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Theorem List for Metamath Proof Explorer - 24801-24900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremlgamgulmlem2 24801* Lemma for lgamgulm 24806. (Contributed by Mario Carneiro, 3-Jul-2017.)
(𝜑𝑅 ∈ ℕ)    &   𝑈 = {𝑥 ∈ ℂ ∣ ((abs‘𝑥) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)))}    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐴𝑈)    &   (𝜑 → (2 · 𝑅) ≤ 𝑁)       (𝜑 → (abs‘((𝐴 / 𝑁) − (log‘((𝐴 / 𝑁) + 1)))) ≤ (𝑅 · ((1 / (𝑁𝑅)) − (1 / 𝑁))))

Theoremlgamgulmlem3 24802* Lemma for lgamgulm 24806. (Contributed by Mario Carneiro, 3-Jul-2017.)
(𝜑𝑅 ∈ ℕ)    &   𝑈 = {𝑥 ∈ ℂ ∣ ((abs‘𝑥) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)))}    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐴𝑈)    &   (𝜑 → (2 · 𝑅) ≤ 𝑁)       (𝜑 → (abs‘((𝐴 · (log‘((𝑁 + 1) / 𝑁))) − (log‘((𝐴 / 𝑁) + 1)))) ≤ (𝑅 · ((2 · (𝑅 + 1)) / (𝑁↑2))))

Theoremlgamgulmlem4 24803* Lemma for lgamgulm 24806. (Contributed by Mario Carneiro, 3-Jul-2017.)
(𝜑𝑅 ∈ ℕ)    &   𝑈 = {𝑥 ∈ ℂ ∣ ((abs‘𝑥) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)))}    &   𝐺 = (𝑚 ∈ ℕ ↦ (𝑧𝑈 ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))))    &   𝑇 = (𝑚 ∈ ℕ ↦ if((2 · 𝑅) ≤ 𝑚, (𝑅 · ((2 · (𝑅 + 1)) / (𝑚↑2))), ((𝑅 · (log‘((𝑚 + 1) / 𝑚))) + ((log‘((𝑅 + 1) · 𝑚)) + π))))       (𝜑 → seq1( + , 𝑇) ∈ dom ⇝ )

Theoremlgamgulmlem5 24804* Lemma for lgamgulm 24806. (Contributed by Mario Carneiro, 3-Jul-2017.)
(𝜑𝑅 ∈ ℕ)    &   𝑈 = {𝑥 ∈ ℂ ∣ ((abs‘𝑥) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)))}    &   𝐺 = (𝑚 ∈ ℕ ↦ (𝑧𝑈 ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))))    &   𝑇 = (𝑚 ∈ ℕ ↦ if((2 · 𝑅) ≤ 𝑚, (𝑅 · ((2 · (𝑅 + 1)) / (𝑚↑2))), ((𝑅 · (log‘((𝑚 + 1) / 𝑚))) + ((log‘((𝑅 + 1) · 𝑚)) + π))))       ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑦𝑈)) → (abs‘((𝐺𝑛)‘𝑦)) ≤ (𝑇𝑛))

Theoremlgamgulmlem6 24805* The series 𝐺 is uniformly convergent on the compact region 𝑈, which describes a circle of radius 𝑅 with holes of size 1 / 𝑅 around the poles of the gamma function. (Contributed by Mario Carneiro, 9-Jul-2017.)
(𝜑𝑅 ∈ ℕ)    &   𝑈 = {𝑥 ∈ ℂ ∣ ((abs‘𝑥) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)))}    &   𝐺 = (𝑚 ∈ ℕ ↦ (𝑧𝑈 ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))))    &   𝑇 = (𝑚 ∈ ℕ ↦ if((2 · 𝑅) ≤ 𝑚, (𝑅 · ((2 · (𝑅 + 1)) / (𝑚↑2))), ((𝑅 · (log‘((𝑚 + 1) / 𝑚))) + ((log‘((𝑅 + 1) · 𝑚)) + π))))       (𝜑 → (seq1( ∘𝑓 + , 𝐺) ∈ dom (⇝𝑢𝑈) ∧ (seq1( ∘𝑓 + , 𝐺)(⇝𝑢𝑈)(𝑧𝑈𝑂) → ∃𝑟 ∈ ℝ ∀𝑧𝑈 (abs‘𝑂) ≤ 𝑟)))

Theoremlgamgulm 24806* The series 𝐺 is uniformly convergent on the compact region 𝑈, which describes a circle of radius 𝑅 with holes of size 1 / 𝑅 around the poles of the gamma function. (Contributed by Mario Carneiro, 3-Jul-2017.)
(𝜑𝑅 ∈ ℕ)    &   𝑈 = {𝑥 ∈ ℂ ∣ ((abs‘𝑥) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)))}    &   𝐺 = (𝑚 ∈ ℕ ↦ (𝑧𝑈 ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))))       (𝜑 → seq1( ∘𝑓 + , 𝐺) ∈ dom (⇝𝑢𝑈))

Theoremlgamgulm2 24807* Rewrite the limit of the sequence 𝐺 in terms of the log-Gamma function. (Contributed by Mario Carneiro, 6-Jul-2017.)
(𝜑𝑅 ∈ ℕ)    &   𝑈 = {𝑥 ∈ ℂ ∣ ((abs‘𝑥) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)))}    &   𝐺 = (𝑚 ∈ ℕ ↦ (𝑧𝑈 ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))))       (𝜑 → (∀𝑧𝑈 (log Γ‘𝑧) ∈ ℂ ∧ seq1( ∘𝑓 + , 𝐺)(⇝𝑢𝑈)(𝑧𝑈 ↦ ((log Γ‘𝑧) + (log‘𝑧)))))

Theoremlgambdd 24808* The log-Gamma function is bounded on the region 𝑈. (Contributed by Mario Carneiro, 9-Jul-2017.)
(𝜑𝑅 ∈ ℕ)    &   𝑈 = {𝑥 ∈ ℂ ∣ ((abs‘𝑥) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)))}    &   𝐺 = (𝑚 ∈ ℕ ↦ (𝑧𝑈 ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))))       (𝜑 → ∃𝑟 ∈ ℝ ∀𝑧𝑈 (abs‘(log Γ‘𝑧)) ≤ 𝑟)

Theoremlgamucov 24809* The 𝑈 regions used in the proof of lgamgulm 24806 have interiors which cover the entire domain of the Gamma function. (Contributed by Mario Carneiro, 6-Jul-2017.)
𝑈 = {𝑥 ∈ ℂ ∣ ((abs‘𝑥) ≤ 𝑟 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑟) ≤ (abs‘(𝑥 + 𝑘)))}    &   (𝜑𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)))    &   𝐽 = (TopOpen‘ℂfld)       (𝜑 → ∃𝑟 ∈ ℕ 𝐴 ∈ ((int‘𝐽)‘𝑈))

Theoremlgamucov2 24810* The 𝑈 regions used in the proof of lgamgulm 24806 have interiors which cover the entire domain of the Gamma function. (Contributed by Mario Carneiro, 8-Jul-2017.)
𝑈 = {𝑥 ∈ ℂ ∣ ((abs‘𝑥) ≤ 𝑟 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑟) ≤ (abs‘(𝑥 + 𝑘)))}    &   (𝜑𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)))       (𝜑 → ∃𝑟 ∈ ℕ 𝐴𝑈)

Theoremlgamcvglem 24811* Lemma for lgamf 24813 and lgamcvg 24825. (Contributed by Mario Carneiro, 8-Jul-2017.)
𝑈 = {𝑥 ∈ ℂ ∣ ((abs‘𝑥) ≤ 𝑟 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑟) ≤ (abs‘(𝑥 + 𝑘)))}    &   (𝜑𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)))    &   𝐺 = (𝑚 ∈ ℕ ↦ ((𝐴 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝐴 / 𝑚) + 1))))       (𝜑 → ((log Γ‘𝐴) ∈ ℂ ∧ seq1( + , 𝐺) ⇝ ((log Γ‘𝐴) + (log‘𝐴))))

Theoremlgamcl 24812 The log-Gamma function is a complex function defined on the whole complex plane except for the negative integers. (Contributed by Mario Carneiro, 8-Jul-2017.)
(𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)) → (log Γ‘𝐴) ∈ ℂ)

Theoremlgamf 24813 The log-Gamma function is a complex function defined on the whole complex plane except for the negative integers. (Contributed by Mario Carneiro, 6-Jul-2017.)
log Γ:(ℂ ∖ (ℤ ∖ ℕ))⟶ℂ

Theoremgamf 24814 The Gamma function is a complex function defined on the whole complex plane except for the negative integers. (Contributed by Mario Carneiro, 6-Jul-2017.)
Γ:(ℂ ∖ (ℤ ∖ ℕ))⟶ℂ

Theoremgamcl 24815 The exponential of the log-Gamma function is the Gamma function (by definition). (Contributed by Mario Carneiro, 8-Jul-2017.)
(𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)) → (Γ‘𝐴) ∈ ℂ)

Theoremeflgam 24816 The exponential of the log-Gamma function is the Gamma function (by definition). (Contributed by Mario Carneiro, 8-Jul-2017.)
(𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)) → (exp‘(log Γ‘𝐴)) = (Γ‘𝐴))

Theoremgamne0 24817 The Gamma function is never zero. (Contributed by Mario Carneiro, 9-Jul-2017.)
(𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)) → (Γ‘𝐴) ≠ 0)

Theoremigamval 24818 Value of the inverse Gamma function. (Contributed by Mario Carneiro, 16-Jul-2017.)
(𝐴 ∈ ℂ → (1/Γ‘𝐴) = if(𝐴 ∈ (ℤ ∖ ℕ), 0, (1 / (Γ‘𝐴))))

Theoremigamz 24819 Value of the inverse Gamma function on nonpositive integers. (Contributed by Mario Carneiro, 16-Jul-2017.)
(𝐴 ∈ (ℤ ∖ ℕ) → (1/Γ‘𝐴) = 0)

Theoremigamgam 24820 Value of the inverse Gamma function in terms of the Gamma function. (Contributed by Mario Carneiro, 16-Jul-2017.)
(𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)) → (1/Γ‘𝐴) = (1 / (Γ‘𝐴)))

Theoremigamlgam 24821 Value of the inverse Gamma function in terms of the log-Gamma function. (Contributed by Mario Carneiro, 16-Jul-2017.)
(𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)) → (1/Γ‘𝐴) = (exp‘-(log Γ‘𝐴)))

Theoremigamf 24822 Closure of the inverse Gamma function. (Contributed by Mario Carneiro, 16-Jul-2017.)
1/Γ:ℂ⟶ℂ

Theoremigamcl 24823 Closure of the inverse Gamma function. (Contributed by Mario Carneiro, 16-Jul-2017.)
(𝐴 ∈ ℂ → (1/Γ‘𝐴) ∈ ℂ)

Theoremgamigam 24824 The Gamma function is the inverse of the inverse Gamma function. (Contributed by Mario Carneiro, 16-Jul-2017.)
(𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)) → (Γ‘𝐴) = (1 / (1/Γ‘𝐴)))

Theoremlgamcvg 24825* The series 𝐺 converges to log Γ(𝐴) + log(𝐴). (Contributed by Mario Carneiro, 6-Jul-2017.)
𝐺 = (𝑚 ∈ ℕ ↦ ((𝐴 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝐴 / 𝑚) + 1))))    &   (𝜑𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)))       (𝜑 → seq1( + , 𝐺) ⇝ ((log Γ‘𝐴) + (log‘𝐴)))

Theoremlgamcvg2 24826* The series 𝐺 converges to log Γ(𝐴 + 1). (Contributed by Mario Carneiro, 9-Jul-2017.)
𝐺 = (𝑚 ∈ ℕ ↦ ((𝐴 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝐴 / 𝑚) + 1))))    &   (𝜑𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)))       (𝜑 → seq1( + , 𝐺) ⇝ (log Γ‘(𝐴 + 1)))

Theoremgamcvg 24827* The pointwise exponential of the series 𝐺 converges to Γ(𝐴) · 𝐴. (Contributed by Mario Carneiro, 6-Jul-2017.)
𝐺 = (𝑚 ∈ ℕ ↦ ((𝐴 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝐴 / 𝑚) + 1))))    &   (𝜑𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)))       (𝜑 → (exp ∘ seq1( + , 𝐺)) ⇝ ((Γ‘𝐴) · 𝐴))

Theoremlgamp1 24828 The functional equation of the (log) Gamma function. (Contributed by Mario Carneiro, 9-Jul-2017.)
(𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)) → (log Γ‘(𝐴 + 1)) = ((log Γ‘𝐴) + (log‘𝐴)))

Theoremgamp1 24829 The functional equation of the Gamma function. (Contributed by Mario Carneiro, 9-Jul-2017.)
(𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)) → (Γ‘(𝐴 + 1)) = ((Γ‘𝐴) · 𝐴))

Theoremgamcvg2lem 24830* Lemma for gamcvg2 24831. (Contributed by Mario Carneiro, 10-Jul-2017.)
𝐹 = (𝑚 ∈ ℕ ↦ ((((𝑚 + 1) / 𝑚)↑𝑐𝐴) / ((𝐴 / 𝑚) + 1)))    &   (𝜑𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)))    &   𝐺 = (𝑚 ∈ ℕ ↦ ((𝐴 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝐴 / 𝑚) + 1))))       (𝜑 → (exp ∘ seq1( + , 𝐺)) = seq1( · , 𝐹))

Theoremgamcvg2 24831* An infinite product expression for the gamma function. (Contributed by Mario Carneiro, 9-Jul-2017.)
𝐹 = (𝑚 ∈ ℕ ↦ ((((𝑚 + 1) / 𝑚)↑𝑐𝐴) / ((𝐴 / 𝑚) + 1)))    &   (𝜑𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)))       (𝜑 → seq1( · , 𝐹) ⇝ ((Γ‘𝐴) · 𝐴))

Theoremregamcl 24832 The Gamma function is real for real input. (Contributed by Mario Carneiro, 9-Jul-2017.)
(𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) → (Γ‘𝐴) ∈ ℝ)

Theoremrelgamcl 24833 The log-Gamma function is real for positive real input. (Contributed by Mario Carneiro, 9-Jul-2017.)
(𝐴 ∈ ℝ+ → (log Γ‘𝐴) ∈ ℝ)

Theoremrpgamcl 24834 The log-Gamma function is positive real for positive real input. (Contributed by Mario Carneiro, 9-Jul-2017.)
(𝐴 ∈ ℝ+ → (Γ‘𝐴) ∈ ℝ+)

Theoremlgam1 24835 The log-Gamma function at one. (Contributed by Mario Carneiro, 9-Jul-2017.)
(log Γ‘1) = 0

Theoremgam1 24836 The log-Gamma function at one. (Contributed by Mario Carneiro, 9-Jul-2017.)
(Γ‘1) = 1

Theoremfacgam 24837 The Gamma function generalizes the factorial. (Contributed by Mario Carneiro, 9-Jul-2017.)
(𝑁 ∈ ℕ0 → (!‘𝑁) = (Γ‘(𝑁 + 1)))

Theoremgamfac 24838 The Gamma function generalizes the factorial. (Contributed by Mario Carneiro, 9-Jul-2017.)
(𝑁 ∈ ℕ → (Γ‘𝑁) = (!‘(𝑁 − 1)))

14.4  Basic number theory

14.4.1  Wilson's theorem

Theoremwilthlem1 24839 The only elements that are equal to their own inverses in the multiplicative group of nonzero elements in ℤ / 𝑃 are 1 and -1≡𝑃 − 1. (Note that from prmdiveq 15538, (𝑁↑(𝑃 − 2)) mod 𝑃 is the modular inverse of 𝑁 in ℤ / 𝑃. (Contributed by Mario Carneiro, 24-Jan-2015.)
((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (𝑁 = ((𝑁↑(𝑃 − 2)) mod 𝑃) ↔ (𝑁 = 1 ∨ 𝑁 = (𝑃 − 1))))

Theoremwilthlem2 24840* Lemma for wilth 24842: induction step. The "hand proof" version of this theorem works by writing out the list of all numbers from 1 to 𝑃 − 1 in pairs such that a number is paired with its inverse. Every number has a unique inverse different from itself except 1 and 𝑃 − 1, and so each pair multiplies to 1, and 1 and 𝑃 − 1≡-1 multiply to -1, so the full product is equal to -1. Here we make this precise by doing the product pair by pair.

The induction hypothesis says that every subset 𝑆 of 1...(𝑃 − 1) that is closed under inverse (i.e. all pairs are matched up) and contains 𝑃 − 1 multiplies to -1 mod 𝑃. Given such a set, we take out one element 𝑧𝑃 − 1. If there are no such elements, then 𝑆 = {𝑃 − 1} which forms the base case. Otherwise, 𝑆 ∖ {𝑧, 𝑧↑-1} is also closed under inverse and contains 𝑃 − 1, so the induction hypothesis says that this equals -1; and the remaining two elements are either equal to each other, in which case wilthlem1 24839 gives that 𝑧 = 1 or 𝑃 − 1, and we've already excluded the second case, so the product gives 1; or 𝑧𝑧↑-1 and their product is 1. In either case the accumulated product is unaffected. (Contributed by Mario Carneiro, 24-Jan-2015.) (Proof shortened by AV, 27-Jul-2019.)

𝑇 = (mulGrp‘ℂfld)    &   𝐴 = {𝑥 ∈ 𝒫 (1...(𝑃 − 1)) ∣ ((𝑃 − 1) ∈ 𝑥 ∧ ∀𝑦𝑥 ((𝑦↑(𝑃 − 2)) mod 𝑃) ∈ 𝑥)}    &   (𝜑𝑃 ∈ ℙ)    &   (𝜑𝑆𝐴)    &   (𝜑 → ∀𝑠𝐴 (𝑠𝑆 → ((𝑇 Σg ( I ↾ 𝑠)) mod 𝑃) = (-1 mod 𝑃)))       (𝜑 → ((𝑇 Σg ( I ↾ 𝑆)) mod 𝑃) = (-1 mod 𝑃))

Theoremwilthlem3 24841* Lemma for wilth 24842. Here we round out the argument of wilthlem2 24840 with the final step of the induction. The induction argument shows that every subset of 1...(𝑃 − 1) that is closed under inverse and contains 𝑃 − 1 multiplies to -1 mod 𝑃, and clearly 1...(𝑃 − 1) itself is such a set. Thus, the product of all the elements is -1, and all that is left is to translate the group sum notation (which we used for its unordered summing capabilities) into an ordered sequence to match the definition of the factorial. (Contributed by Mario Carneiro, 24-Jan-2015.) (Proof shortened by AV, 27-Jul-2019.)
𝑇 = (mulGrp‘ℂfld)    &   𝐴 = {𝑥 ∈ 𝒫 (1...(𝑃 − 1)) ∣ ((𝑃 − 1) ∈ 𝑥 ∧ ∀𝑦𝑥 ((𝑦↑(𝑃 − 2)) mod 𝑃) ∈ 𝑥)}       (𝑃 ∈ ℙ → 𝑃 ∥ ((!‘(𝑃 − 1)) + 1))

Theoremwilth 24842 Wilson's theorem. A number is prime iff it is greater or equal to 2 and (𝑁 − 1)! is congruent to -1, mod 𝑁, or alternatively if 𝑁 divides (𝑁 − 1)! + 1. In this part of the proof we show the relatively simple reverse implication; see wilthlem3 24841 for the forward implication. This is Metamath 100 proof #51. (Contributed by Mario Carneiro, 24-Jan-2015.) (Proof shortened by Fan Zheng, 16-Jun-2016.)
(𝑁 ∈ ℙ ↔ (𝑁 ∈ (ℤ‘2) ∧ 𝑁 ∥ ((!‘(𝑁 − 1)) + 1)))

Theoremwilthimp 24843 The forward implication of Wilson's theorem wilth 24842 (see wilthlem3 24841), expressed using the modulo operation: For any prime 𝑝 we have (𝑝 − 1)!≡ − 1 (mod 𝑝), see theorem 5.24 in [ApostolNT] p. 116. (Contributed by AV, 21-Jul-2021.)
(𝑃 ∈ ℙ → ((!‘(𝑃 − 1)) mod 𝑃) = (-1 mod 𝑃))

14.4.2  The Fundamental Theorem of Algebra

Theoremftalem1 24844* Lemma for fta 24851: "growth lemma". There exists some 𝑟 such that 𝐹 is arbitrarily close in proportion to its dominant term. (Contributed by Mario Carneiro, 14-Sep-2014.)
𝐴 = (coeff‘𝐹)    &   𝑁 = (deg‘𝐹)    &   (𝜑𝐹 ∈ (Poly‘𝑆))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐸 ∈ ℝ+)    &   𝑇 = (Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴𝑘)) / 𝐸)       (𝜑 → ∃𝑟 ∈ ℝ ∀𝑥 ∈ ℂ (𝑟 < (abs‘𝑥) → (abs‘((𝐹𝑥) − ((𝐴𝑁) · (𝑥𝑁)))) < (𝐸 · ((abs‘𝑥)↑𝑁))))

Theoremftalem2 24845* Lemma for fta 24851. There exists some 𝑟 such that 𝐹 has magnitude greater than 𝐹(0) outside the closed ball B(0,r). (Contributed by Mario Carneiro, 14-Sep-2014.)
𝐴 = (coeff‘𝐹)    &   𝑁 = (deg‘𝐹)    &   (𝜑𝐹 ∈ (Poly‘𝑆))    &   (𝜑𝑁 ∈ ℕ)    &   𝑈 = if(if(1 ≤ 𝑠, 𝑠, 1) ≤ 𝑇, 𝑇, if(1 ≤ 𝑠, 𝑠, 1))    &   𝑇 = ((abs‘(𝐹‘0)) / ((abs‘(𝐴𝑁)) / 2))       (𝜑 → ∃𝑟 ∈ ℝ+𝑥 ∈ ℂ (𝑟 < (abs‘𝑥) → (abs‘(𝐹‘0)) < (abs‘(𝐹𝑥))))

Theoremftalem3 24846* Lemma for fta 24851. There exists a global minimum of the function abs ∘ 𝐹. The proof uses a circle of radius 𝑟 where 𝑟 is the value coming from ftalem1 24844; since this is a compact set, the minimum on this disk is achieved, and this must then be the global minimum. (Contributed by Mario Carneiro, 14-Sep-2014.)
𝐴 = (coeff‘𝐹)    &   𝑁 = (deg‘𝐹)    &   (𝜑𝐹 ∈ (Poly‘𝑆))    &   (𝜑𝑁 ∈ ℕ)    &   𝐷 = {𝑦 ∈ ℂ ∣ (abs‘𝑦) ≤ 𝑅}    &   𝐽 = (TopOpen‘ℂfld)    &   (𝜑𝑅 ∈ ℝ+)    &   (𝜑 → ∀𝑥 ∈ ℂ (𝑅 < (abs‘𝑥) → (abs‘(𝐹‘0)) < (abs‘(𝐹𝑥))))       (𝜑 → ∃𝑧 ∈ ℂ ∀𝑥 ∈ ℂ (abs‘(𝐹𝑧)) ≤ (abs‘(𝐹𝑥)))

Theoremftalem4 24847* Lemma for fta 24851: Closure of the auxiliary variables for ftalem5 24848. (Contributed by Mario Carneiro, 20-Sep-2014.) (Revised by AV, 28-Sep-2020.)
𝐴 = (coeff‘𝐹)    &   𝑁 = (deg‘𝐹)    &   (𝜑𝐹 ∈ (Poly‘𝑆))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑 → (𝐹‘0) ≠ 0)    &   𝐾 = inf({𝑛 ∈ ℕ ∣ (𝐴𝑛) ≠ 0}, ℝ, < )    &   𝑇 = (-((𝐹‘0) / (𝐴𝐾))↑𝑐(1 / 𝐾))    &   𝑈 = ((abs‘(𝐹‘0)) / (Σ𝑘 ∈ ((𝐾 + 1)...𝑁)(abs‘((𝐴𝑘) · (𝑇𝑘))) + 1))    &   𝑋 = if(1 ≤ 𝑈, 1, 𝑈)       (𝜑 → ((𝐾 ∈ ℕ ∧ (𝐴𝐾) ≠ 0) ∧ (𝑇 ∈ ℂ ∧ 𝑈 ∈ ℝ+𝑋 ∈ ℝ+)))

Theoremftalem5 24848* Lemma for fta 24851: Main proof. We have already shifted the minimum found in ftalem3 24846 to zero by a change of variables, and now we show that the minimum value is zero. Expanding in a series about the minimum value, let 𝐾 be the lowest term in the polynomial that is nonzero, and let 𝑇 be a 𝐾-th root of -𝐹(0) / 𝐴(𝐾). Then an evaluation of 𝐹(𝑇𝑋) where 𝑋 is a sufficiently small positive number yields 𝐹(0) for the first term and -𝐹(0) · 𝑋𝐾 for the 𝐾-th term, and all higher terms are bounded because 𝑋 is small. Thus, abs(𝐹(𝑇𝑋)) ≤ abs(𝐹(0))(1 − 𝑋𝐾) < abs(𝐹(0)), in contradiction to our choice of 𝐹(0) as the minimum. (Contributed by Mario Carneiro, 14-Sep-2014.) (Revised by AV, 28-Sep-2020.)
𝐴 = (coeff‘𝐹)    &   𝑁 = (deg‘𝐹)    &   (𝜑𝐹 ∈ (Poly‘𝑆))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑 → (𝐹‘0) ≠ 0)    &   𝐾 = inf({𝑛 ∈ ℕ ∣ (𝐴𝑛) ≠ 0}, ℝ, < )    &   𝑇 = (-((𝐹‘0) / (𝐴𝐾))↑𝑐(1 / 𝐾))    &   𝑈 = ((abs‘(𝐹‘0)) / (Σ𝑘 ∈ ((𝐾 + 1)...𝑁)(abs‘((𝐴𝑘) · (𝑇𝑘))) + 1))    &   𝑋 = if(1 ≤ 𝑈, 1, 𝑈)       (𝜑 → ∃𝑥 ∈ ℂ (abs‘(𝐹𝑥)) < (abs‘(𝐹‘0)))

Theoremftalem6 24849* Lemma for fta 24851: Discharge the auxiliary variables in ftalem5 24848. (Contributed by Mario Carneiro, 20-Sep-2014.) (Proof shortened by AV, 28-Sep-2020.)
𝐴 = (coeff‘𝐹)    &   𝑁 = (deg‘𝐹)    &   (𝜑𝐹 ∈ (Poly‘𝑆))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑 → (𝐹‘0) ≠ 0)       (𝜑 → ∃𝑥 ∈ ℂ (abs‘(𝐹𝑥)) < (abs‘(𝐹‘0)))

Theoremftalem7 24850* Lemma for fta 24851. Shift the minimum away from zero by a change of variables. (Contributed by Mario Carneiro, 14-Sep-2014.)
𝐴 = (coeff‘𝐹)    &   𝑁 = (deg‘𝐹)    &   (𝜑𝐹 ∈ (Poly‘𝑆))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑋 ∈ ℂ)    &   (𝜑 → (𝐹𝑋) ≠ 0)       (𝜑 → ¬ ∀𝑥 ∈ ℂ (abs‘(𝐹𝑋)) ≤ (abs‘(𝐹𝑥)))

Theoremfta 24851* The Fundamental Theorem of Algebra. Any polynomial with positive degree (i.e. non-constant) has a root. This is Metamath 100 proof #2. (Contributed by Mario Carneiro, 15-Sep-2014.)
((𝐹 ∈ (Poly‘𝑆) ∧ (deg‘𝐹) ∈ ℕ) → ∃𝑧 ∈ ℂ (𝐹𝑧) = 0)

14.4.3  The Basel problem (ζ(2) = π2/6)

Theorembasellem1 24852 Lemma for basel 24861. Closure of the sequence of roots. (Contributed by Mario Carneiro, 30-Jul-2014.) Replace OLD theorem. (Revised ba Wolf Lammen, 18-Sep-2020.)
𝑁 = ((2 · 𝑀) + 1)       ((𝑀 ∈ ℕ ∧ 𝐾 ∈ (1...𝑀)) → ((𝐾 · π) / 𝑁) ∈ (0(,)(π / 2)))

Theorembasellem2 24853* Lemma for basel 24861. Show that 𝑃 is a polynomial of degree 𝑀, and compute its coefficient function. (Contributed by Mario Carneiro, 30-Jul-2014.)
𝑁 = ((2 · 𝑀) + 1)    &   𝑃 = (𝑡 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑀)(((𝑁C(2 · 𝑗)) · (-1↑(𝑀𝑗))) · (𝑡𝑗)))       (𝑀 ∈ ℕ → (𝑃 ∈ (Poly‘ℂ) ∧ (deg‘𝑃) = 𝑀 ∧ (coeff‘𝑃) = (𝑛 ∈ ℕ0 ↦ ((𝑁C(2 · 𝑛)) · (-1↑(𝑀𝑛))))))

Theorembasellem3 24854* Lemma for basel 24861. Using the binomial theorem and de Moivre's formula, we have the identity e↑i𝑁𝑥 / (sin𝑥)↑𝑛 = Σ𝑚 ∈ (0...𝑁)(𝑁C𝑚)(i↑𝑚)(cot𝑥)↑(𝑁𝑚), so taking imaginary parts yields sin(𝑁𝑥) / (sin𝑥)↑𝑁 = Σ𝑗 ∈ (0...𝑀)(𝑁C2𝑗)(-1)↑(𝑀𝑗) (cot𝑥)↑(-2𝑗) = 𝑃((cot𝑥)↑2), where 𝑁 = 2𝑀 + 1. (Contributed by Mario Carneiro, 30-Jul-2014.)
𝑁 = ((2 · 𝑀) + 1)    &   𝑃 = (𝑡 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑀)(((𝑁C(2 · 𝑗)) · (-1↑(𝑀𝑗))) · (𝑡𝑗)))       ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (𝑃‘((tan‘𝐴)↑-2)) = ((sin‘(𝑁 · 𝐴)) / ((sin‘𝐴)↑𝑁)))

Theorembasellem4 24855* Lemma for basel 24861. By basellem3 24854, the expression 𝑃((cot𝑥)↑2) = sin(𝑁𝑥) / (sin𝑥)↑𝑁 goes to zero whenever 𝑥 = 𝑛π / 𝑁 for some 𝑛 ∈ (1...𝑀), so this function enumerates 𝑀 distinct roots of a degree- 𝑀 polynomial, which must therefore be all the roots by fta1 24108. (Contributed by Mario Carneiro, 28-Jul-2014.)
𝑁 = ((2 · 𝑀) + 1)    &   𝑃 = (𝑡 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑀)(((𝑁C(2 · 𝑗)) · (-1↑(𝑀𝑗))) · (𝑡𝑗)))    &   𝑇 = (𝑛 ∈ (1...𝑀) ↦ ((tan‘((𝑛 · π) / 𝑁))↑-2))       (𝑀 ∈ ℕ → 𝑇:(1...𝑀)–1-1-onto→(𝑃 “ {0}))

Theorembasellem5 24856* Lemma for basel 24861. Using vieta1 24112, we can calculate the sum of the roots of 𝑃 as the quotient of the top two coefficients, and since the function 𝑇 enumerates the roots, we are left with an equation that sums the cot↑2 function at the 𝑀 different roots. (Contributed by Mario Carneiro, 29-Jul-2014.)
𝑁 = ((2 · 𝑀) + 1)    &   𝑃 = (𝑡 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑀)(((𝑁C(2 · 𝑗)) · (-1↑(𝑀𝑗))) · (𝑡𝑗)))    &   𝑇 = (𝑛 ∈ (1...𝑀) ↦ ((tan‘((𝑛 · π) / 𝑁))↑-2))       (𝑀 ∈ ℕ → Σ𝑘 ∈ (1...𝑀)((tan‘((𝑘 · π) / 𝑁))↑-2) = (((2 · 𝑀) · ((2 · 𝑀) − 1)) / 6))

Theorembasellem6 24857 Lemma for basel 24861. The function 𝐺 goes to zero because it is bounded by 1 / 𝑛. (Contributed by Mario Carneiro, 28-Jul-2014.)
𝐺 = (𝑛 ∈ ℕ ↦ (1 / ((2 · 𝑛) + 1)))       𝐺 ⇝ 0

Theorembasellem7 24858 Lemma for basel 24861. The function 1 + 𝐴 · 𝐺 for any fixed 𝐴 goes to 1. (Contributed by Mario Carneiro, 28-Jul-2014.)
𝐺 = (𝑛 ∈ ℕ ↦ (1 / ((2 · 𝑛) + 1)))    &   𝐴 ∈ ℂ       ((ℕ × {1}) ∘𝑓 + ((ℕ × {𝐴}) ∘𝑓 · 𝐺)) ⇝ 1

Theorembasellem8 24859* Lemma for basel 24861. The function 𝐹 of partial sums of the inverse squares is bounded below by 𝐽 and above by 𝐾, obtained by summing the inequality cot↑2𝑥 ≤ 1 / 𝑥↑2 ≤ csc↑2𝑥 = cot↑2𝑥 + 1 over the 𝑀 roots of the polynomial 𝑃, and applying the identity basellem5 24856. (Contributed by Mario Carneiro, 29-Jul-2014.)
𝐺 = (𝑛 ∈ ℕ ↦ (1 / ((2 · 𝑛) + 1)))    &   𝐹 = seq1( + , (𝑛 ∈ ℕ ↦ (𝑛↑-2)))    &   𝐻 = ((ℕ × {((π↑2) / 6)}) ∘𝑓 · ((ℕ × {1}) ∘𝑓𝐺))    &   𝐽 = (𝐻𝑓 · ((ℕ × {1}) ∘𝑓 + ((ℕ × {-2}) ∘𝑓 · 𝐺)))    &   𝐾 = (𝐻𝑓 · ((ℕ × {1}) ∘𝑓 + 𝐺))    &   𝑁 = ((2 · 𝑀) + 1)       (𝑀 ∈ ℕ → ((𝐽𝑀) ≤ (𝐹𝑀) ∧ (𝐹𝑀) ≤ (𝐾𝑀)))

Theorembasellem9 24860* Lemma for basel 24861. Since by basellem8 24859 𝐹 is bounded by two expressions that tend to π↑2 / 6, 𝐹 must also go to π↑2 / 6 by the squeeze theorem climsqz 14415. But the series 𝐹 is exactly the partial sums of 𝑘↑-2, so it follows that this is also the value of the infinite sum Σ𝑘 ∈ ℕ(𝑘↑-2). (Contributed by Mario Carneiro, 28-Jul-2014.)
𝐺 = (𝑛 ∈ ℕ ↦ (1 / ((2 · 𝑛) + 1)))    &   𝐹 = seq1( + , (𝑛 ∈ ℕ ↦ (𝑛↑-2)))    &   𝐻 = ((ℕ × {((π↑2) / 6)}) ∘𝑓 · ((ℕ × {1}) ∘𝑓𝐺))    &   𝐽 = (𝐻𝑓 · ((ℕ × {1}) ∘𝑓 + ((ℕ × {-2}) ∘𝑓 · 𝐺)))    &   𝐾 = (𝐻𝑓 · ((ℕ × {1}) ∘𝑓 + 𝐺))       Σ𝑘 ∈ ℕ (𝑘↑-2) = ((π↑2) / 6)

Theorembasel 24861 The sum of the inverse squares is π↑2 / 6. This is commonly known as the Basel problem, with the first known proof attributed to Euler. See http://en.wikipedia.org/wiki/Basel_problem. This particular proof approach is due to Cauchy (1821). This is Metamath 100 proof #14. (Contributed by Mario Carneiro, 30-Jul-2014.)
Σ𝑘 ∈ ℕ (𝑘↑-2) = ((π↑2) / 6)

14.4.4  Number-theoretical functions

Syntaxccht 24862 Extend class notation with the first Chebyshev function.
class θ

Syntaxcvma 24863 Extend class notation with the von Mangoldt function.
class Λ

Syntaxcchp 24864 Extend class notation with the second Chebyshev function.
class ψ

Syntaxcppi 24865 Extend class notation with the prime-counting function pi.
class π

Syntaxcmu 24866 Extend class notation with the Möbius function.
class μ

Syntaxcsgm 24867 Extend class notation with the divisor function.
class σ

Definitiondf-cht 24868* 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 df-chp 24870. 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))

Definitiondf-vma 24869* Define the von Mangoldt function, which gives the logarithm of the prime at a prime power, and is zero elsewhere, see definition in [ApostolNT] p. 32. (Contributed by Mario Carneiro, 7-Apr-2016.)
Λ = (𝑥 ∈ ℕ ↦ {𝑝 ∈ ℙ ∣ 𝑝𝑥} / 𝑠if((#‘𝑠) = 1, (log‘ 𝑠), 0))

Definitiondf-chp 24870* Define the second Chebyshev function, which adds up the logarithms of the primes corresponding to the prime powers less than 𝑥, see definition in [ApostolNT] p. 75. (Contributed by Mario Carneiro, 7-Apr-2016.)
ψ = (𝑥 ∈ ℝ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))(Λ‘𝑛))

Definitiondf-ppi 24871 Define the prime π function, which counts the number of primes less than or equal to 𝑥, see definition in [ApostolNT] p. 8. (Contributed by Mario Carneiro, 15-Sep-2014.)
π = (𝑥 ∈ ℝ ↦ (#‘((0[,]𝑥) ∩ ℙ)))

Definitiondf-mu 24872* Define the Möbius function, which is zero for non-squarefree numbers and is -1 or 1 for squarefree numbers according as to the number of prime divisors of the number is even or odd, see definition in [ApostolNT] p. 24. (Contributed by Mario Carneiro, 22-Sep-2014.)
μ = (𝑥 ∈ ℕ ↦ if(∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝑥, 0, (-1↑(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑥}))))

Definitiondf-sgm 24873* 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 ∣ 𝑝𝑛} (𝑘𝑐𝑥))

Theoremefnnfsumcl 24874* Finite sum closure in the log-integers. (Contributed by Mario Carneiro, 7-Apr-2016.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℝ)    &   ((𝜑𝑘𝐴) → (exp‘𝐵) ∈ ℕ)       (𝜑 → (exp‘Σ𝑘𝐴 𝐵) ∈ ℕ)

Theoremppisval 24875 The set of primes less than 𝐴 expressed using a finite set of integers. (Contributed by Mario Carneiro, 22-Sep-2014.)
(𝐴 ∈ ℝ → ((0[,]𝐴) ∩ ℙ) = ((2...(⌊‘𝐴)) ∩ ℙ))

Theoremppisval2 24876 The set of primes less than 𝐴 expressed using a finite set of integers. (Contributed by Mario Carneiro, 22-Sep-2014.)
((𝐴 ∈ ℝ ∧ 2 ∈ (ℤ𝑀)) → ((0[,]𝐴) ∩ ℙ) = ((𝑀...(⌊‘𝐴)) ∩ ℙ))

Theoremppifi 24877 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 24878* 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 24879 Domain and range of the Chebyshev function. (Contributed by Mario Carneiro, 15-Sep-2014.)
θ:ℝ⟶ℝ

Theoremchtcl 24880 Real closure of the Chebyshev function. (Contributed by Mario Carneiro, 15-Sep-2014.)
(𝐴 ∈ ℝ → (θ‘𝐴) ∈ ℝ)

Theoremchtval 24881* Value of the Chebyshev function. (Contributed by Mario Carneiro, 15-Sep-2014.)
(𝐴 ∈ ℝ → (θ‘𝐴) = Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)(log‘𝑝))

Theoremefchtcl 24882 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 24883 The Chebyshev function is always positive. (Contributed by Mario Carneiro, 15-Sep-2014.)
(𝐴 ∈ ℝ → 0 ≤ (θ‘𝐴))

Theoremvmaval 24884* 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 24885* 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 24886* 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 24887 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 24888 Value of the von Mangoldt function at a prime. (Contributed by Mario Carneiro, 7-Apr-2016.)
(𝑃 ∈ ℙ → (Λ‘𝑃) = (log‘𝑃))

Theoremvmacl 24889 Closure for the von Mangoldt function. (Contributed by Mario Carneiro, 7-Apr-2016.)
(𝐴 ∈ ℕ → (Λ‘𝐴) ∈ ℝ)

Theoremvmaf 24890 Functionality of the von Mangoldt function. (Contributed by Mario Carneiro, 7-Apr-2016.)
Λ:ℕ⟶ℝ

Theoremefvmacl 24891 The von Mangoldt is closed in the log-integers. (Contributed by Mario Carneiro, 7-Apr-2016.)
(𝐴 ∈ ℕ → (exp‘(Λ‘𝐴)) ∈ ℕ)

Theoremvmage0 24892 The von Mangoldt function is nonnegative. (Contributed by Mario Carneiro, 7-Apr-2016.)
(𝐴 ∈ ℕ → 0 ≤ (Λ‘𝐴))

Theoremchpval 24893* 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 24894 Functionality of the second Chebyshev function. (Contributed by Mario Carneiro, 7-Apr-2016.)
ψ:ℝ⟶ℝ

Theoremchpcl 24895 Closure for the second Chebyshev function. (Contributed by Mario Carneiro, 7-Apr-2016.)
(𝐴 ∈ ℝ → (ψ‘𝐴) ∈ ℝ)

Theoremefchpcl 24896 The second Chebyshev function is closed in the log-integers. (Contributed by Mario Carneiro, 7-Apr-2016.)
(𝐴 ∈ ℝ → (exp‘(ψ‘𝐴)) ∈ ℕ)

Theoremchpge0 24897 The second Chebyshev function is nonnegative. (Contributed by Mario Carneiro, 7-Apr-2016.)
(𝐴 ∈ ℝ → 0 ≤ (ψ‘𝐴))

Theoremppival 24898 Value of the prime-counting function pi. (Contributed by Mario Carneiro, 15-Sep-2014.)
(𝐴 ∈ ℝ → (π𝐴) = (#‘((0[,]𝐴) ∩ ℙ)))

Theoremppival2 24899 Value of the prime-counting function pi. (Contributed by Mario Carneiro, 18-Sep-2014.)
(𝐴 ∈ ℤ → (π𝐴) = (#‘((2...𝐴) ∩ ℙ)))

Theoremppival2g 24900 Value of the prime-counting function pi. (Contributed by Mario Carneiro, 22-Sep-2014.)
((𝐴 ∈ ℤ ∧ 2 ∈ (ℤ𝑀)) → (π𝐴) = (#‘((𝑀...𝐴) ∩ ℙ)))

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