Functionality of the second Chebyshev function. (Contributed by Mario Carneiro, 7-Apr-2016.)
ψ:ℝ⟶ℝ

Theoremchpcl 25719 Closure for the second Chebyshev function. (Contributed by Mario Carneiro, 7-Apr-2016.)
(𝐴 ∈ ℝ → (ψ‘𝐴) ∈ ℝ)

Theoremefchpcl 25720 The second Chebyshev function is closed in the log-integers. (Contributed by Mario Carneiro, 7-Apr-2016.)
(𝐴 ∈ ℝ → (exp‘(ψ‘𝐴)) ∈ ℕ)

Theoremchpge0 25721 The second Chebyshev function is nonnegative. (Contributed by Mario Carneiro, 7-Apr-2016.)
(𝐴 ∈ ℝ → 0 ≤ (ψ‘𝐴))

Theoremppival 25722 Value of the prime-counting function pi. (Contributed by Mario Carneiro, 15-Sep-2014.)
(𝐴 ∈ ℝ → (π𝐴) = (♯‘((0[,]𝐴) ∩ ℙ)))

Theoremppival2 25723 Value of the prime-counting function pi. (Contributed by Mario Carneiro, 18-Sep-2014.)
(𝐴 ∈ ℤ → (π𝐴) = (♯‘((2...𝐴) ∩ ℙ)))

Theoremppival2g 25724 Value of the prime-counting function pi. (Contributed by Mario Carneiro, 22-Sep-2014.)
((𝐴 ∈ ℤ ∧ 2 ∈ (ℤ𝑀)) → (π𝐴) = (♯‘((𝑀...𝐴) ∩ ℙ)))

Theoremppif 25725 Domain and range of the prime-counting function pi. (Contributed by Mario Carneiro, 15-Sep-2014.)
π:ℝ⟶ℕ0

Theoremppicl 25726 Real closure of the prime-counting function pi. (Contributed by Mario Carneiro, 15-Sep-2014.)
(𝐴 ∈ ℝ → (π𝐴) ∈ ℕ0)

Theoremmuval 25727* The value of the Möbius function. (Contributed by Mario Carneiro, 22-Sep-2014.)
(𝐴 ∈ ℕ → (μ‘𝐴) = if(∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝐴, 0, (-1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝𝐴}))))

Theoremmuval1 25728 The value of the Möbius function at a non-squarefree number. (Contributed by Mario Carneiro, 21-Sep-2014.)
((𝐴 ∈ ℕ ∧ 𝑃 ∈ (ℤ‘2) ∧ (𝑃↑2) ∥ 𝐴) → (μ‘𝐴) = 0)

Theoremmuval2 25729* The value of the Möbius function at a squarefree number. (Contributed by Mario Carneiro, 3-Oct-2014.)
((𝐴 ∈ ℕ ∧ (μ‘𝐴) ≠ 0) → (μ‘𝐴) = (-1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝𝐴})))

Theoremisnsqf 25730* Two ways to say that a number is not squarefree. (Contributed by Mario Carneiro, 3-Oct-2014.)
(𝐴 ∈ ℕ → ((μ‘𝐴) = 0 ↔ ∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝐴))

Theoremissqf 25731* Two ways to say that a number is squarefree. (Contributed by Mario Carneiro, 3-Oct-2014.)
(𝐴 ∈ ℕ → ((μ‘𝐴) ≠ 0 ↔ ∀𝑝 ∈ ℙ (𝑝 pCnt 𝐴) ≤ 1))

Theoremsqfpc 25732 The prime count of a squarefree number is at most 1. (Contributed by Mario Carneiro, 1-Jul-2015.)
((𝐴 ∈ ℕ ∧ (μ‘𝐴) ≠ 0 ∧ 𝑃 ∈ ℙ) → (𝑃 pCnt 𝐴) ≤ 1)

Theoremdvdssqf 25733 A divisor of a squarefree number is squarefree. (Contributed by Mario Carneiro, 1-Jul-2015.)
((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐵𝐴) → ((μ‘𝐴) ≠ 0 → (μ‘𝐵) ≠ 0))

Theoremsqf11 25734* A squarefree number is completely determined by the set of its prime divisors. (Contributed by Mario Carneiro, 1-Jul-2015.)
(((𝐴 ∈ ℕ ∧ (μ‘𝐴) ≠ 0) ∧ (𝐵 ∈ ℕ ∧ (μ‘𝐵) ≠ 0)) → (𝐴 = 𝐵 ↔ ∀𝑝 ∈ ℙ (𝑝𝐴𝑝𝐵)))

Theoremmuf 25735 The Möbius function is a function into the integers. (Contributed by Mario Carneiro, 22-Sep-2014.)
μ:ℕ⟶ℤ

Theoremmucl 25736 Closure of the Möbius function. (Contributed by Mario Carneiro, 22-Sep-2014.)
(𝐴 ∈ ℕ → (μ‘𝐴) ∈ ℤ)

Theoremsgmval 25737* 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 ∣ 𝑝𝐵} (𝑘𝑐𝐴))

Theoremsgmval2 25738* The value of the divisor function. (Contributed by Mario Carneiro, 21-Jun-2015.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → (𝐴 σ 𝐵) = Σ𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝𝐵} (𝑘𝐴))

Theorem0sgm 25739* 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 ∣ 𝑝𝐴}))

Theoremsgmf 25740 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

Theoremsgmcl 25741 Closure of the divisor function. (Contributed by Mario Carneiro, 22-Sep-2014.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℕ) → (𝐴 σ 𝐵) ∈ ℂ)

Theoremsgmnncl 25742 Closure of the divisor function. (Contributed by Mario Carneiro, 21-Jun-2015.)
((𝐴 ∈ ℕ0𝐵 ∈ ℕ) → (𝐴 σ 𝐵) ∈ ℕ)

Theoremmule1 25743 The Möbius function takes on values in magnitude at most 1. (Together with mucl 25736, this implies that it takes a value in {-1, 0, 1} for every positive integer.) (Contributed by Mario Carneiro, 22-Sep-2014.)
(𝐴 ∈ ℕ → (abs‘(μ‘𝐴)) ≤ 1)

Theoremchtfl 25744 The Chebyshev function does not change off the integers. (Contributed by Mario Carneiro, 22-Sep-2014.)
(𝐴 ∈ ℝ → (θ‘(⌊‘𝐴)) = (θ‘𝐴))

Theoremchpfl 25745 The second Chebyshev function does not change off the integers. (Contributed by Mario Carneiro, 9-Apr-2016.)
(𝐴 ∈ ℝ → (ψ‘(⌊‘𝐴)) = (ψ‘𝐴))

Theoremppiprm 25746 The prime-counting function π at a prime. (Contributed by Mario Carneiro, 19-Sep-2014.)
((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → (π‘(𝐴 + 1)) = ((π𝐴) + 1))

Theoremppinprm 25747 The prime-counting function π at a non-prime. (Contributed by Mario Carneiro, 19-Sep-2014.)
((𝐴 ∈ ℤ ∧ ¬ (𝐴 + 1) ∈ ℙ) → (π‘(𝐴 + 1)) = (π𝐴))

Theoremchtprm 25748 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))))

Theoremchtnprm 25749 The Chebyshev function at a non-prime. (Contributed by Mario Carneiro, 19-Sep-2014.)
((𝐴 ∈ ℤ ∧ ¬ (𝐴 + 1) ∈ ℙ) → (θ‘(𝐴 + 1)) = (θ‘𝐴))

Theoremchpp1 25750 The second Chebyshev function at a successor. (Contributed by Mario Carneiro, 11-Apr-2016.)
(𝐴 ∈ ℕ0 → (ψ‘(𝐴 + 1)) = ((ψ‘𝐴) + (Λ‘(𝐴 + 1))))

Theoremchtwordi 25751 The Chebyshev function is weakly increasing. (Contributed by Mario Carneiro, 22-Sep-2014.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴𝐵) → (θ‘𝐴) ≤ (θ‘𝐵))

Theoremchpwordi 25752 The second Chebyshev function is weakly increasing. (Contributed by Mario Carneiro, 9-Apr-2016.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴𝐵) → (ψ‘𝐴) ≤ (ψ‘𝐵))

Theoremchtdif 25753* 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 ∈ (ℤ𝑀) → ((θ‘𝑁) − (θ‘𝑀)) = Σ𝑝 ∈ (((𝑀 + 1)...𝑁) ∩ ℙ)(log‘𝑝))

Theoremefchtdvds 25754 The exponentiated Chebyshev function forms a divisibility chain between any two points. (Contributed by Mario Carneiro, 22-Sep-2014.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴𝐵) → (exp‘(θ‘𝐴)) ∥ (exp‘(θ‘𝐵)))

Theoremppifl 25755 The prime-counting function π does not change off the integers. (Contributed by Mario Carneiro, 18-Sep-2014.)
(𝐴 ∈ ℝ → (π‘(⌊‘𝐴)) = (π𝐴))

Theoremppip1le 25756 The prime-counting function π cannot locally increase faster than the identity function. (Contributed by Mario Carneiro, 21-Sep-2014.)
(𝐴 ∈ ℝ → (π‘(𝐴 + 1)) ≤ ((π𝐴) + 1))

Theoremppiwordi 25757 The prime-counting function π is weakly increasing. (Contributed by Mario Carneiro, 19-Sep-2014.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴𝐵) → (π𝐴) ≤ (π𝐵))

Theoremppidif 25758 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 ∈ (ℤ𝑀) → ((π𝑁) − (π𝑀)) = (♯‘(((𝑀 + 1)...𝑁) ∩ ℙ)))

Theoremppi1 25759 The prime-counting function π at 1. (Contributed by Mario Carneiro, 21-Sep-2014.)
(π‘1) = 0

Theoremcht1 25760 The Chebyshev function at 1. (Contributed by Mario Carneiro, 22-Sep-2014.)
(θ‘1) = 0

Theoremvma1 25761 The von Mangoldt function at 1. (Contributed by Mario Carneiro, 9-Apr-2016.)
(Λ‘1) = 0

Theoremchp1 25762 The second Chebyshev function at 1. (Contributed by Mario Carneiro, 9-Apr-2016.)
(ψ‘1) = 0

Theoremppi1i 25763 Inference form of ppiprm 25746. (Contributed by Mario Carneiro, 21-Sep-2014.)
𝑀 ∈ ℕ0    &   𝑁 = (𝑀 + 1)    &   (π𝑀) = 𝐾    &   𝑁 ∈ ℙ       (π𝑁) = (𝐾 + 1)

Theoremppi2i 25764 Inference form of ppinprm 25747. (Contributed by Mario Carneiro, 21-Sep-2014.)
𝑀 ∈ ℕ0    &   𝑁 = (𝑀 + 1)    &   (π𝑀) = 𝐾    &    ¬ 𝑁 ∈ ℙ       (π𝑁) = 𝐾

Theoremppi2 25765 The prime-counting function π at 2. (Contributed by Mario Carneiro, 21-Sep-2014.)
(π‘2) = 1

Theoremppi3 25766 The prime-counting function π at 3. (Contributed by Mario Carneiro, 21-Sep-2014.)
(π‘3) = 2

Theoremcht2 25767 The Chebyshev function at 2. (Contributed by Mario Carneiro, 22-Sep-2014.)
(θ‘2) = (log‘2)

Theoremcht3 25768 The Chebyshev function at 3. (Contributed by Mario Carneiro, 22-Sep-2014.)
(θ‘3) = (log‘6)

Theoremppinncl 25769 Closure of the prime-counting function π in the positive integers. (Contributed by Mario Carneiro, 21-Sep-2014.)
((𝐴 ∈ ℝ ∧ 2 ≤ 𝐴) → (π𝐴) ∈ ℕ)

Theoremchtrpcl 25770 Closure of the Chebyshev function in the positive reals. (Contributed by Mario Carneiro, 22-Sep-2014.)
((𝐴 ∈ ℝ ∧ 2 ≤ 𝐴) → (θ‘𝐴) ∈ ℝ+)

Theoremppieq0 25771 The prime-counting function π is zero iff its argument is less than 2. (Contributed by Mario Carneiro, 22-Sep-2014.)
(𝐴 ∈ ℝ → ((π𝐴) = 0 ↔ 𝐴 < 2))

Theoremppiltx 25772 The prime-counting function π is strictly less than the identity. (Contributed by Mario Carneiro, 22-Sep-2014.)
(𝐴 ∈ ℝ+ → (π𝐴) < 𝐴)

Theoremprmorcht 25773 Relate the primorial (product of the first 𝑛 primes) to the Chebyshev function. (Contributed by Mario Carneiro, 22-Sep-2014.)
𝐹 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, 𝑛, 1))       (𝐴 ∈ ℕ → (exp‘(θ‘𝐴)) = (seq1( · , 𝐹)‘𝐴))

Theoremmumullem1 25774 Lemma for mumul 25776. A multiple of a non-squarefree number is non-squarefree. (Contributed by Mario Carneiro, 3-Oct-2014.)
(((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (μ‘𝐴) = 0) → (μ‘(𝐴 · 𝐵)) = 0)

Theoremmumullem2 25775 Lemma for mumul 25776. The product of two coprime squarefree numbers is squarefree. (Contributed by Mario Carneiro, 3-Oct-2014.)
(((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ ((μ‘𝐴) ≠ 0 ∧ (μ‘𝐵) ≠ 0)) → (μ‘(𝐴 · 𝐵)) ≠ 0)

Theoremmumul 25776 The Möbius function is a multiplicative function. This is one of the primary interests of the Möbius function as an arithmetic function. (Contributed by Mario Carneiro, 3-Oct-2014.)
((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (μ‘(𝐴 · 𝐵)) = ((μ‘𝐴) · (μ‘𝐵)))

Theoremsqff1o 25777* There is a bijection from the squarefree divisors of a number 𝑁 to the powerset of the prime divisors of 𝑁. Among other things, this implies that a number has 2↑𝑘 squarefree divisors where 𝑘 is the number of prime divisors, and a squarefree number has 2↑𝑘 divisors (because all divisors of a squarefree number are squarefree). The inverse function to 𝐹 takes the product of all the primes in some subset of prime divisors of 𝑁. (Contributed by Mario Carneiro, 1-Jul-2015.)
𝑆 = {𝑥 ∈ ℕ ∣ ((μ‘𝑥) ≠ 0 ∧ 𝑥𝑁)}    &   𝐹 = (𝑛𝑆 ↦ {𝑝 ∈ ℙ ∣ 𝑝𝑛})    &   𝐺 = (𝑛 ∈ ℕ ↦ (𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑛)))       (𝑁 ∈ ℕ → 𝐹:𝑆1-1-onto→𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁})

Theoremfsumdvdsdiaglem 25778* A "diagonal commutation" of divisor sums analogous to fsum0diag 15127. (Contributed by Mario Carneiro, 2-Jul-2015.) (Revised by Mario Carneiro, 8-Apr-2016.)
(𝜑𝑁 ∈ ℕ)       (𝜑 → ((𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑁} ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑁 / 𝑗)}) → (𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑁} ∧ 𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑁 / 𝑘)})))

Theoremfsumdvdsdiag 25779* A "diagonal commutation" of divisor sums analogous to fsum0diag 15127. (Contributed by Mario Carneiro, 2-Jul-2015.) (Revised by Mario Carneiro, 8-Apr-2016.)
(𝜑𝑁 ∈ ℕ)    &   ((𝜑 ∧ (𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑁} ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑁 / 𝑗)})) → 𝐴 ∈ ℂ)       (𝜑 → Σ𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑁𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑁 / 𝑗)}𝐴 = Σ𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑁𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑁 / 𝑘)}𝐴)

Theoremfsumdvdscom 25780* A double commutation of divisor sums based on fsumdvdsdiag 25779. Note that 𝐴 depends on both 𝑗 and 𝑘. (Contributed by Mario Carneiro, 13-May-2016.)
(𝜑𝑁 ∈ ℕ)    &   (𝑗 = (𝑘 · 𝑚) → 𝐴 = 𝐵)    &   ((𝜑 ∧ (𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑁} ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑗})) → 𝐴 ∈ ℂ)       (𝜑 → Σ𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑁𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑗}𝐴 = Σ𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑁𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑁 / 𝑘)}𝐵)

Theoremdvdsppwf1o 25781* A bijection from the divisors of a prime power to the integers less than the prime count. (Contributed by Mario Carneiro, 5-May-2016.)
𝐹 = (𝑛 ∈ (0...𝐴) ↦ (𝑃𝑛))       ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ0) → 𝐹:(0...𝐴)–1-1-onto→{𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑃𝐴)})

Theoremdvdsflf1o 25782* A bijection from the numbers less than 𝑁 / 𝐴 to the multiples of 𝐴 less than 𝑁. Useful for some sum manipulations. (Contributed by Mario Carneiro, 3-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝑁 ∈ ℕ)    &   𝐹 = (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑁))) ↦ (𝑁 · 𝑛))       (𝜑𝐹:(1...(⌊‘(𝐴 / 𝑁)))–1-1-onto→{𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑁𝑥})

Theoremdvdsflsumcom 25783* A sum commutation from Σ𝑛𝐴, Σ𝑑𝑛, 𝐵(𝑛, 𝑑) to Σ𝑑𝐴, Σ𝑚𝐴 / 𝑑, 𝐵(𝑛, 𝑑𝑚). (Contributed by Mario Carneiro, 4-May-2016.)
(𝑛 = (𝑑 · 𝑚) → 𝐵 = 𝐶)    &   (𝜑𝐴 ∈ ℝ)    &   ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑛})) → 𝐵 ∈ ℂ)       (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝐴))Σ𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑛}𝐵 = Σ𝑑 ∈ (1...(⌊‘𝐴))Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))𝐶)

Theoremfsumfldivdiaglem 25784* Lemma for fsumfldivdiag 25785. (Contributed by Mario Carneiro, 10-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑 → ((𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛)))) → (𝑚 ∈ (1...(⌊‘𝐴)) ∧ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑚))))))

Theoremfsumfldivdiag 25785* The right-hand side of dvdsflsumcom 25783 is commutative in the variables, because it can be written as the manifestly symmetric sum over those 𝑚, 𝑛 such that 𝑚 · 𝑛𝐴. (Contributed by Mario Carneiro, 4-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → 𝐵 ∈ ℂ)       (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝐴))Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛)))𝐵 = Σ𝑚 ∈ (1...(⌊‘𝐴))Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑚)))𝐵)

Theoremmusum 25786* The sum of the Möbius function over the divisors of 𝑁 gives one if 𝑁 = 1, but otherwise always sums to zero. Theorem 2.1 in [ApostolNT] p. 25. This makes the Möbius function useful for inverting divisor sums; see also muinv 25788. (Contributed by Mario Carneiro, 2-Jul-2015.)
(𝑁 ∈ ℕ → Σ𝑘 ∈ {𝑛 ∈ ℕ ∣ 𝑛𝑁} (μ‘𝑘) = if(𝑁 = 1, 1, 0))

Theoremmusumsum 25787* Evaluate a collapsing sum over the Möbius function. (Contributed by Mario Carneiro, 4-May-2016.)
(𝑚 = 1 → 𝐵 = 𝐶)    &   (𝜑𝐴 ∈ Fin)    &   (𝜑𝐴 ⊆ ℕ)    &   (𝜑 → 1 ∈ 𝐴)    &   ((𝜑𝑚𝐴) → 𝐵 ∈ ℂ)       (𝜑 → Σ𝑚𝐴 Σ𝑘 ∈ {𝑛 ∈ ℕ ∣ 𝑛𝑚} ((μ‘𝑘) · 𝐵) = 𝐶)

Theoremmuinv 25788* The Möbius inversion formula. If 𝐺(𝑛) = Σ𝑘𝑛𝐹(𝑘) for every 𝑛 ∈ ℕ, then 𝐹(𝑛) = Σ𝑘𝑛 μ(𝑘)𝐺(𝑛 / 𝑘) = Σ𝑘𝑛μ(𝑛 / 𝑘)𝐺(𝑘), i.e. the Möbius function is the Dirichlet convolution inverse of the constant function 1. Theorem 2.9 in [ApostolNT] p. 32. (Contributed by Mario Carneiro, 2-Jul-2015.)
(𝜑𝐹:ℕ⟶ℂ)    &   (𝜑𝐺 = (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑛} (𝐹𝑘)))       (𝜑𝐹 = (𝑚 ∈ ℕ ↦ Σ𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑚} ((μ‘𝑗) · (𝐺‘(𝑚 / 𝑗)))))

Theoremdvdsmulf1o 25789* If 𝑀 and 𝑁 are two coprime integers, multiplication forms a bijection from the set of pairs 𝑗, 𝑘 where 𝑗𝑀 and 𝑘𝑁, to the set of divisors of 𝑀 · 𝑁. (Contributed by Mario Carneiro, 2-Jul-2015.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑 → (𝑀 gcd 𝑁) = 1)    &   𝑋 = {𝑥 ∈ ℕ ∣ 𝑥𝑀}    &   𝑌 = {𝑥 ∈ ℕ ∣ 𝑥𝑁}    &   𝑍 = {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑀 · 𝑁)}       (𝜑 → ( · ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)–1-1-onto𝑍)

Theoremfsumdvdsmul 25790* Product of two divisor sums. (This is also the main part of the proof that "Σ𝑘𝑁𝐹(𝑘) is a multiplicative function if 𝐹 is".) (Contributed by Mario Carneiro, 2-Jul-2015.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑 → (𝑀 gcd 𝑁) = 1)    &   𝑋 = {𝑥 ∈ ℕ ∣ 𝑥𝑀}    &   𝑌 = {𝑥 ∈ ℕ ∣ 𝑥𝑁}    &   𝑍 = {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑀 · 𝑁)}    &   ((𝜑𝑗𝑋) → 𝐴 ∈ ℂ)    &   ((𝜑𝑘𝑌) → 𝐵 ∈ ℂ)    &   ((𝜑 ∧ (𝑗𝑋𝑘𝑌)) → (𝐴 · 𝐵) = 𝐷)    &   (𝑖 = (𝑗 · 𝑘) → 𝐶 = 𝐷)       (𝜑 → (Σ𝑗𝑋 𝐴 · Σ𝑘𝑌 𝐵) = Σ𝑖𝑍 𝐶)

Theoremsgmppw 25791* The value of the divisor function at a prime power. (Contributed by Mario Carneiro, 17-May-2016.)
((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ0) → (𝐴 σ (𝑃𝑁)) = Σ𝑘 ∈ (0...𝑁)((𝑃𝑐𝐴)↑𝑘))

Theorem0sgmppw 25792 A prime power 𝑃𝐾 has 𝐾 + 1 divisors. (Contributed by Mario Carneiro, 17-May-2016.)
((𝑃 ∈ ℙ ∧ 𝐾 ∈ ℕ0) → (0 σ (𝑃𝐾)) = (𝐾 + 1))

Theorem1sgmprm 25793 The sum of divisors for a prime is 𝑃 + 1 because the only divisors are 1 and 𝑃. (Contributed by Mario Carneiro, 17-May-2016.)
(𝑃 ∈ ℙ → (1 σ 𝑃) = (𝑃 + 1))

Theorem1sgm2ppw 25794 The sum of the divisors of 2↑(𝑁 − 1). (Contributed by Mario Carneiro, 17-May-2016.)
(𝑁 ∈ ℕ → (1 σ (2↑(𝑁 − 1))) = ((2↑𝑁) − 1))

Theoremsgmmul 25795 The divisor function for fixed parameter 𝐴 is a multiplicative function. (Contributed by Mario Carneiro, 2-Jul-2015.)
((𝐴 ∈ ℂ ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1)) → (𝐴 σ (𝑀 · 𝑁)) = ((𝐴 σ 𝑀) · (𝐴 σ 𝑁)))

Theoremppiublem1 25796 Lemma for ppiub 25798. (Contributed by Mario Carneiro, 12-Mar-2014.)
(𝑁 ≤ 6 ∧ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → ((𝑃 mod 6) ∈ (𝑁...5) → (𝑃 mod 6) ∈ {1, 5})))    &   𝑀 ∈ ℕ0    &   𝑁 = (𝑀 + 1)    &   (2 ∥ 𝑀 ∨ 3 ∥ 𝑀𝑀 ∈ {1, 5})       (𝑀 ≤ 6 ∧ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → ((𝑃 mod 6) ∈ (𝑀...5) → (𝑃 mod 6) ∈ {1, 5})))

Theoremppiublem2 25797 A prime greater than 3 does not divide 2 or 3, so its residue mod 6 is 1 or 5. (Contributed by Mario Carneiro, 12-Mar-2014.)
((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → (𝑃 mod 6) ∈ {1, 5})

Theoremppiub 25798 An upper bound on the prime-counting function π, which counts the number of primes less than 𝑁. (Contributed by Mario Carneiro, 13-Mar-2014.)
((𝑁 ∈ ℝ ∧ 0 ≤ 𝑁) → (π𝑁) ≤ ((𝑁 / 3) + 2))

Theoremvmalelog 25799 The von Mangoldt function is less than the natural log. (Contributed by Mario Carneiro, 7-Apr-2016.)
(𝐴 ∈ ℕ → (Λ‘𝐴) ≤ (log‘𝐴))

Theoremchtlepsi 25800 The first Chebyshev function is less than the second. (Contributed by Mario Carneiro, 7-Apr-2016.)
(𝐴 ∈ ℝ → (θ‘𝐴) ≤ (ψ‘𝐴))

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