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Type | Label | Description |
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Statement | ||
Theorem | cpnres 25301 | The restriction of a 𝓑C𝑛 function is 𝓑C𝑛. (Contributed by Mario Carneiro, 11-Feb-2015.) |
⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C𝑛‘ℂ)‘𝑁)) → (𝐹 ↾ 𝑆) ∈ ((𝓑C𝑛‘𝑆)‘𝑁)) | ||
Theorem | dvaddbr 25302 | The sum rule for derivatives at a point. For the (simpler but more limited) function version, see dvadd 25304. (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 25303 | The product rule for derivatives at a point. For the (simpler but more limited) function version, see dvmul 25305. (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 25304 | The sum rule for derivatives at a point. For the (more general) relation version, see dvaddbr 25302. (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 25305 | The product rule for derivatives at a point. For the (more general) relation version, see dvmulbr 25303. (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 25306 | 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 25307 | 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 25308 | 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 25309 | 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 25310 | The chain rule for derivatives at a point. For the (simpler but more limited) function version, see dvco 25311. (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 25311 | The chain rule for derivatives at a point. For the (more general) relation version, see dvcobr 25310. (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 25312 | 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 25313 | The derivative of the conjugate of a function. For the (simpler but more limited) function version, see dvcj 25314. (This doesn't follow from dvcobr 25310 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 25314 | The derivative of the conjugate of a function. For the (more general) relation version, see dvcjbr 25313. (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 25315 | 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 25316 | 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 25317* | 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 25318* | 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 25319* | 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 25320* | 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 25321* | 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 25322* | 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 25323* | Closure lemma for dvmptcmul 25328 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 25324* | 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 25325* | 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 25326* | 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 25327* | 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 25328* | 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 25329* | 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 25330* | 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 25331* | 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 25332* | 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 25333* | 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 25334* | 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 25335* | 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 25336* | Function-builder for derivative, chain rule. (Contributed by Mario Carneiro, 1-Sep-2014.) |
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) & ⊢ (𝜑 → 𝑇 ∈ {ℝ, ℂ}) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑌) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 𝐶 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 𝐷 ∈ 𝑊) & ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐵)) & ⊢ (𝜑 → (𝑇 D (𝑦 ∈ 𝑌 ↦ 𝐶)) = (𝑦 ∈ 𝑌 ↦ 𝐷)) & ⊢ (𝑦 = 𝐴 → 𝐶 = 𝐸) & ⊢ (𝑦 = 𝐴 → 𝐷 = 𝐹) ⇒ ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐸)) = (𝑥 ∈ 𝑋 ↦ (𝐹 · 𝐵))) | ||
Theorem | dvrecg 25337* | Derivative of the reciprocal of a function. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) & ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ (ℂ ∖ {0})) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐶 ∈ 𝑉) & ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐵)) = (𝑥 ∈ 𝑋 ↦ 𝐶)) ⇒ ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐵))) = (𝑥 ∈ 𝑋 ↦ -((𝐴 · 𝐶) / (𝐵↑2)))) | ||
Theorem | dvmptdiv 25338* | Function-builder for derivative, quotient rule. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐵)) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐶 ∈ (ℂ ∖ {0})) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐷 ∈ ℂ) & ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐶)) = (𝑥 ∈ 𝑋 ↦ 𝐷)) ⇒ ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶))) = (𝑥 ∈ 𝑋 ↦ (((𝐵 · 𝐶) − (𝐷 · 𝐴)) / (𝐶↑2)))) | ||
Theorem | dvmptfsum 25339* | 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 25340 | Lemma for dvcnvre 25383. (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 25341* | A weak version of dvcnvre 25383, 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 25342* | 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 25343 | Derivative of the exponential function at 0. The key step in the proof is eftlub 15991, 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 25344 | 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 25345 | 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 25346 | Derivative of the sine function. (Contributed by Mario Carneiro, 21-May-2016.) |
⊢ (ℂ D sin) = cos | ||
Theorem | dvcos 25347 | Derivative of the cosine function. (Contributed by Mario Carneiro, 21-May-2016.) |
⊢ (ℂ D cos) = (𝑥 ∈ ℂ ↦ -(sin‘𝑥)) | ||
Theorem | dvferm1lem 25348* | Lemma for dvferm 25352. (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 25349* | One-sided version of dvferm 25352. 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 25350* | Lemma for dvferm 25352. (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 25351* | One-sided version of dvferm 25352. 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 25352* | 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 25353* | Lemma for rolle 25354. (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 25354* | 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 25355* | 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 25356* | 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 25357* | A function with derivative bounded by 𝑀 is 𝑀-Lipschitz continuous. (Contributed by Mario Carneiro, 3-Mar-2015.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ)) & ⊢ (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵)) & ⊢ (𝜑 → 𝑀 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝑀) ⇒ ⊢ ((𝜑 ∧ (𝑋 ∈ (𝐴[,]𝐵) ∧ 𝑌 ∈ (𝐴[,]𝐵))) → (abs‘((𝐹‘𝑋) − (𝐹‘𝑌))) ≤ (𝑀 · (abs‘(𝑋 − 𝑌)))) | ||
Theorem | dvlipcn 25358* | A complex function with derivative bounded by 𝑀 on an open ball is 𝑀-Lipschitz continuous. (Contributed by Mario Carneiro, 18-Mar-2015.) |
⊢ (𝜑 → 𝑋 ⊆ ℂ) & ⊢ (𝜑 → 𝐹:𝑋⟶ℂ) & ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝑅 ∈ ℝ*) & ⊢ 𝐵 = (𝐴(ball‘(abs ∘ − ))𝑅) & ⊢ (𝜑 → 𝐵 ⊆ dom (ℂ D 𝐹)) & ⊢ (𝜑 → 𝑀 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (abs‘((ℂ D 𝐹)‘𝑥)) ≤ 𝑀) ⇒ ⊢ ((𝜑 ∧ (𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (abs‘((𝐹‘𝑌) − (𝐹‘𝑍))) ≤ (𝑀 · (abs‘(𝑌 − 𝑍)))) | ||
Theorem | dvlip2 25359* | Combine the results of dvlip 25357 and dvlipcn 25358 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 25360* | Lemma for c1lip1 25361. (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 25361* | C^1 functions are Lipschitz continuous on closed intervals. (Contributed by Stefan O'Rear, 16-Nov-2014.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐹 ∈ (ℂ ↑pm ℝ)) & ⊢ (𝜑 → ((ℝ D 𝐹) ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℝ)) & ⊢ (𝜑 → (𝐹 ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℝ)) ⇒ ⊢ (𝜑 → ∃𝑘 ∈ ℝ ∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥)))) | ||
Theorem | c1lip2 25362* | C^1 functions are Lipschitz continuous on closed intervals. (Contributed by Stefan O'Rear, 16-Nov-2014.) (Revised by Stefan O'Rear, 6-May-2015.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐹 ∈ ((𝓑C𝑛‘ℝ)‘1)) & ⊢ (𝜑 → ran 𝐹 ⊆ ℝ) & ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ dom 𝐹) ⇒ ⊢ (𝜑 → ∃𝑘 ∈ ℝ ∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥)))) | ||
Theorem | c1lip3 25363* | C^1 functions are Lipschitz continuous on closed intervals. (Contributed by Stefan O'Rear, 16-Nov-2014.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → (𝐹 ↾ ℝ) ∈ ((𝓑C𝑛‘ℝ)‘1)) & ⊢ (𝜑 → (𝐹 “ ℝ) ⊆ ℝ) & ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ dom 𝐹) ⇒ ⊢ (𝜑 → ∃𝑘 ∈ ℝ ∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥)))) | ||
Theorem | dveq0 25364 | 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 25365 | 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 25366 | Lemma for dvgt0 25368 and dvlt0 25369. (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 25367* | Lemma for dvgt0 25368 and dvlt0 25369. (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 25368 | 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 25369 | 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 25370 | 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 25371* | 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 25372* | Lemma for dvivth 25374. (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 25373* | Lemma for dvivth 25374. (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 25374 | 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 24822 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 25375 | 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 25376 | 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 25377* | Lemma for lhop1 25378. (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 25378* | 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 25379* | 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 25380* | 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 25381 | Lemma for dvcnvre 25383. (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 25382 | Lemma for dvcnvre 25383. (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 25383* | 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 25384 | 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 25385* | 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 25386* | 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 25387* | 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 25388* | 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 25389* | Lemma for dvfsumrlim 25395. (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 25390* | Lemma for dvfsumrlim 25395. (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 25391* | Lemma for dvfsumrlim 25395. (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 25392* | Lemma for dvfsumrlim 25395. (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 25393* | Lemma for dvfsumrlim 25395. (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 25394* | Lemma for dvfsumrlim 25395. Satisfy the assumption of dvfsumlem4 25393. (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 25395* | 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 25396* | 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 25397* | Conjoin the statements of dvfsumrlim 25395 and dvfsumrlim2 25396. (This is useful as a target for lemmas, because the hypotheses to this theorem are complex, and we don't want to repeat ourselves.) (Contributed by Mario Carneiro, 18-May-2016.) |
⊢ 𝑆 = (𝑇(,)+∞) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐷 ∈ ℝ) & ⊢ (𝜑 → 𝑀 ≤ (𝐷 + 1)) & ⊢ (𝜑 → 𝑇 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐵 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝐵 ∈ ℝ) & ⊢ (𝜑 → (ℝ D (𝑥 ∈ 𝑆 ↦ 𝐴)) = (𝑥 ∈ 𝑆 ↦ 𝐵)) & ⊢ (𝑥 = 𝑘 → 𝐵 = 𝐶) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆) ∧ (𝐷 ≤ 𝑥 ∧ 𝑥 ≤ 𝑘)) → 𝐶 ≤ 𝐵) & ⊢ 𝐺 = (𝑥 ∈ 𝑆 ↦ (Σ𝑘 ∈ (𝑀...(⌊‘𝑥))𝐶 − 𝐴)) & ⊢ (𝜑 → (𝑥 ∈ 𝑆 ↦ 𝐵) ⇝𝑟 0) & ⊢ (𝑥 = 𝑋 → 𝐵 = 𝐸) ⇒ ⊢ (𝜑 → (𝐺:𝑆⟶ℝ ∧ 𝐺 ∈ dom ⇝𝑟 ∧ ((𝐺 ⇝𝑟 𝐿 ∧ 𝑋 ∈ 𝑆 ∧ 𝐷 ≤ 𝑋) → (abs‘((𝐺‘𝑋) − 𝐿)) ≤ 𝐸))) | ||
Theorem | dvfsum2 25398* | The reverse of dvfsumrlim 25395, when comparing a finite sum of increasing terms to an integral. In this case there is no point in stating the limit properties, because the terms of the sum aren't approaching zero, but there is nevertheless still a natural asymptotic statement that can be made. (Contributed by Mario Carneiro, 20-May-2016.) |
⊢ 𝑆 = (𝑇(,)+∞) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐷 ∈ ℝ) & ⊢ (𝜑 → 𝑈 ∈ ℝ*) & ⊢ (𝜑 → 𝑀 ≤ (𝐷 + 1)) & ⊢ (𝜑 → 𝑇 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐵 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝐵 ∈ ℝ) & ⊢ (𝜑 → (ℝ D (𝑥 ∈ 𝑆 ↦ 𝐴)) = (𝑥 ∈ 𝑆 ↦ 𝐵)) & ⊢ (𝑥 = 𝑘 → 𝐵 = 𝐶) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆) ∧ (𝐷 ≤ 𝑥 ∧ 𝑥 ≤ 𝑘 ∧ 𝑘 ≤ 𝑈)) → 𝐵 ≤ 𝐶) & ⊢ 𝐺 = (𝑥 ∈ 𝑆 ↦ (Σ𝑘 ∈ (𝑀...(⌊‘𝑥))𝐶 − 𝐴)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → 0 ≤ 𝐵) & ⊢ (𝜑 → 𝑋 ∈ 𝑆) & ⊢ (𝜑 → 𝑌 ∈ 𝑆) & ⊢ (𝜑 → 𝐷 ≤ 𝑋) & ⊢ (𝜑 → 𝑋 ≤ 𝑌) & ⊢ (𝜑 → 𝑌 ≤ 𝑈) & ⊢ (𝑥 = 𝑌 → 𝐵 = 𝐸) ⇒ ⊢ (𝜑 → (abs‘((𝐺‘𝑌) − (𝐺‘𝑋))) ≤ 𝐸) | ||
Theorem | ftc1lem1 25399* | Lemma for ftc1a 25401 and ftc1 25406. (Contributed by Mario Carneiro, 31-Aug-2014.) |
⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)(𝐹‘𝑡) d𝑡) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) & ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ 𝐷) & ⊢ (𝜑 → 𝐷 ⊆ ℝ) & ⊢ (𝜑 → 𝐹 ∈ 𝐿1) & ⊢ (𝜑 → 𝐹:𝐷⟶ℂ) & ⊢ (𝜑 → 𝑋 ∈ (𝐴[,]𝐵)) & ⊢ (𝜑 → 𝑌 ∈ (𝐴[,]𝐵)) ⇒ ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑌) → ((𝐺‘𝑌) − (𝐺‘𝑋)) = ∫(𝑋(,)𝑌)(𝐹‘𝑡) d𝑡) | ||
Theorem | ftc1lem2 25400* | Lemma for ftc1 25406. (Contributed by Mario Carneiro, 12-Aug-2014.) |
⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)(𝐹‘𝑡) d𝑡) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) & ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ 𝐷) & ⊢ (𝜑 → 𝐷 ⊆ ℝ) & ⊢ (𝜑 → 𝐹 ∈ 𝐿1) & ⊢ (𝜑 → 𝐹:𝐷⟶ℂ) ⇒ ⊢ (𝜑 → 𝐺:(𝐴[,]𝐵)⟶ℂ) |
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