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Theorem List for Metamath Proof Explorer - 23901-24000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnmoi 23901 The operator norm achieves the minimum of the set of upper bounds, if the operator is bounded. (Contributed by Mario Carneiro, 18-Oct-2015.)
𝑁 = (𝑆 normOp 𝑇)    &   𝑉 = (Base‘𝑆)    &   𝐿 = (norm‘𝑆)    &   𝑀 = (norm‘𝑇)       ((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝑋𝑉) → (𝑀‘(𝐹𝑋)) ≤ ((𝑁𝐹) · (𝐿𝑋)))
 
Theoremnmoix 23902 The operator norm is a bound on the size of an operator, even when it is infinite (using extended real multiplication). (Contributed by Mario Carneiro, 18-Oct-2015.)
𝑁 = (𝑆 normOp 𝑇)    &   𝑉 = (Base‘𝑆)    &   𝐿 = (norm‘𝑆)    &   𝑀 = (norm‘𝑇)       (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ 𝑋𝑉) → (𝑀‘(𝐹𝑋)) ≤ ((𝑁𝐹) ·e (𝐿𝑋)))
 
Theoremnmoi2 23903 The operator norm is a bound on the growth of a vector under the action of the operator. (Contributed by Mario Carneiro, 19-Oct-2015.)
𝑁 = (𝑆 normOp 𝑇)    &   𝑉 = (Base‘𝑆)    &   𝐿 = (norm‘𝑆)    &   𝑀 = (norm‘𝑇)    &    0 = (0g𝑆)       (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑋𝑉𝑋0 )) → ((𝑀‘(𝐹𝑋)) / (𝐿𝑋)) ≤ (𝑁𝐹))
 
Theoremnmoleub 23904* The operator norm, defined as an infimum of upper bounds, can also be defined as a supremum of norms of 𝐹(𝑥) away from zero. (Contributed by Mario Carneiro, 18-Oct-2015.)
𝑁 = (𝑆 normOp 𝑇)    &   𝑉 = (Base‘𝑆)    &   𝐿 = (norm‘𝑆)    &   𝑀 = (norm‘𝑇)    &    0 = (0g𝑆)    &   (𝜑𝑆 ∈ NrmGrp)    &   (𝜑𝑇 ∈ NrmGrp)    &   (𝜑𝐹 ∈ (𝑆 GrpHom 𝑇))    &   (𝜑𝐴 ∈ ℝ*)    &   (𝜑 → 0 ≤ 𝐴)       (𝜑 → ((𝑁𝐹) ≤ 𝐴 ↔ ∀𝑥𝑉 (𝑥0 → ((𝑀‘(𝐹𝑥)) / (𝐿𝑥)) ≤ 𝐴)))
 
Theoremnghmrcl1 23905 Reverse closure for a normed group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
(𝐹 ∈ (𝑆 NGHom 𝑇) → 𝑆 ∈ NrmGrp)
 
Theoremnghmrcl2 23906 Reverse closure for a normed group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
(𝐹 ∈ (𝑆 NGHom 𝑇) → 𝑇 ∈ NrmGrp)
 
Theoremnghmghm 23907 A normed group homomorphism is a group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
(𝐹 ∈ (𝑆 NGHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
 
Theoremnmo0 23908 The operator norm of the zero operator. (Contributed by Mario Carneiro, 20-Oct-2015.)
𝑁 = (𝑆 normOp 𝑇)    &   𝑉 = (Base‘𝑆)    &    0 = (0g𝑇)       ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → (𝑁‘(𝑉 × { 0 })) = 0)
 
Theoremnmoeq0 23909 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 = (0g𝑇)       ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) → ((𝑁𝐹) = 0 ↔ 𝐹 = (𝑉 × { 0 })))
 
Theoremnmoco 23910 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‘(𝐹𝐺)) ≤ ((𝐿𝐹) · (𝑀𝐺)))
 
Theoremnghmco 23911 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)) → (𝐹𝐺) ∈ (𝑆 NGHom 𝑈))
 
Theoremnmotri 23912 Triangle inequality for the operator norm. (Contributed by Mario Carneiro, 20-Oct-2015.)
𝑁 = (𝑆 normOp 𝑇)    &    + = (+g𝑇)       ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → (𝑁‘(𝐹f + 𝐺)) ≤ ((𝑁𝐹) + (𝑁𝐺)))
 
Theoremnghmplusg 23913 The sum of two bounded linear operators is bounded linear. (Contributed by Mario Carneiro, 20-Oct-2015.)
+ = (+g𝑇)       ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → (𝐹f + 𝐺) ∈ (𝑆 NGHom 𝑇))
 
Theorem0nghm 23914 The zero operator is a normed group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
𝑉 = (Base‘𝑆)    &    0 = (0g𝑇)       ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → (𝑉 × { 0 }) ∈ (𝑆 NGHom 𝑇))
 
Theoremnmoid 23915 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 = (0g𝑆)       ((𝑆 ∈ NrmGrp ∧ { 0 } ⊊ 𝑉) → (𝑁‘( I ↾ 𝑉)) = 1)
 
Theoremidnghm 23916 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))
 
Theoremnmods 23917 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) ∧ 𝐴𝑉𝐵𝑉) → ((𝐹𝐴)𝐷(𝐹𝐵)) ≤ ((𝑁𝐹) · (𝐴𝐶𝐵)))
 
Theoremnghmcn 23918 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))
 
Theoremisnmhm 23919 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))))
 
Theoremnmhmrcl1 23920 Reverse closure for a normed module homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
(𝐹 ∈ (𝑆 NMHom 𝑇) → 𝑆 ∈ NrmMod)
 
Theoremnmhmrcl2 23921 Reverse closure for a normed module homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
(𝐹 ∈ (𝑆 NMHom 𝑇) → 𝑇 ∈ NrmMod)
 
Theoremnmhmlmhm 23922 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))
 
Theoremnmhmnghm 23923 A normed module homomorphism is a normed group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
(𝐹 ∈ (𝑆 NMHom 𝑇) → 𝐹 ∈ (𝑆 NGHom 𝑇))
 
Theoremnmhmghm 23924 A normed module homomorphism is a group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
(𝐹 ∈ (𝑆 NMHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
 
Theoremisnmhm2 23925 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) ↔ (𝑁𝐹) ∈ ℝ))
 
Theoremnmhmcl 23926 A normed module homomorphism has a real operator norm. (Contributed by Mario Carneiro, 18-Oct-2015.)
𝑁 = (𝑆 normOp 𝑇)       (𝐹 ∈ (𝑆 NMHom 𝑇) → (𝑁𝐹) ∈ ℝ)
 
Theoremidnmhm 23927 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))
 
Theorem0nmhm 23928 The zero operator is a bounded linear operator. (Contributed by Mario Carneiro, 20-Oct-2015.)
𝑉 = (Base‘𝑆)    &    0 = (0g𝑇)    &   𝐹 = (Scalar‘𝑆)    &   𝐺 = (Scalar‘𝑇)       ((𝑆 ∈ NrmMod ∧ 𝑇 ∈ NrmMod ∧ 𝐹 = 𝐺) → (𝑉 × { 0 }) ∈ (𝑆 NMHom 𝑇))
 
Theoremnmhmco 23929 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)) → (𝐹𝐺) ∈ (𝑆 NMHom 𝑈))
 
Theoremnmhmplusg 23930 The sum of two bounded linear operators is bounded linear. (Contributed by Mario Carneiro, 20-Oct-2015.)
+ = (+g𝑇)       ((𝐹 ∈ (𝑆 NMHom 𝑇) ∧ 𝐺 ∈ (𝑆 NMHom 𝑇)) → (𝐹f + 𝐺) ∈ (𝑆 NMHom 𝑇))
 
12.4.10  Topology on the reals
 
Theoremqtopbaslem 23931 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
 
Theoremqtopbas 23932 The set of open intervals with rational endpoints forms a basis for a topology. (Contributed by NM, 8-Mar-2007.)
((,) “ (ℚ × ℚ)) ∈ TopBases
 
Theoremretopbas 23933 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
 
Theoremretop 23934 The standard topology on the reals. (Contributed by FL, 4-Jun-2007.)
(topGen‘ran (,)) ∈ Top
 
Theoremuniretop 23935 The underlying set of the standard topology on the reals is the reals. (Contributed by FL, 4-Jun-2007.)
ℝ = (topGen‘ran (,))
 
Theoremretopon 23936 The standard topology on the reals is a topology on the reals. (Contributed by Mario Carneiro, 28-Aug-2015.)
(topGen‘ran (,)) ∈ (TopOn‘ℝ)
 
Theoremretps 23937 The standard topological space on the reals. (Contributed by NM, 19-Oct-2012.)
𝐾 = {⟨(Base‘ndx), ℝ⟩, ⟨(TopSet‘ndx), (topGen‘ran (,))⟩}       𝐾 ∈ TopSp
 
Theoremiooretop 23938 Open intervals are open sets of the standard topology on the reals . (Contributed by FL, 18-Jun-2007.)
(𝐴(,)𝐵) ∈ (topGen‘ran (,))
 
Theoremicccld 23939 Closed intervals are closed sets of the standard topology on . (Contributed by FL, 14-Sep-2007.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ∈ (Clsd‘(topGen‘ran (,))))
 
Theoremicopnfcld 23940 Right-unbounded closed intervals are closed sets of the standard topology on . (Contributed by Mario Carneiro, 17-Feb-2015.)
(𝐴 ∈ ℝ → (𝐴[,)+∞) ∈ (Clsd‘(topGen‘ran (,))))
 
Theoremiocmnfcld 23941 Left-unbounded closed intervals are closed sets of the standard topology on . (Contributed by Mario Carneiro, 17-Feb-2015.)
(𝐴 ∈ ℝ → (-∞(,]𝐴) ∈ (Clsd‘(topGen‘ran (,))))
 
Theoremqdensere 23942 is dense in the standard topology on . (Contributed by NM, 1-Mar-2007.)
((cls‘(topGen‘ran (,)))‘ℚ) = ℝ
 
Theoremcnmetdval 23943 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‘(𝐴𝐵)))
 
Theoremcnmet 23944 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 23945 The absolute value metric is an extended metric. (Contributed by Mario Carneiro, 28-Aug-2015.)
(abs ∘ − ) ∈ (∞Met‘ℂ)
 
Theoremcnbl0 23946 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 23947* 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})
 
Theoremcnfldms 23948 The complex number field is a metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
fld ∈ MetSp
 
Theoremcnfldxms 23949 The complex number field is a topological space. (Contributed by Mario Carneiro, 28-Aug-2015.)
fld ∈ ∞MetSp
 
Theoremcnfldtps 23950 The complex number field is a topological space. (Contributed by Mario Carneiro, 28-Aug-2015.)
fld ∈ TopSp
 
Theoremcnfldnm 23951 The norm of the field of complex numbers. (Contributed by Mario Carneiro, 4-Oct-2015.)
abs = (norm‘ℂfld)
 
Theoremcnngp 23952 The complex numbers form a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.)
fld ∈ NrmGrp
 
Theoremcnnrg 23953 The complex numbers form a normed ring. (Contributed by Mario Carneiro, 4-Oct-2015.)
fld ∈ NrmRing
 
Theoremcnfldtopn 23954 The topology of the complex numbers. (Contributed by Mario Carneiro, 28-Aug-2015.)
𝐽 = (TopOpen‘ℂfld)       𝐽 = (MetOpen‘(abs ∘ − ))
 
Theoremcnfldtopon 23955 The topology of the complex numbers is a topology. (Contributed by Mario Carneiro, 2-Sep-2015.)
𝐽 = (TopOpen‘ℂfld)       𝐽 ∈ (TopOn‘ℂ)
 
Theoremcnfldtop 23956 The topology of the complex numbers is a topology. (Contributed by Mario Carneiro, 2-Sep-2015.)
𝐽 = (TopOpen‘ℂfld)       𝐽 ∈ Top
 
Theoremcnfldhaus 23957 The topology of the complex numbers is Hausdorff. (Contributed by Mario Carneiro, 8-Sep-2015.)
𝐽 = (TopOpen‘ℂfld)       𝐽 ∈ Haus
 
Theoremunicntop 23958 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)
 
Theoremcnopn 23959 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)
 
Theoremzringnrg 23960 The ring of integers is a normed ring. (Contributed by AV, 13-Jun-2019.)
ring ∈ NrmRing
 
Theoremremetdval 23961 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 23962 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 23963 The absolute value metric is an extended metric. (Contributed by Mario Carneiro, 28-Aug-2015.)
𝐷 = ((abs ∘ − ) ↾ (ℝ × ℝ))       𝐷 ∈ (∞Met‘ℝ)
 
Theorembl2ioo 23964 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 23965 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 23966 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 23967 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 23968 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
 
Theoremqdensere2 23969 is dense in . (Contributed by NM, 24-Aug-2007.)
𝐷 = ((abs ∘ − ) ↾ (ℝ × ℝ))    &   𝐽 = (MetOpen‘𝐷)       ((cls‘𝐽)‘ℚ) = ℝ
 
Theoremblcvx 23970 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*) ∧ (𝐴𝑆𝐵𝑆𝑇 ∈ (0[,]1))) → ((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵)) ∈ 𝑆)
 
Theoremrehaus 23971 The standard topology on the reals is Hausdorff. (Contributed by NM, 8-Mar-2007.)
(topGen‘ran (,)) ∈ Haus
 
Theoremtgqioo 23972 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
 
Theoremre2ndc 23973 The standard topology on the reals is second-countable. (Contributed by Mario Carneiro, 21-Mar-2015.)
(topGen‘ran (,)) ∈ 2ndω
 
Theoremresubmet 23974 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))
 
Theoremtgioo2 23975 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 (,)) = (𝐽t ℝ)
 
Theoremrerest 23976 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 → (𝐽t 𝐴) = (𝑅t 𝐴))
 
Theoremtgioo3 23977 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
 
Theoremxrtgioo 23978 The topology on the extended reals coincides with the standard topology on the reals, when restricted to . (Contributed by Mario Carneiro, 3-Sep-2015.)
𝐽 = ((ordTop‘ ≤ ) ↾t ℝ)       (topGen‘ran (,)) = 𝐽
 
Theoremxrrest 23979 The subspace topology induced by a subset of the reals. (Contributed by Mario Carneiro, 9-Sep-2015.)
𝑋 = (ordTop‘ ≤ )    &   𝑅 = (topGen‘ran (,))       (𝐴 ⊆ ℝ → (𝑋t 𝐴) = (𝑅t 𝐴))
 
Theoremxrrest2 23980 The subspace topology induced by a subset of the reals. (Contributed by Mario Carneiro, 9-Sep-2015.)
𝐽 = (TopOpen‘ℂfld)    &   𝑋 = (ordTop‘ ≤ )       (𝐴 ⊆ ℝ → (𝐽t 𝐴) = (𝑋t 𝐴))
 
Theoremxrsxmet 23981 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*)
 
Theoremxrsdsre 23982 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))
 
Theoremxrsblre 23983 Any ball of the metric of the extended reals centered on an element of is entirely contained in . (Contributed by Mario Carneiro, 4-Sep-2015.)
𝐷 = (dist‘ℝ*𝑠)       ((𝑃 ∈ ℝ ∧ 𝑅 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑅) ⊆ ℝ)
 
Theoremxrsmopn 23984 The metric on the extended reals generates a topology, but this does not match the order topology on *; 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
 
Theoremzcld 23985 The integers are a closed set in the topology on . (Contributed by Mario Carneiro, 17-Feb-2015.)
𝐽 = (topGen‘ran (,))       ℤ ∈ (Clsd‘𝐽)
 
Theoremrecld2 23986 The real numbers are a closed set in the topology on . (Contributed by Mario Carneiro, 17-Feb-2015.)
𝐽 = (TopOpen‘ℂfld)       ℝ ∈ (Clsd‘𝐽)
 
Theoremzcld2 23987 The integers are a closed set in the topology on . (Contributed by Mario Carneiro, 17-Feb-2015.)
𝐽 = (TopOpen‘ℂfld)       ℤ ∈ (Clsd‘𝐽)
 
Theoremzdis 23988 The integers are a discrete set in the topology on . (Contributed by Mario Carneiro, 19-Sep-2015.)
𝐽 = (TopOpen‘ℂfld)       (𝐽t ℤ) = 𝒫 ℤ
 
Theoremsszcld 23989 Every subset of the integers are closed in the topology on . (Contributed by Mario Carneiro, 6-Jul-2017.)
𝐽 = (TopOpen‘ℂfld)       (𝐴 ⊆ ℤ → 𝐴 ∈ (Clsd‘𝐽))
 
Theoremreperflem 23990* 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)    &   ((𝑢𝑆𝑣 ∈ ℝ) → (𝑢 + 𝑣) ∈ 𝑆)    &   𝑆 ⊆ ℂ       (𝐽t 𝑆) ∈ Perf
 
Theoremreperf 23991 The real numbers are a perfect subset of the complex numbers. (Contributed by Mario Carneiro, 26-Dec-2016.)
𝐽 = (TopOpen‘ℂfld)       (𝐽t ℝ) ∈ Perf
 
Theoremcnperf 23992 The complex numbers are a perfect space. (Contributed by Mario Carneiro, 26-Dec-2016.)
𝐽 = (TopOpen‘ℂfld)       𝐽 ∈ Perf
 
Theoremiccntr 23993 The interior of a closed interval in the standard topology on is the corresponding open interval. (Contributed by Mario Carneiro, 1-Sep-2014.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
 
Theoremicccmplem1 23994* Lemma for icccmp 23997. (Contributed by Mario Carneiro, 18-Jun-2014.)
𝐽 = (topGen‘ran (,))    &   𝑇 = (𝐽t (𝐴[,]𝐵))    &   𝐷 = ((abs ∘ − ) ↾ (ℝ × ℝ))    &   𝑆 = {𝑥 ∈ (𝐴[,]𝐵) ∣ ∃𝑧 ∈ (𝒫 𝑈 ∩ Fin)(𝐴[,]𝑥) ⊆ 𝑧}    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴𝐵)    &   (𝜑𝑈𝐽)    &   (𝜑 → (𝐴[,]𝐵) ⊆ 𝑈)       (𝜑 → (𝐴𝑆 ∧ ∀𝑦𝑆 𝑦𝐵))
 
Theoremicccmplem2 23995* Lemma for icccmp 23997. (Contributed by Mario Carneiro, 13-Jun-2014.)
𝐽 = (topGen‘ran (,))    &   𝑇 = (𝐽t (𝐴[,]𝐵))    &   𝐷 = ((abs ∘ − ) ↾ (ℝ × ℝ))    &   𝑆 = {𝑥 ∈ (𝐴[,]𝐵) ∣ ∃𝑧 ∈ (𝒫 𝑈 ∩ Fin)(𝐴[,]𝑥) ⊆ 𝑧}    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴𝐵)    &   (𝜑𝑈𝐽)    &   (𝜑 → (𝐴[,]𝐵) ⊆ 𝑈)    &   (𝜑𝑉𝑈)    &   (𝜑𝐶 ∈ ℝ+)    &   (𝜑 → (𝐺(ball‘𝐷)𝐶) ⊆ 𝑉)    &   𝐺 = sup(𝑆, ℝ, < )    &   𝑅 = if((𝐺 + (𝐶 / 2)) ≤ 𝐵, (𝐺 + (𝐶 / 2)), 𝐵)       (𝜑𝐵𝑆)
 
Theoremicccmplem3 23996* Lemma for icccmp 23997. (Contributed by Mario Carneiro, 13-Jun-2014.)
𝐽 = (topGen‘ran (,))    &   𝑇 = (𝐽t (𝐴[,]𝐵))    &   𝐷 = ((abs ∘ − ) ↾ (ℝ × ℝ))    &   𝑆 = {𝑥 ∈ (𝐴[,]𝐵) ∣ ∃𝑧 ∈ (𝒫 𝑈 ∩ Fin)(𝐴[,]𝑥) ⊆ 𝑧}    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴𝐵)    &   (𝜑𝑈𝐽)    &   (𝜑 → (𝐴[,]𝐵) ⊆ 𝑈)       (𝜑𝐵𝑆)
 
Theoremicccmp 23997 A closed interval in is compact. (Contributed by Mario Carneiro, 13-Jun-2014.)
𝐽 = (topGen‘ran (,))    &   𝑇 = (𝐽t (𝐴[,]𝐵))       ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝑇 ∈ Comp)
 
Theoremreconnlem1 23998 Lemma for reconn 24000. 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) ∧ (𝑋𝐴𝑌𝐴)) → (𝑋[,]𝑌) ⊆ 𝐴)
 
Theoremreconnlem2 23999* Lemma for reconn 24000. (Contributed by Jeff Hankins, 17-Aug-2009.) (Proof shortened by Mario Carneiro, 9-Sep-2015.)
(𝜑𝐴 ⊆ ℝ)    &   (𝜑𝑈 ∈ (topGen‘ran (,)))    &   (𝜑𝑉 ∈ (topGen‘ran (,)))    &   (𝜑 → ∀𝑥𝐴𝑦𝐴 (𝑥[,]𝑦) ⊆ 𝐴)    &   (𝜑𝐵 ∈ (𝑈𝐴))    &   (𝜑𝐶 ∈ (𝑉𝐴))    &   (𝜑 → (𝑈𝑉) ⊆ (ℝ ∖ 𝐴))    &   (𝜑𝐵𝐶)    &   𝑆 = sup((𝑈 ∩ (𝐵[,]𝐶)), ℝ, < )       (𝜑 → ¬ 𝐴 ⊆ (𝑈𝑉))
 
Theoremreconn 24000* 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 ↔ ∀𝑥𝐴𝑦𝐴 (𝑥[,]𝑦) ⊆ 𝐴))
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