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Theorem List for Metamath Proof Explorer - 24201-24300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdv11cn 24201 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 24202 Lemma for dvgt0 24204 and dvlt0 24205. (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 24203* Lemma for dvgt0 24204 and dvlt0 24205. (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 24204 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 24205 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 24206 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 24207* 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 24208* Lemma for dvivth 24210. (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 24209* Lemma for dvivth 24210. (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 24210 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 23662 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 24211 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 24212 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 24213* Lemma for lhop1 24214. (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 24214* 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 24215* 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 24216* 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 24217 Lemma for dvcnvre 24219. (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 24218 Lemma for dvcnvre 24219. (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 24219* 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 24220 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 24221* 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 24222* 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 24223* 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 24224* 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 24225* Lemma for dvfsumrlim 24231. (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 24226* Lemma for dvfsumrlim 24231. (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 24227* Lemma for dvfsumrlim 24231. (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 24228* Lemma for dvfsumrlim 24231. (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 24229* Lemma for dvfsumrlim 24231. (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 24230* Lemma for dvfsumrlim 24231. Satisfy the assumption of dvfsumlem4 24229. (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 24231* 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 24232* 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 24233* Conjoin the statements of dvfsumrlim 24231 and dvfsumrlim2 24232. (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 24234* The reverse of dvfsumrlim 24231, 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 24235* Lemma for ftc1a 24237 and ftc1 24242. (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 24236* Lemma for ftc1 24242. (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 24237* 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 24238* Lemma for ftc1 24242. (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 24239* Lemma for ftc1 24242. (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 24240* Lemma for ftc1 24242. (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 24241* Lemma for ftc1 24242. (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 24242* 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 24243* Strengthen the assumptions of ftc1 24242 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 24244* 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 24245* Lemma for ftc2ditg 24246. (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 24246* Directed integral analogue of ftc2 24244. (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 24247* 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 24248* Lemma for itgsubst 24249. (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 24249* 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 24248, 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)

PART 14  BASIC REAL AND COMPLEX FUNCTIONS

14.1  Polynomials

14.1.1  Polynomial degrees

Syntaxcmdg 24250 Multivariate polynomial degree.
class mDeg

Syntaxcdg1 24251 Univariate polynomial degree.
class deg1

Definitiondf-mdeg 24252* 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 24384. (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 24253 Define the degree of a univariate polynomial. (Contributed by Stefan O'Rear, 23-Mar-2015.)
deg1 = (𝑟 ∈ V ↦ (1o mDeg 𝑟))

Theoremreldmmdeg 24254 Multivariate degree is a binary operation. (Contributed by Stefan O'Rear, 28-Mar-2015.)
Rel dom mDeg

Theoremtdeglem1 24255* 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 24256* 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 24257* 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 24258 Simplification of total degree for the univariate case. (Contributed by Stefan O'Rear, 23-Mar-2015.)
( ∈ (ℕ0𝑚 1o) ↦ (‘∅)) = ( ∈ (ℕ0𝑚 1o) ↦ (ℂfld Σg ))

Theoremmdegfval 24259* 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 24260* 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 24261* 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 24262* 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 24263* 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 24264 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 24265 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 24266 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 24267 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 24268 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 24269 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 24270 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 24271 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 24272 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 24273 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 24274 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 24275* Lemma for mdegmulle2 24276. (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 24276 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 24277 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 24278 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 24279 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 24280 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 24281* 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 24282 Univariate polynomial degree respects protection. (Contributed by Stefan O'Rear, 28-Mar-2015.)
( deg1𝑅) = ( deg1 ‘( I ‘𝑅))

Theoremdeg1propd 24283* 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 24284 Degree of the zero univariate polynomial. (Contributed by Stefan O'Rear, 23-Mar-2015.)
𝐷 = ( deg1𝑅)    &   𝑃 = (Poly1𝑅)    &    0 = (0g𝑃)       (𝑅 ∈ Ring → (𝐷0 ) = -∞)

Theoremdeg1nn0cl 24285 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 24286 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 24287 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 24288 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 24289 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 24290 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 24291 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 24292* 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 24293 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 24294 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 24295 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 24296 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 24297 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 24298 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)))

Theoremdeg1addle2 24299 If both factors have degree bounded by 𝐿, then the sum of the polynomials also has degree bounded by 𝐿. (Contributed by Stefan O'Rear, 1-Apr-2015.)
𝑌 = (Poly1𝑅)    &   𝐷 = ( deg1𝑅)    &   (𝜑𝑅 ∈ Ring)    &   𝐵 = (Base‘𝑌)    &    + = (+g𝑌)    &   (𝜑𝐹𝐵)    &   (𝜑𝐺𝐵)    &   (𝜑𝐿 ∈ ℝ*)    &   (𝜑 → (𝐷𝐹) ≤ 𝐿)    &   (𝜑 → (𝐷𝐺) ≤ 𝐿)       (𝜑 → (𝐷‘(𝐹 + 𝐺)) ≤ 𝐿)

Theoremdeg1add 24300 Exact degree of a sum of two polynomials of unequal degree. (Contributed by Stefan O'Rear, 28-Mar-2015.)
𝑌 = (Poly1𝑅)    &   𝐷 = ( deg1𝑅)    &   (𝜑𝑅 ∈ Ring)    &   𝐵 = (Base‘𝑌)    &    + = (+g𝑌)    &   (𝜑𝐹𝐵)    &   (𝜑𝐺𝐵)    &   (𝜑 → (𝐷𝐺) < (𝐷𝐹))       (𝜑 → (𝐷‘(𝐹 + 𝐺)) = (𝐷𝐹))

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