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Type | Label | Description |
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Statement | ||
Theorem | rlimsqz2 15001* | Convergence of a sequence sandwiched between another converging sequence and its limit. (Contributed by Mario Carneiro, 3-Feb-2014.) (Revised by Mario Carneiro, 20-May-2016.) |
⊢ (𝜑 → 𝐷 ∈ ℝ) & ⊢ (𝜑 → 𝑀 ∈ ℝ) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ⇝𝑟 𝐷) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℝ) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑀 ≤ 𝑥)) → 𝐶 ≤ 𝐵) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑀 ≤ 𝑥)) → 𝐷 ≤ 𝐶) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ⇝𝑟 𝐷) | ||
Theorem | lo1le 15002* | Transfer eventual upper boundedness from a larger function to a smaller function. (Contributed by Mario Carneiro, 26-May-2016.) |
⊢ (𝜑 → 𝑀 ∈ ℝ) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ ≤𝑂(1)) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℝ) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑀 ≤ 𝑥)) → 𝐶 ≤ 𝐵) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ ≤𝑂(1)) | ||
Theorem | o1le 15003* | Transfer eventual boundedness from a larger function to a smaller function. (Contributed by Mario Carneiro, 25-Sep-2014.) (Proof shortened by Mario Carneiro, 26-May-2016.) |
⊢ (𝜑 → 𝑀 ∈ ℝ) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝑂(1)) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℂ) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑀 ≤ 𝑥)) → (abs‘𝐶) ≤ (abs‘𝐵)) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝑂(1)) | ||
Theorem | rlimno1 15004* | A function whose inverse converges to zero is unbounded. (Contributed by Mario Carneiro, 30-May-2016.) |
⊢ (𝜑 → sup(𝐴, ℝ*, < ) = +∞) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (1 / 𝐵)) ⇝𝑟 0) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ≠ 0) ⇒ ⊢ (𝜑 → ¬ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝑂(1)) | ||
Theorem | clim2ser 15005* | The limit of an infinite series with an initial segment removed. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Mario Carneiro, 1-Feb-2014.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑁 ∈ 𝑍) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) & ⊢ (𝜑 → seq𝑀( + , 𝐹) ⇝ 𝐴) ⇒ ⊢ (𝜑 → seq(𝑁 + 1)( + , 𝐹) ⇝ (𝐴 − (seq𝑀( + , 𝐹)‘𝑁))) | ||
Theorem | clim2ser2 15006* | The limit of an infinite series with an initial segment added. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Mario Carneiro, 1-Feb-2014.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑁 ∈ 𝑍) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) & ⊢ (𝜑 → seq(𝑁 + 1)( + , 𝐹) ⇝ 𝐴) ⇒ ⊢ (𝜑 → seq𝑀( + , 𝐹) ⇝ (𝐴 + (seq𝑀( + , 𝐹)‘𝑁))) | ||
Theorem | iserex 15007* | An infinite series converges, if and only if the series does with initial terms removed. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Mario Carneiro, 27-Apr-2014.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑁 ∈ 𝑍) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) ⇒ ⊢ (𝜑 → (seq𝑀( + , 𝐹) ∈ dom ⇝ ↔ seq𝑁( + , 𝐹) ∈ dom ⇝ )) | ||
Theorem | isermulc2 15008* | Multiplication of an infinite series by a constant. (Contributed by Paul Chapman, 14-Nov-2007.) (Revised by Mario Carneiro, 1-Feb-2014.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → seq𝑀( + , 𝐹) ⇝ 𝐴) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = (𝐶 · (𝐹‘𝑘))) ⇒ ⊢ (𝜑 → seq𝑀( + , 𝐺) ⇝ (𝐶 · 𝐴)) | ||
Theorem | climlec2 15009* | Comparison of a constant to the limit of a sequence. (Contributed by NM, 28-Feb-2008.) (Revised by Mario Carneiro, 1-Feb-2014.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐹 ⇝ 𝐵) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ≤ (𝐹‘𝑘)) ⇒ ⊢ (𝜑 → 𝐴 ≤ 𝐵) | ||
Theorem | iserle 15010* | Comparison of the limits of two infinite series. (Contributed by Paul Chapman, 12-Nov-2007.) (Revised by Mario Carneiro, 3-Feb-2014.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → seq𝑀( + , 𝐹) ⇝ 𝐴) & ⊢ (𝜑 → seq𝑀( + , 𝐺) ⇝ 𝐵) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ≤ (𝐺‘𝑘)) ⇒ ⊢ (𝜑 → 𝐴 ≤ 𝐵) | ||
Theorem | iserge0 15011* | The limit of an infinite series of nonnegative reals is nonnegative. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Mario Carneiro, 3-Feb-2014.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → seq𝑀( + , 𝐹) ⇝ 𝐴) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 0 ≤ (𝐹‘𝑘)) ⇒ ⊢ (𝜑 → 0 ≤ 𝐴) | ||
Theorem | climub 15012* | The limit of a monotonic sequence is an upper bound. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 10-Feb-2014.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑁 ∈ 𝑍) & ⊢ (𝜑 → 𝐹 ⇝ 𝐴) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ≤ (𝐹‘(𝑘 + 1))) ⇒ ⊢ (𝜑 → (𝐹‘𝑁) ≤ 𝐴) | ||
Theorem | climserle 15013* | The partial sums of a converging infinite series with nonnegative terms are bounded by its limit. (Contributed by NM, 27-Dec-2005.) (Revised by Mario Carneiro, 9-Feb-2014.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑁 ∈ 𝑍) & ⊢ (𝜑 → seq𝑀( + , 𝐹) ⇝ 𝐴) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 0 ≤ (𝐹‘𝑘)) ⇒ ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) ≤ 𝐴) | ||
Theorem | isershft 15014 | Index shift of the limit of an infinite series. (Contributed by Mario Carneiro, 6-Sep-2013.) (Revised by Mario Carneiro, 5-Nov-2013.) |
⊢ 𝐹 ∈ V ⇒ ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (seq𝑀( + , 𝐹) ⇝ 𝐴 ↔ seq(𝑀 + 𝑁)( + , (𝐹 shift 𝑁)) ⇝ 𝐴)) | ||
Theorem | isercolllem1 15015* | Lemma for isercoll 15018. (Contributed by Mario Carneiro, 6-Apr-2015.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐺:ℕ⟶𝑍) & ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) < (𝐺‘(𝑘 + 1))) ⇒ ⊢ ((𝜑 ∧ 𝑆 ⊆ ℕ) → (𝐺 ↾ 𝑆) Isom < , < (𝑆, (𝐺 “ 𝑆))) | ||
Theorem | isercolllem2 15016* | Lemma for isercoll 15018. (Contributed by Mario Carneiro, 6-Apr-2015.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐺:ℕ⟶𝑍) & ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) < (𝐺‘(𝑘 + 1))) ⇒ ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → (1...(♯‘(𝐺 “ (◡𝐺 “ (𝑀...𝑁))))) = (◡𝐺 “ (𝑀...𝑁))) | ||
Theorem | isercolllem3 15017* | Lemma for isercoll 15018. (Contributed by Mario Carneiro, 6-Apr-2015.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐺:ℕ⟶𝑍) & ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) < (𝐺‘(𝑘 + 1))) & ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑍 ∖ ran 𝐺)) → (𝐹‘𝑛) = 0) & ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐻‘𝑘) = (𝐹‘(𝐺‘𝑘))) ⇒ ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → (seq𝑀( + , 𝐹)‘𝑁) = (seq1( + , 𝐻)‘(♯‘(𝐺 “ (◡𝐺 “ (𝑀...𝑁)))))) | ||
Theorem | isercoll 15018* | Rearrange an infinite series by spacing out the terms using an order isomorphism. (Contributed by Mario Carneiro, 6-Apr-2015.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐺:ℕ⟶𝑍) & ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) < (𝐺‘(𝑘 + 1))) & ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑍 ∖ ran 𝐺)) → (𝐹‘𝑛) = 0) & ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐻‘𝑘) = (𝐹‘(𝐺‘𝑘))) ⇒ ⊢ (𝜑 → (seq1( + , 𝐻) ⇝ 𝐴 ↔ seq𝑀( + , 𝐹) ⇝ 𝐴)) | ||
Theorem | isercoll2 15019* | Generalize isercoll 15018 so that both sequences have arbitrary starting point. (Contributed by Mario Carneiro, 6-Apr-2015.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ 𝑊 = (ℤ≥‘𝑁) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ (𝜑 → 𝐺:𝑍⟶𝑊) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) < (𝐺‘(𝑘 + 1))) & ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑊 ∖ ran 𝐺)) → (𝐹‘𝑛) = 0) & ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑊) → (𝐹‘𝑛) ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐻‘𝑘) = (𝐹‘(𝐺‘𝑘))) ⇒ ⊢ (𝜑 → (seq𝑀( + , 𝐻) ⇝ 𝐴 ↔ seq𝑁( + , 𝐹) ⇝ 𝐴)) | ||
Theorem | climsup 15020* | A bounded monotonic sequence converges to the supremum of its range. Theorem 12-5.1 of [Gleason] p. 180. (Contributed by NM, 13-Mar-2005.) (Revised by Mario Carneiro, 10-Feb-2014.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ≤ (𝐹‘(𝑘 + 1))) & ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ≤ 𝑥) ⇒ ⊢ (𝜑 → 𝐹 ⇝ sup(ran 𝐹, ℝ, < )) | ||
Theorem | climcau 15021* | A converging sequence of complex numbers is a Cauchy sequence. Theorem 12-5.3 of [Gleason] p. 180 (necessity part). (Contributed by NM, 16-Apr-2005.) (Revised by Mario Carneiro, 26-Apr-2014.) |
⊢ 𝑍 = (ℤ≥‘𝑀) ⇒ ⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ dom ⇝ ) → ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥) | ||
Theorem | climbdd 15022* | A converging sequence of complex numbers is bounded. (Contributed by Mario Carneiro, 9-Jul-2017.) |
⊢ 𝑍 = (ℤ≥‘𝑀) ⇒ ⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ dom ⇝ ∧ ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ) → ∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) ≤ 𝑥) | ||
Theorem | caucvgrlem 15023* | Lemma for caurcvgr 15024. (Contributed by Mario Carneiro, 15-Feb-2014.) (Revised by AV, 12-Sep-2020.) |
⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ (𝜑 → 𝐹:𝐴⟶ℝ) & ⊢ (𝜑 → sup(𝐴, ℝ*, < ) = +∞) & ⊢ (𝜑 → ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝐴 ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥)) & ⊢ (𝜑 → 𝑅 ∈ ℝ+) ⇒ ⊢ (𝜑 → ∃𝑗 ∈ 𝐴 ((lim sup‘𝐹) ∈ ℝ ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (lim sup‘𝐹))) < (3 · 𝑅)))) | ||
Theorem | caurcvgr 15024* | A Cauchy sequence of real numbers converges to its limit supremum. The third hypothesis specifies that 𝐹 is a Cauchy sequence. (Contributed by Mario Carneiro, 7-May-2016.) (Revised by AV, 12-Sep-2020.) |
⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ (𝜑 → 𝐹:𝐴⟶ℝ) & ⊢ (𝜑 → sup(𝐴, ℝ*, < ) = +∞) & ⊢ (𝜑 → ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝐴 ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥)) ⇒ ⊢ (𝜑 → 𝐹 ⇝𝑟 (lim sup‘𝐹)) | ||
Theorem | caucvgrlem2 15025* | Lemma for caucvgr 15026. (Contributed by NM, 4-Apr-2005.) (Proof shortened by Mario Carneiro, 8-May-2016.) |
⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) & ⊢ (𝜑 → sup(𝐴, ℝ*, < ) = +∞) & ⊢ (𝜑 → ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝐴 ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥)) & ⊢ 𝐻:ℂ⟶ℝ & ⊢ (((𝐹‘𝑘) ∈ ℂ ∧ (𝐹‘𝑗) ∈ ℂ) → (abs‘((𝐻‘(𝐹‘𝑘)) − (𝐻‘(𝐹‘𝑗)))) ≤ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗)))) ⇒ ⊢ (𝜑 → (𝑛 ∈ 𝐴 ↦ (𝐻‘(𝐹‘𝑛))) ⇝𝑟 ( ⇝𝑟 ‘(𝐻 ∘ 𝐹))) | ||
Theorem | caucvgr 15026* | A Cauchy sequence of complex numbers converges to a complex number. Theorem 12-5.3 of [Gleason] p. 180 (sufficiency part). (Contributed by NM, 20-Dec-2006.) (Revised by Mario Carneiro, 8-May-2016.) |
⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) & ⊢ (𝜑 → sup(𝐴, ℝ*, < ) = +∞) & ⊢ (𝜑 → ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝐴 ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥)) ⇒ ⊢ (𝜑 → 𝐹 ∈ dom ⇝𝑟 ) | ||
Theorem | caurcvg 15027* | A Cauchy sequence of real numbers converges to its limit supremum. The fourth hypothesis specifies that 𝐹 is a Cauchy sequence. (Contributed by NM, 4-Apr-2005.) (Revised by AV, 12-Sep-2020.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ) & ⊢ (𝜑 → ∀𝑥 ∈ ℝ+ ∃𝑚 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑚)(abs‘((𝐹‘𝑘) − (𝐹‘𝑚))) < 𝑥) ⇒ ⊢ (𝜑 → 𝐹 ⇝ (lim sup‘𝐹)) | ||
Theorem | caurcvg2 15028* | A Cauchy sequence of real numbers converges, existence version. (Contributed by NM, 4-Apr-2005.) (Revised by Mario Carneiro, 7-Sep-2014.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹 ∈ 𝑉) & ⊢ (𝜑 → ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℝ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥)) ⇒ ⊢ (𝜑 → 𝐹 ∈ dom ⇝ ) | ||
Theorem | caucvg 15029* | A Cauchy sequence of complex numbers converges to a complex number. Theorem 12-5.3 of [Gleason] p. 180 (sufficiency part). (Contributed by NM, 20-Dec-2006.) (Proof shortened by Mario Carneiro, 15-Feb-2014.) (Revised by Mario Carneiro, 8-May-2016.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) & ⊢ (𝜑 → ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥) & ⊢ (𝜑 → 𝐹 ∈ 𝑉) ⇒ ⊢ (𝜑 → 𝐹 ∈ dom ⇝ ) | ||
Theorem | caucvgb 15030* | A function is convergent if and only if it is Cauchy. Theorem 12-5.3 of [Gleason] p. 180. (Contributed by Mario Carneiro, 15-Feb-2014.) |
⊢ 𝑍 = (ℤ≥‘𝑀) ⇒ ⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ 𝑉) → (𝐹 ∈ dom ⇝ ↔ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥))) | ||
Theorem | serf0 15031* | If an infinite series converges, its underlying sequence converges to zero. (Contributed by NM, 2-Sep-2005.) (Revised by Mario Carneiro, 16-Feb-2014.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐹 ∈ 𝑉) & ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) ⇒ ⊢ (𝜑 → 𝐹 ⇝ 0) | ||
Theorem | iseraltlem1 15032* | Lemma for iseralt 15035. A decreasing sequence with limit zero consists of positive terms. (Contributed by Mario Carneiro, 6-Apr-2015.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐺:𝑍⟶ℝ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘(𝑘 + 1)) ≤ (𝐺‘𝑘)) & ⊢ (𝜑 → 𝐺 ⇝ 0) ⇒ ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍) → 0 ≤ (𝐺‘𝑁)) | ||
Theorem | iseraltlem2 15033* | Lemma for iseralt 15035. The terms of an alternating series form a chain of inequalities in alternate terms, so that for example 𝑆(1) ≤ 𝑆(3) ≤ 𝑆(5) ≤ ... and ... ≤ 𝑆(4) ≤ 𝑆(2) ≤ 𝑆(0) (assuming 𝑀 = 0 so that these terms are defined). (Contributed by Mario Carneiro, 6-Apr-2015.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐺:𝑍⟶ℝ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘(𝑘 + 1)) ≤ (𝐺‘𝑘)) & ⊢ (𝜑 → 𝐺 ⇝ 0) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = ((-1↑𝑘) · (𝐺‘𝑘))) ⇒ ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ0) → ((-1↑𝑁) · (seq𝑀( + , 𝐹)‘(𝑁 + (2 · 𝐾)))) ≤ ((-1↑𝑁) · (seq𝑀( + , 𝐹)‘𝑁))) | ||
Theorem | iseraltlem3 15034* | Lemma for iseralt 15035. From iseraltlem2 15033, we have (-1↑𝑛) · 𝑆(𝑛 + 2𝑘) ≤ (-1↑𝑛) · 𝑆(𝑛) and (-1↑𝑛) · 𝑆(𝑛 + 1) ≤ (-1↑𝑛) · 𝑆(𝑛 + 2𝑘 + 1), and we also have (-1↑𝑛) · 𝑆(𝑛 + 1) = (-1↑𝑛) · 𝑆(𝑛) − 𝐺(𝑛 + 1) for each 𝑛 by the definition of the partial sum 𝑆, so combining the inequalities we get (-1↑𝑛) · 𝑆(𝑛) − 𝐺(𝑛 + 1) = (-1↑𝑛) · 𝑆(𝑛 + 1) ≤ (-1↑𝑛) · 𝑆(𝑛 + 2𝑘 + 1) = (-1↑𝑛) · 𝑆(𝑛 + 2𝑘) − 𝐺(𝑛 + 2𝑘 + 1) ≤ (-1↑𝑛) · 𝑆(𝑛 + 2𝑘) ≤ (-1↑𝑛) · 𝑆(𝑛) ≤ (-1↑𝑛) · 𝑆(𝑛) + 𝐺(𝑛 + 1), so ∣ (-1↑𝑛) · 𝑆(𝑛 + 2𝑘 + 1) − (-1↑𝑛) · 𝑆(𝑛) ∣ = ∣ 𝑆(𝑛 + 2𝑘 + 1) − 𝑆(𝑛) ∣ ≤ 𝐺(𝑛 + 1) and ∣ (-1↑𝑛) · 𝑆(𝑛 + 2𝑘) − (-1↑𝑛) · 𝑆(𝑛) ∣ = ∣ 𝑆(𝑛 + 2𝑘) − 𝑆(𝑛) ∣ ≤ 𝐺(𝑛 + 1). Thus, both even and odd partial sums are Cauchy if 𝐺 converges to 0. (Contributed by Mario Carneiro, 6-Apr-2015.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐺:𝑍⟶ℝ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘(𝑘 + 1)) ≤ (𝐺‘𝑘)) & ⊢ (𝜑 → 𝐺 ⇝ 0) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = ((-1↑𝑘) · (𝐺‘𝑘))) ⇒ ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ0) → ((abs‘((seq𝑀( + , 𝐹)‘(𝑁 + (2 · 𝐾))) − (seq𝑀( + , 𝐹)‘𝑁))) ≤ (𝐺‘(𝑁 + 1)) ∧ (abs‘((seq𝑀( + , 𝐹)‘((𝑁 + (2 · 𝐾)) + 1)) − (seq𝑀( + , 𝐹)‘𝑁))) ≤ (𝐺‘(𝑁 + 1)))) | ||
Theorem | iseralt 15035* | The alternating series test. If 𝐺(𝑘) is a decreasing sequence that converges to 0, then Σ𝑘 ∈ 𝑍(-1↑𝑘) · 𝐺(𝑘) is a convergent series. (Note that the first term is positive if 𝑀 is even, and negative if 𝑀 is odd. If the parity of your series does not match up with this, you will need to post-compose the series with multiplication by -1 using isermulc2 15008.) (Contributed by Mario Carneiro, 7-Apr-2015.) (Proof shortened by AV, 9-Jul-2022.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐺:𝑍⟶ℝ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘(𝑘 + 1)) ≤ (𝐺‘𝑘)) & ⊢ (𝜑 → 𝐺 ⇝ 0) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = ((-1↑𝑘) · (𝐺‘𝑘))) ⇒ ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ ) | ||
Syntax | csu 15036 | Extend class notation to include finite and infinite summations. (An underscore was added to the ASCII token in order to facilitate set.mm text searches, since "sum" is a commonly used word in comments.) |
class Σ𝑘 ∈ 𝐴 𝐵 | ||
Definition | df-sum 15037* | Define the sum of a series with an index set of integers 𝐴. 𝑘 is normally a free variable in 𝐵, i.e. 𝐵 can be thought of as 𝐵(𝑘). This definition is the result of a collection of discussions over the most general definition for a sum that does not need the index set to have a specified ordering. This definition is in two parts, one for finite sums and one for subsets of the upper integers. When summing over a subset of the upper integers, we extend the index set to the upper integers by adding zero outside the domain, and then sum the set in order, setting the result to the limit of the partial sums, if it exists. This means that conditionally convergent sums can be evaluated meaningfully. For finite sums, we are explicitly order-independent, by picking any bijection to a 1-based finite sequence and summing in the induced order. These two methods of summation produce the same result on their common region of definition (i.e. finite sets of integers) by summo 15068. Examples: Σ𝑘 ∈ {1, 2, 4} 𝑘 means 1 + 2 + 4 = 7, and Σ𝑘 ∈ ℕ (1 / (2↑𝑘)) = 1 means 1/2 + 1/4 + 1/8 + ... = 1 (geoihalfsum 15232). (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jun-2019.) |
⊢ Σ𝑘 ∈ 𝐴 𝐵 = (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)))) | ||
Theorem | sumex 15038 | A sum is a set. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jun-2019.) |
⊢ Σ𝑘 ∈ 𝐴 𝐵 ∈ V | ||
Theorem | sumeq1 15039 | Equality theorem for a sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jun-2019.) |
⊢ (𝐴 = 𝐵 → Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐶) | ||
Theorem | nfsum1 15040 | Bound-variable hypothesis builder for sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jun-2019.) |
⊢ Ⅎ𝑘𝐴 ⇒ ⊢ Ⅎ𝑘Σ𝑘 ∈ 𝐴 𝐵 | ||
Theorem | nfsumw 15041* | Bound-variable hypothesis builder for sum: if 𝑥 is (effectively) not free in 𝐴 and 𝐵, it is not free in Σ𝑘 ∈ 𝐴𝐵. Version of nfsum 15042 with a disjoint variable condition, which does not require ax-13 2386. (Contributed by NM, 11-Dec-2005.) (Revised by Gino Giotto, 24-Feb-2024.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ Ⅎ𝑥Σ𝑘 ∈ 𝐴 𝐵 | ||
Theorem | nfsum 15042 | Bound-variable hypothesis builder for sum: if 𝑥 is (effectively) not free in 𝐴 and 𝐵, it is not free in Σ𝑘 ∈ 𝐴𝐵. Usage of this theorem is discouraged because it depends on ax-13 2386. Use the weaker nfsumw 15041 when possible. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jun-2019.) (New usage is discouraged.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ Ⅎ𝑥Σ𝑘 ∈ 𝐴 𝐵 | ||
Theorem | sumeq2w 15043 | Equality theorem for sum, when the class expressions 𝐵 and 𝐶 are equal everywhere. Proved using only Extensionality. (Contributed by Mario Carneiro, 24-Jun-2014.) (Revised by Mario Carneiro, 13-Jun-2019.) |
⊢ (∀𝑘 𝐵 = 𝐶 → Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑘 ∈ 𝐴 𝐶) | ||
Theorem | sumeq2ii 15044* | Equality theorem for sum, with the class expressions 𝐵 and 𝐶 guarded by I to be always sets. (Contributed by Mario Carneiro, 13-Jun-2019.) |
⊢ (∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) → Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑘 ∈ 𝐴 𝐶) | ||
Theorem | sumeq2 15045* | Equality theorem for sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jul-2013.) |
⊢ (∀𝑘 ∈ 𝐴 𝐵 = 𝐶 → Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑘 ∈ 𝐴 𝐶) | ||
Theorem | cbvsum 15046* | Change bound variable in a sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jun-2019.) |
⊢ (𝑗 = 𝑘 → 𝐵 = 𝐶) & ⊢ Ⅎ𝑘𝐴 & ⊢ Ⅎ𝑗𝐴 & ⊢ Ⅎ𝑘𝐵 & ⊢ Ⅎ𝑗𝐶 ⇒ ⊢ Σ𝑗 ∈ 𝐴 𝐵 = Σ𝑘 ∈ 𝐴 𝐶 | ||
Theorem | cbvsumv 15047* | Change bound variable in a sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jul-2013.) |
⊢ (𝑗 = 𝑘 → 𝐵 = 𝐶) ⇒ ⊢ Σ𝑗 ∈ 𝐴 𝐵 = Σ𝑘 ∈ 𝐴 𝐶 | ||
Theorem | cbvsumi 15048* | Change bound variable in a sum. (Contributed by NM, 11-Dec-2005.) |
⊢ Ⅎ𝑘𝐵 & ⊢ Ⅎ𝑗𝐶 & ⊢ (𝑗 = 𝑘 → 𝐵 = 𝐶) ⇒ ⊢ Σ𝑗 ∈ 𝐴 𝐵 = Σ𝑘 ∈ 𝐴 𝐶 | ||
Theorem | sumeq1i 15049* | Equality inference for sum. (Contributed by NM, 2-Jan-2006.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐶 | ||
Theorem | sumeq2i 15050* | Equality inference for sum. (Contributed by NM, 3-Dec-2005.) |
⊢ (𝑘 ∈ 𝐴 → 𝐵 = 𝐶) ⇒ ⊢ Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑘 ∈ 𝐴 𝐶 | ||
Theorem | sumeq12i 15051* | Equality inference for sum. (Contributed by FL, 10-Dec-2006.) |
⊢ 𝐴 = 𝐵 & ⊢ (𝑘 ∈ 𝐴 → 𝐶 = 𝐷) ⇒ ⊢ Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐷 | ||
Theorem | sumeq1d 15052* | Equality deduction for sum. (Contributed by NM, 1-Nov-2005.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐶) | ||
Theorem | sumeq2d 15053* | Equality deduction for sum. Note that unlike sumeq2dv 15054, 𝑘 may occur in 𝜑. (Contributed by NM, 1-Nov-2005.) |
⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑘 ∈ 𝐴 𝐶) | ||
Theorem | sumeq2dv 15054* | Equality deduction for sum. (Contributed by NM, 3-Jan-2006.) (Revised by Mario Carneiro, 31-Jan-2014.) |
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑘 ∈ 𝐴 𝐶) | ||
Theorem | sumeq2sdv 15055* | Equality deduction for sum. (Contributed by NM, 3-Jan-2006.) (Proof shortened by Glauco Siliprandi, 5-Apr-2020.) |
⊢ (𝜑 → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑘 ∈ 𝐴 𝐶) | ||
Theorem | 2sumeq2dv 15056* | Equality deduction for double sum. (Contributed by NM, 3-Jan-2006.) (Revised by Mario Carneiro, 31-Jan-2014.) |
⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵) → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → Σ𝑗 ∈ 𝐴 Σ𝑘 ∈ 𝐵 𝐶 = Σ𝑗 ∈ 𝐴 Σ𝑘 ∈ 𝐵 𝐷) | ||
Theorem | sumeq12dv 15057* | Equality deduction for sum. (Contributed by NM, 1-Dec-2005.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐷) | ||
Theorem | sumeq12rdv 15058* | Equality deduction for sum. (Contributed by NM, 1-Dec-2005.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐷) | ||
Theorem | sum2id 15059* | The second class argument to a sum can be chosen so that it is always a set. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jul-2013.) |
⊢ Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑘 ∈ 𝐴 ( I ‘𝐵) | ||
Theorem | sumfc 15060* | A lemma to facilitate conversions from the function form to the class-variable form of a sum. (Contributed by Mario Carneiro, 12-Aug-2013.) (Revised by Mario Carneiro, 23-Apr-2014.) |
⊢ Σ𝑗 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑗) = Σ𝑘 ∈ 𝐴 𝐵 | ||
Theorem | fz1f1o 15061* | A lemma for working with finite sums. (Contributed by Mario Carneiro, 22-Apr-2014.) |
⊢ (𝐴 ∈ Fin → (𝐴 = ∅ ∨ ((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴))) | ||
Theorem | sumrblem 15062* | Lemma for sumrb 15064. (Contributed by Mario Carneiro, 12-Aug-2013.) |
⊢ 𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 0)) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) ⇒ ⊢ ((𝜑 ∧ 𝐴 ⊆ (ℤ≥‘𝑁)) → (seq𝑀( + , 𝐹) ↾ (ℤ≥‘𝑁)) = seq𝑁( + , 𝐹)) | ||
Theorem | fsumcvg 15063* | The sequence of partial sums of a finite sum converges to the whole sum. (Contributed by Mario Carneiro, 20-Apr-2014.) |
⊢ 𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 0)) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) & ⊢ (𝜑 → 𝐴 ⊆ (𝑀...𝑁)) ⇒ ⊢ (𝜑 → seq𝑀( + , 𝐹) ⇝ (seq𝑀( + , 𝐹)‘𝑁)) | ||
Theorem | sumrb 15064* | Rebase the starting point of a sum. (Contributed by Mario Carneiro, 14-Jul-2013.) (Revised by Mario Carneiro, 9-Apr-2014.) |
⊢ 𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 0)) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ (𝜑 → 𝐴 ⊆ (ℤ≥‘𝑀)) & ⊢ (𝜑 → 𝐴 ⊆ (ℤ≥‘𝑁)) ⇒ ⊢ (𝜑 → (seq𝑀( + , 𝐹) ⇝ 𝐶 ↔ seq𝑁( + , 𝐹) ⇝ 𝐶)) | ||
Theorem | summolem3 15065* | Lemma for summo 15068. (Contributed by Mario Carneiro, 29-Mar-2014.) |
⊢ 𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 0)) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) & ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵) & ⊢ 𝐻 = (𝑛 ∈ ℕ ↦ ⦋(𝐾‘𝑛) / 𝑘⦌𝐵) & ⊢ (𝜑 → (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ)) & ⊢ (𝜑 → 𝑓:(1...𝑀)–1-1-onto→𝐴) & ⊢ (𝜑 → 𝐾:(1...𝑁)–1-1-onto→𝐴) ⇒ ⊢ (𝜑 → (seq1( + , 𝐺)‘𝑀) = (seq1( + , 𝐻)‘𝑁)) | ||
Theorem | summolem2a 15066* | Lemma for summo 15068. (Contributed by Mario Carneiro, 3-Apr-2014.) (Revised by Mario Carneiro, 20-Apr-2014.) |
⊢ 𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 0)) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) & ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵) & ⊢ 𝐻 = (𝑛 ∈ ℕ ↦ ⦋(𝐾‘𝑛) / 𝑘⦌𝐵) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐴 ⊆ (ℤ≥‘𝑀)) & ⊢ (𝜑 → 𝑓:(1...𝑁)–1-1-onto→𝐴) & ⊢ (𝜑 → 𝐾 Isom < , < ((1...(♯‘𝐴)), 𝐴)) ⇒ ⊢ (𝜑 → seq𝑀( + , 𝐹) ⇝ (seq1( + , 𝐺)‘𝑁)) | ||
Theorem | summolem2 15067* | Lemma for summo 15068. (Contributed by Mario Carneiro, 3-Apr-2014.) |
⊢ 𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 0)) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) & ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵) ⇒ ⊢ ((𝜑 ∧ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥)) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , 𝐺)‘𝑚)) → 𝑥 = 𝑦)) | ||
Theorem | summo 15068* | A sum has at most one limit. (Contributed by Mario Carneiro, 3-Apr-2014.) (Revised by Mario Carneiro, 23-Aug-2014.) |
⊢ 𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 0)) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) & ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵) ⇒ ⊢ (𝜑 → ∃*𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , 𝐺)‘𝑚)))) | ||
Theorem | zsum 15069* | Series sum with index set a subset of the upper integers. (Contributed by Mario Carneiro, 13-Jun-2019.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐴 ⊆ 𝑍) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = if(𝑘 ∈ 𝐴, 𝐵, 0)) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 = ( ⇝ ‘seq𝑀( + , 𝐹))) | ||
Theorem | isum 15070* | Series sum with an upper integer index set (i.e. an infinite series). (Contributed by Mario Carneiro, 15-Jul-2013.) (Revised by Mario Carneiro, 7-Apr-2014.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐵) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝑍 𝐵 = ( ⇝ ‘seq𝑀( + , 𝐹))) | ||
Theorem | fsum 15071* | The value of a sum over a nonempty finite set. (Contributed by Mario Carneiro, 20-Apr-2014.) (Revised by Mario Carneiro, 13-Jun-2019.) |
⊢ (𝑘 = (𝐹‘𝑛) → 𝐵 = 𝐶) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝐹:(1...𝑀)–1-1-onto→𝐴) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑀)) → (𝐺‘𝑛) = 𝐶) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 = (seq1( + , 𝐺)‘𝑀)) | ||
Theorem | sum0 15072 | Any sum over the empty set is zero. (Contributed by Mario Carneiro, 12-Aug-2013.) (Revised by Mario Carneiro, 20-Apr-2014.) |
⊢ Σ𝑘 ∈ ∅ 𝐴 = 0 | ||
Theorem | sumz 15073* | Any sum of zero over a summable set is zero. (Contributed by Mario Carneiro, 12-Aug-2013.) (Revised by Mario Carneiro, 20-Apr-2014.) |
⊢ ((𝐴 ⊆ (ℤ≥‘𝑀) ∨ 𝐴 ∈ Fin) → Σ𝑘 ∈ 𝐴 0 = 0) | ||
Theorem | fsumf1o 15074* | Re-index a finite sum using a bijection. (Contributed by Mario Carneiro, 20-Apr-2014.) |
⊢ (𝑘 = 𝐺 → 𝐵 = 𝐷) & ⊢ (𝜑 → 𝐶 ∈ Fin) & ⊢ (𝜑 → 𝐹:𝐶–1-1-onto→𝐴) & ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → (𝐹‘𝑛) = 𝐺) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑛 ∈ 𝐶 𝐷) | ||
Theorem | sumss 15075* | Change the index set to a subset in an upper integer sum. (Contributed by Mario Carneiro, 21-Apr-2014.) |
⊢ (𝜑 → 𝐴 ⊆ 𝐵) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐵 ∖ 𝐴)) → 𝐶 = 0) & ⊢ (𝜑 → 𝐵 ⊆ (ℤ≥‘𝑀)) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐶) | ||
Theorem | fsumss 15076* | Change the index set to a subset in a finite sum. (Contributed by Mario Carneiro, 21-Apr-2014.) |
⊢ (𝜑 → 𝐴 ⊆ 𝐵) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐵 ∖ 𝐴)) → 𝐶 = 0) & ⊢ (𝜑 → 𝐵 ∈ Fin) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐶) | ||
Theorem | sumss2 15077* | Change the index set of a sum by adding zeroes. (Contributed by Mario Carneiro, 15-Jul-2013.) (Revised by Mario Carneiro, 20-Apr-2014.) |
⊢ (((𝐴 ⊆ 𝐵 ∧ ∀𝑘 ∈ 𝐴 𝐶 ∈ ℂ) ∧ (𝐵 ⊆ (ℤ≥‘𝑀) ∨ 𝐵 ∈ Fin)) → Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 if(𝑘 ∈ 𝐴, 𝐶, 0)) | ||
Theorem | fsumcvg2 15078* | The sequence of partial sums of a finite sum converges to the whole sum. (Contributed by Mario Carneiro, 20-Apr-2014.) |
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) = if(𝑘 ∈ 𝐴, 𝐵, 0)) & ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐴 ⊆ (𝑀...𝑁)) ⇒ ⊢ (𝜑 → seq𝑀( + , 𝐹) ⇝ (seq𝑀( + , 𝐹)‘𝑁)) | ||
Theorem | fsumsers 15079* | Special case of series sum over a finite upper integer index set. (Contributed by Mario Carneiro, 26-Jul-2013.) (Revised by Mario Carneiro, 21-Apr-2014.) |
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) = if(𝑘 ∈ 𝐴, 𝐵, 0)) & ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐴 ⊆ (𝑀...𝑁)) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 = (seq𝑀( + , 𝐹)‘𝑁)) | ||
Theorem | fsumcvg3 15080* | A finite sum is convergent. (Contributed by Mario Carneiro, 24-Apr-2014.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐴 ⊆ 𝑍) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = if(𝑘 ∈ 𝐴, 𝐵, 0)) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ ) | ||
Theorem | fsumser 15081* | A finite sum expressed in terms of a partial sum of an infinite series. The recursive definition follows as fsum1 15096 and fsump1i 15118, which should make our notation clear and from which, along with closure fsumcl 15084, we will derive the basic properties of finite sums. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 21-Apr-2014.) |
⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) = 𝐴) & ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) & ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ (𝑀...𝑁)𝐴 = (seq𝑀( + , 𝐹)‘𝑁)) | ||
Theorem | fsumcl2lem 15082* | - Lemma for finite sum closures. (The "-" before "Lemma" forces the math content to be displayed in the Statement List - NM 11-Feb-2008.) (Contributed by Mario Carneiro, 3-Jun-2014.) |
⊢ (𝜑 → 𝑆 ⊆ ℂ) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ 𝑆) & ⊢ (𝜑 → 𝐴 ≠ ∅) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ∈ 𝑆) | ||
Theorem | fsumcllem 15083* | - Lemma for finite sum closures. (The "-" before "Lemma" forces the math content to be displayed in the Statement List - NM 11-Feb-2008.) (Contributed by NM, 9-Nov-2005.) (Revised by Mario Carneiro, 3-Jun-2014.) |
⊢ (𝜑 → 𝑆 ⊆ ℂ) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ 𝑆) & ⊢ (𝜑 → 0 ∈ 𝑆) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ∈ 𝑆) | ||
Theorem | fsumcl 15084* | Closure of a finite sum of complex numbers 𝐴(𝑘). (Contributed by NM, 9-Nov-2005.) (Revised by Mario Carneiro, 22-Apr-2014.) |
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ∈ ℂ) | ||
Theorem | fsumrecl 15085* | Closure of a finite sum of reals. (Contributed by NM, 9-Nov-2005.) (Revised by Mario Carneiro, 22-Apr-2014.) |
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ∈ ℝ) | ||
Theorem | fsumzcl 15086* | Closure of a finite sum of integers. (Contributed by NM, 9-Nov-2005.) (Revised by Mario Carneiro, 22-Apr-2014.) |
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℤ) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ∈ ℤ) | ||
Theorem | fsumnn0cl 15087* | Closure of a finite sum of nonnegative integers. (Contributed by Mario Carneiro, 23-Apr-2015.) |
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℕ0) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ∈ ℕ0) | ||
Theorem | fsumrpcl 15088* | Closure of a finite sum of positive reals. (Contributed by Mario Carneiro, 3-Jun-2014.) |
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐴 ≠ ∅) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ+) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ∈ ℝ+) | ||
Theorem | fsumzcl2 15089* | A finite sum with integer summands is an integer. (Contributed by Alexander van der Vekens, 31-Aug-2018.) |
⊢ ((𝐴 ∈ Fin ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ ℤ) → Σ𝑘 ∈ 𝐴 𝐵 ∈ ℤ) | ||
Theorem | fsumadd 15090* | The sum of two finite sums. (Contributed by NM, 14-Nov-2005.) (Revised by Mario Carneiro, 22-Apr-2014.) |
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 (𝐵 + 𝐶) = (Σ𝑘 ∈ 𝐴 𝐵 + Σ𝑘 ∈ 𝐴 𝐶)) | ||
Theorem | fsumsplit 15091* | Split a sum into two parts. (Contributed by Mario Carneiro, 18-Aug-2013.) (Revised by Mario Carneiro, 22-Apr-2014.) |
⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅) & ⊢ (𝜑 → 𝑈 = (𝐴 ∪ 𝐵)) & ⊢ (𝜑 → 𝑈 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → 𝐶 ∈ ℂ) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝑈 𝐶 = (Σ𝑘 ∈ 𝐴 𝐶 + Σ𝑘 ∈ 𝐵 𝐶)) | ||
Theorem | fsumsplitf 15092* | Split a sum into two parts. A version of fsumsplit 15091 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
⊢ Ⅎ𝑘𝜑 & ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅) & ⊢ (𝜑 → 𝑈 = (𝐴 ∪ 𝐵)) & ⊢ (𝜑 → 𝑈 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → 𝐶 ∈ ℂ) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝑈 𝐶 = (Σ𝑘 ∈ 𝐴 𝐶 + Σ𝑘 ∈ 𝐵 𝐶)) | ||
Theorem | sumsnf 15093* | A sum of a singleton is the term. A version of sumsn 15095 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
⊢ Ⅎ𝑘𝐵 & ⊢ (𝑘 = 𝑀 → 𝐴 = 𝐵) ⇒ ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐵 ∈ ℂ) → Σ𝑘 ∈ {𝑀}𝐴 = 𝐵) | ||
Theorem | fsumsplitsn 15094* | Separate out a term in a finite sum. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
⊢ Ⅎ𝑘𝜑 & ⊢ Ⅎ𝑘𝐷 & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → ¬ 𝐵 ∈ 𝐴) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ) & ⊢ (𝑘 = 𝐵 → 𝐶 = 𝐷) & ⊢ (𝜑 → 𝐷 ∈ ℂ) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ (𝐴 ∪ {𝐵})𝐶 = (Σ𝑘 ∈ 𝐴 𝐶 + 𝐷)) | ||
Theorem | sumsn 15095* | A sum of a singleton is the term. (Contributed by Mario Carneiro, 22-Apr-2014.) |
⊢ (𝑘 = 𝑀 → 𝐴 = 𝐵) ⇒ ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐵 ∈ ℂ) → Σ𝑘 ∈ {𝑀}𝐴 = 𝐵) | ||
Theorem | fsum1 15096* | The finite sum of 𝐴(𝑘) from 𝑘 = 𝑀 to 𝑀 (i.e. a sum with only one term) is 𝐵 i.e. 𝐴(𝑀). (Contributed by NM, 8-Nov-2005.) (Revised by Mario Carneiro, 21-Apr-2014.) |
⊢ (𝑘 = 𝑀 → 𝐴 = 𝐵) ⇒ ⊢ ((𝑀 ∈ ℤ ∧ 𝐵 ∈ ℂ) → Σ𝑘 ∈ (𝑀...𝑀)𝐴 = 𝐵) | ||
Theorem | sumpr 15097* | A sum over a pair is the sum of the elements. (Contributed by Thierry Arnoux, 12-Dec-2016.) |
⊢ (𝑘 = 𝐴 → 𝐶 = 𝐷) & ⊢ (𝑘 = 𝐵 → 𝐶 = 𝐸) & ⊢ (𝜑 → (𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ)) & ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) & ⊢ (𝜑 → 𝐴 ≠ 𝐵) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ {𝐴, 𝐵}𝐶 = (𝐷 + 𝐸)) | ||
Theorem | sumtp 15098* | A sum over a triple is the sum of the elements. (Contributed by AV, 24-Jul-2020.) |
⊢ (𝑘 = 𝐴 → 𝐷 = 𝐸) & ⊢ (𝑘 = 𝐵 → 𝐷 = 𝐹) & ⊢ (𝑘 = 𝐶 → 𝐷 = 𝐺) & ⊢ (𝜑 → (𝐸 ∈ ℂ ∧ 𝐹 ∈ ℂ ∧ 𝐺 ∈ ℂ)) & ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋)) & ⊢ (𝜑 → 𝐴 ≠ 𝐵) & ⊢ (𝜑 → 𝐴 ≠ 𝐶) & ⊢ (𝜑 → 𝐵 ≠ 𝐶) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ {𝐴, 𝐵, 𝐶}𝐷 = ((𝐸 + 𝐹) + 𝐺)) | ||
Theorem | sumsns 15099* | A sum of a singleton is the term. (Contributed by Mario Carneiro, 22-Apr-2014.) |
⊢ ((𝑀 ∈ 𝑉 ∧ ⦋𝑀 / 𝑘⦌𝐴 ∈ ℂ) → Σ𝑘 ∈ {𝑀}𝐴 = ⦋𝑀 / 𝑘⦌𝐴) | ||
Theorem | fsumm1 15100* | Separate out the last term in a finite sum. (Contributed by Mario Carneiro, 26-Apr-2014.) |
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) & ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ) & ⊢ (𝑘 = 𝑁 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ (𝑀...𝑁)𝐴 = (Σ𝑘 ∈ (𝑀...(𝑁 − 1))𝐴 + 𝐵)) |
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