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Type | Label | Description |
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Statement | ||
Theorem | c1lip2 24601* | C^1 functions are Lipschitz continuous on closed intervals. (Contributed by Stefan O'Rear, 16-Nov-2014.) (Revised by Stefan O'Rear, 6-May-2015.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐹 ∈ ((𝓑C𝑛‘ℝ)‘1)) & ⊢ (𝜑 → ran 𝐹 ⊆ ℝ) & ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ dom 𝐹) ⇒ ⊢ (𝜑 → ∃𝑘 ∈ ℝ ∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥)))) | ||
Theorem | c1lip3 24602* | C^1 functions are Lipschitz continuous on closed intervals. (Contributed by Stefan O'Rear, 16-Nov-2014.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → (𝐹 ↾ ℝ) ∈ ((𝓑C𝑛‘ℝ)‘1)) & ⊢ (𝜑 → (𝐹 “ ℝ) ⊆ ℝ) & ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ dom 𝐹) ⇒ ⊢ (𝜑 → ∃𝑘 ∈ ℝ ∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥)))) | ||
Theorem | dveq0 24603 | If a continuous function has zero derivative at all points on the interior of a closed interval, then it must be a constant function. (Contributed by Mario Carneiro, 2-Sep-2014.) (Proof shortened by Mario Carneiro, 3-Mar-2015.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ)) & ⊢ (𝜑 → (ℝ D 𝐹) = ((𝐴(,)𝐵) × {0})) ⇒ ⊢ (𝜑 → 𝐹 = ((𝐴[,]𝐵) × {(𝐹‘𝐴)})) | ||
Theorem | dv11cn 24604 | Two functions defined on a ball whose derivatives are the same and which are equal at any given point 𝐶 in the ball must be equal everywhere. (Contributed by Mario Carneiro, 31-Mar-2015.) |
⊢ 𝑋 = (𝐴(ball‘(abs ∘ − ))𝑅) & ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝑅 ∈ ℝ*) & ⊢ (𝜑 → 𝐹:𝑋⟶ℂ) & ⊢ (𝜑 → 𝐺:𝑋⟶ℂ) & ⊢ (𝜑 → dom (ℂ D 𝐹) = 𝑋) & ⊢ (𝜑 → (ℂ D 𝐹) = (ℂ D 𝐺)) & ⊢ (𝜑 → 𝐶 ∈ 𝑋) & ⊢ (𝜑 → (𝐹‘𝐶) = (𝐺‘𝐶)) ⇒ ⊢ (𝜑 → 𝐹 = 𝐺) | ||
Theorem | dvgt0lem1 24605 | Lemma for dvgt0 24607 and dvlt0 24608. (Contributed by Mario Carneiro, 19-Feb-2015.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ)) & ⊢ (𝜑 → (ℝ D 𝐹):(𝐴(,)𝐵)⟶𝑆) ⇒ ⊢ (((𝜑 ∧ (𝑋 ∈ (𝐴[,]𝐵) ∧ 𝑌 ∈ (𝐴[,]𝐵))) ∧ 𝑋 < 𝑌) → (((𝐹‘𝑌) − (𝐹‘𝑋)) / (𝑌 − 𝑋)) ∈ 𝑆) | ||
Theorem | dvgt0lem2 24606* | Lemma for dvgt0 24607 and dvlt0 24608. (Contributed by Mario Carneiro, 19-Feb-2015.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ)) & ⊢ (𝜑 → (ℝ D 𝐹):(𝐴(,)𝐵)⟶𝑆) & ⊢ 𝑂 Or ℝ & ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝐹‘𝑥)𝑂(𝐹‘𝑦)) ⇒ ⊢ (𝜑 → 𝐹 Isom < , 𝑂 ((𝐴[,]𝐵), ran 𝐹)) | ||
Theorem | dvgt0 24607 | A function on a closed interval with positive derivative is increasing. (Contributed by Mario Carneiro, 19-Feb-2015.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ)) & ⊢ (𝜑 → (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℝ+) ⇒ ⊢ (𝜑 → 𝐹 Isom < , < ((𝐴[,]𝐵), ran 𝐹)) | ||
Theorem | dvlt0 24608 | A function on a closed interval with negative derivative is decreasing. (Contributed by Mario Carneiro, 19-Feb-2015.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ)) & ⊢ (𝜑 → (ℝ D 𝐹):(𝐴(,)𝐵)⟶(-∞(,)0)) ⇒ ⊢ (𝜑 → 𝐹 Isom < , ◡ < ((𝐴[,]𝐵), ran 𝐹)) | ||
Theorem | dvge0 24609 | A function on a closed interval with nonnegative derivative is weakly increasing. (Contributed by Mario Carneiro, 30-Apr-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ)) & ⊢ (𝜑 → (ℝ D 𝐹):(𝐴(,)𝐵)⟶(0[,)+∞)) & ⊢ (𝜑 → 𝑋 ∈ (𝐴[,]𝐵)) & ⊢ (𝜑 → 𝑌 ∈ (𝐴[,]𝐵)) & ⊢ (𝜑 → 𝑋 ≤ 𝑌) ⇒ ⊢ (𝜑 → (𝐹‘𝑋) ≤ (𝐹‘𝑌)) | ||
Theorem | dvle 24610* | If 𝐴(𝑥), 𝐶(𝑥) are differentiable functions and 𝐴‘ ≤ 𝐶‘, then for 𝑥 ≤ 𝑦, 𝐴(𝑦) − 𝐴(𝑥) ≤ 𝐶(𝑦) − 𝐶(𝑥). (Contributed by Mario Carneiro, 16-May-2016.) |
⊢ (𝜑 → 𝑀 ∈ ℝ) & ⊢ (𝜑 → 𝑁 ∈ ℝ) & ⊢ (𝜑 → (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴) ∈ ((𝑀[,]𝑁)–cn→ℝ)) & ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴)) = (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵)) & ⊢ (𝜑 → (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐶) ∈ ((𝑀[,]𝑁)–cn→ℝ)) & ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐶)) = (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐷)) & ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 𝐵 ≤ 𝐷) & ⊢ (𝜑 → 𝑋 ∈ (𝑀[,]𝑁)) & ⊢ (𝜑 → 𝑌 ∈ (𝑀[,]𝑁)) & ⊢ (𝜑 → 𝑋 ≤ 𝑌) & ⊢ (𝑥 = 𝑋 → 𝐴 = 𝑃) & ⊢ (𝑥 = 𝑋 → 𝐶 = 𝑄) & ⊢ (𝑥 = 𝑌 → 𝐴 = 𝑅) & ⊢ (𝑥 = 𝑌 → 𝐶 = 𝑆) ⇒ ⊢ (𝜑 → (𝑅 − 𝑃) ≤ (𝑆 − 𝑄)) | ||
Theorem | dvivthlem1 24611* | Lemma for dvivth 24613. (Contributed by Mario Carneiro, 24-Feb-2015.) |
⊢ (𝜑 → 𝑀 ∈ (𝐴(,)𝐵)) & ⊢ (𝜑 → 𝑁 ∈ (𝐴(,)𝐵)) & ⊢ (𝜑 → 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℝ)) & ⊢ (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵)) & ⊢ (𝜑 → 𝑀 < 𝑁) & ⊢ (𝜑 → 𝐶 ∈ (((ℝ D 𝐹)‘𝑁)[,]((ℝ D 𝐹)‘𝑀))) & ⊢ 𝐺 = (𝑦 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘𝑦) − (𝐶 · 𝑦))) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ (𝑀[,]𝑁)((ℝ D 𝐹)‘𝑥) = 𝐶) | ||
Theorem | dvivthlem2 24612* | Lemma for dvivth 24613. (Contributed by Mario Carneiro, 20-Feb-2015.) |
⊢ (𝜑 → 𝑀 ∈ (𝐴(,)𝐵)) & ⊢ (𝜑 → 𝑁 ∈ (𝐴(,)𝐵)) & ⊢ (𝜑 → 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℝ)) & ⊢ (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵)) & ⊢ (𝜑 → 𝑀 < 𝑁) & ⊢ (𝜑 → 𝐶 ∈ (((ℝ D 𝐹)‘𝑁)[,]((ℝ D 𝐹)‘𝑀))) & ⊢ 𝐺 = (𝑦 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘𝑦) − (𝐶 · 𝑦))) ⇒ ⊢ (𝜑 → 𝐶 ∈ ran (ℝ D 𝐹)) | ||
Theorem | dvivth 24613 | Darboux' theorem, or the intermediate value theorem for derivatives. A differentiable function's derivative satisfies the intermediate value property, even though it may not be continuous (so that ivthicc 24062 does not directly apply). (Contributed by Mario Carneiro, 24-Feb-2015.) |
⊢ (𝜑 → 𝑀 ∈ (𝐴(,)𝐵)) & ⊢ (𝜑 → 𝑁 ∈ (𝐴(,)𝐵)) & ⊢ (𝜑 → 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℝ)) & ⊢ (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵)) ⇒ ⊢ (𝜑 → (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)) ⊆ ran (ℝ D 𝐹)) | ||
Theorem | dvne0 24614 | A function on a closed interval with nonzero derivative is either monotone increasing or monotone decreasing. (Contributed by Mario Carneiro, 19-Feb-2015.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ)) & ⊢ (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵)) & ⊢ (𝜑 → ¬ 0 ∈ ran (ℝ D 𝐹)) ⇒ ⊢ (𝜑 → (𝐹 Isom < , < ((𝐴[,]𝐵), ran 𝐹) ∨ 𝐹 Isom < , ◡ < ((𝐴[,]𝐵), ran 𝐹))) | ||
Theorem | dvne0f1 24615 | A function on a closed interval with nonzero derivative is one-to-one. (Contributed by Mario Carneiro, 19-Feb-2015.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ)) & ⊢ (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵)) & ⊢ (𝜑 → ¬ 0 ∈ ran (ℝ D 𝐹)) ⇒ ⊢ (𝜑 → 𝐹:(𝐴[,]𝐵)–1-1→ℝ) | ||
Theorem | lhop1lem 24616* | Lemma for lhop1 24617. (Contributed by Mario Carneiro, 29-Dec-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐴 < 𝐵) & ⊢ (𝜑 → 𝐹:(𝐴(,)𝐵)⟶ℝ) & ⊢ (𝜑 → 𝐺:(𝐴(,)𝐵)⟶ℝ) & ⊢ (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵)) & ⊢ (𝜑 → dom (ℝ D 𝐺) = (𝐴(,)𝐵)) & ⊢ (𝜑 → 0 ∈ (𝐹 limℂ 𝐴)) & ⊢ (𝜑 → 0 ∈ (𝐺 limℂ 𝐴)) & ⊢ (𝜑 → ¬ 0 ∈ ran 𝐺) & ⊢ (𝜑 → ¬ 0 ∈ ran (ℝ D 𝐺)) & ⊢ (𝜑 → 𝐶 ∈ ((𝑧 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑧) / ((ℝ D 𝐺)‘𝑧))) limℂ 𝐴)) & ⊢ (𝜑 → 𝐸 ∈ ℝ+) & ⊢ (𝜑 → 𝐷 ∈ ℝ) & ⊢ (𝜑 → 𝐷 ≤ 𝐵) & ⊢ (𝜑 → 𝑋 ∈ (𝐴(,)𝐷)) & ⊢ (𝜑 → ∀𝑡 ∈ (𝐴(,)𝐷)(abs‘((((ℝ D 𝐹)‘𝑡) / ((ℝ D 𝐺)‘𝑡)) − 𝐶)) < 𝐸) & ⊢ 𝑅 = (𝐴 + (𝑟 / 2)) ⇒ ⊢ (𝜑 → (abs‘(((𝐹‘𝑋) / (𝐺‘𝑋)) − 𝐶)) < (2 · 𝐸)) | ||
Theorem | lhop1 24617* | L'Hôpital's Rule for limits from the right. If 𝐹 and 𝐺 are differentiable real functions on (𝐴, 𝐵), and 𝐹 and 𝐺 both approach 0 at 𝐴, and 𝐺(𝑥) and 𝐺' (𝑥) are not zero on (𝐴, 𝐵), and the limit of 𝐹' (𝑥) / 𝐺' (𝑥) at 𝐴 is 𝐶, then the limit 𝐹(𝑥) / 𝐺(𝑥) at 𝐴 also exists and equals 𝐶. (Contributed by Mario Carneiro, 29-Dec-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐴 < 𝐵) & ⊢ (𝜑 → 𝐹:(𝐴(,)𝐵)⟶ℝ) & ⊢ (𝜑 → 𝐺:(𝐴(,)𝐵)⟶ℝ) & ⊢ (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵)) & ⊢ (𝜑 → dom (ℝ D 𝐺) = (𝐴(,)𝐵)) & ⊢ (𝜑 → 0 ∈ (𝐹 limℂ 𝐴)) & ⊢ (𝜑 → 0 ∈ (𝐺 limℂ 𝐴)) & ⊢ (𝜑 → ¬ 0 ∈ ran 𝐺) & ⊢ (𝜑 → ¬ 0 ∈ ran (ℝ D 𝐺)) & ⊢ (𝜑 → 𝐶 ∈ ((𝑧 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑧) / ((ℝ D 𝐺)‘𝑧))) limℂ 𝐴)) ⇒ ⊢ (𝜑 → 𝐶 ∈ ((𝑧 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘𝑧) / (𝐺‘𝑧))) limℂ 𝐴)) | ||
Theorem | lhop2 24618* | L'Hôpital's Rule for limits from the left. If 𝐹 and 𝐺 are differentiable real functions on (𝐴, 𝐵), and 𝐹 and 𝐺 both approach 0 at 𝐵, and 𝐺(𝑥) and 𝐺' (𝑥) are not zero on (𝐴, 𝐵), and the limit of 𝐹' (𝑥) / 𝐺' (𝑥) at 𝐵 is 𝐶, then the limit 𝐹(𝑥) / 𝐺(𝑥) at 𝐵 also exists and equals 𝐶. (Contributed by Mario Carneiro, 29-Dec-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 𝐵) & ⊢ (𝜑 → 𝐹:(𝐴(,)𝐵)⟶ℝ) & ⊢ (𝜑 → 𝐺:(𝐴(,)𝐵)⟶ℝ) & ⊢ (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵)) & ⊢ (𝜑 → dom (ℝ D 𝐺) = (𝐴(,)𝐵)) & ⊢ (𝜑 → 0 ∈ (𝐹 limℂ 𝐵)) & ⊢ (𝜑 → 0 ∈ (𝐺 limℂ 𝐵)) & ⊢ (𝜑 → ¬ 0 ∈ ran 𝐺) & ⊢ (𝜑 → ¬ 0 ∈ ran (ℝ D 𝐺)) & ⊢ (𝜑 → 𝐶 ∈ ((𝑧 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑧) / ((ℝ D 𝐺)‘𝑧))) limℂ 𝐵)) ⇒ ⊢ (𝜑 → 𝐶 ∈ ((𝑧 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘𝑧) / (𝐺‘𝑧))) limℂ 𝐵)) | ||
Theorem | lhop 24619* | L'Hôpital's Rule. If 𝐼 is an open set of the reals, 𝐹 and 𝐺 are real functions on 𝐴 containing all of 𝐼 except possibly 𝐵, which are differentiable everywhere on 𝐼 ∖ {𝐵}, 𝐹 and 𝐺 both approach 0, and the limit of 𝐹' (𝑥) / 𝐺' (𝑥) at 𝐵 is 𝐶, then the limit 𝐹(𝑥) / 𝐺(𝑥) at 𝐵 also exists and equals 𝐶. This is Metamath 100 proof #64. (Contributed by Mario Carneiro, 30-Dec-2016.) |
⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ (𝜑 → 𝐹:𝐴⟶ℝ) & ⊢ (𝜑 → 𝐺:𝐴⟶ℝ) & ⊢ (𝜑 → 𝐼 ∈ (topGen‘ran (,))) & ⊢ (𝜑 → 𝐵 ∈ 𝐼) & ⊢ 𝐷 = (𝐼 ∖ {𝐵}) & ⊢ (𝜑 → 𝐷 ⊆ dom (ℝ D 𝐹)) & ⊢ (𝜑 → 𝐷 ⊆ dom (ℝ D 𝐺)) & ⊢ (𝜑 → 0 ∈ (𝐹 limℂ 𝐵)) & ⊢ (𝜑 → 0 ∈ (𝐺 limℂ 𝐵)) & ⊢ (𝜑 → ¬ 0 ∈ (𝐺 “ 𝐷)) & ⊢ (𝜑 → ¬ 0 ∈ ((ℝ D 𝐺) “ 𝐷)) & ⊢ (𝜑 → 𝐶 ∈ ((𝑧 ∈ 𝐷 ↦ (((ℝ D 𝐹)‘𝑧) / ((ℝ D 𝐺)‘𝑧))) limℂ 𝐵)) ⇒ ⊢ (𝜑 → 𝐶 ∈ ((𝑧 ∈ 𝐷 ↦ ((𝐹‘𝑧) / (𝐺‘𝑧))) limℂ 𝐵)) | ||
Theorem | dvcnvrelem1 24620 | Lemma for dvcnvre 24622. (Contributed by Mario Carneiro, 24-Feb-2015.) |
⊢ (𝜑 → 𝐹 ∈ (𝑋–cn→ℝ)) & ⊢ (𝜑 → dom (ℝ D 𝐹) = 𝑋) & ⊢ (𝜑 → ¬ 0 ∈ ran (ℝ D 𝐹)) & ⊢ (𝜑 → 𝐹:𝑋–1-1-onto→𝑌) & ⊢ (𝜑 → 𝐶 ∈ 𝑋) & ⊢ (𝜑 → 𝑅 ∈ ℝ+) & ⊢ (𝜑 → ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) ⊆ 𝑋) ⇒ ⊢ (𝜑 → (𝐹‘𝐶) ∈ ((int‘(topGen‘ran (,)))‘(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))))) | ||
Theorem | dvcnvrelem2 24621 | Lemma for dvcnvre 24622. (Contributed by Mario Carneiro, 19-Feb-2015.) (Revised by Mario Carneiro, 8-Sep-2015.) |
⊢ (𝜑 → 𝐹 ∈ (𝑋–cn→ℝ)) & ⊢ (𝜑 → dom (ℝ D 𝐹) = 𝑋) & ⊢ (𝜑 → ¬ 0 ∈ ran (ℝ D 𝐹)) & ⊢ (𝜑 → 𝐹:𝑋–1-1-onto→𝑌) & ⊢ (𝜑 → 𝐶 ∈ 𝑋) & ⊢ (𝜑 → 𝑅 ∈ ℝ+) & ⊢ (𝜑 → ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) ⊆ 𝑋) & ⊢ 𝑇 = (topGen‘ran (,)) & ⊢ 𝐽 = (TopOpen‘ℂfld) & ⊢ 𝑀 = (𝐽 ↾t 𝑋) & ⊢ 𝑁 = (𝐽 ↾t 𝑌) ⇒ ⊢ (𝜑 → ((𝐹‘𝐶) ∈ ((int‘𝑇)‘𝑌) ∧ ◡𝐹 ∈ ((𝑁 CnP 𝑀)‘(𝐹‘𝐶)))) | ||
Theorem | dvcnvre 24622* | The derivative rule for inverse functions. If 𝐹 is a continuous and differentiable bijective function from 𝑋 to 𝑌 which never has derivative 0, then ◡𝐹 is also differentiable, and its derivative is the reciprocal of the derivative of 𝐹. (Contributed by Mario Carneiro, 24-Feb-2015.) |
⊢ (𝜑 → 𝐹 ∈ (𝑋–cn→ℝ)) & ⊢ (𝜑 → dom (ℝ D 𝐹) = 𝑋) & ⊢ (𝜑 → ¬ 0 ∈ ran (ℝ D 𝐹)) & ⊢ (𝜑 → 𝐹:𝑋–1-1-onto→𝑌) ⇒ ⊢ (𝜑 → (ℝ D ◡𝐹) = (𝑥 ∈ 𝑌 ↦ (1 / ((ℝ D 𝐹)‘(◡𝐹‘𝑥))))) | ||
Theorem | dvcvx 24623 | A real function with strictly increasing derivative is strictly convex. (Contributed by Mario Carneiro, 20-Jun-2015.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 𝐵) & ⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ)) & ⊢ (𝜑 → (ℝ D 𝐹) Isom < , < ((𝐴(,)𝐵), 𝑊)) & ⊢ (𝜑 → 𝑇 ∈ (0(,)1)) & ⊢ 𝐶 = ((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵)) ⇒ ⊢ (𝜑 → (𝐹‘𝐶) < ((𝑇 · (𝐹‘𝐴)) + ((1 − 𝑇) · (𝐹‘𝐵)))) | ||
Theorem | dvfsumle 24624* | Compare a finite sum to an integral (the integral here is given as a function with a known derivative). (Contributed by Mario Carneiro, 14-May-2016.) |
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) & ⊢ (𝜑 → (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴) ∈ ((𝑀[,]𝑁)–cn→ℝ)) & ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴)) = (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵)) & ⊢ (𝑥 = 𝑀 → 𝐴 = 𝐶) & ⊢ (𝑥 = 𝑁 → 𝐴 = 𝐷) & ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑋 ∈ ℝ) & ⊢ ((𝜑 ∧ (𝑘 ∈ (𝑀..^𝑁) ∧ 𝑥 ∈ (𝑘(,)(𝑘 + 1)))) → 𝑋 ≤ 𝐵) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ (𝑀..^𝑁)𝑋 ≤ (𝐷 − 𝐶)) | ||
Theorem | dvfsumge 24625* | Compare a finite sum to an integral (the integral here is given as a function with a known derivative). (Contributed by Mario Carneiro, 14-May-2016.) |
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) & ⊢ (𝜑 → (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴) ∈ ((𝑀[,]𝑁)–cn→ℝ)) & ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴)) = (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵)) & ⊢ (𝑥 = 𝑀 → 𝐴 = 𝐶) & ⊢ (𝑥 = 𝑁 → 𝐴 = 𝐷) & ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑋 ∈ ℝ) & ⊢ ((𝜑 ∧ (𝑘 ∈ (𝑀..^𝑁) ∧ 𝑥 ∈ (𝑘(,)(𝑘 + 1)))) → 𝐵 ≤ 𝑋) ⇒ ⊢ (𝜑 → (𝐷 − 𝐶) ≤ Σ𝑘 ∈ (𝑀..^𝑁)𝑋) | ||
Theorem | dvfsumabs 24626* | Compare a finite sum to an integral (the integral here is given as a function with a known derivative). (Contributed by Mario Carneiro, 14-May-2016.) |
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) & ⊢ (𝜑 → (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴) ∈ ((𝑀[,]𝑁)–cn→ℂ)) & ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴)) = (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵)) & ⊢ (𝑥 = 𝑀 → 𝐴 = 𝐶) & ⊢ (𝑥 = 𝑁 → 𝐴 = 𝐷) & ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑋 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑌 ∈ ℝ) & ⊢ ((𝜑 ∧ (𝑘 ∈ (𝑀..^𝑁) ∧ 𝑥 ∈ (𝑘(,)(𝑘 + 1)))) → (abs‘(𝑋 − 𝐵)) ≤ 𝑌) ⇒ ⊢ (𝜑 → (abs‘(Σ𝑘 ∈ (𝑀..^𝑁)𝑋 − (𝐷 − 𝐶))) ≤ Σ𝑘 ∈ (𝑀..^𝑁)𝑌) | ||
Theorem | dvmptrecl 24627* | Real closure of a derivative. (Contributed by Mario Carneiro, 18-May-2016.) |
⊢ (𝜑 → 𝑆 ⊆ ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → (ℝ D (𝑥 ∈ 𝑆 ↦ 𝐴)) = (𝑥 ∈ 𝑆 ↦ 𝐵)) ⇒ ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐵 ∈ ℝ) | ||
Theorem | dvfsumrlimf 24628* | Lemma for dvfsumrlim 24634. (Contributed by Mario Carneiro, 18-May-2016.) |
⊢ 𝑆 = (𝑇(,)+∞) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐷 ∈ ℝ) & ⊢ (𝜑 → 𝑀 ≤ (𝐷 + 1)) & ⊢ (𝜑 → 𝑇 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐵 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝐵 ∈ ℝ) & ⊢ (𝜑 → (ℝ D (𝑥 ∈ 𝑆 ↦ 𝐴)) = (𝑥 ∈ 𝑆 ↦ 𝐵)) & ⊢ (𝑥 = 𝑘 → 𝐵 = 𝐶) & ⊢ 𝐺 = (𝑥 ∈ 𝑆 ↦ (Σ𝑘 ∈ (𝑀...(⌊‘𝑥))𝐶 − 𝐴)) ⇒ ⊢ (𝜑 → 𝐺:𝑆⟶ℝ) | ||
Theorem | dvfsumlem1 24629* | Lemma for dvfsumrlim 24634. (Contributed by Mario Carneiro, 17-May-2016.) |
⊢ 𝑆 = (𝑇(,)+∞) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐷 ∈ ℝ) & ⊢ (𝜑 → 𝑀 ≤ (𝐷 + 1)) & ⊢ (𝜑 → 𝑇 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐵 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝐵 ∈ ℝ) & ⊢ (𝜑 → (ℝ D (𝑥 ∈ 𝑆 ↦ 𝐴)) = (𝑥 ∈ 𝑆 ↦ 𝐵)) & ⊢ (𝑥 = 𝑘 → 𝐵 = 𝐶) & ⊢ (𝜑 → 𝑈 ∈ ℝ*) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆) ∧ (𝐷 ≤ 𝑥 ∧ 𝑥 ≤ 𝑘 ∧ 𝑘 ≤ 𝑈)) → 𝐶 ≤ 𝐵) & ⊢ 𝐻 = (𝑥 ∈ 𝑆 ↦ (((𝑥 − (⌊‘𝑥)) · 𝐵) + (Σ𝑘 ∈ (𝑀...(⌊‘𝑥))𝐶 − 𝐴))) & ⊢ (𝜑 → 𝑋 ∈ 𝑆) & ⊢ (𝜑 → 𝑌 ∈ 𝑆) & ⊢ (𝜑 → 𝐷 ≤ 𝑋) & ⊢ (𝜑 → 𝑋 ≤ 𝑌) & ⊢ (𝜑 → 𝑌 ≤ 𝑈) & ⊢ (𝜑 → 𝑌 ≤ ((⌊‘𝑋) + 1)) ⇒ ⊢ (𝜑 → (𝐻‘𝑌) = ((((𝑌 − (⌊‘𝑋)) · ⦋𝑌 / 𝑥⦌𝐵) − ⦋𝑌 / 𝑥⦌𝐴) + Σ𝑘 ∈ (𝑀...(⌊‘𝑋))𝐶)) | ||
Theorem | dvfsumlem2 24630* | Lemma for dvfsumrlim 24634. (Contributed by Mario Carneiro, 17-May-2016.) |
⊢ 𝑆 = (𝑇(,)+∞) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐷 ∈ ℝ) & ⊢ (𝜑 → 𝑀 ≤ (𝐷 + 1)) & ⊢ (𝜑 → 𝑇 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐵 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝐵 ∈ ℝ) & ⊢ (𝜑 → (ℝ D (𝑥 ∈ 𝑆 ↦ 𝐴)) = (𝑥 ∈ 𝑆 ↦ 𝐵)) & ⊢ (𝑥 = 𝑘 → 𝐵 = 𝐶) & ⊢ (𝜑 → 𝑈 ∈ ℝ*) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆) ∧ (𝐷 ≤ 𝑥 ∧ 𝑥 ≤ 𝑘 ∧ 𝑘 ≤ 𝑈)) → 𝐶 ≤ 𝐵) & ⊢ 𝐻 = (𝑥 ∈ 𝑆 ↦ (((𝑥 − (⌊‘𝑥)) · 𝐵) + (Σ𝑘 ∈ (𝑀...(⌊‘𝑥))𝐶 − 𝐴))) & ⊢ (𝜑 → 𝑋 ∈ 𝑆) & ⊢ (𝜑 → 𝑌 ∈ 𝑆) & ⊢ (𝜑 → 𝐷 ≤ 𝑋) & ⊢ (𝜑 → 𝑋 ≤ 𝑌) & ⊢ (𝜑 → 𝑌 ≤ 𝑈) & ⊢ (𝜑 → 𝑌 ≤ ((⌊‘𝑋) + 1)) ⇒ ⊢ (𝜑 → ((𝐻‘𝑌) ≤ (𝐻‘𝑋) ∧ ((𝐻‘𝑋) − ⦋𝑋 / 𝑥⦌𝐵) ≤ ((𝐻‘𝑌) − ⦋𝑌 / 𝑥⦌𝐵))) | ||
Theorem | dvfsumlem3 24631* | Lemma for dvfsumrlim 24634. (Contributed by Mario Carneiro, 17-May-2016.) |
⊢ 𝑆 = (𝑇(,)+∞) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐷 ∈ ℝ) & ⊢ (𝜑 → 𝑀 ≤ (𝐷 + 1)) & ⊢ (𝜑 → 𝑇 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐵 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝐵 ∈ ℝ) & ⊢ (𝜑 → (ℝ D (𝑥 ∈ 𝑆 ↦ 𝐴)) = (𝑥 ∈ 𝑆 ↦ 𝐵)) & ⊢ (𝑥 = 𝑘 → 𝐵 = 𝐶) & ⊢ (𝜑 → 𝑈 ∈ ℝ*) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆) ∧ (𝐷 ≤ 𝑥 ∧ 𝑥 ≤ 𝑘 ∧ 𝑘 ≤ 𝑈)) → 𝐶 ≤ 𝐵) & ⊢ 𝐻 = (𝑥 ∈ 𝑆 ↦ (((𝑥 − (⌊‘𝑥)) · 𝐵) + (Σ𝑘 ∈ (𝑀...(⌊‘𝑥))𝐶 − 𝐴))) & ⊢ (𝜑 → 𝑋 ∈ 𝑆) & ⊢ (𝜑 → 𝑌 ∈ 𝑆) & ⊢ (𝜑 → 𝐷 ≤ 𝑋) & ⊢ (𝜑 → 𝑋 ≤ 𝑌) & ⊢ (𝜑 → 𝑌 ≤ 𝑈) ⇒ ⊢ (𝜑 → ((𝐻‘𝑌) ≤ (𝐻‘𝑋) ∧ ((𝐻‘𝑋) − ⦋𝑋 / 𝑥⦌𝐵) ≤ ((𝐻‘𝑌) − ⦋𝑌 / 𝑥⦌𝐵))) | ||
Theorem | dvfsumlem4 24632* | Lemma for dvfsumrlim 24634. (Contributed by Mario Carneiro, 18-May-2016.) |
⊢ 𝑆 = (𝑇(,)+∞) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐷 ∈ ℝ) & ⊢ (𝜑 → 𝑀 ≤ (𝐷 + 1)) & ⊢ (𝜑 → 𝑇 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐵 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝐵 ∈ ℝ) & ⊢ (𝜑 → (ℝ D (𝑥 ∈ 𝑆 ↦ 𝐴)) = (𝑥 ∈ 𝑆 ↦ 𝐵)) & ⊢ (𝑥 = 𝑘 → 𝐵 = 𝐶) & ⊢ (𝜑 → 𝑈 ∈ ℝ*) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆) ∧ (𝐷 ≤ 𝑥 ∧ 𝑥 ≤ 𝑘 ∧ 𝑘 ≤ 𝑈)) → 𝐶 ≤ 𝐵) & ⊢ 𝐺 = (𝑥 ∈ 𝑆 ↦ (Σ𝑘 ∈ (𝑀...(⌊‘𝑥))𝐶 − 𝐴)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥 ∧ 𝑥 ≤ 𝑈)) → 0 ≤ 𝐵) & ⊢ (𝜑 → 𝑋 ∈ 𝑆) & ⊢ (𝜑 → 𝑌 ∈ 𝑆) & ⊢ (𝜑 → 𝐷 ≤ 𝑋) & ⊢ (𝜑 → 𝑋 ≤ 𝑌) & ⊢ (𝜑 → 𝑌 ≤ 𝑈) ⇒ ⊢ (𝜑 → (abs‘((𝐺‘𝑌) − (𝐺‘𝑋))) ≤ ⦋𝑋 / 𝑥⦌𝐵) | ||
Theorem | dvfsumrlimge0 24633* | Lemma for dvfsumrlim 24634. Satisfy the assumption of dvfsumlem4 24632. (Contributed by Mario Carneiro, 18-May-2016.) |
⊢ 𝑆 = (𝑇(,)+∞) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐷 ∈ ℝ) & ⊢ (𝜑 → 𝑀 ≤ (𝐷 + 1)) & ⊢ (𝜑 → 𝑇 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐵 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝐵 ∈ ℝ) & ⊢ (𝜑 → (ℝ D (𝑥 ∈ 𝑆 ↦ 𝐴)) = (𝑥 ∈ 𝑆 ↦ 𝐵)) & ⊢ (𝑥 = 𝑘 → 𝐵 = 𝐶) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆) ∧ (𝐷 ≤ 𝑥 ∧ 𝑥 ≤ 𝑘)) → 𝐶 ≤ 𝐵) & ⊢ 𝐺 = (𝑥 ∈ 𝑆 ↦ (Σ𝑘 ∈ (𝑀...(⌊‘𝑥))𝐶 − 𝐴)) & ⊢ (𝜑 → (𝑥 ∈ 𝑆 ↦ 𝐵) ⇝𝑟 0) ⇒ ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → 0 ≤ 𝐵) | ||
Theorem | dvfsumrlim 24634* | Compare a finite sum to an integral (the integral here is given as a function with a known derivative). The statement here says that if 𝑥 ∈ 𝑆 ↦ 𝐵 is a decreasing function with antiderivative 𝐴 converging to zero, then the difference between Σ𝑘 ∈ (𝑀...(⌊‘𝑥))𝐵(𝑘) and 𝐴(𝑥) = ∫𝑢 ∈ (𝑀[,]𝑥)𝐵(𝑢) d𝑢 converges to a constant limit value, with the remainder term bounded by 𝐵(𝑥). (Contributed by Mario Carneiro, 18-May-2016.) |
⊢ 𝑆 = (𝑇(,)+∞) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐷 ∈ ℝ) & ⊢ (𝜑 → 𝑀 ≤ (𝐷 + 1)) & ⊢ (𝜑 → 𝑇 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐵 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝐵 ∈ ℝ) & ⊢ (𝜑 → (ℝ D (𝑥 ∈ 𝑆 ↦ 𝐴)) = (𝑥 ∈ 𝑆 ↦ 𝐵)) & ⊢ (𝑥 = 𝑘 → 𝐵 = 𝐶) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆) ∧ (𝐷 ≤ 𝑥 ∧ 𝑥 ≤ 𝑘)) → 𝐶 ≤ 𝐵) & ⊢ 𝐺 = (𝑥 ∈ 𝑆 ↦ (Σ𝑘 ∈ (𝑀...(⌊‘𝑥))𝐶 − 𝐴)) & ⊢ (𝜑 → (𝑥 ∈ 𝑆 ↦ 𝐵) ⇝𝑟 0) ⇒ ⊢ (𝜑 → 𝐺 ∈ dom ⇝𝑟 ) | ||
Theorem | dvfsumrlim2 24635* | Compare a finite sum to an integral (the integral here is given as a function with a known derivative). The statement here says that if 𝑥 ∈ 𝑆 ↦ 𝐵 is a decreasing function with antiderivative 𝐴 converging to zero, then the difference between Σ𝑘 ∈ (𝑀...(⌊‘𝑥))𝐵(𝑘) and ∫𝑢 ∈ (𝑀[,]𝑥)𝐵(𝑢) d𝑢 = 𝐴(𝑥) converges to a constant limit value, with the remainder term bounded by 𝐵(𝑥). (Contributed by Mario Carneiro, 18-May-2016.) |
⊢ 𝑆 = (𝑇(,)+∞) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐷 ∈ ℝ) & ⊢ (𝜑 → 𝑀 ≤ (𝐷 + 1)) & ⊢ (𝜑 → 𝑇 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐵 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝐵 ∈ ℝ) & ⊢ (𝜑 → (ℝ D (𝑥 ∈ 𝑆 ↦ 𝐴)) = (𝑥 ∈ 𝑆 ↦ 𝐵)) & ⊢ (𝑥 = 𝑘 → 𝐵 = 𝐶) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆) ∧ (𝐷 ≤ 𝑥 ∧ 𝑥 ≤ 𝑘)) → 𝐶 ≤ 𝐵) & ⊢ 𝐺 = (𝑥 ∈ 𝑆 ↦ (Σ𝑘 ∈ (𝑀...(⌊‘𝑥))𝐶 − 𝐴)) & ⊢ (𝜑 → (𝑥 ∈ 𝑆 ↦ 𝐵) ⇝𝑟 0) & ⊢ (𝜑 → 𝑋 ∈ 𝑆) & ⊢ (𝜑 → 𝐷 ≤ 𝑋) ⇒ ⊢ ((𝜑 ∧ 𝐺 ⇝𝑟 𝐿) → (abs‘((𝐺‘𝑋) − 𝐿)) ≤ ⦋𝑋 / 𝑥⦌𝐵) | ||
Theorem | dvfsumrlim3 24636* | Conjoin the statements of dvfsumrlim 24634 and dvfsumrlim2 24635. (This is useful as a target for lemmas, because the hypotheses to this theorem are complex, and we don't want to repeat ourselves.) (Contributed by Mario Carneiro, 18-May-2016.) |
⊢ 𝑆 = (𝑇(,)+∞) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐷 ∈ ℝ) & ⊢ (𝜑 → 𝑀 ≤ (𝐷 + 1)) & ⊢ (𝜑 → 𝑇 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐵 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝐵 ∈ ℝ) & ⊢ (𝜑 → (ℝ D (𝑥 ∈ 𝑆 ↦ 𝐴)) = (𝑥 ∈ 𝑆 ↦ 𝐵)) & ⊢ (𝑥 = 𝑘 → 𝐵 = 𝐶) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆) ∧ (𝐷 ≤ 𝑥 ∧ 𝑥 ≤ 𝑘)) → 𝐶 ≤ 𝐵) & ⊢ 𝐺 = (𝑥 ∈ 𝑆 ↦ (Σ𝑘 ∈ (𝑀...(⌊‘𝑥))𝐶 − 𝐴)) & ⊢ (𝜑 → (𝑥 ∈ 𝑆 ↦ 𝐵) ⇝𝑟 0) & ⊢ (𝑥 = 𝑋 → 𝐵 = 𝐸) ⇒ ⊢ (𝜑 → (𝐺:𝑆⟶ℝ ∧ 𝐺 ∈ dom ⇝𝑟 ∧ ((𝐺 ⇝𝑟 𝐿 ∧ 𝑋 ∈ 𝑆 ∧ 𝐷 ≤ 𝑋) → (abs‘((𝐺‘𝑋) − 𝐿)) ≤ 𝐸))) | ||
Theorem | dvfsum2 24637* | The reverse of dvfsumrlim 24634, when comparing a finite sum of increasing terms to an integral. In this case there is no point in stating the limit properties, because the terms of the sum aren't approaching zero, but there is nevertheless still a natural asymptotic statement that can be made. (Contributed by Mario Carneiro, 20-May-2016.) |
⊢ 𝑆 = (𝑇(,)+∞) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐷 ∈ ℝ) & ⊢ (𝜑 → 𝑈 ∈ ℝ*) & ⊢ (𝜑 → 𝑀 ≤ (𝐷 + 1)) & ⊢ (𝜑 → 𝑇 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐵 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝐵 ∈ ℝ) & ⊢ (𝜑 → (ℝ D (𝑥 ∈ 𝑆 ↦ 𝐴)) = (𝑥 ∈ 𝑆 ↦ 𝐵)) & ⊢ (𝑥 = 𝑘 → 𝐵 = 𝐶) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆) ∧ (𝐷 ≤ 𝑥 ∧ 𝑥 ≤ 𝑘 ∧ 𝑘 ≤ 𝑈)) → 𝐵 ≤ 𝐶) & ⊢ 𝐺 = (𝑥 ∈ 𝑆 ↦ (Σ𝑘 ∈ (𝑀...(⌊‘𝑥))𝐶 − 𝐴)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → 0 ≤ 𝐵) & ⊢ (𝜑 → 𝑋 ∈ 𝑆) & ⊢ (𝜑 → 𝑌 ∈ 𝑆) & ⊢ (𝜑 → 𝐷 ≤ 𝑋) & ⊢ (𝜑 → 𝑋 ≤ 𝑌) & ⊢ (𝜑 → 𝑌 ≤ 𝑈) & ⊢ (𝑥 = 𝑌 → 𝐵 = 𝐸) ⇒ ⊢ (𝜑 → (abs‘((𝐺‘𝑌) − (𝐺‘𝑋))) ≤ 𝐸) | ||
Theorem | ftc1lem1 24638* | Lemma for ftc1a 24640 and ftc1 24645. (Contributed by Mario Carneiro, 31-Aug-2014.) |
⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)(𝐹‘𝑡) d𝑡) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) & ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ 𝐷) & ⊢ (𝜑 → 𝐷 ⊆ ℝ) & ⊢ (𝜑 → 𝐹 ∈ 𝐿1) & ⊢ (𝜑 → 𝐹:𝐷⟶ℂ) & ⊢ (𝜑 → 𝑋 ∈ (𝐴[,]𝐵)) & ⊢ (𝜑 → 𝑌 ∈ (𝐴[,]𝐵)) ⇒ ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑌) → ((𝐺‘𝑌) − (𝐺‘𝑋)) = ∫(𝑋(,)𝑌)(𝐹‘𝑡) d𝑡) | ||
Theorem | ftc1lem2 24639* | Lemma for ftc1 24645. (Contributed by Mario Carneiro, 12-Aug-2014.) |
⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)(𝐹‘𝑡) d𝑡) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) & ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ 𝐷) & ⊢ (𝜑 → 𝐷 ⊆ ℝ) & ⊢ (𝜑 → 𝐹 ∈ 𝐿1) & ⊢ (𝜑 → 𝐹:𝐷⟶ℂ) ⇒ ⊢ (𝜑 → 𝐺:(𝐴[,]𝐵)⟶ℂ) | ||
Theorem | ftc1a 24640* | The Fundamental Theorem of Calculus, part one. The function 𝐺 formed by varying the right endpoint of an integral of 𝐹 is continuous if 𝐹 is integrable. (Contributed by Mario Carneiro, 1-Sep-2014.) |
⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)(𝐹‘𝑡) d𝑡) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) & ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ 𝐷) & ⊢ (𝜑 → 𝐷 ⊆ ℝ) & ⊢ (𝜑 → 𝐹 ∈ 𝐿1) & ⊢ (𝜑 → 𝐹:𝐷⟶ℂ) ⇒ ⊢ (𝜑 → 𝐺 ∈ ((𝐴[,]𝐵)–cn→ℂ)) | ||
Theorem | ftc1lem3 24641* | Lemma for ftc1 24645. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 8-Sep-2015.) |
⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)(𝐹‘𝑡) d𝑡) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) & ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ 𝐷) & ⊢ (𝜑 → 𝐷 ⊆ ℝ) & ⊢ (𝜑 → 𝐹 ∈ 𝐿1) & ⊢ (𝜑 → 𝐶 ∈ (𝐴(,)𝐵)) & ⊢ (𝜑 → 𝐹 ∈ ((𝐾 CnP 𝐿)‘𝐶)) & ⊢ 𝐽 = (𝐿 ↾t ℝ) & ⊢ 𝐾 = (𝐿 ↾t 𝐷) & ⊢ 𝐿 = (TopOpen‘ℂfld) ⇒ ⊢ (𝜑 → 𝐹:𝐷⟶ℂ) | ||
Theorem | ftc1lem4 24642* | Lemma for ftc1 24645. (Contributed by Mario Carneiro, 31-Aug-2014.) |
⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)(𝐹‘𝑡) d𝑡) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) & ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ 𝐷) & ⊢ (𝜑 → 𝐷 ⊆ ℝ) & ⊢ (𝜑 → 𝐹 ∈ 𝐿1) & ⊢ (𝜑 → 𝐶 ∈ (𝐴(,)𝐵)) & ⊢ (𝜑 → 𝐹 ∈ ((𝐾 CnP 𝐿)‘𝐶)) & ⊢ 𝐽 = (𝐿 ↾t ℝ) & ⊢ 𝐾 = (𝐿 ↾t 𝐷) & ⊢ 𝐿 = (TopOpen‘ℂfld) & ⊢ 𝐻 = (𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝐶}) ↦ (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))) & ⊢ (𝜑 → 𝐸 ∈ ℝ+) & ⊢ (𝜑 → 𝑅 ∈ ℝ+) & ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → ((abs‘(𝑦 − 𝐶)) < 𝑅 → (abs‘((𝐹‘𝑦) − (𝐹‘𝐶))) < 𝐸)) & ⊢ (𝜑 → 𝑋 ∈ (𝐴[,]𝐵)) & ⊢ (𝜑 → (abs‘(𝑋 − 𝐶)) < 𝑅) & ⊢ (𝜑 → 𝑌 ∈ (𝐴[,]𝐵)) & ⊢ (𝜑 → (abs‘(𝑌 − 𝐶)) < 𝑅) ⇒ ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → (abs‘((((𝐺‘𝑌) − (𝐺‘𝑋)) / (𝑌 − 𝑋)) − (𝐹‘𝐶))) < 𝐸) | ||
Theorem | ftc1lem5 24643* | Lemma for ftc1 24645. (Contributed by Mario Carneiro, 14-Aug-2014.) (Revised by Mario Carneiro, 28-Dec-2016.) |
⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)(𝐹‘𝑡) d𝑡) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) & ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ 𝐷) & ⊢ (𝜑 → 𝐷 ⊆ ℝ) & ⊢ (𝜑 → 𝐹 ∈ 𝐿1) & ⊢ (𝜑 → 𝐶 ∈ (𝐴(,)𝐵)) & ⊢ (𝜑 → 𝐹 ∈ ((𝐾 CnP 𝐿)‘𝐶)) & ⊢ 𝐽 = (𝐿 ↾t ℝ) & ⊢ 𝐾 = (𝐿 ↾t 𝐷) & ⊢ 𝐿 = (TopOpen‘ℂfld) & ⊢ 𝐻 = (𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝐶}) ↦ (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))) & ⊢ (𝜑 → 𝐸 ∈ ℝ+) & ⊢ (𝜑 → 𝑅 ∈ ℝ+) & ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → ((abs‘(𝑦 − 𝐶)) < 𝑅 → (abs‘((𝐹‘𝑦) − (𝐹‘𝐶))) < 𝐸)) & ⊢ (𝜑 → 𝑋 ∈ (𝐴[,]𝐵)) & ⊢ (𝜑 → (abs‘(𝑋 − 𝐶)) < 𝑅) ⇒ ⊢ ((𝜑 ∧ 𝑋 ≠ 𝐶) → (abs‘((𝐻‘𝑋) − (𝐹‘𝐶))) < 𝐸) | ||
Theorem | ftc1lem6 24644* | Lemma for ftc1 24645. (Contributed by Mario Carneiro, 14-Aug-2014.) (Proof shortened by Mario Carneiro, 28-Dec-2016.) |
⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)(𝐹‘𝑡) d𝑡) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) & ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ 𝐷) & ⊢ (𝜑 → 𝐷 ⊆ ℝ) & ⊢ (𝜑 → 𝐹 ∈ 𝐿1) & ⊢ (𝜑 → 𝐶 ∈ (𝐴(,)𝐵)) & ⊢ (𝜑 → 𝐹 ∈ ((𝐾 CnP 𝐿)‘𝐶)) & ⊢ 𝐽 = (𝐿 ↾t ℝ) & ⊢ 𝐾 = (𝐿 ↾t 𝐷) & ⊢ 𝐿 = (TopOpen‘ℂfld) & ⊢ 𝐻 = (𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝐶}) ↦ (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))) ⇒ ⊢ (𝜑 → (𝐹‘𝐶) ∈ (𝐻 limℂ 𝐶)) | ||
Theorem | ftc1 24645* | The Fundamental Theorem of Calculus, part one. The function formed by varying the right endpoint of an integral is differentiable at 𝐶 with derivative 𝐹(𝐶) if the original function is continuous at 𝐶. This is part of Metamath 100 proof #15. (Contributed by Mario Carneiro, 1-Sep-2014.) |
⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)(𝐹‘𝑡) d𝑡) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) & ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ 𝐷) & ⊢ (𝜑 → 𝐷 ⊆ ℝ) & ⊢ (𝜑 → 𝐹 ∈ 𝐿1) & ⊢ (𝜑 → 𝐶 ∈ (𝐴(,)𝐵)) & ⊢ (𝜑 → 𝐹 ∈ ((𝐾 CnP 𝐿)‘𝐶)) & ⊢ 𝐽 = (𝐿 ↾t ℝ) & ⊢ 𝐾 = (𝐿 ↾t 𝐷) & ⊢ 𝐿 = (TopOpen‘ℂfld) ⇒ ⊢ (𝜑 → 𝐶(ℝ D 𝐺)(𝐹‘𝐶)) | ||
Theorem | ftc1cn 24646* | Strengthen the assumptions of ftc1 24645 to when the function 𝐹 is continuous on the entire interval (𝐴, 𝐵); in this case we can calculate D 𝐺 exactly. (Contributed by Mario Carneiro, 1-Sep-2014.) |
⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)(𝐹‘𝑡) d𝑡) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) & ⊢ (𝜑 → 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ)) & ⊢ (𝜑 → 𝐹 ∈ 𝐿1) ⇒ ⊢ (𝜑 → (ℝ D 𝐺) = 𝐹) | ||
Theorem | ftc2 24647* | The Fundamental Theorem of Calculus, part two. If 𝐹 is a function continuous on [𝐴, 𝐵] and continuously differentiable on (𝐴, 𝐵), then the integral of the derivative of 𝐹 is equal to 𝐹(𝐵) − 𝐹(𝐴). This is part of Metamath 100 proof #15. (Contributed by Mario Carneiro, 2-Sep-2014.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) & ⊢ (𝜑 → (ℝ D 𝐹) ∈ ((𝐴(,)𝐵)–cn→ℂ)) & ⊢ (𝜑 → (ℝ D 𝐹) ∈ 𝐿1) & ⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ)) ⇒ ⊢ (𝜑 → ∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 = ((𝐹‘𝐵) − (𝐹‘𝐴))) | ||
Theorem | ftc2ditglem 24648* | Lemma for ftc2ditg 24649. (Contributed by Mario Carneiro, 3-Sep-2014.) |
⊢ (𝜑 → 𝑋 ∈ ℝ) & ⊢ (𝜑 → 𝑌 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ∈ (𝑋[,]𝑌)) & ⊢ (𝜑 → 𝐵 ∈ (𝑋[,]𝑌)) & ⊢ (𝜑 → (ℝ D 𝐹) ∈ ((𝑋(,)𝑌)–cn→ℂ)) & ⊢ (𝜑 → (ℝ D 𝐹) ∈ 𝐿1) & ⊢ (𝜑 → 𝐹 ∈ ((𝑋[,]𝑌)–cn→ℂ)) ⇒ ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ⨜[𝐴 → 𝐵]((ℝ D 𝐹)‘𝑡) d𝑡 = ((𝐹‘𝐵) − (𝐹‘𝐴))) | ||
Theorem | ftc2ditg 24649* | Directed integral analogue of ftc2 24647. (Contributed by Mario Carneiro, 3-Sep-2014.) |
⊢ (𝜑 → 𝑋 ∈ ℝ) & ⊢ (𝜑 → 𝑌 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ∈ (𝑋[,]𝑌)) & ⊢ (𝜑 → 𝐵 ∈ (𝑋[,]𝑌)) & ⊢ (𝜑 → (ℝ D 𝐹) ∈ ((𝑋(,)𝑌)–cn→ℂ)) & ⊢ (𝜑 → (ℝ D 𝐹) ∈ 𝐿1) & ⊢ (𝜑 → 𝐹 ∈ ((𝑋[,]𝑌)–cn→ℂ)) ⇒ ⊢ (𝜑 → ⨜[𝐴 → 𝐵]((ℝ D 𝐹)‘𝑡) d𝑡 = ((𝐹‘𝐵) − (𝐹‘𝐴))) | ||
Theorem | itgparts 24650* | Integration by parts. If 𝐵(𝑥) is the derivative of 𝐴(𝑥) and 𝐷(𝑥) is the derivative of 𝐶(𝑥), and 𝐸 = (𝐴 · 𝐵)(𝑋) and 𝐹 = (𝐴 · 𝐵)(𝑌), then under suitable integrability and differentiability assumptions, the integral of 𝐴 · 𝐷 from 𝑋 to 𝑌 is equal to 𝐹 − 𝐸 minus the integral of 𝐵 · 𝐶. (Contributed by Mario Carneiro, 3-Sep-2014.) |
⊢ (𝜑 → 𝑋 ∈ ℝ) & ⊢ (𝜑 → 𝑌 ∈ ℝ) & ⊢ (𝜑 → 𝑋 ≤ 𝑌) & ⊢ (𝜑 → (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴) ∈ ((𝑋[,]𝑌)–cn→ℂ)) & ⊢ (𝜑 → (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐶) ∈ ((𝑋[,]𝑌)–cn→ℂ)) & ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐵) ∈ ((𝑋(,)𝑌)–cn→ℂ)) & ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐷) ∈ ((𝑋(,)𝑌)–cn→ℂ)) & ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ (𝐴 · 𝐷)) ∈ 𝐿1) & ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ (𝐵 · 𝐶)) ∈ 𝐿1) & ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)) = (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐵)) & ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐶)) = (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐷)) & ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝐴 · 𝐶) = 𝐸) & ⊢ ((𝜑 ∧ 𝑥 = 𝑌) → (𝐴 · 𝐶) = 𝐹) ⇒ ⊢ (𝜑 → ∫(𝑋(,)𝑌)(𝐴 · 𝐷) d𝑥 = ((𝐹 − 𝐸) − ∫(𝑋(,)𝑌)(𝐵 · 𝐶) d𝑥)) | ||
Theorem | itgsubstlem 24651* | Lemma for itgsubst 24652. (Contributed by Mario Carneiro, 12-Sep-2014.) |
⊢ (𝜑 → 𝑋 ∈ ℝ) & ⊢ (𝜑 → 𝑌 ∈ ℝ) & ⊢ (𝜑 → 𝑋 ≤ 𝑌) & ⊢ (𝜑 → 𝑍 ∈ ℝ*) & ⊢ (𝜑 → 𝑊 ∈ ℝ*) & ⊢ (𝜑 → (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴) ∈ ((𝑋[,]𝑌)–cn→(𝑍(,)𝑊))) & ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐵) ∈ (((𝑋(,)𝑌)–cn→ℂ) ∩ 𝐿1)) & ⊢ (𝜑 → (𝑢 ∈ (𝑍(,)𝑊) ↦ 𝐶) ∈ ((𝑍(,)𝑊)–cn→ℂ)) & ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)) = (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐵)) & ⊢ (𝑢 = 𝐴 → 𝐶 = 𝐸) & ⊢ (𝑥 = 𝑋 → 𝐴 = 𝐾) & ⊢ (𝑥 = 𝑌 → 𝐴 = 𝐿) & ⊢ (𝜑 → 𝑀 ∈ (𝑍(,)𝑊)) & ⊢ (𝜑 → 𝑁 ∈ (𝑍(,)𝑊)) & ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋[,]𝑌)) → 𝐴 ∈ (𝑀(,)𝑁)) ⇒ ⊢ (𝜑 → ⨜[𝐾 → 𝐿]𝐶 d𝑢 = ⨜[𝑋 → 𝑌](𝐸 · 𝐵) d𝑥) | ||
Theorem | itgsubst 24652* | Integration by 𝑢-substitution. If 𝐴(𝑥) is a continuous, differentiable function from [𝑋, 𝑌] to (𝑍, 𝑊), whose derivative is continuous and integrable, and 𝐶(𝑢) is a continuous function on (𝑍, 𝑊), then the integral of 𝐶(𝑢) from 𝐾 = 𝐴(𝑋) to 𝐿 = 𝐴(𝑌) is equal to the integral of 𝐶(𝐴(𝑥)) D 𝐴(𝑥) from 𝑋 to 𝑌. In this part of the proof we discharge the assumptions in itgsubstlem 24651, which use the fact that (𝑍, 𝑊) is open to shrink the interval a little to (𝑀, 𝑁) where 𝑍 < 𝑀 < 𝑁 < 𝑊- this is possible because 𝐴(𝑥) is a continuous function on a closed interval, so its range is in fact a closed interval, and we have some wiggle room on the edges. (Contributed by Mario Carneiro, 7-Sep-2014.) |
⊢ (𝜑 → 𝑋 ∈ ℝ) & ⊢ (𝜑 → 𝑌 ∈ ℝ) & ⊢ (𝜑 → 𝑋 ≤ 𝑌) & ⊢ (𝜑 → 𝑍 ∈ ℝ*) & ⊢ (𝜑 → 𝑊 ∈ ℝ*) & ⊢ (𝜑 → (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴) ∈ ((𝑋[,]𝑌)–cn→(𝑍(,)𝑊))) & ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐵) ∈ (((𝑋(,)𝑌)–cn→ℂ) ∩ 𝐿1)) & ⊢ (𝜑 → (𝑢 ∈ (𝑍(,)𝑊) ↦ 𝐶) ∈ ((𝑍(,)𝑊)–cn→ℂ)) & ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)) = (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐵)) & ⊢ (𝑢 = 𝐴 → 𝐶 = 𝐸) & ⊢ (𝑥 = 𝑋 → 𝐴 = 𝐾) & ⊢ (𝑥 = 𝑌 → 𝐴 = 𝐿) ⇒ ⊢ (𝜑 → ⨜[𝐾 → 𝐿]𝐶 d𝑢 = ⨜[𝑋 → 𝑌](𝐸 · 𝐵) d𝑥) | ||
Theorem | itgpowd 24653* | The integral of a monomial on a closed bounded interval of the real line. Co-authors TA and MC. (Contributed by Jon Pennant, 31-May-2019.) (Revised by Thierry Arnoux, 14-Jun-2019.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) ⇒ ⊢ (𝜑 → ∫(𝐴[,]𝐵)(𝑥↑𝑁) d𝑥 = (((𝐵↑(𝑁 + 1)) − (𝐴↑(𝑁 + 1))) / (𝑁 + 1))) | ||
Syntax | cmdg 24654 | Multivariate polynomial degree. |
class mDeg | ||
Syntax | cdg1 24655 | Univariate polynomial degree. |
class deg1 | ||
Definition | df-mdeg 24656* | Define the degree of a polynomial. Note (SO): as an experiment I am using a definition which makes the degree of the zero polynomial -∞, contrary to the convention used in df-dgr 24788. (Contributed by Stefan O'Rear, 19-Mar-2015.) (Revised by AV, 25-Jun-2019.) |
⊢ mDeg = (𝑖 ∈ V, 𝑟 ∈ V ↦ (𝑓 ∈ (Base‘(𝑖 mPoly 𝑟)) ↦ sup(ran (ℎ ∈ (𝑓 supp (0g‘𝑟)) ↦ (ℂfld Σg ℎ)), ℝ*, < ))) | ||
Definition | df-deg1 24657 | Define the degree of a univariate polynomial. (Contributed by Stefan O'Rear, 23-Mar-2015.) |
⊢ deg1 = (𝑟 ∈ V ↦ (1o mDeg 𝑟)) | ||
Theorem | reldmmdeg 24658 | Multivariate degree is a binary operation. (Contributed by Stefan O'Rear, 28-Mar-2015.) |
⊢ Rel dom mDeg | ||
Theorem | tdeglem1 24659* | Functionality of the total degree helper function. (Contributed by Stefan O'Rear, 19-Mar-2015.) (Proof shortened by AV, 27-Jul-2019.) |
⊢ 𝐴 = {𝑚 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑚 “ ℕ) ∈ Fin} & ⊢ 𝐻 = (ℎ ∈ 𝐴 ↦ (ℂfld Σg ℎ)) ⇒ ⊢ (𝐼 ∈ 𝑉 → 𝐻:𝐴⟶ℕ0) | ||
Theorem | tdeglem3 24660* | Additivity of the total degree helper function. (Contributed by Stefan O'Rear, 26-Mar-2015.) (Proof shortened by AV, 27-Jul-2019.) |
⊢ 𝐴 = {𝑚 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑚 “ ℕ) ∈ Fin} & ⊢ 𝐻 = (ℎ ∈ 𝐴 ↦ (ℂfld Σg ℎ)) ⇒ ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝐻‘(𝑋 ∘f + 𝑌)) = ((𝐻‘𝑋) + (𝐻‘𝑌))) | ||
Theorem | tdeglem4 24661* | There is only one multi-index with total degree 0. (Contributed by Stefan O'Rear, 29-Mar-2015.) |
⊢ 𝐴 = {𝑚 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑚 “ ℕ) ∈ Fin} & ⊢ 𝐻 = (ℎ ∈ 𝐴 ↦ (ℂfld Σg ℎ)) ⇒ ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → ((𝐻‘𝑋) = 0 ↔ 𝑋 = (𝐼 × {0}))) | ||
Theorem | tdeglem2 24662 | Simplification of total degree for the univariate case. (Contributed by Stefan O'Rear, 23-Mar-2015.) |
⊢ (ℎ ∈ (ℕ0 ↑m 1o) ↦ (ℎ‘∅)) = (ℎ ∈ (ℕ0 ↑m 1o) ↦ (ℂfld Σg ℎ)) | ||
Theorem | mdegfval 24663* | Value of the multivariate degree function. (Contributed by Stefan O'Rear, 19-Mar-2015.) (Revised by AV, 25-Jun-2019.) |
⊢ 𝐷 = (𝐼 mDeg 𝑅) & ⊢ 𝑃 = (𝐼 mPoly 𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐴 = {𝑚 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑚 “ ℕ) ∈ Fin} & ⊢ 𝐻 = (ℎ ∈ 𝐴 ↦ (ℂfld Σg ℎ)) ⇒ ⊢ 𝐷 = (𝑓 ∈ 𝐵 ↦ sup((𝐻 “ (𝑓 supp 0 )), ℝ*, < )) | ||
Theorem | mdegval 24664* | Value of the multivariate degree function at some particular polynomial. (Contributed by Stefan O'Rear, 19-Mar-2015.) (Revised by AV, 25-Jun-2019.) |
⊢ 𝐷 = (𝐼 mDeg 𝑅) & ⊢ 𝑃 = (𝐼 mPoly 𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐴 = {𝑚 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑚 “ ℕ) ∈ Fin} & ⊢ 𝐻 = (ℎ ∈ 𝐴 ↦ (ℂfld Σg ℎ)) ⇒ ⊢ (𝐹 ∈ 𝐵 → (𝐷‘𝐹) = sup((𝐻 “ (𝐹 supp 0 )), ℝ*, < )) | ||
Theorem | mdegleb 24665* | Property of being of limited degree. (Contributed by Stefan O'Rear, 19-Mar-2015.) |
⊢ 𝐷 = (𝐼 mDeg 𝑅) & ⊢ 𝑃 = (𝐼 mPoly 𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐴 = {𝑚 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑚 “ ℕ) ∈ Fin} & ⊢ 𝐻 = (ℎ ∈ 𝐴 ↦ (ℂfld Σg ℎ)) ⇒ ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ ℝ*) → ((𝐷‘𝐹) ≤ 𝐺 ↔ ∀𝑥 ∈ 𝐴 (𝐺 < (𝐻‘𝑥) → (𝐹‘𝑥) = 0 ))) | ||
Theorem | mdeglt 24666* | If there is an upper limit on the degree of a polynomial that is lower than the degree of some exponent bag, then that exponent bag is unrepresented in the polynomial. (Contributed by Stefan O'Rear, 26-Mar-2015.) (Proof shortened by AV, 27-Jul-2019.) |
⊢ 𝐷 = (𝐼 mDeg 𝑅) & ⊢ 𝑃 = (𝐼 mPoly 𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐴 = {𝑚 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑚 “ ℕ) ∈ Fin} & ⊢ 𝐻 = (ℎ ∈ 𝐴 ↦ (ℂfld Σg ℎ)) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → (𝐷‘𝐹) < (𝐻‘𝑋)) ⇒ ⊢ (𝜑 → (𝐹‘𝑋) = 0 ) | ||
Theorem | mdegldg 24667* | A nonzero polynomial has some coefficient which witnesses its degree. (Contributed by Stefan O'Rear, 23-Mar-2015.) |
⊢ 𝐷 = (𝐼 mDeg 𝑅) & ⊢ 𝑃 = (𝐼 mPoly 𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐴 = {𝑚 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑚 “ ℕ) ∈ Fin} & ⊢ 𝐻 = (ℎ ∈ 𝐴 ↦ (ℂfld Σg ℎ)) & ⊢ 𝑌 = (0g‘𝑃) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 𝑌) → ∃𝑥 ∈ 𝐴 ((𝐹‘𝑥) ≠ 0 ∧ (𝐻‘𝑥) = (𝐷‘𝐹))) | ||
Theorem | mdegxrcl 24668 | Closure of polynomial degree in the extended reals. (Contributed by Stefan O'Rear, 19-Mar-2015.) (Proof shortened by AV, 27-Jul-2019.) |
⊢ 𝐷 = (𝐼 mDeg 𝑅) & ⊢ 𝑃 = (𝐼 mPoly 𝑅) & ⊢ 𝐵 = (Base‘𝑃) ⇒ ⊢ (𝐹 ∈ 𝐵 → (𝐷‘𝐹) ∈ ℝ*) | ||
Theorem | mdegxrf 24669 | Functionality of polynomial degree in the extended reals. (Contributed by Stefan O'Rear, 19-Mar-2015.) (Proof shortened by AV, 27-Jul-2019.) |
⊢ 𝐷 = (𝐼 mDeg 𝑅) & ⊢ 𝑃 = (𝐼 mPoly 𝑅) & ⊢ 𝐵 = (Base‘𝑃) ⇒ ⊢ 𝐷:𝐵⟶ℝ* | ||
Theorem | mdegcl 24670 | Sharp closure for multivariate polynomials. (Contributed by Stefan O'Rear, 23-Mar-2015.) |
⊢ 𝐷 = (𝐼 mDeg 𝑅) & ⊢ 𝑃 = (𝐼 mPoly 𝑅) & ⊢ 𝐵 = (Base‘𝑃) ⇒ ⊢ (𝐹 ∈ 𝐵 → (𝐷‘𝐹) ∈ (ℕ0 ∪ {-∞})) | ||
Theorem | mdeg0 24671 | Degree of the zero polynomial. (Contributed by Stefan O'Rear, 20-Mar-2015.) (Proof shortened by AV, 27-Jul-2019.) |
⊢ 𝐷 = (𝐼 mDeg 𝑅) & ⊢ 𝑃 = (𝐼 mPoly 𝑅) & ⊢ 0 = (0g‘𝑃) ⇒ ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → (𝐷‘ 0 ) = -∞) | ||
Theorem | mdegnn0cl 24672 | Degree of a nonzero polynomial. (Contributed by Stefan O'Rear, 23-Mar-2015.) |
⊢ 𝐷 = (𝐼 mDeg 𝑅) & ⊢ 𝑃 = (𝐼 mPoly 𝑅) & ⊢ 0 = (0g‘𝑃) & ⊢ 𝐵 = (Base‘𝑃) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ) → (𝐷‘𝐹) ∈ ℕ0) | ||
Theorem | degltlem1 24673 | Theorem on arithmetic of extended reals useful for degrees. (Contributed by Stefan O'Rear, 23-Mar-2015.) |
⊢ ((𝑋 ∈ (ℕ0 ∪ {-∞}) ∧ 𝑌 ∈ ℤ) → (𝑋 < 𝑌 ↔ 𝑋 ≤ (𝑌 − 1))) | ||
Theorem | degltp1le 24674 | Theorem on arithmetic of extended reals useful for degrees. (Contributed by Stefan O'Rear, 1-Apr-2015.) |
⊢ ((𝑋 ∈ (ℕ0 ∪ {-∞}) ∧ 𝑌 ∈ ℤ) → (𝑋 < (𝑌 + 1) ↔ 𝑋 ≤ 𝑌)) | ||
Theorem | mdegaddle 24675 | The degree of a sum is at most the maximum of the degrees of the factors. (Contributed by Stefan O'Rear, 26-Mar-2015.) |
⊢ 𝑌 = (𝐼 mPoly 𝑅) & ⊢ 𝐷 = (𝐼 mDeg 𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ + = (+g‘𝑌) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝐺 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐷‘(𝐹 + 𝐺)) ≤ if((𝐷‘𝐹) ≤ (𝐷‘𝐺), (𝐷‘𝐺), (𝐷‘𝐹))) | ||
Theorem | mdegvscale 24676 | The degree of a scalar multiple of a polynomial is at most the degree of the original polynomial. (Contributed by Stefan O'Rear, 26-Mar-2015.) |
⊢ 𝑌 = (𝐼 mPoly 𝑅) & ⊢ 𝐷 = (𝐼 mDeg 𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ 𝐾 = (Base‘𝑅) & ⊢ · = ( ·𝑠 ‘𝑌) & ⊢ (𝜑 → 𝐹 ∈ 𝐾) & ⊢ (𝜑 → 𝐺 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐷‘(𝐹 · 𝐺)) ≤ (𝐷‘𝐺)) | ||
Theorem | mdegvsca 24677 | The degree of a scalar multiple of a polynomial is exactly the degree of the original polynomial when the multiple is a nonzero-divisor. (Contributed by Stefan O'Rear, 28-Mar-2015.) (Proof shortened by AV, 27-Jul-2019.) |
⊢ 𝑌 = (𝐼 mPoly 𝑅) & ⊢ 𝐷 = (𝐼 mDeg 𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ 𝐸 = (RLReg‘𝑅) & ⊢ · = ( ·𝑠 ‘𝑌) & ⊢ (𝜑 → 𝐹 ∈ 𝐸) & ⊢ (𝜑 → 𝐺 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐷‘(𝐹 · 𝐺)) = (𝐷‘𝐺)) | ||
Theorem | mdegle0 24678 | A polynomial has nonpositive degree iff it is a constant. (Contributed by Stefan O'Rear, 29-Mar-2015.) |
⊢ 𝑌 = (𝐼 mPoly 𝑅) & ⊢ 𝐷 = (𝐼 mDeg 𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ 𝐴 = (algSc‘𝑌) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((𝐷‘𝐹) ≤ 0 ↔ 𝐹 = (𝐴‘(𝐹‘(𝐼 × {0}))))) | ||
Theorem | mdegmullem 24679* | Lemma for mdegmulle2 24680. (Contributed by Stefan O'Rear, 26-Mar-2015.) |
⊢ 𝑌 = (𝐼 mPoly 𝑅) & ⊢ 𝐷 = (𝐼 mDeg 𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ · = (.r‘𝑌) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝐺 ∈ 𝐵) & ⊢ (𝜑 → 𝐽 ∈ ℕ0) & ⊢ (𝜑 → 𝐾 ∈ ℕ0) & ⊢ (𝜑 → (𝐷‘𝐹) ≤ 𝐽) & ⊢ (𝜑 → (𝐷‘𝐺) ≤ 𝐾) & ⊢ 𝐴 = {𝑎 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} & ⊢ 𝐻 = (𝑏 ∈ 𝐴 ↦ (ℂfld Σg 𝑏)) ⇒ ⊢ (𝜑 → (𝐷‘(𝐹 · 𝐺)) ≤ (𝐽 + 𝐾)) | ||
Theorem | mdegmulle2 24680 | The multivariate degree of a product of polynomials is at most the sum of the degrees of the polynomials. (Contributed by Stefan O'Rear, 26-Mar-2015.) |
⊢ 𝑌 = (𝐼 mPoly 𝑅) & ⊢ 𝐷 = (𝐼 mDeg 𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ · = (.r‘𝑌) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝐺 ∈ 𝐵) & ⊢ (𝜑 → 𝐽 ∈ ℕ0) & ⊢ (𝜑 → 𝐾 ∈ ℕ0) & ⊢ (𝜑 → (𝐷‘𝐹) ≤ 𝐽) & ⊢ (𝜑 → (𝐷‘𝐺) ≤ 𝐾) ⇒ ⊢ (𝜑 → (𝐷‘(𝐹 · 𝐺)) ≤ (𝐽 + 𝐾)) | ||
Theorem | deg1fval 24681 | Relate univariate polynomial degree to multivariate. (Contributed by Stefan O'Rear, 23-Mar-2015.) (Revised by Mario Carneiro, 7-Oct-2015.) |
⊢ 𝐷 = ( deg1 ‘𝑅) ⇒ ⊢ 𝐷 = (1o mDeg 𝑅) | ||
Theorem | deg1xrf 24682 | Functionality of univariate polynomial degree, weak range. (Contributed by Stefan O'Rear, 23-Mar-2015.) |
⊢ 𝐷 = ( deg1 ‘𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) ⇒ ⊢ 𝐷:𝐵⟶ℝ* | ||
Theorem | deg1xrcl 24683 | Closure of univariate polynomial degree in extended reals. (Contributed by Stefan O'Rear, 23-Mar-2015.) |
⊢ 𝐷 = ( deg1 ‘𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) ⇒ ⊢ (𝐹 ∈ 𝐵 → (𝐷‘𝐹) ∈ ℝ*) | ||
Theorem | deg1cl 24684 | Sharp closure of univariate polynomial degree. (Contributed by Stefan O'Rear, 23-Mar-2015.) |
⊢ 𝐷 = ( deg1 ‘𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) ⇒ ⊢ (𝐹 ∈ 𝐵 → (𝐷‘𝐹) ∈ (ℕ0 ∪ {-∞})) | ||
Theorem | mdegpropd 24685* | Property deduction for polynomial degree. (Contributed by Stefan O'Rear, 28-Mar-2015.) (Proof shortened by AV, 27-Jul-2019.) |
⊢ (𝜑 → 𝐵 = (Base‘𝑅)) & ⊢ (𝜑 → 𝐵 = (Base‘𝑆)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝑅)𝑦) = (𝑥(+g‘𝑆)𝑦)) ⇒ ⊢ (𝜑 → (𝐼 mDeg 𝑅) = (𝐼 mDeg 𝑆)) | ||
Theorem | deg1fvi 24686 | Univariate polynomial degree respects protection. (Contributed by Stefan O'Rear, 28-Mar-2015.) |
⊢ ( deg1 ‘𝑅) = ( deg1 ‘( I ‘𝑅)) | ||
Theorem | deg1propd 24687* | Property deduction for polynomial degree. (Contributed by Stefan O'Rear, 28-Mar-2015.) |
⊢ (𝜑 → 𝐵 = (Base‘𝑅)) & ⊢ (𝜑 → 𝐵 = (Base‘𝑆)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝑅)𝑦) = (𝑥(+g‘𝑆)𝑦)) ⇒ ⊢ (𝜑 → ( deg1 ‘𝑅) = ( deg1 ‘𝑆)) | ||
Theorem | deg1z 24688 | Degree of the zero univariate polynomial. (Contributed by Stefan O'Rear, 23-Mar-2015.) |
⊢ 𝐷 = ( deg1 ‘𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 0 = (0g‘𝑃) ⇒ ⊢ (𝑅 ∈ Ring → (𝐷‘ 0 ) = -∞) | ||
Theorem | deg1nn0cl 24689 | Degree of a nonzero univariate polynomial. (Contributed by Stefan O'Rear, 23-Mar-2015.) (Revised by Mario Carneiro, 7-Oct-2015.) |
⊢ 𝐷 = ( deg1 ‘𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 0 = (0g‘𝑃) & ⊢ 𝐵 = (Base‘𝑃) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ) → (𝐷‘𝐹) ∈ ℕ0) | ||
Theorem | deg1n0ima 24690 | Degree image of a set of polynomials which does not include zero. (Contributed by Stefan O'Rear, 28-Mar-2015.) |
⊢ 𝐷 = ( deg1 ‘𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 0 = (0g‘𝑃) & ⊢ 𝐵 = (Base‘𝑃) ⇒ ⊢ (𝑅 ∈ Ring → (𝐷 “ (𝐵 ∖ { 0 })) ⊆ ℕ0) | ||
Theorem | deg1nn0clb 24691 | A polynomial is nonzero iff it has definite degree. (Contributed by Stefan O'Rear, 23-Mar-2015.) |
⊢ 𝐷 = ( deg1 ‘𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 0 = (0g‘𝑃) & ⊢ 𝐵 = (Base‘𝑃) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵) → (𝐹 ≠ 0 ↔ (𝐷‘𝐹) ∈ ℕ0)) | ||
Theorem | deg1lt0 24692 | A polynomial is zero iff it has negative degree. (Contributed by Stefan O'Rear, 1-Apr-2015.) |
⊢ 𝐷 = ( deg1 ‘𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 0 = (0g‘𝑃) & ⊢ 𝐵 = (Base‘𝑃) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵) → ((𝐷‘𝐹) < 0 ↔ 𝐹 = 0 )) | ||
Theorem | deg1ldg 24693 | A nonzero univariate polynomial always has a nonzero leading coefficient. (Contributed by Stefan O'Rear, 23-Mar-2015.) |
⊢ 𝐷 = ( deg1 ‘𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 0 = (0g‘𝑃) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝑌 = (0g‘𝑅) & ⊢ 𝐴 = (coe1‘𝐹) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ) → (𝐴‘(𝐷‘𝐹)) ≠ 𝑌) | ||
Theorem | deg1ldgn 24694 | An index at which a polynomial is zero, cannot be its degree. (Contributed by Stefan O'Rear, 23-Mar-2015.) |
⊢ 𝐷 = ( deg1 ‘𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 0 = (0g‘𝑃) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝑌 = (0g‘𝑅) & ⊢ 𝐴 = (coe1‘𝐹) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝑋 ∈ ℕ0) & ⊢ (𝜑 → (𝐴‘𝑋) = 𝑌) ⇒ ⊢ (𝜑 → (𝐷‘𝐹) ≠ 𝑋) | ||
Theorem | deg1ldgdomn 24695 | A nonzero univariate polynomial over a domain always has a nonzero-divisor leading coefficient. (Contributed by Stefan O'Rear, 28-Mar-2015.) |
⊢ 𝐷 = ( deg1 ‘𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 0 = (0g‘𝑃) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝐸 = (RLReg‘𝑅) & ⊢ 𝐴 = (coe1‘𝐹) ⇒ ⊢ ((𝑅 ∈ Domn ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ) → (𝐴‘(𝐷‘𝐹)) ∈ 𝐸) | ||
Theorem | deg1leb 24696* | Property of being of limited degree. (Contributed by Stefan O'Rear, 23-Mar-2015.) |
⊢ 𝐷 = ( deg1 ‘𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐴 = (coe1‘𝐹) ⇒ ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ ℝ*) → ((𝐷‘𝐹) ≤ 𝐺 ↔ ∀𝑥 ∈ ℕ0 (𝐺 < 𝑥 → (𝐴‘𝑥) = 0 ))) | ||
Theorem | deg1val 24697 | Value of the univariate degree as a supremum. (Contributed by Stefan O'Rear, 29-Mar-2015.) (Revised by AV, 25-Jul-2019.) |
⊢ 𝐷 = ( deg1 ‘𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐴 = (coe1‘𝐹) ⇒ ⊢ (𝐹 ∈ 𝐵 → (𝐷‘𝐹) = sup((𝐴 supp 0 ), ℝ*, < )) | ||
Theorem | deg1lt 24698 | If the degree of a univariate polynomial is less than some index, then that coefficient must be zero. (Contributed by Stefan O'Rear, 23-Mar-2015.) |
⊢ 𝐷 = ( deg1 ‘𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐴 = (coe1‘𝐹) ⇒ ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ ℕ0 ∧ (𝐷‘𝐹) < 𝐺) → (𝐴‘𝐺) = 0 ) | ||
Theorem | deg1ge 24699 | Conversely, a nonzero coefficient sets a lower bound on the degree. (Contributed by Stefan O'Rear, 23-Mar-2015.) |
⊢ 𝐷 = ( deg1 ‘𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐴 = (coe1‘𝐹) ⇒ ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ ℕ0 ∧ (𝐴‘𝐺) ≠ 0 ) → 𝐺 ≤ (𝐷‘𝐹)) | ||
Theorem | coe1mul3 24700 | The coefficient vector of multiplication in the univariate polynomial ring, at indices high enough that at most one component can be active in the sum. (Contributed by Stefan O'Rear, 25-Mar-2015.) |
⊢ 𝑌 = (Poly1‘𝑅) & ⊢ ∙ = (.r‘𝑌) & ⊢ · = (.r‘𝑅) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ 𝐷 = ( deg1 ‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝐼 ∈ ℕ0) & ⊢ (𝜑 → (𝐷‘𝐹) ≤ 𝐼) & ⊢ (𝜑 → 𝐺 ∈ 𝐵) & ⊢ (𝜑 → 𝐽 ∈ ℕ0) & ⊢ (𝜑 → (𝐷‘𝐺) ≤ 𝐽) ⇒ ⊢ (𝜑 → ((coe1‘(𝐹 ∙ 𝐺))‘(𝐼 + 𝐽)) = (((coe1‘𝐹)‘𝐼) · ((coe1‘𝐺)‘𝐽))) |
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