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Theorem List for Metamath Proof Explorer - 23701-23800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdvcmul 23701 The product rule when one argument is a constant. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝐹:𝑋⟶ℂ)    &   (𝜑𝐴 ∈ ℂ)    &   (𝜑𝑋𝑆)    &   (𝜑𝐶 ∈ dom (𝑆 D 𝐹))       (𝜑 → ((𝑆 D ((𝑆 × {𝐴}) ∘𝑓 · 𝐹))‘𝐶) = (𝐴 · ((𝑆 D 𝐹)‘𝐶)))

Theoremdvcmulf 23702 The product rule when one argument is a constant. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝐹:𝑋⟶ℂ)    &   (𝜑𝐴 ∈ ℂ)    &   (𝜑 → dom (𝑆 D 𝐹) = 𝑋)       (𝜑 → (𝑆 D ((𝑆 × {𝐴}) ∘𝑓 · 𝐹)) = ((𝑆 × {𝐴}) ∘𝑓 · (𝑆 D 𝐹)))

Theoremdvcobr 23703 The chain rule for derivatives at a point. For the (simpler but more limited) function version, see dvco 23704. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 28-Dec-2016.)
(𝜑𝐹:𝑋⟶ℂ)    &   (𝜑𝑋𝑆)    &   (𝜑𝐺:𝑌𝑋)    &   (𝜑𝑌𝑇)    &   (𝜑𝑆 ⊆ ℂ)    &   (𝜑𝑇 ⊆ ℂ)    &   (𝜑𝐾𝑉)    &   (𝜑𝐿𝑉)    &   (𝜑 → (𝐺𝐶)(𝑆 D 𝐹)𝐾)    &   (𝜑𝐶(𝑇 D 𝐺)𝐿)    &   𝐽 = (TopOpen‘ℂfld)       (𝜑𝐶(𝑇 D (𝐹𝐺))(𝐾 · 𝐿))

Theoremdvco 23704 The chain rule for derivatives at a point. For the (more general) relation version, see dvcobr 23703. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)
(𝜑𝐹:𝑋⟶ℂ)    &   (𝜑𝑋𝑆)    &   (𝜑𝐺:𝑌𝑋)    &   (𝜑𝑌𝑇)    &   (𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝑇 ∈ {ℝ, ℂ})    &   (𝜑 → (𝐺𝐶) ∈ dom (𝑆 D 𝐹))    &   (𝜑𝐶 ∈ dom (𝑇 D 𝐺))       (𝜑 → ((𝑇 D (𝐹𝐺))‘𝐶) = (((𝑆 D 𝐹)‘(𝐺𝐶)) · ((𝑇 D 𝐺)‘𝐶)))

Theoremdvcof 23705 The chain rule for everywhere-differentiable functions. (Contributed by Mario Carneiro, 10-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝑇 ∈ {ℝ, ℂ})    &   (𝜑𝐹:𝑋⟶ℂ)    &   (𝜑𝐺:𝑌𝑋)    &   (𝜑 → dom (𝑆 D 𝐹) = 𝑋)    &   (𝜑 → dom (𝑇 D 𝐺) = 𝑌)       (𝜑 → (𝑇 D (𝐹𝐺)) = (((𝑆 D 𝐹) ∘ 𝐺) ∘𝑓 · (𝑇 D 𝐺)))

Theoremdvcjbr 23706 The derivative of the conjugate of a function. For the (simpler but more limited) function version, see dvcj 23707. (This doesn't follow from dvcobr 23703 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.)
(𝜑𝐹:𝑋⟶ℂ)    &   (𝜑𝑋 ⊆ ℝ)    &   (𝜑𝐶 ∈ dom (ℝ D 𝐹))       (𝜑𝐶(ℝ D (∗ ∘ 𝐹))(∗‘((ℝ D 𝐹)‘𝐶)))

Theoremdvcj 23707 The derivative of the conjugate of a function. For the (more general) relation version, see dvcjbr 23706. (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)))

Theoremdvfre 23708 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)

Theoremdvnfre 23709 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)

Theoremdvexp 23710* 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 ↦ (𝑥𝑁))) = (𝑥 ∈ ℂ ↦ (𝑁 · (𝑥↑(𝑁 − 1)))))

Theoremdvexp2 23711* 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 ↦ (𝑥𝑁))) = (𝑥 ∈ ℂ ↦ if(𝑁 = 0, 0, (𝑁 · (𝑥↑(𝑁 − 1))))))

Theoremdvrec 23712* 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))))

Theoremdvmptres3 23713* Function-builder for derivative: restrict a derivative to a subset. (Contributed by Mario Carneiro, 11-Feb-2015.)
𝐽 = (TopOpen‘ℂfld)    &   (𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝑋𝐽)    &   (𝜑 → (𝑆𝑋) = 𝑌)    &   ((𝜑𝑥𝑋) → 𝐴 ∈ ℂ)    &   ((𝜑𝑥𝑋) → 𝐵𝑉)    &   (𝜑 → (ℂ D (𝑥𝑋𝐴)) = (𝑥𝑋𝐵))       (𝜑 → (𝑆 D (𝑥𝑌𝐴)) = (𝑥𝑌𝐵))

Theoremdvmptid 23714* Function-builder for derivative: derivative of the identity. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
(𝜑𝑆 ∈ {ℝ, ℂ})       (𝜑 → (𝑆 D (𝑥𝑆𝑥)) = (𝑥𝑆 ↦ 1))

Theoremdvmptc 23715* Function-builder for derivative: derivative of a constant. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝐴 ∈ ℂ)       (𝜑 → (𝑆 D (𝑥𝑆𝐴)) = (𝑥𝑆 ↦ 0))

Theoremdvmptcl 23716* Closure lemma for dvmptcmul 23721 and other related theorems. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   ((𝜑𝑥𝑋) → 𝐴 ∈ ℂ)    &   ((𝜑𝑥𝑋) → 𝐵𝑉)    &   (𝜑 → (𝑆 D (𝑥𝑋𝐴)) = (𝑥𝑋𝐵))       ((𝜑𝑥𝑋) → 𝐵 ∈ ℂ)

Theoremdvmptadd 23717* Function-builder for derivative, addition rule. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   ((𝜑𝑥𝑋) → 𝐴 ∈ ℂ)    &   ((𝜑𝑥𝑋) → 𝐵𝑉)    &   (𝜑 → (𝑆 D (𝑥𝑋𝐴)) = (𝑥𝑋𝐵))    &   ((𝜑𝑥𝑋) → 𝐶 ∈ ℂ)    &   ((𝜑𝑥𝑋) → 𝐷𝑊)    &   (𝜑 → (𝑆 D (𝑥𝑋𝐶)) = (𝑥𝑋𝐷))       (𝜑 → (𝑆 D (𝑥𝑋 ↦ (𝐴 + 𝐶))) = (𝑥𝑋 ↦ (𝐵 + 𝐷)))

Theoremdvmptmul 23718* Function-builder for derivative, product rule. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   ((𝜑𝑥𝑋) → 𝐴 ∈ ℂ)    &   ((𝜑𝑥𝑋) → 𝐵𝑉)    &   (𝜑 → (𝑆 D (𝑥𝑋𝐴)) = (𝑥𝑋𝐵))    &   ((𝜑𝑥𝑋) → 𝐶 ∈ ℂ)    &   ((𝜑𝑥𝑋) → 𝐷𝑊)    &   (𝜑 → (𝑆 D (𝑥𝑋𝐶)) = (𝑥𝑋𝐷))       (𝜑 → (𝑆 D (𝑥𝑋 ↦ (𝐴 · 𝐶))) = (𝑥𝑋 ↦ ((𝐵 · 𝐶) + (𝐷 · 𝐴))))

Theoremdvmptres2 23719* Function-builder for derivative: restrict a derivative to a subset. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   ((𝜑𝑥𝑋) → 𝐴 ∈ ℂ)    &   ((𝜑𝑥𝑋) → 𝐵𝑉)    &   (𝜑 → (𝑆 D (𝑥𝑋𝐴)) = (𝑥𝑋𝐵))    &   (𝜑𝑍𝑋)    &   𝐽 = (𝐾t 𝑆)    &   𝐾 = (TopOpen‘ℂfld)    &   (𝜑 → ((int‘𝐽)‘𝑍) = 𝑌)       (𝜑 → (𝑆 D (𝑥𝑍𝐴)) = (𝑥𝑌𝐵))

Theoremdvmptres 23720* 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.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   ((𝜑𝑥𝑋) → 𝐴 ∈ ℂ)    &   ((𝜑𝑥𝑋) → 𝐵𝑉)    &   (𝜑 → (𝑆 D (𝑥𝑋𝐴)) = (𝑥𝑋𝐵))    &   (𝜑𝑌𝑋)    &   𝐽 = (𝐾t 𝑆)    &   𝐾 = (TopOpen‘ℂfld)    &   (𝜑𝑌𝐽)       (𝜑 → (𝑆 D (𝑥𝑌𝐴)) = (𝑥𝑌𝐵))

Theoremdvmptcmul 23721* Function-builder for derivative, product rule for constant multiplier. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   ((𝜑𝑥𝑋) → 𝐴 ∈ ℂ)    &   ((𝜑𝑥𝑋) → 𝐵𝑉)    &   (𝜑 → (𝑆 D (𝑥𝑋𝐴)) = (𝑥𝑋𝐵))    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → (𝑆 D (𝑥𝑋 ↦ (𝐶 · 𝐴))) = (𝑥𝑋 ↦ (𝐶 · 𝐵)))

Theoremdvmptdivc 23722* Function-builder for derivative, division rule for constant divisor. (Contributed by Mario Carneiro, 18-May-2016.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   ((𝜑𝑥𝑋) → 𝐴 ∈ ℂ)    &   ((𝜑𝑥𝑋) → 𝐵𝑉)    &   (𝜑 → (𝑆 D (𝑥𝑋𝐴)) = (𝑥𝑋𝐵))    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐶 ≠ 0)       (𝜑 → (𝑆 D (𝑥𝑋 ↦ (𝐴 / 𝐶))) = (𝑥𝑋 ↦ (𝐵 / 𝐶)))

Theoremdvmptneg 23723* Function-builder for derivative, product rule for negatives. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   ((𝜑𝑥𝑋) → 𝐴 ∈ ℂ)    &   ((𝜑𝑥𝑋) → 𝐵𝑉)    &   (𝜑 → (𝑆 D (𝑥𝑋𝐴)) = (𝑥𝑋𝐵))       (𝜑 → (𝑆 D (𝑥𝑋 ↦ -𝐴)) = (𝑥𝑋 ↦ -𝐵))

Theoremdvmptsub 23724* Function-builder for derivative, subtraction rule. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   ((𝜑𝑥𝑋) → 𝐴 ∈ ℂ)    &   ((𝜑𝑥𝑋) → 𝐵𝑉)    &   (𝜑 → (𝑆 D (𝑥𝑋𝐴)) = (𝑥𝑋𝐵))    &   ((𝜑𝑥𝑋) → 𝐶 ∈ ℂ)    &   ((𝜑𝑥𝑋) → 𝐷𝑊)    &   (𝜑 → (𝑆 D (𝑥𝑋𝐶)) = (𝑥𝑋𝐷))       (𝜑 → (𝑆 D (𝑥𝑋 ↦ (𝐴𝐶))) = (𝑥𝑋 ↦ (𝐵𝐷)))

Theoremdvmptcj 23725* Function-builder for derivative, conjugate rule. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
((𝜑𝑥𝑋) → 𝐴 ∈ ℂ)    &   ((𝜑𝑥𝑋) → 𝐵𝑉)    &   (𝜑 → (ℝ D (𝑥𝑋𝐴)) = (𝑥𝑋𝐵))       (𝜑 → (ℝ D (𝑥𝑋 ↦ (∗‘𝐴))) = (𝑥𝑋 ↦ (∗‘𝐵)))

Theoremdvmptre 23726* Function-builder for derivative, real part. (Contributed by Mario Carneiro, 1-Sep-2014.)
((𝜑𝑥𝑋) → 𝐴 ∈ ℂ)    &   ((𝜑𝑥𝑋) → 𝐵𝑉)    &   (𝜑 → (ℝ D (𝑥𝑋𝐴)) = (𝑥𝑋𝐵))       (𝜑 → (ℝ D (𝑥𝑋 ↦ (ℜ‘𝐴))) = (𝑥𝑋 ↦ (ℜ‘𝐵)))

Theoremdvmptim 23727* Function-builder for derivative, imaginary part. (Contributed by Mario Carneiro, 1-Sep-2014.)
((𝜑𝑥𝑋) → 𝐴 ∈ ℂ)    &   ((𝜑𝑥𝑋) → 𝐵𝑉)    &   (𝜑 → (ℝ D (𝑥𝑋𝐴)) = (𝑥𝑋𝐵))       (𝜑 → (ℝ D (𝑥𝑋 ↦ (ℑ‘𝐴))) = (𝑥𝑋 ↦ (ℑ‘𝐵)))

Theoremdvmptntr 23728* 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.)
(𝜑𝑆 ⊆ ℂ)    &   (𝜑𝑋𝑆)    &   ((𝜑𝑥𝑋) → 𝐴 ∈ ℂ)    &   𝐽 = (𝐾t 𝑆)    &   𝐾 = (TopOpen‘ℂfld)    &   (𝜑 → ((int‘𝐽)‘𝑋) = 𝑌)       (𝜑 → (𝑆 D (𝑥𝑋𝐴)) = (𝑆 D (𝑥𝑌𝐴)))

Theoremdvmptco 23729* Function-builder for derivative, chain rule. (Contributed by Mario Carneiro, 1-Sep-2014.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝑇 ∈ {ℝ, ℂ})    &   ((𝜑𝑥𝑋) → 𝐴𝑌)    &   ((𝜑𝑥𝑋) → 𝐵𝑉)    &   ((𝜑𝑦𝑌) → 𝐶 ∈ ℂ)    &   ((𝜑𝑦𝑌) → 𝐷𝑊)    &   (𝜑 → (𝑆 D (𝑥𝑋𝐴)) = (𝑥𝑋𝐵))    &   (𝜑 → (𝑇 D (𝑦𝑌𝐶)) = (𝑦𝑌𝐷))    &   (𝑦 = 𝐴𝐶 = 𝐸)    &   (𝑦 = 𝐴𝐷 = 𝐹)       (𝜑 → (𝑆 D (𝑥𝑋𝐸)) = (𝑥𝑋 ↦ (𝐹 · 𝐵)))

Theoremdvrecg 23730* Derivative of the reciprocal of a function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝐴 ∈ ℂ)    &   ((𝜑𝑥𝑋) → 𝐵 ∈ (ℂ ∖ {0}))    &   ((𝜑𝑥𝑋) → 𝐶𝑉)    &   (𝜑 → (𝑆 D (𝑥𝑋𝐵)) = (𝑥𝑋𝐶))       (𝜑 → (𝑆 D (𝑥𝑋 ↦ (𝐴 / 𝐵))) = (𝑥𝑋 ↦ -((𝐴 · 𝐶) / (𝐵↑2))))

Theoremdvmptdiv 23731* Function-builder for derivative, quotient rule. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   ((𝜑𝑥𝑋) → 𝐴 ∈ ℂ)    &   ((𝜑𝑥𝑋) → 𝐵𝑉)    &   (𝜑 → (𝑆 D (𝑥𝑋𝐴)) = (𝑥𝑋𝐵))    &   ((𝜑𝑥𝑋) → 𝐶 ∈ (ℂ ∖ {0}))    &   ((𝜑𝑥𝑋) → 𝐷 ∈ ℂ)    &   (𝜑 → (𝑆 D (𝑥𝑋𝐶)) = (𝑥𝑋𝐷))       (𝜑 → (𝑆 D (𝑥𝑋 ↦ (𝐴 / 𝐶))) = (𝑥𝑋 ↦ (((𝐵 · 𝐶) − (𝐷 · 𝐴)) / (𝐶↑2))))

Theoremdvmptfsum 23732* Function-builder for derivative, finite sums rule. (Contributed by Stefan O'Rear, 12-Nov-2014.)
𝐽 = (𝐾t 𝑆)    &   𝐾 = (TopOpen‘ℂfld)    &   (𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝑋𝐽)    &   (𝜑𝐼 ∈ Fin)    &   ((𝜑𝑖𝐼𝑥𝑋) → 𝐴 ∈ ℂ)    &   ((𝜑𝑖𝐼𝑥𝑋) → 𝐵 ∈ ℂ)    &   ((𝜑𝑖𝐼) → (𝑆 D (𝑥𝑋𝐴)) = (𝑥𝑋𝐵))       (𝜑 → (𝑆 D (𝑥𝑋 ↦ Σ𝑖𝐼 𝐴)) = (𝑥𝑋 ↦ Σ𝑖𝐼 𝐵))

Theoremdvcnvlem 23733 Lemma for dvcnvre 23776. (Contributed by Mario Carneiro, 25-Feb-2015.) (Revised by Mario Carneiro, 8-Sep-2015.)
𝐽 = (TopOpen‘ℂfld)    &   𝐾 = (𝐽t 𝑆)    &   (𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝑌𝐾)    &   (𝜑𝐹:𝑋1-1-onto𝑌)    &   (𝜑𝐹 ∈ (𝑌cn𝑋))    &   (𝜑 → dom (𝑆 D 𝐹) = 𝑋)    &   (𝜑 → ¬ 0 ∈ ran (𝑆 D 𝐹))    &   (𝜑𝐶𝑋)       (𝜑 → (𝐹𝐶)(𝑆 D 𝐹)(1 / ((𝑆 D 𝐹)‘𝐶)))

Theoremdvcnv 23734* A weak version of dvcnvre 23776, 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 = (𝐽t 𝑆)    &   (𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝑌𝐾)    &   (𝜑𝐹:𝑋1-1-onto𝑌)    &   (𝜑𝐹 ∈ (𝑌cn𝑋))    &   (𝜑 → dom (𝑆 D 𝐹) = 𝑋)    &   (𝜑 → ¬ 0 ∈ ran (𝑆 D 𝐹))       (𝜑 → (𝑆 D 𝐹) = (𝑥𝑌 ↦ (1 / ((𝑆 D 𝐹)‘(𝐹𝑥)))))

Theoremdvexp3 23735* Derivative of an exponential of integer exponent. (Contributed by Mario Carneiro, 26-Feb-2015.)
(𝑁 ∈ ℤ → (ℂ D (𝑥 ∈ (ℂ ∖ {0}) ↦ (𝑥𝑁))) = (𝑥 ∈ (ℂ ∖ {0}) ↦ (𝑁 · (𝑥↑(𝑁 − 1)))))

Theoremdveflem 23736 Derivative of the exponential function at 0. The key step in the proof is eftlub 14833, 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

Theoremdvef 23737 Derivative of the exponential function. (Contributed by Mario Carneiro, 9-Aug-2014.) (Proof shortened by Mario Carneiro, 10-Feb-2015.)
(ℂ D exp) = exp

Theoremdvsincos 23738 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)))

Theoremdvsin 23739 Derivative of the sine function. (Contributed by Mario Carneiro, 21-May-2016.)
(ℂ D sin) = cos

Theoremdvcos 23740 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

Theoremdvferm1lem 23741* Lemma for dvferm 23745. (Contributed by Mario Carneiro, 24-Feb-2015.)
(𝜑𝐹:𝑋⟶ℝ)    &   (𝜑𝑋 ⊆ ℝ)    &   (𝜑𝑈 ∈ (𝐴(,)𝐵))    &   (𝜑 → (𝐴(,)𝐵) ⊆ 𝑋)    &   (𝜑𝑈 ∈ dom (ℝ D 𝐹))    &   (𝜑 → ∀𝑦 ∈ (𝑈(,)𝐵)(𝐹𝑦) ≤ (𝐹𝑈))    &   (𝜑 → 0 < ((ℝ D 𝐹)‘𝑈))    &   (𝜑𝑇 ∈ ℝ+)    &   (𝜑 → ∀𝑧 ∈ (𝑋 ∖ {𝑈})((𝑧𝑈 ∧ (abs‘(𝑧𝑈)) < 𝑇) → (abs‘((((𝐹𝑧) − (𝐹𝑈)) / (𝑧𝑈)) − ((ℝ D 𝐹)‘𝑈))) < ((ℝ D 𝐹)‘𝑈)))    &   𝑆 = ((𝑈 + if(𝐵 ≤ (𝑈 + 𝑇), 𝐵, (𝑈 + 𝑇))) / 2)        ¬ 𝜑

Theoremdvferm1 23742* One-sided version of dvferm 23745. 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.)
(𝜑𝐹:𝑋⟶ℝ)    &   (𝜑𝑋 ⊆ ℝ)    &   (𝜑𝑈 ∈ (𝐴(,)𝐵))    &   (𝜑 → (𝐴(,)𝐵) ⊆ 𝑋)    &   (𝜑𝑈 ∈ dom (ℝ D 𝐹))    &   (𝜑 → ∀𝑦 ∈ (𝑈(,)𝐵)(𝐹𝑦) ≤ (𝐹𝑈))       (𝜑 → ((ℝ D 𝐹)‘𝑈) ≤ 0)

Theoremdvferm2lem 23743* Lemma for dvferm 23745. (Contributed by Mario Carneiro, 24-Feb-2015.)
(𝜑𝐹:𝑋⟶ℝ)    &   (𝜑𝑋 ⊆ ℝ)    &   (𝜑𝑈 ∈ (𝐴(,)𝐵))    &   (𝜑 → (𝐴(,)𝐵) ⊆ 𝑋)    &   (𝜑𝑈 ∈ dom (ℝ D 𝐹))    &   (𝜑 → ∀𝑦 ∈ (𝐴(,)𝑈)(𝐹𝑦) ≤ (𝐹𝑈))    &   (𝜑 → ((ℝ D 𝐹)‘𝑈) < 0)    &   (𝜑𝑇 ∈ ℝ+)    &   (𝜑 → ∀𝑧 ∈ (𝑋 ∖ {𝑈})((𝑧𝑈 ∧ (abs‘(𝑧𝑈)) < 𝑇) → (abs‘((((𝐹𝑧) − (𝐹𝑈)) / (𝑧𝑈)) − ((ℝ D 𝐹)‘𝑈))) < -((ℝ D 𝐹)‘𝑈)))    &   𝑆 = ((if(𝐴 ≤ (𝑈𝑇), (𝑈𝑇), 𝐴) + 𝑈) / 2)        ¬ 𝜑

Theoremdvferm2 23744* One-sided version of dvferm 23745. 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.)
(𝜑𝐹:𝑋⟶ℝ)    &   (𝜑𝑋 ⊆ ℝ)    &   (𝜑𝑈 ∈ (𝐴(,)𝐵))    &   (𝜑 → (𝐴(,)𝐵) ⊆ 𝑋)    &   (𝜑𝑈 ∈ dom (ℝ D 𝐹))    &   (𝜑 → ∀𝑦 ∈ (𝐴(,)𝑈)(𝐹𝑦) ≤ (𝐹𝑈))       (𝜑 → 0 ≤ ((ℝ D 𝐹)‘𝑈))

Theoremdvferm 23745* 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.)
(𝜑𝐹:𝑋⟶ℝ)    &   (𝜑𝑋 ⊆ ℝ)    &   (𝜑𝑈 ∈ (𝐴(,)𝐵))    &   (𝜑 → (𝐴(,)𝐵) ⊆ 𝑋)    &   (𝜑𝑈 ∈ dom (ℝ D 𝐹))    &   (𝜑 → ∀𝑦 ∈ (𝐴(,)𝐵)(𝐹𝑦) ≤ (𝐹𝑈))       (𝜑 → ((ℝ D 𝐹)‘𝑈) = 0)

Theoremrollelem 23746* Lemma for rolle 23747. (Contributed by Mario Carneiro, 1-Sep-2014.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   (𝜑𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ))    &   (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵))    &   (𝜑 → ∀𝑦 ∈ (𝐴[,]𝐵)(𝐹𝑦) ≤ (𝐹𝑈))    &   (𝜑𝑈 ∈ (𝐴[,]𝐵))    &   (𝜑 → ¬ 𝑈 ∈ {𝐴, 𝐵})       (𝜑 → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0)

Theoremrolle 23747* 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.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   (𝜑𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ))    &   (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵))    &   (𝜑 → (𝐹𝐴) = (𝐹𝐵))       (𝜑 → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0)

Theoremcmvth 23748* 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.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   (𝜑𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ))    &   (𝜑𝐺 ∈ ((𝐴[,]𝐵)–cn→ℝ))    &   (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵))    &   (𝜑 → dom (ℝ D 𝐺) = (𝐴(,)𝐵))       (𝜑 → ∃𝑥 ∈ (𝐴(,)𝐵)(((𝐹𝐵) − (𝐹𝐴)) · ((ℝ D 𝐺)‘𝑥)) = (((𝐺𝐵) − (𝐺𝐴)) · ((ℝ D 𝐹)‘𝑥)))

Theoremmvth 23749* 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.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   (𝜑𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ))    &   (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵))       (𝜑 → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = (((𝐹𝐵) − (𝐹𝐴)) / (𝐵𝐴)))

Theoremdvlip 23750* A function with derivative bounded by 𝑀 is Lipschitz continuous with Lipschitz constant equal to 𝑀. (Contributed by Mario Carneiro, 3-Mar-2015.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ))    &   (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵))    &   (𝜑𝑀 ∈ ℝ)    &   ((𝜑𝑥 ∈ (𝐴(,)𝐵)) → (abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝑀)       ((𝜑 ∧ (𝑋 ∈ (𝐴[,]𝐵) ∧ 𝑌 ∈ (𝐴[,]𝐵))) → (abs‘((𝐹𝑋) − (𝐹𝑌))) ≤ (𝑀 · (abs‘(𝑋𝑌))))

Theoremdvlipcn 23751* A complex function with derivative bounded by 𝑀 on an open ball is Lipschitz continuous with Lipschitz constant equal to 𝑀. (Contributed by Mario Carneiro, 18-Mar-2015.)
(𝜑𝑋 ⊆ ℂ)    &   (𝜑𝐹:𝑋⟶ℂ)    &   (𝜑𝐴 ∈ ℂ)    &   (𝜑𝑅 ∈ ℝ*)    &   𝐵 = (𝐴(ball‘(abs ∘ − ))𝑅)    &   (𝜑𝐵 ⊆ dom (ℂ D 𝐹))    &   (𝜑𝑀 ∈ ℝ)    &   ((𝜑𝑥𝐵) → (abs‘((ℂ D 𝐹)‘𝑥)) ≤ 𝑀)       ((𝜑 ∧ (𝑌𝐵𝑍𝐵)) → (abs‘((𝐹𝑌) − (𝐹𝑍))) ≤ (𝑀 · (abs‘(𝑌𝑍))))

Theoremdvlip2 23752* Combine the results of dvlip 23750 and dvlipcn 23751 into one. (Contributed by Mario Carneiro, 18-Mar-2015.) (Revised by Mario Carneiro, 8-Sep-2015.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   𝐽 = ((abs ∘ − ) ↾ (𝑆 × 𝑆))    &   (𝜑𝑋𝑆)    &   (𝜑𝐹:𝑋⟶ℂ)    &   (𝜑𝐴𝑆)    &   (𝜑𝑅 ∈ ℝ*)    &   𝐵 = (𝐴(ball‘𝐽)𝑅)    &   (𝜑𝐵 ⊆ dom (𝑆 D 𝐹))    &   (𝜑𝑀 ∈ ℝ)    &   ((𝜑𝑥𝐵) → (abs‘((𝑆 D 𝐹)‘𝑥)) ≤ 𝑀)       ((𝜑 ∧ (𝑌𝐵𝑍𝐵)) → (abs‘((𝐹𝑌) − (𝐹𝑍))) ≤ (𝑀 · (abs‘(𝑌𝑍))))

Theoremc1liplem1 23753* Lemma for c1lip1 23754. (Contributed by Stefan O'Rear, 15-Nov-2014.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴𝐵)    &   (𝜑𝐹 ∈ (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‘((𝐹𝑦) − (𝐹𝑥))) ≤ (𝐾 · (abs‘(𝑦𝑥))))))

Theoremc1lip1 23754* C1 functions are Lipschitz continuous on closed intervals. (Contributed by Stefan O'Rear, 16-Nov-2014.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐹 ∈ (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‘((𝐹𝑦) − (𝐹𝑥))) ≤ (𝑘 · (abs‘(𝑦𝑥))))

Theoremc1lip2 23755* C1 functions are Lipschitz continuous on closed intervals. (Contributed by Stefan O'Rear, 16-Nov-2014.) (Revised by Stefan O'Rear, 6-May-2015.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐹 ∈ ((Cn‘ℝ)‘1))    &   (𝜑 → ran 𝐹 ⊆ ℝ)    &   (𝜑 → (𝐴[,]𝐵) ⊆ dom 𝐹)       (𝜑 → ∃𝑘 ∈ ℝ ∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(abs‘((𝐹𝑦) − (𝐹𝑥))) ≤ (𝑘 · (abs‘(𝑦𝑥))))

Theoremc1lip3 23756* C1 functions are Lipschitz continuous on closed intervals. (Contributed by Stefan O'Rear, 16-Nov-2014.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → (𝐹 ↾ ℝ) ∈ ((Cn‘ℝ)‘1))    &   (𝜑 → (𝐹 “ ℝ) ⊆ ℝ)    &   (𝜑 → (𝐴[,]𝐵) ⊆ dom 𝐹)       (𝜑 → ∃𝑘 ∈ ℝ ∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(abs‘((𝐹𝑦) − (𝐹𝑥))) ≤ (𝑘 · (abs‘(𝑦𝑥))))

Theoremdveq0 23757 If a continuous function has zero derivative at all points on the interior of a closed interval, then it must be a constant function. (Contributed by Mario Carneiro, 2-Sep-2014.) (Proof shortened by Mario Carneiro, 3-Mar-2015.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ))    &   (𝜑 → (ℝ D 𝐹) = ((𝐴(,)𝐵) × {0}))       (𝜑𝐹 = ((𝐴[,]𝐵) × {(𝐹𝐴)}))

Theoremdv11cn 23758 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 23759 Lemma for dvgt0 23761 and dvlt0 23762. (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 23760* Lemma for dvgt0 23761 and dvlt0 23762. (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 23761 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 23762 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 23763 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 23764* 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 23765* Lemma for dvivth 23767. (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 23766* Lemma for dvivth 23767. (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 23767 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 23221 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 23768 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 23769 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 23770* Lemma for lhop1 23771. (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 23771* 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 23772* 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 23773* 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 23774 Lemma for dvcnvre 23776. (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 23775 Lemma for dvcnvre 23776. (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 23776* 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 23777 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 23778* 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 23779* 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 23780* 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 23781* 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 23782* Lemma for dvfsumrlim 23788. (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 23783* Lemma for dvfsumrlim 23788. (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 23784* Lemma for dvfsumrlim 23788. (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 23785* Lemma for dvfsumrlim 23788. (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 23786* Lemma for dvfsumrlim 23788. (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 23787* Lemma for dvfsumrlim 23788. Satisfy the assumption of dvfsumlem4 23786. (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 23788* 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 23789* 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 23790* Conjoin the statements of dvfsumrlim 23788 and dvfsumrlim2 23789. (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 23791* The reverse of dvfsumrlim 23788, 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 23792* Lemma for ftc1a 23794 and ftc1 23799. (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 23793* Lemma for ftc1 23799. (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 23794* 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 23795* Lemma for ftc1 23799. (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 23796* Lemma for ftc1 23799. (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 23797* Lemma for ftc1 23799. (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 23798* Lemma for ftc1 23799. (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 23799* 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 23800* Strengthen the assumptions of ftc1 23799 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)

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