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	dvco 25901	The chain rule for derivatives at a point. For the (more general) relation version, see dvcobr 25899. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)
⊢ (𝜑 → 𝐹:𝑋⟶ℂ)    &   ⊢ (𝜑 → 𝑋 ⊆ 𝑆)    &   ⊢ (𝜑 → 𝐺:𝑌⟶𝑋)    &   ⊢ (𝜑 → 𝑌 ⊆ 𝑇)    &   ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})    &   ⊢ (𝜑 → 𝑇 ∈ {ℝ, ℂ})    &   ⊢ (𝜑 → (𝐺‘𝐶) ∈ dom (𝑆 D 𝐹))    &   ⊢ (𝜑 → 𝐶 ∈ dom (𝑇 D 𝐺))    ⇒   ⊢ (𝜑 → ((𝑇 D (𝐹 ∘ 𝐺))‘𝐶) = (((𝑆 D 𝐹)‘(𝐺‘𝐶)) · ((𝑇 D 𝐺)‘𝐶)))
Theorem	dvcof 25902	The chain rule for everywhere-differentiable functions. (Contributed by Mario Carneiro, 10-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})    &   ⊢ (𝜑 → 𝑇 ∈ {ℝ, ℂ})    &   ⊢ (𝜑 → 𝐹:𝑋⟶ℂ)    &   ⊢ (𝜑 → 𝐺:𝑌⟶𝑋)    &   ⊢ (𝜑 → dom (𝑆 D 𝐹) = 𝑋)    &   ⊢ (𝜑 → dom (𝑇 D 𝐺) = 𝑌)    ⇒   ⊢ (𝜑 → (𝑇 D (𝐹 ∘ 𝐺)) = (((𝑆 D 𝐹) ∘ 𝐺) ∘f · (𝑇 D 𝐺)))
Theorem	dvcjbr 25903	The derivative of the conjugate of a function. For the (simpler but more limited) function version, see dvcj 25904. (This doesn't follow from dvcobr 25899 because ∗ is not a function on the reals, and even if we used complex derivatives, ∗ is not complex-differentiable.) (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)
⊢ (𝜑 → 𝐹:𝑋⟶ℂ)    &   ⊢ (𝜑 → 𝑋 ⊆ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ dom (ℝ D 𝐹))    ⇒   ⊢ (𝜑 → 𝐶(ℝ D (∗ ∘ 𝐹))(∗‘((ℝ D 𝐹)‘𝐶)))
Theorem	dvcj 25904	The derivative of the conjugate of a function. For the (more general) relation version, see dvcjbr 25903. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)
⊢ ((𝐹:𝑋⟶ℂ ∧ 𝑋 ⊆ ℝ) → (ℝ D (∗ ∘ 𝐹)) = (∗ ∘ (ℝ D 𝐹)))
Theorem	dvfre 25905	The derivative of a real function is real. (Contributed by Mario Carneiro, 1-Sep-2014.)
⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ)
Theorem	dvnfre 25906	The 𝑁-th derivative of a real function is real. (Contributed by Mario Carneiro, 1-Jan-2017.)
⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ ∧ 𝑁 ∈ ℕ0) → ((ℝ D𝑛 𝐹)‘𝑁):dom ((ℝ D𝑛 𝐹)‘𝑁)⟶ℝ)
Theorem	dvexp 25907*	Derivative of a power function. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)
⊢ (𝑁 ∈ ℕ → (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑𝑁))) = (𝑥 ∈ ℂ ↦ (𝑁 · (𝑥↑(𝑁 − 1)))))
Theorem	dvexp2 25908*	Derivative of an exponential, possibly zero power. (Contributed by Stefan O'Rear, 13-Nov-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)
⊢ (𝑁 ∈ ℕ0 → (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑𝑁))) = (𝑥 ∈ ℂ ↦ if(𝑁 = 0, 0, (𝑁 · (𝑥↑(𝑁 − 1))))))
Theorem	dvrec 25909*	Derivative of the reciprocal function. (Contributed by Mario Carneiro, 25-Feb-2015.) (Revised by Mario Carneiro, 28-Dec-2016.)
⊢ (𝐴 ∈ ℂ → (ℂ D (𝑥 ∈ (ℂ ∖ {0}) ↦ (𝐴 / 𝑥))) = (𝑥 ∈ (ℂ ∖ {0}) ↦ -(𝐴 / (𝑥↑2))))
Theorem	dvmptres3 25910*	Function-builder for derivative: restrict a derivative to a subset. (Contributed by Mario Carneiro, 11-Feb-2015.)
⊢ 𝐽 = (TopOpen‘ℂfld)    &   ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})    &   ⊢ (𝜑 → 𝑋 ∈ 𝐽)    &   ⊢ (𝜑 → (𝑆 ∩ 𝑋) = 𝑌)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → (ℂ D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐵))    ⇒   ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑌 ↦ 𝐴)) = (𝑥 ∈ 𝑌 ↦ 𝐵))
Theorem	dvmptid 25911*	Function-builder for derivative: derivative of the identity. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})    ⇒   ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑆 ↦ 𝑥)) = (𝑥 ∈ 𝑆 ↦ 1))
Theorem	dvmptc 25912*	Function-builder for derivative: derivative of a constant. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})    &   ⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑆 ↦ 𝐴)) = (𝑥 ∈ 𝑆 ↦ 0))
Theorem	dvmptcl 25913*	Closure lemma for dvmptcmul 25918 and other related theorems. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐵))    ⇒   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ ℂ)
Theorem	dvmptadd 25914*	Function-builder for derivative, addition rule. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐵))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐶 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐷 ∈ 𝑊)    &   ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐶)) = (𝑥 ∈ 𝑋 ↦ 𝐷))    ⇒   ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ (𝐴 + 𝐶))) = (𝑥 ∈ 𝑋 ↦ (𝐵 + 𝐷)))
Theorem	dvmptmul 25915*	Function-builder for derivative, product rule. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐵))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐶 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐷 ∈ 𝑊)    &   ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐶)) = (𝑥 ∈ 𝑋 ↦ 𝐷))    ⇒   ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐶))) = (𝑥 ∈ 𝑋 ↦ ((𝐵 · 𝐶) + (𝐷 · 𝐴))))
Theorem	dvmptres2 25916*	Function-builder for derivative: restrict a derivative to a subset. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐵))    &   ⊢ (𝜑 → 𝑍 ⊆ 𝑋)    &   ⊢ 𝐽 = (𝐾 ↾t 𝑆)    &   ⊢ 𝐾 = (TopOpen‘ℂfld)    &   ⊢ (𝜑 → ((int‘𝐽)‘𝑍) = 𝑌)    ⇒   ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑍 ↦ 𝐴)) = (𝑥 ∈ 𝑌 ↦ 𝐵))
Theorem	dvmptres 25917*	Function-builder for derivative: restrict a derivative to an open subset. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐵))    &   ⊢ (𝜑 → 𝑌 ⊆ 𝑋)    &   ⊢ 𝐽 = (𝐾 ↾t 𝑆)    &   ⊢ 𝐾 = (TopOpen‘ℂfld)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐽)    ⇒   ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑌 ↦ 𝐴)) = (𝑥 ∈ 𝑌 ↦ 𝐵))
Theorem	dvmptcmul 25918*	Function-builder for derivative, product rule for constant multiplier. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐵))    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ (𝐶 · 𝐴))) = (𝑥 ∈ 𝑋 ↦ (𝐶 · 𝐵)))
Theorem	dvmptdivc 25919*	Function-builder for derivative, division rule for constant divisor. (Contributed by Mario Carneiro, 18-May-2016.)
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐵))    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ≠ 0)    ⇒   ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶))) = (𝑥 ∈ 𝑋 ↦ (𝐵 / 𝐶)))
Theorem	dvmptneg 25920*	Function-builder for derivative, product rule for negatives. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐵))    ⇒   ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ -𝐴)) = (𝑥 ∈ 𝑋 ↦ -𝐵))
Theorem	dvmptsub 25921*	Function-builder for derivative, subtraction rule. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐵))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐶 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐷 ∈ 𝑊)    &   ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐶)) = (𝑥 ∈ 𝑋 ↦ 𝐷))    ⇒   ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ (𝐴 − 𝐶))) = (𝑥 ∈ 𝑋 ↦ (𝐵 − 𝐷)))
Theorem	dvmptcj 25922*	Function-builder for derivative, conjugate rule. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → (ℝ D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐵))    ⇒   ⊢ (𝜑 → (ℝ D (𝑥 ∈ 𝑋 ↦ (∗‘𝐴))) = (𝑥 ∈ 𝑋 ↦ (∗‘𝐵)))
Theorem	dvmptre 25923*	Function-builder for derivative, real part. (Contributed by Mario Carneiro, 1-Sep-2014.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → (ℝ D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐵))    ⇒   ⊢ (𝜑 → (ℝ D (𝑥 ∈ 𝑋 ↦ (ℜ‘𝐴))) = (𝑥 ∈ 𝑋 ↦ (ℜ‘𝐵)))
Theorem	dvmptim 25924*	Function-builder for derivative, imaginary part. (Contributed by Mario Carneiro, 1-Sep-2014.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → (ℝ D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐵))    ⇒   ⊢ (𝜑 → (ℝ D (𝑥 ∈ 𝑋 ↦ (ℑ‘𝐴))) = (𝑥 ∈ 𝑋 ↦ (ℑ‘𝐵)))
Theorem	dvmptntr 25925*	Function-builder for derivative: expand the function from an open set to its closure. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
⊢ (𝜑 → 𝑆 ⊆ ℂ)    &   ⊢ (𝜑 → 𝑋 ⊆ 𝑆)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ)    &   ⊢ 𝐽 = (𝐾 ↾t 𝑆)    &   ⊢ 𝐾 = (TopOpen‘ℂfld)    &   ⊢ (𝜑 → ((int‘𝐽)‘𝑋) = 𝑌)    ⇒   ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑆 D (𝑥 ∈ 𝑌 ↦ 𝐴)))
Theorem	dvmptco 25926*	Function-builder for derivative, chain rule. (Contributed by Mario Carneiro, 1-Sep-2014.)
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})    &   ⊢ (𝜑 → 𝑇 ∈ {ℝ, ℂ})    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑌)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 𝐶 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 𝐷 ∈ 𝑊)    &   ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐵))    &   ⊢ (𝜑 → (𝑇 D (𝑦 ∈ 𝑌 ↦ 𝐶)) = (𝑦 ∈ 𝑌 ↦ 𝐷))    &   ⊢ (𝑦 = 𝐴 → 𝐶 = 𝐸)    &   ⊢ (𝑦 = 𝐴 → 𝐷 = 𝐹)    ⇒   ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐸)) = (𝑥 ∈ 𝑋 ↦ (𝐹 · 𝐵)))
Theorem	dvrecg 25927*	Derivative of the reciprocal of a function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})    &   ⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ (ℂ ∖ {0}))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐵)) = (𝑥 ∈ 𝑋 ↦ 𝐶))    ⇒   ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐵))) = (𝑥 ∈ 𝑋 ↦ -((𝐴 · 𝐶) / (𝐵↑2))))
Theorem	dvmptdiv 25928*	Function-builder for derivative, quotient rule. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐵))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐶 ∈ (ℂ ∖ {0}))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐷 ∈ ℂ)    &   ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐶)) = (𝑥 ∈ 𝑋 ↦ 𝐷))    ⇒   ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶))) = (𝑥 ∈ 𝑋 ↦ (((𝐵 · 𝐶) − (𝐷 · 𝐴)) / (𝐶↑2))))
Theorem	dvmptfsum 25929*	Function-builder for derivative, finite sums rule. (Contributed by Stefan O'Rear, 12-Nov-2014.)
⊢ 𝐽 = (𝐾 ↾t 𝑆)    &   ⊢ 𝐾 = (TopOpen‘ℂfld)    &   ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})    &   ⊢ (𝜑 → 𝑋 ∈ 𝐽)    &   ⊢ (𝜑 → 𝐼 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐵))    ⇒   ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝐼 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝐼 𝐵))
Theorem	dvcnvlem 25930	Lemma for dvcnvre 25974. (Contributed by Mario Carneiro, 25-Feb-2015.) (Revised by Mario Carneiro, 8-Sep-2015.)
⊢ 𝐽 = (TopOpen‘ℂfld)    &   ⊢ 𝐾 = (𝐽 ↾t 𝑆)    &   ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})    &   ⊢ (𝜑 → 𝑌 ∈ 𝐾)    &   ⊢ (𝜑 → 𝐹:𝑋–1-1-onto→𝑌)    &   ⊢ (𝜑 → ◡𝐹 ∈ (𝑌–cn→𝑋))    &   ⊢ (𝜑 → dom (𝑆 D 𝐹) = 𝑋)    &   ⊢ (𝜑 → ¬ 0 ∈ ran (𝑆 D 𝐹))    &   ⊢ (𝜑 → 𝐶 ∈ 𝑋)    ⇒   ⊢ (𝜑 → (𝐹‘𝐶)(𝑆 D ◡𝐹)(1 / ((𝑆 D 𝐹)‘𝐶)))
Theorem	dvcnv 25931*	A weak version of dvcnvre 25974, valid for both real and complex domains but under the hypothesis that the inverse function is already known to be continuous, and the image set is known to be open. A more advanced proof can show that these conditions are unnecessary. (Contributed by Mario Carneiro, 25-Feb-2015.) (Revised by Mario Carneiro, 8-Sep-2015.)
⊢ 𝐽 = (TopOpen‘ℂfld)    &   ⊢ 𝐾 = (𝐽 ↾t 𝑆)    &   ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})    &   ⊢ (𝜑 → 𝑌 ∈ 𝐾)    &   ⊢ (𝜑 → 𝐹:𝑋–1-1-onto→𝑌)    &   ⊢ (𝜑 → ◡𝐹 ∈ (𝑌–cn→𝑋))    &   ⊢ (𝜑 → dom (𝑆 D 𝐹) = 𝑋)    &   ⊢ (𝜑 → ¬ 0 ∈ ran (𝑆 D 𝐹))    ⇒   ⊢ (𝜑 → (𝑆 D ◡𝐹) = (𝑥 ∈ 𝑌 ↦ (1 / ((𝑆 D 𝐹)‘(◡𝐹‘𝑥)))))
Theorem	dvexp3 25932*	Derivative of an exponential of integer exponent. (Contributed by Mario Carneiro, 26-Feb-2015.)
⊢ (𝑁 ∈ ℤ → (ℂ D (𝑥 ∈ (ℂ ∖ {0}) ↦ (𝑥↑𝑁))) = (𝑥 ∈ (ℂ ∖ {0}) ↦ (𝑁 · (𝑥↑(𝑁 − 1)))))
Theorem	dveflem 25933	Derivative of the exponential function at 0. The key step in the proof is eftlub 16125, to show that abs(exp(𝑥) − 1 − 𝑥) ≤ abs(𝑥)↑2 · (3 / 4). (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 28-Dec-2016.)
⊢ 0(ℂ D exp)1
Theorem	dvef 25934	Derivative of the exponential function. (Contributed by Mario Carneiro, 9-Aug-2014.) (Proof shortened by Mario Carneiro, 10-Feb-2015.)
⊢ (ℂ D exp) = exp
Theorem	dvsincos 25935	Derivative of the sine and cosine functions. (Contributed by Mario Carneiro, 21-May-2016.)
⊢ ((ℂ D sin) = cos ∧ (ℂ D cos) = (𝑥 ∈ ℂ ↦ -(sin‘𝑥)))
Theorem	dvsin 25936	Derivative of the sine function. (Contributed by Mario Carneiro, 21-May-2016.)
⊢ (ℂ D sin) = cos
Theorem	dvcos 25937	Derivative of the cosine function. (Contributed by Mario Carneiro, 21-May-2016.)
⊢ (ℂ D cos) = (𝑥 ∈ ℂ ↦ -(sin‘𝑥))
13.3.1.2  Results on real differentiation
Theorem	dvferm1lem 25938*	Lemma for dvferm 25942. (Contributed by Mario Carneiro, 24-Feb-2015.)
⊢ (𝜑 → 𝐹:𝑋⟶ℝ)    &   ⊢ (𝜑 → 𝑋 ⊆ ℝ)    &   ⊢ (𝜑 → 𝑈 ∈ (𝐴(,)𝐵))    &   ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ 𝑋)    &   ⊢ (𝜑 → 𝑈 ∈ dom (ℝ D 𝐹))    &   ⊢ (𝜑 → ∀𝑦 ∈ (𝑈(,)𝐵)(𝐹‘𝑦) ≤ (𝐹‘𝑈))    &   ⊢ (𝜑 → 0 < ((ℝ D 𝐹)‘𝑈))    &   ⊢ (𝜑 → 𝑇 ∈ ℝ+)    &   ⊢ (𝜑 → ∀𝑧 ∈ (𝑋 ∖ {𝑈})((𝑧 ≠ 𝑈 ∧ (abs‘(𝑧 − 𝑈)) < 𝑇) → (abs‘((((𝐹‘𝑧) − (𝐹‘𝑈)) / (𝑧 − 𝑈)) − ((ℝ D 𝐹)‘𝑈))) < ((ℝ D 𝐹)‘𝑈)))    &   ⊢ 𝑆 = ((𝑈 + if(𝐵 ≤ (𝑈 + 𝑇), 𝐵, (𝑈 + 𝑇))) / 2)    ⇒   ⊢ ¬ 𝜑
Theorem	dvferm1 25939*	One-sided version of dvferm 25942. A point 𝑈 which is the local maximum of its right neighborhood has derivative at most zero. (Contributed by Mario Carneiro, 24-Feb-2015.) (Proof shortened by Mario Carneiro, 28-Dec-2016.)
⊢ (𝜑 → 𝐹:𝑋⟶ℝ)    &   ⊢ (𝜑 → 𝑋 ⊆ ℝ)    &   ⊢ (𝜑 → 𝑈 ∈ (𝐴(,)𝐵))    &   ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ 𝑋)    &   ⊢ (𝜑 → 𝑈 ∈ dom (ℝ D 𝐹))    &   ⊢ (𝜑 → ∀𝑦 ∈ (𝑈(,)𝐵)(𝐹‘𝑦) ≤ (𝐹‘𝑈))    ⇒   ⊢ (𝜑 → ((ℝ D 𝐹)‘𝑈) ≤ 0)
Theorem	dvferm2lem 25940*	Lemma for dvferm 25942. (Contributed by Mario Carneiro, 24-Feb-2015.)
⊢ (𝜑 → 𝐹:𝑋⟶ℝ)    &   ⊢ (𝜑 → 𝑋 ⊆ ℝ)    &   ⊢ (𝜑 → 𝑈 ∈ (𝐴(,)𝐵))    &   ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ 𝑋)    &   ⊢ (𝜑 → 𝑈 ∈ dom (ℝ D 𝐹))    &   ⊢ (𝜑 → ∀𝑦 ∈ (𝐴(,)𝑈)(𝐹‘𝑦) ≤ (𝐹‘𝑈))    &   ⊢ (𝜑 → ((ℝ D 𝐹)‘𝑈) < 0)    &   ⊢ (𝜑 → 𝑇 ∈ ℝ+)    &   ⊢ (𝜑 → ∀𝑧 ∈ (𝑋 ∖ {𝑈})((𝑧 ≠ 𝑈 ∧ (abs‘(𝑧 − 𝑈)) < 𝑇) → (abs‘((((𝐹‘𝑧) − (𝐹‘𝑈)) / (𝑧 − 𝑈)) − ((ℝ D 𝐹)‘𝑈))) < -((ℝ D 𝐹)‘𝑈)))    &   ⊢ 𝑆 = ((if(𝐴 ≤ (𝑈 − 𝑇), (𝑈 − 𝑇), 𝐴) + 𝑈) / 2)    ⇒   ⊢ ¬ 𝜑
Theorem	dvferm2 25941*	One-sided version of dvferm 25942. A point 𝑈 which is the local maximum of its left neighborhood has derivative at least zero. (Contributed by Mario Carneiro, 24-Feb-2015.) (Proof shortened by Mario Carneiro, 28-Dec-2016.)
⊢ (𝜑 → 𝐹:𝑋⟶ℝ)    &   ⊢ (𝜑 → 𝑋 ⊆ ℝ)    &   ⊢ (𝜑 → 𝑈 ∈ (𝐴(,)𝐵))    &   ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ 𝑋)    &   ⊢ (𝜑 → 𝑈 ∈ dom (ℝ D 𝐹))    &   ⊢ (𝜑 → ∀𝑦 ∈ (𝐴(,)𝑈)(𝐹‘𝑦) ≤ (𝐹‘𝑈))    ⇒   ⊢ (𝜑 → 0 ≤ ((ℝ D 𝐹)‘𝑈))
Theorem	dvferm 25942*	Fermat's theorem on stationary points. A point 𝑈 which is a local maximum has derivative equal to zero. (Contributed by Mario Carneiro, 1-Sep-2014.)
⊢ (𝜑 → 𝐹:𝑋⟶ℝ)    &   ⊢ (𝜑 → 𝑋 ⊆ ℝ)    &   ⊢ (𝜑 → 𝑈 ∈ (𝐴(,)𝐵))    &   ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ 𝑋)    &   ⊢ (𝜑 → 𝑈 ∈ dom (ℝ D 𝐹))    &   ⊢ (𝜑 → ∀𝑦 ∈ (𝐴(,)𝐵)(𝐹‘𝑦) ≤ (𝐹‘𝑈))    ⇒   ⊢ (𝜑 → ((ℝ D 𝐹)‘𝑈) = 0)
Theorem	rollelem 25943*	Lemma for rolle 25944. (Contributed by Mario Carneiro, 1-Sep-2014.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 < 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ))    &   ⊢ (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵))    &   ⊢ (𝜑 → ∀𝑦 ∈ (𝐴[,]𝐵)(𝐹‘𝑦) ≤ (𝐹‘𝑈))    &   ⊢ (𝜑 → 𝑈 ∈ (𝐴[,]𝐵))    &   ⊢ (𝜑 → ¬ 𝑈 ∈ {𝐴, 𝐵})    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0)
Theorem	rolle 25944*	Rolle's theorem. If 𝐹 is a real continuous function on [𝐴, 𝐵] which is differentiable on (𝐴, 𝐵), and 𝐹(𝐴) = 𝐹(𝐵), then there is some 𝑥 ∈ (𝐴, 𝐵) such that (ℝ D 𝐹)‘𝑥 = 0. (Contributed by Mario Carneiro, 1-Sep-2014.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 < 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ))    &   ⊢ (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵))    &   ⊢ (𝜑 → (𝐹‘𝐴) = (𝐹‘𝐵))    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0)
Theorem	cmvth 25945*	Cauchy's Mean Value Theorem. If 𝐹, 𝐺 are real continuous functions on [𝐴, 𝐵] differentiable on (𝐴, 𝐵), then there is some 𝑥 ∈ (𝐴, 𝐵) such that 𝐹' (𝑥) / 𝐺' (𝑥) = (𝐹(𝐴) − 𝐹(𝐵)) / (𝐺(𝐴) − 𝐺(𝐵)). (We express the condition without division, so that we need no nonzero constraints.) (Contributed by Mario Carneiro, 29-Dec-2016.) Avoid ax-mulf 11207. (Revised by GG, 16-Mar-2025.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 < 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ))    &   ⊢ (𝜑 → 𝐺 ∈ ((𝐴[,]𝐵)–cn→ℝ))    &   ⊢ (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵))    &   ⊢ (𝜑 → dom (ℝ D 𝐺) = (𝐴(,)𝐵))    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ (𝐴(,)𝐵)(((𝐹‘𝐵) − (𝐹‘𝐴)) · ((ℝ D 𝐺)‘𝑥)) = (((𝐺‘𝐵) − (𝐺‘𝐴)) · ((ℝ D 𝐹)‘𝑥)))
Theorem	cmvthOLD 25946*	Obsolete version of cmvth 25945 as of 16-Apr-2025. (Contributed by Mario Carneiro, 29-Dec-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 < 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ))    &   ⊢ (𝜑 → 𝐺 ∈ ((𝐴[,]𝐵)–cn→ℝ))    &   ⊢ (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵))    &   ⊢ (𝜑 → dom (ℝ D 𝐺) = (𝐴(,)𝐵))    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ (𝐴(,)𝐵)(((𝐹‘𝐵) − (𝐹‘𝐴)) · ((ℝ D 𝐺)‘𝑥)) = (((𝐺‘𝐵) − (𝐺‘𝐴)) · ((ℝ D 𝐹)‘𝑥)))
Theorem	mvth 25947*	The Mean Value Theorem. If 𝐹 is a real continuous function on [𝐴, 𝐵] which is differentiable on (𝐴, 𝐵), then there is some 𝑥 ∈ (𝐴, 𝐵) such that (ℝ D 𝐹)‘𝑥 is equal to the average slope over [𝐴, 𝐵]. This is Metamath 100 proof #75. (Contributed by Mario Carneiro, 1-Sep-2014.) (Proof shortened by Mario Carneiro, 29-Dec-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 < 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ))    &   ⊢ (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵))    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = (((𝐹‘𝐵) − (𝐹‘𝐴)) / (𝐵 − 𝐴)))
Theorem	dvlip 25948*	A function with derivative bounded by 𝑀 is 𝑀-Lipschitz continuous. (Contributed by Mario Carneiro, 3-Mar-2015.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ))    &   ⊢ (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵))    &   ⊢ (𝜑 → 𝑀 ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝑀)    ⇒   ⊢ ((𝜑 ∧ (𝑋 ∈ (𝐴[,]𝐵) ∧ 𝑌 ∈ (𝐴[,]𝐵))) → (abs‘((𝐹‘𝑋) − (𝐹‘𝑌))) ≤ (𝑀 · (abs‘(𝑋 − 𝑌))))
Theorem	dvlipcn 25949*	A complex function with derivative bounded by 𝑀 on an open ball is 𝑀-Lipschitz continuous. (Contributed by Mario Carneiro, 18-Mar-2015.)
⊢ (𝜑 → 𝑋 ⊆ ℂ)    &   ⊢ (𝜑 → 𝐹:𝑋⟶ℂ)    &   ⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝑅 ∈ ℝ*)    &   ⊢ 𝐵 = (𝐴(ball‘(abs ∘ − ))𝑅)    &   ⊢ (𝜑 → 𝐵 ⊆ dom (ℂ D 𝐹))    &   ⊢ (𝜑 → 𝑀 ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (abs‘((ℂ D 𝐹)‘𝑥)) ≤ 𝑀)    ⇒   ⊢ ((𝜑 ∧ (𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (abs‘((𝐹‘𝑌) − (𝐹‘𝑍))) ≤ (𝑀 · (abs‘(𝑌 − 𝑍))))
Theorem	dvlip2 25950*	Combine the results of dvlip 25948 and dvlipcn 25949 into one. (Contributed by Mario Carneiro, 18-Mar-2015.) (Revised by Mario Carneiro, 8-Sep-2015.)
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})    &   ⊢ 𝐽 = ((abs ∘ − ) ↾ (𝑆 × 𝑆))    &   ⊢ (𝜑 → 𝑋 ⊆ 𝑆)    &   ⊢ (𝜑 → 𝐹:𝑋⟶ℂ)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑅 ∈ ℝ*)    &   ⊢ 𝐵 = (𝐴(ball‘𝐽)𝑅)    &   ⊢ (𝜑 → 𝐵 ⊆ dom (𝑆 D 𝐹))    &   ⊢ (𝜑 → 𝑀 ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (abs‘((𝑆 D 𝐹)‘𝑥)) ≤ 𝑀)    ⇒   ⊢ ((𝜑 ∧ (𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (abs‘((𝐹‘𝑌) − (𝐹‘𝑍))) ≤ (𝑀 · (abs‘(𝑌 − 𝑍))))
Theorem	c1liplem1 25951*	Lemma for c1lip1 25952. (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 25952*	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 25953*	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 25954*	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 25955	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 25956	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 25957	Lemma for dvgt0 25959 and dvlt0 25960. (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 25958*	Lemma for dvgt0 25959 and dvlt0 25960. (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 25959	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 25960	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 25961	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 25962*	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 25963*	Lemma for dvivth 25965. (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 25964*	Lemma for dvivth 25965. (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 25965	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 25409 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 25966	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 25967	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 25968*	Lemma for lhop1 25969. (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 25969*	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 25970*	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 25971*	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 25972	Lemma for dvcnvre 25974. (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 25973	Lemma for dvcnvre 25974. (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 25974*	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 25975	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 25976*	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 11207. (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 25977*	Obsolete version of dvfsumle 25976 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 25978*	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 25979*	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 25980*	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 25981*	Lemma for dvfsumrlim 25988. (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 25982*	Lemma for dvfsumrlim 25988. (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 25983*	Lemma for dvfsumrlim 25988. (Contributed by Mario Carneiro, 17-May-2016.) Avoid ax-mulf 11207. (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 25984*	Obsolete version of dvfsumlem2 25983 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 25985*	Lemma for dvfsumrlim 25988. (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 25986*	Lemma for dvfsumrlim 25988. (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 25987*	Lemma for dvfsumrlim 25988. Satisfy the assumption of dvfsumlem4 25986. (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 25988*	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 25989*	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 25990*	Conjoin the statements of dvfsumrlim 25988 and dvfsumrlim2 25989. (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 25991*	The reverse of dvfsumrlim 25988, 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 25992*	Lemma for ftc1a 25994 and ftc1 25999. (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 25993*	Lemma for ftc1 25999. (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 25994*	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 25995*	Lemma for ftc1 25999. (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 25996*	Lemma for ftc1 25999. (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 25997*	Lemma for ftc1 25999. (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 25998*	Lemma for ftc1 25999. (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 25999*	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 26000*	Strengthen the assumptions of ftc1 25999 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)