HomeHome Intuitionistic Logic Explorer
Theorem List (p. 144 of 152)
< Previous  Next >
Bad symbols? Try the
GIF version.

Mirrors  >  Metamath Home Page  >  ILE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Theorem List for Intuitionistic Logic Explorer - 14301-14400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcnmet 14301 The absolute value metric determines a metric space on the complex numbers. This theorem provides a link between complex numbers and metrics spaces, making metric space theorems available for use with complex numbers. (Contributed by FL, 9-Oct-2006.)
(abs ∘ − ) ∈ (Met‘ℂ)
 
Theoremcnxmet 14302 The absolute value metric is an extended metric. (Contributed by Mario Carneiro, 28-Aug-2015.)
(abs ∘ − ) ∈ (∞Met‘ℂ)
 
Theoremcntoptopon 14303 The topology of the complex numbers is a topology. (Contributed by Jim Kingdon, 6-Jun-2023.)
𝐽 = (MetOpen‘(abs ∘ − ))       𝐽 ∈ (TopOn‘ℂ)
 
Theoremcntoptop 14304 The topology of the complex numbers is a topology. (Contributed by Jim Kingdon, 6-Jun-2023.)
𝐽 = (MetOpen‘(abs ∘ − ))       𝐽 ∈ Top
 
Theoremcnbl0 14305 Two ways to write the open ball centered at zero. (Contributed by Mario Carneiro, 8-Sep-2015.)
𝐷 = (abs ∘ − )       (𝑅 ∈ ℝ* → (abs “ (0[,)𝑅)) = (0(ball‘𝐷)𝑅))
 
Theoremcnblcld 14306* Two ways to write the closed ball centered at zero. (Contributed by Mario Carneiro, 8-Sep-2015.)
𝐷 = (abs ∘ − )       (𝑅 ∈ ℝ* → (abs “ (0[,]𝑅)) = {𝑥 ∈ ℂ ∣ (0𝐷𝑥) ≤ 𝑅})
 
Theoremunicntopcntop 14307 The underlying set of the standard topology on the complex numbers is the set of complex numbers. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Revised by Jim Kingdon, 12-Dec-2023.)
ℂ = (MetOpen‘(abs ∘ − ))
 
Theoremcnopncntop 14308 The set of complex numbers is open with respect to the standard topology on complex numbers. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Revised by Jim Kingdon, 12-Dec-2023.)
ℂ ∈ (MetOpen‘(abs ∘ − ))
 
Theoremreopnap 14309* The real numbers apart from a given real number form an open set. (Contributed by Jim Kingdon, 13-Dec-2023.)
(𝐴 ∈ ℝ → {𝑤 ∈ ℝ ∣ 𝑤 # 𝐴} ∈ (topGen‘ran (,)))
 
Theoremremetdval 14310 Value of the distance function of the metric space of real numbers. (Contributed by NM, 16-May-2007.)
𝐷 = ((abs ∘ − ) ↾ (ℝ × ℝ))       ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴𝐷𝐵) = (abs‘(𝐴𝐵)))
 
Theoremremet 14311 The absolute value metric determines a metric space on the reals. (Contributed by NM, 10-Feb-2007.)
𝐷 = ((abs ∘ − ) ↾ (ℝ × ℝ))       𝐷 ∈ (Met‘ℝ)
 
Theoremrexmet 14312 The absolute value metric is an extended metric. (Contributed by Mario Carneiro, 28-Aug-2015.)
𝐷 = ((abs ∘ − ) ↾ (ℝ × ℝ))       𝐷 ∈ (∞Met‘ℝ)
 
Theorembl2ioo 14313 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 14314 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 14315 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 14316 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 14317 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 14318 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 14319 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 14320 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 14321 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 14322* Lemma for addcncntop 14323, subcncntop 14324, and mulcncntop 14325. (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 14323 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 14324 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 14325 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 14326* 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 14327* 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))
 
8.2.7  Topological definitions using the reals
 
Syntaxccncf 14328 Extend class notation to include the operation which returns a class of continuous complex functions.
class cn
 
Definitiondf-cncf 14329* 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 14330* 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 14331* 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 14332* Version of elcncf 14331 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 14333 Reverse closure of the continuous function predicate. (Contributed by Mario Carneiro, 25-Aug-2014.)
(𝐹 ∈ (𝐴cn𝐵) → 𝐴 ⊆ ℂ)
 
Theoremcncfrss2 14334 Reverse closure of the continuous function predicate. (Contributed by Mario Carneiro, 25-Aug-2014.)
(𝐹 ∈ (𝐴cn𝐵) → 𝐵 ⊆ ℂ)
 
Theoremcncff 14335 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 14336* 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 14337* 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 14338* 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 14339 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)))
 
Theoremcncfcdm 14340 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 14341 The set of continuous functions is expanded when the codomain is expanded. (Contributed by Mario Carneiro, 30-Aug-2014.)
((𝐵𝐶𝐶 ⊆ ℂ) → (𝐴cn𝐵) ⊆ (𝐴cn𝐶))
 
Theoremclimcncf 14342 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 14343 Absolute value is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
abs ∈ (ℂ–cn→ℝ)
 
Theoremrecncf 14344 Real part is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
ℜ ∈ (ℂ–cn→ℝ)
 
Theoremimcncf 14345 Imaginary part is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
ℑ ∈ (ℂ–cn→ℝ)
 
Theoremcjcncf 14346 Complex conjugate is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
∗ ∈ (ℂ–cn→ℂ)
 
Theoremmulc1cncf 14347* 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 14348* 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 14349 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 14350 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 14351 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 14352 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 14353* 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 14354* 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 14355* Composition of continuous functions. cn analogue of cnmpt11f 14055. (Contributed by Mario Carneiro, 3-Sep-2014.)
(𝜑𝐹 ∈ (ℂ–cn→ℂ))    &   (𝜑 → (𝑥𝑋𝐴) ∈ (𝑋cn→ℂ))       (𝜑 → (𝑥𝑋 ↦ (𝐹𝐴)) ∈ (𝑋cn→ℂ))
 
Theoremcncfmpt2fcntop 14356* Composition of continuous functions. cn analogue of cnmpt12f 14057. (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))
 
Theoremaddccncf 14357* Adding a constant is a continuous function. (Contributed by Jeff Madsen, 2-Sep-2009.)
𝐹 = (𝑥 ∈ ℂ ↦ (𝑥 + 𝐴))       (𝐴 ∈ ℂ → 𝐹 ∈ (ℂ–cn→ℂ))
 
Theoremcdivcncfap 14358* 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 14359* The negative function is continuous. (Contributed by Mario Carneiro, 30-Dec-2016.)
𝐹 = (𝑥𝐴 ↦ -𝑥)       (𝐴 ⊆ ℂ → 𝐹 ∈ (𝐴cn→ℂ))
 
Theoremnegfcncf 14360* 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 14361* Lemma for mulcncf 14362. (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 14362* 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 14363* 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 14364* The canonical bijection from (ℝ × ℝ) to described in cnref1o 9663 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 14365* 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 9  BASIC REAL AND COMPLEX ANALYSIS
 
9.0.1  Dedekind cuts
 
Theoremdedekindeulemuub 14366* Lemma for dedekindeu 14372. 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 14367* Lemma for dedekindeu 14372. 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 14368* Lemma for dedekindeu 14372. 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 14369* Lemma for dedekindeu 14372. 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 14370* Lemma for dedekindeu 14372. 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 14371* Lemma for dedekindeu 14372. 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 14372* A Dedekind cut identifies a unique real number. Similar to df-inp 7478 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 14373* An inhabited, bounded-above, located set of reals in a closed interval has a supremum. A similar theorem is axsuploc 8043 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 14374* An inhabited, bounded-above, located set of reals in a closed interval has a supremum. A similar theorem is axsuploc 8043 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 14375* Lemma for dedekindicc 14382. 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 14376* Lemma for dedekindicc 14382. 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 14377* Lemma for dedekindicc 14382. 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 14378* Lemma for dedekindicc 14382. 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 14379* Lemma for dedekindicc 14382. 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 14380* Lemma for dedekindicc 14382. 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 14381* Lemma for dedekindicc 14382. Same as dedekindicc 14382, 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 14382* A Dedekind cut identifies a unique real number. Similar to df-inp 7478 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))
 
9.0.2  Intermediate value theorem
 
Theoremivthinclemlm 14383* Lemma for ivthinc 14392. 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 14384* Lemma for ivthinc 14392. 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 14385* Lemma for ivthinc 14392. 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 14386* Lemma for ivthinc 14392. 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 14387* Lemma for ivthinc 14392. 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 14388* Lemma for ivthinc 14392. 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 14389* Lemma for ivthinc 14392. 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 14390* Lemma for ivthinc 14392. 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 14391* Lemma for ivthinc 14392. 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 14392* 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 14393* 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)
 
9.1  Derivatives
 
9.1.1  Real and complex differentiation
 
9.1.1.1  Derivatives of functions of one complex or real variable
 
Syntaxclimc 14394 The limit operator.
class lim
 
Syntaxcdv 14395 The derivative operator.
class D
 
Definitiondf-limced 14396* 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 14397* 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 14398 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 14399 Closure of the limit operator. (Contributed by Mario Carneiro, 25-Dec-2016.)
(𝐹 lim 𝐵) ⊆ ℂ
 
Theoremellimc3apf 14400* 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))))
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15104
  Copyright terms: Public domain < Previous  Next >