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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | dmlogdmgm 25601 | If 𝐴 is in the continuous domain of the logarithm, then it is in the domain of the Gamma function. (Contributed by Mario Carneiro, 8-Jul-2017.) |
⊢ (𝐴 ∈ (ℂ ∖ (-∞(,]0)) → 𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ))) | ||
Theorem | rpdmgm 25602 | A positive real number is in the domain of the Gamma function. (Contributed by Mario Carneiro, 9-Jul-2017.) |
⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ))) | ||
Theorem | dmgmn0 25603 | If 𝐴 is not a nonpositive integer, then 𝐴 is nonzero. (Contributed by Mario Carneiro, 3-Jul-2017.) |
⊢ (𝜑 → 𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ))) ⇒ ⊢ (𝜑 → 𝐴 ≠ 0) | ||
Theorem | dmgmaddnn0 25604 | If 𝐴 is not a nonpositive integer and 𝑁 is a nonnegative integer, then 𝐴 + 𝑁 is also not a nonpositive integer. (Contributed by Mario Carneiro, 6-Jul-2017.) |
⊢ (𝜑 → 𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ))) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) ⇒ ⊢ (𝜑 → (𝐴 + 𝑁) ∈ (ℂ ∖ (ℤ ∖ ℕ))) | ||
Theorem | dmgmdivn0 25605 | Lemma for lgamf 25619. (Contributed by Mario Carneiro, 3-Jul-2017.) |
⊢ (𝜑 → 𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ))) & ⊢ (𝜑 → 𝑀 ∈ ℕ) ⇒ ⊢ (𝜑 → ((𝐴 / 𝑀) + 1) ≠ 0) | ||
Theorem | lgamgulmlem1 25606* | Lemma for lgamgulm 25612. (Contributed by Mario Carneiro, 3-Jul-2017.) |
⊢ (𝜑 → 𝑅 ∈ ℕ) & ⊢ 𝑈 = {𝑥 ∈ ℂ ∣ ((abs‘𝑥) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)))} ⇒ ⊢ (𝜑 → 𝑈 ⊆ (ℂ ∖ (ℤ ∖ ℕ))) | ||
Theorem | lgamgulmlem2 25607* | Lemma for lgamgulm 25612. (Contributed by Mario Carneiro, 3-Jul-2017.) |
⊢ (𝜑 → 𝑅 ∈ ℕ) & ⊢ 𝑈 = {𝑥 ∈ ℂ ∣ ((abs‘𝑥) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)))} & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐴 ∈ 𝑈) & ⊢ (𝜑 → (2 · 𝑅) ≤ 𝑁) ⇒ ⊢ (𝜑 → (abs‘((𝐴 / 𝑁) − (log‘((𝐴 / 𝑁) + 1)))) ≤ (𝑅 · ((1 / (𝑁 − 𝑅)) − (1 / 𝑁)))) | ||
Theorem | lgamgulmlem3 25608* | Lemma for lgamgulm 25612. (Contributed by Mario Carneiro, 3-Jul-2017.) |
⊢ (𝜑 → 𝑅 ∈ ℕ) & ⊢ 𝑈 = {𝑥 ∈ ℂ ∣ ((abs‘𝑥) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)))} & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐴 ∈ 𝑈) & ⊢ (𝜑 → (2 · 𝑅) ≤ 𝑁) ⇒ ⊢ (𝜑 → (abs‘((𝐴 · (log‘((𝑁 + 1) / 𝑁))) − (log‘((𝐴 / 𝑁) + 1)))) ≤ (𝑅 · ((2 · (𝑅 + 1)) / (𝑁↑2)))) | ||
Theorem | lgamgulmlem4 25609* | Lemma for lgamgulm 25612. (Contributed by Mario Carneiro, 3-Jul-2017.) |
⊢ (𝜑 → 𝑅 ∈ ℕ) & ⊢ 𝑈 = {𝑥 ∈ ℂ ∣ ((abs‘𝑥) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)))} & ⊢ 𝐺 = (𝑚 ∈ ℕ ↦ (𝑧 ∈ 𝑈 ↦ ((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 ⇝ ) | ||
Theorem | lgamgulmlem5 25610* | Lemma for lgamgulm 25612. (Contributed by Mario Carneiro, 3-Jul-2017.) |
⊢ (𝜑 → 𝑅 ∈ ℕ) & ⊢ 𝑈 = {𝑥 ∈ ℂ ∣ ((abs‘𝑥) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)))} & ⊢ 𝐺 = (𝑚 ∈ ℕ ↦ (𝑧 ∈ 𝑈 ↦ ((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 ∧ 𝑦 ∈ 𝑈)) → (abs‘((𝐺‘𝑛)‘𝑦)) ≤ (𝑇‘𝑛)) | ||
Theorem | lgamgulmlem6 25611* | 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.) |
⊢ (𝜑 → 𝑅 ∈ ℕ) & ⊢ 𝑈 = {𝑥 ∈ ℂ ∣ ((abs‘𝑥) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)))} & ⊢ 𝐺 = (𝑚 ∈ ℕ ↦ (𝑧 ∈ 𝑈 ↦ ((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 (⇝𝑢‘𝑈) ∧ (seq1( ∘f + , 𝐺)(⇝𝑢‘𝑈)(𝑧 ∈ 𝑈 ↦ 𝑂) → ∃𝑟 ∈ ℝ ∀𝑧 ∈ 𝑈 (abs‘𝑂) ≤ 𝑟))) | ||
Theorem | lgamgulm 25612* | 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.) |
⊢ (𝜑 → 𝑅 ∈ ℕ) & ⊢ 𝑈 = {𝑥 ∈ ℂ ∣ ((abs‘𝑥) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)))} & ⊢ 𝐺 = (𝑚 ∈ ℕ ↦ (𝑧 ∈ 𝑈 ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))))) ⇒ ⊢ (𝜑 → seq1( ∘f + , 𝐺) ∈ dom (⇝𝑢‘𝑈)) | ||
Theorem | lgamgulm2 25613* | Rewrite the limit of the sequence 𝐺 in terms of the log-Gamma function. (Contributed by Mario Carneiro, 6-Jul-2017.) |
⊢ (𝜑 → 𝑅 ∈ ℕ) & ⊢ 𝑈 = {𝑥 ∈ ℂ ∣ ((abs‘𝑥) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)))} & ⊢ 𝐺 = (𝑚 ∈ ℕ ↦ (𝑧 ∈ 𝑈 ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))))) ⇒ ⊢ (𝜑 → (∀𝑧 ∈ 𝑈 (log Γ‘𝑧) ∈ ℂ ∧ seq1( ∘f + , 𝐺)(⇝𝑢‘𝑈)(𝑧 ∈ 𝑈 ↦ ((log Γ‘𝑧) + (log‘𝑧))))) | ||
Theorem | lgambdd 25614* | The log-Gamma function is bounded on the region 𝑈. (Contributed by Mario Carneiro, 9-Jul-2017.) |
⊢ (𝜑 → 𝑅 ∈ ℕ) & ⊢ 𝑈 = {𝑥 ∈ ℂ ∣ ((abs‘𝑥) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)))} & ⊢ 𝐺 = (𝑚 ∈ ℕ ↦ (𝑧 ∈ 𝑈 ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))))) ⇒ ⊢ (𝜑 → ∃𝑟 ∈ ℝ ∀𝑧 ∈ 𝑈 (abs‘(log Γ‘𝑧)) ≤ 𝑟) | ||
Theorem | lgamucov 25615* | The 𝑈 regions used in the proof of lgamgulm 25612 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)))} & ⊢ (𝜑 → 𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ))) & ⊢ 𝐽 = (TopOpen‘ℂfld) ⇒ ⊢ (𝜑 → ∃𝑟 ∈ ℕ 𝐴 ∈ ((int‘𝐽)‘𝑈)) | ||
Theorem | lgamucov2 25616* | The 𝑈 regions used in the proof of lgamgulm 25612 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)))} & ⊢ (𝜑 → 𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ))) ⇒ ⊢ (𝜑 → ∃𝑟 ∈ ℕ 𝐴 ∈ 𝑈) | ||
Theorem | lgamcvglem 25617* | Lemma for lgamf 25619 and lgamcvg 25631. (Contributed by Mario Carneiro, 8-Jul-2017.) |
⊢ 𝑈 = {𝑥 ∈ ℂ ∣ ((abs‘𝑥) ≤ 𝑟 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑟) ≤ (abs‘(𝑥 + 𝑘)))} & ⊢ (𝜑 → 𝐴 ∈ (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)))) | ||
Theorem | lgamcl 25618 | 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) | ||
Theorem | lgamf 25619 | 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 | ||
Theorem | gamf 25620 | 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 | ||
Theorem | gamcl 25621 | 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) | ||
Theorem | eflgam 25622 | 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)) | ||
Theorem | gamne0 25623 | The Gamma function is never zero. (Contributed by Mario Carneiro, 9-Jul-2017.) |
⊢ (𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)) → (Γ‘𝐴) ≠ 0) | ||
Theorem | igamval 25624 | 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)))) | ||
Theorem | igamz 25625 | Value of the inverse Gamma function on nonpositive integers. (Contributed by Mario Carneiro, 16-Jul-2017.) |
⊢ (𝐴 ∈ (ℤ ∖ ℕ) → (1/Γ‘𝐴) = 0) | ||
Theorem | igamgam 25626 | 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))) | ||
Theorem | igamlgam 25627 | 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))) | ||
Theorem | igamf 25628 | Closure of the inverse Gamma function. (Contributed by Mario Carneiro, 16-Jul-2017.) |
⊢ 1/Γ:ℂ⟶ℂ | ||
Theorem | igamcl 25629 | Closure of the inverse Gamma function. (Contributed by Mario Carneiro, 16-Jul-2017.) |
⊢ (𝐴 ∈ ℂ → (1/Γ‘𝐴) ∈ ℂ) | ||
Theorem | gamigam 25630 | 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))) | ||
Theorem | lgamcvg 25631* | 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)))) & ⊢ (𝜑 → 𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ))) ⇒ ⊢ (𝜑 → seq1( + , 𝐺) ⇝ ((log Γ‘𝐴) + (log‘𝐴))) | ||
Theorem | lgamcvg2 25632* | 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)))) & ⊢ (𝜑 → 𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ))) ⇒ ⊢ (𝜑 → seq1( + , 𝐺) ⇝ (log Γ‘(𝐴 + 1))) | ||
Theorem | gamcvg 25633* | 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)))) & ⊢ (𝜑 → 𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ))) ⇒ ⊢ (𝜑 → (exp ∘ seq1( + , 𝐺)) ⇝ ((Γ‘𝐴) · 𝐴)) | ||
Theorem | lgamp1 25634 | 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))) | ||
Theorem | gamp1 25635 | The functional equation of the Gamma function. (Contributed by Mario Carneiro, 9-Jul-2017.) |
⊢ (𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)) → (Γ‘(𝐴 + 1)) = ((Γ‘𝐴) · 𝐴)) | ||
Theorem | gamcvg2lem 25636* | Lemma for gamcvg2 25637. (Contributed by Mario Carneiro, 10-Jul-2017.) |
⊢ 𝐹 = (𝑚 ∈ ℕ ↦ ((((𝑚 + 1) / 𝑚)↑𝑐𝐴) / ((𝐴 / 𝑚) + 1))) & ⊢ (𝜑 → 𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ))) & ⊢ 𝐺 = (𝑚 ∈ ℕ ↦ ((𝐴 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝐴 / 𝑚) + 1)))) ⇒ ⊢ (𝜑 → (exp ∘ seq1( + , 𝐺)) = seq1( · , 𝐹)) | ||
Theorem | gamcvg2 25637* | 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))) & ⊢ (𝜑 → 𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ))) ⇒ ⊢ (𝜑 → seq1( · , 𝐹) ⇝ ((Γ‘𝐴) · 𝐴)) | ||
Theorem | regamcl 25638 | The Gamma function is real for real input. (Contributed by Mario Carneiro, 9-Jul-2017.) |
⊢ (𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) → (Γ‘𝐴) ∈ ℝ) | ||
Theorem | relgamcl 25639 | The log-Gamma function is real for positive real input. (Contributed by Mario Carneiro, 9-Jul-2017.) |
⊢ (𝐴 ∈ ℝ+ → (log Γ‘𝐴) ∈ ℝ) | ||
Theorem | rpgamcl 25640 | The log-Gamma function is positive real for positive real input. (Contributed by Mario Carneiro, 9-Jul-2017.) |
⊢ (𝐴 ∈ ℝ+ → (Γ‘𝐴) ∈ ℝ+) | ||
Theorem | lgam1 25641 | The log-Gamma function at one. (Contributed by Mario Carneiro, 9-Jul-2017.) |
⊢ (log Γ‘1) = 0 | ||
Theorem | gam1 25642 | The log-Gamma function at one. (Contributed by Mario Carneiro, 9-Jul-2017.) |
⊢ (Γ‘1) = 1 | ||
Theorem | facgam 25643 | The Gamma function generalizes the factorial. (Contributed by Mario Carneiro, 9-Jul-2017.) |
⊢ (𝑁 ∈ ℕ0 → (!‘𝑁) = (Γ‘(𝑁 + 1))) | ||
Theorem | gamfac 25644 | The Gamma function generalizes the factorial. (Contributed by Mario Carneiro, 9-Jul-2017.) |
⊢ (𝑁 ∈ ℕ → (Γ‘𝑁) = (!‘(𝑁 − 1))) | ||
Theorem | wilthlem1 25645 | 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 16123, (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)))) | ||
Theorem | wilthlem2 25646* |
Lemma for wilth 25648: 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 𝑧 ≠ 𝑃 − 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 25645 gives that 𝑧 = 1 or 𝑃 − 1, and we've already excluded the second case, so the product gives 1; or 𝑧 ≠ 𝑧↑-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 ∧ ∀𝑦 ∈ 𝑥 ((𝑦↑(𝑃 − 2)) mod 𝑃) ∈ 𝑥)} & ⊢ (𝜑 → 𝑃 ∈ ℙ) & ⊢ (𝜑 → 𝑆 ∈ 𝐴) & ⊢ (𝜑 → ∀𝑠 ∈ 𝐴 (𝑠 ⊊ 𝑆 → ((𝑇 Σg ( I ↾ 𝑠)) mod 𝑃) = (-1 mod 𝑃))) ⇒ ⊢ (𝜑 → ((𝑇 Σg ( I ↾ 𝑆)) mod 𝑃) = (-1 mod 𝑃)) | ||
Theorem | wilthlem3 25647* | Lemma for wilth 25648. Here we round out the argument of wilthlem2 25646 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 ∧ ∀𝑦 ∈ 𝑥 ((𝑦↑(𝑃 − 2)) mod 𝑃) ∈ 𝑥)} ⇒ ⊢ (𝑃 ∈ ℙ → 𝑃 ∥ ((!‘(𝑃 − 1)) + 1)) | ||
Theorem | wilth 25648 | Wilson's theorem. A number is prime iff it is greater than 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 25647 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))) | ||
Theorem | wilthimp 25649 | The forward implication of Wilson's theorem wilth 25648 (see wilthlem3 25647), 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)) | ||
Theorem | ftalem1 25650* | Lemma for fta 25657: "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) & ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆)) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐸 ∈ ℝ+) & ⊢ 𝑇 = (Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴‘𝑘)) / 𝐸) ⇒ ⊢ (𝜑 → ∃𝑟 ∈ ℝ ∀𝑥 ∈ ℂ (𝑟 < (abs‘𝑥) → (abs‘((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁)))) < (𝐸 · ((abs‘𝑥)↑𝑁)))) | ||
Theorem | ftalem2 25651* | Lemma for fta 25657. 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) & ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆)) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ 𝑈 = if(if(1 ≤ 𝑠, 𝑠, 1) ≤ 𝑇, 𝑇, if(1 ≤ 𝑠, 𝑠, 1)) & ⊢ 𝑇 = ((abs‘(𝐹‘0)) / ((abs‘(𝐴‘𝑁)) / 2)) ⇒ ⊢ (𝜑 → ∃𝑟 ∈ ℝ+ ∀𝑥 ∈ ℂ (𝑟 < (abs‘𝑥) → (abs‘(𝐹‘0)) < (abs‘(𝐹‘𝑥)))) | ||
Theorem | ftalem3 25652* | Lemma for fta 25657. 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 25650; 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) & ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆)) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ 𝐷 = {𝑦 ∈ ℂ ∣ (abs‘𝑦) ≤ 𝑅} & ⊢ 𝐽 = (TopOpen‘ℂfld) & ⊢ (𝜑 → 𝑅 ∈ ℝ+) & ⊢ (𝜑 → ∀𝑥 ∈ ℂ (𝑅 < (abs‘𝑥) → (abs‘(𝐹‘0)) < (abs‘(𝐹‘𝑥)))) ⇒ ⊢ (𝜑 → ∃𝑧 ∈ ℂ ∀𝑥 ∈ ℂ (abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑥))) | ||
Theorem | ftalem4 25653* | Lemma for fta 25657: Closure of the auxiliary variables for ftalem5 25654. (Contributed by Mario Carneiro, 20-Sep-2014.) (Revised by AV, 28-Sep-2020.) |
⊢ 𝐴 = (coeff‘𝐹) & ⊢ 𝑁 = (deg‘𝐹) & ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆)) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → (𝐹‘0) ≠ 0) & ⊢ 𝐾 = inf({𝑛 ∈ ℕ ∣ (𝐴‘𝑛) ≠ 0}, ℝ, < ) & ⊢ 𝑇 = (-((𝐹‘0) / (𝐴‘𝐾))↑𝑐(1 / 𝐾)) & ⊢ 𝑈 = ((abs‘(𝐹‘0)) / (Σ𝑘 ∈ ((𝐾 + 1)...𝑁)(abs‘((𝐴‘𝑘) · (𝑇↑𝑘))) + 1)) & ⊢ 𝑋 = if(1 ≤ 𝑈, 1, 𝑈) ⇒ ⊢ (𝜑 → ((𝐾 ∈ ℕ ∧ (𝐴‘𝐾) ≠ 0) ∧ (𝑇 ∈ ℂ ∧ 𝑈 ∈ ℝ+ ∧ 𝑋 ∈ ℝ+))) | ||
Theorem | ftalem5 25654* | Lemma for fta 25657: Main proof. We have already shifted the minimum found in ftalem3 25652 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) · 𝑋↑𝐾 for the 𝐾-th term, and all higher terms are bounded because 𝑋 is small. Thus, abs(𝐹(𝑇𝑋)) ≤ abs(𝐹(0))(1 − 𝑋↑𝐾) < 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) & ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆)) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → (𝐹‘0) ≠ 0) & ⊢ 𝐾 = inf({𝑛 ∈ ℕ ∣ (𝐴‘𝑛) ≠ 0}, ℝ, < ) & ⊢ 𝑇 = (-((𝐹‘0) / (𝐴‘𝐾))↑𝑐(1 / 𝐾)) & ⊢ 𝑈 = ((abs‘(𝐹‘0)) / (Σ𝑘 ∈ ((𝐾 + 1)...𝑁)(abs‘((𝐴‘𝑘) · (𝑇↑𝑘))) + 1)) & ⊢ 𝑋 = if(1 ≤ 𝑈, 1, 𝑈) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ ℂ (abs‘(𝐹‘𝑥)) < (abs‘(𝐹‘0))) | ||
Theorem | ftalem6 25655* | Lemma for fta 25657: Discharge the auxiliary variables in ftalem5 25654. (Contributed by Mario Carneiro, 20-Sep-2014.) (Proof shortened by AV, 28-Sep-2020.) |
⊢ 𝐴 = (coeff‘𝐹) & ⊢ 𝑁 = (deg‘𝐹) & ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆)) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → (𝐹‘0) ≠ 0) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ ℂ (abs‘(𝐹‘𝑥)) < (abs‘(𝐹‘0))) | ||
Theorem | ftalem7 25656* | Lemma for fta 25657. Shift the minimum away from zero by a change of variables. (Contributed by Mario Carneiro, 14-Sep-2014.) |
⊢ 𝐴 = (coeff‘𝐹) & ⊢ 𝑁 = (deg‘𝐹) & ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆)) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑋 ∈ ℂ) & ⊢ (𝜑 → (𝐹‘𝑋) ≠ 0) ⇒ ⊢ (𝜑 → ¬ ∀𝑥 ∈ ℂ (abs‘(𝐹‘𝑋)) ≤ (abs‘(𝐹‘𝑥))) | ||
Theorem | fta 25657* | 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 (𝐹‘𝑧) = 0) | ||
Theorem | basellem1 25658 | Lemma for basel 25667. 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))) | ||
Theorem | basellem2 25659* | Lemma for basel 25667. 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↑(𝑀 − 𝑗))) · (𝑡↑𝑗))) ⇒ ⊢ (𝑀 ∈ ℕ → (𝑃 ∈ (Poly‘ℂ) ∧ (deg‘𝑃) = 𝑀 ∧ (coeff‘𝑃) = (𝑛 ∈ ℕ0 ↦ ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛)))))) | ||
Theorem | basellem3 25660* | Lemma for basel 25667. Using the binomial theorem and de Moivre's formula, we have the identity e↑i𝑁𝑥 / (sin𝑥)↑𝑛 = Σ𝑚 ∈ (0...𝑁)(𝑁C𝑚)(i↑𝑚)(cot𝑥)↑(𝑁 − 𝑚), so taking imaginary parts yields sin(𝑁𝑥) / (sin𝑥)↑𝑁 = Σ𝑗 ∈ (0...𝑀)(𝑁C2𝑗)(-1)↑(𝑀 − 𝑗) (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↑(𝑀 − 𝑗))) · (𝑡↑𝑗))) ⇒ ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (𝑃‘((tan‘𝐴)↑-2)) = ((sin‘(𝑁 · 𝐴)) / ((sin‘𝐴)↑𝑁))) | ||
Theorem | basellem4 25661* | Lemma for basel 25667. By basellem3 25660, 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 24897. (Contributed by Mario Carneiro, 28-Jul-2014.) |
⊢ 𝑁 = ((2 · 𝑀) + 1) & ⊢ 𝑃 = (𝑡 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑀)(((𝑁C(2 · 𝑗)) · (-1↑(𝑀 − 𝑗))) · (𝑡↑𝑗))) & ⊢ 𝑇 = (𝑛 ∈ (1...𝑀) ↦ ((tan‘((𝑛 · π) / 𝑁))↑-2)) ⇒ ⊢ (𝑀 ∈ ℕ → 𝑇:(1...𝑀)–1-1-onto→(◡𝑃 “ {0})) | ||
Theorem | basellem5 25662* | Lemma for basel 25667. Using vieta1 24901, 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↑(𝑀 − 𝑗))) · (𝑡↑𝑗))) & ⊢ 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)) | ||
Theorem | basellem6 25663 | Lemma for basel 25667. 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 | ||
Theorem | basellem7 25664 | Lemma for basel 25667. 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 | ||
Theorem | basellem8 25665* | Lemma for basel 25667. 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 25662. (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}) ∘f − 𝐺)) & ⊢ 𝐽 = (𝐻 ∘f · ((ℕ × {1}) ∘f + ((ℕ × {-2}) ∘f · 𝐺))) & ⊢ 𝐾 = (𝐻 ∘f · ((ℕ × {1}) ∘f + 𝐺)) & ⊢ 𝑁 = ((2 · 𝑀) + 1) ⇒ ⊢ (𝑀 ∈ ℕ → ((𝐽‘𝑀) ≤ (𝐹‘𝑀) ∧ (𝐹‘𝑀) ≤ (𝐾‘𝑀))) | ||
Theorem | basellem9 25666* | Lemma for basel 25667. Since by basellem8 25665 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 14997. 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}) ∘f − 𝐺)) & ⊢ 𝐽 = (𝐻 ∘f · ((ℕ × {1}) ∘f + ((ℕ × {-2}) ∘f · 𝐺))) & ⊢ 𝐾 = (𝐻 ∘f · ((ℕ × {1}) ∘f + 𝐺)) ⇒ ⊢ Σ𝑘 ∈ ℕ (𝑘↑-2) = ((π↑2) / 6) | ||
Theorem | basel 25667 | 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) | ||
Syntax | ccht 25668 | Extend class notation with the first Chebyshev function. |
class θ | ||
Syntax | cvma 25669 | Extend class notation with the von Mangoldt function. |
class Λ | ||
Syntax | cchp 25670 | Extend class notation with the second Chebyshev function. |
class ψ | ||
Syntax | cppi 25671 | Extend class notation with the prime-counting function pi. |
class π | ||
Syntax | cmu 25672 | Extend class notation with the Möbius function. |
class μ | ||
Syntax | csgm 25673 | Extend class notation with the divisor function. |
class σ | ||
Definition | df-cht 25674* | 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 25676. See https://en.wikipedia.org/wiki/Chebyshev_function 25676 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-vma 25675* | 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 ∣ 𝑝 ∥ 𝑥} / 𝑠⦌if((♯‘𝑠) = 1, (log‘∪ 𝑠), 0)) | ||
Definition | df-chp 25676* | 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)) | ||
Definition | df-ppi 25677 | 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))) | ||
Definition | df-mu 25678* | 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 ∣ 𝑝 ∥ 𝑥})))) | ||
Definition | df-sgm 25679* | 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 25680* | 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 | ppisval 25681 | 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)) | ||
Theorem | ppisval2 25682 | The set of primes less than 𝐴 expressed using a finite set of integers. (Contributed by Mario Carneiro, 22-Sep-2014.) |
⊢ ((𝐴 ∈ ℝ ∧ 2 ∈ (ℤ≥‘𝑀)) → ((0[,]𝐴) ∩ ℙ) = ((𝑀...(⌊‘𝐴)) ∩ ℙ)) | ||
Theorem | ppifi 25683 | The set of primes less than 𝐴 is a finite set. (Contributed by Mario Carneiro, 15-Sep-2014.) |
⊢ (𝐴 ∈ ℝ → ((0[,]𝐴) ∩ ℙ) ∈ Fin) | ||
Theorem | prmdvdsfi 25684* | 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 | chtf 25685 | Domain and range of the Chebyshev function. (Contributed by Mario Carneiro, 15-Sep-2014.) |
⊢ θ:ℝ⟶ℝ | ||
Theorem | chtcl 25686 | Real closure of the Chebyshev function. (Contributed by Mario Carneiro, 15-Sep-2014.) |
⊢ (𝐴 ∈ ℝ → (θ‘𝐴) ∈ ℝ) | ||
Theorem | chtval 25687* | Value of the Chebyshev function. (Contributed by Mario Carneiro, 15-Sep-2014.) |
⊢ (𝐴 ∈ ℝ → (θ‘𝐴) = Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)(log‘𝑝)) | ||
Theorem | efchtcl 25688 | 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) | ||
Theorem | chtge0 25689 | The Chebyshev function is always positive. (Contributed by Mario Carneiro, 15-Sep-2014.) |
⊢ (𝐴 ∈ ℝ → 0 ≤ (θ‘𝐴)) | ||
Theorem | vmaval 25690* | Value of the von Mangoldt function. (Contributed by Mario Carneiro, 7-Apr-2016.) |
⊢ 𝑆 = {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ⇒ ⊢ (𝐴 ∈ ℕ → (Λ‘𝐴) = if((♯‘𝑆) = 1, (log‘∪ 𝑆), 0)) | ||
Theorem | isppw 25691* | Two ways to say that 𝐴 is a prime power. (Contributed by Mario Carneiro, 7-Apr-2016.) |
⊢ (𝐴 ∈ ℕ → ((Λ‘𝐴) ≠ 0 ↔ ∃!𝑝 ∈ ℙ 𝑝 ∥ 𝐴)) | ||
Theorem | isppw2 25692* | 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 = (𝑝↑𝑘))) | ||
Theorem | vmappw 25693 | Value of the von Mangoldt function at a prime power. (Contributed by Mario Carneiro, 7-Apr-2016.) |
⊢ ((𝑃 ∈ ℙ ∧ 𝐾 ∈ ℕ) → (Λ‘(𝑃↑𝐾)) = (log‘𝑃)) | ||
Theorem | vmaprm 25694 | Value of the von Mangoldt function at a prime. (Contributed by Mario Carneiro, 7-Apr-2016.) |
⊢ (𝑃 ∈ ℙ → (Λ‘𝑃) = (log‘𝑃)) | ||
Theorem | vmacl 25695 | Closure for the von Mangoldt function. (Contributed by Mario Carneiro, 7-Apr-2016.) |
⊢ (𝐴 ∈ ℕ → (Λ‘𝐴) ∈ ℝ) | ||
Theorem | vmaf 25696 | Functionality of the von Mangoldt function. (Contributed by Mario Carneiro, 7-Apr-2016.) |
⊢ Λ:ℕ⟶ℝ | ||
Theorem | efvmacl 25697 | The von Mangoldt is closed in the log-integers. (Contributed by Mario Carneiro, 7-Apr-2016.) |
⊢ (𝐴 ∈ ℕ → (exp‘(Λ‘𝐴)) ∈ ℕ) | ||
Theorem | vmage0 25698 | The von Mangoldt function is nonnegative. (Contributed by Mario Carneiro, 7-Apr-2016.) |
⊢ (𝐴 ∈ ℕ → 0 ≤ (Λ‘𝐴)) | ||
Theorem | chpval 25699* | 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)) | ||
Theorem | chpf 25700 | Functionality of the second Chebyshev function. (Contributed by Mario Carneiro, 7-Apr-2016.) |
⊢ ψ:ℝ⟶ℝ |
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