P. 260 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 25901-26000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	c1liplem1 25901*	Lemma for c1lip1 25902. (Contributed by Stefan O'Rear, 15-Nov-2014.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ (ℂ ↑pm ℝ))    &   ⊢ (𝜑 → ((ℝ D 𝐹) ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℝ))    &   ⊢ (𝜑 → (𝐹 ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℝ))    &   ⊢ 𝐾 = sup((abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))), ℝ, < )    ⇒   ⊢ (𝜑 → (𝐾 ∈ ℝ ∧ ∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(𝑥 < 𝑦 → (abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝐾 · (abs‘(𝑦 − 𝑥))))))
Theorem	c1lip1 25902*	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 ∈ (ℂ ↑pm ℝ))    &   ⊢ (𝜑 → ((ℝ D 𝐹) ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℝ))    &   ⊢ (𝜑 → (𝐹 ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℝ))    ⇒   ⊢ (𝜑 → ∃𝑘 ∈ ℝ ∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥))))
Theorem	c1lip2 25903*	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 25904*	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 25905	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 25906	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 25907	Lemma for dvgt0 25909 and dvlt0 25910. (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 25908*	Lemma for dvgt0 25909 and dvlt0 25910. (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 25909	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 25910	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 25911	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 25912*	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 25913*	Lemma for dvivth 25915. (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 25914*	Lemma for dvivth 25915. (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 25915	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 25359 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 25916	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 25917	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 25918*	Lemma for lhop1 25919. (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 25919*	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 25920*	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 25921*	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 25922	Lemma for dvcnvre 25924. (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 25923	Lemma for dvcnvre 25924. (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 25924*	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 25925	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 25926*	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.) Avoid ax-mulf 11148. (Revised by GG, 16-Mar-2025.)
⊢ (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	dvfsumleOLD 25927*	Obsolete version of dvfsumle 25926 as of 17-Apr-2025. (Contributed by Mario Carneiro, 14-May-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (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 25928*	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 25929*	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 25930*	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 25931*	Lemma for dvfsumrlim 25938. (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 25932*	Lemma for dvfsumrlim 25938. (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 25933*	Lemma for dvfsumrlim 25938. (Contributed by Mario Carneiro, 17-May-2016.) Avoid ax-mulf 11148. (Revised by GG, 16-Mar-2025.)
⊢ 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	dvfsumlem2OLD 25934*	Obsolete version of dvfsumlem2 25933 as of 17-Apr-2025. (Contributed by Mario Carneiro, 17-May-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 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 25935*	Lemma for dvfsumrlim 25938. (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 25936*	Lemma for dvfsumrlim 25938. (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 25937*	Lemma for dvfsumrlim 25938. Satisfy the assumption of dvfsumlem4 25936. (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 25938*	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 25939*	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 25940*	Conjoin the statements of dvfsumrlim 25938 and dvfsumrlim2 25939. (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 25941*	The reverse of dvfsumrlim 25938, 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 25942*	Lemma for ftc1a 25944 and ftc1 25949. (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 25943*	Lemma for ftc1 25949. (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 25944*	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 25945*	Lemma for ftc1 25949. (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 25946*	Lemma for ftc1 25949. (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 25947*	Lemma for ftc1 25949. (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 25948*	Lemma for ftc1 25949. (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 25949*	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 25950*	Strengthen the assumptions of ftc1 25949 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 25951*	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 25952*	Lemma for ftc2ditg 25953. (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 25953*	Directed integral analogue of ftc2 25951. (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 25954*	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 25955*	Lemma for itgsubst 25956. (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 25956*	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 25955, 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 25957*	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)))
PART 14  BASIC REAL AND COMPLEX FUNCTIONS
14.1  Polynomials
14.1.1  Polynomial degrees
Syntax	cmdg 25958	Multivariate polynomial degree.
class mDeg
Syntax	cdg1 25959	Univariate polynomial degree.
class deg1
Definition	df-mdeg 25960*	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 26096. (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 25961	Define the degree of a univariate polynomial. (Contributed by Stefan O'Rear, 23-Mar-2015.)
⊢ deg1 = (𝑟 ∈ V ↦ (1o mDeg 𝑟))
Theorem	reldmmdeg 25962	Multivariate degree is a binary operation. (Contributed by Stefan O'Rear, 28-Mar-2015.)
⊢ Rel dom mDeg
Theorem	tdeglem1 25963*	Functionality of the total degree helper function. (Contributed by Stefan O'Rear, 19-Mar-2015.) (Proof shortened by AV, 27-Jul-2019.) Remove sethood antecedent. (Revised by SN, 7-Aug-2024.)
⊢ 𝐴 = {𝑚 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑚 “ ℕ) ∈ Fin}    &   ⊢ 𝐻 = (ℎ ∈ 𝐴 ↦ (ℂfld Σg ℎ))    ⇒   ⊢ 𝐻:𝐴⟶ℕ0
Theorem	tdeglem3 25964*	Additivity of the total degree helper function. (Contributed by Stefan O'Rear, 26-Mar-2015.) (Proof shortened by AV, 27-Jul-2019.) Remove a sethood antecedent. (Revised by SN, 7-Aug-2024.)
⊢ 𝐴 = {𝑚 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑚 “ ℕ) ∈ Fin}    &   ⊢ 𝐻 = (ℎ ∈ 𝐴 ↦ (ℂfld Σg ℎ))    ⇒   ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝐻‘(𝑋 ∘f + 𝑌)) = ((𝐻‘𝑋) + (𝐻‘𝑌)))
Theorem	tdeglem4 25965*	There is only one multi-index with total degree 0. (Contributed by Stefan O'Rear, 29-Mar-2015.) Remove a sethood antecedent. (Revised by SN, 7-Aug-2024.)
⊢ 𝐴 = {𝑚 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑚 “ ℕ) ∈ Fin}    &   ⊢ 𝐻 = (ℎ ∈ 𝐴 ↦ (ℂfld Σg ℎ))    ⇒   ⊢ (𝑋 ∈ 𝐴 → ((𝐻‘𝑋) = 0 ↔ 𝑋 = (𝐼 × {0})))
Theorem	tdeglem2 25966	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 25967*	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 25968*	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 25969*	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 25970*	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 25971*	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 25972	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 25973	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 25974	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 25975	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 25976	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 25977	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 25978	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 25979	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 25980	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 25981	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 25982	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 25983*	Lemma for mdegmulle2 25984. (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 25984	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 25985	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 25986	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 25987	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 25988	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 25989*	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 25990	Univariate polynomial degree respects protection. (Contributed by Stefan O'Rear, 28-Mar-2015.)
⊢ (deg1‘𝑅) = (deg1‘( I ‘𝑅))
Theorem	deg1propd 25991*	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 25992	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 25993	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 25994	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 25995	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 25996	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 25997	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 25998	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 25999	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 26000*	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 )))