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Theorem List for Metamath Proof Explorer - 23501-23600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcpnfval 23501* Condition for n-times continuous differentiability. (Contributed by Stefan O'Rear, 15-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
(𝑆 ⊆ ℂ → (Cn𝑆) = (𝑛 ∈ ℕ0 ↦ {𝑓 ∈ (ℂ ↑pm 𝑆) ∣ ((𝑆 D𝑛 𝑓)‘𝑛) ∈ (dom 𝑓cn→ℂ)}))

Theoremfncpn 23502 The Cn object is a function. (Contributed by Stefan O'Rear, 16-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
(𝑆 ⊆ ℂ → (Cn𝑆) Fn ℕ0)

Theoremelcpn 23503 Condition for n-times continuous differentiability. (Contributed by Stefan O'Rear, 15-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
((𝑆 ⊆ ℂ ∧ 𝑁 ∈ ℕ0) → (𝐹 ∈ ((Cn𝑆)‘𝑁) ↔ (𝐹 ∈ (ℂ ↑pm 𝑆) ∧ ((𝑆 D𝑛 𝐹)‘𝑁) ∈ (dom 𝐹cn→ℂ))))

Theoremcpnord 23504 Cn conditions are ordered by strength. (Contributed by Stefan O'Rear, 16-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
((𝑆 ∈ {ℝ, ℂ} ∧ 𝑀 ∈ ℕ0𝑁 ∈ (ℤ𝑀)) → ((Cn𝑆)‘𝑁) ⊆ ((Cn𝑆)‘𝑀))

Theoremcpncn 23505 A Cn function is continuous. (Contributed by Mario Carneiro, 11-Feb-2015.)
((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((Cn𝑆)‘𝑁)) → 𝐹 ∈ (dom 𝐹cn→ℂ))

Theoremcpnres 23506 The restriction of a Cn function is Cn. (Contributed by Mario Carneiro, 11-Feb-2015.)
((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((Cn‘ℂ)‘𝑁)) → (𝐹𝑆) ∈ ((Cn𝑆)‘𝑁))

Theoremdvaddbr 23507 The sum rule for derivatives at a point. For the (simpler but more limited) function version, see dvadd 23509. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 28-Dec-2016.)
(𝜑𝐹:𝑋⟶ℂ)    &   (𝜑𝑋𝑆)    &   (𝜑𝐺:𝑌⟶ℂ)    &   (𝜑𝑌𝑆)    &   (𝜑𝑆 ⊆ ℂ)    &   (𝜑𝐾𝑉)    &   (𝜑𝐿𝑉)    &   (𝜑𝐶(𝑆 D 𝐹)𝐾)    &   (𝜑𝐶(𝑆 D 𝐺)𝐿)    &   𝐽 = (TopOpen‘ℂfld)       (𝜑𝐶(𝑆 D (𝐹𝑓 + 𝐺))(𝐾 + 𝐿))

Theoremdvmulbr 23508 The product rule for derivatives at a point. For the (simpler but more limited) function version, see dvmul 23510. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 28-Dec-2016.)
(𝜑𝐹:𝑋⟶ℂ)    &   (𝜑𝑋𝑆)    &   (𝜑𝐺:𝑌⟶ℂ)    &   (𝜑𝑌𝑆)    &   (𝜑𝑆 ⊆ ℂ)    &   (𝜑𝐾𝑉)    &   (𝜑𝐿𝑉)    &   (𝜑𝐶(𝑆 D 𝐹)𝐾)    &   (𝜑𝐶(𝑆 D 𝐺)𝐿)    &   𝐽 = (TopOpen‘ℂfld)       (𝜑𝐶(𝑆 D (𝐹𝑓 · 𝐺))((𝐾 · (𝐺𝐶)) + (𝐿 · (𝐹𝐶))))

Theoremdvadd 23509 The sum rule for derivatives at a point. For the (more general) relation version, see dvaddbr 23507. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)
(𝜑𝐹:𝑋⟶ℂ)    &   (𝜑𝑋𝑆)    &   (𝜑𝐺:𝑌⟶ℂ)    &   (𝜑𝑌𝑆)    &   (𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝐶 ∈ dom (𝑆 D 𝐹))    &   (𝜑𝐶 ∈ dom (𝑆 D 𝐺))       (𝜑 → ((𝑆 D (𝐹𝑓 + 𝐺))‘𝐶) = (((𝑆 D 𝐹)‘𝐶) + ((𝑆 D 𝐺)‘𝐶)))

Theoremdvmul 23510 The product rule for derivatives at a point. For the (more general) relation version, see dvmulbr 23508. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)
(𝜑𝐹:𝑋⟶ℂ)    &   (𝜑𝑋𝑆)    &   (𝜑𝐺:𝑌⟶ℂ)    &   (𝜑𝑌𝑆)    &   (𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝐶 ∈ dom (𝑆 D 𝐹))    &   (𝜑𝐶 ∈ dom (𝑆 D 𝐺))       (𝜑 → ((𝑆 D (𝐹𝑓 · 𝐺))‘𝐶) = ((((𝑆 D 𝐹)‘𝐶) · (𝐺𝐶)) + (((𝑆 D 𝐺)‘𝐶) · (𝐹𝐶))))

Theoremdvaddf 23511 The sum rule for everywhere-differentiable functions. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝐹:𝑋⟶ℂ)    &   (𝜑𝐺:𝑋⟶ℂ)    &   (𝜑 → dom (𝑆 D 𝐹) = 𝑋)    &   (𝜑 → dom (𝑆 D 𝐺) = 𝑋)       (𝜑 → (𝑆 D (𝐹𝑓 + 𝐺)) = ((𝑆 D 𝐹) ∘𝑓 + (𝑆 D 𝐺)))

Theoremdvmulf 23512 The product rule for everywhere-differentiable functions. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝐹:𝑋⟶ℂ)    &   (𝜑𝐺:𝑋⟶ℂ)    &   (𝜑 → dom (𝑆 D 𝐹) = 𝑋)    &   (𝜑 → dom (𝑆 D 𝐺) = 𝑋)       (𝜑 → (𝑆 D (𝐹𝑓 · 𝐺)) = (((𝑆 D 𝐹) ∘𝑓 · 𝐺) ∘𝑓 + ((𝑆 D 𝐺) ∘𝑓 · 𝐹)))

Theoremdvcmul 23513 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 23514 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 23515 The chain rule for derivatives at a point. For the (simpler but more limited) function version, see dvco 23516. (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 23516 The chain rule for derivatives at a point. For the (more general) relation version, see dvcobr 23515. (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 23517 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 23518 The derivative of the conjugate of a function. For the (simpler but more limited) function version, see dvcj 23519. (This doesn't follow from dvcobr 23515 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 23519 The derivative of the conjugate of a function. For the (more general) relation version, see dvcjbr 23518. (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 23520 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 23521 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 23522* 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 23523* 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 23524* 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 23525* 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 23526* 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 23527* 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 23528* Closure lemma for dvmptcmul 23533 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 23529* 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 23530* 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 23531* 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 23532* 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 23533* 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 23534* 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 23535* 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 23536* 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 23537* 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 23538* 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 23539* 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 23540* 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 23541* 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)))

Theoremdvmptfsum 23542* 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 23543 Lemma for dvcnvre 23586. (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 23544* A weak version of dvcnvre 23586, 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 23545* 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 23546 Derivative of the exponential function at 0. The key step in the proof is eftlub 14678, 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 23547 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 23548 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 23549 Derivative of the sine function. (Contributed by Mario Carneiro, 21-May-2016.)
(ℂ D sin) = cos

Theoremdvcos 23550 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 23551* Lemma for dvferm 23555. (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 23552* One-sided version of dvferm 23555. 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 23553* Lemma for dvferm 23555. (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 23554* One-sided version of dvferm 23555. 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 23555* 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 23556* Lemma for rolle 23557. (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 23557* 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 23558* 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 23559* 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 23560* 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 23561* 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 23562* Combine the results of dvlip 23560 and dvlipcn 23561 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 23563* Lemma for c1lip1 23564. (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 23564* 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 23565* 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 23566* 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 23567 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 23568 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 23569 Lemma for dvgt0 23571 and dvlt0 23572. (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 23570* Lemma for dvgt0 23571 and dvlt0 23572. (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 23571 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 23572 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 23573 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 23574* 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 23575* Lemma for dvivth 23577. (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 23576* Lemma for dvivth 23577. (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 23577 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 23034 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 23578 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 23579 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 23580* Lemma for lhop1 23581. (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 23581* 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 23582* 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 23583* 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 23584 Lemma for dvcnvre 23586. (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 23585 Lemma for dvcnvre 23586. (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 23586* 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 23587 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 23588* 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 23589* 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 23590* 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 23591* 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 23592* Lemma for dvfsumrlim 23598. (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 23593* Lemma for dvfsumrlim 23598. (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 23594* Lemma for dvfsumrlim 23598. (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 23595* Lemma for dvfsumrlim 23598. (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 23596* Lemma for dvfsumrlim 23598. (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 23597* Lemma for dvfsumrlim 23598. Satisfy the assumption of dvfsumlem4 23596. (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 23598* 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 23599* 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 23600* Conjoin the statements of dvfsumrlim 23598 and dvfsumrlim2 23599. (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)))

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78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42360
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