Type | Label | Description |
Statement |
|
Theorem | cnmet 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) |
|
Theorem | cnxmet 14302 |
The absolute value metric is an extended metric. (Contributed by Mario
Carneiro, 28-Aug-2015.)
|
⊢ (abs ∘ − ) ∈
(∞Met‘ℂ) |
|
Theorem | cntoptopon 14303 |
The topology of the complex numbers is a topology. (Contributed by Jim
Kingdon, 6-Jun-2023.)
|
⊢ 𝐽 = (MetOpen‘(abs ∘ −
)) ⇒ ⊢ 𝐽 ∈
(TopOn‘ℂ) |
|
Theorem | cntoptop 14304 |
The topology of the complex numbers is a topology. (Contributed by Jim
Kingdon, 6-Jun-2023.)
|
⊢ 𝐽 = (MetOpen‘(abs ∘ −
)) ⇒ ⊢ 𝐽 ∈ Top |
|
Theorem | cnbl0 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)) |
|
Theorem | cnblcld 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}) |
|
Theorem | unicntopcntop 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 ∘ − )) |
|
Theorem | cnopncntop 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 ∘
− )) |
|
Theorem | reopnap 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
(,))) |
|
Theorem | remetdval 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‘(𝐴 − 𝐵))) |
|
Theorem | remet 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) |
|
Theorem | rexmet 14312 |
The absolute value metric is an extended metric. (Contributed by Mario
Carneiro, 28-Aug-2015.)
|
⊢ 𝐷 = ((abs ∘ − ) ↾ (ℝ
× ℝ)) ⇒ ⊢ 𝐷 ∈
(∞Met‘ℝ) |
|
Theorem | bl2ioo 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) = ((𝐴 − 𝐵)(,)(𝐴 + 𝐵))) |
|
Theorem | ioo2bl 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)((𝐵 − 𝐴) / 2))) |
|
Theorem | ioo2blex 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)) |
|
Theorem | blssioo 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
(,) |
|
Theorem | tgioo 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 |
|
Theorem | tgqioo 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 |
|
Theorem | resubmet 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 = (𝑅 ↾t 𝐴)) |
|
Theorem | tgioo2cntop 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 (,)) = (𝐽 ↾t
ℝ) |
|
Theorem | rerestcntop 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 → (𝐽 ↾t 𝐴) = (𝑅 ↾t 𝐴)) |
|
Theorem | addcncntoplem 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‘(𝑢 −
𝑏)) < 𝑦 ∧ (abs‘(𝑣 − 𝑐)) < 𝑧) → (abs‘((𝑢 + 𝑣) − (𝑏 + 𝑐))) < 𝑎)) ⇒ ⊢ + ∈ ((𝐽 ×t 𝐽) Cn 𝐽) |
|
Theorem | addcncntop 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) |
|
Theorem | subcncntop 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) |
|
Theorem | mulcncntop 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) |
|
Theorem | divcnap 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 = (𝐽 ↾t {𝑥 ∈ ℂ ∣ 𝑥 # 0}) ⇒ ⊢ (𝑦 ∈ ℂ, 𝑧 ∈ {𝑥 ∈ ℂ ∣ 𝑥 # 0} ↦ (𝑦 / 𝑧)) ∈ ((𝐽 ×t 𝐾) Cn 𝐽) |
|
Theorem | fsumcncntop 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 ∘ −
))
& ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐾)) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝐴 𝐵) ∈ (𝐽 Cn 𝐾)) |
|
8.2.7 Topological definitions using the
reals
|
|
Syntax | ccncf 14328 |
Extend class notation to include the operation which returns a class of
continuous complex functions.
|
class –cn→ |
|
Definition | df-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 ∈ (𝑏 ↑𝑚
𝑎) ∣ ∀𝑥 ∈ 𝑎 ∀𝑒 ∈ ℝ+ ∃𝑑 ∈ ℝ+
∀𝑦 ∈ 𝑎 ((abs‘(𝑥 − 𝑦)) < 𝑑 → (abs‘((𝑓‘𝑥) − (𝑓‘𝑦))) < 𝑒)}) |
|
Theorem | cncfval 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) → (𝐴–cn→𝐵) = {𝑓 ∈ (𝐵 ↑𝑚 𝐴) ∣ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+
∀𝑤 ∈ 𝐴 ((abs‘(𝑥 − 𝑤)) < 𝑧 → (abs‘((𝑓‘𝑥) − (𝑓‘𝑤))) < 𝑦)}) |
|
Theorem | elcncf 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 ∈ (𝐴–cn→𝐵) ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+
∀𝑤 ∈ 𝐴 ((abs‘(𝑥 − 𝑤)) < 𝑧 → (abs‘((𝐹‘𝑥) − (𝐹‘𝑤))) < 𝑦)))) |
|
Theorem | elcncf2 14332* |
Version of elcncf 14331 with arguments commuted. (Contributed by
Mario
Carneiro, 28-Apr-2014.)
|
⊢ ((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → (𝐹 ∈ (𝐴–cn→𝐵) ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+
∀𝑤 ∈ 𝐴 ((abs‘(𝑤 − 𝑥)) < 𝑧 → (abs‘((𝐹‘𝑤) − (𝐹‘𝑥))) < 𝑦)))) |
|
Theorem | cncfrss 14333 |
Reverse closure of the continuous function predicate. (Contributed by
Mario Carneiro, 25-Aug-2014.)
|
⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐴 ⊆ ℂ) |
|
Theorem | cncfrss2 14334 |
Reverse closure of the continuous function predicate. (Contributed by
Mario Carneiro, 25-Aug-2014.)
|
⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐵 ⊆ ℂ) |
|
Theorem | cncff 14335 |
A continuous complex function's domain and codomain. (Contributed by
Paul Chapman, 17-Jan-2008.) (Revised by Mario Carneiro,
25-Aug-2014.)
|
⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐹:𝐴⟶𝐵) |
|
Theorem | cncfi 14336* |
Defining property of a continuous function. (Contributed by Mario
Carneiro, 30-Apr-2014.) (Revised by Mario Carneiro, 25-Aug-2014.)
|
⊢ ((𝐹 ∈ (𝐴–cn→𝐵) ∧ 𝐶 ∈ 𝐴 ∧ 𝑅 ∈ ℝ+) →
∃𝑧 ∈
ℝ+ ∀𝑤 ∈ 𝐴 ((abs‘(𝑤 − 𝐶)) < 𝑧 → (abs‘((𝐹‘𝑤) − (𝐹‘𝐶))) < 𝑅)) |
|
Theorem | elcncf1di 14337* |
Membership in the set of continuous complex functions from 𝐴 to
𝐵. (Contributed by Paul Chapman,
26-Nov-2007.)
|
⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
& ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ℝ+) → 𝑍 ∈
ℝ+))
& ⊢ (𝜑 → (((𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ 𝑦 ∈ ℝ+) →
((abs‘(𝑥 −
𝑤)) < 𝑍 → (abs‘((𝐹‘𝑥) − (𝐹‘𝑤))) < 𝑦))) ⇒ ⊢ (𝜑 → ((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → 𝐹 ∈ (𝐴–cn→𝐵))) |
|
Theorem | elcncf1ii 14338* |
Membership in the set of continuous complex functions from 𝐴 to
𝐵. (Contributed by Paul Chapman,
26-Nov-2007.)
|
⊢ 𝐹:𝐴⟶𝐵
& ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ℝ+) → 𝑍 ∈
ℝ+)
& ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ 𝑦 ∈ ℝ+) →
((abs‘(𝑥 −
𝑤)) < 𝑍 → (abs‘((𝐹‘𝑥) − (𝐹‘𝑤))) < 𝑦)) ⇒ ⊢ ((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → 𝐹 ∈ (𝐴–cn→𝐵)) |
|
Theorem | rescncf 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.)
|
⊢ (𝐶 ⊆ 𝐴 → (𝐹 ∈ (𝐴–cn→𝐵) → (𝐹 ↾ 𝐶) ∈ (𝐶–cn→𝐵))) |
|
Theorem | cncfcdm 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 ∈ (𝐴–cn→𝐵)) → (𝐹 ∈ (𝐴–cn→𝐶) ↔ 𝐹:𝐴⟶𝐶)) |
|
Theorem | cncfss 14341 |
The set of continuous functions is expanded when the codomain is
expanded. (Contributed by Mario Carneiro, 30-Aug-2014.)
|
⊢ ((𝐵 ⊆ 𝐶 ∧ 𝐶 ⊆ ℂ) → (𝐴–cn→𝐵) ⊆ (𝐴–cn→𝐶)) |
|
Theorem | climcncf 14342 |
Image of a limit under a continuous map. (Contributed by Mario
Carneiro, 7-Apr-2015.)
|
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐹 ∈ (𝐴–cn→𝐵)) & ⊢ (𝜑 → 𝐺:𝑍⟶𝐴)
& ⊢ (𝜑 → 𝐺 ⇝ 𝐷)
& ⊢ (𝜑 → 𝐷 ∈ 𝐴) ⇒ ⊢ (𝜑 → (𝐹 ∘ 𝐺) ⇝ (𝐹‘𝐷)) |
|
Theorem | abscncf 14343 |
Absolute value is continuous. (Contributed by Paul Chapman,
21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
|
⊢ abs ∈ (ℂ–cn→ℝ) |
|
Theorem | recncf 14344 |
Real part is continuous. (Contributed by Paul Chapman, 21-Oct-2007.)
(Revised by Mario Carneiro, 28-Apr-2014.)
|
⊢ ℜ ∈ (ℂ–cn→ℝ) |
|
Theorem | imcncf 14345 |
Imaginary part is continuous. (Contributed by Paul Chapman,
21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
|
⊢ ℑ ∈ (ℂ–cn→ℝ) |
|
Theorem | cjcncf 14346 |
Complex conjugate is continuous. (Contributed by Paul Chapman,
21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
|
⊢ ∗ ∈ (ℂ–cn→ℂ) |
|
Theorem | mulc1cncf 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)) |
|
Theorem | divccncfap 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)) |
|
Theorem | cncfco 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.)
|
⊢ (𝜑 → 𝐹 ∈ (𝐴–cn→𝐵)) & ⊢ (𝜑 → 𝐺 ∈ (𝐵–cn→𝐶)) ⇒ ⊢ (𝜑 → (𝐺 ∘ 𝐹) ∈ (𝐴–cn→𝐶)) |
|
Theorem | cncfmet 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) → (𝐴–cn→𝐵) = (𝐽 Cn 𝐾)) |
|
Theorem | cncfcncntop 14351 |
Relate complex function continuity to topological continuity.
(Contributed by Mario Carneiro, 17-Feb-2015.)
|
⊢ 𝐽 = (MetOpen‘(abs ∘ −
))
& ⊢ 𝐾 = (𝐽 ↾t 𝐴)
& ⊢ 𝐿 = (𝐽 ↾t 𝐵) ⇒ ⊢ ((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → (𝐴–cn→𝐵) = (𝐾 Cn 𝐿)) |
|
Theorem | cncfcn1cntop 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) |
|
Theorem | cncfmptc 14353* |
A constant function is a continuous function on ℂ. (Contributed
by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro,
7-Sep-2015.)
|
⊢ ((𝐴 ∈ 𝑇 ∧ 𝑆 ⊆ ℂ ∧ 𝑇 ⊆ ℂ) → (𝑥 ∈ 𝑆 ↦ 𝐴) ∈ (𝑆–cn→𝑇)) |
|
Theorem | cncfmptid 14354* |
The identity function is a continuous function on ℂ. (Contributed
by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro,
17-May-2016.)
|
⊢ ((𝑆 ⊆ 𝑇 ∧ 𝑇 ⊆ ℂ) → (𝑥 ∈ 𝑆 ↦ 𝑥) ∈ (𝑆–cn→𝑇)) |
|
Theorem | cncfmpt1f 14355* |
Composition of continuous functions. –cn→ analogue of cnmpt11f 14055.
(Contributed by Mario Carneiro, 3-Sep-2014.)
|
⊢ (𝜑 → 𝐹 ∈ (ℂ–cn→ℂ)) & ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝑋–cn→ℂ)) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐹‘𝐴)) ∈ (𝑋–cn→ℂ)) |
|
Theorem | cncfmpt2fcntop 14356* |
Composition of continuous functions. –cn→ analogue of cnmpt12f 14057.
(Contributed by Mario Carneiro, 3-Sep-2014.)
|
⊢ 𝐽 = (MetOpen‘(abs ∘ −
))
& ⊢ (𝜑 → 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) & ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝑋–cn→ℂ)) & ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝑋–cn→ℂ)) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴𝐹𝐵)) ∈ (𝑋–cn→ℂ)) |
|
Theorem | addccncf 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)) |
|
Theorem | cdivcncfap 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)) |
|
Theorem | negcncf 14359* |
The negative function is continuous. (Contributed by Mario Carneiro,
30-Dec-2016.)
|
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ -𝑥) ⇒ ⊢ (𝐴 ⊆ ℂ → 𝐹 ∈ (𝐴–cn→ℂ)) |
|
Theorem | negfcncf 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 = (𝑥 ∈ 𝐴 ↦ -(𝐹‘𝑥)) ⇒ ⊢ (𝐹 ∈ (𝐴–cn→ℂ) → 𝐺 ∈ (𝐴–cn→ℂ)) |
|
Theorem | mulcncflem 14361* |
Lemma for mulcncf 14362. (Contributed by Jim Kingdon, 29-May-2023.)
|
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝑋–cn→ℂ)) & ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝑋–cn→ℂ)) & ⊢ (𝜑 → 𝑉 ∈ 𝑋)
& ⊢ (𝜑 → 𝐸 ∈ ℝ+) & ⊢ (𝜑 → 𝐹 ∈ ℝ+) & ⊢ (𝜑 → 𝐺 ∈ ℝ+) & ⊢ (𝜑 → 𝑆 ∈ ℝ+) & ⊢ (𝜑 → 𝑇 ∈ ℝ+) & ⊢ (𝜑 → ∀𝑢 ∈ 𝑋 ((abs‘(𝑢 − 𝑉)) < 𝑆 → (abs‘(((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑢) − ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑉))) < 𝐹)) & ⊢ (𝜑 → ∀𝑢 ∈ 𝑋 ((abs‘(𝑢 − 𝑉)) < 𝑇 → (abs‘(((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑢) − ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑉))) < 𝐺)) & ⊢ (𝜑 → ∀𝑢 ∈ 𝑋 (((abs‘(⦋𝑢 / 𝑥⦌𝐴 − ⦋𝑉 / 𝑥⦌𝐴)) < 𝐹 ∧ (abs‘(⦋𝑢 / 𝑥⦌𝐵 − ⦋𝑉 / 𝑥⦌𝐵)) < 𝐺) → (abs‘((⦋𝑢 / 𝑥⦌𝐴 · ⦋𝑢 / 𝑥⦌𝐵) − (⦋𝑉 / 𝑥⦌𝐴 · ⦋𝑉 / 𝑥⦌𝐵))) < 𝐸)) ⇒ ⊢ (𝜑 → ∃𝑑 ∈ ℝ+ ∀𝑢 ∈ 𝑋 ((abs‘(𝑢 − 𝑉)) < 𝑑 → (abs‘(((𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵))‘𝑢) − ((𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵))‘𝑉))) < 𝐸)) |
|
Theorem | mulcncf 14362* |
The multiplication of two continuous complex functions is continuous.
(Contributed by Glauco Siliprandi, 29-Jun-2017.)
|
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝑋–cn→ℂ)) & ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝑋–cn→ℂ)) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)) ∈ (𝑋–cn→ℂ)) |
|
Theorem | expcncf 14363* |
The power function on complex numbers, for fixed exponent N, is
continuous. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
|
⊢ (𝑁 ∈ ℕ0 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∈ (ℂ–cn→ℂ)) |
|
Theorem | cnrehmeocntop 14364* |
The canonical bijection from (ℝ × ℝ)
to ℂ 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) |
|
Theorem | cnopnap 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
|
|
Theorem | dedekindeulemuub 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.)
|
⊢ (𝜑 → 𝐿 ⊆ ℝ) & ⊢ (𝜑 → 𝑈 ⊆ ℝ) & ⊢ (𝜑 → ∃𝑞 ∈ ℝ 𝑞 ∈ 𝐿)
& ⊢ (𝜑 → ∃𝑟 ∈ ℝ 𝑟 ∈ 𝑈)
& ⊢ (𝜑 → ∀𝑞 ∈ ℝ (𝑞 ∈ 𝐿 ↔ ∃𝑟 ∈ 𝐿 𝑞 < 𝑟))
& ⊢ (𝜑 → ∀𝑟 ∈ ℝ (𝑟 ∈ 𝑈 ↔ ∃𝑞 ∈ 𝑈 𝑞 < 𝑟))
& ⊢ (𝜑 → (𝐿 ∩ 𝑈) = ∅) & ⊢ (𝜑 → ∀𝑞 ∈ ℝ ∀𝑟 ∈ ℝ (𝑞 < 𝑟 → (𝑞 ∈ 𝐿 ∨ 𝑟 ∈ 𝑈))) & ⊢ (𝜑 → 𝐴 ∈ 𝑈) ⇒ ⊢ (𝜑 → ∀𝑧 ∈ 𝐿 𝑧 < 𝐴) |
|
Theorem | dedekindeulemub 14367* |
Lemma for dedekindeu 14372. The lower cut has an upper bound.
(Contributed by Jim Kingdon, 31-Jan-2024.)
|
⊢ (𝜑 → 𝐿 ⊆ ℝ) & ⊢ (𝜑 → 𝑈 ⊆ ℝ) & ⊢ (𝜑 → ∃𝑞 ∈ ℝ 𝑞 ∈ 𝐿)
& ⊢ (𝜑 → ∃𝑟 ∈ ℝ 𝑟 ∈ 𝑈)
& ⊢ (𝜑 → ∀𝑞 ∈ ℝ (𝑞 ∈ 𝐿 ↔ ∃𝑟 ∈ 𝐿 𝑞 < 𝑟))
& ⊢ (𝜑 → ∀𝑟 ∈ ℝ (𝑟 ∈ 𝑈 ↔ ∃𝑞 ∈ 𝑈 𝑞 < 𝑟))
& ⊢ (𝜑 → (𝐿 ∩ 𝑈) = ∅) & ⊢ (𝜑 → ∀𝑞 ∈ ℝ ∀𝑟 ∈ ℝ (𝑞 < 𝑟 → (𝑞 ∈ 𝐿 ∨ 𝑟 ∈ 𝑈))) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐿 𝑦 < 𝑥) |
|
Theorem | dedekindeulemloc 14368* |
Lemma for dedekindeu 14372. The set L is located. (Contributed by Jim
Kingdon, 31-Jan-2024.)
|
⊢ (𝜑 → 𝐿 ⊆ ℝ) & ⊢ (𝜑 → 𝑈 ⊆ ℝ) & ⊢ (𝜑 → ∃𝑞 ∈ ℝ 𝑞 ∈ 𝐿)
& ⊢ (𝜑 → ∃𝑟 ∈ ℝ 𝑟 ∈ 𝑈)
& ⊢ (𝜑 → ∀𝑞 ∈ ℝ (𝑞 ∈ 𝐿 ↔ ∃𝑟 ∈ 𝐿 𝑞 < 𝑟))
& ⊢ (𝜑 → ∀𝑟 ∈ ℝ (𝑟 ∈ 𝑈 ↔ ∃𝑞 ∈ 𝑈 𝑞 < 𝑟))
& ⊢ (𝜑 → (𝐿 ∩ 𝑈) = ∅) & ⊢ (𝜑 → ∀𝑞 ∈ ℝ ∀𝑟 ∈ ℝ (𝑞 < 𝑟 → (𝑞 ∈ 𝐿 ∨ 𝑟 ∈ 𝑈))) ⇒ ⊢ (𝜑 → ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → (∃𝑧 ∈ 𝐿 𝑥 < 𝑧 ∨ ∀𝑧 ∈ 𝐿 𝑧 < 𝑦))) |
|
Theorem | dedekindeulemlub 14369* |
Lemma for dedekindeu 14372. The set L has a least upper bound.
(Contributed by Jim Kingdon, 31-Jan-2024.)
|
⊢ (𝜑 → 𝐿 ⊆ ℝ) & ⊢ (𝜑 → 𝑈 ⊆ ℝ) & ⊢ (𝜑 → ∃𝑞 ∈ ℝ 𝑞 ∈ 𝐿)
& ⊢ (𝜑 → ∃𝑟 ∈ ℝ 𝑟 ∈ 𝑈)
& ⊢ (𝜑 → ∀𝑞 ∈ ℝ (𝑞 ∈ 𝐿 ↔ ∃𝑟 ∈ 𝐿 𝑞 < 𝑟))
& ⊢ (𝜑 → ∀𝑟 ∈ ℝ (𝑟 ∈ 𝑈 ↔ ∃𝑞 ∈ 𝑈 𝑞 < 𝑟))
& ⊢ (𝜑 → (𝐿 ∩ 𝑈) = ∅) & ⊢ (𝜑 → ∀𝑞 ∈ ℝ ∀𝑟 ∈ ℝ (𝑞 < 𝑟 → (𝑞 ∈ 𝐿 ∨ 𝑟 ∈ 𝑈))) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐿 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐿 𝑦 < 𝑧))) |
|
Theorem | dedekindeulemlu 14370* |
Lemma for dedekindeu 14372. There is a number which separates the
lower
and upper cuts. (Contributed by Jim Kingdon, 31-Jan-2024.)
|
⊢ (𝜑 → 𝐿 ⊆ ℝ) & ⊢ (𝜑 → 𝑈 ⊆ ℝ) & ⊢ (𝜑 → ∃𝑞 ∈ ℝ 𝑞 ∈ 𝐿)
& ⊢ (𝜑 → ∃𝑟 ∈ ℝ 𝑟 ∈ 𝑈)
& ⊢ (𝜑 → ∀𝑞 ∈ ℝ (𝑞 ∈ 𝐿 ↔ ∃𝑟 ∈ 𝐿 𝑞 < 𝑟))
& ⊢ (𝜑 → ∀𝑟 ∈ ℝ (𝑟 ∈ 𝑈 ↔ ∃𝑞 ∈ 𝑈 𝑞 < 𝑟))
& ⊢ (𝜑 → (𝐿 ∩ 𝑈) = ∅) & ⊢ (𝜑 → ∀𝑞 ∈ ℝ ∀𝑟 ∈ ℝ (𝑞 < 𝑟 → (𝑞 ∈ 𝐿 ∨ 𝑟 ∈ 𝑈))) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ ℝ (∀𝑞 ∈ 𝐿 𝑞 < 𝑥 ∧ ∀𝑟 ∈ 𝑈 𝑥 < 𝑟)) |
|
Theorem | dedekindeulemeu 14371* |
Lemma for dedekindeu 14372. Part of proving uniqueness. (Contributed
by
Jim Kingdon, 31-Jan-2024.)
|
⊢ (𝜑 → 𝐿 ⊆ ℝ) & ⊢ (𝜑 → 𝑈 ⊆ ℝ) & ⊢ (𝜑 → ∃𝑞 ∈ ℝ 𝑞 ∈ 𝐿)
& ⊢ (𝜑 → ∃𝑟 ∈ ℝ 𝑟 ∈ 𝑈)
& ⊢ (𝜑 → ∀𝑞 ∈ ℝ (𝑞 ∈ 𝐿 ↔ ∃𝑟 ∈ 𝐿 𝑞 < 𝑟))
& ⊢ (𝜑 → ∀𝑟 ∈ ℝ (𝑟 ∈ 𝑈 ↔ ∃𝑞 ∈ 𝑈 𝑞 < 𝑟))
& ⊢ (𝜑 → (𝐿 ∩ 𝑈) = ∅) & ⊢ (𝜑 → ∀𝑞 ∈ ℝ ∀𝑟 ∈ ℝ (𝑞 < 𝑟 → (𝑞 ∈ 𝐿 ∨ 𝑟 ∈ 𝑈))) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → (∀𝑞 ∈ 𝐿 𝑞 < 𝐴 ∧ ∀𝑟 ∈ 𝑈 𝐴 < 𝑟))
& ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → (∀𝑞 ∈ 𝐿 𝑞 < 𝐵 ∧ ∀𝑟 ∈ 𝑈 𝐵 < 𝑟))
& ⊢ (𝜑 → 𝐴 < 𝐵) ⇒ ⊢ (𝜑 → ⊥) |
|
Theorem | dedekindeu 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.)
|
⊢ (𝜑 → 𝐿 ⊆ ℝ) & ⊢ (𝜑 → 𝑈 ⊆ ℝ) & ⊢ (𝜑 → ∃𝑞 ∈ ℝ 𝑞 ∈ 𝐿)
& ⊢ (𝜑 → ∃𝑟 ∈ ℝ 𝑟 ∈ 𝑈)
& ⊢ (𝜑 → ∀𝑞 ∈ ℝ (𝑞 ∈ 𝐿 ↔ ∃𝑟 ∈ 𝐿 𝑞 < 𝑟))
& ⊢ (𝜑 → ∀𝑟 ∈ ℝ (𝑟 ∈ 𝑈 ↔ ∃𝑞 ∈ 𝑈 𝑞 < 𝑟))
& ⊢ (𝜑 → (𝐿 ∩ 𝑈) = ∅) & ⊢ (𝜑 → ∀𝑞 ∈ ℝ ∀𝑟 ∈ ℝ (𝑞 < 𝑟 → (𝑞 ∈ 𝐿 ∨ 𝑟 ∈ 𝑈))) ⇒ ⊢ (𝜑 → ∃!𝑥 ∈ ℝ (∀𝑞 ∈ 𝐿 𝑞 < 𝑥 ∧ ∀𝑟 ∈ 𝑈 𝑥 < 𝑟)) |
|
Theorem | suplociccreex 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.)
|
⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐵 < 𝐶)
& ⊢ (𝜑 → 𝐴 ⊆ (𝐵[,]𝐶)) & ⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐴)
& ⊢ (𝜑 → ∀𝑥 ∈ (𝐵[,]𝐶)∀𝑦 ∈ (𝐵[,]𝐶)(𝑥 < 𝑦 → (∃𝑧 ∈ 𝐴 𝑥 < 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧 < 𝑦))) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) |
|
Theorem | suplociccex 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.)
|
⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐵 < 𝐶)
& ⊢ (𝜑 → 𝐴 ⊆ (𝐵[,]𝐶)) & ⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐴)
& ⊢ (𝜑 → ∀𝑥 ∈ (𝐵[,]𝐶)∀𝑦 ∈ (𝐵[,]𝐶)(𝑥 < 𝑦 → (∃𝑧 ∈ 𝐴 𝑥 < 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧 < 𝑦))) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ (𝐵[,]𝐶)(∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ (𝐵[,]𝐶)(𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) |
|
Theorem | dedekindicclemuub 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.)
|
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐿 ⊆ (𝐴[,]𝐵)) & ⊢ (𝜑 → 𝑈 ⊆ (𝐴[,]𝐵)) & ⊢ (𝜑 → ∃𝑞 ∈ (𝐴[,]𝐵)𝑞 ∈ 𝐿)
& ⊢ (𝜑 → ∃𝑟 ∈ (𝐴[,]𝐵)𝑟 ∈ 𝑈)
& ⊢ (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)(𝑞 ∈ 𝐿 ↔ ∃𝑟 ∈ 𝐿 𝑞 < 𝑟))
& ⊢ (𝜑 → ∀𝑟 ∈ (𝐴[,]𝐵)(𝑟 ∈ 𝑈 ↔ ∃𝑞 ∈ 𝑈 𝑞 < 𝑟))
& ⊢ (𝜑 → (𝐿 ∩ 𝑈) = ∅) & ⊢ (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)∀𝑟 ∈ (𝐴[,]𝐵)(𝑞 < 𝑟 → (𝑞 ∈ 𝐿 ∨ 𝑟 ∈ 𝑈))) & ⊢ (𝜑 → 𝐶 ∈ 𝑈) ⇒ ⊢ (𝜑 → ∀𝑧 ∈ 𝐿 𝑧 < 𝐶) |
|
Theorem | dedekindicclemub 14376* |
Lemma for dedekindicc 14382. The lower cut has an upper bound.
(Contributed by Jim Kingdon, 15-Feb-2024.)
|
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐿 ⊆ (𝐴[,]𝐵)) & ⊢ (𝜑 → 𝑈 ⊆ (𝐴[,]𝐵)) & ⊢ (𝜑 → ∃𝑞 ∈ (𝐴[,]𝐵)𝑞 ∈ 𝐿)
& ⊢ (𝜑 → ∃𝑟 ∈ (𝐴[,]𝐵)𝑟 ∈ 𝑈)
& ⊢ (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)(𝑞 ∈ 𝐿 ↔ ∃𝑟 ∈ 𝐿 𝑞 < 𝑟))
& ⊢ (𝜑 → ∀𝑟 ∈ (𝐴[,]𝐵)(𝑟 ∈ 𝑈 ↔ ∃𝑞 ∈ 𝑈 𝑞 < 𝑟))
& ⊢ (𝜑 → (𝐿 ∩ 𝑈) = ∅) & ⊢ (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)∀𝑟 ∈ (𝐴[,]𝐵)(𝑞 < 𝑟 → (𝑞 ∈ 𝐿 ∨ 𝑟 ∈ 𝑈))) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ 𝐿 𝑦 < 𝑥) |
|
Theorem | dedekindicclemloc 14377* |
Lemma for dedekindicc 14382. The set L is located. (Contributed by Jim
Kingdon, 15-Feb-2024.)
|
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐿 ⊆ (𝐴[,]𝐵)) & ⊢ (𝜑 → 𝑈 ⊆ (𝐴[,]𝐵)) & ⊢ (𝜑 → ∃𝑞 ∈ (𝐴[,]𝐵)𝑞 ∈ 𝐿)
& ⊢ (𝜑 → ∃𝑟 ∈ (𝐴[,]𝐵)𝑟 ∈ 𝑈)
& ⊢ (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)(𝑞 ∈ 𝐿 ↔ ∃𝑟 ∈ 𝐿 𝑞 < 𝑟))
& ⊢ (𝜑 → ∀𝑟 ∈ (𝐴[,]𝐵)(𝑟 ∈ 𝑈 ↔ ∃𝑞 ∈ 𝑈 𝑞 < 𝑟))
& ⊢ (𝜑 → (𝐿 ∩ 𝑈) = ∅) & ⊢ (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)∀𝑟 ∈ (𝐴[,]𝐵)(𝑞 < 𝑟 → (𝑞 ∈ 𝐿 ∨ 𝑟 ∈ 𝑈))) ⇒ ⊢ (𝜑 → ∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(𝑥 < 𝑦 → (∃𝑧 ∈ 𝐿 𝑥 < 𝑧 ∨ ∀𝑧 ∈ 𝐿 𝑧 < 𝑦))) |
|
Theorem | dedekindicclemlub 14378* |
Lemma for dedekindicc 14382. The set L has a least upper bound.
(Contributed by Jim Kingdon, 15-Feb-2024.)
|
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐿 ⊆ (𝐴[,]𝐵)) & ⊢ (𝜑 → 𝑈 ⊆ (𝐴[,]𝐵)) & ⊢ (𝜑 → ∃𝑞 ∈ (𝐴[,]𝐵)𝑞 ∈ 𝐿)
& ⊢ (𝜑 → ∃𝑟 ∈ (𝐴[,]𝐵)𝑟 ∈ 𝑈)
& ⊢ (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)(𝑞 ∈ 𝐿 ↔ ∃𝑟 ∈ 𝐿 𝑞 < 𝑟))
& ⊢ (𝜑 → ∀𝑟 ∈ (𝐴[,]𝐵)(𝑟 ∈ 𝑈 ↔ ∃𝑞 ∈ 𝑈 𝑞 < 𝑟))
& ⊢ (𝜑 → (𝐿 ∩ 𝑈) = ∅) & ⊢ (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)∀𝑟 ∈ (𝐴[,]𝐵)(𝑞 < 𝑟 → (𝑞 ∈ 𝐿 ∨ 𝑟 ∈ 𝑈))) & ⊢ (𝜑 → 𝐴 < 𝐵) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ (𝐴[,]𝐵)(∀𝑦 ∈ 𝐿 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ (𝐴[,]𝐵)(𝑦 < 𝑥 → ∃𝑧 ∈ 𝐿 𝑦 < 𝑧))) |
|
Theorem | dedekindicclemlu 14379* |
Lemma for dedekindicc 14382. There is a number which separates the
lower
and upper cuts. (Contributed by Jim Kingdon, 15-Feb-2024.)
|
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐿 ⊆ (𝐴[,]𝐵)) & ⊢ (𝜑 → 𝑈 ⊆ (𝐴[,]𝐵)) & ⊢ (𝜑 → ∃𝑞 ∈ (𝐴[,]𝐵)𝑞 ∈ 𝐿)
& ⊢ (𝜑 → ∃𝑟 ∈ (𝐴[,]𝐵)𝑟 ∈ 𝑈)
& ⊢ (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)(𝑞 ∈ 𝐿 ↔ ∃𝑟 ∈ 𝐿 𝑞 < 𝑟))
& ⊢ (𝜑 → ∀𝑟 ∈ (𝐴[,]𝐵)(𝑟 ∈ 𝑈 ↔ ∃𝑞 ∈ 𝑈 𝑞 < 𝑟))
& ⊢ (𝜑 → (𝐿 ∩ 𝑈) = ∅) & ⊢ (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)∀𝑟 ∈ (𝐴[,]𝐵)(𝑞 < 𝑟 → (𝑞 ∈ 𝐿 ∨ 𝑟 ∈ 𝑈))) & ⊢ (𝜑 → 𝐴 < 𝐵) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ (𝐴[,]𝐵)(∀𝑞 ∈ 𝐿 𝑞 < 𝑥 ∧ ∀𝑟 ∈ 𝑈 𝑥 < 𝑟)) |
|
Theorem | dedekindicclemeu 14380* |
Lemma for dedekindicc 14382. Part of proving uniqueness. (Contributed
by Jim Kingdon, 15-Feb-2024.)
|
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐿 ⊆ (𝐴[,]𝐵)) & ⊢ (𝜑 → 𝑈 ⊆ (𝐴[,]𝐵)) & ⊢ (𝜑 → ∃𝑞 ∈ (𝐴[,]𝐵)𝑞 ∈ 𝐿)
& ⊢ (𝜑 → ∃𝑟 ∈ (𝐴[,]𝐵)𝑟 ∈ 𝑈)
& ⊢ (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)(𝑞 ∈ 𝐿 ↔ ∃𝑟 ∈ 𝐿 𝑞 < 𝑟))
& ⊢ (𝜑 → ∀𝑟 ∈ (𝐴[,]𝐵)(𝑟 ∈ 𝑈 ↔ ∃𝑞 ∈ 𝑈 𝑞 < 𝑟))
& ⊢ (𝜑 → (𝐿 ∩ 𝑈) = ∅) & ⊢ (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)∀𝑟 ∈ (𝐴[,]𝐵)(𝑞 < 𝑟 → (𝑞 ∈ 𝐿 ∨ 𝑟 ∈ 𝑈))) & ⊢ (𝜑 → 𝐴 < 𝐵)
& ⊢ (𝜑 → 𝐶 ∈ (𝐴[,]𝐵)) & ⊢ (𝜑 → (∀𝑞 ∈ 𝐿 𝑞 < 𝐶 ∧ ∀𝑟 ∈ 𝑈 𝐶 < 𝑟))
& ⊢ (𝜑 → 𝐷 ∈ (𝐴[,]𝐵)) & ⊢ (𝜑 → (∀𝑞 ∈ 𝐿 𝑞 < 𝐷 ∧ ∀𝑟 ∈ 𝑈 𝐷 < 𝑟))
& ⊢ (𝜑 → 𝐶 < 𝐷) ⇒ ⊢ (𝜑 → ⊥) |
|
Theorem | dedekindicclemicc 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.)
|
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐿 ⊆ (𝐴[,]𝐵)) & ⊢ (𝜑 → 𝑈 ⊆ (𝐴[,]𝐵)) & ⊢ (𝜑 → ∃𝑞 ∈ (𝐴[,]𝐵)𝑞 ∈ 𝐿)
& ⊢ (𝜑 → ∃𝑟 ∈ (𝐴[,]𝐵)𝑟 ∈ 𝑈)
& ⊢ (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)(𝑞 ∈ 𝐿 ↔ ∃𝑟 ∈ 𝐿 𝑞 < 𝑟))
& ⊢ (𝜑 → ∀𝑟 ∈ (𝐴[,]𝐵)(𝑟 ∈ 𝑈 ↔ ∃𝑞 ∈ 𝑈 𝑞 < 𝑟))
& ⊢ (𝜑 → (𝐿 ∩ 𝑈) = ∅) & ⊢ (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)∀𝑟 ∈ (𝐴[,]𝐵)(𝑞 < 𝑟 → (𝑞 ∈ 𝐿 ∨ 𝑟 ∈ 𝑈))) & ⊢ (𝜑 → 𝐴 < 𝐵) ⇒ ⊢ (𝜑 → ∃!𝑥 ∈ (𝐴[,]𝐵)(∀𝑞 ∈ 𝐿 𝑞 < 𝑥 ∧ ∀𝑟 ∈ 𝑈 𝑥 < 𝑟)) |
|
Theorem | dedekindicc 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.)
|
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐿 ⊆ (𝐴[,]𝐵)) & ⊢ (𝜑 → 𝑈 ⊆ (𝐴[,]𝐵)) & ⊢ (𝜑 → ∃𝑞 ∈ (𝐴[,]𝐵)𝑞 ∈ 𝐿)
& ⊢ (𝜑 → ∃𝑟 ∈ (𝐴[,]𝐵)𝑟 ∈ 𝑈)
& ⊢ (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)(𝑞 ∈ 𝐿 ↔ ∃𝑟 ∈ 𝐿 𝑞 < 𝑟))
& ⊢ (𝜑 → ∀𝑟 ∈ (𝐴[,]𝐵)(𝑟 ∈ 𝑈 ↔ ∃𝑞 ∈ 𝑈 𝑞 < 𝑟))
& ⊢ (𝜑 → (𝐿 ∩ 𝑈) = ∅) & ⊢ (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)∀𝑟 ∈ (𝐴[,]𝐵)(𝑞 < 𝑟 → (𝑞 ∈ 𝐿 ∨ 𝑟 ∈ 𝑈))) & ⊢ (𝜑 → 𝐴 < 𝐵) ⇒ ⊢ (𝜑 → ∃!𝑥 ∈ (𝐴(,)𝐵)(∀𝑞 ∈ 𝐿 𝑞 < 𝑥 ∧ ∀𝑟 ∈ 𝑈 𝑥 < 𝑟)) |
|
9.0.2 Intermediate value theorem
|
|
Theorem | ivthinclemlm 14383* |
Lemma for ivthinc 14392. The lower cut is bounded. (Contributed by
Jim Kingdon, 18-Feb-2024.)
|
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝑈 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 𝐵)
& ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)
& ⊢ (𝜑 → 𝐹 ∈ (𝐷–cn→ℂ)) & ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℝ) & ⊢ (𝜑 → ((𝐹‘𝐴) < 𝑈 ∧ 𝑈 < (𝐹‘𝐵))) & ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝑥 < 𝑦)) → (𝐹‘𝑥) < (𝐹‘𝑦))
& ⊢ 𝐿 = {𝑤 ∈ (𝐴[,]𝐵) ∣ (𝐹‘𝑤) < 𝑈}
& ⊢ 𝑅 = {𝑤 ∈ (𝐴[,]𝐵) ∣ 𝑈 < (𝐹‘𝑤)} ⇒ ⊢ (𝜑 → ∃𝑞 ∈ (𝐴[,]𝐵)𝑞 ∈ 𝐿) |
|
Theorem | ivthinclemum 14384* |
Lemma for ivthinc 14392. The upper cut is bounded. (Contributed by
Jim Kingdon, 18-Feb-2024.)
|
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝑈 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 𝐵)
& ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)
& ⊢ (𝜑 → 𝐹 ∈ (𝐷–cn→ℂ)) & ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℝ) & ⊢ (𝜑 → ((𝐹‘𝐴) < 𝑈 ∧ 𝑈 < (𝐹‘𝐵))) & ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝑥 < 𝑦)) → (𝐹‘𝑥) < (𝐹‘𝑦))
& ⊢ 𝐿 = {𝑤 ∈ (𝐴[,]𝐵) ∣ (𝐹‘𝑤) < 𝑈}
& ⊢ 𝑅 = {𝑤 ∈ (𝐴[,]𝐵) ∣ 𝑈 < (𝐹‘𝑤)} ⇒ ⊢ (𝜑 → ∃𝑟 ∈ (𝐴[,]𝐵)𝑟 ∈ 𝑅) |
|
Theorem | ivthinclemlopn 14385* |
Lemma for ivthinc 14392. The lower cut is open. (Contributed by
Jim
Kingdon, 6-Feb-2024.)
|
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝑈 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 𝐵)
& ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)
& ⊢ (𝜑 → 𝐹 ∈ (𝐷–cn→ℂ)) & ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℝ) & ⊢ (𝜑 → ((𝐹‘𝐴) < 𝑈 ∧ 𝑈 < (𝐹‘𝐵))) & ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝑥 < 𝑦)) → (𝐹‘𝑥) < (𝐹‘𝑦))
& ⊢ 𝐿 = {𝑤 ∈ (𝐴[,]𝐵) ∣ (𝐹‘𝑤) < 𝑈}
& ⊢ 𝑅 = {𝑤 ∈ (𝐴[,]𝐵) ∣ 𝑈 < (𝐹‘𝑤)}
& ⊢ (𝜑 → 𝑄 ∈ 𝐿) ⇒ ⊢ (𝜑 → ∃𝑟 ∈ 𝐿 𝑄 < 𝑟) |
|
Theorem | ivthinclemlr 14386* |
Lemma for ivthinc 14392. The lower cut is rounded. (Contributed by
Jim Kingdon, 18-Feb-2024.)
|
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝑈 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 𝐵)
& ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)
& ⊢ (𝜑 → 𝐹 ∈ (𝐷–cn→ℂ)) & ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℝ) & ⊢ (𝜑 → ((𝐹‘𝐴) < 𝑈 ∧ 𝑈 < (𝐹‘𝐵))) & ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝑥 < 𝑦)) → (𝐹‘𝑥) < (𝐹‘𝑦))
& ⊢ 𝐿 = {𝑤 ∈ (𝐴[,]𝐵) ∣ (𝐹‘𝑤) < 𝑈}
& ⊢ 𝑅 = {𝑤 ∈ (𝐴[,]𝐵) ∣ 𝑈 < (𝐹‘𝑤)} ⇒ ⊢ (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)(𝑞 ∈ 𝐿 ↔ ∃𝑟 ∈ 𝐿 𝑞 < 𝑟)) |
|
Theorem | ivthinclemuopn 14387* |
Lemma for ivthinc 14392. The upper cut is open. (Contributed by
Jim
Kingdon, 19-Feb-2024.)
|
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝑈 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 𝐵)
& ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)
& ⊢ (𝜑 → 𝐹 ∈ (𝐷–cn→ℂ)) & ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℝ) & ⊢ (𝜑 → ((𝐹‘𝐴) < 𝑈 ∧ 𝑈 < (𝐹‘𝐵))) & ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝑥 < 𝑦)) → (𝐹‘𝑥) < (𝐹‘𝑦))
& ⊢ 𝐿 = {𝑤 ∈ (𝐴[,]𝐵) ∣ (𝐹‘𝑤) < 𝑈}
& ⊢ 𝑅 = {𝑤 ∈ (𝐴[,]𝐵) ∣ 𝑈 < (𝐹‘𝑤)}
& ⊢ (𝜑 → 𝑆 ∈ 𝑅) ⇒ ⊢ (𝜑 → ∃𝑞 ∈ 𝑅 𝑞 < 𝑆) |
|
Theorem | ivthinclemur 14388* |
Lemma for ivthinc 14392. The upper cut is rounded. (Contributed by
Jim Kingdon, 18-Feb-2024.)
|
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝑈 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 𝐵)
& ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)
& ⊢ (𝜑 → 𝐹 ∈ (𝐷–cn→ℂ)) & ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℝ) & ⊢ (𝜑 → ((𝐹‘𝐴) < 𝑈 ∧ 𝑈 < (𝐹‘𝐵))) & ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝑥 < 𝑦)) → (𝐹‘𝑥) < (𝐹‘𝑦))
& ⊢ 𝐿 = {𝑤 ∈ (𝐴[,]𝐵) ∣ (𝐹‘𝑤) < 𝑈}
& ⊢ 𝑅 = {𝑤 ∈ (𝐴[,]𝐵) ∣ 𝑈 < (𝐹‘𝑤)} ⇒ ⊢ (𝜑 → ∀𝑟 ∈ (𝐴[,]𝐵)(𝑟 ∈ 𝑅 ↔ ∃𝑞 ∈ 𝑅 𝑞 < 𝑟)) |
|
Theorem | ivthinclemdisj 14389* |
Lemma for ivthinc 14392. The lower and upper cuts are disjoint.
(Contributed by Jim Kingdon, 18-Feb-2024.)
|
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝑈 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 𝐵)
& ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)
& ⊢ (𝜑 → 𝐹 ∈ (𝐷–cn→ℂ)) & ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℝ) & ⊢ (𝜑 → ((𝐹‘𝐴) < 𝑈 ∧ 𝑈 < (𝐹‘𝐵))) & ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝑥 < 𝑦)) → (𝐹‘𝑥) < (𝐹‘𝑦))
& ⊢ 𝐿 = {𝑤 ∈ (𝐴[,]𝐵) ∣ (𝐹‘𝑤) < 𝑈}
& ⊢ 𝑅 = {𝑤 ∈ (𝐴[,]𝐵) ∣ 𝑈 < (𝐹‘𝑤)} ⇒ ⊢ (𝜑 → (𝐿 ∩ 𝑅) = ∅) |
|
Theorem | ivthinclemloc 14390* |
Lemma for ivthinc 14392. Locatedness. (Contributed by Jim Kingdon,
18-Feb-2024.)
|
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝑈 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 𝐵)
& ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)
& ⊢ (𝜑 → 𝐹 ∈ (𝐷–cn→ℂ)) & ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℝ) & ⊢ (𝜑 → ((𝐹‘𝐴) < 𝑈 ∧ 𝑈 < (𝐹‘𝐵))) & ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝑥 < 𝑦)) → (𝐹‘𝑥) < (𝐹‘𝑦))
& ⊢ 𝐿 = {𝑤 ∈ (𝐴[,]𝐵) ∣ (𝐹‘𝑤) < 𝑈}
& ⊢ 𝑅 = {𝑤 ∈ (𝐴[,]𝐵) ∣ 𝑈 < (𝐹‘𝑤)} ⇒ ⊢ (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)∀𝑟 ∈ (𝐴[,]𝐵)(𝑞 < 𝑟 → (𝑞 ∈ 𝐿 ∨ 𝑟 ∈ 𝑅))) |
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Theorem | ivthinclemex 14391* |
Lemma for ivthinc 14392. Existence of a number between the lower
cut
and the upper cut. (Contributed by Jim Kingdon, 20-Feb-2024.)
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⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝑈 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 𝐵)
& ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)
& ⊢ (𝜑 → 𝐹 ∈ (𝐷–cn→ℂ)) & ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℝ) & ⊢ (𝜑 → ((𝐹‘𝐴) < 𝑈 ∧ 𝑈 < (𝐹‘𝐵))) & ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝑥 < 𝑦)) → (𝐹‘𝑥) < (𝐹‘𝑦))
& ⊢ 𝐿 = {𝑤 ∈ (𝐴[,]𝐵) ∣ (𝐹‘𝑤) < 𝑈}
& ⊢ 𝑅 = {𝑤 ∈ (𝐴[,]𝐵) ∣ 𝑈 < (𝐹‘𝑤)} ⇒ ⊢ (𝜑 → ∃!𝑧 ∈ (𝐴(,)𝐵)(∀𝑞 ∈ 𝐿 𝑞 < 𝑧 ∧ ∀𝑟 ∈ 𝑅 𝑧 < 𝑟)) |
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Theorem | ivthinc 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.)
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⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝑈 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 𝐵)
& ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)
& ⊢ (𝜑 → 𝐹 ∈ (𝐷–cn→ℂ)) & ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℝ) & ⊢ (𝜑 → ((𝐹‘𝐴) < 𝑈 ∧ 𝑈 < (𝐹‘𝐵))) & ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝑥 < 𝑦)) → (𝐹‘𝑥) < (𝐹‘𝑦)) ⇒ ⊢ (𝜑 → ∃𝑐 ∈ (𝐴(,)𝐵)(𝐹‘𝑐) = 𝑈) |
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Theorem | ivthdec 14393* |
The intermediate value theorem, decreasing case, for a strictly
monotonic function. (Contributed by Jim Kingdon, 20-Feb-2024.)
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⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝑈 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 𝐵)
& ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)
& ⊢ (𝜑 → 𝐹 ∈ (𝐷–cn→ℂ)) & ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℝ) & ⊢ (𝜑 → ((𝐹‘𝐵) < 𝑈 ∧ 𝑈 < (𝐹‘𝐴))) & ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝑥 < 𝑦)) → (𝐹‘𝑦) < (𝐹‘𝑥)) ⇒ ⊢ (𝜑 → ∃𝑐 ∈ (𝐴(,)𝐵)(𝐹‘𝑐) = 𝑈) |
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9.1 Derivatives
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9.1.1 Real and complex
differentiation
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9.1.1.1 Derivatives of functions of one complex
or real variable
|
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Syntax | climc 14394 |
The limit operator.
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class limℂ |
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Syntax | cdv 14395 |
The derivative operator.
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class D |
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Definition | df-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.)
|
⊢ limℂ = (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‘(𝑧 − 𝑥)) < 𝑑) → (abs‘((𝑓‘𝑧) − 𝑦)) < 𝑒)))}) |
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Definition | df-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 ℂ
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.)
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⊢ D = (𝑠 ∈ 𝒫 ℂ, 𝑓 ∈ (ℂ
↑pm 𝑠) ↦ ∪ 𝑥 ∈ ((int‘((MetOpen‘(abs
∘ − )) ↾t 𝑠))‘dom 𝑓)({𝑥} × ((𝑧 ∈ {𝑤 ∈ dom 𝑓 ∣ 𝑤 # 𝑥} ↦ (((𝑓‘𝑧) − (𝑓‘𝑥)) / (𝑧 − 𝑥))) limℂ 𝑥))) |
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Theorem | limcrcl 14398 |
Reverse closure for the limit operator. (Contributed by Mario Carneiro,
28-Dec-2016.)
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⊢ (𝐶 ∈ (𝐹 limℂ 𝐵) → (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ ∧ 𝐵 ∈ ℂ)) |
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Theorem | limccl 14399 |
Closure of the limit operator. (Contributed by Mario Carneiro,
25-Dec-2016.)
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⊢ (𝐹 limℂ 𝐵) ⊆ ℂ |
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Theorem | ellimc3apf 14400* |
Write the epsilon-delta definition of a limit. (Contributed by Mario
Carneiro, 28-Dec-2016.) (Revised by Jim Kingdon, 4-Nov-2023.)
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⊢ (𝜑 → 𝐹:𝐴⟶ℂ) & ⊢ (𝜑 → 𝐴 ⊆ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢
Ⅎ𝑧𝐹 ⇒ ⊢ (𝜑 → (𝐶 ∈ (𝐹 limℂ 𝐵) ↔ (𝐶 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑦 ∈
ℝ+ ∀𝑧 ∈ 𝐴 ((𝑧 # 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦) → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥)))) |