HomeTheorem List (p. 160 of 507)
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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | hash2iun 15901* | The cardinality of a nested disjoint indexed union. (Contributed by AV, 9-Jan-2022.) |
| ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝐶 ∈ Fin) & ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 𝐶) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → Disj 𝑦 ∈ 𝐵 𝐶) ⇒ ⊢ (𝜑 → (♯‘∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 𝐶) = Σ𝑥 ∈ 𝐴 Σ𝑦 ∈ 𝐵 (♯‘𝐶)) | ||
| Theorem | hash2iun1dif1 15902* | The cardinality of a nested disjoint indexed union. (Contributed by AV, 9-Jan-2022.) |
| ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ 𝐵 = (𝐴 ∖ {𝑥}) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝐶 ∈ Fin) & ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 𝐶) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → Disj 𝑦 ∈ 𝐵 𝐶) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (♯‘𝐶) = 1) ⇒ ⊢ (𝜑 → (♯‘∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 𝐶) = ((♯‘𝐴) · ((♯‘𝐴) − 1))) | ||
| Theorem | hashrabrex 15903* | The number of elements in a class abstraction with a restricted existential quantification. (Contributed by Alexander van der Vekens, 29-Jul-2018.) |
| ⊢ (𝜑 → 𝑌 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → {𝑥 ∈ 𝑋 ∣ 𝜓} ∈ Fin) & ⊢ (𝜑 → Disj 𝑦 ∈ 𝑌 {𝑥 ∈ 𝑋 ∣ 𝜓}) ⇒ ⊢ (𝜑 → (♯‘{𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑌 𝜓}) = Σ𝑦 ∈ 𝑌 (♯‘{𝑥 ∈ 𝑋 ∣ 𝜓})) | ||
| Theorem | hashuni 15904* | The cardinality of a disjoint union. (Contributed by Mario Carneiro, 24-Jan-2015.) |
| ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐴 ⊆ Fin) & ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 𝑥) ⇒ ⊢ (𝜑 → (♯‘∪ 𝐴) = Σ𝑥 ∈ 𝐴 (♯‘𝑥)) | ||
| Theorem | qshash 15905* | The cardinality of a set with an equivalence relation is the sum of the cardinalities of its equivalence classes. (Contributed by Mario Carneiro, 16-Jan-2015.) |
| ⊢ (𝜑 → ∼ Er 𝐴) & ⊢ (𝜑 → 𝐴 ∈ Fin) ⇒ ⊢ (𝜑 → (♯‘𝐴) = Σ𝑥 ∈ (𝐴 / ∼ )(♯‘𝑥)) | ||
| Theorem | indsum 15906* | Finite sum of a product with the indicator function / Cartesian product with the indicator function. Note: this theorem cannot be efficiently shortened using sumss2 15803, unless there are some additional auxiliary theorems like (if(𝑥 ∈ 𝐴, 1, 0) · 𝐵) = if(𝑥 ∈ 𝐴, 𝐵, 0). (Contributed by Thierry Arnoux, 14-Aug-2017.) (Proof shortened by AV, 11-Apr-2026.) |
| ⊢ (𝜑 → 𝑂 ∈ Fin) & ⊢ (𝜑 → 𝐴 ⊆ 𝑂) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑂) → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → Σ𝑥 ∈ 𝑂 ((((𝟭‘𝑂)‘𝐴)‘𝑥) · 𝐵) = Σ𝑥 ∈ 𝐴 𝐵) | ||
| Theorem | indsumhash 15907* | The finite sum of the indicator function is the number of elements of the corresponding subset. (Contributed by AV, 10-Apr-2026.) |
| ⊢ 1 = ((𝟭‘𝑂)‘𝐴) ⇒ ⊢ ((𝑂 ∈ Fin ∧ 𝐴 ⊆ 𝑂) → Σ𝑘 ∈ 𝑂 ( 1 ‘𝑘) = (♯‘𝐴)) | ||
| Theorem | ackbijnn 15908* | Translate the Ackermann bijection ackbij1 10239 onto the positive integers. (Contributed by Mario Carneiro, 16-Jan-2015.) |
| ⊢ 𝐹 = (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (2↑𝑦)) ⇒ ⊢ 𝐹:(𝒫 ℕ0 ∩ Fin)–1-1-onto→ℕ0 | ||
| Theorem | binomlem 15909* | Lemma for binom 15910 (binomial theorem). Inductive step. (Contributed by NM, 6-Dec-2005.) (Revised by Mario Carneiro, 24-Apr-2014.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜓 → ((𝐴 + 𝐵)↑𝑁) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘)))) ⇒ ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + 𝐵)↑(𝑁 + 1)) = Σ𝑘 ∈ (0...(𝑁 + 1))(((𝑁 + 1)C𝑘) · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘)))) | ||
| Theorem | binom 15910* | The binomial theorem: (𝐴 + 𝐵)↑𝑁 is the sum from 𝑘 = 0 to 𝑁 of (𝑁C𝑘) · ((𝐴↑𝑘) · (𝐵↑(𝑁 − 𝑘)). Theorem 15-2.8 of [Gleason] p. 296. This part of the proof sets up the induction and does the base case, with the bulk of the work (the induction step) in binomlem 15909. This is Metamath 100 proof #44. (Contributed by NM, 7-Dec-2005.) (Proof shortened by Mario Carneiro, 24-Apr-2014.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → ((𝐴 + 𝐵)↑𝑁) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘)))) | ||
| Theorem | binom1p 15911* | Special case of the binomial theorem for (1 + 𝐴)↑𝑁. (Contributed by Paul Chapman, 10-May-2007.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → ((1 + 𝐴)↑𝑁) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · (𝐴↑𝑘))) | ||
| Theorem | binom11 15912* | Special case of the binomial theorem for 2↑𝑁. (Contributed by Mario Carneiro, 13-Mar-2014.) |
| ⊢ (𝑁 ∈ ℕ0 → (2↑𝑁) = Σ𝑘 ∈ (0...𝑁)(𝑁C𝑘)) | ||
| Theorem | binom1dif 15913* | A summation for the difference between ((𝐴 + 1)↑𝑁) and (𝐴↑𝑁). (Contributed by Scott Fenton, 9-Apr-2014.) (Revised by Mario Carneiro, 22-May-2014.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (((𝐴 + 1)↑𝑁) − (𝐴↑𝑁)) = Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · (𝐴↑𝑘))) | ||
| Theorem | bcxmaslem1 15914 | Lemma for bcxmas 15915. (Contributed by Paul Chapman, 18-May-2007.) |
| ⊢ (𝐴 = 𝐵 → ((𝑁 + 𝐴)C𝐴) = ((𝑁 + 𝐵)C𝐵)) | ||
| Theorem | bcxmas 15915* | Parallel summation (Christmas Stocking) theorem for Pascal's Triangle. (Contributed by Paul Chapman, 18-May-2007.) (Revised by Mario Carneiro, 24-Apr-2014.) |
| ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0) → (((𝑁 + 1) + 𝑀)C𝑀) = Σ𝑗 ∈ (0...𝑀)((𝑁 + 𝑗)C𝑗)) | ||
| Theorem | incexclem 15916* | Lemma for incexc 15917. (Contributed by Mario Carneiro, 7-Aug-2017.) |
| ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((♯‘𝐵) − (♯‘(𝐵 ∩ ∪ 𝐴))) = Σ𝑠 ∈ 𝒫 𝐴((-1↑(♯‘𝑠)) · (♯‘(𝐵 ∩ ∩ 𝑠)))) | ||
| Theorem | incexc 15917* | The inclusion/exclusion principle for counting the elements of a finite union of finite sets. This is Metamath 100 proof #96. (Contributed by Mario Carneiro, 7-Aug-2017.) |
| ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → (♯‘∪ 𝐴) = Σ𝑠 ∈ (𝒫 𝐴 ∖ {∅})((-1↑((♯‘𝑠) − 1)) · (♯‘∩ 𝑠))) | ||
| Theorem | incexc2 15918* | The inclusion/exclusion principle for counting the elements of a finite union of finite sets. (Contributed by Mario Carneiro, 7-Aug-2017.) |
| ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → (♯‘∪ 𝐴) = Σ𝑛 ∈ (1...(♯‘𝐴))((-1↑(𝑛 − 1)) · Σ𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛} (♯‘∩ 𝑠))) | ||
| Theorem | isumshft 15919* | Index shift of an infinite sum. (Contributed by Paul Chapman, 31-Oct-2007.) (Revised by Mario Carneiro, 24-Apr-2014.) |
| ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ 𝑊 = (ℤ≥‘(𝑀 + 𝐾)) & ⊢ (𝑗 = (𝐾 + 𝑘) → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐾 ∈ ℤ) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑊) → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → Σ𝑗 ∈ 𝑊 𝐴 = Σ𝑘 ∈ 𝑍 𝐵) | ||
| Theorem | isumsplit 15920* | Split off the first 𝑁 terms of an infinite sum. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Mario Carneiro, 24-Apr-2014.) |
| ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ 𝑊 = (ℤ≥‘𝑁) & ⊢ (𝜑 → 𝑁 ∈ 𝑍) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ) & ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ ) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝑍 𝐴 = (Σ𝑘 ∈ (𝑀...(𝑁 − 1))𝐴 + Σ𝑘 ∈ 𝑊 𝐴)) | ||
| Theorem | isum1p 15921* | The infinite sum of a converging infinite series equals the first term plus the infinite sum of the rest of it. (Contributed by NM, 2-Jan-2006.) (Revised by Mario Carneiro, 24-Apr-2014.) |
| ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ) & ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ ) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝑍 𝐴 = ((𝐹‘𝑀) + Σ𝑘 ∈ (ℤ≥‘(𝑀 + 1))𝐴)) | ||
| Theorem | isumnn0nn 15922* | Sum from 0 to infinity in terms of sum from 1 to infinity. (Contributed by NM, 2-Jan-2006.) (Revised by Mario Carneiro, 24-Apr-2014.) |
| ⊢ (𝑘 = 0 → 𝐴 = 𝐵) & ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘𝑘) = 𝐴) & ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝐴 ∈ ℂ) & ⊢ (𝜑 → seq0( + , 𝐹) ∈ dom ⇝ ) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ ℕ0 𝐴 = (𝐵 + Σ𝑘 ∈ ℕ 𝐴)) | ||
| Theorem | isumrpcl 15923* | The infinite sum of positive reals is positive. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Mario Carneiro, 24-Apr-2014.) |
| ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ 𝑊 = (ℤ≥‘𝑁) & ⊢ (𝜑 → 𝑁 ∈ 𝑍) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℝ+) & ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ ) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝑊 𝐴 ∈ ℝ+) | ||
| Theorem | isumle 15924* | Comparison of two infinite sums. (Contributed by Paul Chapman, 13-Nov-2007.) (Revised by Mario Carneiro, 24-Apr-2014.) |
| ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = 𝐵) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ≤ 𝐵) & ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ ) & ⊢ (𝜑 → seq𝑀( + , 𝐺) ∈ dom ⇝ ) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝑍 𝐴 ≤ Σ𝑘 ∈ 𝑍 𝐵) | ||
| Theorem | isumless 15925* | A finite sum of nonnegative numbers is less than or equal to its limit. (Contributed by Mario Carneiro, 24-Apr-2014.) |
| ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐴 ⊆ 𝑍) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐵) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 0 ≤ 𝐵) & ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ ) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ≤ Σ𝑘 ∈ 𝑍 𝐵) | ||
| Theorem | isumsup2 15926* | An infinite sum of nonnegative terms is equal to the supremum of the partial sums. (Contributed by Mario Carneiro, 12-Jun-2014.) |
| ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ 𝐺 = seq𝑀( + , 𝐹) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 0 ≤ 𝐴) & ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 (𝐺‘𝑗) ≤ 𝑥) ⇒ ⊢ (𝜑 → 𝐺 ⇝ sup(ran 𝐺, ℝ, < )) | ||
| Theorem | isumsup 15927* | An infinite sum of nonnegative terms is equal to the supremum of the partial sums. (Contributed by Mario Carneiro, 12-Jun-2014.) |
| ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ 𝐺 = seq𝑀( + , 𝐹) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 0 ≤ 𝐴) & ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 (𝐺‘𝑗) ≤ 𝑥) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝑍 𝐴 = sup(ran 𝐺, ℝ, < )) | ||
| Theorem | isumltss 15928* | A partial sum of a series with positive terms is less than the infinite sum. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 12-Mar-2015.) |
| ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐴 ⊆ 𝑍) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐵) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℝ+) & ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ ) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 < Σ𝑘 ∈ 𝑍 𝐵) | ||
| Theorem | climcndslem1 15929* | Lemma for climcnds 15931: bound the original series by the condensed series. (Contributed by Mario Carneiro, 18-Jul-2014.) |
| ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 0 ≤ (𝐹‘𝑘)) & ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘)) & ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (𝐺‘𝑛) = ((2↑𝑛) · (𝐹‘(2↑𝑛)))) ⇒ ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0) → (seq1( + , 𝐹)‘((2↑(𝑁 + 1)) − 1)) ≤ (seq0( + , 𝐺)‘𝑁)) | ||
| Theorem | climcndslem2 15930* | Lemma for climcnds 15931: bound the condensed series by the original series. (Contributed by Mario Carneiro, 18-Jul-2014.) (Proof shortened by AV, 10-Jul-2022.) |
| ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 0 ≤ (𝐹‘𝑘)) & ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘)) & ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (𝐺‘𝑛) = ((2↑𝑛) · (𝐹‘(2↑𝑛)))) ⇒ ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → (seq1( + , 𝐺)‘𝑁) ≤ (2 · (seq1( + , 𝐹)‘(2↑𝑁)))) | ||
| Theorem | climcnds 15931* | The Cauchy condensation test. If 𝑎(𝑘) is a decreasing sequence of nonnegative terms, then Σ𝑘 ∈ ℕ𝑎(𝑘) converges iff Σ𝑛 ∈ ℕ02↑𝑛 · 𝑎(2↑𝑛) converges. (Contributed by Mario Carneiro, 18-Jul-2014.) (Proof shortened by AV, 10-Jul-2022.) |
| ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 0 ≤ (𝐹‘𝑘)) & ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘)) & ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (𝐺‘𝑛) = ((2↑𝑛) · (𝐹‘(2↑𝑛)))) ⇒ ⊢ (𝜑 → (seq1( + , 𝐹) ∈ dom ⇝ ↔ seq0( + , 𝐺) ∈ dom ⇝ )) | ||
| Theorem | divrcnv 15932* | The sequence of reciprocals of real numbers, multiplied by the factor 𝐴, converges to zero. (Contributed by Mario Carneiro, 18-Sep-2014.) |
| ⊢ (𝐴 ∈ ℂ → (𝑛 ∈ ℝ+ ↦ (𝐴 / 𝑛)) ⇝𝑟 0) | ||
| Theorem | divcnv 15933* | The sequence of reciprocals of positive integers, multiplied by the factor 𝐴, converges to zero. (Contributed by NM, 6-Feb-2008.) (Revised by Mario Carneiro, 18-Sep-2014.) |
| ⊢ (𝐴 ∈ ℂ → (𝑛 ∈ ℕ ↦ (𝐴 / 𝑛)) ⇝ 0) | ||
| Theorem | flo1 15934 | The floor function satisfies ⌊(𝑥) = 𝑥 + 𝑂(1). (Contributed by Mario Carneiro, 21-May-2016.) |
| ⊢ (𝑥 ∈ ℝ ↦ (𝑥 − (⌊‘𝑥))) ∈ 𝑂(1) | ||
| Theorem | divcnvshft 15935* | Limit of a ratio function. (Contributed by Scott Fenton, 16-Dec-2017.) |
| ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℤ) & ⊢ (𝜑 → 𝐹 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = (𝐴 / (𝑘 + 𝐵))) ⇒ ⊢ (𝜑 → 𝐹 ⇝ 0) | ||
| Theorem | supcvg 15936* | Extract a sequence 𝑓 in 𝑋 such that the image of the points in the bounded set 𝐴 converges to the supremum 𝑆 of the set. Similar to Equation 4 of [Kreyszig] p. 144. The proof uses countable choice ax-cc 10437. (Contributed by Mario Carneiro, 15-Feb-2013.) (Proof shortened by Mario Carneiro, 26-Apr-2014.) |
| ⊢ 𝑋 ∈ V & ⊢ 𝑆 = sup(𝐴, ℝ, < ) & ⊢ 𝑅 = (𝑛 ∈ ℕ ↦ (𝑆 − (1 / 𝑛))) & ⊢ (𝜑 → 𝑋 ≠ ∅) & ⊢ (𝜑 → 𝐹:𝑋–onto→𝐴) & ⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ⇒ ⊢ (𝜑 → ∃𝑓(𝑓:ℕ⟶𝑋 ∧ (𝐹 ∘ 𝑓) ⇝ 𝑆)) | ||
| Theorem | infcvgaux1i 15937* | Auxiliary theorem for applications of supcvg 15936. Hypothesis for several supremum theorems. (Contributed by NM, 8-Feb-2008.) |
| ⊢ 𝑅 = {𝑥 ∣ ∃𝑦 ∈ 𝑋 𝑥 = -𝐴} & ⊢ (𝑦 ∈ 𝑋 → 𝐴 ∈ ℝ) & ⊢ 𝑍 ∈ 𝑋 & ⊢ ∃𝑧 ∈ ℝ ∀𝑤 ∈ 𝑅 𝑤 ≤ 𝑧 ⇒ ⊢ (𝑅 ⊆ ℝ ∧ 𝑅 ≠ ∅ ∧ ∃𝑧 ∈ ℝ ∀𝑤 ∈ 𝑅 𝑤 ≤ 𝑧) | ||
| Theorem | infcvgaux2i 15938* | Auxiliary theorem for applications of supcvg 15936. (Contributed by NM, 4-Mar-2008.) |
| ⊢ 𝑅 = {𝑥 ∣ ∃𝑦 ∈ 𝑋 𝑥 = -𝐴} & ⊢ (𝑦 ∈ 𝑋 → 𝐴 ∈ ℝ) & ⊢ 𝑍 ∈ 𝑋 & ⊢ ∃𝑧 ∈ ℝ ∀𝑤 ∈ 𝑅 𝑤 ≤ 𝑧 & ⊢ 𝑆 = -sup(𝑅, ℝ, < ) & ⊢ (𝑦 = 𝐶 → 𝐴 = 𝐵) ⇒ ⊢ (𝐶 ∈ 𝑋 → 𝑆 ≤ 𝐵) | ||
| Theorem | harmonic 15939 | The harmonic series 𝐻 diverges. This fact follows from the stronger emcl 27204, which establishes that the harmonic series grows as log𝑛 + γ + o(1), but this uses a more elementary method, attributed to Nicole Oresme (1323-1382). This is Metamath 100 proof #34. (Contributed by Mario Carneiro, 11-Jul-2014.) |
| ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ (1 / 𝑛)) & ⊢ 𝐻 = seq1( + , 𝐹) ⇒ ⊢ ¬ 𝐻 ∈ dom ⇝ | ||
| Theorem | arisum 15940* | Arithmetic series sum of the first 𝑁 positive integers. This is Metamath 100 proof #68. (Contributed by FL, 16-Nov-2006.) (Proof shortened by Mario Carneiro, 22-May-2014.) |
| ⊢ (𝑁 ∈ ℕ0 → Σ𝑘 ∈ (1...𝑁)𝑘 = (((𝑁↑2) + 𝑁) / 2)) | ||
| Theorem | arisum2 15941* | Arithmetic series sum of the first 𝑁 nonnegative integers. (Contributed by Mario Carneiro, 17-Apr-2015.) (Proof shortened by AV, 2-Aug-2021.) |
| ⊢ (𝑁 ∈ ℕ0 → Σ𝑘 ∈ (0...(𝑁 − 1))𝑘 = (((𝑁↑2) − 𝑁) / 2)) | ||
| Theorem | trireciplem 15942 | Lemma for trirecip 15943. Show that the sum converges. (Contributed by Scott Fenton, 22-Apr-2014.) (Revised by Mario Carneiro, 22-May-2014.) |
| ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ (1 / (𝑛 · (𝑛 + 1)))) ⇒ ⊢ seq1( + , 𝐹) ⇝ 1 | ||
| Theorem | trirecip 15943 | The sum of the reciprocals of the triangle numbers converge to two. This is Metamath 100 proof #42. (Contributed by Scott Fenton, 23-Apr-2014.) (Revised by Mario Carneiro, 22-May-2014.) |
| ⊢ Σ𝑘 ∈ ℕ (2 / (𝑘 · (𝑘 + 1))) = 2 | ||
| Theorem | expcnv 15944* | A sequence of powers of a complex number 𝐴 with absolute value less than 1 converges to zero. (Contributed by NM, 8-May-2006.) (Proof shortened by Mario Carneiro, 26-Apr-2014.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → (abs‘𝐴) < 1) ⇒ ⊢ (𝜑 → (𝑛 ∈ ℕ0 ↦ (𝐴↑𝑛)) ⇝ 0) | ||
| Theorem | explecnv 15945* | A sequence of terms converges to zero when it is less than powers of a number 𝐴 whose absolute value is less than 1. (Contributed by NM, 19-Jul-2008.) (Revised by Mario Carneiro, 26-Apr-2014.) |
| ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹 ∈ 𝑉) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → (abs‘𝐴) < 1) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (abs‘(𝐹‘𝑘)) ≤ (𝐴↑𝑘)) ⇒ ⊢ (𝜑 → 𝐹 ⇝ 0) | ||
| Theorem | geoserg 15946* | The value of the finite geometric series 𝐴↑𝑀 + 𝐴↑(𝑀 + 1) +... + 𝐴↑(𝑁 − 1). (Contributed by Mario Carneiro, 2-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐴 ≠ 1) & ⊢ (𝜑 → 𝑀 ∈ ℕ0) & ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ (𝑀..^𝑁)(𝐴↑𝑘) = (((𝐴↑𝑀) − (𝐴↑𝑁)) / (1 − 𝐴))) | ||
| Theorem | geoser 15947* | The value of the finite geometric series 1 + 𝐴↑1 + 𝐴↑2 +... + 𝐴↑(𝑁 − 1). This is Metamath 100 proof #66. (Contributed by NM, 12-May-2006.) (Proof shortened by Mario Carneiro, 15-Jun-2014.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐴 ≠ 1) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ (0...(𝑁 − 1))(𝐴↑𝑘) = ((1 − (𝐴↑𝑁)) / (1 − 𝐴))) | ||
| Theorem | pwdif 15948* | The difference of two numbers to the same power is the difference of the two numbers multiplied with a finite sum. Generalization of subsq 14266. See Wikipedia "Fermat number", section "Other theorems about Fermat numbers", https://en.wikipedia.org/wiki/Fermat_number 14266, 5-Aug-2021. (Contributed by AV, 6-Aug-2021.) (Revised by AV, 19-Aug-2021.) |
| ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴↑𝑁) − (𝐵↑𝑁)) = ((𝐴 − 𝐵) · Σ𝑘 ∈ (0..^𝑁)((𝐴↑𝑘) · (𝐵↑((𝑁 − 𝑘) − 1))))) | ||
| Theorem | pwm1geoser 15949* | The n-th power of a number decreased by 1 expressed by the finite geometric series 1 + 𝐴↑1 + 𝐴↑2 +... + 𝐴↑(𝑁 − 1). (Contributed by AV, 14-Aug-2021.) (Proof shortened by AV, 19-Aug-2021.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) ⇒ ⊢ (𝜑 → ((𝐴↑𝑁) − 1) = ((𝐴 − 1) · Σ𝑘 ∈ (0...(𝑁 − 1))(𝐴↑𝑘))) | ||
| Theorem | geolim 15950* | The partial sums in the infinite series 1 + 𝐴↑1 + 𝐴↑2... converge to (1 / (1 − 𝐴)). (Contributed by NM, 15-May-2006.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → (abs‘𝐴) < 1) & ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘𝑘) = (𝐴↑𝑘)) ⇒ ⊢ (𝜑 → seq0( + , 𝐹) ⇝ (1 / (1 − 𝐴))) | ||
| Theorem | geolim2 15951* | The partial sums in the geometric series 𝐴↑𝑀 + 𝐴↑(𝑀 + 1)... converge to ((𝐴↑𝑀) / (1 − 𝐴)). (Contributed by NM, 6-Jun-2006.) (Revised by Mario Carneiro, 26-Apr-2014.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → (abs‘𝐴) < 1) & ⊢ (𝜑 → 𝑀 ∈ ℕ0) & ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) = (𝐴↑𝑘)) ⇒ ⊢ (𝜑 → seq𝑀( + , 𝐹) ⇝ ((𝐴↑𝑀) / (1 − 𝐴))) | ||
| Theorem | georeclim 15952* | The limit of a geometric series of reciprocals. (Contributed by Paul Chapman, 28-Dec-2007.) (Revised by Mario Carneiro, 26-Apr-2014.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 1 < (abs‘𝐴)) & ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘𝑘) = ((1 / 𝐴)↑𝑘)) ⇒ ⊢ (𝜑 → seq0( + , 𝐹) ⇝ (𝐴 / (𝐴 − 1))) | ||
| Theorem | geo2sum 15953* | The value of the finite geometric series 2↑-1 + 2↑-2 +... + 2↑-𝑁, multiplied by a constant. (Contributed by Mario Carneiro, 17-Mar-2014.) (Revised by Mario Carneiro, 26-Apr-2014.) |
| ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℂ) → Σ𝑘 ∈ (1...𝑁)(𝐴 / (2↑𝑘)) = (𝐴 − (𝐴 / (2↑𝑁)))) | ||
| Theorem | geo2sum2 15954* | The value of the finite geometric series 1 + 2 + 4 + 8 +... + 2↑(𝑁 − 1). (Contributed by Mario Carneiro, 7-Sep-2016.) |
| ⊢ (𝑁 ∈ ℕ0 → Σ𝑘 ∈ (0..^𝑁)(2↑𝑘) = ((2↑𝑁) − 1)) | ||
| Theorem | geo2lim 15955* | The value of the infinite geometric series 2↑-1 + 2↑-2 +... , multiplied by a constant. (Contributed by Mario Carneiro, 15-Jun-2014.) |
| ⊢ 𝐹 = (𝑘 ∈ ℕ ↦ (𝐴 / (2↑𝑘))) ⇒ ⊢ (𝐴 ∈ ℂ → seq1( + , 𝐹) ⇝ 𝐴) | ||
| Theorem | geomulcvg 15956* | The geometric series converges even if it is multiplied by 𝑘 to result in the larger series 𝑘 · 𝐴↑𝑘. (Contributed by Mario Carneiro, 27-Mar-2015.) |
| ⊢ 𝐹 = (𝑘 ∈ ℕ0 ↦ (𝑘 · (𝐴↑𝑘))) ⇒ ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → seq0( + , 𝐹) ∈ dom ⇝ ) | ||
| Theorem | geoisum 15957* | The infinite sum of 1 + 𝐴↑1 + 𝐴↑2... is (1 / (1 − 𝐴)). (Contributed by NM, 15-May-2006.) (Revised by Mario Carneiro, 26-Apr-2014.) |
| ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → Σ𝑘 ∈ ℕ0 (𝐴↑𝑘) = (1 / (1 − 𝐴))) | ||
| Theorem | geoisumr 15958* | The infinite sum of reciprocals 1 + (1 / 𝐴)↑1 + (1 / 𝐴)↑2... is 𝐴 / (𝐴 − 1). (Contributed by rpenner, 3-Nov-2007.) (Revised by Mario Carneiro, 26-Apr-2014.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 1 < (abs‘𝐴)) → Σ𝑘 ∈ ℕ0 ((1 / 𝐴)↑𝑘) = (𝐴 / (𝐴 − 1))) | ||
| Theorem | geoisum1 15959* | The infinite sum of 𝐴↑1 + 𝐴↑2... is (𝐴 / (1 − 𝐴)). (Contributed by NM, 1-Nov-2007.) (Revised by Mario Carneiro, 26-Apr-2014.) |
| ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → Σ𝑘 ∈ ℕ (𝐴↑𝑘) = (𝐴 / (1 − 𝐴))) | ||
| Theorem | geoisum1c 15960* | The infinite sum of 𝐴 · (𝑅↑1) + 𝐴 · (𝑅↑2)... is (𝐴 · 𝑅) / (1 − 𝑅). (Contributed by NM, 2-Nov-2007.) (Revised by Mario Carneiro, 26-Apr-2014.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝑅 ∈ ℂ ∧ (abs‘𝑅) < 1) → Σ𝑘 ∈ ℕ (𝐴 · (𝑅↑𝑘)) = ((𝐴 · 𝑅) / (1 − 𝑅))) | ||
| Theorem | 0.999... 15961 | The recurring decimal 0.999..., which is defined as the infinite sum 0.9 + 0.09 + 0.009 + ... i.e. 9 / 10↑1 + 9 / 10↑2 + 9 / 10↑3 + ..., is exactly equal to 1, according to ZF set theory. Interestingly, about 40% of the people responding to a poll at http://forum.physorg.com/index.php?showtopic=13177 disagree. (Contributed by NM, 2-Nov-2007.) (Revised by AV, 8-Sep-2021.) |
| ⊢ Σ𝑘 ∈ ℕ (9 / (;10↑𝑘)) = 1 | ||
| Theorem | geoihalfsum 15962 | Prove that the infinite geometric series of 1/2, 1/2 + 1/4 + 1/8 + ... = 1. Uses geoisum1 15959. This is a representation of .111... in binary with an infinite number of 1's. Theorem 0.999... 15961 proves a similar claim for .999... in base 10. (Contributed by David A. Wheeler, 4-Jan-2017.) (Proof shortened by AV, 9-Jul-2022.) |
| ⊢ Σ𝑘 ∈ ℕ (1 / (2↑𝑘)) = 1 | ||
| Theorem | cvgrat 15963* | Ratio test for convergence of a complex infinite series. If the ratio 𝐴 of the absolute values of successive terms in an infinite sequence 𝐹 is less than 1 for all terms beyond some index 𝐵, then the infinite sum of the terms of 𝐹 converges to a complex number. Equivalent to first part of Exercise 4 of [Gleason] p. 182. (Contributed by NM, 26-Apr-2005.) (Proof shortened by Mario Carneiro, 27-Apr-2014.) |
| ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ 𝑊 = (ℤ≥‘𝑁) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 1) & ⊢ (𝜑 → 𝑁 ∈ 𝑍) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (abs‘(𝐹‘(𝑘 + 1))) ≤ (𝐴 · (abs‘(𝐹‘𝑘)))) ⇒ ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ ) | ||
| Theorem | mertenslem1 15964* | Lemma for mertens 15966. (Contributed by Mario Carneiro, 29-Apr-2014.) |
| ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → (𝐹‘𝑗) = 𝐴) & ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → (𝐾‘𝑗) = (abs‘𝐴)) & ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → 𝐴 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐺‘𝑘) = 𝐵) & ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝐵 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐻‘𝑘) = Σ𝑗 ∈ (0...𝑘)(𝐴 · (𝐺‘(𝑘 − 𝑗)))) & ⊢ (𝜑 → seq0( + , 𝐾) ∈ dom ⇝ ) & ⊢ (𝜑 → seq0( + , 𝐺) ∈ dom ⇝ ) & ⊢ (𝜑 → 𝐸 ∈ ℝ+) & ⊢ 𝑇 = {𝑧 ∣ ∃𝑛 ∈ (0...(𝑠 − 1))𝑧 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘))} & ⊢ (𝜓 ↔ (𝑠 ∈ ℕ ∧ ∀𝑛 ∈ (ℤ≥‘𝑠)(abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)) < ((𝐸 / 2) / (Σ𝑗 ∈ ℕ0 (𝐾‘𝑗) + 1)))) & ⊢ (𝜑 → (𝜓 ∧ (𝑡 ∈ ℕ0 ∧ ∀𝑚 ∈ (ℤ≥‘𝑡)(𝐾‘𝑚) < (((𝐸 / 2) / 𝑠) / (sup(𝑇, ℝ, < ) + 1))))) & ⊢ (𝜑 → (0 ≤ sup(𝑇, ℝ, < ) ∧ (𝑇 ⊆ ℝ ∧ 𝑇 ≠ ∅ ∧ ∃𝑧 ∈ ℝ ∀𝑤 ∈ 𝑇 𝑤 ≤ 𝑧))) ⇒ ⊢ (𝜑 → ∃𝑦 ∈ ℕ0 ∀𝑚 ∈ (ℤ≥‘𝑦)(abs‘Σ𝑗 ∈ (0...𝑚)(𝐴 · Σ𝑘 ∈ (ℤ≥‘((𝑚 − 𝑗) + 1))𝐵)) < 𝐸) | ||
| Theorem | mertenslem2 15965* | Lemma for mertens 15966. (Contributed by Mario Carneiro, 28-Apr-2014.) |
| ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → (𝐹‘𝑗) = 𝐴) & ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → (𝐾‘𝑗) = (abs‘𝐴)) & ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → 𝐴 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐺‘𝑘) = 𝐵) & ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝐵 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐻‘𝑘) = Σ𝑗 ∈ (0...𝑘)(𝐴 · (𝐺‘(𝑘 − 𝑗)))) & ⊢ (𝜑 → seq0( + , 𝐾) ∈ dom ⇝ ) & ⊢ (𝜑 → seq0( + , 𝐺) ∈ dom ⇝ ) & ⊢ (𝜑 → 𝐸 ∈ ℝ+) & ⊢ 𝑇 = {𝑧 ∣ ∃𝑛 ∈ (0...(𝑠 − 1))𝑧 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘))} & ⊢ (𝜓 ↔ (𝑠 ∈ ℕ ∧ ∀𝑛 ∈ (ℤ≥‘𝑠)(abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)) < ((𝐸 / 2) / (Σ𝑗 ∈ ℕ0 (𝐾‘𝑗) + 1)))) ⇒ ⊢ (𝜑 → ∃𝑦 ∈ ℕ0 ∀𝑚 ∈ (ℤ≥‘𝑦)(abs‘Σ𝑗 ∈ (0...𝑚)(𝐴 · Σ𝑘 ∈ (ℤ≥‘((𝑚 − 𝑗) + 1))𝐵)) < 𝐸) | ||
| Theorem | mertens 15966* | Mertens' theorem. If 𝐴(𝑗) is an absolutely convergent series and 𝐵(𝑘) is convergent, then (Σ𝑗 ∈ ℕ0𝐴(𝑗) · Σ𝑘 ∈ ℕ0𝐵(𝑘)) = Σ𝑘 ∈ ℕ0Σ𝑗 ∈ (0...𝑘)(𝐴(𝑗) · 𝐵(𝑘 − 𝑗)) (and this latter series is convergent). This latter sum is commonly known as the Cauchy product of the sequences. The proof follows the outline at http://en.wikipedia.org/wiki/Cauchy_product#Proof_of_Mertens.27_theorem. (Contributed by Mario Carneiro, 29-Apr-2014.) |
| ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → (𝐹‘𝑗) = 𝐴) & ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → (𝐾‘𝑗) = (abs‘𝐴)) & ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → 𝐴 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐺‘𝑘) = 𝐵) & ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝐵 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐻‘𝑘) = Σ𝑗 ∈ (0...𝑘)(𝐴 · (𝐺‘(𝑘 − 𝑗)))) & ⊢ (𝜑 → seq0( + , 𝐾) ∈ dom ⇝ ) & ⊢ (𝜑 → seq0( + , 𝐺) ∈ dom ⇝ ) ⇒ ⊢ (𝜑 → seq0( + , 𝐻) ⇝ (Σ𝑗 ∈ ℕ0 𝐴 · Σ𝑘 ∈ ℕ0 𝐵)) | ||
| Theorem | prodf 15967* | An infinite product of complex terms is a function from an upper set of integers to ℂ. (Contributed by Scott Fenton, 4-Dec-2017.) |
| ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) ⇒ ⊢ (𝜑 → seq𝑀( · , 𝐹):𝑍⟶ℂ) | ||
| Theorem | clim2prod 15968* | The limit of an infinite product with an initial segment added. (Contributed by Scott Fenton, 18-Dec-2017.) |
| ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑁 ∈ 𝑍) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) & ⊢ (𝜑 → seq(𝑁 + 1)( · , 𝐹) ⇝ 𝐴) ⇒ ⊢ (𝜑 → seq𝑀( · , 𝐹) ⇝ ((seq𝑀( · , 𝐹)‘𝑁) · 𝐴)) | ||
| Theorem | clim2div 15969* | The limit of an infinite product with an initial segment removed. (Contributed by Scott Fenton, 20-Dec-2017.) |
| ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑁 ∈ 𝑍) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) & ⊢ (𝜑 → seq𝑀( · , 𝐹) ⇝ 𝐴) & ⊢ (𝜑 → (seq𝑀( · , 𝐹)‘𝑁) ≠ 0) ⇒ ⊢ (𝜑 → seq(𝑁 + 1)( · , 𝐹) ⇝ (𝐴 / (seq𝑀( · , 𝐹)‘𝑁))) | ||
| Theorem | prodfmul 15970* | The product of two infinite products. (Contributed by Scott Fenton, 18-Dec-2017.) |
| ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) & ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐺‘𝑘) ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐻‘𝑘) = ((𝐹‘𝑘) · (𝐺‘𝑘))) ⇒ ⊢ (𝜑 → (seq𝑀( · , 𝐻)‘𝑁) = ((seq𝑀( · , 𝐹)‘𝑁) · (seq𝑀( · , 𝐺)‘𝑁))) | ||
| Theorem | prodf1 15971 | The value of the partial products in a one-valued infinite product. (Contributed by Scott Fenton, 5-Dec-2017.) |
| ⊢ 𝑍 = (ℤ≥‘𝑀) ⇒ ⊢ (𝑁 ∈ 𝑍 → (seq𝑀( · , (𝑍 × {1}))‘𝑁) = 1) | ||
| Theorem | prodf1f 15972 | A one-valued infinite product is equal to the constant one function. (Contributed by Scott Fenton, 5-Dec-2017.) |
| ⊢ 𝑍 = (ℤ≥‘𝑀) ⇒ ⊢ (𝑀 ∈ ℤ → seq𝑀( · , (𝑍 × {1})) = (𝑍 × {1})) | ||
| Theorem | prodfclim1 15973 | The constant one product converges to one. (Contributed by Scott Fenton, 5-Dec-2017.) |
| ⊢ 𝑍 = (ℤ≥‘𝑀) ⇒ ⊢ (𝑀 ∈ ℤ → seq𝑀( · , (𝑍 × {1})) ⇝ 1) | ||
| Theorem | prodfn0 15974* | No term of a nonzero infinite product is zero. (Contributed by Scott Fenton, 14-Jan-2018.) |
| ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) & ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ≠ 0) ⇒ ⊢ (𝜑 → (seq𝑀( · , 𝐹)‘𝑁) ≠ 0) | ||
| Theorem | prodfrec 15975* | The reciprocal of an infinite product. (Contributed by Scott Fenton, 15-Jan-2018.) |
| ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) & ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ≠ 0) & ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐺‘𝑘) = (1 / (𝐹‘𝑘))) ⇒ ⊢ (𝜑 → (seq𝑀( · , 𝐺)‘𝑁) = (1 / (seq𝑀( · , 𝐹)‘𝑁))) | ||
| Theorem | prodfdiv 15976* | The quotient of two infinite products. (Contributed by Scott Fenton, 15-Jan-2018.) |
| ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) & ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐺‘𝑘) ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐺‘𝑘) ≠ 0) & ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐻‘𝑘) = ((𝐹‘𝑘) / (𝐺‘𝑘))) ⇒ ⊢ (𝜑 → (seq𝑀( · , 𝐻)‘𝑁) = ((seq𝑀( · , 𝐹)‘𝑁) / (seq𝑀( · , 𝐺)‘𝑁))) | ||
| Theorem | ntrivcvg 15977* | A non-trivially converging infinite product converges. (Contributed by Scott Fenton, 18-Dec-2017.) |
| ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → ∃𝑛 ∈ 𝑍 ∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦)) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) ⇒ ⊢ (𝜑 → seq𝑀( · , 𝐹) ∈ dom ⇝ ) | ||
| Theorem | ntrivcvgn0 15978* | A product that converges to a nonzero value converges non-trivially. (Contributed by Scott Fenton, 18-Dec-2017.) |
| ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → seq𝑀( · , 𝐹) ⇝ 𝑋) & ⊢ (𝜑 → 𝑋 ≠ 0) ⇒ ⊢ (𝜑 → ∃𝑛 ∈ 𝑍 ∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦)) | ||
| Theorem | ntrivcvgfvn0 15979* | Any value of a product sequence that converges to a nonzero value is itself nonzero. (Contributed by Scott Fenton, 20-Dec-2017.) |
| ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑁 ∈ 𝑍) & ⊢ (𝜑 → seq𝑀( · , 𝐹) ⇝ 𝑋) & ⊢ (𝜑 → 𝑋 ≠ 0) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) ⇒ ⊢ (𝜑 → (seq𝑀( · , 𝐹)‘𝑁) ≠ 0) | ||
| Theorem | ntrivcvgtail 15980* | A tail of a non-trivially convergent sequence converges non-trivially. (Contributed by Scott Fenton, 20-Dec-2017.) |
| ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑁 ∈ 𝑍) & ⊢ (𝜑 → seq𝑀( · , 𝐹) ⇝ 𝑋) & ⊢ (𝜑 → 𝑋 ≠ 0) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) ⇒ ⊢ (𝜑 → (( ⇝ ‘seq𝑁( · , 𝐹)) ≠ 0 ∧ seq𝑁( · , 𝐹) ⇝ ( ⇝ ‘seq𝑁( · , 𝐹)))) | ||
| Theorem | ntrivcvgmullem 15981* | Lemma for ntrivcvgmul 15982. (Contributed by Scott Fenton, 19-Dec-2017.) |
| ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑁 ∈ 𝑍) & ⊢ (𝜑 → 𝑃 ∈ 𝑍) & ⊢ (𝜑 → 𝑋 ≠ 0) & ⊢ (𝜑 → 𝑌 ≠ 0) & ⊢ (𝜑 → seq𝑁( · , 𝐹) ⇝ 𝑋) & ⊢ (𝜑 → seq𝑃( · , 𝐺) ⇝ 𝑌) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ ℂ) & ⊢ (𝜑 → 𝑁 ≤ 𝑃) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐻‘𝑘) = ((𝐹‘𝑘) · (𝐺‘𝑘))) ⇒ ⊢ (𝜑 → ∃𝑞 ∈ 𝑍 ∃𝑤(𝑤 ≠ 0 ∧ seq𝑞( · , 𝐻) ⇝ 𝑤)) | ||
| Theorem | ntrivcvgmul 15982* | The product of two non-trivially converging products converges non-trivially. (Contributed by Scott Fenton, 18-Dec-2017.) |
| ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → ∃𝑛 ∈ 𝑍 ∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦)) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) & ⊢ (𝜑 → ∃𝑚 ∈ 𝑍 ∃𝑧(𝑧 ≠ 0 ∧ seq𝑚( · , 𝐺) ⇝ 𝑧)) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐻‘𝑘) = ((𝐹‘𝑘) · (𝐺‘𝑘))) ⇒ ⊢ (𝜑 → ∃𝑝 ∈ 𝑍 ∃𝑤(𝑤 ≠ 0 ∧ seq𝑝( · , 𝐻) ⇝ 𝑤)) | ||
| Syntax | cprod 15983 | Extend class notation to include complex products. |
| class ∏𝑘 ∈ 𝐴 𝐵 | ||
| Definition | df-prod 15984* | Define the product of a series with an index set of integers 𝐴. This definition takes most of the aspects of df-sum 15764 and adapts them for multiplication instead of addition. However, we insist that in the infinite case, there is a nonzero tail of the sequence. This ensures that the convergence criteria match those of infinite sums. (Contributed by Scott Fenton, 4-Dec-2017.) |
| ⊢ ∏𝑘 ∈ 𝐴 𝐵 = (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)))) | ||
| Theorem | prodex 15985 | A product is a set. (Contributed by Scott Fenton, 4-Dec-2017.) |
| ⊢ ∏𝑘 ∈ 𝐴 𝐵 ∈ V | ||
| Theorem | prodeq1f 15986 | Equality theorem for a product. (Contributed by Scott Fenton, 1-Dec-2017.) |
| ⊢ Ⅎ𝑘𝐴 & ⊢ Ⅎ𝑘𝐵 ⇒ ⊢ (𝐴 = 𝐵 → ∏𝑘 ∈ 𝐴 𝐶 = ∏𝑘 ∈ 𝐵 𝐶) | ||
| Theorem | prodeq1 15987* | Equality theorem for a product. (Contributed by Scott Fenton, 1-Dec-2017.) |
| ⊢ (𝐴 = 𝐵 → ∏𝑘 ∈ 𝐴 𝐶 = ∏𝑘 ∈ 𝐵 𝐶) | ||
| Theorem | nfcprod1 15988* | Bound-variable hypothesis builder for product. (Contributed by Scott Fenton, 4-Dec-2017.) |
| ⊢ Ⅎ𝑘𝐴 ⇒ ⊢ Ⅎ𝑘∏𝑘 ∈ 𝐴 𝐵 | ||
| Theorem | nfcprod 15989* | Bound-variable hypothesis builder for product: if 𝑥 is (effectively) not free in 𝐴 and 𝐵, it is not free in ∏𝑘 ∈ 𝐴𝐵. (Contributed by Scott Fenton, 1-Dec-2017.) |
| ⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ Ⅎ𝑥∏𝑘 ∈ 𝐴 𝐵 | ||
| Theorem | prodeq2w 15990* | Equality theorem for product, when the class expressions 𝐵 and 𝐶 are equal everywhere. Proved using only Extensionality. (Contributed by Scott Fenton, 4-Dec-2017.) |
| ⊢ (∀𝑘 𝐵 = 𝐶 → ∏𝑘 ∈ 𝐴 𝐵 = ∏𝑘 ∈ 𝐴 𝐶) | ||
| Theorem | prodeq2ii 15991* | Equality theorem for product, with the class expressions 𝐵 and 𝐶 guarded by I to be always sets. (Contributed by Scott Fenton, 4-Dec-2017.) |
| ⊢ (∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) → ∏𝑘 ∈ 𝐴 𝐵 = ∏𝑘 ∈ 𝐴 𝐶) | ||
| Theorem | prodeq2 15992* | Equality theorem for product. (Contributed by Scott Fenton, 4-Dec-2017.) |
| ⊢ (∀𝑘 ∈ 𝐴 𝐵 = 𝐶 → ∏𝑘 ∈ 𝐴 𝐵 = ∏𝑘 ∈ 𝐴 𝐶) | ||
| Theorem | cbvprod 15993* | Change bound variable in a product. (Contributed by Scott Fenton, 4-Dec-2017.) |
| ⊢ (𝑗 = 𝑘 → 𝐵 = 𝐶) & ⊢ Ⅎ𝑘𝐴 & ⊢ Ⅎ𝑗𝐴 & ⊢ Ⅎ𝑘𝐵 & ⊢ Ⅎ𝑗𝐶 ⇒ ⊢ ∏𝑗 ∈ 𝐴 𝐵 = ∏𝑘 ∈ 𝐴 𝐶 | ||
| Theorem | cbvprodv 15994* | Change bound variable in a product. (Contributed by Scott Fenton, 4-Dec-2017.) |
| ⊢ (𝑗 = 𝑘 → 𝐵 = 𝐶) ⇒ ⊢ ∏𝑗 ∈ 𝐴 𝐵 = ∏𝑘 ∈ 𝐴 𝐶 | ||
| Theorem | cbvprodi 15995* | Change bound variable in a product. (Contributed by Scott Fenton, 4-Dec-2017.) |
| ⊢ Ⅎ𝑘𝐵 & ⊢ Ⅎ𝑗𝐶 & ⊢ (𝑗 = 𝑘 → 𝐵 = 𝐶) ⇒ ⊢ ∏𝑗 ∈ 𝐴 𝐵 = ∏𝑘 ∈ 𝐴 𝐶 | ||
| Theorem | prodeq1i 15996 | Equality inference for product. (Contributed by Scott Fenton, 4-Dec-2017.) Remove DV conditions. (Revised by GG, 1-Sep-2025.) |
| ⊢ 𝐴 = 𝐵 ⇒ ⊢ ∏𝑘 ∈ 𝐴 𝐶 = ∏𝑘 ∈ 𝐵 𝐶 | ||
| Theorem | prodeq1iOLD 15997* | Obsolete version of prodeq1i 15996 as of 1-Sep-2025. (Contributed by Scott Fenton, 4-Dec-2017.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ 𝐴 = 𝐵 ⇒ ⊢ ∏𝑘 ∈ 𝐴 𝐶 = ∏𝑘 ∈ 𝐵 𝐶 | ||
| Theorem | prodeq2i 15998* | Equality inference for product. (Contributed by Scott Fenton, 4-Dec-2017.) |
| ⊢ (𝑘 ∈ 𝐴 → 𝐵 = 𝐶) ⇒ ⊢ ∏𝑘 ∈ 𝐴 𝐵 = ∏𝑘 ∈ 𝐴 𝐶 | ||
| Theorem | prodeq12i 15999* | Equality inference for product. (Contributed by Scott Fenton, 4-Dec-2017.) |
| ⊢ 𝐴 = 𝐵 & ⊢ (𝑘 ∈ 𝐴 → 𝐶 = 𝐷) ⇒ ⊢ ∏𝑘 ∈ 𝐴 𝐶 = ∏𝑘 ∈ 𝐵 𝐷 | ||
| Theorem | prodeq1d 16000* | Equality deduction for product. (Contributed by Scott Fenton, 4-Dec-2017.) |
| ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐶 = ∏𝑘 ∈ 𝐵 𝐶) | ||
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