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Theorem List for Metamath Proof Explorer - 24601-24700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdveq0 24601 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.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ))    &   (𝜑 → (ℝ D 𝐹) = ((𝐴(,)𝐵) × {0}))       (𝜑𝐹 = ((𝐴[,]𝐵) × {(𝐹𝐴)}))
 
Theoremdv11cn 24602 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)    &   (𝜑𝐴 ∈ ℂ)    &   (𝜑𝑅 ∈ ℝ*)    &   (𝜑𝐹:𝑋⟶ℂ)    &   (𝜑𝐺:𝑋⟶ℂ)    &   (𝜑 → dom (ℂ D 𝐹) = 𝑋)    &   (𝜑 → (ℂ D 𝐹) = (ℂ D 𝐺))    &   (𝜑𝐶𝑋)    &   (𝜑 → (𝐹𝐶) = (𝐺𝐶))       (𝜑𝐹 = 𝐺)
 
Theoremdvgt0lem1 24603 Lemma for dvgt0 24605 and dvlt0 24606. (Contributed by Mario Carneiro, 19-Feb-2015.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ))    &   (𝜑 → (ℝ D 𝐹):(𝐴(,)𝐵)⟶𝑆)       (((𝜑 ∧ (𝑋 ∈ (𝐴[,]𝐵) ∧ 𝑌 ∈ (𝐴[,]𝐵))) ∧ 𝑋 < 𝑌) → (((𝐹𝑌) − (𝐹𝑋)) / (𝑌𝑋)) ∈ 𝑆)
 
Theoremdvgt0lem2 24604* Lemma for dvgt0 24605 and dvlt0 24606. (Contributed by Mario Carneiro, 19-Feb-2015.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ))    &   (𝜑 → (ℝ D 𝐹):(𝐴(,)𝐵)⟶𝑆)    &   𝑂 Or ℝ    &   (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝐹𝑥)𝑂(𝐹𝑦))       (𝜑𝐹 Isom < , 𝑂 ((𝐴[,]𝐵), ran 𝐹))
 
Theoremdvgt0 24605 A function on a closed interval with positive derivative is increasing. (Contributed by Mario Carneiro, 19-Feb-2015.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ))    &   (𝜑 → (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℝ+)       (𝜑𝐹 Isom < , < ((𝐴[,]𝐵), ran 𝐹))
 
Theoremdvlt0 24606 A function on a closed interval with negative derivative is decreasing. (Contributed by Mario Carneiro, 19-Feb-2015.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ))    &   (𝜑 → (ℝ D 𝐹):(𝐴(,)𝐵)⟶(-∞(,)0))       (𝜑𝐹 Isom < , < ((𝐴[,]𝐵), ran 𝐹))
 
Theoremdvge0 24607 A function on a closed interval with nonnegative derivative is weakly increasing. (Contributed by Mario Carneiro, 30-Apr-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ))    &   (𝜑 → (ℝ D 𝐹):(𝐴(,)𝐵)⟶(0[,)+∞))    &   (𝜑𝑋 ∈ (𝐴[,]𝐵))    &   (𝜑𝑌 ∈ (𝐴[,]𝐵))    &   (𝜑𝑋𝑌)       (𝜑 → (𝐹𝑋) ≤ (𝐹𝑌))
 
Theoremdvle 24608* If 𝐴(𝑥), 𝐶(𝑥) are differentiable functions and 𝐴‘ ≤ 𝐶, then for 𝑥𝑦, 𝐴(𝑦) − 𝐴(𝑥) ≤ 𝐶(𝑦) − 𝐶(𝑥). (Contributed by Mario Carneiro, 16-May-2016.)
(𝜑𝑀 ∈ ℝ)    &   (𝜑𝑁 ∈ 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))    &   ((𝜑𝑥 ∈ (𝑀(,)𝑁)) → 𝐵𝐷)    &   (𝜑𝑋 ∈ (𝑀[,]𝑁))    &   (𝜑𝑌 ∈ (𝑀[,]𝑁))    &   (𝜑𝑋𝑌)    &   (𝑥 = 𝑋𝐴 = 𝑃)    &   (𝑥 = 𝑋𝐶 = 𝑄)    &   (𝑥 = 𝑌𝐴 = 𝑅)    &   (𝑥 = 𝑌𝐶 = 𝑆)       (𝜑 → (𝑅𝑃) ≤ (𝑆𝑄))
 
Theoremdvivthlem1 24609* Lemma for dvivth 24611. (Contributed by Mario Carneiro, 24-Feb-2015.)
(𝜑𝑀 ∈ (𝐴(,)𝐵))    &   (𝜑𝑁 ∈ (𝐴(,)𝐵))    &   (𝜑𝐹 ∈ ((𝐴(,)𝐵)–cn→ℝ))    &   (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵))    &   (𝜑𝑀 < 𝑁)    &   (𝜑𝐶 ∈ (((ℝ D 𝐹)‘𝑁)[,]((ℝ D 𝐹)‘𝑀)))    &   𝐺 = (𝑦 ∈ (𝐴(,)𝐵) ↦ ((𝐹𝑦) − (𝐶 · 𝑦)))       (𝜑 → ∃𝑥 ∈ (𝑀[,]𝑁)((ℝ D 𝐹)‘𝑥) = 𝐶)
 
Theoremdvivthlem2 24610* Lemma for dvivth 24611. (Contributed by Mario Carneiro, 20-Feb-2015.)
(𝜑𝑀 ∈ (𝐴(,)𝐵))    &   (𝜑𝑁 ∈ (𝐴(,)𝐵))    &   (𝜑𝐹 ∈ ((𝐴(,)𝐵)–cn→ℝ))    &   (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵))    &   (𝜑𝑀 < 𝑁)    &   (𝜑𝐶 ∈ (((ℝ D 𝐹)‘𝑁)[,]((ℝ D 𝐹)‘𝑀)))    &   𝐺 = (𝑦 ∈ (𝐴(,)𝐵) ↦ ((𝐹𝑦) − (𝐶 · 𝑦)))       (𝜑𝐶 ∈ ran (ℝ D 𝐹))
 
Theoremdvivth 24611 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 24060 does not directly apply). (Contributed by Mario Carneiro, 24-Feb-2015.)
(𝜑𝑀 ∈ (𝐴(,)𝐵))    &   (𝜑𝑁 ∈ (𝐴(,)𝐵))    &   (𝜑𝐹 ∈ ((𝐴(,)𝐵)–cn→ℝ))    &   (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵))       (𝜑 → (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)) ⊆ ran (ℝ D 𝐹))
 
Theoremdvne0 24612 A function on a closed interval with nonzero derivative is either monotone increasing or monotone decreasing. (Contributed by Mario Carneiro, 19-Feb-2015.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐹 ∈ ((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)))
 
Theoremdvne0f1 24613 A function on a closed interval with nonzero derivative is one-to-one. (Contributed by Mario Carneiro, 19-Feb-2015.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ))    &   (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵))    &   (𝜑 → ¬ 0 ∈ ran (ℝ D 𝐹))       (𝜑𝐹:(𝐴[,]𝐵)–1-1→ℝ)
 
Theoremlhop1lem 24614* Lemma for lhop1 24615. (Contributed by Mario Carneiro, 29-Dec-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐴 < 𝐵)    &   (𝜑𝐹:(𝐴(,)𝐵)⟶ℝ)    &   (𝜑𝐺:(𝐴(,)𝐵)⟶ℝ)    &   (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵))    &   (𝜑 → dom (ℝ D 𝐺) = (𝐴(,)𝐵))    &   (𝜑 → 0 ∈ (𝐹 lim 𝐴))    &   (𝜑 → 0 ∈ (𝐺 lim 𝐴))    &   (𝜑 → ¬ 0 ∈ ran 𝐺)    &   (𝜑 → ¬ 0 ∈ ran (ℝ D 𝐺))    &   (𝜑𝐶 ∈ ((𝑧 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑧) / ((ℝ D 𝐺)‘𝑧))) lim 𝐴))    &   (𝜑𝐸 ∈ ℝ+)    &   (𝜑𝐷 ∈ ℝ)    &   (𝜑𝐷𝐵)    &   (𝜑𝑋 ∈ (𝐴(,)𝐷))    &   (𝜑 → ∀𝑡 ∈ (𝐴(,)𝐷)(abs‘((((ℝ D 𝐹)‘𝑡) / ((ℝ D 𝐺)‘𝑡)) − 𝐶)) < 𝐸)    &   𝑅 = (𝐴 + (𝑟 / 2))       (𝜑 → (abs‘(((𝐹𝑋) / (𝐺𝑋)) − 𝐶)) < (2 · 𝐸))
 
Theoremlhop1 24615* 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.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐴 < 𝐵)    &   (𝜑𝐹:(𝐴(,)𝐵)⟶ℝ)    &   (𝜑𝐺:(𝐴(,)𝐵)⟶ℝ)    &   (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵))    &   (𝜑 → dom (ℝ D 𝐺) = (𝐴(,)𝐵))    &   (𝜑 → 0 ∈ (𝐹 lim 𝐴))    &   (𝜑 → 0 ∈ (𝐺 lim 𝐴))    &   (𝜑 → ¬ 0 ∈ ran 𝐺)    &   (𝜑 → ¬ 0 ∈ ran (ℝ D 𝐺))    &   (𝜑𝐶 ∈ ((𝑧 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑧) / ((ℝ D 𝐺)‘𝑧))) lim 𝐴))       (𝜑𝐶 ∈ ((𝑧 ∈ (𝐴(,)𝐵) ↦ ((𝐹𝑧) / (𝐺𝑧))) lim 𝐴))
 
Theoremlhop2 24616* 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.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   (𝜑𝐹:(𝐴(,)𝐵)⟶ℝ)    &   (𝜑𝐺:(𝐴(,)𝐵)⟶ℝ)    &   (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵))    &   (𝜑 → dom (ℝ D 𝐺) = (𝐴(,)𝐵))    &   (𝜑 → 0 ∈ (𝐹 lim 𝐵))    &   (𝜑 → 0 ∈ (𝐺 lim 𝐵))    &   (𝜑 → ¬ 0 ∈ ran 𝐺)    &   (𝜑 → ¬ 0 ∈ ran (ℝ D 𝐺))    &   (𝜑𝐶 ∈ ((𝑧 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑧) / ((ℝ D 𝐺)‘𝑧))) lim 𝐵))       (𝜑𝐶 ∈ ((𝑧 ∈ (𝐴(,)𝐵) ↦ ((𝐹𝑧) / (𝐺𝑧))) lim 𝐵))
 
Theoremlhop 24617* 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.)
(𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐹:𝐴⟶ℝ)    &   (𝜑𝐺:𝐴⟶ℝ)    &   (𝜑𝐼 ∈ (topGen‘ran (,)))    &   (𝜑𝐵𝐼)    &   𝐷 = (𝐼 ∖ {𝐵})    &   (𝜑𝐷 ⊆ dom (ℝ D 𝐹))    &   (𝜑𝐷 ⊆ dom (ℝ D 𝐺))    &   (𝜑 → 0 ∈ (𝐹 lim 𝐵))    &   (𝜑 → 0 ∈ (𝐺 lim 𝐵))    &   (𝜑 → ¬ 0 ∈ (𝐺𝐷))    &   (𝜑 → ¬ 0 ∈ ((ℝ D 𝐺) “ 𝐷))    &   (𝜑𝐶 ∈ ((𝑧𝐷 ↦ (((ℝ D 𝐹)‘𝑧) / ((ℝ D 𝐺)‘𝑧))) lim 𝐵))       (𝜑𝐶 ∈ ((𝑧𝐷 ↦ ((𝐹𝑧) / (𝐺𝑧))) lim 𝐵))
 
Theoremdvcnvrelem1 24618 Lemma for dvcnvre 24620. (Contributed by Mario Carneiro, 24-Feb-2015.)
(𝜑𝐹 ∈ (𝑋cn→ℝ))    &   (𝜑 → dom (ℝ D 𝐹) = 𝑋)    &   (𝜑 → ¬ 0 ∈ ran (ℝ D 𝐹))    &   (𝜑𝐹:𝑋1-1-onto𝑌)    &   (𝜑𝐶𝑋)    &   (𝜑𝑅 ∈ ℝ+)    &   (𝜑 → ((𝐶𝑅)[,](𝐶 + 𝑅)) ⊆ 𝑋)       (𝜑 → (𝐹𝐶) ∈ ((int‘(topGen‘ran (,)))‘(𝐹 “ ((𝐶𝑅)[,](𝐶 + 𝑅)))))
 
Theoremdvcnvrelem2 24619 Lemma for dvcnvre 24620. (Contributed by Mario Carneiro, 19-Feb-2015.) (Revised by Mario Carneiro, 8-Sep-2015.)
(𝜑𝐹 ∈ (𝑋cn→ℝ))    &   (𝜑 → dom (ℝ D 𝐹) = 𝑋)    &   (𝜑 → ¬ 0 ∈ ran (ℝ D 𝐹))    &   (𝜑𝐹:𝑋1-1-onto𝑌)    &   (𝜑𝐶𝑋)    &   (𝜑𝑅 ∈ ℝ+)    &   (𝜑 → ((𝐶𝑅)[,](𝐶 + 𝑅)) ⊆ 𝑋)    &   𝑇 = (topGen‘ran (,))    &   𝐽 = (TopOpen‘ℂfld)    &   𝑀 = (𝐽t 𝑋)    &   𝑁 = (𝐽t 𝑌)       (𝜑 → ((𝐹𝐶) ∈ ((int‘𝑇)‘𝑌) ∧ 𝐹 ∈ ((𝑁 CnP 𝑀)‘(𝐹𝐶))))
 
Theoremdvcnvre 24620* 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.)
(𝜑𝐹 ∈ (𝑋cn→ℝ))    &   (𝜑 → dom (ℝ D 𝐹) = 𝑋)    &   (𝜑 → ¬ 0 ∈ ran (ℝ D 𝐹))    &   (𝜑𝐹:𝑋1-1-onto𝑌)       (𝜑 → (ℝ D 𝐹) = (𝑥𝑌 ↦ (1 / ((ℝ D 𝐹)‘(𝐹𝑥)))))
 
Theoremdvcvx 24621 A real function with strictly increasing derivative is strictly convex. (Contributed by Mario Carneiro, 20-Jun-2015.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   (𝜑𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ))    &   (𝜑 → (ℝ D 𝐹) Isom < , < ((𝐴(,)𝐵), 𝑊))    &   (𝜑𝑇 ∈ (0(,)1))    &   𝐶 = ((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵))       (𝜑 → (𝐹𝐶) < ((𝑇 · (𝐹𝐴)) + ((1 − 𝑇) · (𝐹𝐵))))
 
Theoremdvfsumle 24622* 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.)
(𝜑𝑁 ∈ (ℤ𝑀))    &   (𝜑 → (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴) ∈ ((𝑀[,]𝑁)–cn→ℝ))    &   ((𝜑𝑥 ∈ (𝑀(,)𝑁)) → 𝐵𝑉)    &   (𝜑 → (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴)) = (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵))    &   (𝑥 = 𝑀𝐴 = 𝐶)    &   (𝑥 = 𝑁𝐴 = 𝐷)    &   ((𝜑𝑘 ∈ (𝑀..^𝑁)) → 𝑋 ∈ ℝ)    &   ((𝜑 ∧ (𝑘 ∈ (𝑀..^𝑁) ∧ 𝑥 ∈ (𝑘(,)(𝑘 + 1)))) → 𝑋𝐵)       (𝜑 → Σ𝑘 ∈ (𝑀..^𝑁)𝑋 ≤ (𝐷𝐶))
 
Theoremdvfsumge 24623* 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.)
(𝜑𝑁 ∈ (ℤ𝑀))    &   (𝜑 → (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴) ∈ ((𝑀[,]𝑁)–cn→ℝ))    &   ((𝜑𝑥 ∈ (𝑀(,)𝑁)) → 𝐵𝑉)    &   (𝜑 → (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴)) = (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵))    &   (𝑥 = 𝑀𝐴 = 𝐶)    &   (𝑥 = 𝑁𝐴 = 𝐷)    &   ((𝜑𝑘 ∈ (𝑀..^𝑁)) → 𝑋 ∈ ℝ)    &   ((𝜑 ∧ (𝑘 ∈ (𝑀..^𝑁) ∧ 𝑥 ∈ (𝑘(,)(𝑘 + 1)))) → 𝐵𝑋)       (𝜑 → (𝐷𝐶) ≤ Σ𝑘 ∈ (𝑀..^𝑁)𝑋)
 
Theoremdvfsumabs 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.)
(𝜑𝑁 ∈ (ℤ𝑀))    &   (𝜑 → (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴) ∈ ((𝑀[,]𝑁)–cn→ℂ))    &   ((𝜑𝑥 ∈ (𝑀(,)𝑁)) → 𝐵𝑉)    &   (𝜑 → (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴)) = (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵))    &   (𝑥 = 𝑀𝐴 = 𝐶)    &   (𝑥 = 𝑁𝐴 = 𝐷)    &   ((𝜑𝑘 ∈ (𝑀..^𝑁)) → 𝑋 ∈ ℂ)    &   ((𝜑𝑘 ∈ (𝑀..^𝑁)) → 𝑌 ∈ ℝ)    &   ((𝜑 ∧ (𝑘 ∈ (𝑀..^𝑁) ∧ 𝑥 ∈ (𝑘(,)(𝑘 + 1)))) → (abs‘(𝑋𝐵)) ≤ 𝑌)       (𝜑 → (abs‘(Σ𝑘 ∈ (𝑀..^𝑁)𝑋 − (𝐷𝐶))) ≤ Σ𝑘 ∈ (𝑀..^𝑁)𝑌)
 
Theoremdvmptrecl 24625* Real closure of a derivative. (Contributed by Mario Carneiro, 18-May-2016.)
(𝜑𝑆 ⊆ ℝ)    &   ((𝜑𝑥𝑆) → 𝐴 ∈ ℝ)    &   ((𝜑𝑥𝑆) → 𝐵𝑉)    &   (𝜑 → (ℝ D (𝑥𝑆𝐴)) = (𝑥𝑆𝐵))       ((𝜑𝑥𝑆) → 𝐵 ∈ ℝ)
 
Theoremdvfsumrlimf 24626* Lemma for dvfsumrlim 24632. (Contributed by Mario Carneiro, 18-May-2016.)
𝑆 = (𝑇(,)+∞)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐷 ∈ ℝ)    &   (𝜑𝑀 ≤ (𝐷 + 1))    &   (𝜑𝑇 ∈ ℝ)    &   ((𝜑𝑥𝑆) → 𝐴 ∈ ℝ)    &   ((𝜑𝑥𝑆) → 𝐵𝑉)    &   ((𝜑𝑥𝑍) → 𝐵 ∈ ℝ)    &   (𝜑 → (ℝ D (𝑥𝑆𝐴)) = (𝑥𝑆𝐵))    &   (𝑥 = 𝑘𝐵 = 𝐶)    &   𝐺 = (𝑥𝑆 ↦ (Σ𝑘 ∈ (𝑀...(⌊‘𝑥))𝐶𝐴))       (𝜑𝐺:𝑆⟶ℝ)
 
Theoremdvfsumlem1 24627* Lemma for dvfsumrlim 24632. (Contributed by Mario Carneiro, 17-May-2016.)
𝑆 = (𝑇(,)+∞)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐷 ∈ ℝ)    &   (𝜑𝑀 ≤ (𝐷 + 1))    &   (𝜑𝑇 ∈ ℝ)    &   ((𝜑𝑥𝑆) → 𝐴 ∈ ℝ)    &   ((𝜑𝑥𝑆) → 𝐵𝑉)    &   ((𝜑𝑥𝑍) → 𝐵 ∈ ℝ)    &   (𝜑 → (ℝ D (𝑥𝑆𝐴)) = (𝑥𝑆𝐵))    &   (𝑥 = 𝑘𝐵 = 𝐶)    &   (𝜑𝑈 ∈ ℝ*)    &   ((𝜑 ∧ (𝑥𝑆𝑘𝑆) ∧ (𝐷𝑥𝑥𝑘𝑘𝑈)) → 𝐶𝐵)    &   𝐻 = (𝑥𝑆 ↦ (((𝑥 − (⌊‘𝑥)) · 𝐵) + (Σ𝑘 ∈ (𝑀...(⌊‘𝑥))𝐶𝐴)))    &   (𝜑𝑋𝑆)    &   (𝜑𝑌𝑆)    &   (𝜑𝐷𝑋)    &   (𝜑𝑋𝑌)    &   (𝜑𝑌𝑈)    &   (𝜑𝑌 ≤ ((⌊‘𝑋) + 1))       (𝜑 → (𝐻𝑌) = ((((𝑌 − (⌊‘𝑋)) · 𝑌 / 𝑥𝐵) − 𝑌 / 𝑥𝐴) + Σ𝑘 ∈ (𝑀...(⌊‘𝑋))𝐶))
 
Theoremdvfsumlem2 24628* Lemma for dvfsumrlim 24632. (Contributed by Mario Carneiro, 17-May-2016.)
𝑆 = (𝑇(,)+∞)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐷 ∈ ℝ)    &   (𝜑𝑀 ≤ (𝐷 + 1))    &   (𝜑𝑇 ∈ ℝ)    &   ((𝜑𝑥𝑆) → 𝐴 ∈ ℝ)    &   ((𝜑𝑥𝑆) → 𝐵𝑉)    &   ((𝜑𝑥𝑍) → 𝐵 ∈ ℝ)    &   (𝜑 → (ℝ D (𝑥𝑆𝐴)) = (𝑥𝑆𝐵))    &   (𝑥 = 𝑘𝐵 = 𝐶)    &   (𝜑𝑈 ∈ ℝ*)    &   ((𝜑 ∧ (𝑥𝑆𝑘𝑆) ∧ (𝐷𝑥𝑥𝑘𝑘𝑈)) → 𝐶𝐵)    &   𝐻 = (𝑥𝑆 ↦ (((𝑥 − (⌊‘𝑥)) · 𝐵) + (Σ𝑘 ∈ (𝑀...(⌊‘𝑥))𝐶𝐴)))    &   (𝜑𝑋𝑆)    &   (𝜑𝑌𝑆)    &   (𝜑𝐷𝑋)    &   (𝜑𝑋𝑌)    &   (𝜑𝑌𝑈)    &   (𝜑𝑌 ≤ ((⌊‘𝑋) + 1))       (𝜑 → ((𝐻𝑌) ≤ (𝐻𝑋) ∧ ((𝐻𝑋) − 𝑋 / 𝑥𝐵) ≤ ((𝐻𝑌) − 𝑌 / 𝑥𝐵)))
 
Theoremdvfsumlem3 24629* Lemma for dvfsumrlim 24632. (Contributed by Mario Carneiro, 17-May-2016.)
𝑆 = (𝑇(,)+∞)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐷 ∈ ℝ)    &   (𝜑𝑀 ≤ (𝐷 + 1))    &   (𝜑𝑇 ∈ ℝ)    &   ((𝜑𝑥𝑆) → 𝐴 ∈ ℝ)    &   ((𝜑𝑥𝑆) → 𝐵𝑉)    &   ((𝜑𝑥𝑍) → 𝐵 ∈ ℝ)    &   (𝜑 → (ℝ D (𝑥𝑆𝐴)) = (𝑥𝑆𝐵))    &   (𝑥 = 𝑘𝐵 = 𝐶)    &   (𝜑𝑈 ∈ ℝ*)    &   ((𝜑 ∧ (𝑥𝑆𝑘𝑆) ∧ (𝐷𝑥𝑥𝑘𝑘𝑈)) → 𝐶𝐵)    &   𝐻 = (𝑥𝑆 ↦ (((𝑥 − (⌊‘𝑥)) · 𝐵) + (Σ𝑘 ∈ (𝑀...(⌊‘𝑥))𝐶𝐴)))    &   (𝜑𝑋𝑆)    &   (𝜑𝑌𝑆)    &   (𝜑𝐷𝑋)    &   (𝜑𝑋𝑌)    &   (𝜑𝑌𝑈)       (𝜑 → ((𝐻𝑌) ≤ (𝐻𝑋) ∧ ((𝐻𝑋) − 𝑋 / 𝑥𝐵) ≤ ((𝐻𝑌) − 𝑌 / 𝑥𝐵)))
 
Theoremdvfsumlem4 24630* Lemma for dvfsumrlim 24632. (Contributed by Mario Carneiro, 18-May-2016.)
𝑆 = (𝑇(,)+∞)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐷 ∈ ℝ)    &   (𝜑𝑀 ≤ (𝐷 + 1))    &   (𝜑𝑇 ∈ ℝ)    &   ((𝜑𝑥𝑆) → 𝐴 ∈ ℝ)    &   ((𝜑𝑥𝑆) → 𝐵𝑉)    &   ((𝜑𝑥𝑍) → 𝐵 ∈ ℝ)    &   (𝜑 → (ℝ D (𝑥𝑆𝐴)) = (𝑥𝑆𝐵))    &   (𝑥 = 𝑘𝐵 = 𝐶)    &   (𝜑𝑈 ∈ ℝ*)    &   ((𝜑 ∧ (𝑥𝑆𝑘𝑆) ∧ (𝐷𝑥𝑥𝑘𝑘𝑈)) → 𝐶𝐵)    &   𝐺 = (𝑥𝑆 ↦ (Σ𝑘 ∈ (𝑀...(⌊‘𝑥))𝐶𝐴))    &   ((𝜑 ∧ (𝑥𝑆𝐷𝑥𝑥𝑈)) → 0 ≤ 𝐵)    &   (𝜑𝑋𝑆)    &   (𝜑𝑌𝑆)    &   (𝜑𝐷𝑋)    &   (𝜑𝑋𝑌)    &   (𝜑𝑌𝑈)       (𝜑 → (abs‘((𝐺𝑌) − (𝐺𝑋))) ≤ 𝑋 / 𝑥𝐵)
 
Theoremdvfsumrlimge0 24631* Lemma for dvfsumrlim 24632. Satisfy the assumption of dvfsumlem4 24630. (Contributed by Mario Carneiro, 18-May-2016.)
𝑆 = (𝑇(,)+∞)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐷 ∈ ℝ)    &   (𝜑𝑀 ≤ (𝐷 + 1))    &   (𝜑𝑇 ∈ ℝ)    &   ((𝜑𝑥𝑆) → 𝐴 ∈ ℝ)    &   ((𝜑𝑥𝑆) → 𝐵𝑉)    &   ((𝜑𝑥𝑍) → 𝐵 ∈ ℝ)    &   (𝜑 → (ℝ D (𝑥𝑆𝐴)) = (𝑥𝑆𝐵))    &   (𝑥 = 𝑘𝐵 = 𝐶)    &   ((𝜑 ∧ (𝑥𝑆𝑘𝑆) ∧ (𝐷𝑥𝑥𝑘)) → 𝐶𝐵)    &   𝐺 = (𝑥𝑆 ↦ (Σ𝑘 ∈ (𝑀...(⌊‘𝑥))𝐶𝐴))    &   (𝜑 → (𝑥𝑆𝐵) ⇝𝑟 0)       ((𝜑 ∧ (𝑥𝑆𝐷𝑥)) → 0 ≤ 𝐵)
 
Theoremdvfsumrlim 24632* 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 𝑥𝑆𝐵 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 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐷 ∈ ℝ)    &   (𝜑𝑀 ≤ (𝐷 + 1))    &   (𝜑𝑇 ∈ ℝ)    &   ((𝜑𝑥𝑆) → 𝐴 ∈ ℝ)    &   ((𝜑𝑥𝑆) → 𝐵𝑉)    &   ((𝜑𝑥𝑍) → 𝐵 ∈ ℝ)    &   (𝜑 → (ℝ D (𝑥𝑆𝐴)) = (𝑥𝑆𝐵))    &   (𝑥 = 𝑘𝐵 = 𝐶)    &   ((𝜑 ∧ (𝑥𝑆𝑘𝑆) ∧ (𝐷𝑥𝑥𝑘)) → 𝐶𝐵)    &   𝐺 = (𝑥𝑆 ↦ (Σ𝑘 ∈ (𝑀...(⌊‘𝑥))𝐶𝐴))    &   (𝜑 → (𝑥𝑆𝐵) ⇝𝑟 0)       (𝜑𝐺 ∈ dom ⇝𝑟 )
 
Theoremdvfsumrlim2 24633* 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 𝑥𝑆𝐵 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 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐷 ∈ ℝ)    &   (𝜑𝑀 ≤ (𝐷 + 1))    &   (𝜑𝑇 ∈ ℝ)    &   ((𝜑𝑥𝑆) → 𝐴 ∈ ℝ)    &   ((𝜑𝑥𝑆) → 𝐵𝑉)    &   ((𝜑𝑥𝑍) → 𝐵 ∈ ℝ)    &   (𝜑 → (ℝ D (𝑥𝑆𝐴)) = (𝑥𝑆𝐵))    &   (𝑥 = 𝑘𝐵 = 𝐶)    &   ((𝜑 ∧ (𝑥𝑆𝑘𝑆) ∧ (𝐷𝑥𝑥𝑘)) → 𝐶𝐵)    &   𝐺 = (𝑥𝑆 ↦ (Σ𝑘 ∈ (𝑀...(⌊‘𝑥))𝐶𝐴))    &   (𝜑 → (𝑥𝑆𝐵) ⇝𝑟 0)    &   (𝜑𝑋𝑆)    &   (𝜑𝐷𝑋)       ((𝜑𝐺𝑟 𝐿) → (abs‘((𝐺𝑋) − 𝐿)) ≤ 𝑋 / 𝑥𝐵)
 
Theoremdvfsumrlim3 24634* Conjoin the statements of dvfsumrlim 24632 and dvfsumrlim2 24633. (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 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐷 ∈ ℝ)    &   (𝜑𝑀 ≤ (𝐷 + 1))    &   (𝜑𝑇 ∈ ℝ)    &   ((𝜑𝑥𝑆) → 𝐴 ∈ ℝ)    &   ((𝜑𝑥𝑆) → 𝐵𝑉)    &   ((𝜑𝑥𝑍) → 𝐵 ∈ ℝ)    &   (𝜑 → (ℝ D (𝑥𝑆𝐴)) = (𝑥𝑆𝐵))    &   (𝑥 = 𝑘𝐵 = 𝐶)    &   ((𝜑 ∧ (𝑥𝑆𝑘𝑆) ∧ (𝐷𝑥𝑥𝑘)) → 𝐶𝐵)    &   𝐺 = (𝑥𝑆 ↦ (Σ𝑘 ∈ (𝑀...(⌊‘𝑥))𝐶𝐴))    &   (𝜑 → (𝑥𝑆𝐵) ⇝𝑟 0)    &   (𝑥 = 𝑋𝐵 = 𝐸)       (𝜑 → (𝐺:𝑆⟶ℝ ∧ 𝐺 ∈ dom ⇝𝑟 ∧ ((𝐺𝑟 𝐿𝑋𝑆𝐷𝑋) → (abs‘((𝐺𝑋) − 𝐿)) ≤ 𝐸)))
 
Theoremdvfsum2 24635* The reverse of dvfsumrlim 24632, 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 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐷 ∈ ℝ)    &   (𝜑𝑈 ∈ ℝ*)    &   (𝜑𝑀 ≤ (𝐷 + 1))    &   (𝜑𝑇 ∈ ℝ)    &   ((𝜑𝑥𝑆) → 𝐴 ∈ ℝ)    &   ((𝜑𝑥𝑆) → 𝐵𝑉)    &   ((𝜑𝑥𝑍) → 𝐵 ∈ ℝ)    &   (𝜑 → (ℝ D (𝑥𝑆𝐴)) = (𝑥𝑆𝐵))    &   (𝑥 = 𝑘𝐵 = 𝐶)    &   ((𝜑 ∧ (𝑥𝑆𝑘𝑆) ∧ (𝐷𝑥𝑥𝑘𝑘𝑈)) → 𝐵𝐶)    &   𝐺 = (𝑥𝑆 ↦ (Σ𝑘 ∈ (𝑀...(⌊‘𝑥))𝐶𝐴))    &   ((𝜑 ∧ (𝑥𝑆𝐷𝑥)) → 0 ≤ 𝐵)    &   (𝜑𝑋𝑆)    &   (𝜑𝑌𝑆)    &   (𝜑𝐷𝑋)    &   (𝜑𝑋𝑌)    &   (𝜑𝑌𝑈)    &   (𝑥 = 𝑌𝐵 = 𝐸)       (𝜑 → (abs‘((𝐺𝑌) − (𝐺𝑋))) ≤ 𝐸)
 
Theoremftc1lem1 24636* Lemma for ftc1a 24638 and ftc1 24643. (Contributed by Mario Carneiro, 31-Aug-2014.)
𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)(𝐹𝑡) d𝑡)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴𝐵)    &   (𝜑 → (𝐴(,)𝐵) ⊆ 𝐷)    &   (𝜑𝐷 ⊆ ℝ)    &   (𝜑𝐹 ∈ 𝐿1)    &   (𝜑𝐹:𝐷⟶ℂ)    &   (𝜑𝑋 ∈ (𝐴[,]𝐵))    &   (𝜑𝑌 ∈ (𝐴[,]𝐵))       ((𝜑𝑋𝑌) → ((𝐺𝑌) − (𝐺𝑋)) = ∫(𝑋(,)𝑌)(𝐹𝑡) d𝑡)
 
Theoremftc1lem2 24637* Lemma for ftc1 24643. (Contributed by Mario Carneiro, 12-Aug-2014.)
𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)(𝐹𝑡) d𝑡)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴𝐵)    &   (𝜑 → (𝐴(,)𝐵) ⊆ 𝐷)    &   (𝜑𝐷 ⊆ ℝ)    &   (𝜑𝐹 ∈ 𝐿1)    &   (𝜑𝐹:𝐷⟶ℂ)       (𝜑𝐺:(𝐴[,]𝐵)⟶ℂ)
 
Theoremftc1a 24638* 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)(𝐹𝑡) d𝑡)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴𝐵)    &   (𝜑 → (𝐴(,)𝐵) ⊆ 𝐷)    &   (𝜑𝐷 ⊆ ℝ)    &   (𝜑𝐹 ∈ 𝐿1)    &   (𝜑𝐹:𝐷⟶ℂ)       (𝜑𝐺 ∈ ((𝐴[,]𝐵)–cn→ℂ))
 
Theoremftc1lem3 24639* Lemma for ftc1 24643. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 8-Sep-2015.)
𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)(𝐹𝑡) d𝑡)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴𝐵)    &   (𝜑 → (𝐴(,)𝐵) ⊆ 𝐷)    &   (𝜑𝐷 ⊆ ℝ)    &   (𝜑𝐹 ∈ 𝐿1)    &   (𝜑𝐶 ∈ (𝐴(,)𝐵))    &   (𝜑𝐹 ∈ ((𝐾 CnP 𝐿)‘𝐶))    &   𝐽 = (𝐿t ℝ)    &   𝐾 = (𝐿t 𝐷)    &   𝐿 = (TopOpen‘ℂfld)       (𝜑𝐹:𝐷⟶ℂ)
 
Theoremftc1lem4 24640* Lemma for ftc1 24643. (Contributed by Mario Carneiro, 31-Aug-2014.)
𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)(𝐹𝑡) d𝑡)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴𝐵)    &   (𝜑 → (𝐴(,)𝐵) ⊆ 𝐷)    &   (𝜑𝐷 ⊆ ℝ)    &   (𝜑𝐹 ∈ 𝐿1)    &   (𝜑𝐶 ∈ (𝐴(,)𝐵))    &   (𝜑𝐹 ∈ ((𝐾 CnP 𝐿)‘𝐶))    &   𝐽 = (𝐿t ℝ)    &   𝐾 = (𝐿t 𝐷)    &   𝐿 = (TopOpen‘ℂfld)    &   𝐻 = (𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝐶}) ↦ (((𝐺𝑧) − (𝐺𝐶)) / (𝑧𝐶)))    &   (𝜑𝐸 ∈ ℝ+)    &   (𝜑𝑅 ∈ ℝ+)    &   ((𝜑𝑦𝐷) → ((abs‘(𝑦𝐶)) < 𝑅 → (abs‘((𝐹𝑦) − (𝐹𝐶))) < 𝐸))    &   (𝜑𝑋 ∈ (𝐴[,]𝐵))    &   (𝜑 → (abs‘(𝑋𝐶)) < 𝑅)    &   (𝜑𝑌 ∈ (𝐴[,]𝐵))    &   (𝜑 → (abs‘(𝑌𝐶)) < 𝑅)       ((𝜑𝑋 < 𝑌) → (abs‘((((𝐺𝑌) − (𝐺𝑋)) / (𝑌𝑋)) − (𝐹𝐶))) < 𝐸)
 
Theoremftc1lem5 24641* Lemma for ftc1 24643. (Contributed by Mario Carneiro, 14-Aug-2014.) (Revised by Mario Carneiro, 28-Dec-2016.)
𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)(𝐹𝑡) d𝑡)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴𝐵)    &   (𝜑 → (𝐴(,)𝐵) ⊆ 𝐷)    &   (𝜑𝐷 ⊆ ℝ)    &   (𝜑𝐹 ∈ 𝐿1)    &   (𝜑𝐶 ∈ (𝐴(,)𝐵))    &   (𝜑𝐹 ∈ ((𝐾 CnP 𝐿)‘𝐶))    &   𝐽 = (𝐿t ℝ)    &   𝐾 = (𝐿t 𝐷)    &   𝐿 = (TopOpen‘ℂfld)    &   𝐻 = (𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝐶}) ↦ (((𝐺𝑧) − (𝐺𝐶)) / (𝑧𝐶)))    &   (𝜑𝐸 ∈ ℝ+)    &   (𝜑𝑅 ∈ ℝ+)    &   ((𝜑𝑦𝐷) → ((abs‘(𝑦𝐶)) < 𝑅 → (abs‘((𝐹𝑦) − (𝐹𝐶))) < 𝐸))    &   (𝜑𝑋 ∈ (𝐴[,]𝐵))    &   (𝜑 → (abs‘(𝑋𝐶)) < 𝑅)       ((𝜑𝑋𝐶) → (abs‘((𝐻𝑋) − (𝐹𝐶))) < 𝐸)
 
Theoremftc1lem6 24642* Lemma for ftc1 24643. (Contributed by Mario Carneiro, 14-Aug-2014.) (Proof shortened by Mario Carneiro, 28-Dec-2016.)
𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)(𝐹𝑡) d𝑡)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴𝐵)    &   (𝜑 → (𝐴(,)𝐵) ⊆ 𝐷)    &   (𝜑𝐷 ⊆ ℝ)    &   (𝜑𝐹 ∈ 𝐿1)    &   (𝜑𝐶 ∈ (𝐴(,)𝐵))    &   (𝜑𝐹 ∈ ((𝐾 CnP 𝐿)‘𝐶))    &   𝐽 = (𝐿t ℝ)    &   𝐾 = (𝐿t 𝐷)    &   𝐿 = (TopOpen‘ℂfld)    &   𝐻 = (𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝐶}) ↦ (((𝐺𝑧) − (𝐺𝐶)) / (𝑧𝐶)))       (𝜑 → (𝐹𝐶) ∈ (𝐻 lim 𝐶))
 
Theoremftc1 24643* 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)(𝐹𝑡) d𝑡)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴𝐵)    &   (𝜑 → (𝐴(,)𝐵) ⊆ 𝐷)    &   (𝜑𝐷 ⊆ ℝ)    &   (𝜑𝐹 ∈ 𝐿1)    &   (𝜑𝐶 ∈ (𝐴(,)𝐵))    &   (𝜑𝐹 ∈ ((𝐾 CnP 𝐿)‘𝐶))    &   𝐽 = (𝐿t ℝ)    &   𝐾 = (𝐿t 𝐷)    &   𝐿 = (TopOpen‘ℂfld)       (𝜑𝐶(ℝ D 𝐺)(𝐹𝐶))
 
Theoremftc1cn 24644* Strengthen the assumptions of ftc1 24643 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)(𝐹𝑡) d𝑡)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴𝐵)    &   (𝜑𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ))    &   (𝜑𝐹 ∈ 𝐿1)       (𝜑 → (ℝ D 𝐺) = 𝐹)
 
Theoremftc2 24645* 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.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴𝐵)    &   (𝜑 → (ℝ D 𝐹) ∈ ((𝐴(,)𝐵)–cn→ℂ))    &   (𝜑 → (ℝ D 𝐹) ∈ 𝐿1)    &   (𝜑𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ))       (𝜑 → ∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 = ((𝐹𝐵) − (𝐹𝐴)))
 
Theoremftc2ditglem 24646* Lemma for ftc2ditg 24647. (Contributed by Mario Carneiro, 3-Sep-2014.)
(𝜑𝑋 ∈ ℝ)    &   (𝜑𝑌 ∈ ℝ)    &   (𝜑𝐴 ∈ (𝑋[,]𝑌))    &   (𝜑𝐵 ∈ (𝑋[,]𝑌))    &   (𝜑 → (ℝ D 𝐹) ∈ ((𝑋(,)𝑌)–cn→ℂ))    &   (𝜑 → (ℝ D 𝐹) ∈ 𝐿1)    &   (𝜑𝐹 ∈ ((𝑋[,]𝑌)–cn→ℂ))       ((𝜑𝐴𝐵) → ⨜[𝐴𝐵]((ℝ D 𝐹)‘𝑡) d𝑡 = ((𝐹𝐵) − (𝐹𝐴)))
 
Theoremftc2ditg 24647* Directed integral analogue of ftc2 24645. (Contributed by Mario Carneiro, 3-Sep-2014.)
(𝜑𝑋 ∈ ℝ)    &   (𝜑𝑌 ∈ ℝ)    &   (𝜑𝐴 ∈ (𝑋[,]𝑌))    &   (𝜑𝐵 ∈ (𝑋[,]𝑌))    &   (𝜑 → (ℝ D 𝐹) ∈ ((𝑋(,)𝑌)–cn→ℂ))    &   (𝜑 → (ℝ D 𝐹) ∈ 𝐿1)    &   (𝜑𝐹 ∈ ((𝑋[,]𝑌)–cn→ℂ))       (𝜑 → ⨜[𝐴𝐵]((ℝ D 𝐹)‘𝑡) d𝑡 = ((𝐹𝐵) − (𝐹𝐴)))
 
Theoremitgparts 24648* 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 𝐹𝐸 minus the integral of 𝐵 · 𝐶. (Contributed by Mario Carneiro, 3-Sep-2014.)
(𝜑𝑋 ∈ ℝ)    &   (𝜑𝑌 ∈ ℝ)    &   (𝜑𝑋𝑌)    &   (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))    &   ((𝜑𝑥 = 𝑋) → (𝐴 · 𝐶) = 𝐸)    &   ((𝜑𝑥 = 𝑌) → (𝐴 · 𝐶) = 𝐹)       (𝜑 → ∫(𝑋(,)𝑌)(𝐴 · 𝐷) d𝑥 = ((𝐹𝐸) − ∫(𝑋(,)𝑌)(𝐵 · 𝐶) d𝑥))
 
Theoremitgsubstlem 24649* Lemma for itgsubst 24650. (Contributed by Mario Carneiro, 12-Sep-2014.)
(𝜑𝑋 ∈ ℝ)    &   (𝜑𝑌 ∈ ℝ)    &   (𝜑𝑋𝑌)    &   (𝜑𝑍 ∈ ℝ*)    &   (𝜑𝑊 ∈ 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 = 𝐴𝐶 = 𝐸)    &   (𝑥 = 𝑋𝐴 = 𝐾)    &   (𝑥 = 𝑌𝐴 = 𝐿)    &   (𝜑𝑀 ∈ (𝑍(,)𝑊))    &   (𝜑𝑁 ∈ (𝑍(,)𝑊))    &   ((𝜑𝑥 ∈ (𝑋[,]𝑌)) → 𝐴 ∈ (𝑀(,)𝑁))       (𝜑 → ⨜[𝐾𝐿]𝐶 d𝑢 = ⨜[𝑋𝑌](𝐸 · 𝐵) d𝑥)
 
Theoremitgsubst 24650* 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 24649, 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.)
(𝜑𝑋 ∈ ℝ)    &   (𝜑𝑌 ∈ ℝ)    &   (𝜑𝑋𝑌)    &   (𝜑𝑍 ∈ ℝ*)    &   (𝜑𝑊 ∈ 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 = 𝐴𝐶 = 𝐸)    &   (𝑥 = 𝑋𝐴 = 𝐾)    &   (𝑥 = 𝑌𝐴 = 𝐿)       (𝜑 → ⨜[𝐾𝐿]𝐶 d𝑢 = ⨜[𝑋𝑌](𝐸 · 𝐵) d𝑥)
 
Theoremitgpowd 24651* 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.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴𝐵)    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → ∫(𝐴[,]𝐵)(𝑥𝑁) d𝑥 = (((𝐵↑(𝑁 + 1)) − (𝐴↑(𝑁 + 1))) / (𝑁 + 1)))
 
PART 14  BASIC REAL AND COMPLEX FUNCTIONS
 
14.1  Polynomials
 
14.1.1  Polynomial degrees
 
Syntaxcmdg 24652 Multivariate polynomial degree.
class mDeg
 
Syntaxcdg1 24653 Univariate polynomial degree.
class deg1
 
Definitiondf-mdeg 24654* 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 24786. (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 ( ∈ (𝑓 supp (0g𝑟)) ↦ (ℂfld Σg )), ℝ*, < )))
 
Definitiondf-deg1 24655 Define the degree of a univariate polynomial. (Contributed by Stefan O'Rear, 23-Mar-2015.)
deg1 = (𝑟 ∈ V ↦ (1o mDeg 𝑟))
 
Theoremreldmmdeg 24656 Multivariate degree is a binary operation. (Contributed by Stefan O'Rear, 28-Mar-2015.)
Rel dom mDeg
 
Theoremtdeglem1 24657* Functionality of the total degree helper function. (Contributed by Stefan O'Rear, 19-Mar-2015.) (Proof shortened by AV, 27-Jul-2019.)
𝐴 = {𝑚 ∈ (ℕ0m 𝐼) ∣ (𝑚 “ ℕ) ∈ Fin}    &   𝐻 = (𝐴 ↦ (ℂfld Σg ))       (𝐼𝑉𝐻:𝐴⟶ℕ0)
 
Theoremtdeglem3 24658* Additivity of the total degree helper function. (Contributed by Stefan O'Rear, 26-Mar-2015.) (Proof shortened by AV, 27-Jul-2019.)
𝐴 = {𝑚 ∈ (ℕ0m 𝐼) ∣ (𝑚 “ ℕ) ∈ Fin}    &   𝐻 = (𝐴 ↦ (ℂfld Σg ))       ((𝐼𝑉𝑋𝐴𝑌𝐴) → (𝐻‘(𝑋f + 𝑌)) = ((𝐻𝑋) + (𝐻𝑌)))
 
Theoremtdeglem4 24659* There is only one multi-index with total degree 0. (Contributed by Stefan O'Rear, 29-Mar-2015.)
𝐴 = {𝑚 ∈ (ℕ0m 𝐼) ∣ (𝑚 “ ℕ) ∈ Fin}    &   𝐻 = (𝐴 ↦ (ℂfld Σg ))       ((𝐼𝑉𝑋𝐴) → ((𝐻𝑋) = 0 ↔ 𝑋 = (𝐼 × {0})))
 
Theoremtdeglem2 24660 Simplification of total degree for the univariate case. (Contributed by Stefan O'Rear, 23-Mar-2015.)
( ∈ (ℕ0m 1o) ↦ (‘∅)) = ( ∈ (ℕ0m 1o) ↦ (ℂfld Σg ))
 
Theoremmdegfval 24661* 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 = (0g𝑅)    &   𝐴 = {𝑚 ∈ (ℕ0m 𝐼) ∣ (𝑚 “ ℕ) ∈ Fin}    &   𝐻 = (𝐴 ↦ (ℂfld Σg ))       𝐷 = (𝑓𝐵 ↦ sup((𝐻 “ (𝑓 supp 0 )), ℝ*, < ))
 
Theoremmdegval 24662* 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 = (0g𝑅)    &   𝐴 = {𝑚 ∈ (ℕ0m 𝐼) ∣ (𝑚 “ ℕ) ∈ Fin}    &   𝐻 = (𝐴 ↦ (ℂfld Σg ))       (𝐹𝐵 → (𝐷𝐹) = sup((𝐻 “ (𝐹 supp 0 )), ℝ*, < ))
 
Theoremmdegleb 24663* 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 = (0g𝑅)    &   𝐴 = {𝑚 ∈ (ℕ0m 𝐼) ∣ (𝑚 “ ℕ) ∈ Fin}    &   𝐻 = (𝐴 ↦ (ℂfld Σg ))       ((𝐹𝐵𝐺 ∈ ℝ*) → ((𝐷𝐹) ≤ 𝐺 ↔ ∀𝑥𝐴 (𝐺 < (𝐻𝑥) → (𝐹𝑥) = 0 )))
 
Theoremmdeglt 24664* 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 = (0g𝑅)    &   𝐴 = {𝑚 ∈ (ℕ0m 𝐼) ∣ (𝑚 “ ℕ) ∈ Fin}    &   𝐻 = (𝐴 ↦ (ℂfld Σg ))    &   (𝜑𝐹𝐵)    &   (𝜑𝑋𝐴)    &   (𝜑 → (𝐷𝐹) < (𝐻𝑋))       (𝜑 → (𝐹𝑋) = 0 )
 
Theoremmdegldg 24665* 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 = (0g𝑅)    &   𝐴 = {𝑚 ∈ (ℕ0m 𝐼) ∣ (𝑚 “ ℕ) ∈ Fin}    &   𝐻 = (𝐴 ↦ (ℂfld Σg ))    &   𝑌 = (0g𝑃)       ((𝑅 ∈ Ring ∧ 𝐹𝐵𝐹𝑌) → ∃𝑥𝐴 ((𝐹𝑥) ≠ 0 ∧ (𝐻𝑥) = (𝐷𝐹)))
 
Theoremmdegxrcl 24666 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)       (𝐹𝐵 → (𝐷𝐹) ∈ ℝ*)
 
Theoremmdegxrf 24667 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*
 
Theoremmdegcl 24668 Sharp closure for multivariate polynomials. (Contributed by Stefan O'Rear, 23-Mar-2015.)
𝐷 = (𝐼 mDeg 𝑅)    &   𝑃 = (𝐼 mPoly 𝑅)    &   𝐵 = (Base‘𝑃)       (𝐹𝐵 → (𝐷𝐹) ∈ (ℕ0 ∪ {-∞}))
 
Theoremmdeg0 24669 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 = (0g𝑃)       ((𝐼𝑉𝑅 ∈ Ring) → (𝐷0 ) = -∞)
 
Theoremmdegnn0cl 24670 Degree of a nonzero polynomial. (Contributed by Stefan O'Rear, 23-Mar-2015.)
𝐷 = (𝐼 mDeg 𝑅)    &   𝑃 = (𝐼 mPoly 𝑅)    &    0 = (0g𝑃)    &   𝐵 = (Base‘𝑃)       ((𝑅 ∈ Ring ∧ 𝐹𝐵𝐹0 ) → (𝐷𝐹) ∈ ℕ0)
 
Theoremdegltlem1 24671 Theorem on arithmetic of extended reals useful for degrees. (Contributed by Stefan O'Rear, 23-Mar-2015.)
((𝑋 ∈ (ℕ0 ∪ {-∞}) ∧ 𝑌 ∈ ℤ) → (𝑋 < 𝑌𝑋 ≤ (𝑌 − 1)))
 
Theoremdegltp1le 24672 Theorem on arithmetic of extended reals useful for degrees. (Contributed by Stefan O'Rear, 1-Apr-2015.)
((𝑋 ∈ (ℕ0 ∪ {-∞}) ∧ 𝑌 ∈ ℤ) → (𝑋 < (𝑌 + 1) ↔ 𝑋𝑌))
 
Theoremmdegaddle 24673 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)    &   (𝜑𝐼𝑉)    &   (𝜑𝑅 ∈ Ring)    &   𝐵 = (Base‘𝑌)    &    + = (+g𝑌)    &   (𝜑𝐹𝐵)    &   (𝜑𝐺𝐵)       (𝜑 → (𝐷‘(𝐹 + 𝐺)) ≤ if((𝐷𝐹) ≤ (𝐷𝐺), (𝐷𝐺), (𝐷𝐹)))
 
Theoremmdegvscale 24674 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)    &   (𝜑𝐼𝑉)    &   (𝜑𝑅 ∈ Ring)    &   𝐵 = (Base‘𝑌)    &   𝐾 = (Base‘𝑅)    &    · = ( ·𝑠𝑌)    &   (𝜑𝐹𝐾)    &   (𝜑𝐺𝐵)       (𝜑 → (𝐷‘(𝐹 · 𝐺)) ≤ (𝐷𝐺))
 
Theoremmdegvsca 24675 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)    &   (𝜑𝐼𝑉)    &   (𝜑𝑅 ∈ Ring)    &   𝐵 = (Base‘𝑌)    &   𝐸 = (RLReg‘𝑅)    &    · = ( ·𝑠𝑌)    &   (𝜑𝐹𝐸)    &   (𝜑𝐺𝐵)       (𝜑 → (𝐷‘(𝐹 · 𝐺)) = (𝐷𝐺))
 
Theoremmdegle0 24676 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)    &   (𝜑𝐼𝑉)    &   (𝜑𝑅 ∈ Ring)    &   𝐵 = (Base‘𝑌)    &   𝐴 = (algSc‘𝑌)    &   (𝜑𝐹𝐵)       (𝜑 → ((𝐷𝐹) ≤ 0 ↔ 𝐹 = (𝐴‘(𝐹‘(𝐼 × {0})))))
 
Theoremmdegmullem 24677* Lemma for mdegmulle2 24678. (Contributed by Stefan O'Rear, 26-Mar-2015.)
𝑌 = (𝐼 mPoly 𝑅)    &   𝐷 = (𝐼 mDeg 𝑅)    &   (𝜑𝐼𝑉)    &   (𝜑𝑅 ∈ Ring)    &   𝐵 = (Base‘𝑌)    &    · = (.r𝑌)    &   (𝜑𝐹𝐵)    &   (𝜑𝐺𝐵)    &   (𝜑𝐽 ∈ ℕ0)    &   (𝜑𝐾 ∈ ℕ0)    &   (𝜑 → (𝐷𝐹) ≤ 𝐽)    &   (𝜑 → (𝐷𝐺) ≤ 𝐾)    &   𝐴 = {𝑎 ∈ (ℕ0m 𝐼) ∣ (𝑎 “ ℕ) ∈ Fin}    &   𝐻 = (𝑏𝐴 ↦ (ℂfld Σg 𝑏))       (𝜑 → (𝐷‘(𝐹 · 𝐺)) ≤ (𝐽 + 𝐾))
 
Theoremmdegmulle2 24678 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)    &   (𝜑𝐼𝑉)    &   (𝜑𝑅 ∈ Ring)    &   𝐵 = (Base‘𝑌)    &    · = (.r𝑌)    &   (𝜑𝐹𝐵)    &   (𝜑𝐺𝐵)    &   (𝜑𝐽 ∈ ℕ0)    &   (𝜑𝐾 ∈ ℕ0)    &   (𝜑 → (𝐷𝐹) ≤ 𝐽)    &   (𝜑 → (𝐷𝐺) ≤ 𝐾)       (𝜑 → (𝐷‘(𝐹 · 𝐺)) ≤ (𝐽 + 𝐾))
 
Theoremdeg1fval 24679 Relate univariate polynomial degree to multivariate. (Contributed by Stefan O'Rear, 23-Mar-2015.) (Revised by Mario Carneiro, 7-Oct-2015.)
𝐷 = ( deg1𝑅)       𝐷 = (1o mDeg 𝑅)
 
Theoremdeg1xrf 24680 Functionality of univariate polynomial degree, weak range. (Contributed by Stefan O'Rear, 23-Mar-2015.)
𝐷 = ( deg1𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)       𝐷:𝐵⟶ℝ*
 
Theoremdeg1xrcl 24681 Closure of univariate polynomial degree in extended reals. (Contributed by Stefan O'Rear, 23-Mar-2015.)
𝐷 = ( deg1𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)       (𝐹𝐵 → (𝐷𝐹) ∈ ℝ*)
 
Theoremdeg1cl 24682 Sharp closure of univariate polynomial degree. (Contributed by Stefan O'Rear, 23-Mar-2015.)
𝐷 = ( deg1𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)       (𝐹𝐵 → (𝐷𝐹) ∈ (ℕ0 ∪ {-∞}))
 
Theoremmdegpropd 24683* Property deduction for polynomial degree. (Contributed by Stefan O'Rear, 28-Mar-2015.) (Proof shortened by AV, 27-Jul-2019.)
(𝜑𝐵 = (Base‘𝑅))    &   (𝜑𝐵 = (Base‘𝑆))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝑅)𝑦) = (𝑥(+g𝑆)𝑦))       (𝜑 → (𝐼 mDeg 𝑅) = (𝐼 mDeg 𝑆))
 
Theoremdeg1fvi 24684 Univariate polynomial degree respects protection. (Contributed by Stefan O'Rear, 28-Mar-2015.)
( deg1𝑅) = ( deg1 ‘( I ‘𝑅))
 
Theoremdeg1propd 24685* Property deduction for polynomial degree. (Contributed by Stefan O'Rear, 28-Mar-2015.)
(𝜑𝐵 = (Base‘𝑅))    &   (𝜑𝐵 = (Base‘𝑆))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝑅)𝑦) = (𝑥(+g𝑆)𝑦))       (𝜑 → ( deg1𝑅) = ( deg1𝑆))
 
Theoremdeg1z 24686 Degree of the zero univariate polynomial. (Contributed by Stefan O'Rear, 23-Mar-2015.)
𝐷 = ( deg1𝑅)    &   𝑃 = (Poly1𝑅)    &    0 = (0g𝑃)       (𝑅 ∈ Ring → (𝐷0 ) = -∞)
 
Theoremdeg1nn0cl 24687 Degree of a nonzero univariate polynomial. (Contributed by Stefan O'Rear, 23-Mar-2015.) (Revised by Mario Carneiro, 7-Oct-2015.)
𝐷 = ( deg1𝑅)    &   𝑃 = (Poly1𝑅)    &    0 = (0g𝑃)    &   𝐵 = (Base‘𝑃)       ((𝑅 ∈ Ring ∧ 𝐹𝐵𝐹0 ) → (𝐷𝐹) ∈ ℕ0)
 
Theoremdeg1n0ima 24688 Degree image of a set of polynomials which does not include zero. (Contributed by Stefan O'Rear, 28-Mar-2015.)
𝐷 = ( deg1𝑅)    &   𝑃 = (Poly1𝑅)    &    0 = (0g𝑃)    &   𝐵 = (Base‘𝑃)       (𝑅 ∈ Ring → (𝐷 “ (𝐵 ∖ { 0 })) ⊆ ℕ0)
 
Theoremdeg1nn0clb 24689 A polynomial is nonzero iff it has definite degree. (Contributed by Stefan O'Rear, 23-Mar-2015.)
𝐷 = ( deg1𝑅)    &   𝑃 = (Poly1𝑅)    &    0 = (0g𝑃)    &   𝐵 = (Base‘𝑃)       ((𝑅 ∈ Ring ∧ 𝐹𝐵) → (𝐹0 ↔ (𝐷𝐹) ∈ ℕ0))
 
Theoremdeg1lt0 24690 A polynomial is zero iff it has negative degree. (Contributed by Stefan O'Rear, 1-Apr-2015.)
𝐷 = ( deg1𝑅)    &   𝑃 = (Poly1𝑅)    &    0 = (0g𝑃)    &   𝐵 = (Base‘𝑃)       ((𝑅 ∈ Ring ∧ 𝐹𝐵) → ((𝐷𝐹) < 0 ↔ 𝐹 = 0 ))
 
Theoremdeg1ldg 24691 A nonzero univariate polynomial always has a nonzero leading coefficient. (Contributed by Stefan O'Rear, 23-Mar-2015.)
𝐷 = ( deg1𝑅)    &   𝑃 = (Poly1𝑅)    &    0 = (0g𝑃)    &   𝐵 = (Base‘𝑃)    &   𝑌 = (0g𝑅)    &   𝐴 = (coe1𝐹)       ((𝑅 ∈ Ring ∧ 𝐹𝐵𝐹0 ) → (𝐴‘(𝐷𝐹)) ≠ 𝑌)
 
Theoremdeg1ldgn 24692 An index at which a polynomial is zero, cannot be its degree. (Contributed by Stefan O'Rear, 23-Mar-2015.)
𝐷 = ( deg1𝑅)    &   𝑃 = (Poly1𝑅)    &    0 = (0g𝑃)    &   𝐵 = (Base‘𝑃)    &   𝑌 = (0g𝑅)    &   𝐴 = (coe1𝐹)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐹𝐵)    &   (𝜑𝑋 ∈ ℕ0)    &   (𝜑 → (𝐴𝑋) = 𝑌)       (𝜑 → (𝐷𝐹) ≠ 𝑋)
 
Theoremdeg1ldgdomn 24693 A nonzero univariate polynomial over a domain always has a nonzero-divisor leading coefficient. (Contributed by Stefan O'Rear, 28-Mar-2015.)
𝐷 = ( deg1𝑅)    &   𝑃 = (Poly1𝑅)    &    0 = (0g𝑃)    &   𝐵 = (Base‘𝑃)    &   𝐸 = (RLReg‘𝑅)    &   𝐴 = (coe1𝐹)       ((𝑅 ∈ Domn ∧ 𝐹𝐵𝐹0 ) → (𝐴‘(𝐷𝐹)) ∈ 𝐸)
 
Theoremdeg1leb 24694* Property of being of limited degree. (Contributed by Stefan O'Rear, 23-Mar-2015.)
𝐷 = ( deg1𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &    0 = (0g𝑅)    &   𝐴 = (coe1𝐹)       ((𝐹𝐵𝐺 ∈ ℝ*) → ((𝐷𝐹) ≤ 𝐺 ↔ ∀𝑥 ∈ ℕ0 (𝐺 < 𝑥 → (𝐴𝑥) = 0 )))
 
Theoremdeg1val 24695 Value of the univariate degree as a supremum. (Contributed by Stefan O'Rear, 29-Mar-2015.) (Revised by AV, 25-Jul-2019.)
𝐷 = ( deg1𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &    0 = (0g𝑅)    &   𝐴 = (coe1𝐹)       (𝐹𝐵 → (𝐷𝐹) = sup((𝐴 supp 0 ), ℝ*, < ))
 
Theoremdeg1lt 24696 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 = ( deg1𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &    0 = (0g𝑅)    &   𝐴 = (coe1𝐹)       ((𝐹𝐵𝐺 ∈ ℕ0 ∧ (𝐷𝐹) < 𝐺) → (𝐴𝐺) = 0 )
 
Theoremdeg1ge 24697 Conversely, a nonzero coefficient sets a lower bound on the degree. (Contributed by Stefan O'Rear, 23-Mar-2015.)
𝐷 = ( deg1𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &    0 = (0g𝑅)    &   𝐴 = (coe1𝐹)       ((𝐹𝐵𝐺 ∈ ℕ0 ∧ (𝐴𝐺) ≠ 0 ) → 𝐺 ≤ (𝐷𝐹))
 
Theoremcoe1mul3 24698 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 = (Poly1𝑅)    &    = (.r𝑌)    &    · = (.r𝑅)    &   𝐵 = (Base‘𝑌)    &   𝐷 = ( deg1𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐹𝐵)    &   (𝜑𝐼 ∈ ℕ0)    &   (𝜑 → (𝐷𝐹) ≤ 𝐼)    &   (𝜑𝐺𝐵)    &   (𝜑𝐽 ∈ ℕ0)    &   (𝜑 → (𝐷𝐺) ≤ 𝐽)       (𝜑 → ((coe1‘(𝐹 𝐺))‘(𝐼 + 𝐽)) = (((coe1𝐹)‘𝐼) · ((coe1𝐺)‘𝐽)))
 
Theoremcoe1mul4 24699 Value of the "leading" coefficient of a product of two nonzero polynomials. This will fail to actually be the leading coefficient only if it is zero (requiring the basic ring to contain zero divisors). (Contributed by Stefan O'Rear, 25-Mar-2015.)
𝑌 = (Poly1𝑅)    &    = (.r𝑌)    &    · = (.r𝑅)    &   𝐵 = (Base‘𝑌)    &   𝐷 = ( deg1𝑅)    &    0 = (0g𝑌)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐹𝐵)    &   (𝜑𝐹0 )    &   (𝜑𝐺𝐵)    &   (𝜑𝐺0 )       (𝜑 → ((coe1‘(𝐹 𝐺))‘((𝐷𝐹) + (𝐷𝐺))) = (((coe1𝐹)‘(𝐷𝐹)) · ((coe1𝐺)‘(𝐷𝐺))))
 
Theoremdeg1addle 24700 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 = (Poly1𝑅)    &   𝐷 = ( deg1𝑅)    &   (𝜑𝑅 ∈ Ring)    &   𝐵 = (Base‘𝑌)    &    + = (+g𝑌)    &   (𝜑𝐹𝐵)    &   (𝜑𝐺𝐵)       (𝜑 → (𝐷‘(𝐹 + 𝐺)) ≤ if((𝐷𝐹) ≤ (𝐷𝐺), (𝐷𝐺), (𝐷𝐹)))
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78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44900 450 44901-45000 451 45001-45100 452 45101-45187
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