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Theorem List for Intuitionistic Logic Explorer - 12901-13000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremrexmet 12901 The absolute value metric is an extended metric. (Contributed by Mario Carneiro, 28-Aug-2015.)
𝐷 = ((abs ∘ − ) ↾ (ℝ × ℝ))       𝐷 ∈ (∞Met‘ℝ)

Theorembl2ioo 12902 A ball in terms of an open interval of reals. (Contributed by NM, 18-May-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)
𝐷 = ((abs ∘ − ) ↾ (ℝ × ℝ))       ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴(ball‘𝐷)𝐵) = ((𝐴𝐵)(,)(𝐴 + 𝐵)))

Theoremioo2bl 12903 An open interval of reals in terms of a ball. (Contributed by NM, 18-May-2007.) (Revised by Mario Carneiro, 28-Aug-2015.)
𝐷 = ((abs ∘ − ) ↾ (ℝ × ℝ))       ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴(,)𝐵) = (((𝐴 + 𝐵) / 2)(ball‘𝐷)((𝐵𝐴) / 2)))

Theoremioo2blex 12904 An open interval of reals in terms of a ball. (Contributed by Mario Carneiro, 14-Nov-2013.)
𝐷 = ((abs ∘ − ) ↾ (ℝ × ℝ))       ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴(,)𝐵) ∈ ran (ball‘𝐷))

Theoremblssioo 12905 The balls of the standard real metric space are included in the open real intervals. (Contributed by NM, 8-May-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)
𝐷 = ((abs ∘ − ) ↾ (ℝ × ℝ))       ran (ball‘𝐷) ⊆ ran (,)

Theoremtgioo 12906 The topology generated by open intervals of reals is the same as the open sets of the standard metric space on the reals. (Contributed by NM, 7-May-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)
𝐷 = ((abs ∘ − ) ↾ (ℝ × ℝ))    &   𝐽 = (MetOpen‘𝐷)       (topGen‘ran (,)) = 𝐽

Theoremtgqioo 12907 The topology generated by open intervals of reals with rational endpoints is the same as the open sets of the standard metric space on the reals. In particular, this proves that the standard topology on the reals is second-countable. (Contributed by Mario Carneiro, 17-Jun-2014.)
𝑄 = (topGen‘((,) “ (ℚ × ℚ)))       (topGen‘ran (,)) = 𝑄

Theoremresubmet 12908 The subspace topology induced by a subset of the reals. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 13-Aug-2014.)
𝑅 = (topGen‘ran (,))    &   𝐽 = (MetOpen‘((abs ∘ − ) ↾ (𝐴 × 𝐴)))       (𝐴 ⊆ ℝ → 𝐽 = (𝑅t 𝐴))

Theoremtgioo2cntop 12909 The standard topology on the reals is a subspace of the complex metric topology. (Contributed by Mario Carneiro, 13-Aug-2014.) (Revised by Jim Kingdon, 6-Aug-2023.)
𝐽 = (MetOpen‘(abs ∘ − ))       (topGen‘ran (,)) = (𝐽t ℝ)

Theoremrerestcntop 12910 The subspace topology induced by a subset of the reals. (Contributed by Mario Carneiro, 13-Aug-2014.) (Revised by Jim Kingdon, 6-Aug-2023.)
𝐽 = (MetOpen‘(abs ∘ − ))    &   𝑅 = (topGen‘ran (,))       (𝐴 ⊆ ℝ → (𝐽t 𝐴) = (𝑅t 𝐴))

Theoremaddcncntoplem 12911* Lemma for addcncntop 12912, subcncntop 12913, and mulcncntop 12914. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Jim Kingdon, 22-Oct-2023.)
𝐽 = (MetOpen‘(abs ∘ − ))    &    + :(ℂ × ℂ)⟶ℂ    &   ((𝑎 ∈ ℝ+𝑏 ∈ ℂ ∧ 𝑐 ∈ ℂ) → ∃𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑢 ∈ ℂ ∀𝑣 ∈ ℂ (((abs‘(𝑢𝑏)) < 𝑦 ∧ (abs‘(𝑣𝑐)) < 𝑧) → (abs‘((𝑢 + 𝑣) − (𝑏 + 𝑐))) < 𝑎))        + ∈ ((𝐽 ×t 𝐽) Cn 𝐽)

Theoremaddcncntop 12912 Complex number addition is a continuous function. Part of Proposition 14-4.16 of [Gleason] p. 243. (Contributed by NM, 30-Jul-2007.) (Proof shortened by Mario Carneiro, 5-May-2014.)
𝐽 = (MetOpen‘(abs ∘ − ))        + ∈ ((𝐽 ×t 𝐽) Cn 𝐽)

Theoremsubcncntop 12913 Complex number subtraction is a continuous function. Part of Proposition 14-4.16 of [Gleason] p. 243. (Contributed by NM, 4-Aug-2007.) (Proof shortened by Mario Carneiro, 5-May-2014.)
𝐽 = (MetOpen‘(abs ∘ − ))        − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)

Theoremmulcncntop 12914 Complex number multiplication is a continuous function. Part of Proposition 14-4.16 of [Gleason] p. 243. (Contributed by NM, 30-Jul-2007.) (Proof shortened by Mario Carneiro, 5-May-2014.)
𝐽 = (MetOpen‘(abs ∘ − ))        · ∈ ((𝐽 ×t 𝐽) Cn 𝐽)

Theoremdivcnap 12915* Complex number division is a continuous function, when the second argument is apart from zero. (Contributed by Mario Carneiro, 12-Aug-2014.) (Revised by Jim Kingdon, 25-Oct-2023.)
𝐽 = (MetOpen‘(abs ∘ − ))    &   𝐾 = (𝐽t {𝑥 ∈ ℂ ∣ 𝑥 # 0})       (𝑦 ∈ ℂ, 𝑧 ∈ {𝑥 ∈ ℂ ∣ 𝑥 # 0} ↦ (𝑦 / 𝑧)) ∈ ((𝐽 ×t 𝐾) Cn 𝐽)

Theoremfsumcncntop 12916* A finite sum of functions to complex numbers from a common topological space is continuous. The class expression for 𝐵 normally contains free variables 𝑘 and 𝑥 to index it. (Contributed by NM, 8-Aug-2007.) (Revised by Mario Carneiro, 23-Aug-2014.)
𝐾 = (MetOpen‘(abs ∘ − ))    &   (𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → (𝑥𝑋𝐵) ∈ (𝐽 Cn 𝐾))       (𝜑 → (𝑥𝑋 ↦ Σ𝑘𝐴 𝐵) ∈ (𝐽 Cn 𝐾))

7.2.7  Topological definitions using the reals

Syntaxccncf 12917 Extend class notation to include the operation which returns a class of continuous complex functions.
class cn

Definitiondf-cncf 12918* Define the operation whose value is a class of continuous complex functions. (Contributed by Paul Chapman, 11-Oct-2007.)
cn→ = (𝑎 ∈ 𝒫 ℂ, 𝑏 ∈ 𝒫 ℂ ↦ {𝑓 ∈ (𝑏𝑚 𝑎) ∣ ∀𝑥𝑎𝑒 ∈ ℝ+𝑑 ∈ ℝ+𝑦𝑎 ((abs‘(𝑥𝑦)) < 𝑑 → (abs‘((𝑓𝑥) − (𝑓𝑦))) < 𝑒)})

Theoremcncfval 12919* The value of the continuous complex function operation is the set of continuous functions from 𝐴 to 𝐵. (Contributed by Paul Chapman, 11-Oct-2007.) (Revised by Mario Carneiro, 9-Nov-2013.)
((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → (𝐴cn𝐵) = {𝑓 ∈ (𝐵𝑚 𝐴) ∣ ∀𝑥𝐴𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤𝐴 ((abs‘(𝑥𝑤)) < 𝑧 → (abs‘((𝑓𝑥) − (𝑓𝑤))) < 𝑦)})

Theoremelcncf 12920* Membership in the set of continuous complex functions from 𝐴 to 𝐵. (Contributed by Paul Chapman, 11-Oct-2007.) (Revised by Mario Carneiro, 9-Nov-2013.)
((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → (𝐹 ∈ (𝐴cn𝐵) ↔ (𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤𝐴 ((abs‘(𝑥𝑤)) < 𝑧 → (abs‘((𝐹𝑥) − (𝐹𝑤))) < 𝑦))))

Theoremelcncf2 12921* Version of elcncf 12920 with arguments commuted. (Contributed by Mario Carneiro, 28-Apr-2014.)
((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → (𝐹 ∈ (𝐴cn𝐵) ↔ (𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤𝐴 ((abs‘(𝑤𝑥)) < 𝑧 → (abs‘((𝐹𝑤) − (𝐹𝑥))) < 𝑦))))

Theoremcncfrss 12922 Reverse closure of the continuous function predicate. (Contributed by Mario Carneiro, 25-Aug-2014.)
(𝐹 ∈ (𝐴cn𝐵) → 𝐴 ⊆ ℂ)

Theoremcncfrss2 12923 Reverse closure of the continuous function predicate. (Contributed by Mario Carneiro, 25-Aug-2014.)
(𝐹 ∈ (𝐴cn𝐵) → 𝐵 ⊆ ℂ)

Theoremcncff 12924 A continuous complex function's domain and codomain. (Contributed by Paul Chapman, 17-Jan-2008.) (Revised by Mario Carneiro, 25-Aug-2014.)
(𝐹 ∈ (𝐴cn𝐵) → 𝐹:𝐴𝐵)

Theoremcncfi 12925* Defining property of a continuous function. (Contributed by Mario Carneiro, 30-Apr-2014.) (Revised by Mario Carneiro, 25-Aug-2014.)
((𝐹 ∈ (𝐴cn𝐵) ∧ 𝐶𝐴𝑅 ∈ ℝ+) → ∃𝑧 ∈ ℝ+𝑤𝐴 ((abs‘(𝑤𝐶)) < 𝑧 → (abs‘((𝐹𝑤) − (𝐹𝐶))) < 𝑅))

Theoremelcncf1di 12926* Membership in the set of continuous complex functions from 𝐴 to 𝐵. (Contributed by Paul Chapman, 26-Nov-2007.)
(𝜑𝐹:𝐴𝐵)    &   (𝜑 → ((𝑥𝐴𝑦 ∈ ℝ+) → 𝑍 ∈ ℝ+))    &   (𝜑 → (((𝑥𝐴𝑤𝐴) ∧ 𝑦 ∈ ℝ+) → ((abs‘(𝑥𝑤)) < 𝑍 → (abs‘((𝐹𝑥) − (𝐹𝑤))) < 𝑦)))       (𝜑 → ((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → 𝐹 ∈ (𝐴cn𝐵)))

Theoremelcncf1ii 12927* Membership in the set of continuous complex functions from 𝐴 to 𝐵. (Contributed by Paul Chapman, 26-Nov-2007.)
𝐹:𝐴𝐵    &   ((𝑥𝐴𝑦 ∈ ℝ+) → 𝑍 ∈ ℝ+)    &   (((𝑥𝐴𝑤𝐴) ∧ 𝑦 ∈ ℝ+) → ((abs‘(𝑥𝑤)) < 𝑍 → (abs‘((𝐹𝑥) − (𝐹𝑤))) < 𝑦))       ((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → 𝐹 ∈ (𝐴cn𝐵))

Theoremrescncf 12928 A continuous complex function restricted to a subset is continuous. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 25-Aug-2014.)
(𝐶𝐴 → (𝐹 ∈ (𝐴cn𝐵) → (𝐹𝐶) ∈ (𝐶cn𝐵)))

Theoremcncffvrn 12929 Change the codomain of a continuous complex function. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 1-May-2015.)
((𝐶 ⊆ ℂ ∧ 𝐹 ∈ (𝐴cn𝐵)) → (𝐹 ∈ (𝐴cn𝐶) ↔ 𝐹:𝐴𝐶))

Theoremcncfss 12930 The set of continuous functions is expanded when the range is expanded. (Contributed by Mario Carneiro, 30-Aug-2014.)
((𝐵𝐶𝐶 ⊆ ℂ) → (𝐴cn𝐵) ⊆ (𝐴cn𝐶))

Theoremclimcncf 12931 Image of a limit under a continuous map. (Contributed by Mario Carneiro, 7-Apr-2015.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹 ∈ (𝐴cn𝐵))    &   (𝜑𝐺:𝑍𝐴)    &   (𝜑𝐺𝐷)    &   (𝜑𝐷𝐴)       (𝜑 → (𝐹𝐺) ⇝ (𝐹𝐷))

Theoremabscncf 12932 Absolute value is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
abs ∈ (ℂ–cn→ℝ)

Theoremrecncf 12933 Real part is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
ℜ ∈ (ℂ–cn→ℝ)

Theoremimcncf 12934 Imaginary part is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
ℑ ∈ (ℂ–cn→ℝ)

Theoremcjcncf 12935 Complex conjugate is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
∗ ∈ (ℂ–cn→ℂ)

Theoremmulc1cncf 12936* Multiplication by a constant is continuous. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Mario Carneiro, 30-Apr-2014.)
𝐹 = (𝑥 ∈ ℂ ↦ (𝐴 · 𝑥))       (𝐴 ∈ ℂ → 𝐹 ∈ (ℂ–cn→ℂ))

Theoremdivccncfap 12937* Division by a constant is continuous. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Jim Kingdon, 9-Jan-2023.)
𝐹 = (𝑥 ∈ ℂ ↦ (𝑥 / 𝐴))       ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → 𝐹 ∈ (ℂ–cn→ℂ))

Theoremcncfco 12938 The composition of two continuous maps on complex numbers is also continuous. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 25-Aug-2014.)
(𝜑𝐹 ∈ (𝐴cn𝐵))    &   (𝜑𝐺 ∈ (𝐵cn𝐶))       (𝜑 → (𝐺𝐹) ∈ (𝐴cn𝐶))

Theoremcncfmet 12939 Relate complex function continuity to metric space continuity. (Contributed by Paul Chapman, 26-Nov-2007.) (Revised by Mario Carneiro, 7-Sep-2015.)
𝐶 = ((abs ∘ − ) ↾ (𝐴 × 𝐴))    &   𝐷 = ((abs ∘ − ) ↾ (𝐵 × 𝐵))    &   𝐽 = (MetOpen‘𝐶)    &   𝐾 = (MetOpen‘𝐷)       ((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → (𝐴cn𝐵) = (𝐽 Cn 𝐾))

Theoremcncfcncntop 12940 Relate complex function continuity to topological continuity. (Contributed by Mario Carneiro, 17-Feb-2015.)
𝐽 = (MetOpen‘(abs ∘ − ))    &   𝐾 = (𝐽t 𝐴)    &   𝐿 = (𝐽t 𝐵)       ((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → (𝐴cn𝐵) = (𝐾 Cn 𝐿))

Theoremcncfcn1cntop 12941 Relate complex function continuity to topological continuity. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Mario Carneiro, 7-Sep-2015.) (Revised by Jim Kingdon, 16-Jun-2023.)
𝐽 = (MetOpen‘(abs ∘ − ))       (ℂ–cn→ℂ) = (𝐽 Cn 𝐽)

Theoremcncfmptc 12942* A constant function is a continuous function on . (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 7-Sep-2015.)
((𝐴𝑇𝑆 ⊆ ℂ ∧ 𝑇 ⊆ ℂ) → (𝑥𝑆𝐴) ∈ (𝑆cn𝑇))

Theoremcncfmptid 12943* The identity function is a continuous function on . (Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro, 17-May-2016.)
((𝑆𝑇𝑇 ⊆ ℂ) → (𝑥𝑆𝑥) ∈ (𝑆cn𝑇))

Theoremcncfmpt1f 12944* Composition of continuous functions. cn analogue of cnmpt11f 12644. (Contributed by Mario Carneiro, 3-Sep-2014.)
(𝜑𝐹 ∈ (ℂ–cn→ℂ))    &   (𝜑 → (𝑥𝑋𝐴) ∈ (𝑋cn→ℂ))       (𝜑 → (𝑥𝑋 ↦ (𝐹𝐴)) ∈ (𝑋cn→ℂ))

Theoremcncfmpt2fcntop 12945* Composition of continuous functions. cn analogue of cnmpt12f 12646. (Contributed by Mario Carneiro, 3-Sep-2014.)
𝐽 = (MetOpen‘(abs ∘ − ))    &   (𝜑𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽))    &   (𝜑 → (𝑥𝑋𝐴) ∈ (𝑋cn→ℂ))    &   (𝜑 → (𝑥𝑋𝐵) ∈ (𝑋cn→ℂ))       (𝜑 → (𝑥𝑋 ↦ (𝐴𝐹𝐵)) ∈ (𝑋cn→ℂ))

𝐹 = (𝑥 ∈ ℂ ↦ (𝑥 + 𝐴))       (𝐴 ∈ ℂ → 𝐹 ∈ (ℂ–cn→ℂ))

Theoremcdivcncfap 12947* Division with a constant numerator is continuous. (Contributed by Mario Carneiro, 28-Dec-2016.) (Revised by Jim Kingdon, 26-May-2023.)
𝐹 = (𝑥 ∈ {𝑦 ∈ ℂ ∣ 𝑦 # 0} ↦ (𝐴 / 𝑥))       (𝐴 ∈ ℂ → 𝐹 ∈ ({𝑦 ∈ ℂ ∣ 𝑦 # 0}–cn→ℂ))

Theoremnegcncf 12948* The negative function is continuous. (Contributed by Mario Carneiro, 30-Dec-2016.)
𝐹 = (𝑥𝐴 ↦ -𝑥)       (𝐴 ⊆ ℂ → 𝐹 ∈ (𝐴cn→ℂ))

Theoremnegfcncf 12949* The negative of a continuous complex function is continuous. (Contributed by Paul Chapman, 21-Jan-2008.) (Revised by Mario Carneiro, 25-Aug-2014.)
𝐺 = (𝑥𝐴 ↦ -(𝐹𝑥))       (𝐹 ∈ (𝐴cn→ℂ) → 𝐺 ∈ (𝐴cn→ℂ))

Theoremmulcncflem 12950* Lemma for mulcncf 12951. (Contributed by Jim Kingdon, 29-May-2023.)
(𝜑 → (𝑥𝑋𝐴) ∈ (𝑋cn→ℂ))    &   (𝜑 → (𝑥𝑋𝐵) ∈ (𝑋cn→ℂ))    &   (𝜑𝑉𝑋)    &   (𝜑𝐸 ∈ ℝ+)    &   (𝜑𝐹 ∈ ℝ+)    &   (𝜑𝐺 ∈ ℝ+)    &   (𝜑𝑆 ∈ ℝ+)    &   (𝜑𝑇 ∈ ℝ+)    &   (𝜑 → ∀𝑢𝑋 ((abs‘(𝑢𝑉)) < 𝑆 → (abs‘(((𝑥𝑋𝐴)‘𝑢) − ((𝑥𝑋𝐴)‘𝑉))) < 𝐹))    &   (𝜑 → ∀𝑢𝑋 ((abs‘(𝑢𝑉)) < 𝑇 → (abs‘(((𝑥𝑋𝐵)‘𝑢) − ((𝑥𝑋𝐵)‘𝑉))) < 𝐺))    &   (𝜑 → ∀𝑢𝑋 (((abs‘(𝑢 / 𝑥𝐴𝑉 / 𝑥𝐴)) < 𝐹 ∧ (abs‘(𝑢 / 𝑥𝐵𝑉 / 𝑥𝐵)) < 𝐺) → (abs‘((𝑢 / 𝑥𝐴 · 𝑢 / 𝑥𝐵) − (𝑉 / 𝑥𝐴 · 𝑉 / 𝑥𝐵))) < 𝐸))       (𝜑 → ∃𝑑 ∈ ℝ+𝑢𝑋 ((abs‘(𝑢𝑉)) < 𝑑 → (abs‘(((𝑥𝑋 ↦ (𝐴 · 𝐵))‘𝑢) − ((𝑥𝑋 ↦ (𝐴 · 𝐵))‘𝑉))) < 𝐸))

Theoremmulcncf 12951* The multiplication of two continuous complex functions is continuous. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
(𝜑 → (𝑥𝑋𝐴) ∈ (𝑋cn→ℂ))    &   (𝜑 → (𝑥𝑋𝐵) ∈ (𝑋cn→ℂ))       (𝜑 → (𝑥𝑋 ↦ (𝐴 · 𝐵)) ∈ (𝑋cn→ℂ))

Theoremexpcncf 12952* The power function on complex numbers, for fixed exponent N, is continuous. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
(𝑁 ∈ ℕ0 → (𝑥 ∈ ℂ ↦ (𝑥𝑁)) ∈ (ℂ–cn→ℂ))

Theoremcnrehmeocntop 12953* The canonical bijection from (ℝ × ℝ) to described in cnref1o 9541 is in fact a homeomorphism of the usual topologies on these sets. (It is also an isometry, if (ℝ × ℝ) is metrized with the l<SUP>2</SUP> norm.) (Contributed by Mario Carneiro, 25-Aug-2014.)
𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + (i · 𝑦)))    &   𝐽 = (topGen‘ran (,))    &   𝐾 = (MetOpen‘(abs ∘ − ))       𝐹 ∈ ((𝐽 ×t 𝐽)Homeo𝐾)

Theoremcnopnap 12954* The complex numbers apart from a given complex number form an open set. (Contributed by Jim Kingdon, 14-Dec-2023.)
(𝐴 ∈ ℂ → {𝑤 ∈ ℂ ∣ 𝑤 # 𝐴} ∈ (MetOpen‘(abs ∘ − )))

PART 8  BASIC REAL AND COMPLEX ANALYSIS

8.0.1  Dedekind cuts

Theoremdedekindeulemuub 12955* Lemma for dedekindeu 12961. Any element of the upper cut is an upper bound for the lower cut. (Contributed by Jim Kingdon, 2-Feb-2024.)
(𝜑𝐿 ⊆ ℝ)    &   (𝜑𝑈 ⊆ ℝ)    &   (𝜑 → ∃𝑞 ∈ ℝ 𝑞𝐿)    &   (𝜑 → ∃𝑟 ∈ ℝ 𝑟𝑈)    &   (𝜑 → ∀𝑞 ∈ ℝ (𝑞𝐿 ↔ ∃𝑟𝐿 𝑞 < 𝑟))    &   (𝜑 → ∀𝑟 ∈ ℝ (𝑟𝑈 ↔ ∃𝑞𝑈 𝑞 < 𝑟))    &   (𝜑 → (𝐿𝑈) = ∅)    &   (𝜑 → ∀𝑞 ∈ ℝ ∀𝑟 ∈ ℝ (𝑞 < 𝑟 → (𝑞𝐿𝑟𝑈)))    &   (𝜑𝐴𝑈)       (𝜑 → ∀𝑧𝐿 𝑧 < 𝐴)

Theoremdedekindeulemub 12956* Lemma for dedekindeu 12961. The lower cut has an upper bound. (Contributed by Jim Kingdon, 31-Jan-2024.)
(𝜑𝐿 ⊆ ℝ)    &   (𝜑𝑈 ⊆ ℝ)    &   (𝜑 → ∃𝑞 ∈ ℝ 𝑞𝐿)    &   (𝜑 → ∃𝑟 ∈ ℝ 𝑟𝑈)    &   (𝜑 → ∀𝑞 ∈ ℝ (𝑞𝐿 ↔ ∃𝑟𝐿 𝑞 < 𝑟))    &   (𝜑 → ∀𝑟 ∈ ℝ (𝑟𝑈 ↔ ∃𝑞𝑈 𝑞 < 𝑟))    &   (𝜑 → (𝐿𝑈) = ∅)    &   (𝜑 → ∀𝑞 ∈ ℝ ∀𝑟 ∈ ℝ (𝑞 < 𝑟 → (𝑞𝐿𝑟𝑈)))       (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦𝐿 𝑦 < 𝑥)

Theoremdedekindeulemloc 12957* Lemma for dedekindeu 12961. The set L is located. (Contributed by Jim Kingdon, 31-Jan-2024.)
(𝜑𝐿 ⊆ ℝ)    &   (𝜑𝑈 ⊆ ℝ)    &   (𝜑 → ∃𝑞 ∈ ℝ 𝑞𝐿)    &   (𝜑 → ∃𝑟 ∈ ℝ 𝑟𝑈)    &   (𝜑 → ∀𝑞 ∈ ℝ (𝑞𝐿 ↔ ∃𝑟𝐿 𝑞 < 𝑟))    &   (𝜑 → ∀𝑟 ∈ ℝ (𝑟𝑈 ↔ ∃𝑞𝑈 𝑞 < 𝑟))    &   (𝜑 → (𝐿𝑈) = ∅)    &   (𝜑 → ∀𝑞 ∈ ℝ ∀𝑟 ∈ ℝ (𝑞 < 𝑟 → (𝑞𝐿𝑟𝑈)))       (𝜑 → ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → (∃𝑧𝐿 𝑥 < 𝑧 ∨ ∀𝑧𝐿 𝑧 < 𝑦)))

Theoremdedekindeulemlub 12958* Lemma for dedekindeu 12961. The set L has a least upper bound. (Contributed by Jim Kingdon, 31-Jan-2024.)
(𝜑𝐿 ⊆ ℝ)    &   (𝜑𝑈 ⊆ ℝ)    &   (𝜑 → ∃𝑞 ∈ ℝ 𝑞𝐿)    &   (𝜑 → ∃𝑟 ∈ ℝ 𝑟𝑈)    &   (𝜑 → ∀𝑞 ∈ ℝ (𝑞𝐿 ↔ ∃𝑟𝐿 𝑞 < 𝑟))    &   (𝜑 → ∀𝑟 ∈ ℝ (𝑟𝑈 ↔ ∃𝑞𝑈 𝑞 < 𝑟))    &   (𝜑 → (𝐿𝑈) = ∅)    &   (𝜑 → ∀𝑞 ∈ ℝ ∀𝑟 ∈ ℝ (𝑞 < 𝑟 → (𝑞𝐿𝑟𝑈)))       (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦𝐿 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧𝐿 𝑦 < 𝑧)))

Theoremdedekindeulemlu 12959* Lemma for dedekindeu 12961. There is a number which separates the lower and upper cuts. (Contributed by Jim Kingdon, 31-Jan-2024.)
(𝜑𝐿 ⊆ ℝ)    &   (𝜑𝑈 ⊆ ℝ)    &   (𝜑 → ∃𝑞 ∈ ℝ 𝑞𝐿)    &   (𝜑 → ∃𝑟 ∈ ℝ 𝑟𝑈)    &   (𝜑 → ∀𝑞 ∈ ℝ (𝑞𝐿 ↔ ∃𝑟𝐿 𝑞 < 𝑟))    &   (𝜑 → ∀𝑟 ∈ ℝ (𝑟𝑈 ↔ ∃𝑞𝑈 𝑞 < 𝑟))    &   (𝜑 → (𝐿𝑈) = ∅)    &   (𝜑 → ∀𝑞 ∈ ℝ ∀𝑟 ∈ ℝ (𝑞 < 𝑟 → (𝑞𝐿𝑟𝑈)))       (𝜑 → ∃𝑥 ∈ ℝ (∀𝑞𝐿 𝑞 < 𝑥 ∧ ∀𝑟𝑈 𝑥 < 𝑟))

Theoremdedekindeulemeu 12960* Lemma for dedekindeu 12961. Part of proving uniqueness. (Contributed by Jim Kingdon, 31-Jan-2024.)
(𝜑𝐿 ⊆ ℝ)    &   (𝜑𝑈 ⊆ ℝ)    &   (𝜑 → ∃𝑞 ∈ ℝ 𝑞𝐿)    &   (𝜑 → ∃𝑟 ∈ ℝ 𝑟𝑈)    &   (𝜑 → ∀𝑞 ∈ ℝ (𝑞𝐿 ↔ ∃𝑟𝐿 𝑞 < 𝑟))    &   (𝜑 → ∀𝑟 ∈ ℝ (𝑟𝑈 ↔ ∃𝑞𝑈 𝑞 < 𝑟))    &   (𝜑 → (𝐿𝑈) = ∅)    &   (𝜑 → ∀𝑞 ∈ ℝ ∀𝑟 ∈ ℝ (𝑞 < 𝑟 → (𝑞𝐿𝑟𝑈)))    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑 → (∀𝑞𝐿 𝑞 < 𝐴 ∧ ∀𝑟𝑈 𝐴 < 𝑟))    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → (∀𝑞𝐿 𝑞 < 𝐵 ∧ ∀𝑟𝑈 𝐵 < 𝑟))    &   (𝜑𝐴 < 𝐵)       (𝜑 → ⊥)

Theoremdedekindeu 12961* A Dedekind cut identifies a unique real number. Similar to df-inp 7369 except that the the Dedekind cut is formed by sets of reals (rather than positive rationals). But in both cases the defining property of a Dedekind cut is that it is inhabited (bounded), rounded, disjoint, and located. (Contributed by Jim Kingdon, 5-Jan-2024.)
(𝜑𝐿 ⊆ ℝ)    &   (𝜑𝑈 ⊆ ℝ)    &   (𝜑 → ∃𝑞 ∈ ℝ 𝑞𝐿)    &   (𝜑 → ∃𝑟 ∈ ℝ 𝑟𝑈)    &   (𝜑 → ∀𝑞 ∈ ℝ (𝑞𝐿 ↔ ∃𝑟𝐿 𝑞 < 𝑟))    &   (𝜑 → ∀𝑟 ∈ ℝ (𝑟𝑈 ↔ ∃𝑞𝑈 𝑞 < 𝑟))    &   (𝜑 → (𝐿𝑈) = ∅)    &   (𝜑 → ∀𝑞 ∈ ℝ ∀𝑟 ∈ ℝ (𝑞 < 𝑟 → (𝑞𝐿𝑟𝑈)))       (𝜑 → ∃!𝑥 ∈ ℝ (∀𝑞𝐿 𝑞 < 𝑥 ∧ ∀𝑟𝑈 𝑥 < 𝑟))

Theoremsuplociccreex 12962* An inhabited, bounded-above, located set of reals in a closed interval has a supremum. A similar theorem is axsuploc 7933 but that one is for the entire real line rather than a closed interval. (Contributed by Jim Kingdon, 14-Feb-2024.)
(𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐵 < 𝐶)    &   (𝜑𝐴 ⊆ (𝐵[,]𝐶))    &   (𝜑 → ∃𝑥 𝑥𝐴)    &   (𝜑 → ∀𝑥 ∈ (𝐵[,]𝐶)∀𝑦 ∈ (𝐵[,]𝐶)(𝑥 < 𝑦 → (∃𝑧𝐴 𝑥 < 𝑧 ∨ ∀𝑧𝐴 𝑧 < 𝑦)))       (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧)))

Theoremsuplociccex 12963* An inhabited, bounded-above, located set of reals in a closed interval has a supremum. A similar theorem is axsuploc 7933 but that one is for the entire real line rather than a closed interval. (Contributed by Jim Kingdon, 14-Feb-2024.)
(𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐵 < 𝐶)    &   (𝜑𝐴 ⊆ (𝐵[,]𝐶))    &   (𝜑 → ∃𝑥 𝑥𝐴)    &   (𝜑 → ∀𝑥 ∈ (𝐵[,]𝐶)∀𝑦 ∈ (𝐵[,]𝐶)(𝑥 < 𝑦 → (∃𝑧𝐴 𝑥 < 𝑧 ∨ ∀𝑧𝐴 𝑧 < 𝑦)))       (𝜑 → ∃𝑥 ∈ (𝐵[,]𝐶)(∀𝑦𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ (𝐵[,]𝐶)(𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧)))

Theoremdedekindicclemuub 12964* Lemma for dedekindicc 12971. Any element of the upper cut is an upper bound for the lower cut. (Contributed by Jim Kingdon, 15-Feb-2024.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐿 ⊆ (𝐴[,]𝐵))    &   (𝜑𝑈 ⊆ (𝐴[,]𝐵))    &   (𝜑 → ∃𝑞 ∈ (𝐴[,]𝐵)𝑞𝐿)    &   (𝜑 → ∃𝑟 ∈ (𝐴[,]𝐵)𝑟𝑈)    &   (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)(𝑞𝐿 ↔ ∃𝑟𝐿 𝑞 < 𝑟))    &   (𝜑 → ∀𝑟 ∈ (𝐴[,]𝐵)(𝑟𝑈 ↔ ∃𝑞𝑈 𝑞 < 𝑟))    &   (𝜑 → (𝐿𝑈) = ∅)    &   (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)∀𝑟 ∈ (𝐴[,]𝐵)(𝑞 < 𝑟 → (𝑞𝐿𝑟𝑈)))    &   (𝜑𝐶𝑈)       (𝜑 → ∀𝑧𝐿 𝑧 < 𝐶)

Theoremdedekindicclemub 12965* Lemma for dedekindicc 12971. The lower cut has an upper bound. (Contributed by Jim Kingdon, 15-Feb-2024.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐿 ⊆ (𝐴[,]𝐵))    &   (𝜑𝑈 ⊆ (𝐴[,]𝐵))    &   (𝜑 → ∃𝑞 ∈ (𝐴[,]𝐵)𝑞𝐿)    &   (𝜑 → ∃𝑟 ∈ (𝐴[,]𝐵)𝑟𝑈)    &   (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)(𝑞𝐿 ↔ ∃𝑟𝐿 𝑞 < 𝑟))    &   (𝜑 → ∀𝑟 ∈ (𝐴[,]𝐵)(𝑟𝑈 ↔ ∃𝑞𝑈 𝑞 < 𝑟))    &   (𝜑 → (𝐿𝑈) = ∅)    &   (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)∀𝑟 ∈ (𝐴[,]𝐵)(𝑞 < 𝑟 → (𝑞𝐿𝑟𝑈)))       (𝜑 → ∃𝑥 ∈ (𝐴[,]𝐵)∀𝑦𝐿 𝑦 < 𝑥)

Theoremdedekindicclemloc 12966* Lemma for dedekindicc 12971. The set L is located. (Contributed by Jim Kingdon, 15-Feb-2024.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐿 ⊆ (𝐴[,]𝐵))    &   (𝜑𝑈 ⊆ (𝐴[,]𝐵))    &   (𝜑 → ∃𝑞 ∈ (𝐴[,]𝐵)𝑞𝐿)    &   (𝜑 → ∃𝑟 ∈ (𝐴[,]𝐵)𝑟𝑈)    &   (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)(𝑞𝐿 ↔ ∃𝑟𝐿 𝑞 < 𝑟))    &   (𝜑 → ∀𝑟 ∈ (𝐴[,]𝐵)(𝑟𝑈 ↔ ∃𝑞𝑈 𝑞 < 𝑟))    &   (𝜑 → (𝐿𝑈) = ∅)    &   (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)∀𝑟 ∈ (𝐴[,]𝐵)(𝑞 < 𝑟 → (𝑞𝐿𝑟𝑈)))       (𝜑 → ∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(𝑥 < 𝑦 → (∃𝑧𝐿 𝑥 < 𝑧 ∨ ∀𝑧𝐿 𝑧 < 𝑦)))

Theoremdedekindicclemlub 12967* Lemma for dedekindicc 12971. The set L has a least upper bound. (Contributed by Jim Kingdon, 15-Feb-2024.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐿 ⊆ (𝐴[,]𝐵))    &   (𝜑𝑈 ⊆ (𝐴[,]𝐵))    &   (𝜑 → ∃𝑞 ∈ (𝐴[,]𝐵)𝑞𝐿)    &   (𝜑 → ∃𝑟 ∈ (𝐴[,]𝐵)𝑟𝑈)    &   (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)(𝑞𝐿 ↔ ∃𝑟𝐿 𝑞 < 𝑟))    &   (𝜑 → ∀𝑟 ∈ (𝐴[,]𝐵)(𝑟𝑈 ↔ ∃𝑞𝑈 𝑞 < 𝑟))    &   (𝜑 → (𝐿𝑈) = ∅)    &   (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)∀𝑟 ∈ (𝐴[,]𝐵)(𝑞 < 𝑟 → (𝑞𝐿𝑟𝑈)))    &   (𝜑𝐴 < 𝐵)       (𝜑 → ∃𝑥 ∈ (𝐴[,]𝐵)(∀𝑦𝐿 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ (𝐴[,]𝐵)(𝑦 < 𝑥 → ∃𝑧𝐿 𝑦 < 𝑧)))

Theoremdedekindicclemlu 12968* Lemma for dedekindicc 12971. There is a number which separates the lower and upper cuts. (Contributed by Jim Kingdon, 15-Feb-2024.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐿 ⊆ (𝐴[,]𝐵))    &   (𝜑𝑈 ⊆ (𝐴[,]𝐵))    &   (𝜑 → ∃𝑞 ∈ (𝐴[,]𝐵)𝑞𝐿)    &   (𝜑 → ∃𝑟 ∈ (𝐴[,]𝐵)𝑟𝑈)    &   (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)(𝑞𝐿 ↔ ∃𝑟𝐿 𝑞 < 𝑟))    &   (𝜑 → ∀𝑟 ∈ (𝐴[,]𝐵)(𝑟𝑈 ↔ ∃𝑞𝑈 𝑞 < 𝑟))    &   (𝜑 → (𝐿𝑈) = ∅)    &   (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)∀𝑟 ∈ (𝐴[,]𝐵)(𝑞 < 𝑟 → (𝑞𝐿𝑟𝑈)))    &   (𝜑𝐴 < 𝐵)       (𝜑 → ∃𝑥 ∈ (𝐴[,]𝐵)(∀𝑞𝐿 𝑞 < 𝑥 ∧ ∀𝑟𝑈 𝑥 < 𝑟))

Theoremdedekindicclemeu 12969* Lemma for dedekindicc 12971. Part of proving uniqueness. (Contributed by Jim Kingdon, 15-Feb-2024.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐿 ⊆ (𝐴[,]𝐵))    &   (𝜑𝑈 ⊆ (𝐴[,]𝐵))    &   (𝜑 → ∃𝑞 ∈ (𝐴[,]𝐵)𝑞𝐿)    &   (𝜑 → ∃𝑟 ∈ (𝐴[,]𝐵)𝑟𝑈)    &   (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)(𝑞𝐿 ↔ ∃𝑟𝐿 𝑞 < 𝑟))    &   (𝜑 → ∀𝑟 ∈ (𝐴[,]𝐵)(𝑟𝑈 ↔ ∃𝑞𝑈 𝑞 < 𝑟))    &   (𝜑 → (𝐿𝑈) = ∅)    &   (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)∀𝑟 ∈ (𝐴[,]𝐵)(𝑞 < 𝑟 → (𝑞𝐿𝑟𝑈)))    &   (𝜑𝐴 < 𝐵)    &   (𝜑𝐶 ∈ (𝐴[,]𝐵))    &   (𝜑 → (∀𝑞𝐿 𝑞 < 𝐶 ∧ ∀𝑟𝑈 𝐶 < 𝑟))    &   (𝜑𝐷 ∈ (𝐴[,]𝐵))    &   (𝜑 → (∀𝑞𝐿 𝑞 < 𝐷 ∧ ∀𝑟𝑈 𝐷 < 𝑟))    &   (𝜑𝐶 < 𝐷)       (𝜑 → ⊥)

Theoremdedekindicclemicc 12970* Lemma for dedekindicc 12971. Same as dedekindicc 12971, except that we merely show 𝑥 to be an element of (𝐴[,]𝐵). Later we will strengthen that to (𝐴(,)𝐵). (Contributed by Jim Kingdon, 5-Jan-2024.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐿 ⊆ (𝐴[,]𝐵))    &   (𝜑𝑈 ⊆ (𝐴[,]𝐵))    &   (𝜑 → ∃𝑞 ∈ (𝐴[,]𝐵)𝑞𝐿)    &   (𝜑 → ∃𝑟 ∈ (𝐴[,]𝐵)𝑟𝑈)    &   (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)(𝑞𝐿 ↔ ∃𝑟𝐿 𝑞 < 𝑟))    &   (𝜑 → ∀𝑟 ∈ (𝐴[,]𝐵)(𝑟𝑈 ↔ ∃𝑞𝑈 𝑞 < 𝑟))    &   (𝜑 → (𝐿𝑈) = ∅)    &   (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)∀𝑟 ∈ (𝐴[,]𝐵)(𝑞 < 𝑟 → (𝑞𝐿𝑟𝑈)))    &   (𝜑𝐴 < 𝐵)       (𝜑 → ∃!𝑥 ∈ (𝐴[,]𝐵)(∀𝑞𝐿 𝑞 < 𝑥 ∧ ∀𝑟𝑈 𝑥 < 𝑟))

Theoremdedekindicc 12971* A Dedekind cut identifies a unique real number. Similar to df-inp 7369 except that the Dedekind cut is formed by sets of reals (rather than positive rationals). But in both cases the defining property of a Dedekind cut is that it is inhabited (bounded), rounded, disjoint, and located. (Contributed by Jim Kingdon, 19-Feb-2024.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐿 ⊆ (𝐴[,]𝐵))    &   (𝜑𝑈 ⊆ (𝐴[,]𝐵))    &   (𝜑 → ∃𝑞 ∈ (𝐴[,]𝐵)𝑞𝐿)    &   (𝜑 → ∃𝑟 ∈ (𝐴[,]𝐵)𝑟𝑈)    &   (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)(𝑞𝐿 ↔ ∃𝑟𝐿 𝑞 < 𝑟))    &   (𝜑 → ∀𝑟 ∈ (𝐴[,]𝐵)(𝑟𝑈 ↔ ∃𝑞𝑈 𝑞 < 𝑟))    &   (𝜑 → (𝐿𝑈) = ∅)    &   (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)∀𝑟 ∈ (𝐴[,]𝐵)(𝑞 < 𝑟 → (𝑞𝐿𝑟𝑈)))    &   (𝜑𝐴 < 𝐵)       (𝜑 → ∃!𝑥 ∈ (𝐴(,)𝐵)(∀𝑞𝐿 𝑞 < 𝑥 ∧ ∀𝑟𝑈 𝑥 < 𝑟))

8.0.2  Intermediate value theorem

Theoremivthinclemlm 12972* Lemma for ivthinc 12981. The lower cut is bounded. (Contributed by Jim Kingdon, 18-Feb-2024.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝑈 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)    &   (𝜑𝐹 ∈ (𝐷cn→ℂ))    &   ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (𝐹𝑥) ∈ ℝ)    &   (𝜑 → ((𝐹𝐴) < 𝑈𝑈 < (𝐹𝐵)))    &   (((𝜑𝑥 ∈ (𝐴[,]𝐵)) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝑥 < 𝑦)) → (𝐹𝑥) < (𝐹𝑦))    &   𝐿 = {𝑤 ∈ (𝐴[,]𝐵) ∣ (𝐹𝑤) < 𝑈}    &   𝑅 = {𝑤 ∈ (𝐴[,]𝐵) ∣ 𝑈 < (𝐹𝑤)}       (𝜑 → ∃𝑞 ∈ (𝐴[,]𝐵)𝑞𝐿)

Theoremivthinclemum 12973* Lemma for ivthinc 12981. The upper cut is bounded. (Contributed by Jim Kingdon, 18-Feb-2024.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝑈 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)    &   (𝜑𝐹 ∈ (𝐷cn→ℂ))    &   ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (𝐹𝑥) ∈ ℝ)    &   (𝜑 → ((𝐹𝐴) < 𝑈𝑈 < (𝐹𝐵)))    &   (((𝜑𝑥 ∈ (𝐴[,]𝐵)) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝑥 < 𝑦)) → (𝐹𝑥) < (𝐹𝑦))    &   𝐿 = {𝑤 ∈ (𝐴[,]𝐵) ∣ (𝐹𝑤) < 𝑈}    &   𝑅 = {𝑤 ∈ (𝐴[,]𝐵) ∣ 𝑈 < (𝐹𝑤)}       (𝜑 → ∃𝑟 ∈ (𝐴[,]𝐵)𝑟𝑅)

Theoremivthinclemlopn 12974* Lemma for ivthinc 12981. The lower cut is open. (Contributed by Jim Kingdon, 6-Feb-2024.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝑈 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)    &   (𝜑𝐹 ∈ (𝐷cn→ℂ))    &   ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (𝐹𝑥) ∈ ℝ)    &   (𝜑 → ((𝐹𝐴) < 𝑈𝑈 < (𝐹𝐵)))    &   (((𝜑𝑥 ∈ (𝐴[,]𝐵)) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝑥 < 𝑦)) → (𝐹𝑥) < (𝐹𝑦))    &   𝐿 = {𝑤 ∈ (𝐴[,]𝐵) ∣ (𝐹𝑤) < 𝑈}    &   𝑅 = {𝑤 ∈ (𝐴[,]𝐵) ∣ 𝑈 < (𝐹𝑤)}    &   (𝜑𝑄𝐿)       (𝜑 → ∃𝑟𝐿 𝑄 < 𝑟)

Theoremivthinclemlr 12975* Lemma for ivthinc 12981. The lower cut is rounded. (Contributed by Jim Kingdon, 18-Feb-2024.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝑈 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)    &   (𝜑𝐹 ∈ (𝐷cn→ℂ))    &   ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (𝐹𝑥) ∈ ℝ)    &   (𝜑 → ((𝐹𝐴) < 𝑈𝑈 < (𝐹𝐵)))    &   (((𝜑𝑥 ∈ (𝐴[,]𝐵)) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝑥 < 𝑦)) → (𝐹𝑥) < (𝐹𝑦))    &   𝐿 = {𝑤 ∈ (𝐴[,]𝐵) ∣ (𝐹𝑤) < 𝑈}    &   𝑅 = {𝑤 ∈ (𝐴[,]𝐵) ∣ 𝑈 < (𝐹𝑤)}       (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)(𝑞𝐿 ↔ ∃𝑟𝐿 𝑞 < 𝑟))

Theoremivthinclemuopn 12976* Lemma for ivthinc 12981. The upper cut is open. (Contributed by Jim Kingdon, 19-Feb-2024.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝑈 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)    &   (𝜑𝐹 ∈ (𝐷cn→ℂ))    &   ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (𝐹𝑥) ∈ ℝ)    &   (𝜑 → ((𝐹𝐴) < 𝑈𝑈 < (𝐹𝐵)))    &   (((𝜑𝑥 ∈ (𝐴[,]𝐵)) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝑥 < 𝑦)) → (𝐹𝑥) < (𝐹𝑦))    &   𝐿 = {𝑤 ∈ (𝐴[,]𝐵) ∣ (𝐹𝑤) < 𝑈}    &   𝑅 = {𝑤 ∈ (𝐴[,]𝐵) ∣ 𝑈 < (𝐹𝑤)}    &   (𝜑𝑆𝑅)       (𝜑 → ∃𝑞𝑅 𝑞 < 𝑆)

Theoremivthinclemur 12977* Lemma for ivthinc 12981. The upper cut is rounded. (Contributed by Jim Kingdon, 18-Feb-2024.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝑈 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)    &   (𝜑𝐹 ∈ (𝐷cn→ℂ))    &   ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (𝐹𝑥) ∈ ℝ)    &   (𝜑 → ((𝐹𝐴) < 𝑈𝑈 < (𝐹𝐵)))    &   (((𝜑𝑥 ∈ (𝐴[,]𝐵)) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝑥 < 𝑦)) → (𝐹𝑥) < (𝐹𝑦))    &   𝐿 = {𝑤 ∈ (𝐴[,]𝐵) ∣ (𝐹𝑤) < 𝑈}    &   𝑅 = {𝑤 ∈ (𝐴[,]𝐵) ∣ 𝑈 < (𝐹𝑤)}       (𝜑 → ∀𝑟 ∈ (𝐴[,]𝐵)(𝑟𝑅 ↔ ∃𝑞𝑅 𝑞 < 𝑟))

Theoremivthinclemdisj 12978* Lemma for ivthinc 12981. The lower and upper cuts are disjoint. (Contributed by Jim Kingdon, 18-Feb-2024.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝑈 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)    &   (𝜑𝐹 ∈ (𝐷cn→ℂ))    &   ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (𝐹𝑥) ∈ ℝ)    &   (𝜑 → ((𝐹𝐴) < 𝑈𝑈 < (𝐹𝐵)))    &   (((𝜑𝑥 ∈ (𝐴[,]𝐵)) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝑥 < 𝑦)) → (𝐹𝑥) < (𝐹𝑦))    &   𝐿 = {𝑤 ∈ (𝐴[,]𝐵) ∣ (𝐹𝑤) < 𝑈}    &   𝑅 = {𝑤 ∈ (𝐴[,]𝐵) ∣ 𝑈 < (𝐹𝑤)}       (𝜑 → (𝐿𝑅) = ∅)

Theoremivthinclemloc 12979* Lemma for ivthinc 12981. Locatedness. (Contributed by Jim Kingdon, 18-Feb-2024.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝑈 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)    &   (𝜑𝐹 ∈ (𝐷cn→ℂ))    &   ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (𝐹𝑥) ∈ ℝ)    &   (𝜑 → ((𝐹𝐴) < 𝑈𝑈 < (𝐹𝐵)))    &   (((𝜑𝑥 ∈ (𝐴[,]𝐵)) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝑥 < 𝑦)) → (𝐹𝑥) < (𝐹𝑦))    &   𝐿 = {𝑤 ∈ (𝐴[,]𝐵) ∣ (𝐹𝑤) < 𝑈}    &   𝑅 = {𝑤 ∈ (𝐴[,]𝐵) ∣ 𝑈 < (𝐹𝑤)}       (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)∀𝑟 ∈ (𝐴[,]𝐵)(𝑞 < 𝑟 → (𝑞𝐿𝑟𝑅)))

Theoremivthinclemex 12980* Lemma for ivthinc 12981. Existence of a number between the lower cut and the upper cut. (Contributed by Jim Kingdon, 20-Feb-2024.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝑈 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)    &   (𝜑𝐹 ∈ (𝐷cn→ℂ))    &   ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (𝐹𝑥) ∈ ℝ)    &   (𝜑 → ((𝐹𝐴) < 𝑈𝑈 < (𝐹𝐵)))    &   (((𝜑𝑥 ∈ (𝐴[,]𝐵)) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝑥 < 𝑦)) → (𝐹𝑥) < (𝐹𝑦))    &   𝐿 = {𝑤 ∈ (𝐴[,]𝐵) ∣ (𝐹𝑤) < 𝑈}    &   𝑅 = {𝑤 ∈ (𝐴[,]𝐵) ∣ 𝑈 < (𝐹𝑤)}       (𝜑 → ∃!𝑧 ∈ (𝐴(,)𝐵)(∀𝑞𝐿 𝑞 < 𝑧 ∧ ∀𝑟𝑅 𝑧 < 𝑟))

Theoremivthinc 12981* The intermediate value theorem, increasing case, for a strictly monotonic function. Theorem 5.5 of [Bauer], p. 494. This is Metamath 100 proof #79. (Contributed by Jim Kingdon, 5-Feb-2024.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝑈 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)    &   (𝜑𝐹 ∈ (𝐷cn→ℂ))    &   ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (𝐹𝑥) ∈ ℝ)    &   (𝜑 → ((𝐹𝐴) < 𝑈𝑈 < (𝐹𝐵)))    &   (((𝜑𝑥 ∈ (𝐴[,]𝐵)) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝑥 < 𝑦)) → (𝐹𝑥) < (𝐹𝑦))       (𝜑 → ∃𝑐 ∈ (𝐴(,)𝐵)(𝐹𝑐) = 𝑈)

Theoremivthdec 12982* The intermediate value theorem, decreasing case, for a strictly monotonic function. (Contributed by Jim Kingdon, 20-Feb-2024.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝑈 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)    &   (𝜑𝐹 ∈ (𝐷cn→ℂ))    &   ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (𝐹𝑥) ∈ ℝ)    &   (𝜑 → ((𝐹𝐵) < 𝑈𝑈 < (𝐹𝐴)))    &   (((𝜑𝑥 ∈ (𝐴[,]𝐵)) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝑥 < 𝑦)) → (𝐹𝑦) < (𝐹𝑥))       (𝜑 → ∃𝑐 ∈ (𝐴(,)𝐵)(𝐹𝑐) = 𝑈)

8.1  Derivatives

8.1.1  Real and complex differentiation

8.1.1.1  Derivatives of functions of one complex or real variable

Syntaxclimc 12983 The limit operator.
class lim

Syntaxcdv 12984 The derivative operator.
class D

Definitiondf-limced 12985* Define the set of limits of a complex function at a point. Under normal circumstances, this will be a singleton or empty, depending on whether the limit exists. (Contributed by Mario Carneiro, 24-Dec-2016.) (Revised by Jim Kingdon, 3-Jun-2023.)
lim = (𝑓 ∈ (ℂ ↑pm ℂ), 𝑥 ∈ ℂ ↦ {𝑦 ∈ ℂ ∣ ((𝑓:dom 𝑓⟶ℂ ∧ dom 𝑓 ⊆ ℂ) ∧ (𝑥 ∈ ℂ ∧ ∀𝑒 ∈ ℝ+𝑑 ∈ ℝ+𝑧 ∈ dom 𝑓((𝑧 # 𝑥 ∧ (abs‘(𝑧𝑥)) < 𝑑) → (abs‘((𝑓𝑧) − 𝑦)) < 𝑒)))})

Definitiondf-dvap 12986* Define the derivative operator. This acts on functions to produce a function that is defined where the original function is differentiable, with value the derivative of the function at these points. The set 𝑠 here is the ambient topological space under which we are evaluating the continuity of the difference quotient. Although the definition is valid for any subset of and is well-behaved when 𝑠 contains no isolated points, we will restrict our attention to the cases 𝑠 = ℝ or 𝑠 = ℂ for the majority of the development, these corresponding respectively to real and complex differentiation. (Contributed by Mario Carneiro, 7-Aug-2014.) (Revised by Jim Kingdon, 25-Jun-2023.)
D = (𝑠 ∈ 𝒫 ℂ, 𝑓 ∈ (ℂ ↑pm 𝑠) ↦ 𝑥 ∈ ((int‘((MetOpen‘(abs ∘ − )) ↾t 𝑠))‘dom 𝑓)({𝑥} × ((𝑧 ∈ {𝑤 ∈ dom 𝑓𝑤 # 𝑥} ↦ (((𝑓𝑧) − (𝑓𝑥)) / (𝑧𝑥))) lim 𝑥)))

Theoremlimcrcl 12987 Reverse closure for the limit operator. (Contributed by Mario Carneiro, 28-Dec-2016.)
(𝐶 ∈ (𝐹 lim 𝐵) → (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ ∧ 𝐵 ∈ ℂ))

Theoremlimccl 12988 Closure of the limit operator. (Contributed by Mario Carneiro, 25-Dec-2016.)
(𝐹 lim 𝐵) ⊆ ℂ

Theoremellimc3apf 12989* Write the epsilon-delta definition of a limit. (Contributed by Mario Carneiro, 28-Dec-2016.) (Revised by Jim Kingdon, 4-Nov-2023.)
(𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐴 ⊆ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   𝑧𝐹       (𝜑 → (𝐶 ∈ (𝐹 lim 𝐵) ↔ (𝐶 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ℝ+𝑧𝐴 ((𝑧 # 𝐵 ∧ (abs‘(𝑧𝐵)) < 𝑦) → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥))))

Theoremellimc3ap 12990* Write the epsilon-delta definition of a limit. (Contributed by Mario Carneiro, 28-Dec-2016.) Use apartness. (Revised by Jim Kingdon, 3-Jun-2023.)
(𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐴 ⊆ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (𝐶 ∈ (𝐹 lim 𝐵) ↔ (𝐶 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ℝ+𝑧𝐴 ((𝑧 # 𝐵 ∧ (abs‘(𝑧𝐵)) < 𝑦) → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥))))

Theoremlimcdifap 12991* It suffices to consider functions which are not defined at 𝐵 to define the limit of a function. In particular, the value of the original function 𝐹 at 𝐵 does not affect the limit of 𝐹. (Contributed by Mario Carneiro, 25-Dec-2016.) (Revised by Jim Kingdon, 3-Jun-2023.)
(𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐴 ⊆ ℂ)       (𝜑 → (𝐹 lim 𝐵) = ((𝐹 ↾ {𝑥𝐴𝑥 # 𝐵}) lim 𝐵))

Theoremlimcmpted 12992* Express the limit operator for a function defined by a mapping, via epsilon-delta. (Contributed by Jim Kingdon, 3-Nov-2023.)
(𝜑𝐴 ⊆ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   ((𝜑𝑧𝐴) → 𝐷 ∈ ℂ)       (𝜑 → (𝐶 ∈ ((𝑧𝐴𝐷) lim 𝐵) ↔ (𝐶 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ℝ+𝑧𝐴 ((𝑧 # 𝐵 ∧ (abs‘(𝑧𝐵)) < 𝑦) → (abs‘(𝐷𝐶)) < 𝑥))))

Theoremlimcimolemlt 12993* Lemma for limcimo 12994. (Contributed by Jim Kingdon, 3-Jul-2023.)
(𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐴 ⊆ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵𝐶)    &   (𝜑𝐵𝑆)    &   (𝜑𝐶 ∈ (𝐾t 𝑆))    &   (𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑 → {𝑞𝐶𝑞 # 𝐵} ⊆ 𝐴)    &   𝐾 = (MetOpen‘(abs ∘ − ))    &   (𝜑𝐷 ∈ ℝ+)    &   (𝜑𝑋 ∈ (𝐹 lim 𝐵))    &   (𝜑𝑌 ∈ (𝐹 lim 𝐵))    &   (𝜑 → ∀𝑧𝐴 ((𝑧 # 𝐵 ∧ (abs‘(𝑧𝐵)) < 𝐷) → (abs‘((𝐹𝑧) − 𝑋)) < ((abs‘(𝑋𝑌)) / 2)))    &   (𝜑𝐺 ∈ ℝ+)    &   (𝜑 → ∀𝑤𝐴 ((𝑤 # 𝐵 ∧ (abs‘(𝑤𝐵)) < 𝐺) → (abs‘((𝐹𝑤) − 𝑌)) < ((abs‘(𝑋𝑌)) / 2)))       (𝜑 → (abs‘(𝑋𝑌)) < (abs‘(𝑋𝑌)))

Theoremlimcimo 12994* 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.)
(𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐴 ⊆ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵𝐶)    &   (𝜑𝐵𝑆)    &   (𝜑𝐶 ∈ (𝐾t 𝑆))    &   (𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑 → {𝑞𝐶𝑞 # 𝐵} ⊆ 𝐴)    &   𝐾 = (MetOpen‘(abs ∘ − ))       (𝜑 → ∃*𝑥 𝑥 ∈ (𝐹 lim 𝐵))

Theoremlimcresi 12995 Any limit of 𝐹 is also a limit of the restriction of 𝐹. (Contributed by Mario Carneiro, 28-Dec-2016.)
(𝐹 lim 𝐵) ⊆ ((𝐹𝐶) lim 𝐵)

Theoremcnplimcim 12996 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 = (𝐾t 𝐴)       ((𝐴 ⊆ ℂ ∧ 𝐵𝐴) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐵) → (𝐹:𝐴⟶ℂ ∧ (𝐹𝐵) ∈ (𝐹 lim 𝐵))))

Theoremcnplimclemle 12997 Lemma for cnplimccntop 12999. Satisfying the epsilon condition for continuity. (Contributed by Mario Carneiro and Jim Kingdon, 17-Nov-2023.)
𝐾 = (MetOpen‘(abs ∘ − ))    &   𝐽 = (𝐾t 𝐴)    &   (𝜑𝐴 ⊆ ℂ)    &   (𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐵𝐴)    &   (𝜑 → (𝐹𝐵) ∈ (𝐹 lim 𝐵))    &   (𝜑𝐸 ∈ ℝ+)    &   (𝜑𝐷 ∈ ℝ+)    &   (𝜑𝑍𝐴)    &   ((𝜑𝑍 # 𝐵 ∧ (abs‘(𝑍𝐵)) < 𝐷) → (abs‘((𝐹𝑍) − (𝐹𝐵))) < (𝐸 / 2))    &   (𝜑 → (abs‘(𝑍𝐵)) < 𝐷)       (𝜑 → (abs‘((𝐹𝑍) − (𝐹𝐵))) < 𝐸)

Theoremcnplimclemr 12998 Lemma for cnplimccntop 12999. The reverse direction. (Contributed by Mario Carneiro and Jim Kingdon, 17-Nov-2023.)
𝐾 = (MetOpen‘(abs ∘ − ))    &   𝐽 = (𝐾t 𝐴)    &   (𝜑𝐴 ⊆ ℂ)    &   (𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐵𝐴)    &   (𝜑 → (𝐹𝐵) ∈ (𝐹 lim 𝐵))       (𝜑𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐵))

Theoremcnplimccntop 12999 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 = (𝐾t 𝐴)       ((𝐴 ⊆ ℂ ∧ 𝐵𝐴) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐵) ↔ (𝐹:𝐴⟶ℂ ∧ (𝐹𝐵) ∈ (𝐹 lim 𝐵))))

Theoremcnlimcim 13000* 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 ∈ (𝐴cn→ℂ) → (𝐹:𝐴⟶ℂ ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ (𝐹 lim 𝑥))))

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