P. 150 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 14901-15000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	limcimo 14901*	Conditions which ensure there is at most one limit value of 𝐹 at 𝐵. (Contributed by Mario Carneiro, 25-Dec-2016.) (Revised by Jim Kingdon, 8-Jul-2023.)
⊢ (𝜑 → 𝐹:𝐴⟶ℂ)    &   ⊢ (𝜑 → 𝐴 ⊆ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐶)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐶 ∈ (𝐾 ↾t 𝑆))    &   ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})    &   ⊢ (𝜑 → {𝑞 ∈ 𝐶 ∣ 𝑞 # 𝐵} ⊆ 𝐴)    &   ⊢ 𝐾 = (MetOpen‘(abs ∘ − ))    ⇒   ⊢ (𝜑 → ∃*𝑥 𝑥 ∈ (𝐹 limℂ 𝐵))
Theorem	limcresi 14902	Any limit of 𝐹 is also a limit of the restriction of 𝐹. (Contributed by Mario Carneiro, 28-Dec-2016.)
⊢ (𝐹 limℂ 𝐵) ⊆ ((𝐹 ↾ 𝐶) limℂ 𝐵)
Theorem	cnplimcim 14903	If a function is continuous at 𝐵, its limit at 𝐵 equals the value of the function there. (Contributed by Mario Carneiro, 28-Dec-2016.) (Revised by Jim Kingdon, 14-Jun-2023.)
⊢ 𝐾 = (MetOpen‘(abs ∘ − ))    &   ⊢ 𝐽 = (𝐾 ↾t 𝐴)    ⇒   ⊢ ((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐵) → (𝐹:𝐴⟶ℂ ∧ (𝐹‘𝐵) ∈ (𝐹 limℂ 𝐵))))
Theorem	cnplimclemle 14904	Lemma for cnplimccntop 14906. Satisfying the epsilon condition for continuity. (Contributed by Mario Carneiro and Jim Kingdon, 17-Nov-2023.)
⊢ 𝐾 = (MetOpen‘(abs ∘ − ))    &   ⊢ 𝐽 = (𝐾 ↾t 𝐴)    &   ⊢ (𝜑 → 𝐴 ⊆ ℂ)    &   ⊢ (𝜑 → 𝐹:𝐴⟶ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐴)    &   ⊢ (𝜑 → (𝐹‘𝐵) ∈ (𝐹 limℂ 𝐵))    &   ⊢ (𝜑 → 𝐸 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐷 ∈ ℝ+)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐴)    &   ⊢ ((𝜑 ∧ 𝑍 # 𝐵 ∧ (abs‘(𝑍 − 𝐵)) < 𝐷) → (abs‘((𝐹‘𝑍) − (𝐹‘𝐵))) < (𝐸 / 2))    &   ⊢ (𝜑 → (abs‘(𝑍 − 𝐵)) < 𝐷)    ⇒   ⊢ (𝜑 → (abs‘((𝐹‘𝑍) − (𝐹‘𝐵))) < 𝐸)
Theorem	cnplimclemr 14905	Lemma for cnplimccntop 14906. The reverse direction. (Contributed by Mario Carneiro and Jim Kingdon, 17-Nov-2023.)
⊢ 𝐾 = (MetOpen‘(abs ∘ − ))    &   ⊢ 𝐽 = (𝐾 ↾t 𝐴)    &   ⊢ (𝜑 → 𝐴 ⊆ ℂ)    &   ⊢ (𝜑 → 𝐹:𝐴⟶ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐴)    &   ⊢ (𝜑 → (𝐹‘𝐵) ∈ (𝐹 limℂ 𝐵))    ⇒   ⊢ (𝜑 → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐵))
Theorem	cnplimccntop 14906	A function is continuous at 𝐵 iff its limit at 𝐵 equals the value of the function there. (Contributed by Mario Carneiro, 28-Dec-2016.)
⊢ 𝐾 = (MetOpen‘(abs ∘ − ))    &   ⊢ 𝐽 = (𝐾 ↾t 𝐴)    ⇒   ⊢ ((𝐴 ⊆ ℂ ∧ 𝐵 ∈ 𝐴) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐵) ↔ (𝐹:𝐴⟶ℂ ∧ (𝐹‘𝐵) ∈ (𝐹 limℂ 𝐵))))
Theorem	cnlimcim 14907*	If 𝐹 is a continuous function, the limit of the function at each point equals the value of the function. (Contributed by Mario Carneiro, 28-Dec-2016.) (Revised by Jim Kingdon, 16-Jun-2023.)
⊢ (𝐴 ⊆ ℂ → (𝐹 ∈ (𝐴–cn→ℂ) → (𝐹:𝐴⟶ℂ ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ (𝐹 limℂ 𝑥))))
Theorem	cnlimc 14908*	𝐹 is a continuous function iff the limit of the function at each point equals the value of the function. (Contributed by Mario Carneiro, 28-Dec-2016.)
⊢ (𝐴 ⊆ ℂ → (𝐹 ∈ (𝐴–cn→ℂ) ↔ (𝐹:𝐴⟶ℂ ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ (𝐹 limℂ 𝑥))))
Theorem	cnlimci 14909	If 𝐹 is a continuous function, then the limit of the function at any point equals its value. (Contributed by Mario Carneiro, 28-Dec-2016.)
⊢ (𝜑 → 𝐹 ∈ (𝐴–cn→𝐷))    &   ⊢ (𝜑 → 𝐵 ∈ 𝐴)    ⇒   ⊢ (𝜑 → (𝐹‘𝐵) ∈ (𝐹 limℂ 𝐵))
Theorem	cnmptlimc 14910*	If 𝐹 is a continuous function, then the limit of the function at any point equals its value. (Contributed by Mario Carneiro, 28-Dec-2016.)
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝑋) ∈ (𝐴–cn→𝐷))    &   ⊢ (𝜑 → 𝐵 ∈ 𝐴)    &   ⊢ (𝑥 = 𝐵 → 𝑋 = 𝑌)    ⇒   ⊢ (𝜑 → 𝑌 ∈ ((𝑥 ∈ 𝐴 ↦ 𝑋) limℂ 𝐵))
Theorem	limccnpcntop 14911	If the limit of 𝐹 at 𝐵 is 𝐶 and 𝐺 is continuous at 𝐶, then the limit of 𝐺 ∘ 𝐹 at 𝐵 is 𝐺(𝐶). (Contributed by Mario Carneiro, 28-Dec-2016.) (Revised by Jim Kingdon, 18-Jun-2023.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐷)    &   ⊢ (𝜑 → 𝐷 ⊆ ℂ)    &   ⊢ 𝐾 = (MetOpen‘(abs ∘ − ))    &   ⊢ 𝐽 = (𝐾 ↾t 𝐷)    &   ⊢ (𝜑 → 𝐶 ∈ (𝐹 limℂ 𝐵))    &   ⊢ (𝜑 → 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐶))    ⇒   ⊢ (𝜑 → (𝐺‘𝐶) ∈ ((𝐺 ∘ 𝐹) limℂ 𝐵))
Theorem	limccnp2lem 14912*	Lemma for limccnp2cntop 14913. This is most of the result, expressed in epsilon-delta form, with a large number of hypotheses so that lengthy expressions do not need to be repeated. (Contributed by Jim Kingdon, 9-Nov-2023.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑅 ∈ 𝑋)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑆 ∈ 𝑌)    &   ⊢ (𝜑 → 𝑋 ⊆ ℂ)    &   ⊢ (𝜑 → 𝑌 ⊆ ℂ)    &   ⊢ 𝐾 = (MetOpen‘(abs ∘ − ))    &   ⊢ 𝐽 = ((𝐾 ×t 𝐾) ↾t (𝑋 × 𝑌))    &   ⊢ (𝜑 → 𝐶 ∈ ((𝑥 ∈ 𝐴 ↦ 𝑅) limℂ 𝐵))    &   ⊢ (𝜑 → 𝐷 ∈ ((𝑥 ∈ 𝐴 ↦ 𝑆) limℂ 𝐵))    &   ⊢ (𝜑 → 𝐻 ∈ ((𝐽 CnP 𝐾)‘⟨𝐶, 𝐷⟩))    &   ⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → 𝐸 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐿 ∈ ℝ+)    &   ⊢ (𝜑 → ∀𝑟 ∈ 𝑋 ∀𝑠 ∈ 𝑌 (((𝐶((abs ∘ − ) ↾ (𝑋 × 𝑋))𝑟) < 𝐿 ∧ (𝐷((abs ∘ − ) ↾ (𝑌 × 𝑌))𝑠) < 𝐿) → ((𝐶𝐻𝐷)(abs ∘ − )(𝑟𝐻𝑠)) < 𝐸))    &   ⊢ (𝜑 → 𝐹 ∈ ℝ+)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ((𝑥 # 𝐵 ∧ (abs‘(𝑥 − 𝐵)) < 𝐹) → (abs‘(𝑅 − 𝐶)) < 𝐿))    &   ⊢ (𝜑 → 𝐺 ∈ ℝ+)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ((𝑥 # 𝐵 ∧ (abs‘(𝑥 − 𝐵)) < 𝐺) → (abs‘(𝑆 − 𝐷)) < 𝐿))    ⇒   ⊢ (𝜑 → ∃𝑑 ∈ ℝ+ ∀𝑥 ∈ 𝐴 ((𝑥 # 𝐵 ∧ (abs‘(𝑥 − 𝐵)) < 𝑑) → (abs‘((𝑅𝐻𝑆) − (𝐶𝐻𝐷))) < 𝐸))
Theorem	limccnp2cntop 14913*	The image of a convergent sequence under a continuous map is convergent to the image of the original point. Binary operation version. (Contributed by Mario Carneiro, 28-Dec-2016.) (Revised by Jim Kingdon, 14-Nov-2023.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑅 ∈ 𝑋)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑆 ∈ 𝑌)    &   ⊢ (𝜑 → 𝑋 ⊆ ℂ)    &   ⊢ (𝜑 → 𝑌 ⊆ ℂ)    &   ⊢ 𝐾 = (MetOpen‘(abs ∘ − ))    &   ⊢ 𝐽 = ((𝐾 ×t 𝐾) ↾t (𝑋 × 𝑌))    &   ⊢ (𝜑 → 𝐶 ∈ ((𝑥 ∈ 𝐴 ↦ 𝑅) limℂ 𝐵))    &   ⊢ (𝜑 → 𝐷 ∈ ((𝑥 ∈ 𝐴 ↦ 𝑆) limℂ 𝐵))    &   ⊢ (𝜑 → 𝐻 ∈ ((𝐽 CnP 𝐾)‘⟨𝐶, 𝐷⟩))    ⇒   ⊢ (𝜑 → (𝐶𝐻𝐷) ∈ ((𝑥 ∈ 𝐴 ↦ (𝑅𝐻𝑆)) limℂ 𝐵))
Theorem	limccoap 14914*	Composition of two limits. This theorem is only usable in the case where 𝑥 # 𝑋 implies R(x) # 𝐶 so it is less general than might appear at first. (Contributed by Mario Carneiro, 29-Dec-2016.) (Revised by Jim Kingdon, 18-Dec-2023.)
⊢ ((𝜑 ∧ 𝑥 ∈ {𝑤 ∈ 𝐴 ∣ 𝑤 # 𝑋}) → 𝑅 ∈ {𝑤 ∈ 𝐵 ∣ 𝑤 # 𝐶})    &   ⊢ ((𝜑 ∧ 𝑦 ∈ {𝑤 ∈ 𝐵 ∣ 𝑤 # 𝐶}) → 𝑆 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ((𝑥 ∈ {𝑤 ∈ 𝐴 ∣ 𝑤 # 𝑋} ↦ 𝑅) limℂ 𝑋))    &   ⊢ (𝜑 → 𝐷 ∈ ((𝑦 ∈ {𝑤 ∈ 𝐵 ∣ 𝑤 # 𝐶} ↦ 𝑆) limℂ 𝐶))    &   ⊢ (𝑦 = 𝑅 → 𝑆 = 𝑇)    ⇒   ⊢ (𝜑 → 𝐷 ∈ ((𝑥 ∈ {𝑤 ∈ 𝐴 ∣ 𝑤 # 𝑋} ↦ 𝑇) limℂ 𝑋))
Theorem	reldvg 14915	The derivative function is a relation. (Contributed by Mario Carneiro, 7-Aug-2014.) (Revised by Jim Kingdon, 25-Jun-2023.)
⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) → Rel (𝑆 D 𝐹))
Theorem	dvlemap 14916*	Closure for a difference quotient. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Jim Kingdon, 27-Jun-2023.)
⊢ (𝜑 → 𝐹:𝐷⟶ℂ)    &   ⊢ (𝜑 → 𝐷 ⊆ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐷)    ⇒   ⊢ ((𝜑 ∧ 𝐴 ∈ {𝑤 ∈ 𝐷 ∣ 𝑤 # 𝐵}) → (((𝐹‘𝐴) − (𝐹‘𝐵)) / (𝐴 − 𝐵)) ∈ ℂ)
Theorem	dvfvalap 14917*	Value and set bounds on the derivative operator. (Contributed by Mario Carneiro, 7-Aug-2014.) (Revised by Jim Kingdon, 27-Jun-2023.)
⊢ 𝑇 = (𝐾 ↾t 𝑆)    &   ⊢ 𝐾 = (MetOpen‘(abs ∘ − ))    ⇒   ⊢ ((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ 𝑆) → ((𝑆 D 𝐹) = ∪ 𝑥 ∈ ((int‘𝑇)‘𝐴)({𝑥} × ((𝑧 ∈ {𝑤 ∈ 𝐴 ∣ 𝑤 # 𝑥} ↦ (((𝐹‘𝑧) − (𝐹‘𝑥)) / (𝑧 − 𝑥))) limℂ 𝑥)) ∧ (𝑆 D 𝐹) ⊆ (((int‘𝑇)‘𝐴) × ℂ)))
Theorem	eldvap 14918*	The differentiable predicate. A function 𝐹 is differentiable at 𝐵 with derivative 𝐶 iff 𝐹 is defined in a neighborhood of 𝐵 and the difference quotient has limit 𝐶 at 𝐵. (Contributed by Mario Carneiro, 7-Aug-2014.) (Revised by Jim Kingdon, 27-Jun-2023.)
⊢ 𝑇 = (𝐾 ↾t 𝑆)    &   ⊢ 𝐾 = (MetOpen‘(abs ∘ − ))    &   ⊢ 𝐺 = (𝑧 ∈ {𝑤 ∈ 𝐴 ∣ 𝑤 # 𝐵} ↦ (((𝐹‘𝑧) − (𝐹‘𝐵)) / (𝑧 − 𝐵)))    &   ⊢ (𝜑 → 𝑆 ⊆ ℂ)    &   ⊢ (𝜑 → 𝐹:𝐴⟶ℂ)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝑆)    ⇒   ⊢ (𝜑 → (𝐵(𝑆 D 𝐹)𝐶 ↔ (𝐵 ∈ ((int‘𝑇)‘𝐴) ∧ 𝐶 ∈ (𝐺 limℂ 𝐵))))
Theorem	dvcl 14919	The derivative function takes values in the complex numbers. (Contributed by Mario Carneiro, 7-Aug-2014.) (Revised by Mario Carneiro, 9-Feb-2015.)
⊢ (𝜑 → 𝑆 ⊆ ℂ)    &   ⊢ (𝜑 → 𝐹:𝐴⟶ℂ)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝑆)    ⇒   ⊢ ((𝜑 ∧ 𝐵(𝑆 D 𝐹)𝐶) → 𝐶 ∈ ℂ)
Theorem	dvbssntrcntop 14920	The set of differentiable points is a subset of the interior of the domain of the function. (Contributed by Mario Carneiro, 7-Aug-2014.) (Revised by Jim Kingdon, 27-Jun-2023.)
⊢ (𝜑 → 𝑆 ⊆ ℂ)    &   ⊢ (𝜑 → 𝐹:𝐴⟶ℂ)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝑆)    &   ⊢ 𝐽 = (𝐾 ↾t 𝑆)    &   ⊢ 𝐾 = (MetOpen‘(abs ∘ − ))    ⇒   ⊢ (𝜑 → dom (𝑆 D 𝐹) ⊆ ((int‘𝐽)‘𝐴))
Theorem	dvbss 14921	The set of differentiable points is a subset of the domain of the function. (Contributed by Mario Carneiro, 6-Aug-2014.) (Revised by Mario Carneiro, 9-Feb-2015.)
⊢ (𝜑 → 𝑆 ⊆ ℂ)    &   ⊢ (𝜑 → 𝐹:𝐴⟶ℂ)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝑆)    ⇒   ⊢ (𝜑 → dom (𝑆 D 𝐹) ⊆ 𝐴)
Theorem	dvbsssg 14922	The set of differentiable points is a subset of the ambient topology. (Contributed by Mario Carneiro, 18-Mar-2015.) (Revised by Jim Kingdon, 28-Jun-2023.)
⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) → dom (𝑆 D 𝐹) ⊆ 𝑆)
Theorem	recnprss 14923	Both ℝ and ℂ are subsets of ℂ. (Contributed by Mario Carneiro, 10-Feb-2015.)
⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ)
Theorem	dvfgg 14924	Explicitly write out the functionality condition on derivative for 𝑆 = ℝ and ℂ. (Contributed by Mario Carneiro, 9-Feb-2015.) (Revised by Jim Kingdon, 28-Jun-2023.)
⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) → (𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ)
Theorem	dvfpm 14925	The derivative is a function. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Jim Kingdon, 28-Jul-2023.)
⊢ (𝐹 ∈ (ℂ ↑pm ℝ) → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℂ)
Theorem	dvfcnpm 14926	The derivative is a function. (Contributed by Mario Carneiro, 9-Feb-2015.) (Revised by Jim Kingdon, 28-Jul-2023.)
⊢ (𝐹 ∈ (ℂ ↑pm ℂ) → (ℂ D 𝐹):dom (ℂ D 𝐹)⟶ℂ)
Theorem	dvidlemap 14927*	Lemma for dvid 14931 and dvconst 14930. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Jim Kingdon, 2-Aug-2023.)
⊢ (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	dvidrelem 14928*	Lemma for dvidre 14933 and dvconstre 14932. Analogue of dvidlemap 14927 for real numbers rather than complex numbers. (Contributed by Jim Kingdon, 3-Oct-2025.)
⊢ (𝜑 → 𝐹:ℝ⟶ℂ)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ 𝑧 # 𝑥)) → (((𝐹‘𝑧) − (𝐹‘𝑥)) / (𝑧 − 𝑥)) = 𝐵)    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ (𝜑 → (ℝ D 𝐹) = (ℝ × {𝐵}))
Theorem	dvidsslem 14929*	Lemma for dvconstss 14934. Analogue of dvidlemap 14927 where 𝐹 is defined on an open subset of the real or complex numbers. (Contributed by Jim Kingdon, 3-Oct-2025.)
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})    &   ⊢ 𝐽 = (𝐾 ↾t 𝑆)    &   ⊢ 𝐾 = (MetOpen‘(abs ∘ − ))    &   ⊢ (𝜑 → 𝐹:𝑋⟶ℂ)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐽)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝑧 # 𝑥)) → (((𝐹‘𝑧) − (𝐹‘𝑥)) / (𝑧 − 𝑥)) = 𝐵)    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ (𝜑 → (𝑆 D 𝐹) = (𝑋 × {𝐵}))
Theorem	dvconst 14930	Derivative of a constant function. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Jim Kingdon, 2-Aug-2023.)
⊢ (𝐴 ∈ ℂ → (ℂ D (ℂ × {𝐴})) = (ℂ × {0}))
Theorem	dvid 14931	Derivative of the identity function. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Jim Kingdon, 2-Aug-2023.)
⊢ (ℂ D ( I ↾ ℂ)) = (ℂ × {1})
Theorem	dvconstre 14932	Real derivative of a constant function. (Contributed by Jim Kingdon, 3-Oct-2025.)
⊢ (𝐴 ∈ ℂ → (ℝ D (ℝ × {𝐴})) = (ℝ × {0}))
Theorem	dvidre 14933	Real derivative of the identity function. (Contributed by Jim Kingdon, 3-Oct-2025.)
⊢ (ℝ D ( I ↾ ℝ)) = (ℝ × {1})
Theorem	dvconstss 14934	Derivative of a constant function defined on an open set. (Contributed by Jim Kingdon, 6-Oct-2025.)
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})    &   ⊢ 𝐽 = (𝐾 ↾t 𝑆)    &   ⊢ 𝐾 = (MetOpen‘(abs ∘ − ))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐽)    &   ⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝑆 D (𝑋 × {𝐴})) = (𝑋 × {0}))
Theorem	dvcnp2cntop 14935	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.)
⊢ 𝐽 = (𝐾 ↾t 𝐴)    &   ⊢ 𝐾 = (MetOpen‘(abs ∘ − ))    ⇒   ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ 𝑆) ∧ 𝐵 ∈ dom (𝑆 D 𝐹)) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐵))
Theorem	dvcn 14936	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	dvaddxxbr 14937	The sum rule for derivatives at a point. That is, if the derivative of 𝐹 at 𝐶 is 𝐾 and the derivative of 𝐺 at 𝐶 is 𝐿, then the derivative of the pointwise sum of those two functions at 𝐶 is 𝐾 + 𝐿. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Jim Kingdon, 25-Nov-2023.)
⊢ (𝜑 → 𝐹:𝑋⟶ℂ)    &   ⊢ (𝜑 → 𝑋 ⊆ 𝑆)    &   ⊢ (𝜑 → 𝐺:𝑋⟶ℂ)    &   ⊢ (𝜑 → 𝑆 ⊆ ℂ)    &   ⊢ (𝜑 → 𝐶(𝑆 D 𝐹)𝐾)    &   ⊢ (𝜑 → 𝐶(𝑆 D 𝐺)𝐿)    &   ⊢ 𝐽 = (MetOpen‘(abs ∘ − ))    ⇒   ⊢ (𝜑 → 𝐶(𝑆 D (𝐹 ∘𝑓 + 𝐺))(𝐾 + 𝐿))
Theorem	dvmulxxbr 14938	The product rule for derivatives at a point. For the (simpler but more limited) function version, see dvmulxx 14940. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Jim Kingdon, 1-Dec-2023.)
⊢ (𝜑 → 𝐹:𝑋⟶ℂ)    &   ⊢ (𝜑 → 𝑋 ⊆ 𝑆)    &   ⊢ (𝜑 → 𝐺:𝑋⟶ℂ)    &   ⊢ (𝜑 → 𝑆 ⊆ ℂ)    &   ⊢ (𝜑 → 𝐶(𝑆 D 𝐹)𝐾)    &   ⊢ (𝜑 → 𝐶(𝑆 D 𝐺)𝐿)    &   ⊢ 𝐽 = (MetOpen‘(abs ∘ − ))    ⇒   ⊢ (𝜑 → 𝐶(𝑆 D (𝐹 ∘𝑓 · 𝐺))((𝐾 · (𝐺‘𝐶)) + (𝐿 · (𝐹‘𝐶))))
Theorem	dvaddxx 14939	The sum rule for derivatives at a point. For the (more general) relation version, see dvaddxxbr 14937. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Jim Kingdon, 25-Nov-2023.)
⊢ (𝜑 → 𝐹:𝑋⟶ℂ)    &   ⊢ (𝜑 → 𝑋 ⊆ 𝑆)    &   ⊢ (𝜑 → 𝐺:𝑋⟶ℂ)    &   ⊢ (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	dvmulxx 14940	The product rule for derivatives at a point. For the (more general) relation version, see dvmulxxbr 14938. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Jim Kingdon, 2-Dec-2023.)
⊢ (𝜑 → 𝐹:𝑋⟶ℂ)    &   ⊢ (𝜑 → 𝑋 ⊆ 𝑆)    &   ⊢ (𝜑 → 𝐺:𝑋⟶ℂ)    &   ⊢ (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	dviaddf 14941	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 → 𝑋 ⊆ 𝑆)    &   ⊢ (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	dvimulf 14942	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 → 𝑋 ⊆ 𝑆)    &   ⊢ (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	dvcoapbr 14943*	The chain rule for derivatives at a point. The 𝑢 # 𝐶 → (𝐺‘𝑢) # (𝐺‘𝐶) hypothesis constrains what functions work for 𝐺. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Jim Kingdon, 21-Dec-2023.)
⊢ (𝜑 → 𝐹:𝑋⟶ℂ)    &   ⊢ (𝜑 → 𝑋 ⊆ 𝑆)    &   ⊢ (𝜑 → 𝐺:𝑌⟶𝑋)    &   ⊢ (𝜑 → 𝑌 ⊆ 𝑇)    &   ⊢ (𝜑 → ∀𝑢 ∈ 𝑌 (𝑢 # 𝐶 → (𝐺‘𝑢) # (𝐺‘𝐶)))    &   ⊢ (𝜑 → 𝑆 ⊆ ℂ)    &   ⊢ (𝜑 → 𝑇 ⊆ ℂ)    &   ⊢ (𝜑 → (𝐺‘𝐶)(𝑆 D 𝐹)𝐾)    &   ⊢ (𝜑 → 𝐶(𝑇 D 𝐺)𝐿)    &   ⊢ 𝐽 = (MetOpen‘(abs ∘ − ))    ⇒   ⊢ (𝜑 → 𝐶(𝑇 D (𝐹 ∘ 𝐺))(𝐾 · 𝐿))
Theorem	dvcjbr 14944	The derivative of the conjugate of a function. For the (simpler but more limited) function version, see dvcj 14945. (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 14945	The derivative of the conjugate of a function. For the (more general) relation version, see dvcjbr 14944. (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 14946	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	dvexp 14947*	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 14948*	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	dvrecap 14949*	Derivative of the reciprocal function. (Contributed by Mario Carneiro, 25-Feb-2015.) (Revised by Mario Carneiro, 28-Dec-2016.)
⊢ (𝐴 ∈ ℂ → (ℂ D (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ (𝐴 / 𝑥))) = (𝑥 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ↦ -(𝐴 / (𝑥↑2))))
Theorem	dvmptidcn 14950	Function-builder for derivative: derivative of the identity. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Jim Kingdon, 30-Dec-2023.)
⊢ (ℂ D (𝑥 ∈ ℂ ↦ 𝑥)) = (𝑥 ∈ ℂ ↦ 1)
Theorem	dvmptccn 14951*	Function-builder for derivative: derivative of a constant. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Jim Kingdon, 30-Dec-2023.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (ℂ D (𝑥 ∈ ℂ ↦ 𝐴)) = (𝑥 ∈ ℂ ↦ 0))
Theorem	dvmptid 14952*	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 14953*	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	dvmptclx 14954*	Closure lemma for dvmptmulx 14956 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 ⊆ 𝑆)    ⇒   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ ℂ)
Theorem	dvmptaddx 14955*	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 ⊆ 𝑆)    &   ⊢ ((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	dvmptmulx 14956*	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 ⊆ 𝑆)    &   ⊢ ((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	dvmptcmulcn 14957*	Function-builder for derivative, product rule for constant multiplier. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Jim Kingdon, 31-Dec-2023.)
⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝐴 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → (ℂ D (𝑥 ∈ ℂ ↦ 𝐴)) = (𝑥 ∈ ℂ ↦ 𝐵))    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    ⇒   ⊢ (𝜑 → (ℂ D (𝑥 ∈ ℂ ↦ (𝐶 · 𝐴))) = (𝑥 ∈ ℂ ↦ (𝐶 · 𝐵)))
Theorem	dvmptnegcn 14958*	Function-builder for derivative, product rule for negatives. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Jim Kingdon, 31-Dec-2023.)
⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝐴 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → (ℂ D (𝑥 ∈ ℂ ↦ 𝐴)) = (𝑥 ∈ ℂ ↦ 𝐵))    ⇒   ⊢ (𝜑 → (ℂ D (𝑥 ∈ ℂ ↦ -𝐴)) = (𝑥 ∈ ℂ ↦ -𝐵))
Theorem	dvmptsubcn 14959*	Function-builder for derivative, subtraction rule. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Jim Kingdon, 31-Dec-2023.)
⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝐴 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → (ℂ D (𝑥 ∈ ℂ ↦ 𝐴)) = (𝑥 ∈ ℂ ↦ 𝐵))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝐶 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝐷 ∈ 𝑊)    &   ⊢ (𝜑 → (ℂ D (𝑥 ∈ ℂ ↦ 𝐶)) = (𝑥 ∈ ℂ ↦ 𝐷))    ⇒   ⊢ (𝜑 → (ℂ D (𝑥 ∈ ℂ ↦ (𝐴 − 𝐶))) = (𝑥 ∈ ℂ ↦ (𝐵 − 𝐷)))
Theorem	dvmptcjx 14960*	Function-builder for derivative, conjugate rule. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Jim Kingdon, 24-May-2024.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → (ℝ D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐵))    &   ⊢ (𝜑 → 𝑋 ⊆ ℝ)    ⇒   ⊢ (𝜑 → (ℝ D (𝑥 ∈ 𝑋 ↦ (∗‘𝐴))) = (𝑥 ∈ 𝑋 ↦ (∗‘𝐵)))
Theorem	dvmptfsum 14961*	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	dveflem 14962	Derivative of the exponential function at 0. The key step in the proof is eftlub 11855, 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 14963	Derivative of the exponential function. (Contributed by Mario Carneiro, 9-Aug-2014.) (Proof shortened by Mario Carneiro, 10-Feb-2015.)
⊢ (ℂ D exp) = exp
PART 11  BASIC REAL AND COMPLEX FUNCTIONS
11.1  Polynomials
11.1.1  Elementary properties of complex polynomials
Syntax	cply 14964	Extend class notation to include the set of complex polynomials.
class Poly
Syntax	cidp 14965	Extend class notation to include the identity polynomial.
class Xp
Definition	df-ply 14966*	Define the set of polynomials on the complex numbers with coefficients in the given subset. (Contributed by Mario Carneiro, 17-Jul-2014.)
⊢ Poly = (𝑥 ∈ 𝒫 ℂ ↦ {𝑓 ∣ ∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ((𝑥 ∪ {0}) ↑𝑚 ℕ0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))})
Definition	df-idp 14967	Define the identity polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)
⊢ Xp = ( I ↾ ℂ)
Theorem	plyval 14968*	Value of the polynomial set function. (Contributed by Mario Carneiro, 17-Jul-2014.)
⊢ (𝑆 ⊆ ℂ → (Poly‘𝑆) = {𝑓 ∣ ∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))})
Theorem	plybss 14969	Reverse closure of the parameter 𝑆 of the polynomial set function. (Contributed by Mario Carneiro, 22-Jul-2014.)
⊢ (𝐹 ∈ (Poly‘𝑆) → 𝑆 ⊆ ℂ)
Theorem	elply 14970*	Definition of a polynomial with coefficients in 𝑆. (Contributed by Mario Carneiro, 17-Jul-2014.)
⊢ (𝐹 ∈ (Poly‘𝑆) ↔ (𝑆 ⊆ ℂ ∧ ∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0)𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))))
Theorem	elply2 14971*	The coefficient function can be assumed to have zeroes outside 0...𝑛. (Contributed by Mario Carneiro, 20-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
⊢ (𝐹 ∈ (Poly‘𝑆) ↔ (𝑆 ⊆ ℂ ∧ ∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0)((𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))))))
Theorem	plyun0 14972	The set of polynomials is unaffected by the addition of zero. (This is built into the definition because all higher powers of a polynomial are effectively zero, so we require that the coefficient field contain zero to simplify some of our closure theorems.) (Contributed by Mario Carneiro, 17-Jul-2014.)
⊢ (Poly‘(𝑆 ∪ {0})) = (Poly‘𝑆)
Theorem	plyf 14973	A polynomial is a function on the complex numbers. (Contributed by Mario Carneiro, 22-Jul-2014.)
⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ)
Theorem	plyss 14974	The polynomial set function preserves the subset relation. (Contributed by Mario Carneiro, 17-Jul-2014.)
⊢ ((𝑆 ⊆ 𝑇 ∧ 𝑇 ⊆ ℂ) → (Poly‘𝑆) ⊆ (Poly‘𝑇))
Theorem	plyssc 14975	Every polynomial ring is contained in the ring of polynomials over ℂ. (Contributed by Mario Carneiro, 22-Jul-2014.)
⊢ (Poly‘𝑆) ⊆ (Poly‘ℂ)
Theorem	elplyr 14976*	Sufficient condition for elementhood in the set of polynomials. (Contributed by Mario Carneiro, 17-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
⊢ ((𝑆 ⊆ ℂ ∧ 𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶𝑆) → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · (𝑧↑𝑘))) ∈ (Poly‘𝑆))
Theorem	elplyd 14977*	Sufficient condition for elementhood in the set of polynomials. (Contributed by Mario Carneiro, 17-Jul-2014.)
⊢ (𝜑 → 𝑆 ⊆ ℂ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝐴 ∈ 𝑆)    ⇒   ⊢ (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)(𝐴 · (𝑧↑𝑘))) ∈ (Poly‘𝑆))
Theorem	ply1termlem 14978*	Lemma for ply1term 14979. (Contributed by Mario Carneiro, 26-Jul-2014.)
⊢ 𝐹 = (𝑧 ∈ ℂ ↦ (𝐴 · (𝑧↑𝑁)))    ⇒   ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)(if(𝑘 = 𝑁, 𝐴, 0) · (𝑧↑𝑘))))
Theorem	ply1term 14979*	A one-term polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)
⊢ 𝐹 = (𝑧 ∈ ℂ ↦ (𝐴 · (𝑧↑𝑁)))    ⇒   ⊢ ((𝑆 ⊆ ℂ ∧ 𝐴 ∈ 𝑆 ∧ 𝑁 ∈ ℕ0) → 𝐹 ∈ (Poly‘𝑆))
Theorem	plypow 14980*	A power is a polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)
⊢ ((𝑆 ⊆ ℂ ∧ 1 ∈ 𝑆 ∧ 𝑁 ∈ ℕ0) → (𝑧 ∈ ℂ ↦ (𝑧↑𝑁)) ∈ (Poly‘𝑆))
Theorem	plyconst 14981	A constant function is a polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)
⊢ ((𝑆 ⊆ ℂ ∧ 𝐴 ∈ 𝑆) → (ℂ × {𝐴}) ∈ (Poly‘𝑆))
Theorem	plyid 14982	The identity function is a polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)
⊢ ((𝑆 ⊆ ℂ ∧ 1 ∈ 𝑆) → Xp ∈ (Poly‘𝑆))
Theorem	plyaddlem1 14983*	Derive the coefficient function for the sum of two polynomials. (Contributed by Mario Carneiro, 23-Jul-2014.)
⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆))    &   ⊢ (𝜑 → 𝐺 ∈ (Poly‘𝑆))    &   ⊢ (𝜑 → 𝑀 ∈ ℕ0)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐴:ℕ0⟶ℂ)    &   ⊢ (𝜑 → 𝐵:ℕ0⟶ℂ)    &   ⊢ (𝜑 → (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0})    &   ⊢ (𝜑 → (𝐵 “ (ℤ≥‘(𝑁 + 1))) = {0})    &   ⊢ (𝜑 → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑧↑𝑘))))    &   ⊢ (𝜑 → 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐵‘𝑘) · (𝑧↑𝑘))))    ⇒   ⊢ (𝜑 → (𝐹 ∘𝑓 + 𝐺) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀))(((𝐴 ∘𝑓 + 𝐵)‘𝑘) · (𝑧↑𝑘))))
Theorem	plymullem1 14984*	Derive the coefficient function for the product of two polynomials. (Contributed by Mario Carneiro, 23-Jul-2014.)
⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆))    &   ⊢ (𝜑 → 𝐺 ∈ (Poly‘𝑆))    &   ⊢ (𝜑 → 𝑀 ∈ ℕ0)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐴:ℕ0⟶ℂ)    &   ⊢ (𝜑 → 𝐵:ℕ0⟶ℂ)    &   ⊢ (𝜑 → (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0})    &   ⊢ (𝜑 → (𝐵 “ (ℤ≥‘(𝑁 + 1))) = {0})    &   ⊢ (𝜑 → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑧↑𝑘))))    &   ⊢ (𝜑 → 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐵‘𝑘) · (𝑧↑𝑘))))    ⇒   ⊢ (𝜑 → (𝐹 ∘𝑓 · 𝐺) = (𝑧 ∈ ℂ ↦ Σ𝑛 ∈ (0...(𝑀 + 𝑁))(Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘))) · (𝑧↑𝑛))))
Theorem	plyaddlem 14985*	Lemma for plyadd 14987. (Contributed by Mario Carneiro, 21-Jul-2014.)
⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆))    &   ⊢ (𝜑 → 𝐺 ∈ (Poly‘𝑆))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ⊢ (𝜑 → 𝑀 ∈ ℕ0)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐴 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0))    &   ⊢ (𝜑 → 𝐵 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0))    &   ⊢ (𝜑 → (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0})    &   ⊢ (𝜑 → (𝐵 “ (ℤ≥‘(𝑁 + 1))) = {0})    &   ⊢ (𝜑 → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑧↑𝑘))))    &   ⊢ (𝜑 → 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐵‘𝑘) · (𝑧↑𝑘))))    ⇒   ⊢ (𝜑 → (𝐹 ∘𝑓 + 𝐺) ∈ (Poly‘𝑆))
Theorem	plymullem 14986*	Lemma for plymul 14988. (Contributed by Mario Carneiro, 21-Jul-2014.)
⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆))    &   ⊢ (𝜑 → 𝐺 ∈ (Poly‘𝑆))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ⊢ (𝜑 → 𝑀 ∈ ℕ0)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐴 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0))    &   ⊢ (𝜑 → 𝐵 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0))    &   ⊢ (𝜑 → (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0})    &   ⊢ (𝜑 → (𝐵 “ (ℤ≥‘(𝑁 + 1))) = {0})    &   ⊢ (𝜑 → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑧↑𝑘))))    &   ⊢ (𝜑 → 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐵‘𝑘) · (𝑧↑𝑘))))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 · 𝑦) ∈ 𝑆)    ⇒   ⊢ (𝜑 → (𝐹 ∘𝑓 · 𝐺) ∈ (Poly‘𝑆))
Theorem	plyadd 14987*	The sum of two polynomials is a polynomial. (Contributed by Mario Carneiro, 21-Jul-2014.)
⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆))    &   ⊢ (𝜑 → 𝐺 ∈ (Poly‘𝑆))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    ⇒   ⊢ (𝜑 → (𝐹 ∘𝑓 + 𝐺) ∈ (Poly‘𝑆))
Theorem	plymul 14988*	The product of two polynomials is a polynomial. (Contributed by Mario Carneiro, 21-Jul-2014.)
⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆))    &   ⊢ (𝜑 → 𝐺 ∈ (Poly‘𝑆))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 · 𝑦) ∈ 𝑆)    ⇒   ⊢ (𝜑 → (𝐹 ∘𝑓 · 𝐺) ∈ (Poly‘𝑆))
Theorem	plysub 14989*	The difference of two polynomials is a polynomial. (Contributed by Mario Carneiro, 21-Jul-2014.)
⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆))    &   ⊢ (𝜑 → 𝐺 ∈ (Poly‘𝑆))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 · 𝑦) ∈ 𝑆)    &   ⊢ (𝜑 → -1 ∈ 𝑆)    ⇒   ⊢ (𝜑 → (𝐹 ∘𝑓 − 𝐺) ∈ (Poly‘𝑆))
Theorem	plyaddcl 14990	The sum of two polynomials is a polynomial. (Contributed by Mario Carneiro, 24-Jul-2014.)
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹 ∘𝑓 + 𝐺) ∈ (Poly‘ℂ))
Theorem	plymulcl 14991	The product of two polynomials is a polynomial. (Contributed by Mario Carneiro, 24-Jul-2014.)
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹 ∘𝑓 · 𝐺) ∈ (Poly‘ℂ))
Theorem	plysubcl 14992	The difference of two polynomials is a polynomial. (Contributed by Mario Carneiro, 24-Jul-2014.)
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹 ∘𝑓 − 𝐺) ∈ (Poly‘ℂ))
Theorem	plycoeid3 14993*	Reconstruct a polynomial as an explicit sum of the coefficient function up to an index no smaller than the degree of the polynomial. (Contributed by Jim Kingdon, 17-Oct-2025.)
⊢ (𝜑 → 𝐷 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐴:ℕ0⟶ℂ)    &   ⊢ (𝜑 → (𝐴 “ (ℤ≥‘(𝐷 + 1))) = {0})    &   ⊢ (𝜑 → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝐷)((𝐴‘𝑘) · (𝑧↑𝑘))))    &   ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘𝐷))    &   ⊢ (𝜑 → 𝑋 ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝐹‘𝑋) = Σ𝑗 ∈ (0...𝑀)((𝐴‘𝑗) · (𝑋↑𝑗)))
Theorem	plycolemc 14994*	Lemma for plyco 14995. The result expressed as a sum, with a degree and coefficients for 𝐹 specified as hypotheses. (Contributed by Jim Kingdon, 20-Sep-2025.)
⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆))    &   ⊢ (𝜑 → 𝐺 ∈ (Poly‘𝑆))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 · 𝑦) ∈ 𝑆)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐴:ℕ0⟶(𝑆 ∪ {0}))    &   ⊢ (𝜑 → (𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0})    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · (𝑥↑𝑘))))    ⇒   ⊢ (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · ((𝐺‘𝑧)↑𝑘))) ∈ (Poly‘𝑆))
Theorem	plyco 14995*	The composition of two polynomials is a polynomial. (Contributed by Mario Carneiro, 23-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆))    &   ⊢ (𝜑 → 𝐺 ∈ (Poly‘𝑆))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 · 𝑦) ∈ 𝑆)    ⇒   ⊢ (𝜑 → (𝐹 ∘ 𝐺) ∈ (Poly‘𝑆))
Theorem	plycjlemc 14996*	Lemma for plycj 14997. (Contributed by Mario Carneiro, 24-Jul-2014.) (Revised by Jim Kingdon, 22-Sep-2025.)
⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ 𝐺 = ((∗ ∘ 𝐹) ∘ ∗)    &   ⊢ (𝜑 → 𝐴:ℕ0⟶(𝑆 ∪ {0}))    &   ⊢ (𝜑 → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · (𝑧↑𝑘))))    &   ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆))    ⇒   ⊢ (𝜑 → 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)(((∗ ∘ 𝐴)‘𝑘) · (𝑧↑𝑘))))
Theorem	plycj 14997*	The double conjugation of a polynomial is a polynomial. (The single conjugation is not because our definition of polynomial includes only holomorphic functions, i.e. no dependence on (∗‘𝑧) independently of 𝑧.) (Contributed by Mario Carneiro, 24-Jul-2014.)
⊢ 𝐺 = ((∗ ∘ 𝐹) ∘ ∗)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (∗‘𝑥) ∈ 𝑆)    &   ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆))    ⇒   ⊢ (𝜑 → 𝐺 ∈ (Poly‘𝑆))
Theorem	plycn 14998	A polynomial is a continuous function. (Contributed by Mario Carneiro, 23-Jul-2014.) Avoid ax-mulf 8002. (Revised by GG, 16-Mar-2025.)
⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹 ∈ (ℂ–cn→ℂ))
Theorem	plyrecj 14999	A polynomial with real coefficients distributes under conjugation. (Contributed by Mario Carneiro, 24-Jul-2014.)
⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) → (∗‘(𝐹‘𝐴)) = (𝐹‘(∗‘𝐴)))
Theorem	plyreres 15000	Real-coefficient polynomials restrict to real functions. (Contributed by Stefan O'Rear, 16-Nov-2014.)
⊢ (𝐹 ∈ (Poly‘ℝ) → (𝐹 ↾ ℝ):ℝ⟶ℝ)