P. 248 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 24701-24800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	nmoeq0 24701	The operator norm is zero only for the zero operator. (Contributed by Mario Carneiro, 20-Oct-2015.)
⊢ 𝑁 = (𝑆 normOp 𝑇)    &   ⊢ 𝑉 = (Base‘𝑆)    &   ⊢ 0 = (0g‘𝑇)    ⇒   ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) → ((𝑁‘𝐹) = 0 ↔ 𝐹 = (𝑉 × { 0 })))
Theorem	nmoco 24702	An upper bound on the operator norm of a composition. (Contributed by Mario Carneiro, 20-Oct-2015.)
⊢ 𝑁 = (𝑆 normOp 𝑈)    &   ⊢ 𝐿 = (𝑇 normOp 𝑈)    &   ⊢ 𝑀 = (𝑆 normOp 𝑇)    ⇒   ⊢ ((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → (𝑁‘(𝐹 ∘ 𝐺)) ≤ ((𝐿‘𝐹) · (𝑀‘𝐺)))
Theorem	nghmco 24703	The composition of normed group homomorphisms is a normed group homomorphism. (Contributed by Mario Carneiro, 20-Oct-2015.)
⊢ ((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → (𝐹 ∘ 𝐺) ∈ (𝑆 NGHom 𝑈))
Theorem	nmotri 24704	Triangle inequality for the operator norm. (Contributed by Mario Carneiro, 20-Oct-2015.)
⊢ 𝑁 = (𝑆 normOp 𝑇)    &   ⊢ + = (+g‘𝑇)    ⇒   ⊢ ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → (𝑁‘(𝐹 ∘f + 𝐺)) ≤ ((𝑁‘𝐹) + (𝑁‘𝐺)))
Theorem	nghmplusg 24705	The sum of two bounded linear operators is bounded linear. (Contributed by Mario Carneiro, 20-Oct-2015.)
⊢ + = (+g‘𝑇)    ⇒   ⊢ ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → (𝐹 ∘f + 𝐺) ∈ (𝑆 NGHom 𝑇))
Theorem	0nghm 24706	The zero operator is a normed group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
⊢ 𝑉 = (Base‘𝑆)    &   ⊢ 0 = (0g‘𝑇)    ⇒   ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → (𝑉 × { 0 }) ∈ (𝑆 NGHom 𝑇))
Theorem	nmoid 24707	The operator norm of the identity function on a nontrivial group. (Contributed by Mario Carneiro, 20-Oct-2015.)
⊢ 𝑁 = (𝑆 normOp 𝑆)    &   ⊢ 𝑉 = (Base‘𝑆)    &   ⊢ 0 = (0g‘𝑆)    ⇒   ⊢ ((𝑆 ∈ NrmGrp ∧ { 0 } ⊊ 𝑉) → (𝑁‘( I ↾ 𝑉)) = 1)
Theorem	idnghm 24708	The identity operator is a normed group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
⊢ 𝑉 = (Base‘𝑆)    ⇒   ⊢ (𝑆 ∈ NrmGrp → ( I ↾ 𝑉) ∈ (𝑆 NGHom 𝑆))
Theorem	nmods 24709	Upper bound for the distance between the values of a bounded linear operator. (Contributed by Mario Carneiro, 22-Oct-2015.)
⊢ 𝑁 = (𝑆 normOp 𝑇)    &   ⊢ 𝑉 = (Base‘𝑆)    &   ⊢ 𝐶 = (dist‘𝑆)    &   ⊢ 𝐷 = (dist‘𝑇)    ⇒   ⊢ ((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ((𝐹‘𝐴)𝐷(𝐹‘𝐵)) ≤ ((𝑁‘𝐹) · (𝐴𝐶𝐵)))
Theorem	nghmcn 24710	A normed group homomorphism is a continuous function. (Contributed by Mario Carneiro, 20-Oct-2015.)
⊢ 𝐽 = (TopOpen‘𝑆)    &   ⊢ 𝐾 = (TopOpen‘𝑇)    ⇒   ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝐹 ∈ (𝐽 Cn 𝐾))
Theorem	isnmhm 24711	A normed module homomorphism is a left module homomorphism which is also a normed group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
⊢ (𝐹 ∈ (𝑆 NMHom 𝑇) ↔ ((𝑆 ∈ NrmMod ∧ 𝑇 ∈ NrmMod) ∧ (𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹 ∈ (𝑆 NGHom 𝑇))))
Theorem	nmhmrcl1 24712	Reverse closure for a normed module homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
⊢ (𝐹 ∈ (𝑆 NMHom 𝑇) → 𝑆 ∈ NrmMod)
Theorem	nmhmrcl2 24713	Reverse closure for a normed module homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
⊢ (𝐹 ∈ (𝑆 NMHom 𝑇) → 𝑇 ∈ NrmMod)
Theorem	nmhmlmhm 24714	A normed module homomorphism is a left module homomorphism (a linear operator). (Contributed by Mario Carneiro, 18-Oct-2015.)
⊢ (𝐹 ∈ (𝑆 NMHom 𝑇) → 𝐹 ∈ (𝑆 LMHom 𝑇))
Theorem	nmhmnghm 24715	A normed module homomorphism is a normed group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
⊢ (𝐹 ∈ (𝑆 NMHom 𝑇) → 𝐹 ∈ (𝑆 NGHom 𝑇))
Theorem	nmhmghm 24716	A normed module homomorphism is a group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
⊢ (𝐹 ∈ (𝑆 NMHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
Theorem	isnmhm2 24717	A normed module homomorphism is a left module homomorphism with bounded norm (a bounded linear operator). (Contributed by Mario Carneiro, 18-Oct-2015.)
⊢ 𝑁 = (𝑆 normOp 𝑇)    ⇒   ⊢ ((𝑆 ∈ NrmMod ∧ 𝑇 ∈ NrmMod ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) → (𝐹 ∈ (𝑆 NMHom 𝑇) ↔ (𝑁‘𝐹) ∈ ℝ))
Theorem	nmhmcl 24718	A normed module homomorphism has a real operator norm. (Contributed by Mario Carneiro, 18-Oct-2015.)
⊢ 𝑁 = (𝑆 normOp 𝑇)    ⇒   ⊢ (𝐹 ∈ (𝑆 NMHom 𝑇) → (𝑁‘𝐹) ∈ ℝ)
Theorem	idnmhm 24719	The identity operator is a bounded linear operator. (Contributed by Mario Carneiro, 20-Oct-2015.)
⊢ 𝑉 = (Base‘𝑆)    ⇒   ⊢ (𝑆 ∈ NrmMod → ( I ↾ 𝑉) ∈ (𝑆 NMHom 𝑆))
Theorem	0nmhm 24720	The zero operator is a bounded linear operator. (Contributed by Mario Carneiro, 20-Oct-2015.)
⊢ 𝑉 = (Base‘𝑆)    &   ⊢ 0 = (0g‘𝑇)    &   ⊢ 𝐹 = (Scalar‘𝑆)    &   ⊢ 𝐺 = (Scalar‘𝑇)    ⇒   ⊢ ((𝑆 ∈ NrmMod ∧ 𝑇 ∈ NrmMod ∧ 𝐹 = 𝐺) → (𝑉 × { 0 }) ∈ (𝑆 NMHom 𝑇))
Theorem	nmhmco 24721	The composition of bounded linear operators is a bounded linear operator. (Contributed by Mario Carneiro, 20-Oct-2015.)
⊢ ((𝐹 ∈ (𝑇 NMHom 𝑈) ∧ 𝐺 ∈ (𝑆 NMHom 𝑇)) → (𝐹 ∘ 𝐺) ∈ (𝑆 NMHom 𝑈))
Theorem	nmhmplusg 24722	The sum of two bounded linear operators is bounded linear. (Contributed by Mario Carneiro, 20-Oct-2015.)
⊢ + = (+g‘𝑇)    ⇒   ⊢ ((𝐹 ∈ (𝑆 NMHom 𝑇) ∧ 𝐺 ∈ (𝑆 NMHom 𝑇)) → (𝐹 ∘f + 𝐺) ∈ (𝑆 NMHom 𝑇))
12.4.10  Topology on the reals
Theorem	qtopbaslem 24723	The set of open intervals with endpoints in a subset forms a basis for a topology. (Contributed by Mario Carneiro, 17-Jun-2014.)
⊢ 𝑆 ⊆ ℝ*    ⇒   ⊢ ((,) “ (𝑆 × 𝑆)) ∈ TopBases
Theorem	qtopbas 24724	The set of open intervals with rational endpoints forms a basis for a topology. (Contributed by NM, 8-Mar-2007.)
⊢ ((,) “ (ℚ × ℚ)) ∈ TopBases
Theorem	retopbas 24725	A basis for the standard topology on the reals. (Contributed by NM, 6-Feb-2007.) (Proof shortened by Mario Carneiro, 17-Jun-2014.)
⊢ ran (,) ∈ TopBases
Theorem	retop 24726	The standard topology on the reals. (Contributed by FL, 4-Jun-2007.)
⊢ (topGen‘ran (,)) ∈ Top
Theorem	uniretop 24727	The underlying set of the standard topology on the reals is the reals. (Contributed by FL, 4-Jun-2007.)
⊢ ℝ = ∪ (topGen‘ran (,))
Theorem	retopon 24728	The standard topology on the reals is a topology on the reals. (Contributed by Mario Carneiro, 28-Aug-2015.)
⊢ (topGen‘ran (,)) ∈ (TopOn‘ℝ)
Theorem	retps 24729	The standard topological space on the reals. (Contributed by NM, 19-Oct-2012.)
⊢ 𝐾 = {⟨(Base‘ndx), ℝ⟩, ⟨(TopSet‘ndx), (topGen‘ran (,))⟩}    ⇒   ⊢ 𝐾 ∈ TopSp
Theorem	iooretop 24730	Open intervals are open sets of the standard topology on the reals . (Contributed by FL, 18-Jun-2007.)
⊢ (𝐴(,)𝐵) ∈ (topGen‘ran (,))
Theorem	icccld 24731	Closed intervals are closed sets of the standard topology on ℝ. (Contributed by FL, 14-Sep-2007.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ∈ (Clsd‘(topGen‘ran (,))))
Theorem	icopnfcld 24732	Right-unbounded closed intervals are closed sets of the standard topology on ℝ. (Contributed by Mario Carneiro, 17-Feb-2015.)
⊢ (𝐴 ∈ ℝ → (𝐴[,)+∞) ∈ (Clsd‘(topGen‘ran (,))))
Theorem	iocmnfcld 24733	Left-unbounded closed intervals are closed sets of the standard topology on ℝ. (Contributed by Mario Carneiro, 17-Feb-2015.)
⊢ (𝐴 ∈ ℝ → (-∞(,]𝐴) ∈ (Clsd‘(topGen‘ran (,))))
Theorem	qdensere 24734	ℚ is dense in the standard topology on ℝ. (Contributed by NM, 1-Mar-2007.)
⊢ ((cls‘(topGen‘ran (,)))‘ℚ) = ℝ
Theorem	cnmetdval 24735	Value of the distance function of the metric space of complex numbers. (Contributed by NM, 9-Dec-2006.) (Revised by Mario Carneiro, 27-Dec-2014.)
⊢ 𝐷 = (abs ∘ − )    ⇒   ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴𝐷𝐵) = (abs‘(𝐴 − 𝐵)))
Theorem	cnmet 24736	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 24737	The absolute value metric is an extended metric. (Contributed by Mario Carneiro, 28-Aug-2015.)
⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
Theorem	cnbl0 24738	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 24739*	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	cnfldms 24740	The complex number field is a metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
⊢ ℂfld ∈ MetSp
Theorem	cnfldxms 24741	The complex number field is a topological space. (Contributed by Mario Carneiro, 28-Aug-2015.)
⊢ ℂfld ∈ ∞MetSp
Theorem	cnfldtps 24742	The complex number field is a topological space. (Contributed by Mario Carneiro, 28-Aug-2015.)
⊢ ℂfld ∈ TopSp
Theorem	cnfldnm 24743	The norm of the field of complex numbers. (Contributed by Mario Carneiro, 4-Oct-2015.)
⊢ abs = (norm‘ℂfld)
Theorem	cnngp 24744	The complex numbers form a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.)
⊢ ℂfld ∈ NrmGrp
Theorem	cnnrg 24745	The complex numbers form a normed ring. (Contributed by Mario Carneiro, 4-Oct-2015.)
⊢ ℂfld ∈ NrmRing
Theorem	cnfldtopn 24746	The topology of the complex numbers. (Contributed by Mario Carneiro, 28-Aug-2015.)
⊢ 𝐽 = (TopOpen‘ℂfld)    ⇒   ⊢ 𝐽 = (MetOpen‘(abs ∘ − ))
Theorem	cnfldtopon 24747	The topology of the complex numbers is a topology. (Contributed by Mario Carneiro, 2-Sep-2015.)
⊢ 𝐽 = (TopOpen‘ℂfld)    ⇒   ⊢ 𝐽 ∈ (TopOn‘ℂ)
Theorem	cnfldtop 24748	The topology of the complex numbers is a topology. (Contributed by Mario Carneiro, 2-Sep-2015.)
⊢ 𝐽 = (TopOpen‘ℂfld)    ⇒   ⊢ 𝐽 ∈ Top
Theorem	cnfldhaus 24749	The topology of the complex numbers is Hausdorff. (Contributed by Mario Carneiro, 8-Sep-2015.)
⊢ 𝐽 = (TopOpen‘ℂfld)    ⇒   ⊢ 𝐽 ∈ Haus
Theorem	unicntop 24750	The underlying set of the standard topology on the complex numbers is the set of complex numbers. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ℂ = ∪ (TopOpen‘ℂfld)
Theorem	cnopn 24751	The set of complex numbers is open with respect to the standard topology on complex numbers. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ℂ ∈ (TopOpen‘ℂfld)
Theorem	cnn0opn 24752	The set of nonzero complex numbers is open with respect to the standard topology on complex numbers. (Contributed by SN, 7-Oct-2025.)
⊢ (ℂ ∖ {0}) ∈ (TopOpen‘ℂfld)
Theorem	zringnrg 24753	The ring of integers is a normed ring. (Contributed by AV, 13-Jun-2019.)
⊢ ℤring ∈ NrmRing
Theorem	remetdval 24754	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 24755	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 24756	The absolute value metric is an extended metric. (Contributed by Mario Carneiro, 28-Aug-2015.)
⊢ 𝐷 = ((abs ∘ − ) ↾ (ℝ × ℝ))    ⇒   ⊢ 𝐷 ∈ (∞Met‘ℝ)
Theorem	bl2ioo 24757	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 24758	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 24759	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 24760	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 24761	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	qdensere2 24762	ℚ is dense in ℝ. (Contributed by NM, 24-Aug-2007.)
⊢ 𝐷 = ((abs ∘ − ) ↾ (ℝ × ℝ))    &   ⊢ 𝐽 = (MetOpen‘𝐷)    ⇒   ⊢ ((cls‘𝐽)‘ℚ) = ℝ
Theorem	blcvx 24763	An open ball in the complex numbers is a convex set. (Contributed by Mario Carneiro, 12-Feb-2015.) (Revised by Mario Carneiro, 8-Sep-2015.)
⊢ 𝑆 = (𝑃(ball‘(abs ∘ − ))𝑅)    ⇒   ⊢ (((𝑃 ∈ ℂ ∧ 𝑅 ∈ ℝ*) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝑇 ∈ (0[,]1))) → ((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵)) ∈ 𝑆)
Theorem	rehaus 24764	The standard topology on the reals is Hausdorff. (Contributed by NM, 8-Mar-2007.)
⊢ (topGen‘ran (,)) ∈ Haus
Theorem	tgqioo 24765	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	re2ndc 24766	The standard topology on the reals is second-countable. (Contributed by Mario Carneiro, 21-Mar-2015.)
⊢ (topGen‘ran (,)) ∈ 2ndω
Theorem	resubmet 24767	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	tgioo2 24768	The standard topology on the reals is a subspace of the complex metric topology. (Contributed by Mario Carneiro, 13-Aug-2014.)
⊢ 𝐽 = (TopOpen‘ℂfld)    ⇒   ⊢ (topGen‘ran (,)) = (𝐽 ↾t ℝ)
Theorem	rerest 24769	The subspace topology induced by a subset of the reals. (Contributed by Mario Carneiro, 13-Aug-2014.)
⊢ 𝐽 = (TopOpen‘ℂfld)    &   ⊢ 𝑅 = (topGen‘ran (,))    ⇒   ⊢ (𝐴 ⊆ ℝ → (𝐽 ↾t 𝐴) = (𝑅 ↾t 𝐴))
Theorem	tgioo4 24770	The standard topology on the reals is a subspace of the complex metric topology. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
⊢ (topGen‘ran (,)) = ((TopOpen‘ℂfld) ↾t ℝ)
Theorem	tgioo3 24771	The standard topology on the reals is a subspace of the complex metric topology. (Contributed by Mario Carneiro, 13-Aug-2014.) (Revised by Thierry Arnoux, 3-Jul-2019.)
⊢ 𝐽 = (TopOpen‘ℝfld)    ⇒   ⊢ (topGen‘ran (,)) = 𝐽
Theorem	xrtgioo 24772	The topology on the extended reals coincides with the standard topology on the reals, when restricted to ℝ. (Contributed by Mario Carneiro, 3-Sep-2015.)
⊢ 𝐽 = ((ordTop‘ ≤ ) ↾t ℝ)    ⇒   ⊢ (topGen‘ran (,)) = 𝐽
Theorem	xrrest 24773	The subspace topology induced by a subset of the reals. (Contributed by Mario Carneiro, 9-Sep-2015.)
⊢ 𝑋 = (ordTop‘ ≤ )    &   ⊢ 𝑅 = (topGen‘ran (,))    ⇒   ⊢ (𝐴 ⊆ ℝ → (𝑋 ↾t 𝐴) = (𝑅 ↾t 𝐴))
Theorem	xrrest2 24774	The subspace topology induced by a subset of the reals. (Contributed by Mario Carneiro, 9-Sep-2015.)
⊢ 𝐽 = (TopOpen‘ℂfld)    &   ⊢ 𝑋 = (ordTop‘ ≤ )    ⇒   ⊢ (𝐴 ⊆ ℝ → (𝐽 ↾t 𝐴) = (𝑋 ↾t 𝐴))
Theorem	xrsxmet 24775	The metric on the extended reals is a proper extended metric. (Contributed by Mario Carneiro, 4-Sep-2015.)
⊢ 𝐷 = (dist‘ℝ*𝑠)    ⇒   ⊢ 𝐷 ∈ (∞Met‘ℝ*)
Theorem	xrsdsre 24776	The metric on the extended reals coincides with the usual metric on the reals. (Contributed by Mario Carneiro, 4-Sep-2015.)
⊢ 𝐷 = (dist‘ℝ*𝑠)    ⇒   ⊢ (𝐷 ↾ (ℝ × ℝ)) = ((abs ∘ − ) ↾ (ℝ × ℝ))
Theorem	xrsblre 24777	Any ball of the metric of the extended reals centered on an element of ℝ is entirely contained in ℝ. (Contributed by Mario Carneiro, 4-Sep-2015.)
⊢ 𝐷 = (dist‘ℝ*𝑠)    ⇒   ⊢ ((𝑃 ∈ ℝ ∧ 𝑅 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑅) ⊆ ℝ)
Theorem	xrsmopn 24778	The metric on the extended reals generates a topology, but this does not match the order topology on ℝ*; for example {+∞} is open in the metric topology, but not the order topology. However, the metric topology is finer than the order topology, meaning that all open intervals are open in the metric topology. (Contributed by Mario Carneiro, 4-Sep-2015.)
⊢ 𝐷 = (dist‘ℝ*𝑠)    &   ⊢ 𝐽 = (MetOpen‘𝐷)    ⇒   ⊢ (ordTop‘ ≤ ) ⊆ 𝐽
Theorem	zcld 24779	The integers are a closed set in the topology on ℝ. (Contributed by Mario Carneiro, 17-Feb-2015.)
⊢ 𝐽 = (topGen‘ran (,))    ⇒   ⊢ ℤ ∈ (Clsd‘𝐽)
Theorem	recld2 24780	The real numbers are a closed set in the topology on ℂ. (Contributed by Mario Carneiro, 17-Feb-2015.)
⊢ 𝐽 = (TopOpen‘ℂfld)    ⇒   ⊢ ℝ ∈ (Clsd‘𝐽)
Theorem	zcld2 24781	The integers are a closed set in the topology on ℂ. (Contributed by Mario Carneiro, 17-Feb-2015.)
⊢ 𝐽 = (TopOpen‘ℂfld)    ⇒   ⊢ ℤ ∈ (Clsd‘𝐽)
Theorem	zdis 24782	The integers are a discrete set in the topology on ℂ. (Contributed by Mario Carneiro, 19-Sep-2015.)
⊢ 𝐽 = (TopOpen‘ℂfld)    ⇒   ⊢ (𝐽 ↾t ℤ) = 𝒫 ℤ
Theorem	sszcld 24783	Every subset of the integers are closed in the topology on ℂ. (Contributed by Mario Carneiro, 6-Jul-2017.)
⊢ 𝐽 = (TopOpen‘ℂfld)    ⇒   ⊢ (𝐴 ⊆ ℤ → 𝐴 ∈ (Clsd‘𝐽))
Theorem	reperflem 24784*	A subset of the real numbers that is closed under addition with real numbers is perfect. (Contributed by Mario Carneiro, 26-Dec-2016.)
⊢ 𝐽 = (TopOpen‘ℂfld)    &   ⊢ ((𝑢 ∈ 𝑆 ∧ 𝑣 ∈ ℝ) → (𝑢 + 𝑣) ∈ 𝑆)    &   ⊢ 𝑆 ⊆ ℂ    ⇒   ⊢ (𝐽 ↾t 𝑆) ∈ Perf
Theorem	reperf 24785	The real numbers are a perfect subset of the complex numbers. (Contributed by Mario Carneiro, 26-Dec-2016.)
⊢ 𝐽 = (TopOpen‘ℂfld)    ⇒   ⊢ (𝐽 ↾t ℝ) ∈ Perf
Theorem	cnperf 24786	The complex numbers are a perfect space. (Contributed by Mario Carneiro, 26-Dec-2016.)
⊢ 𝐽 = (TopOpen‘ℂfld)    ⇒   ⊢ 𝐽 ∈ Perf
Theorem	iccntr 24787	The interior of a closed interval in the standard topology on ℝ is the corresponding open interval. (Contributed by Mario Carneiro, 1-Sep-2014.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
Theorem	icccmplem1 24788*	Lemma for icccmp 24791. (Contributed by Mario Carneiro, 18-Jun-2014.)
⊢ 𝐽 = (topGen‘ran (,))    &   ⊢ 𝑇 = (𝐽 ↾t (𝐴[,]𝐵))    &   ⊢ 𝐷 = ((abs ∘ − ) ↾ (ℝ × ℝ))    &   ⊢ 𝑆 = {𝑥 ∈ (𝐴[,]𝐵) ∣ ∃𝑧 ∈ (𝒫 𝑈 ∩ Fin)(𝐴[,]𝑥) ⊆ ∪ 𝑧}    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    &   ⊢ (𝜑 → 𝑈 ⊆ 𝐽)    &   ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ∪ 𝑈)    ⇒   ⊢ (𝜑 → (𝐴 ∈ 𝑆 ∧ ∀𝑦 ∈ 𝑆 𝑦 ≤ 𝐵))
Theorem	icccmplem2 24789*	Lemma for icccmp 24791. (Contributed by Mario Carneiro, 13-Jun-2014.)
⊢ 𝐽 = (topGen‘ran (,))    &   ⊢ 𝑇 = (𝐽 ↾t (𝐴[,]𝐵))    &   ⊢ 𝐷 = ((abs ∘ − ) ↾ (ℝ × ℝ))    &   ⊢ 𝑆 = {𝑥 ∈ (𝐴[,]𝐵) ∣ ∃𝑧 ∈ (𝒫 𝑈 ∩ Fin)(𝐴[,]𝑥) ⊆ ∪ 𝑧}    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    &   ⊢ (𝜑 → 𝑈 ⊆ 𝐽)    &   ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ∪ 𝑈)    &   ⊢ (𝜑 → 𝑉 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ+)    &   ⊢ (𝜑 → (𝐺(ball‘𝐷)𝐶) ⊆ 𝑉)    &   ⊢ 𝐺 = sup(𝑆, ℝ, < )    &   ⊢ 𝑅 = if((𝐺 + (𝐶 / 2)) ≤ 𝐵, (𝐺 + (𝐶 / 2)), 𝐵)    ⇒   ⊢ (𝜑 → 𝐵 ∈ 𝑆)
Theorem	icccmplem3 24790*	Lemma for icccmp 24791. (Contributed by Mario Carneiro, 13-Jun-2014.)
⊢ 𝐽 = (topGen‘ran (,))    &   ⊢ 𝑇 = (𝐽 ↾t (𝐴[,]𝐵))    &   ⊢ 𝐷 = ((abs ∘ − ) ↾ (ℝ × ℝ))    &   ⊢ 𝑆 = {𝑥 ∈ (𝐴[,]𝐵) ∣ ∃𝑧 ∈ (𝒫 𝑈 ∩ Fin)(𝐴[,]𝑥) ⊆ ∪ 𝑧}    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    &   ⊢ (𝜑 → 𝑈 ⊆ 𝐽)    &   ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ∪ 𝑈)    ⇒   ⊢ (𝜑 → 𝐵 ∈ 𝑆)
Theorem	icccmp 24791	A closed interval in ℝ is compact. (Contributed by Mario Carneiro, 13-Jun-2014.)
⊢ 𝐽 = (topGen‘ran (,))    &   ⊢ 𝑇 = (𝐽 ↾t (𝐴[,]𝐵))    ⇒   ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝑇 ∈ Comp)
Theorem	reconnlem1 24792	Lemma for reconn 24794. Connectedness in the reals-easy direction. (Contributed by Jeff Hankins, 13-Jul-2009.) (Proof shortened by Mario Carneiro, 9-Sep-2015.)
⊢ (((𝐴 ⊆ ℝ ∧ ((topGen‘ran (,)) ↾t 𝐴) ∈ Conn) ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴)) → (𝑋[,]𝑌) ⊆ 𝐴)
Theorem	reconnlem2 24793*	Lemma for reconn 24794. (Contributed by Jeff Hankins, 17-Aug-2009.) (Proof shortened by Mario Carneiro, 9-Sep-2015.)
⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ (𝜑 → 𝑈 ∈ (topGen‘ran (,)))    &   ⊢ (𝜑 → 𝑉 ∈ (topGen‘ran (,)))    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥[,]𝑦) ⊆ 𝐴)    &   ⊢ (𝜑 → 𝐵 ∈ (𝑈 ∩ 𝐴))    &   ⊢ (𝜑 → 𝐶 ∈ (𝑉 ∩ 𝐴))    &   ⊢ (𝜑 → (𝑈 ∩ 𝑉) ⊆ (ℝ ∖ 𝐴))    &   ⊢ (𝜑 → 𝐵 ≤ 𝐶)    &   ⊢ 𝑆 = sup((𝑈 ∩ (𝐵[,]𝐶)), ℝ, < )    ⇒   ⊢ (𝜑 → ¬ 𝐴 ⊆ (𝑈 ∪ 𝑉))
Theorem	reconn 24794*	A subset of the reals is connected iff it has the interval property. (Contributed by Jeff Hankins, 15-Jul-2009.) (Proof shortened by Mario Carneiro, 9-Sep-2015.)
⊢ (𝐴 ⊆ ℝ → (((topGen‘ran (,)) ↾t 𝐴) ∈ Conn ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥[,]𝑦) ⊆ 𝐴))
Theorem	retopconn 24795	Corollary of reconn 24794. The set of real numbers is connected. (Contributed by Jeff Hankins, 17-Aug-2009.)
⊢ (topGen‘ran (,)) ∈ Conn
Theorem	iccconn 24796	A closed interval is connected. (Contributed by Jeff Hankins, 17-Aug-2009.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((topGen‘ran (,)) ↾t (𝐴[,]𝐵)) ∈ Conn)
Theorem	opnreen 24797	Every nonempty open set is uncountable. (Contributed by Mario Carneiro, 26-Jul-2014.) (Revised by Mario Carneiro, 20-Feb-2015.)
⊢ ((𝐴 ∈ (topGen‘ran (,)) ∧ 𝐴 ≠ ∅) → 𝐴 ≈ 𝒫 ℕ)
Theorem	rectbntr0 24798	A countable subset of the reals has empty interior. (Contributed by Mario Carneiro, 26-Jul-2014.)
⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≼ ℕ) → ((int‘(topGen‘ran (,)))‘𝐴) = ∅)
Theorem	xrge0gsumle 24799	A finite sum in the nonnegative extended reals is monotonic in the support. (Contributed by Mario Carneiro, 13-Sep-2015.)
⊢ 𝐺 = (ℝ*𝑠 ↾s (0[,]+∞))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴⟶(0[,]+∞))    &   ⊢ (𝜑 → 𝐵 ∈ (𝒫 𝐴 ∩ Fin))    &   ⊢ (𝜑 → 𝐶 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ 𝐶)) ≤ (𝐺 Σg (𝐹 ↾ 𝐵)))
Theorem	xrge0tsms 24800*	Any finite or infinite sum in the nonnegative extended reals is uniquely convergent to the supremum of all finite sums. (Contributed by Mario Carneiro, 13-Sep-2015.) (Proof shortened by AV, 26-Jul-2019.)
⊢ 𝐺 = (ℝ*𝑠 ↾s (0[,]+∞))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴⟶(0[,]+∞))    &   ⊢ 𝑆 = sup(ran (𝑠 ∈ (𝒫 𝐴 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑠))), ℝ*, < )    ⇒   ⊢ (𝜑 → (𝐺 tsums 𝐹) = {𝑆})