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Theorem List for Metamath Proof Explorer - 24701-24800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremchtrpcl 24701 Closure of the Chebyshev function in the positive reals. (Contributed by Mario Carneiro, 22-Sep-2014.)
((𝐴 ∈ ℝ ∧ 2 ≤ 𝐴) → (θ‘𝐴) ∈ ℝ+)

Theoremppieq0 24702 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 24703 The prime-counting function π is strictly less than the identity. (Contributed by Mario Carneiro, 22-Sep-2014.)
(𝐴 ∈ ℝ+ → (π𝐴) < 𝐴)

Theoremprmorcht 24704 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 24705 Lemma for mumul 24707. 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 24706 Lemma for mumul 24707. 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 24707 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 24708* 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 24709* A "diagonal commutation" of divisor sums analogous to fsum0diag 14351. (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 24710* A "diagonal commutation" of divisor sums analogous to fsum0diag 14351. (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 24711* A double commutation of divisor sums based on fsumdvdsdiag 24710. 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 24712* 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 24713* 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 24714* 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 24715* Lemma for fsumfldivdiag 24716. (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 24716* The right-hand side of dvdsflsumcom 24714 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 24717* 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 24719. (Contributed by Mario Carneiro, 2-Jul-2015.)
(𝑁 ∈ ℕ → Σ𝑘 ∈ {𝑛 ∈ ℕ ∣ 𝑛𝑁} (μ‘𝑘) = if(𝑁 = 1, 1, 0))

Theoremmusumsum 24718* 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 24719* 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 24720* 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 24721* 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 24722* 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 24723 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 24724 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 24725 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 24726 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 24727 Lemma for ppiub 24729. (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 24728 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 24729 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 24730 The von Mangoldt function is less than the natural log. (Contributed by Mario Carneiro, 7-Apr-2016.)
(𝐴 ∈ ℕ → (Λ‘𝐴) ≤ (log‘𝐴))

Theoremchtlepsi 24731 The first Chebyshev function is less than the second. (Contributed by Mario Carneiro, 7-Apr-2016.)
(𝐴 ∈ ℝ → (θ‘𝐴) ≤ (ψ‘𝐴))

Theoremchprpcl 24732 Closure of the second Chebyshev function in the positive reals. (Contributed by Mario Carneiro, 8-Apr-2016.)
((𝐴 ∈ ℝ ∧ 2 ≤ 𝐴) → (ψ‘𝐴) ∈ ℝ+)

Theoremchpeq0 24733 The second Chebyshev function is zero iff its argument is less than 2. (Contributed by Mario Carneiro, 9-Apr-2016.)
(𝐴 ∈ ℝ → ((ψ‘𝐴) = 0 ↔ 𝐴 < 2))

Theoremchteq0 24734 The first Chebyshev function is zero iff its argument is less than 2. (Contributed by Mario Carneiro, 9-Apr-2016.)
(𝐴 ∈ ℝ → ((θ‘𝐴) = 0 ↔ 𝐴 < 2))

Theoremchtleppi 24735 Upper bound on the θ function. (Contributed by Mario Carneiro, 22-Sep-2014.)
(𝐴 ∈ ℝ+ → (θ‘𝐴) ≤ ((π𝐴) · (log‘𝐴)))

Theoremchtublem 24736 Lemma for chtub 24737. (Contributed by Mario Carneiro, 13-Mar-2014.)
(𝑁 ∈ ℕ → (θ‘((2 · 𝑁) − 1)) ≤ ((θ‘𝑁) + ((log‘4) · (𝑁 − 1))))

Theoremchtub 24737 An upper bound on the Chebyshev function. (Contributed by Mario Carneiro, 13-Mar-2014.) (Revised 22-Sep-2014.)
((𝑁 ∈ ℝ ∧ 2 < 𝑁) → (θ‘𝑁) < ((log‘2) · ((2 · 𝑁) − 3)))

Theoremfsumvma 24738* Rewrite a sum over the von Mangoldt function as a sum over prime powers. (Contributed by Mario Carneiro, 15-Apr-2016.)
(𝑥 = (𝑝𝑘) → 𝐵 = 𝐶)    &   (𝜑𝐴 ∈ Fin)    &   (𝜑𝐴 ⊆ ℕ)    &   (𝜑𝑃 ∈ Fin)    &   (𝜑 → ((𝑝𝑃𝑘𝐾) ↔ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ (𝑝𝑘) ∈ 𝐴)))    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℂ)    &   ((𝜑 ∧ (𝑥𝐴 ∧ (Λ‘𝑥) = 0)) → 𝐵 = 0)       (𝜑 → Σ𝑥𝐴 𝐵 = Σ𝑝𝑃 Σ𝑘𝐾 𝐶)

Theoremfsumvma2 24739* Apply fsumvma 24738 for the common case of all numbers less than a real number 𝐴. (Contributed by Mario Carneiro, 30-Apr-2016.)
(𝑥 = (𝑝𝑘) → 𝐵 = 𝐶)    &   (𝜑𝐴 ∈ ℝ)    &   ((𝜑𝑥 ∈ (1...(⌊‘𝐴))) → 𝐵 ∈ ℂ)    &   ((𝜑 ∧ (𝑥 ∈ (1...(⌊‘𝐴)) ∧ (Λ‘𝑥) = 0)) → 𝐵 = 0)       (𝜑 → Σ𝑥 ∈ (1...(⌊‘𝐴))𝐵 = Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)Σ𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))𝐶)

Theorempclogsum 24740* The logarithmic analogue of pcprod 15437. The sum of the logarithms of the primes dividing 𝐴 multiplied by their powers yields the logarithm of 𝐴. (Contributed by Mario Carneiro, 15-Apr-2016.)
(𝐴 ∈ ℕ → Σ𝑝 ∈ ((1...𝐴) ∩ ℙ)((𝑝 pCnt 𝐴) · (log‘𝑝)) = (log‘𝐴))

Theoremvmasum 24741* The sum of the von Mangoldt function over the divisors of 𝑛. Equation 9.2.4 of [Shapiro], p. 328 and theorem 2.10 in [ApostolNT] p. 32. (Contributed by Mario Carneiro, 15-Apr-2016.)
(𝐴 ∈ ℕ → Σ𝑛 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝐴} (Λ‘𝑛) = (log‘𝐴))

Theoremlogfac2 24742* Another expression for the logarithm of a factorial, in terms of the von Mangoldt function. Equation 9.2.7 of [Shapiro], p. 329. (Contributed by Mario Carneiro, 15-Apr-2016.) (Revised by Mario Carneiro, 3-May-2016.)
((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (log‘(!‘(⌊‘𝐴))) = Σ𝑘 ∈ (1...(⌊‘𝐴))((Λ‘𝑘) · (⌊‘(𝐴 / 𝑘))))

Theoremchpval2 24743* Express the second Chebyshev function directly as a sum over the primes less than 𝐴 (instead of indirectly through the von Mangoldt function). (Contributed by Mario Carneiro, 8-Apr-2016.)
(𝐴 ∈ ℝ → (ψ‘𝐴) = Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)((log‘𝑝) · (⌊‘((log‘𝐴) / (log‘𝑝)))))

Theoremchpchtsum 24744* The second Chebyshev function is the sum of the theta function at arguments quickly approaching zero. (This is usually stated as an infinite sum, but after a certain point, the terms are all zero, and it is easier for us to use an explicit finite sum.) (Contributed by Mario Carneiro, 7-Apr-2016.)
(𝐴 ∈ ℝ → (ψ‘𝐴) = Σ𝑘 ∈ (1...(⌊‘𝐴))(θ‘(𝐴𝑐(1 / 𝑘))))

Theoremchpub 24745 An upper bound on the second Chebyshev function. (Contributed by Mario Carneiro, 8-Apr-2016.)
((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (ψ‘𝐴) ≤ ((θ‘𝐴) + ((√‘𝐴) · (log‘𝐴))))

Theoremlogfacubnd 24746 A simple upper bound on the logarithm of a factorial. (Contributed by Mario Carneiro, 16-Apr-2016.)
((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (log‘(!‘(⌊‘𝐴))) ≤ (𝐴 · (log‘𝐴)))

Theoremlogfaclbnd 24747 A lower bound on the logarithm of a factorial. (Contributed by Mario Carneiro, 16-Apr-2016.)
(𝐴 ∈ ℝ+ → (𝐴 · ((log‘𝐴) − 2)) ≤ (log‘(!‘(⌊‘𝐴))))

Theoremlogfacbnd3 24748 Show the stronger statement log(𝑥!) = 𝑥log𝑥𝑥 + 𝑂(log𝑥) alluded to in logfacrlim 24749. (Contributed by Mario Carneiro, 20-May-2016.)
((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (abs‘((log‘(!‘(⌊‘𝐴))) − (𝐴 · ((log‘𝐴) − 1)))) ≤ ((log‘𝐴) + 1))

Theoremlogfacrlim 24749 Combine the estimates logfacubnd 24746 and logfaclbnd 24747, to get log(𝑥!) = 𝑥log𝑥 + 𝑂(𝑥). Equation 9.2.9 of [Shapiro], p. 329. This is a weak form of the even stronger statement, log(𝑥!) = 𝑥log𝑥𝑥 + 𝑂(log𝑥). (Contributed by Mario Carneiro, 16-Apr-2016.) (Revised by Mario Carneiro, 21-May-2016.)
(𝑥 ∈ ℝ+ ↦ ((log‘𝑥) − ((log‘(!‘(⌊‘𝑥))) / 𝑥))) ⇝𝑟 1

Theoremlogexprlim 24750* The sum Σ𝑛𝑥, log↑𝑁(𝑥 / 𝑛) has the asymptotic expansion (𝑁!)𝑥 + 𝑜(𝑥). (More precisely, the omitted term has order 𝑂(log↑𝑁(𝑥) / 𝑥).) (Contributed by Mario Carneiro, 22-May-2016.)
(𝑁 ∈ ℕ0 → (𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑𝑁) / 𝑥)) ⇝𝑟 (!‘𝑁))

Theoremlogfacrlim2 24751* Write out logfacrlim 24749 as a sum of logs. (Contributed by Mario Carneiro, 18-May-2016.) (Revised by Mario Carneiro, 22-May-2016.)
(𝑥 ∈ ℝ+ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛)) / 𝑥)) ⇝𝑟 1

14.4.5  Perfect Number Theorem

Theoremmersenne 24752 A Mersenne prime is a prime number of the form 2↑𝑃 − 1. This theorem shows that the 𝑃 in this expression is necessarily also prime. (Contributed by Mario Carneiro, 17-May-2016.)
((𝑃 ∈ ℤ ∧ ((2↑𝑃) − 1) ∈ ℙ) → 𝑃 ∈ ℙ)

Theoremperfect1 24753 Euclid's contribution to the Euclid-Euler theorem. A number of the form 2↑(𝑝 − 1) · (2↑𝑝 − 1) is a perfect number. (Contributed by Mario Carneiro, 17-May-2016.)
((𝑃 ∈ ℤ ∧ ((2↑𝑃) − 1) ∈ ℙ) → (1 σ ((2↑(𝑃 − 1)) · ((2↑𝑃) − 1))) = ((2↑𝑃) · ((2↑𝑃) − 1)))

Theoremperfectlem1 24754 Lemma for perfect 24756. (Contributed by Mario Carneiro, 7-Jun-2016.)
(𝜑𝐴 ∈ ℕ)    &   (𝜑𝐵 ∈ ℕ)    &   (𝜑 → ¬ 2 ∥ 𝐵)    &   (𝜑 → (1 σ ((2↑𝐴) · 𝐵)) = (2 · ((2↑𝐴) · 𝐵)))       (𝜑 → ((2↑(𝐴 + 1)) ∈ ℕ ∧ ((2↑(𝐴 + 1)) − 1) ∈ ℕ ∧ (𝐵 / ((2↑(𝐴 + 1)) − 1)) ∈ ℕ))

Theoremperfectlem2 24755 Lemma for perfect 24756. (Contributed by Mario Carneiro, 17-May-2016.) Replace OLD theorem. (Revised by Wolf Lammen, 17-Sep-2020.)
(𝜑𝐴 ∈ ℕ)    &   (𝜑𝐵 ∈ ℕ)    &   (𝜑 → ¬ 2 ∥ 𝐵)    &   (𝜑 → (1 σ ((2↑𝐴) · 𝐵)) = (2 · ((2↑𝐴) · 𝐵)))       (𝜑 → (𝐵 ∈ ℙ ∧ 𝐵 = ((2↑(𝐴 + 1)) − 1)))

Theoremperfect 24756* The Euclid-Euler theorem, or Perfect Number theorem. A positive even integer 𝑁 is a perfect number (that is, its divisor sum is 2𝑁) if and only if it is of the form 2↑(𝑝 − 1) · (2↑𝑝 − 1), where 2↑𝑝 − 1 is prime (a Mersenne prime). (It follows from this that 𝑝 is also prime.) This is Metamath 100 proof #70. (Contributed by Mario Carneiro, 17-May-2016.)
((𝑁 ∈ ℕ ∧ 2 ∥ 𝑁) → ((1 σ 𝑁) = (2 · 𝑁) ↔ ∃𝑝 ∈ ℤ (((2↑𝑝) − 1) ∈ ℙ ∧ 𝑁 = ((2↑(𝑝 − 1)) · ((2↑𝑝) − 1)))))

14.4.6  Characters of Z/nZ

Syntaxcdchr 24757 Extend class notation with the group of Dirichlet characters.
class DChr

Definitiondf-dchr 24758* The group of Dirichlet characters mod 𝑛 is the set of monoid homomorphisms from ℤ / 𝑛 to the multiplicative monoid of the complex numbers, equipped with the group operation of pointwise multiplication. (Contributed by Mario Carneiro, 18-Apr-2016.)
DChr = (𝑛 ∈ ℕ ↦ (ℤ/nℤ‘𝑛) / 𝑧{𝑥 ∈ ((mulGrp‘𝑧) MndHom (mulGrp‘ℂfld)) ∣ (((Base‘𝑧) ∖ (Unit‘𝑧)) × {0}) ⊆ 𝑥} / 𝑏{⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), ( ∘𝑓 · ↾ (𝑏 × 𝑏))⟩})

Theoremdchrval 24759* Value of the group of Dirichlet characters. (Contributed by Mario Carneiro, 18-Apr-2016.)
𝐺 = (DChr‘𝑁)    &   𝑍 = (ℤ/nℤ‘𝑁)    &   𝐵 = (Base‘𝑍)    &   𝑈 = (Unit‘𝑍)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐷 = {𝑥 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∣ ((𝐵𝑈) × {0}) ⊆ 𝑥})       (𝜑𝐺 = {⟨(Base‘ndx), 𝐷⟩, ⟨(+g‘ndx), ( ∘𝑓 · ↾ (𝐷 × 𝐷))⟩})

Theoremdchrbas 24760* Base set of the group of Dirichlet characters. (Contributed by Mario Carneiro, 18-Apr-2016.)
𝐺 = (DChr‘𝑁)    &   𝑍 = (ℤ/nℤ‘𝑁)    &   𝐵 = (Base‘𝑍)    &   𝑈 = (Unit‘𝑍)    &   (𝜑𝑁 ∈ ℕ)    &   𝐷 = (Base‘𝐺)       (𝜑𝐷 = {𝑥 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∣ ((𝐵𝑈) × {0}) ⊆ 𝑥})

Theoremdchrelbas 24761 A Dirichlet character is a monoid homomorphism from the multiplicative monoid on ℤ/n to the multiplicative monoid of , which is zero off the group of units of ℤ/n. (Contributed by Mario Carneiro, 18-Apr-2016.)
𝐺 = (DChr‘𝑁)    &   𝑍 = (ℤ/nℤ‘𝑁)    &   𝐵 = (Base‘𝑍)    &   𝑈 = (Unit‘𝑍)    &   (𝜑𝑁 ∈ ℕ)    &   𝐷 = (Base‘𝐺)       (𝜑 → (𝑋𝐷 ↔ (𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ ((𝐵𝑈) × {0}) ⊆ 𝑋)))

Theoremdchrelbas2 24762* A Dirichlet character is a monoid homomorphism from the multiplicative monoid on ℤ/n to the multiplicative monoid of , which is zero off the group of units of ℤ/n. (Contributed by Mario Carneiro, 18-Apr-2016.)
𝐺 = (DChr‘𝑁)    &   𝑍 = (ℤ/nℤ‘𝑁)    &   𝐵 = (Base‘𝑍)    &   𝑈 = (Unit‘𝑍)    &   (𝜑𝑁 ∈ ℕ)    &   𝐷 = (Base‘𝐺)       (𝜑 → (𝑋𝐷 ↔ (𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ ∀𝑥𝐵 ((𝑋𝑥) ≠ 0 → 𝑥𝑈))))

Theoremdchrelbas3 24763* A Dirichlet character is a monoid homomorphism from the multiplicative monoid on ℤ/n to the multiplicative monoid of , which is zero off the group of units of ℤ/n. (Contributed by Mario Carneiro, 19-Apr-2016.)
𝐺 = (DChr‘𝑁)    &   𝑍 = (ℤ/nℤ‘𝑁)    &   𝐵 = (Base‘𝑍)    &   𝑈 = (Unit‘𝑍)    &   (𝜑𝑁 ∈ ℕ)    &   𝐷 = (Base‘𝐺)       (𝜑 → (𝑋𝐷 ↔ (𝑋:𝐵⟶ℂ ∧ (∀𝑥𝑈𝑦𝑈 (𝑋‘(𝑥(.r𝑍)𝑦)) = ((𝑋𝑥) · (𝑋𝑦)) ∧ (𝑋‘(1r𝑍)) = 1 ∧ ∀𝑥𝐵 ((𝑋𝑥) ≠ 0 → 𝑥𝑈)))))

Theoremdchrelbasd 24764* A Dirichlet character is a monoid homomorphism from the multiplicative monoid on ℤ/n to the multiplicative monoid of , which is zero off the group of units of ℤ/n. (Contributed by Mario Carneiro, 28-Apr-2016.)
𝐺 = (DChr‘𝑁)    &   𝑍 = (ℤ/nℤ‘𝑁)    &   𝐵 = (Base‘𝑍)    &   𝑈 = (Unit‘𝑍)    &   (𝜑𝑁 ∈ ℕ)    &   𝐷 = (Base‘𝐺)    &   (𝑘 = 𝑥𝑋 = 𝐴)    &   (𝑘 = 𝑦𝑋 = 𝐶)    &   (𝑘 = (𝑥(.r𝑍)𝑦) → 𝑋 = 𝐸)    &   (𝑘 = (1r𝑍) → 𝑋 = 𝑌)    &   ((𝜑𝑘𝑈) → 𝑋 ∈ ℂ)    &   ((𝜑 ∧ (𝑥𝑈𝑦𝑈)) → 𝐸 = (𝐴 · 𝐶))    &   (𝜑𝑌 = 1)       (𝜑 → (𝑘𝐵 ↦ if(𝑘𝑈, 𝑋, 0)) ∈ 𝐷)

Theoremdchrrcl 24765 Reverse closure for a Dirichlet character. (Contributed by Mario Carneiro, 12-May-2016.)
𝐺 = (DChr‘𝑁)    &   𝐷 = (Base‘𝐺)       (𝑋𝐷𝑁 ∈ ℕ)

Theoremdchrmhm 24766 A Dirichlet character is a monoid homomorphism. (Contributed by Mario Carneiro, 19-Apr-2016.)
𝐺 = (DChr‘𝑁)    &   𝑍 = (ℤ/nℤ‘𝑁)    &   𝐷 = (Base‘𝐺)       𝐷 ⊆ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld))

Theoremdchrf 24767 A Dirichlet character is a function. (Contributed by Mario Carneiro, 18-Apr-2016.)
𝐺 = (DChr‘𝑁)    &   𝑍 = (ℤ/nℤ‘𝑁)    &   𝐷 = (Base‘𝐺)    &   𝐵 = (Base‘𝑍)    &   (𝜑𝑋𝐷)       (𝜑𝑋:𝐵⟶ℂ)

Theoremdchrelbas4 24768* A Dirichlet character is a monoid homomorphism from the multiplicative monoid on ℤ/n to the multiplicative monoid of , which is zero off the group of units of ℤ/n. (Contributed by Mario Carneiro, 18-Apr-2016.)
𝐺 = (DChr‘𝑁)    &   𝑍 = (ℤ/nℤ‘𝑁)    &   𝐷 = (Base‘𝐺)    &   𝐿 = (ℤRHom‘𝑍)       (𝑋𝐷 ↔ (𝑁 ∈ ℕ ∧ 𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ ∀𝑥 ∈ ℤ (1 < (𝑥 gcd 𝑁) → (𝑋‘(𝐿𝑥)) = 0)))

Theoremdchrzrh1 24769 Value of a Dirichlet character at one. (Contributed by Mario Carneiro, 4-May-2016.)
𝐺 = (DChr‘𝑁)    &   𝑍 = (ℤ/nℤ‘𝑁)    &   𝐷 = (Base‘𝐺)    &   𝐿 = (ℤRHom‘𝑍)    &   (𝜑𝑋𝐷)       (𝜑 → (𝑋‘(𝐿‘1)) = 1)

Theoremdchrzrhcl 24770 A Dirichlet character takes values in the complex numbers. (Contributed by Mario Carneiro, 12-May-2016.)
𝐺 = (DChr‘𝑁)    &   𝑍 = (ℤ/nℤ‘𝑁)    &   𝐷 = (Base‘𝐺)    &   𝐿 = (ℤRHom‘𝑍)    &   (𝜑𝑋𝐷)    &   (𝜑𝐴 ∈ ℤ)       (𝜑 → (𝑋‘(𝐿𝐴)) ∈ ℂ)

Theoremdchrzrhmul 24771 A Dirichlet character is completely multiplicative. (Contributed by Mario Carneiro, 4-May-2016.)
𝐺 = (DChr‘𝑁)    &   𝑍 = (ℤ/nℤ‘𝑁)    &   𝐷 = (Base‘𝐺)    &   𝐿 = (ℤRHom‘𝑍)    &   (𝜑𝑋𝐷)    &   (𝜑𝐴 ∈ ℤ)    &   (𝜑𝐶 ∈ ℤ)       (𝜑 → (𝑋‘(𝐿‘(𝐴 · 𝐶))) = ((𝑋‘(𝐿𝐴)) · (𝑋‘(𝐿𝐶))))

Theoremdchrplusg 24772 Group operation on the group of Dirichlet characters. (Contributed by Mario Carneiro, 18-Apr-2016.)
𝐺 = (DChr‘𝑁)    &   𝑍 = (ℤ/nℤ‘𝑁)    &   𝐷 = (Base‘𝐺)    &    · = (+g𝐺)    &   (𝜑𝑁 ∈ ℕ)       (𝜑· = ( ∘𝑓 · ↾ (𝐷 × 𝐷)))

Theoremdchrmul 24773 Group operation on the group of Dirichlet characters. (Contributed by Mario Carneiro, 18-Apr-2016.)
𝐺 = (DChr‘𝑁)    &   𝑍 = (ℤ/nℤ‘𝑁)    &   𝐷 = (Base‘𝐺)    &    · = (+g𝐺)    &   (𝜑𝑋𝐷)    &   (𝜑𝑌𝐷)       (𝜑 → (𝑋 · 𝑌) = (𝑋𝑓 · 𝑌))

Theoremdchrmulcl 24774 Closure of the group operation on Dirichlet characters. (Contributed by Mario Carneiro, 18-Apr-2016.)
𝐺 = (DChr‘𝑁)    &   𝑍 = (ℤ/nℤ‘𝑁)    &   𝐷 = (Base‘𝐺)    &    · = (+g𝐺)    &   (𝜑𝑋𝐷)    &   (𝜑𝑌𝐷)       (𝜑 → (𝑋 · 𝑌) ∈ 𝐷)

Theoremdchrn0 24775 A Dirichlet character is nonzero on the units of ℤ/n. (Contributed by Mario Carneiro, 18-Apr-2016.)
𝐺 = (DChr‘𝑁)    &   𝑍 = (ℤ/nℤ‘𝑁)    &   𝐷 = (Base‘𝐺)    &   𝐵 = (Base‘𝑍)    &   𝑈 = (Unit‘𝑍)    &   (𝜑𝑋𝐷)    &   (𝜑𝐴𝐵)       (𝜑 → ((𝑋𝐴) ≠ 0 ↔ 𝐴𝑈))

Theoremdchr1cl 24776* Closure of the principal Dirichlet character. (Contributed by Mario Carneiro, 18-Apr-2016.)
𝐺 = (DChr‘𝑁)    &   𝑍 = (ℤ/nℤ‘𝑁)    &   𝐷 = (Base‘𝐺)    &   𝐵 = (Base‘𝑍)    &   𝑈 = (Unit‘𝑍)    &    1 = (𝑘𝐵 ↦ if(𝑘𝑈, 1, 0))    &   (𝜑𝑁 ∈ ℕ)       (𝜑1𝐷)

Theoremdchrmulid2 24777* Left identity for the principal Dirichlet character. (Contributed by Mario Carneiro, 18-Apr-2016.)
𝐺 = (DChr‘𝑁)    &   𝑍 = (ℤ/nℤ‘𝑁)    &   𝐷 = (Base‘𝐺)    &   𝐵 = (Base‘𝑍)    &   𝑈 = (Unit‘𝑍)    &    1 = (𝑘𝐵 ↦ if(𝑘𝑈, 1, 0))    &    · = (+g𝐺)    &   (𝜑𝑋𝐷)       (𝜑 → ( 1 · 𝑋) = 𝑋)

Theoremdchrinvcl 24778* Closure of the group inverse operation on Dirichlet characters. (Contributed by Mario Carneiro, 19-Apr-2016.)
𝐺 = (DChr‘𝑁)    &   𝑍 = (ℤ/nℤ‘𝑁)    &   𝐷 = (Base‘𝐺)    &   𝐵 = (Base‘𝑍)    &   𝑈 = (Unit‘𝑍)    &    1 = (𝑘𝐵 ↦ if(𝑘𝑈, 1, 0))    &    · = (+g𝐺)    &   (𝜑𝑋𝐷)    &   𝐾 = (𝑘𝐵 ↦ if(𝑘𝑈, (1 / (𝑋𝑘)), 0))       (𝜑 → (𝐾𝐷 ∧ (𝐾 · 𝑋) = 1 ))

Theoremdchrabl 24779 The set of Dirichlet characters is an Abelian group. (Contributed by Mario Carneiro, 19-Apr-2016.)
𝐺 = (DChr‘𝑁)       (𝑁 ∈ ℕ → 𝐺 ∈ Abel)

Theoremdchrfi 24780 The group of Dirichlet characters is a finite group. (Contributed by Mario Carneiro, 19-Apr-2016.)
𝐺 = (DChr‘𝑁)    &   𝐷 = (Base‘𝐺)       (𝑁 ∈ ℕ → 𝐷 ∈ Fin)

Theoremdchrghm 24781 A Dirichlet character restricted to the unit group of ℤ/n is a group homomorphism into the multiplicative group of nonzero complex numbers. (Contributed by Mario Carneiro, 21-Apr-2016.)
𝐺 = (DChr‘𝑁)    &   𝑍 = (ℤ/nℤ‘𝑁)    &   𝐷 = (Base‘𝐺)    &   𝑈 = (Unit‘𝑍)    &   𝐻 = ((mulGrp‘𝑍) ↾s 𝑈)    &   𝑀 = ((mulGrp‘ℂfld) ↾s (ℂ ∖ {0}))    &   (𝜑𝑋𝐷)       (𝜑 → (𝑋𝑈) ∈ (𝐻 GrpHom 𝑀))

Theoremdchr1 24782 Value of the principal Dirichlet character. (Contributed by Mario Carneiro, 28-Apr-2016.)
𝐺 = (DChr‘𝑁)    &   𝑍 = (ℤ/nℤ‘𝑁)    &    1 = (0g𝐺)    &   𝑈 = (Unit‘𝑍)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐴𝑈)       (𝜑 → ( 1𝐴) = 1)

Theoremdchreq 24783* A Dirichlet character is determined by its values on the unit group. (Contributed by Mario Carneiro, 28-Apr-2016.)
𝐺 = (DChr‘𝑁)    &   𝑍 = (ℤ/nℤ‘𝑁)    &   𝐷 = (Base‘𝐺)    &   𝑈 = (Unit‘𝑍)    &   (𝜑𝑋𝐷)    &   (𝜑𝑌𝐷)       (𝜑 → (𝑋 = 𝑌 ↔ ∀𝑘𝑈 (𝑋𝑘) = (𝑌𝑘)))

Theoremdchrresb 24784 A Dirichlet character is determined by its values on the unit group. (Contributed by Mario Carneiro, 28-Apr-2016.)
𝐺 = (DChr‘𝑁)    &   𝑍 = (ℤ/nℤ‘𝑁)    &   𝐷 = (Base‘𝐺)    &   𝑈 = (Unit‘𝑍)    &   (𝜑𝑋𝐷)    &   (𝜑𝑌𝐷)       (𝜑 → ((𝑋𝑈) = (𝑌𝑈) ↔ 𝑋 = 𝑌))

Theoremdchrabs 24785 A Dirichlet character takes values on the unit circle. (Contributed by Mario Carneiro, 28-Apr-2016.)
𝐺 = (DChr‘𝑁)    &   𝐷 = (Base‘𝐺)    &   (𝜑𝑋𝐷)    &   𝑍 = (ℤ/nℤ‘𝑁)    &   𝑈 = (Unit‘𝑍)    &   (𝜑𝐴𝑈)       (𝜑 → (abs‘(𝑋𝐴)) = 1)

Theoremdchrinv 24786 The inverse of a Dirichlet character is the conjugate (which is also the multiplicative inverse, because the values of 𝑋 are unimodular). (Contributed by Mario Carneiro, 28-Apr-2016.)
𝐺 = (DChr‘𝑁)    &   𝐷 = (Base‘𝐺)    &   (𝜑𝑋𝐷)    &   𝐼 = (invg𝐺)       (𝜑 → (𝐼𝑋) = (∗ ∘ 𝑋))

Theoremdchrabs2 24787 A Dirichlet character takes values inside the unit circle. (Contributed by Mario Carneiro, 3-May-2016.)
𝐺 = (DChr‘𝑁)    &   𝐷 = (Base‘𝐺)    &   𝑍 = (ℤ/nℤ‘𝑁)    &   𝐵 = (Base‘𝑍)    &   (𝜑𝑋𝐷)    &   (𝜑𝐴𝐵)       (𝜑 → (abs‘(𝑋𝐴)) ≤ 1)

Theoremdchr1re 24788 The principal Dirichlet character is a real character. (Contributed by Mario Carneiro, 2-May-2016.)
𝐺 = (DChr‘𝑁)    &   𝑍 = (ℤ/nℤ‘𝑁)    &    1 = (0g𝐺)    &   𝐵 = (Base‘𝑍)    &   (𝜑𝑁 ∈ ℕ)       (𝜑1 :𝐵⟶ℝ)

Theoremdchrptlem1 24789* Lemma for dchrpt 24792. (Contributed by Mario Carneiro, 28-Apr-2016.)
𝐺 = (DChr‘𝑁)    &   𝑍 = (ℤ/nℤ‘𝑁)    &   𝐷 = (Base‘𝐺)    &   𝐵 = (Base‘𝑍)    &    1 = (1r𝑍)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐴1 )    &   𝑈 = (Unit‘𝑍)    &   𝐻 = ((mulGrp‘𝑍) ↾s 𝑈)    &    · = (.g𝐻)    &   𝑆 = (𝑘 ∈ dom 𝑊 ↦ ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑊𝑘))))    &   (𝜑𝐴𝑈)    &   (𝜑𝑊 ∈ Word 𝑈)    &   (𝜑𝐻dom DProd 𝑆)    &   (𝜑 → (𝐻 DProd 𝑆) = 𝑈)    &   𝑃 = (𝐻dProj𝑆)    &   𝑂 = (od‘𝐻)    &   𝑇 = (-1↑𝑐(2 / (𝑂‘(𝑊𝐼))))    &   (𝜑𝐼 ∈ dom 𝑊)    &   (𝜑 → ((𝑃𝐼)‘𝐴) ≠ 1 )    &   𝑋 = (𝑢𝑈 ↦ (℩𝑚 ∈ ℤ (((𝑃𝐼)‘𝑢) = (𝑚 · (𝑊𝐼)) ∧ = (𝑇𝑚))))       (((𝜑𝐶𝑈) ∧ (𝑀 ∈ ℤ ∧ ((𝑃𝐼)‘𝐶) = (𝑀 · (𝑊𝐼)))) → (𝑋𝐶) = (𝑇𝑀))

Theoremdchrptlem2 24790* Lemma for dchrpt 24792. (Contributed by Mario Carneiro, 28-Apr-2016.)
𝐺 = (DChr‘𝑁)    &   𝑍 = (ℤ/nℤ‘𝑁)    &   𝐷 = (Base‘𝐺)    &   𝐵 = (Base‘𝑍)    &    1 = (1r𝑍)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐴1 )    &   𝑈 = (Unit‘𝑍)    &   𝐻 = ((mulGrp‘𝑍) ↾s 𝑈)    &    · = (.g𝐻)    &   𝑆 = (𝑘 ∈ dom 𝑊 ↦ ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑊𝑘))))    &   (𝜑𝐴𝑈)    &   (𝜑𝑊 ∈ Word 𝑈)    &   (𝜑𝐻dom DProd 𝑆)    &   (𝜑 → (𝐻 DProd 𝑆) = 𝑈)    &   𝑃 = (𝐻dProj𝑆)    &   𝑂 = (od‘𝐻)    &   𝑇 = (-1↑𝑐(2 / (𝑂‘(𝑊𝐼))))    &   (𝜑𝐼 ∈ dom 𝑊)    &   (𝜑 → ((𝑃𝐼)‘𝐴) ≠ 1 )    &   𝑋 = (𝑢𝑈 ↦ (℩𝑚 ∈ ℤ (((𝑃𝐼)‘𝑢) = (𝑚 · (𝑊𝐼)) ∧ = (𝑇𝑚))))       (𝜑 → ∃𝑥𝐷 (𝑥𝐴) ≠ 1)

Theoremdchrptlem3 24791* Lemma for dchrpt 24792. (Contributed by Mario Carneiro, 28-Apr-2016.)
𝐺 = (DChr‘𝑁)    &   𝑍 = (ℤ/nℤ‘𝑁)    &   𝐷 = (Base‘𝐺)    &   𝐵 = (Base‘𝑍)    &    1 = (1r𝑍)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐴1 )    &   𝑈 = (Unit‘𝑍)    &   𝐻 = ((mulGrp‘𝑍) ↾s 𝑈)    &    · = (.g𝐻)    &   𝑆 = (𝑘 ∈ dom 𝑊 ↦ ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑊𝑘))))    &   (𝜑𝐴𝑈)    &   (𝜑𝑊 ∈ Word 𝑈)    &   (𝜑𝐻dom DProd 𝑆)    &   (𝜑 → (𝐻 DProd 𝑆) = 𝑈)       (𝜑 → ∃𝑥𝐷 (𝑥𝐴) ≠ 1)

Theoremdchrpt 24792* For any element other than 1, there is a Dirichlet character that is not one at the given element. (Contributed by Mario Carneiro, 28-Apr-2016.)
𝐺 = (DChr‘𝑁)    &   𝑍 = (ℤ/nℤ‘𝑁)    &   𝐷 = (Base‘𝐺)    &   𝐵 = (Base‘𝑍)    &    1 = (1r𝑍)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐴1 )    &   (𝜑𝐴𝐵)       (𝜑 → ∃𝑥𝐷 (𝑥𝐴) ≠ 1)

Theoremdchrsum2 24793* An orthogonality relation for Dirichlet characters: the sum of all the values of a Dirichlet character 𝑋 is 0 if 𝑋 is non-principal and ϕ(𝑛) otherwise. Part of Theorem 6.5.1 of [Shapiro] p. 230. (Contributed by Mario Carneiro, 28-Apr-2016.)
𝐺 = (DChr‘𝑁)    &   𝑍 = (ℤ/nℤ‘𝑁)    &   𝐷 = (Base‘𝐺)    &    1 = (0g𝐺)    &   (𝜑𝑋𝐷)    &   𝑈 = (Unit‘𝑍)       (𝜑 → Σ𝑎𝑈 (𝑋𝑎) = if(𝑋 = 1 , (ϕ‘𝑁), 0))

Theoremdchrsum 24794* An orthogonality relation for Dirichlet characters: the sum of all the values of a Dirichlet character 𝑋 is 0 if 𝑋 is non-principal and ϕ(𝑛) otherwise. Part of Theorem 6.5.1 of [Shapiro] p. 230. (Contributed by Mario Carneiro, 28-Apr-2016.)
𝐺 = (DChr‘𝑁)    &   𝑍 = (ℤ/nℤ‘𝑁)    &   𝐷 = (Base‘𝐺)    &    1 = (0g𝐺)    &   (𝜑𝑋𝐷)    &   𝐵 = (Base‘𝑍)       (𝜑 → Σ𝑎𝐵 (𝑋𝑎) = if(𝑋 = 1 , (ϕ‘𝑁), 0))

Theoremsumdchr2 24795* Lemma for sumdchr 24797. (Contributed by Mario Carneiro, 28-Apr-2016.)
𝐺 = (DChr‘𝑁)    &   𝐷 = (Base‘𝐺)    &   𝑍 = (ℤ/nℤ‘𝑁)    &    1 = (1r𝑍)    &   𝐵 = (Base‘𝑍)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐴𝐵)       (𝜑 → Σ𝑥𝐷 (𝑥𝐴) = if(𝐴 = 1 , (#‘𝐷), 0))

Theoremdchrhash 24796 There are exactly ϕ(𝑁) Dirichlet characters modulo 𝑁. Part of Theorem 6.5.1 of [Shapiro] p. 230. (Contributed by Mario Carneiro, 28-Apr-2016.)
𝐺 = (DChr‘𝑁)    &   𝐷 = (Base‘𝐺)       (𝑁 ∈ ℕ → (#‘𝐷) = (ϕ‘𝑁))

Theoremsumdchr 24797* An orthogonality relation for Dirichlet characters: the sum of 𝑥(𝐴) for fixed 𝐴 and all 𝑥 is 0 if 𝐴 = 1 and ϕ(𝑛) otherwise. Theorem 6.5.1 of [Shapiro] p. 230. (Contributed by Mario Carneiro, 28-Apr-2016.)
𝐺 = (DChr‘𝑁)    &   𝐷 = (Base‘𝐺)    &   𝑍 = (ℤ/nℤ‘𝑁)    &    1 = (1r𝑍)    &   𝐵 = (Base‘𝑍)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐴𝐵)       (𝜑 → Σ𝑥𝐷 (𝑥𝐴) = if(𝐴 = 1 , (ϕ‘𝑁), 0))

Theoremdchr2sum 24798* An orthogonality relation for Dirichlet characters: the sum of 𝑋(𝑎) · ∗𝑌(𝑎) over all 𝑎 is nonzero only when 𝑋 = 𝑌. Part of Theorem 6.5.2 of [Shapiro] p. 232. (Contributed by Mario Carneiro, 28-Apr-2016.)
𝐺 = (DChr‘𝑁)    &   𝑍 = (ℤ/nℤ‘𝑁)    &   𝐷 = (Base‘𝐺)    &   𝐵 = (Base‘𝑍)    &   (𝜑𝑋𝐷)    &   (𝜑𝑌𝐷)       (𝜑 → Σ𝑎𝐵 ((𝑋𝑎) · (∗‘(𝑌𝑎))) = if(𝑋 = 𝑌, (ϕ‘𝑁), 0))

Theoremsum2dchr 24799* An orthogonality relation for Dirichlet characters: the sum of 𝑥(𝐴) for fixed 𝐴 and all 𝑥 is 0 if 𝐴 = 1 and ϕ(𝑛) otherwise. Part of Theorem 6.5.2 of [Shapiro] p. 232. (Contributed by Mario Carneiro, 28-Apr-2016.)
𝐺 = (DChr‘𝑁)    &   𝐷 = (Base‘𝐺)    &   𝑍 = (ℤ/nℤ‘𝑁)    &   𝐵 = (Base‘𝑍)    &   𝑈 = (Unit‘𝑍)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐴𝐵)    &   (𝜑𝐶𝑈)       (𝜑 → Σ𝑥𝐷 ((𝑥𝐴) · (∗‘(𝑥𝐶))) = if(𝐴 = 𝐶, (ϕ‘𝑁), 0))

14.4.7  Bertrand's postulate

Theorembcctr 24800 Value of the central binomial coefficient. (Contributed by Mario Carneiro, 13-Mar-2014.)
(𝑁 ∈ ℕ0 → ((2 · 𝑁)C𝑁) = ((!‘(2 · 𝑁)) / ((!‘𝑁) · (!‘𝑁))))

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