P. 262 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 26101-26200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	recnperf 26101	Both ℝ and ℂ are perfect subsets of ℂ. (Contributed by Mario Carneiro, 28-Dec-2016.)
⊢ 𝐾 = (TopOpen‘ℂfld)    ⇒   ⊢ (𝑆 ∈ {ℝ, ℂ} → (𝐾 ↾t 𝑆) ∈ Perf)
Theorem	dvfg 26102	Explicitly write out the functionality condition on derivative for 𝑆 = ℝ and ℂ. (Contributed by Mario Carneiro, 9-Feb-2015.)
⊢ (𝑆 ∈ {ℝ, ℂ} → (𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ)
Theorem	dvf 26103	The derivative is a function. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Mario Carneiro, 9-Feb-2015.)
⊢ (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℂ
Theorem	dvfcn 26104	The derivative is a function. (Contributed by Mario Carneiro, 9-Feb-2015.)
⊢ (ℂ D 𝐹):dom (ℂ D 𝐹)⟶ℂ
Theorem	dvreslem 26105*	Lemma for dvres 26107. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Mario Carneiro, 28-Dec-2016.) Commute the consequent and shorten proof. (Revised by Peter Mazsa, 2-Oct-2022.)
⊢ 𝐾 = (TopOpen‘ℂfld)    &   ⊢ 𝑇 = (𝐾 ↾t 𝑆)    &   ⊢ 𝐺 = (𝑧 ∈ (𝐴 ∖ {𝑥}) ↦ (((𝐹‘𝑧) − (𝐹‘𝑥)) / (𝑧 − 𝑥)))    &   ⊢ (𝜑 → 𝑆 ⊆ ℂ)    &   ⊢ (𝜑 → 𝐹:𝐴⟶ℂ)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝑆)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝑆)    &   ⊢ (𝜑 → 𝑦 ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝑥(𝑆 D (𝐹 ↾ 𝐵))𝑦 ↔ (𝑥 ∈ ((int‘𝑇)‘𝐵) ∧ 𝑥(𝑆 D 𝐹)𝑦)))
Theorem	dvres2lem 26106*	Lemma for dvres2 26108. (Contributed by Mario Carneiro, 9-Feb-2015.) (Revised by Mario Carneiro, 28-Dec-2016.)
⊢ 𝐾 = (TopOpen‘ℂfld)    &   ⊢ 𝑇 = (𝐾 ↾t 𝑆)    &   ⊢ 𝐺 = (𝑧 ∈ (𝐴 ∖ {𝑥}) ↦ (((𝐹‘𝑧) − (𝐹‘𝑥)) / (𝑧 − 𝑥)))    &   ⊢ (𝜑 → 𝑆 ⊆ ℂ)    &   ⊢ (𝜑 → 𝐹:𝐴⟶ℂ)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝑆)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝑆)    &   ⊢ (𝜑 → 𝑦 ∈ ℂ)    &   ⊢ (𝜑 → 𝑥(𝑆 D 𝐹)𝑦)    &   ⊢ (𝜑 → 𝑥 ∈ 𝐵)    ⇒   ⊢ (𝜑 → 𝑥(𝐵 D (𝐹 ↾ 𝐵))𝑦)
Theorem	dvres 26107	Restriction of a derivative. Note that our definition of derivative df-dv 26063 would still make sense if we demanded that 𝑥 be an element of the domain instead of an interior point of the domain, but then it is possible for a non-differentiable function to have two different derivatives at a single point 𝑥 when restricted to different subsets containing 𝑥; a classic example is the absolute value function restricted to [0, +∞) and (-∞, 0]. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Mario Carneiro, 9-Feb-2015.)
⊢ 𝐾 = (TopOpen‘ℂfld)    &   ⊢ 𝑇 = (𝐾 ↾t 𝑆)    ⇒   ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆)) → (𝑆 D (𝐹 ↾ 𝐵)) = ((𝑆 D 𝐹) ↾ ((int‘𝑇)‘𝐵)))
Theorem	dvres2 26108	Restriction of the base set of a derivative. The primary application of this theorem says that if a function is complex-differentiable then it is also real-differentiable. Unlike dvres 26107, there is no simple reverse relation relating real-differentiable functions to complex differentiability, and indeed there are functions like ℜ(𝑥) which are everywhere real-differentiable but nowhere complex-differentiable.) (Contributed by Mario Carneiro, 9-Feb-2015.)
⊢ (((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆)) → ((𝑆 D 𝐹) ↾ 𝐵) ⊆ (𝐵 D (𝐹 ↾ 𝐵)))
Theorem	dvres3 26109	Restriction of a complex differentiable function to the reals. (Contributed by Mario Carneiro, 10-Feb-2015.)
⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ ℂ ∧ 𝑆 ⊆ dom (ℂ D 𝐹))) → (𝑆 D (𝐹 ↾ 𝑆)) = ((ℂ D 𝐹) ↾ 𝑆))
Theorem	dvres3a 26110	Restriction of a complex differentiable function to the reals. This version of dvres3 26109 assumes that 𝐹 is differentiable on its domain, but does not require 𝐹 to be differentiable on the whole real line. (Contributed by Mario Carneiro, 11-Feb-2015.)
⊢ 𝐽 = (TopOpen‘ℂfld)    ⇒   ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ∈ 𝐽 ∧ dom (ℂ D 𝐹) = 𝐴)) → (𝑆 D (𝐹 ↾ 𝑆)) = ((ℂ D 𝐹) ↾ 𝑆))
Theorem	dvidlem 26111*	Lemma for dvid 26114 and dvconst 26113. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Mario Carneiro, 9-Feb-2015.)
⊢ (𝜑 → 𝐹:ℂ⟶ℂ)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑧 ≠ 𝑥)) → (((𝐹‘𝑧) − (𝐹‘𝑥)) / (𝑧 − 𝑥)) = 𝐵)    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ (𝜑 → (ℂ D 𝐹) = (ℂ × {𝐵}))
Theorem	dvmptresicc 26112*	Derivative of a function restricted to a closed interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ 𝐹 = (𝑥 ∈ ℂ ↦ 𝐴)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → (ℂ D 𝐹) = (𝑥 ∈ ℂ ↦ 𝐵))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐷 ∈ ℝ)    ⇒   ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐶[,]𝐷) ↦ 𝐴)) = (𝑥 ∈ (𝐶(,)𝐷) ↦ 𝐵))
Theorem	dvconst 26113	Derivative of a constant function. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Mario Carneiro, 9-Feb-2015.)
⊢ (𝐴 ∈ ℂ → (ℂ D (ℂ × {𝐴})) = (ℂ × {0}))
Theorem	dvid 26114	Derivative of the identity function. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Mario Carneiro, 9-Feb-2015.)
⊢ (ℂ D ( I ↾ ℂ)) = (ℂ × {1})
Theorem	dvcnp 26115*	The difference quotient is continuous at 𝐵 when the original function is differentiable at 𝐵. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Mario Carneiro, 28-Dec-2016.)
⊢ 𝐽 = (𝐾 ↾t 𝐴)    &   ⊢ 𝐾 = (TopOpen‘ℂfld)    &   ⊢ 𝐺 = (𝑧 ∈ 𝐴 ↦ if(𝑧 = 𝐵, ((𝑆 D 𝐹)‘𝐵), (((𝐹‘𝑧) − (𝐹‘𝐵)) / (𝑧 − 𝐵))))    ⇒   ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ 𝑆) ∧ 𝐵 ∈ dom (𝑆 D 𝐹)) → 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐵))
Theorem	dvcnp2 26116	A function is continuous at each point for which it is differentiable. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 28-Dec-2016.) Avoid ax-mulf 11198. (Revised by GG, 16-Mar-2025.)
⊢ 𝐽 = (𝐾 ↾t 𝐴)    &   ⊢ 𝐾 = (TopOpen‘ℂfld)    ⇒   ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ 𝑆) ∧ 𝐵 ∈ dom (𝑆 D 𝐹)) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐵))
Theorem	dvcn 26117	A differentiable function is continuous. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by Mario Carneiro, 7-Sep-2015.)
⊢ (((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ 𝑆) ∧ dom (𝑆 D 𝐹) = 𝐴) → 𝐹 ∈ (𝐴–cn→ℂ))
Theorem	dvnfval 26118*	Value of the iterated derivative. (Contributed by Mario Carneiro, 11-Feb-2015.)
⊢ 𝐺 = (𝑥 ∈ V ↦ (𝑆 D 𝑥))    ⇒   ⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) → (𝑆 D𝑛 𝐹) = seq0((𝐺 ∘ 1st ), (ℕ0 × {𝐹})))
Theorem	dvnff 26119	The iterated derivative is a function. (Contributed by Mario Carneiro, 11-Feb-2015.)
⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) → (𝑆 D𝑛 𝐹):ℕ0⟶(ℂ ↑pm dom 𝐹))
Theorem	dvn0 26120	Zero times iterated derivative. (Contributed by Stefan O'Rear, 15-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) → ((𝑆 D𝑛 𝐹)‘0) = 𝐹)
Theorem	dvnp1 26121	Successor iterated derivative. (Contributed by Stefan O'Rear, 15-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆) ∧ 𝑁 ∈ ℕ0) → ((𝑆 D𝑛 𝐹)‘(𝑁 + 1)) = (𝑆 D ((𝑆 D𝑛 𝐹)‘𝑁)))
Theorem	dvn1 26122	One times iterated derivative. (Contributed by Mario Carneiro, 1-Jan-2017.)
⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) → ((𝑆 D𝑛 𝐹)‘1) = (𝑆 D 𝐹))
Theorem	dvnf 26123	The N-times derivative is a function. (Contributed by Stefan O'Rear, 16-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm 𝑆) ∧ 𝑁 ∈ ℕ0) → ((𝑆 D𝑛 𝐹)‘𝑁):dom ((𝑆 D𝑛 𝐹)‘𝑁)⟶ℂ)
Theorem	dvnbss 26124	The set of N-times differentiable points is a subset of the domain of the function. (Contributed by Stefan O'Rear, 16-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm 𝑆) ∧ 𝑁 ∈ ℕ0) → dom ((𝑆 D𝑛 𝐹)‘𝑁) ⊆ dom 𝐹)
Theorem	dvnadd 26125	The 𝑁-th derivative of the 𝑀-th derivative of 𝐹 is the same as the 𝑀 + 𝑁-th derivative of 𝐹. (Contributed by Mario Carneiro, 11-Feb-2015.)
⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) ∧ (𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0)) → ((𝑆 D𝑛 ((𝑆 D𝑛 𝐹)‘𝑀))‘𝑁) = ((𝑆 D𝑛 𝐹)‘(𝑀 + 𝑁)))
Theorem	dvn2bss 26126	An N-times differentiable point is an M-times differentiable point, if 𝑀 ≤ 𝑁. (Contributed by Mario Carneiro, 30-Dec-2016.)
⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm 𝑆) ∧ 𝑀 ∈ (0...𝑁)) → dom ((𝑆 D𝑛 𝐹)‘𝑁) ⊆ dom ((𝑆 D𝑛 𝐹)‘𝑀))
Theorem	dvnres 26127	Multiple derivative version of dvres3a 26110. (Contributed by Mario Carneiro, 11-Feb-2015.)
⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ) ∧ 𝑁 ∈ ℕ0) ∧ dom ((ℂ D𝑛 𝐹)‘𝑁) = dom 𝐹) → ((𝑆 D𝑛 (𝐹 ↾ 𝑆))‘𝑁) = (((ℂ D𝑛 𝐹)‘𝑁) ↾ 𝑆))
Theorem	cpnfval 26128*	Condition for n-times continuous differentiability. (Contributed by Stefan O'Rear, 15-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
⊢ (𝑆 ⊆ ℂ → (𝓑C𝑛‘𝑆) = (𝑛 ∈ ℕ0 ↦ {𝑓 ∈ (ℂ ↑pm 𝑆) ∣ ((𝑆 D𝑛 𝑓)‘𝑛) ∈ (dom 𝑓–cn→ℂ)}))
Theorem	fncpn 26129	The 𝓑C𝑛 object is a function. (Contributed by Stefan O'Rear, 16-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
⊢ (𝑆 ⊆ ℂ → (𝓑C𝑛‘𝑆) Fn ℕ0)
Theorem	elcpn 26130	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 ∈ ((𝓑C𝑛‘𝑆)‘𝑁) ↔ (𝐹 ∈ (ℂ ↑pm 𝑆) ∧ ((𝑆 D𝑛 𝐹)‘𝑁) ∈ (dom 𝐹–cn→ℂ))))
Theorem	cpnord 26131	𝓑C𝑛 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)) → ((𝓑C𝑛‘𝑆)‘𝑁) ⊆ ((𝓑C𝑛‘𝑆)‘𝑀))
Theorem	cpncn 26132	A 𝓑C𝑛 function is continuous. (Contributed by Mario Carneiro, 11-Feb-2015.)
⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C𝑛‘𝑆)‘𝑁)) → 𝐹 ∈ (dom 𝐹–cn→ℂ))
Theorem	cpnres 26133	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 26134	The sum rule for derivatives at a point. For the (simpler but more limited) function version, see dvadd 26136. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 28-Dec-2016.) Remove unnecessary hypotheses. (Revised by GG, 10-Apr-2025.)
⊢ (𝜑 → 𝐹:𝑋⟶ℂ)    &   ⊢ (𝜑 → 𝑋 ⊆ 𝑆)    &   ⊢ (𝜑 → 𝐺:𝑌⟶ℂ)    &   ⊢ (𝜑 → 𝑌 ⊆ 𝑆)    &   ⊢ (𝜑 → 𝑆 ⊆ ℂ)    &   ⊢ (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 26135	The product rule for derivatives at a point. For the (simpler but more limited) function version, see dvmul 26137. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 28-Dec-2016.) Avoid ax-mulf 11198 and remove unnecessary hypotheses. (Revised by GG, 16-Mar-2025.)
⊢ (𝜑 → 𝐹:𝑋⟶ℂ)    &   ⊢ (𝜑 → 𝑋 ⊆ 𝑆)    &   ⊢ (𝜑 → 𝐺:𝑌⟶ℂ)    &   ⊢ (𝜑 → 𝑌 ⊆ 𝑆)    &   ⊢ (𝜑 → 𝑆 ⊆ ℂ)    &   ⊢ (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 26136	The sum rule for derivatives at a point. For the (more general) relation version, see dvaddbr 26134. (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 26137	The product rule for derivatives at a point. For the (more general) relation version, see dvmulbr 26135. (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 26138	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 26139	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 26140	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 26141	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 26142	The chain rule for derivatives at a point. For the (simpler but more limited) function version, see dvco 26143. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 28-Dec-2016.) Avoid ax-mulf 11198 and remove unnecessary hypotheses. (Revised by GG, 16-Mar-2025.)
⊢ (𝜑 → 𝐹:𝑋⟶ℂ)    &   ⊢ (𝜑 → 𝑋 ⊆ 𝑆)    &   ⊢ (𝜑 → 𝐺:𝑌⟶𝑋)    &   ⊢ (𝜑 → 𝑌 ⊆ 𝑇)    &   ⊢ (𝜑 → 𝑆 ⊆ ℂ)    &   ⊢ (𝜑 → 𝑇 ⊆ ℂ)    &   ⊢ (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 26143	The chain rule for derivatives at a point. For the (more general) relation version, see dvcobr 26142. (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 26144	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 26145	The derivative of the conjugate of a function. For the (simpler but more limited) function version, see dvcj 26146. (This doesn't follow from dvcobr 26142 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 26146	The derivative of the conjugate of a function. For the (more general) relation version, see dvcjbr 26145. (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 26147	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 26148	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 26149*	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 26150*	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 26151*	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 26152*	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 26153*	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 26154*	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 26155*	Closure lemma for dvmptcmul 26160 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 26156*	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 26157*	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 26158*	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 26159*	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 26160*	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 26161*	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 26162*	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 26163*	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 26164*	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 26165*	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 26166*	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 26167*	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 26168*	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 26169*	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 26170*	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 26171*	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 26172	Lemma for dvcnvre 26215. (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 26173*	A weak version of dvcnvre 26215, 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 26174*	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 26175	Derivative of the exponential function at 0. The key step in the proof is eftlub 16190, 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 26176	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 26177	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 26178	Derivative of the sine function. (Contributed by Mario Carneiro, 21-May-2016.)
⊢ (ℂ D sin) = cos
Theorem	dvcos 26179	Derivative of the cosine function. (Contributed by Mario Carneiro, 21-May-2016.)
⊢ (ℂ D cos) = (𝑥 ∈ ℂ ↦ -(sin‘𝑥))
13.3.1.2  Results on real differentiation
Theorem	dvferm1lem 26180*	Lemma for dvferm 26184. (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 26181*	One-sided version of dvferm 26184. 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 26182*	Lemma for dvferm 26184. (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 26183*	One-sided version of dvferm 26184. 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 26184*	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 26185*	Lemma for rolle 26186. (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 26186*	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 26187*	Cauchy's Mean Value Theorem. If 𝐹, 𝐺 are real continuous functions on [𝐴, 𝐵] differentiable on (𝐴, 𝐵), then there is some 𝑥 ∈ (𝐴, 𝐵) such that 𝐹' (𝑥) / 𝐺' (𝑥) = (𝐹(𝐴) − 𝐹(𝐵)) / (𝐺(𝐴) − 𝐺(𝐵)). (We express the condition without division, so that we need no nonzero constraints.) (Contributed by Mario Carneiro, 29-Dec-2016.) Avoid ax-mulf 11198. (Revised by GG, 16-Mar-2025.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 < 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ))    &   ⊢ (𝜑 → 𝐺 ∈ ((𝐴[,]𝐵)–cn→ℝ))    &   ⊢ (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵))    &   ⊢ (𝜑 → dom (ℝ D 𝐺) = (𝐴(,)𝐵))    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ (𝐴(,)𝐵)(((𝐹‘𝐵) − (𝐹‘𝐴)) · ((ℝ D 𝐺)‘𝑥)) = (((𝐺‘𝐵) − (𝐺‘𝐴)) · ((ℝ D 𝐹)‘𝑥)))
Theorem	mvth 26188*	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 26189*	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 26190*	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 26191*	Combine the results of dvlip 26189 and dvlipcn 26190 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 26192*	Lemma for c1lip1 26193. (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 26193*	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 26194*	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 26195*	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 26196	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 26197	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 26198	Lemma for dvgt0 26200 and dvlt0 26201. (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 26199*	Lemma for dvgt0 26200 and dvlt0 26201. (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 26200	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))