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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | cpnfval 23501* | Condition for n-times continuous differentiability. (Contributed by Stefan O'Rear, 15-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.) |
⊢ (𝑆 ⊆ ℂ → (Cn‘𝑆) = (𝑛 ∈ ℕ0 ↦ {𝑓 ∈ (ℂ ↑pm 𝑆) ∣ ((𝑆 D𝑛 𝑓)‘𝑛) ∈ (dom 𝑓–cn→ℂ)})) | ||
Theorem | fncpn 23502 | The Cn object is a function. (Contributed by Stefan O'Rear, 16-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.) |
⊢ (𝑆 ⊆ ℂ → (Cn‘𝑆) Fn ℕ0) | ||
Theorem | elcpn 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 ∈ ((Cn‘𝑆)‘𝑁) ↔ (𝐹 ∈ (ℂ ↑pm 𝑆) ∧ ((𝑆 D𝑛 𝐹)‘𝑁) ∈ (dom 𝐹–cn→ℂ)))) | ||
Theorem | cpnord 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 ∈ ℕ0 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → ((Cn‘𝑆)‘𝑁) ⊆ ((Cn‘𝑆)‘𝑀)) | ||
Theorem | cpncn 23505 | A Cn function is continuous. (Contributed by Mario Carneiro, 11-Feb-2015.) |
⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((Cn‘𝑆)‘𝑁)) → 𝐹 ∈ (dom 𝐹–cn→ℂ)) | ||
Theorem | cpnres 23506 | The restriction of a Cn function is Cn. (Contributed by Mario Carneiro, 11-Feb-2015.) |
⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((Cn‘ℂ)‘𝑁)) → (𝐹 ↾ 𝑆) ∈ ((Cn‘𝑆)‘𝑁)) | ||
Theorem | dvaddbr 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.) |
⊢ (𝜑 → 𝐹:𝑋⟶ℂ) & ⊢ (𝜑 → 𝑋 ⊆ 𝑆) & ⊢ (𝜑 → 𝐺:𝑌⟶ℂ) & ⊢ (𝜑 → 𝑌 ⊆ 𝑆) & ⊢ (𝜑 → 𝑆 ⊆ ℂ) & ⊢ (𝜑 → 𝐾 ∈ 𝑉) & ⊢ (𝜑 → 𝐿 ∈ 𝑉) & ⊢ (𝜑 → 𝐶(𝑆 D 𝐹)𝐾) & ⊢ (𝜑 → 𝐶(𝑆 D 𝐺)𝐿) & ⊢ 𝐽 = (TopOpen‘ℂfld) ⇒ ⊢ (𝜑 → 𝐶(𝑆 D (𝐹 ∘𝑓 + 𝐺))(𝐾 + 𝐿)) | ||
Theorem | dvmulbr 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.) |
⊢ (𝜑 → 𝐹:𝑋⟶ℂ) & ⊢ (𝜑 → 𝑋 ⊆ 𝑆) & ⊢ (𝜑 → 𝐺:𝑌⟶ℂ) & ⊢ (𝜑 → 𝑌 ⊆ 𝑆) & ⊢ (𝜑 → 𝑆 ⊆ ℂ) & ⊢ (𝜑 → 𝐾 ∈ 𝑉) & ⊢ (𝜑 → 𝐿 ∈ 𝑉) & ⊢ (𝜑 → 𝐶(𝑆 D 𝐹)𝐾) & ⊢ (𝜑 → 𝐶(𝑆 D 𝐺)𝐿) & ⊢ 𝐽 = (TopOpen‘ℂfld) ⇒ ⊢ (𝜑 → 𝐶(𝑆 D (𝐹 ∘𝑓 · 𝐺))((𝐾 · (𝐺‘𝐶)) + (𝐿 · (𝐹‘𝐶)))) | ||
Theorem | dvadd 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.) |
⊢ (𝜑 → 𝐹:𝑋⟶ℂ) & ⊢ (𝜑 → 𝑋 ⊆ 𝑆) & ⊢ (𝜑 → 𝐺:𝑌⟶ℂ) & ⊢ (𝜑 → 𝑌 ⊆ 𝑆) & ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) & ⊢ (𝜑 → 𝐶 ∈ dom (𝑆 D 𝐹)) & ⊢ (𝜑 → 𝐶 ∈ dom (𝑆 D 𝐺)) ⇒ ⊢ (𝜑 → ((𝑆 D (𝐹 ∘𝑓 + 𝐺))‘𝐶) = (((𝑆 D 𝐹)‘𝐶) + ((𝑆 D 𝐺)‘𝐶))) | ||
Theorem | dvmul 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.) |
⊢ (𝜑 → 𝐹:𝑋⟶ℂ) & ⊢ (𝜑 → 𝑋 ⊆ 𝑆) & ⊢ (𝜑 → 𝐺:𝑌⟶ℂ) & ⊢ (𝜑 → 𝑌 ⊆ 𝑆) & ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) & ⊢ (𝜑 → 𝐶 ∈ dom (𝑆 D 𝐹)) & ⊢ (𝜑 → 𝐶 ∈ dom (𝑆 D 𝐺)) ⇒ ⊢ (𝜑 → ((𝑆 D (𝐹 ∘𝑓 · 𝐺))‘𝐶) = ((((𝑆 D 𝐹)‘𝐶) · (𝐺‘𝐶)) + (((𝑆 D 𝐺)‘𝐶) · (𝐹‘𝐶)))) | ||
Theorem | dvaddf 23511 | The sum rule for everywhere-differentiable functions. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.) |
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) & ⊢ (𝜑 → 𝐹:𝑋⟶ℂ) & ⊢ (𝜑 → 𝐺:𝑋⟶ℂ) & ⊢ (𝜑 → dom (𝑆 D 𝐹) = 𝑋) & ⊢ (𝜑 → dom (𝑆 D 𝐺) = 𝑋) ⇒ ⊢ (𝜑 → (𝑆 D (𝐹 ∘𝑓 + 𝐺)) = ((𝑆 D 𝐹) ∘𝑓 + (𝑆 D 𝐺))) | ||
Theorem | dvmulf 23512 | The product rule for everywhere-differentiable functions. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.) |
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) & ⊢ (𝜑 → 𝐹:𝑋⟶ℂ) & ⊢ (𝜑 → 𝐺:𝑋⟶ℂ) & ⊢ (𝜑 → dom (𝑆 D 𝐹) = 𝑋) & ⊢ (𝜑 → dom (𝑆 D 𝐺) = 𝑋) ⇒ ⊢ (𝜑 → (𝑆 D (𝐹 ∘𝑓 · 𝐺)) = (((𝑆 D 𝐹) ∘𝑓 · 𝐺) ∘𝑓 + ((𝑆 D 𝐺) ∘𝑓 · 𝐹))) | ||
Theorem | dvcmul 23513 | The product rule when one argument is a constant. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.) |
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) & ⊢ (𝜑 → 𝐹:𝑋⟶ℂ) & ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝑋 ⊆ 𝑆) & ⊢ (𝜑 → 𝐶 ∈ dom (𝑆 D 𝐹)) ⇒ ⊢ (𝜑 → ((𝑆 D ((𝑆 × {𝐴}) ∘𝑓 · 𝐹))‘𝐶) = (𝐴 · ((𝑆 D 𝐹)‘𝐶))) | ||
Theorem | dvcmulf 23514 | The product rule when one argument is a constant. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.) |
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) & ⊢ (𝜑 → 𝐹:𝑋⟶ℂ) & ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → dom (𝑆 D 𝐹) = 𝑋) ⇒ ⊢ (𝜑 → (𝑆 D ((𝑆 × {𝐴}) ∘𝑓 · 𝐹)) = ((𝑆 × {𝐴}) ∘𝑓 · (𝑆 D 𝐹))) | ||
Theorem | dvcobr 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.) |
⊢ (𝜑 → 𝐹:𝑋⟶ℂ) & ⊢ (𝜑 → 𝑋 ⊆ 𝑆) & ⊢ (𝜑 → 𝐺:𝑌⟶𝑋) & ⊢ (𝜑 → 𝑌 ⊆ 𝑇) & ⊢ (𝜑 → 𝑆 ⊆ ℂ) & ⊢ (𝜑 → 𝑇 ⊆ ℂ) & ⊢ (𝜑 → 𝐾 ∈ 𝑉) & ⊢ (𝜑 → 𝐿 ∈ 𝑉) & ⊢ (𝜑 → (𝐺‘𝐶)(𝑆 D 𝐹)𝐾) & ⊢ (𝜑 → 𝐶(𝑇 D 𝐺)𝐿) & ⊢ 𝐽 = (TopOpen‘ℂfld) ⇒ ⊢ (𝜑 → 𝐶(𝑇 D (𝐹 ∘ 𝐺))(𝐾 · 𝐿)) | ||
Theorem | dvco 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.) |
⊢ (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 23517 | 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 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.) |
⊢ (𝜑 → 𝐹:𝑋⟶ℂ) & ⊢ (𝜑 → 𝑋 ⊆ ℝ) & ⊢ (𝜑 → 𝐶 ∈ dom (ℝ D 𝐹)) ⇒ ⊢ (𝜑 → 𝐶(ℝ D (∗ ∘ 𝐹))(∗‘((ℝ D 𝐹)‘𝐶))) | ||
Theorem | dvcj 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))) | ||
Theorem | dvfre 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) | ||
Theorem | dvnfre 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) | ||
Theorem | dvexp 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 ↦ (𝑥↑𝑁))) = (𝑥 ∈ ℂ ↦ (𝑁 · (𝑥↑(𝑁 − 1))))) | ||
Theorem | dvexp2 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 ↦ (𝑥↑𝑁))) = (𝑥 ∈ ℂ ↦ if(𝑁 = 0, 0, (𝑁 · (𝑥↑(𝑁 − 1)))))) | ||
Theorem | dvrec 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)))) | ||
Theorem | dvmptres3 23525* | 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 23526* | 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 23527* | 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 23528* | Closure lemma for dvmptcmul 23533 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 23529* | 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 23530* | 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 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.) |
⊢ (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 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.) |
⊢ (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 23533* | 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 23534* | 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 23535* | 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 23536* | 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 23537* | 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 23538* | 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 23539* | 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 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.) |
⊢ (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 23541* | 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 | dvmptfsum 23542* | 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 23543 | Lemma for dvcnvre 23586. (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 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 = (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 23545* | 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 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 | ||
Theorem | dvef 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 | ||
Theorem | dvsincos 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))) | ||
Theorem | dvsin 23549 | Derivative of the sine function. (Contributed by Mario Carneiro, 21-May-2016.) |
⊢ (ℂ D sin) = cos | ||
Theorem | dvcos 23550 | Derivative of the cosine function. (Contributed by Mario Carneiro, 21-May-2016.) |
⊢ (ℂ D cos) = (𝑥 ∈ ℂ ↦ -(sin‘𝑥)) | ||
Theorem | dvferm1lem 23551* | Lemma for dvferm 23555. (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 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.) |
⊢ (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 23553* | Lemma for dvferm 23555. (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 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.) |
⊢ (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 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.) |
⊢ (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 23556* | Lemma for rolle 23557. (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 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.) |
⊢ (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 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.) |
⊢ (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 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.) |
⊢ (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 23560* | A function with derivative bounded by 𝑀 is Lipschitz continuous with Lipschitz constant equal to 𝑀. (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 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.) |
⊢ (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 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.) |
⊢ (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 23563* | Lemma for c1lip1 23564. (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 23564* | C1 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 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.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐹 ∈ ((Cn‘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 23566* | C1 functions are Lipschitz continuous on closed intervals. (Contributed by Stefan O'Rear, 16-Nov-2014.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ 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‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥)))) | ||
Theorem | dveq0 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.) |
⊢ (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 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) & ⊢ (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 23569 | Lemma for dvgt0 23571 and dvlt0 23572. (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 23570* | Lemma for dvgt0 23571 and dvlt0 23572. (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 23571 | 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 23572 | 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 23573 | 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 23574* | 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 23575* | Lemma for dvivth 23577. (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 23576* | Lemma for dvivth 23577. (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 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.) |
⊢ (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 23578 | 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 23579 | 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 23580* | Lemma for lhop1 23581. (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 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.) |
⊢ (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 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.) |
⊢ (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 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.) |
⊢ (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 23584 | Lemma for dvcnvre 23586. (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 23585 | Lemma for dvcnvre 23586. (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 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.) |
⊢ (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 23587 | 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 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.) |
⊢ (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 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.) |
⊢ (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 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.) |
⊢ (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 23591* | 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 23592* | Lemma for dvfsumrlim 23598. (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 23593* | Lemma for dvfsumrlim 23598. (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 23594* | Lemma for dvfsumrlim 23598. (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) ≤ (𝐻‘𝑋) ∧ ((𝐻‘𝑋) − ⦋𝑋 / 𝑥⦌𝐵) ≤ ((𝐻‘𝑌) − ⦋𝑌 / 𝑥⦌𝐵))) | ||
Theorem | dvfsumlem3 23595* | Lemma for dvfsumrlim 23598. (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 23596* | Lemma for dvfsumrlim 23598. (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 23597* | Lemma for dvfsumrlim 23598. Satisfy the assumption of dvfsumlem4 23596. (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 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 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 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 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 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 = (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))) |
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