P. 249 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 24801-24900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	expcnOLD 24801*	Obsolete version of expcn 24799 as of 6-Apr-2025. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 23-Aug-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐽 = (TopOpen‘ℂfld)    ⇒   ⊢ (𝑁 ∈ ℕ0 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∈ (𝐽 Cn 𝐽))
Theorem	divccnOLD 24802*	Obsolete version of divccn 24800 as of 6-Apr-2025. (Contributed by Mario Carneiro, 5-May-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐽 = (TopOpen‘ℂfld)    ⇒   ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (𝑥 ∈ ℂ ↦ (𝑥 / 𝐴)) ∈ (𝐽 Cn 𝐽))
Theorem	sqcn 24803*	The square function on complex numbers is continuous. (Contributed by NM, 13-Jun-2007.) (Proof shortened by Mario Carneiro, 5-May-2014.)
⊢ 𝐽 = (TopOpen‘ℂfld)    ⇒   ⊢ (𝑥 ∈ ℂ ↦ (𝑥↑2)) ∈ (𝐽 Cn 𝐽)
12.4.11  Topological definitions using the reals
Syntax	cii 24804	Extend class notation with the unit interval.
class II
Syntax	ccncf 24805	Extend class notation to include the operation which returns a class of continuous complex functions.
class –cn→
Definition	df-ii 24806	Define the unit interval with the Euclidean topology. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 3-Sep-2015.)
⊢ II = (MetOpen‘((abs ∘ − ) ↾ ((0[,]1) × (0[,]1))))
Definition	df-cncf 24807*	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	iitopon 24808	The unit interval is a topological space. (Contributed by Mario Carneiro, 3-Sep-2015.)
⊢ II ∈ (TopOn‘(0[,]1))
Theorem	iitop 24809	The unit interval is a topological space. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ II ∈ Top
Theorem	iiuni 24810	The base set of the unit interval. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 15-Jan-2014.)
⊢ (0[,]1) = ∪ II
Theorem	dfii2 24811	Alternate definition of the unit interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ II = ((topGen‘ran (,)) ↾t (0[,]1))
Theorem	dfii3 24812	Alternate definition of the unit interval. (Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro, 3-Sep-2015.)
⊢ 𝐽 = (TopOpen‘ℂfld)    ⇒   ⊢ II = (𝐽 ↾t (0[,]1))
Theorem	dfii4 24813	Alternate definition of the unit interval. (Contributed by Mario Carneiro, 3-Sep-2015.)
⊢ 𝐼 = (ℂfld ↾s (0[,]1))    ⇒   ⊢ II = (TopOpen‘𝐼)
Theorem	dfii5 24814	The unit interval expressed as an order topology. (Contributed by Mario Carneiro, 9-Sep-2015.)
⊢ II = (ordTop‘( ≤ ∩ ((0[,]1) × (0[,]1))))
Theorem	iicmp 24815	The unit interval is compact. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 13-Jun-2014.)
⊢ II ∈ Comp
Theorem	iiconn 24816	The unit interval is connected. (Contributed by Mario Carneiro, 11-Feb-2015.)
⊢ II ∈ Conn
Theorem	cncfval 24817*	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 24818*	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 24819*	Version of elcncf 24818 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 24820	Reverse closure of the continuous function predicate. (Contributed by Mario Carneiro, 25-Aug-2014.)
⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐴 ⊆ ℂ)
Theorem	cncfrss2 24821	Reverse closure of the continuous function predicate. (Contributed by Mario Carneiro, 25-Aug-2014.)
⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐵 ⊆ ℂ)
Theorem	cncff 24822	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 24823*	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 24824*	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 24825*	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 24826	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 24827	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 24828	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 24829	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 24830	Absolute value is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
⊢ abs ∈ (ℂ–cn→ℝ)
Theorem	recncf 24831	Real part is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
⊢ ℜ ∈ (ℂ–cn→ℝ)
Theorem	imcncf 24832	Imaginary part is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
⊢ ℑ ∈ (ℂ–cn→ℝ)
Theorem	cjcncf 24833	Complex conjugate is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
⊢ ∗ ∈ (ℂ–cn→ℂ)
Theorem	mulc1cncf 24834*	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	divccncf 24835*	Division by a constant is continuous. (Contributed by Paul Chapman, 28-Nov-2007.)
⊢ 𝐹 = (𝑥 ∈ ℂ ↦ (𝑥 / 𝐴))    ⇒   ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → 𝐹 ∈ (ℂ–cn→ℂ))
Theorem	cncfco 24836	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	cncfcompt2 24837*	Composition of continuous functions. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝑅) ∈ (𝐴–cn→𝐵))    &   ⊢ (𝜑 → (𝑦 ∈ 𝐶 ↦ 𝑆) ∈ (𝐶–cn→𝐸))    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐶)    &   ⊢ (𝑦 = 𝑅 → 𝑆 = 𝑇)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝑇) ∈ (𝐴–cn→𝐸))
Theorem	cncfmet 24838	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	cncfcn 24839	Relate complex function continuity to topological continuity. (Contributed by Mario Carneiro, 17-Feb-2015.)
⊢ 𝐽 = (TopOpen‘ℂfld)    &   ⊢ 𝐾 = (𝐽 ↾t 𝐴)    &   ⊢ 𝐿 = (𝐽 ↾t 𝐵)    ⇒   ⊢ ((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → (𝐴–cn→𝐵) = (𝐾 Cn 𝐿))
Theorem	cncfcn1 24840	Relate complex function continuity to topological continuity. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Mario Carneiro, 7-Sep-2015.)
⊢ 𝐽 = (TopOpen‘ℂfld)    ⇒   ⊢ (ℂ–cn→ℂ) = (𝐽 Cn 𝐽)
Theorem	cncfmptc 24841*	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 24842*	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 24843*	Composition of continuous functions. –cn→ analogue of cnmpt11f 23587. (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	cncfmpt2f 24844*	Composition of continuous functions. –cn→ analogue of cnmpt12f 23589. (Contributed by Mario Carneiro, 3-Sep-2014.)
⊢ 𝐽 = (TopOpen‘ℂfld)    &   ⊢ (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	cncfmpt2ss 24845*	Composition of continuous functions in a subset. (Contributed by Mario Carneiro, 17-May-2016.)
⊢ 𝐽 = (TopOpen‘ℂfld)    &   ⊢ 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽)    &   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝑋–cn→𝑆))    &   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝑋–cn→𝑆))    &   ⊢ 𝑆 ⊆ ℂ    &   ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴𝐹𝐵) ∈ 𝑆)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴𝐹𝐵)) ∈ (𝑋–cn→𝑆))
Theorem	addccncf 24846*	Adding a constant is a continuous function. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)
⊢ 𝐹 = (𝑥 ∈ ℂ ↦ (𝑥 + 𝐴))    ⇒   ⊢ (𝐴 ∈ ℂ → 𝐹 ∈ (ℂ–cn→ℂ))
Theorem	idcncf 24847	The identity function is a continuous function on ℂ. (Contributed by Jeff Madsen, 11-Jun-2010.) (Moved into main set.mm as cncfmptid 24842 and may be deleted by mathbox owner, JM. --MC 12-Sep-2015.) (Revised by Mario Carneiro, 12-Sep-2015.)
⊢ 𝐹 = (𝑥 ∈ ℂ ↦ 𝑥)    ⇒   ⊢ 𝐹 ∈ (ℂ–cn→ℂ)
Theorem	sub1cncf 24848*	Subtracting a constant is a continuous function. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)
⊢ 𝐹 = (𝑥 ∈ ℂ ↦ (𝑥 − 𝐴))    ⇒   ⊢ (𝐴 ∈ ℂ → 𝐹 ∈ (ℂ–cn→ℂ))
Theorem	sub2cncf 24849*	Subtraction from a constant is a continuous function. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)
⊢ 𝐹 = (𝑥 ∈ ℂ ↦ (𝐴 − 𝑥))    ⇒   ⊢ (𝐴 ∈ ℂ → 𝐹 ∈ (ℂ–cn→ℂ))
Theorem	cdivcncf 24850*	Division with a constant numerator is continuous. (Contributed by Mario Carneiro, 28-Dec-2016.)
⊢ 𝐹 = (𝑥 ∈ (ℂ ∖ {0}) ↦ (𝐴 / 𝑥))    ⇒   ⊢ (𝐴 ∈ ℂ → 𝐹 ∈ ((ℂ ∖ {0})–cn→ℂ))
Theorem	negcncf 24851*	The negative function is continuous. (Contributed by Mario Carneiro, 30-Dec-2016.) Avoid ax-mulf 11201. (Revised by GG, 16-Mar-2025.)
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ -𝑥)    ⇒   ⊢ (𝐴 ⊆ ℂ → 𝐹 ∈ (𝐴–cn→ℂ))
Theorem	negcncfOLD 24852*	Obsolete version of negcncf 24851 as of 9-Apr-2025. (Contributed by Mario Carneiro, 30-Dec-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ -𝑥)    ⇒   ⊢ (𝐴 ⊆ ℂ → 𝐹 ∈ (𝐴–cn→ℂ))
Theorem	negfcncf 24853*	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	abscncfALT 24854	Absolute value is continuous. Alternate proof of abscncf 24830. (Contributed by NM, 6-Jun-2008.) (Revised by Mario Carneiro, 10-Sep-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ abs ∈ (ℂ–cn→ℝ)
Theorem	cncfcnvcn 24855	Rewrite cmphaushmeo 23723 for functions on the complex numbers. (Contributed by Mario Carneiro, 19-Feb-2015.)
⊢ 𝐽 = (TopOpen‘ℂfld)    &   ⊢ 𝐾 = (𝐽 ↾t 𝑋)    ⇒   ⊢ ((𝐾 ∈ Comp ∧ 𝐹 ∈ (𝑋–cn→𝑌)) → (𝐹:𝑋–1-1-onto→𝑌 ↔ ◡𝐹 ∈ (𝑌–cn→𝑋)))
Theorem	expcncf 24856*	The power function on complex numbers, for fixed exponent N, is continuous. Similar to expcn 24799. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
⊢ (𝑁 ∈ ℕ0 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∈ (ℂ–cn→ℂ))
Theorem	cnmptre 24857*	Lemma for iirevcn 24860 and related functions. (Contributed by Mario Carneiro, 6-Jun-2014.)
⊢ 𝑅 = (TopOpen‘ℂfld)    &   ⊢ 𝐽 = ((topGen‘ran (,)) ↾t 𝐴)    &   ⊢ 𝐾 = ((topGen‘ran (,)) ↾t 𝐵)    &   ⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ (𝜑 → 𝐵 ⊆ ℝ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐹 ∈ 𝐵)    &   ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ 𝐹) ∈ (𝑅 Cn 𝑅))    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐹) ∈ (𝐽 Cn 𝐾))
Theorem	cnmpopc 24858*	Piecewise definition of a continuous function on a real interval. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 5-Jun-2014.)
⊢ 𝑅 = (topGen‘ran (,))    &   ⊢ 𝑀 = (𝑅 ↾t (𝐴[,]𝐵))    &   ⊢ 𝑁 = (𝑅 ↾t (𝐵[,]𝐶))    &   ⊢ 𝑂 = (𝑅 ↾t (𝐴[,]𝐶))    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ (𝐴[,]𝐶))    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ ((𝜑 ∧ (𝑥 = 𝐵 ∧ 𝑦 ∈ 𝑋)) → 𝐷 = 𝐸)    &   ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵), 𝑦 ∈ 𝑋 ↦ 𝐷) ∈ ((𝑀 ×t 𝐽) Cn 𝐾))    &   ⊢ (𝜑 → (𝑥 ∈ (𝐵[,]𝐶), 𝑦 ∈ 𝑋 ↦ 𝐸) ∈ ((𝑁 ×t 𝐽) Cn 𝐾))    ⇒   ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐶), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)) ∈ ((𝑂 ×t 𝐽) Cn 𝐾))
Theorem	iirev 24859	Reverse the unit interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ (𝑋 ∈ (0[,]1) → (1 − 𝑋) ∈ (0[,]1))
Theorem	iirevcn 24860	The reversion function is a continuous map of the unit interval. (Contributed by Mario Carneiro, 6-Jun-2014.)
⊢ (𝑥 ∈ (0[,]1) ↦ (1 − 𝑥)) ∈ (II Cn II)
Theorem	iihalf1 24861	Map the first half of II into II. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ (𝑋 ∈ (0[,](1 / 2)) → (2 · 𝑋) ∈ (0[,]1))
Theorem	iihalf1cn 24862	The first half function is a continuous map. (Contributed by Mario Carneiro, 6-Jun-2014.) Avoid ax-mulf 11201. (Revised by GG, 16-Mar-2025.)
⊢ 𝐽 = ((topGen‘ran (,)) ↾t (0[,](1 / 2)))    ⇒   ⊢ (𝑥 ∈ (0[,](1 / 2)) ↦ (2 · 𝑥)) ∈ (𝐽 Cn II)
Theorem	iihalf1cnOLD 24863	Obsolete version of iihalf1cn 24862 as of 9-Apr-2025. (Contributed by Mario Carneiro, 6-Jun-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐽 = ((topGen‘ran (,)) ↾t (0[,](1 / 2)))    ⇒   ⊢ (𝑥 ∈ (0[,](1 / 2)) ↦ (2 · 𝑥)) ∈ (𝐽 Cn II)
Theorem	iihalf2 24864	Map the second half of II into II. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ (𝑋 ∈ ((1 / 2)[,]1) → ((2 · 𝑋) − 1) ∈ (0[,]1))
Theorem	iihalf2cn 24865	The second half function is a continuous map. (Contributed by Mario Carneiro, 6-Jun-2014.) Avoid ax-mulf 11201. (Revised by GG, 16-Mar-2025.)
⊢ 𝐽 = ((topGen‘ran (,)) ↾t ((1 / 2)[,]1))    ⇒   ⊢ (𝑥 ∈ ((1 / 2)[,]1) ↦ ((2 · 𝑥) − 1)) ∈ (𝐽 Cn II)
Theorem	iihalf2cnOLD 24866	Obsolete version of iihalf2cn 24865 as of 9-Apr-2025. (Contributed by Mario Carneiro, 6-Jun-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐽 = ((topGen‘ran (,)) ↾t ((1 / 2)[,]1))    ⇒   ⊢ (𝑥 ∈ ((1 / 2)[,]1) ↦ ((2 · 𝑥) − 1)) ∈ (𝐽 Cn II)
Theorem	elii1 24867	Divide the unit interval into two pieces. (Contributed by Mario Carneiro, 7-Jun-2014.)
⊢ (𝑋 ∈ (0[,](1 / 2)) ↔ (𝑋 ∈ (0[,]1) ∧ 𝑋 ≤ (1 / 2)))
Theorem	elii2 24868	Divide the unit interval into two pieces. (Contributed by Mario Carneiro, 7-Jun-2014.)
⊢ ((𝑋 ∈ (0[,]1) ∧ ¬ 𝑋 ≤ (1 / 2)) → 𝑋 ∈ ((1 / 2)[,]1))
Theorem	iimulcl 24869	The unit interval is closed under multiplication. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ ((𝐴 ∈ (0[,]1) ∧ 𝐵 ∈ (0[,]1)) → (𝐴 · 𝐵) ∈ (0[,]1))
Theorem	iimulcn 24870*	Multiplication is a continuous function on the unit interval. (Contributed by Mario Carneiro, 8-Jun-2014.) Avoid ax-mulf 11201. (Revised by GG, 16-Mar-2025.)
⊢ (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝑥 · 𝑦)) ∈ ((II ×t II) Cn II)
Theorem	iimulcnOLD 24871*	Obsolete version of iimulcn 24870 as of 9-Apr-2025. (Contributed by Mario Carneiro, 8-Jun-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝑥 · 𝑦)) ∈ ((II ×t II) Cn II)
Theorem	icoopnst 24872	A half-open interval starting at 𝐴 is open in the closed interval from 𝐴 to 𝐵. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 15-Dec-2013.)
⊢ 𝐽 = (MetOpen‘((abs ∘ − ) ↾ ((𝐴[,]𝐵) × (𝐴[,]𝐵))))    ⇒   ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐶 ∈ (𝐴(,]𝐵) → (𝐴[,)𝐶) ∈ 𝐽))
Theorem	iocopnst 24873	A half-open interval ending at 𝐵 is open in the closed interval from 𝐴 to 𝐵. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 15-Dec-2013.)
⊢ 𝐽 = (MetOpen‘((abs ∘ − ) ↾ ((𝐴[,]𝐵) × (𝐴[,]𝐵))))    ⇒   ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐶 ∈ (𝐴[,)𝐵) → (𝐶(,]𝐵) ∈ 𝐽))
Theorem	icchmeo 24874*	The natural bijection from [0, 1] to an arbitrary nontrivial closed interval [𝐴, 𝐵] is a homeomorphism. (Contributed by Mario Carneiro, 8-Sep-2015.) Avoid ax-mulf 11201. (Revised by GG, 16-Mar-2025.)
⊢ 𝐽 = (TopOpen‘ℂfld)    &   ⊢ 𝐹 = (𝑥 ∈ (0[,]1) ↦ ((𝑥 · 𝐵) + ((1 − 𝑥) · 𝐴)))    ⇒   ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐹 ∈ (IIHomeo(𝐽 ↾t (𝐴[,]𝐵))))
Theorem	icchmeoOLD 24875*	Obsolete version of icchmeo 24874 as of 9-Apr-2025. (Contributed by Mario Carneiro, 8-Sep-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐽 = (TopOpen‘ℂfld)    &   ⊢ 𝐹 = (𝑥 ∈ (0[,]1) ↦ ((𝑥 · 𝐵) + ((1 − 𝑥) · 𝐴)))    ⇒   ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐹 ∈ (IIHomeo(𝐽 ↾t (𝐴[,]𝐵))))
Theorem	icopnfcnv 24876*	Define a bijection from [0, 1) to [0, +∞). (Contributed by Mario Carneiro, 9-Sep-2015.)
⊢ 𝐹 = (𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥)))    ⇒   ⊢ (𝐹:(0[,)1)–1-1-onto→(0[,)+∞) ∧ ◡𝐹 = (𝑦 ∈ (0[,)+∞) ↦ (𝑦 / (1 + 𝑦))))
Theorem	icopnfhmeo 24877*	The defined bijection from [0, 1) to [0, +∞) is an order isomorphism and a homeomorphism. (Contributed by Mario Carneiro, 9-Sep-2015.)
⊢ 𝐹 = (𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥)))    &   ⊢ 𝐽 = (TopOpen‘ℂfld)    ⇒   ⊢ (𝐹 Isom < , < ((0[,)1), (0[,)+∞)) ∧ 𝐹 ∈ ((𝐽 ↾t (0[,)1))Homeo(𝐽 ↾t (0[,)+∞))))
Theorem	iccpnfcnv 24878*	Define a bijection from [0, 1] to [0, +∞]. (Contributed by Mario Carneiro, 9-Sep-2015.)
⊢ 𝐹 = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥))))    ⇒   ⊢ (𝐹:(0[,]1)–1-1-onto→(0[,]+∞) ∧ ◡𝐹 = (𝑦 ∈ (0[,]+∞) ↦ if(𝑦 = +∞, 1, (𝑦 / (1 + 𝑦)))))
Theorem	iccpnfhmeo 24879	The defined bijection from [0, 1] to [0, +∞] is an order isomorphism and a homeomorphism. (Contributed by Mario Carneiro, 8-Sep-2015.)
⊢ 𝐹 = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥))))    &   ⊢ 𝐾 = ((ordTop‘ ≤ ) ↾t (0[,]+∞))    ⇒   ⊢ (𝐹 Isom < , < ((0[,]1), (0[,]+∞)) ∧ 𝐹 ∈ (IIHomeo𝐾))
Theorem	xrhmeo 24880*	The bijection from [-1, 1] to the extended reals is an order isomorphism and a homeomorphism. (Contributed by Mario Carneiro, 9-Sep-2015.)
⊢ 𝐹 = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥))))    &   ⊢ 𝐺 = (𝑦 ∈ (-1[,]1) ↦ if(0 ≤ 𝑦, (𝐹‘𝑦), -𝑒(𝐹‘-𝑦)))    &   ⊢ 𝐽 = (TopOpen‘ℂfld)    ⇒   ⊢ (𝐺 Isom < , < ((-1[,]1), ℝ*) ∧ 𝐺 ∈ ((𝐽 ↾t (-1[,]1))Homeo(ordTop‘ ≤ )))
Theorem	xrhmph 24881	The extended reals are homeomorphic to the interval [0, 1]. (Contributed by Mario Carneiro, 9-Sep-2015.)
⊢ II ≃ (ordTop‘ ≤ )
Theorem	xrcmp 24882	The topology of the extended reals is compact. Since the topology of the extended reals extends the topology of the reals (by xrtgioo 24731), this means that ℝ* is a compactification of ℝ. (Contributed by Mario Carneiro, 9-Sep-2015.)
⊢ (ordTop‘ ≤ ) ∈ Comp
Theorem	xrconn 24883	The topology of the extended reals is connected. (Contributed by Mario Carneiro, 9-Sep-2015.)
⊢ (ordTop‘ ≤ ) ∈ Conn
Theorem	icccvx 24884	A linear combination of two reals lies in the interval between them. Equivalently, a closed interval is a convex set. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐶 ∈ (𝐴[,]𝐵) ∧ 𝐷 ∈ (𝐴[,]𝐵) ∧ 𝑇 ∈ (0[,]1)) → (((1 − 𝑇) · 𝐶) + (𝑇 · 𝐷)) ∈ (𝐴[,]𝐵)))
Theorem	oprpiece1res1 24885*	Restriction to the first part of a piecewise defined function. (Contributed by Jeff Madsen, 11-Jun-2010.) (Proof shortened by Mario Carneiro, 3-Sep-2015.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    &   ⊢ 𝐴 ≤ 𝐵    &   ⊢ 𝑅 ∈ V    &   ⊢ 𝑆 ∈ V    &   ⊢ 𝐾 ∈ (𝐴[,]𝐵)    &   ⊢ 𝐹 = (𝑥 ∈ (𝐴[,]𝐵), 𝑦 ∈ 𝐶 ↦ if(𝑥 ≤ 𝐾, 𝑅, 𝑆))    &   ⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐾), 𝑦 ∈ 𝐶 ↦ 𝑅)    ⇒   ⊢ (𝐹 ↾ ((𝐴[,]𝐾) × 𝐶)) = 𝐺
Theorem	oprpiece1res2 24886*	Restriction to the second part of a piecewise defined function. (Contributed by Jeff Madsen, 11-Jun-2010.) (Proof shortened by Mario Carneiro, 3-Sep-2015.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    &   ⊢ 𝐴 ≤ 𝐵    &   ⊢ 𝑅 ∈ V    &   ⊢ 𝑆 ∈ V    &   ⊢ 𝐾 ∈ (𝐴[,]𝐵)    &   ⊢ 𝐹 = (𝑥 ∈ (𝐴[,]𝐵), 𝑦 ∈ 𝐶 ↦ if(𝑥 ≤ 𝐾, 𝑅, 𝑆))    &   ⊢ (𝑥 = 𝐾 → 𝑅 = 𝑃)    &   ⊢ (𝑥 = 𝐾 → 𝑆 = 𝑄)    &   ⊢ (𝑦 ∈ 𝐶 → 𝑃 = 𝑄)    &   ⊢ 𝐺 = (𝑥 ∈ (𝐾[,]𝐵), 𝑦 ∈ 𝐶 ↦ 𝑆)    ⇒   ⊢ (𝐹 ↾ ((𝐾[,]𝐵) × 𝐶)) = 𝐺
Theorem	cnrehmeo 24887*	The canonical bijection from (ℝ × ℝ) to ℂ described in cnref1o 12993 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.) Avoid ax-mulf 11201. (Revised by GG, 16-Mar-2025.)
⊢ 𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + (i · 𝑦)))    &   ⊢ 𝐽 = (topGen‘ran (,))    &   ⊢ 𝐾 = (TopOpen‘ℂfld)    ⇒   ⊢ 𝐹 ∈ ((𝐽 ×t 𝐽)Homeo𝐾)
Theorem	cnrehmeoOLD 24888*	Obsolete version of cnrehmeo 24887 as of 9-Apr-2025. (Contributed by Mario Carneiro, 25-Aug-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + (i · 𝑦)))    &   ⊢ 𝐽 = (topGen‘ran (,))    &   ⊢ 𝐾 = (TopOpen‘ℂfld)    ⇒   ⊢ 𝐹 ∈ ((𝐽 ×t 𝐽)Homeo𝐾)
Theorem	cnheiborlem 24889*	Lemma for cnheibor 24890. (Contributed by Mario Carneiro, 14-Sep-2014.)
⊢ 𝐽 = (TopOpen‘ℂfld)    &   ⊢ 𝑇 = (𝐽 ↾t 𝑋)    &   ⊢ 𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + (i · 𝑦)))    &   ⊢ 𝑌 = (𝐹 “ ((-𝑅[,]𝑅) × (-𝑅[,]𝑅)))    ⇒   ⊢ ((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) → 𝑇 ∈ Comp)
Theorem	cnheibor 24890*	Heine-Borel theorem for complex numbers. A subset of ℂ is compact iff it is closed and bounded. (Contributed by Mario Carneiro, 14-Sep-2014.)
⊢ 𝐽 = (TopOpen‘ℂfld)    &   ⊢ 𝑇 = (𝐽 ↾t 𝑋)    ⇒   ⊢ (𝑋 ⊆ ℂ → (𝑇 ∈ Comp ↔ (𝑋 ∈ (Clsd‘𝐽) ∧ ∃𝑟 ∈ ℝ ∀𝑥 ∈ 𝑋 (abs‘𝑥) ≤ 𝑟)))
Theorem	cnllycmp 24891	The topology on the complex numbers is locally compact. (Contributed by Mario Carneiro, 2-Mar-2015.)
⊢ 𝐽 = (TopOpen‘ℂfld)    ⇒   ⊢ 𝐽 ∈ 𝑛-Locally Comp
Theorem	rellycmp 24892	The topology on the reals is locally compact. (Contributed by Mario Carneiro, 2-Mar-2015.)
⊢ (topGen‘ran (,)) ∈ 𝑛-Locally Comp
Theorem	bndth 24893*	The Boundedness Theorem. A continuous function from a compact topological space to the reals is bounded (above). (Boundedness below is obtained by applying this theorem to -𝐹.) (Contributed by Mario Carneiro, 12-Aug-2014.)
⊢ 𝑋 = ∪ 𝐽    &   ⊢ 𝐾 = (topGen‘ran (,))    &   ⊢ (𝜑 → 𝐽 ∈ Comp)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑋 (𝐹‘𝑦) ≤ 𝑥)
Theorem	evth 24894*	The Extreme Value Theorem. A continuous function from a nonempty compact topological space to the reals attains its maximum at some point in the domain. (Contributed by Mario Carneiro, 12-Aug-2014.)
⊢ 𝑋 = ∪ 𝐽    &   ⊢ 𝐾 = (topGen‘ran (,))    &   ⊢ (𝜑 → 𝐽 ∈ Comp)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))    &   ⊢ (𝜑 → 𝑋 ≠ ∅)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐹‘𝑦) ≤ (𝐹‘𝑥))
Theorem	evth2 24895*	The Extreme Value Theorem, minimum version. A continuous function from a nonempty compact topological space to the reals attains its minimum at some point in the domain. (Contributed by Mario Carneiro, 12-Aug-2014.)
⊢ 𝑋 = ∪ 𝐽    &   ⊢ 𝐾 = (topGen‘ran (,))    &   ⊢ (𝜑 → 𝐽 ∈ Comp)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))    &   ⊢ (𝜑 → 𝑋 ≠ ∅)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐹‘𝑥) ≤ (𝐹‘𝑦))
Theorem	lebnumlem1 24896*	Lemma for lebnum 24899. The function 𝐹 measures the sum of all of the distances to escape the sets of the cover. Since by assumption it is a cover, there is at least one set which covers a given point, and since it is open, the point is a positive distance from the edge of the set. Thus, the sum is a strictly positive number. (Contributed by Mario Carneiro, 14-Feb-2015.) (Revised by AV, 30-Sep-2020.)
⊢ 𝐽 = (MetOpen‘𝐷)    &   ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋))    &   ⊢ (𝜑 → 𝐽 ∈ Comp)    &   ⊢ (𝜑 → 𝑈 ⊆ 𝐽)    &   ⊢ (𝜑 → 𝑋 = ∪ 𝑈)    &   ⊢ (𝜑 → 𝑈 ∈ Fin)    &   ⊢ (𝜑 → ¬ 𝑋 ∈ 𝑈)    &   ⊢ 𝐹 = (𝑦 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑈 inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ))    ⇒   ⊢ (𝜑 → 𝐹:𝑋⟶ℝ+)
Theorem	lebnumlem2 24897*	Lemma for lebnum 24899. As a finite sum of point-to-set distance functions, which are continuous by metdscn 24781, the function 𝐹 is also continuous. (Contributed by Mario Carneiro, 14-Feb-2015.) (Revised by AV, 30-Sep-2020.)
⊢ 𝐽 = (MetOpen‘𝐷)    &   ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋))    &   ⊢ (𝜑 → 𝐽 ∈ Comp)    &   ⊢ (𝜑 → 𝑈 ⊆ 𝐽)    &   ⊢ (𝜑 → 𝑋 = ∪ 𝑈)    &   ⊢ (𝜑 → 𝑈 ∈ Fin)    &   ⊢ (𝜑 → ¬ 𝑋 ∈ 𝑈)    &   ⊢ 𝐹 = (𝑦 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑈 inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ))    &   ⊢ 𝐾 = (topGen‘ran (,))    ⇒   ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))
Theorem	lebnumlem3 24898*	Lemma for lebnum 24899. By the previous lemmas, 𝐹 is continuous and positive on a compact set, so it has a positive minimum 𝑟. Then setting 𝑑 = 𝑟 / ♯(𝑈), since for each 𝑢 ∈ 𝑈 we have ball(𝑥, 𝑑) ⊆ 𝑢 iff 𝑑 ≤ 𝑑(𝑥, 𝑋 ∖ 𝑢), if ¬ ball(𝑥, 𝑑) ⊆ 𝑢 for all 𝑢 then summing over 𝑢 yields Σ𝑢 ∈ 𝑈𝑑(𝑥, 𝑋 ∖ 𝑢) = 𝐹(𝑥) < Σ𝑢 ∈ 𝑈𝑑 = 𝑟, in contradiction to the assumption that 𝑟 is the minimum of 𝐹. (Contributed by Mario Carneiro, 14-Feb-2015.) (Revised by Mario Carneiro, 5-Sep-2015.) (Revised by AV, 30-Sep-2020.)
⊢ 𝐽 = (MetOpen‘𝐷)    &   ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋))    &   ⊢ (𝜑 → 𝐽 ∈ Comp)    &   ⊢ (𝜑 → 𝑈 ⊆ 𝐽)    &   ⊢ (𝜑 → 𝑋 = ∪ 𝑈)    &   ⊢ (𝜑 → 𝑈 ∈ Fin)    &   ⊢ (𝜑 → ¬ 𝑋 ∈ 𝑈)    &   ⊢ 𝐹 = (𝑦 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑈 inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ))    &   ⊢ 𝐾 = (topGen‘ran (,))    ⇒   ⊢ (𝜑 → ∃𝑑 ∈ ℝ+ ∀𝑥 ∈ 𝑋 ∃𝑢 ∈ 𝑈 (𝑥(ball‘𝐷)𝑑) ⊆ 𝑢)
Theorem	lebnum 24899*	The Lebesgue number lemma, or Lebesgue covering lemma. If 𝑋 is a compact metric space and 𝑈 is an open cover of 𝑋, then there exists a positive real number 𝑑 such that every ball of size 𝑑 (and every subset of a ball of size 𝑑, including every subset of diameter less than 𝑑) is a subset of some member of the cover. (Contributed by Mario Carneiro, 14-Feb-2015.) (Proof shortened by Mario Carneiro, 5-Sep-2015.) (Proof shortened by AV, 30-Sep-2020.)
⊢ 𝐽 = (MetOpen‘𝐷)    &   ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋))    &   ⊢ (𝜑 → 𝐽 ∈ Comp)    &   ⊢ (𝜑 → 𝑈 ⊆ 𝐽)    &   ⊢ (𝜑 → 𝑋 = ∪ 𝑈)    ⇒   ⊢ (𝜑 → ∃𝑑 ∈ ℝ+ ∀𝑥 ∈ 𝑋 ∃𝑢 ∈ 𝑈 (𝑥(ball‘𝐷)𝑑) ⊆ 𝑢)
Theorem	xlebnum 24900*	Generalize lebnum 24899 to extended metrics. (Contributed by Mario Carneiro, 5-Sep-2015.)
⊢ 𝐽 = (MetOpen‘𝐷)    &   ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋))    &   ⊢ (𝜑 → 𝐽 ∈ Comp)    &   ⊢ (𝜑 → 𝑈 ⊆ 𝐽)    &   ⊢ (𝜑 → 𝑋 = ∪ 𝑈)    ⇒   ⊢ (𝜑 → ∃𝑑 ∈ ℝ+ ∀𝑥 ∈ 𝑋 ∃𝑢 ∈ 𝑈 (𝑥(ball‘𝐷)𝑑) ⊆ 𝑢)