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Theorem List for Intuitionistic Logic Explorer - 12101-12200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremlmodstrd 12101 A constructed left module or left vector space is a structure. (Contributed by Mario Carneiro, 1-Oct-2013.) (Revised by Jim Kingdon, 5-Feb-2023.)
𝑊 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝐹⟩} ∪ {⟨( ·𝑠 ‘ndx), · ⟩})    &   (𝜑𝐵𝑉)    &   (𝜑+𝑋)    &   (𝜑𝐹𝑌)    &   (𝜑·𝑍)       (𝜑𝑊 Struct ⟨1, 6⟩)

Theoremlmodbased 12102 The base set of a constructed left vector space. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Jim Kingdon, 6-Feb-2023.)
𝑊 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝐹⟩} ∪ {⟨( ·𝑠 ‘ndx), · ⟩})    &   (𝜑𝐵𝑉)    &   (𝜑+𝑋)    &   (𝜑𝐹𝑌)    &   (𝜑·𝑍)       (𝜑𝐵 = (Base‘𝑊))

Theoremlmodplusgd 12103 The additive operation of a constructed left vector space. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Jim Kingdon, 6-Feb-2023.)
𝑊 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝐹⟩} ∪ {⟨( ·𝑠 ‘ndx), · ⟩})    &   (𝜑𝐵𝑉)    &   (𝜑+𝑋)    &   (𝜑𝐹𝑌)    &   (𝜑·𝑍)       (𝜑+ = (+g𝑊))

Theoremlmodscad 12104 The set of scalars of a constructed left vector space. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Jim Kingdon, 6-Feb-2023.)
𝑊 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝐹⟩} ∪ {⟨( ·𝑠 ‘ndx), · ⟩})    &   (𝜑𝐵𝑉)    &   (𝜑+𝑋)    &   (𝜑𝐹𝑌)    &   (𝜑·𝑍)       (𝜑𝐹 = (Scalar‘𝑊))

Theoremlmodvscad 12105 The scalar product operation of a constructed left vector space. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Jim Kingdon, 7-Feb-2023.)
𝑊 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝐹⟩} ∪ {⟨( ·𝑠 ‘ndx), · ⟩})    &   (𝜑𝐵𝑉)    &   (𝜑+𝑋)    &   (𝜑𝐹𝑌)    &   (𝜑·𝑍)       (𝜑· = ( ·𝑠𝑊))

Theoremipndx 12106 Index value of the df-ip 12048 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
(·𝑖‘ndx) = 8

Theoremipid 12107 Utility theorem: index-independent form of df-ip 12048. (Contributed by Mario Carneiro, 6-Oct-2013.)
·𝑖 = Slot (·𝑖‘ndx)

Theoremipslid 12108 Slot property of ·𝑖. (Contributed by Jim Kingdon, 7-Feb-2023.)
(·𝑖 = Slot (·𝑖‘ndx) ∧ (·𝑖‘ndx) ∈ ℕ)

Theoremipsstrd 12109 A constructed inner product space is a structure. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon, 7-Feb-2023.)
𝐴 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), 𝐼⟩})    &   (𝜑𝐵𝑉)    &   (𝜑+𝑊)    &   (𝜑×𝑋)    &   (𝜑𝑆𝑌)    &   (𝜑·𝑄)    &   (𝜑𝐼𝑍)       (𝜑𝐴 Struct ⟨1, 8⟩)

Theoremipsbased 12110 The base set of a constructed inner product space. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon, 7-Feb-2023.)
𝐴 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), 𝐼⟩})    &   (𝜑𝐵𝑉)    &   (𝜑+𝑊)    &   (𝜑×𝑋)    &   (𝜑𝑆𝑌)    &   (𝜑·𝑄)    &   (𝜑𝐼𝑍)       (𝜑𝐵 = (Base‘𝐴))

Theoremipsaddgd 12111 The additive operation of a constructed inner product space. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon, 7-Feb-2023.)
𝐴 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), 𝐼⟩})    &   (𝜑𝐵𝑉)    &   (𝜑+𝑊)    &   (𝜑×𝑋)    &   (𝜑𝑆𝑌)    &   (𝜑·𝑄)    &   (𝜑𝐼𝑍)       (𝜑+ = (+g𝐴))

Theoremipsmulrd 12112 The multiplicative operation of a constructed inner product space. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon, 7-Feb-2023.)
𝐴 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), 𝐼⟩})    &   (𝜑𝐵𝑉)    &   (𝜑+𝑊)    &   (𝜑×𝑋)    &   (𝜑𝑆𝑌)    &   (𝜑·𝑄)    &   (𝜑𝐼𝑍)       (𝜑× = (.r𝐴))

Theoremipsscad 12113 The set of scalars of a constructed inner product space. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon, 8-Feb-2023.)
𝐴 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), 𝐼⟩})    &   (𝜑𝐵𝑉)    &   (𝜑+𝑊)    &   (𝜑×𝑋)    &   (𝜑𝑆𝑌)    &   (𝜑·𝑄)    &   (𝜑𝐼𝑍)       (𝜑𝑆 = (Scalar‘𝐴))

Theoremipsvscad 12114 The scalar product operation of a constructed inner product space. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon, 8-Feb-2023.)
𝐴 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), 𝐼⟩})    &   (𝜑𝐵𝑉)    &   (𝜑+𝑊)    &   (𝜑×𝑋)    &   (𝜑𝑆𝑌)    &   (𝜑·𝑄)    &   (𝜑𝐼𝑍)       (𝜑· = ( ·𝑠𝐴))

Theoremipsipd 12115 The multiplicative operation of a constructed inner product space. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon, 8-Feb-2023.)
𝐴 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), 𝐼⟩})    &   (𝜑𝐵𝑉)    &   (𝜑+𝑊)    &   (𝜑×𝑋)    &   (𝜑𝑆𝑌)    &   (𝜑·𝑄)    &   (𝜑𝐼𝑍)       (𝜑𝐼 = (·𝑖𝐴))

Theoremtsetndx 12116 Index value of the df-tset 12049 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
(TopSet‘ndx) = 9

Theoremtsetid 12117 Utility theorem: index-independent form of df-tset 12049. (Contributed by NM, 20-Oct-2012.)
TopSet = Slot (TopSet‘ndx)

Theoremtsetslid 12118 Slot property of TopSet. (Contributed by Jim Kingdon, 9-Feb-2023.)
(TopSet = Slot (TopSet‘ndx) ∧ (TopSet‘ndx) ∈ ℕ)

Theoremtopgrpstrd 12119 A constructed topological group is a structure. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 9-Feb-2023.)
𝑊 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩}    &   (𝜑𝐵𝑉)    &   (𝜑+𝑊)    &   (𝜑𝐽𝑋)       (𝜑𝑊 Struct ⟨1, 9⟩)

Theoremtopgrpbasd 12120 The base set of a constructed topological group. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 9-Feb-2023.)
𝑊 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩}    &   (𝜑𝐵𝑉)    &   (𝜑+𝑊)    &   (𝜑𝐽𝑋)       (𝜑𝐵 = (Base‘𝑊))

Theoremtopgrpplusgd 12121 The additive operation of a constructed topological group. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 9-Feb-2023.)
𝑊 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩}    &   (𝜑𝐵𝑉)    &   (𝜑+𝑊)    &   (𝜑𝐽𝑋)       (𝜑+ = (+g𝑊))

Theoremtopgrptsetd 12122 The topology of a constructed topological group. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 9-Feb-2023.)
𝑊 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩}    &   (𝜑𝐵𝑉)    &   (𝜑+𝑊)    &   (𝜑𝐽𝑋)       (𝜑𝐽 = (TopSet‘𝑊))

Theoremplendx 12123 Index value of the df-ple 12050 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by AV, 9-Sep-2021.)
(le‘ndx) = 10

Theorempleid 12124 Utility theorem: self-referencing, index-independent form of df-ple 12050. (Contributed by NM, 9-Nov-2012.) (Revised by AV, 9-Sep-2021.)
le = Slot (le‘ndx)

Theorempleslid 12125 Slot property of le. (Contributed by Jim Kingdon, 9-Feb-2023.)
(le = Slot (le‘ndx) ∧ (le‘ndx) ∈ ℕ)

Theoremdsndx 12126 Index value of the df-ds 12052 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
(dist‘ndx) = 12

Theoremdsid 12127 Utility theorem: index-independent form of df-ds 12052. (Contributed by Mario Carneiro, 23-Dec-2013.)
dist = Slot (dist‘ndx)

Theoremdsslid 12128 Slot property of dist. (Contributed by Jim Kingdon, 6-May-2023.)
(dist = Slot (dist‘ndx) ∧ (dist‘ndx) ∈ ℕ)

6.1.3  Definition of the structure product

Syntaxcrest 12129 Extend class notation with the function returning a subspace topology.
class t

Syntaxctopn 12130 Extend class notation with the topology extractor function.
class TopOpen

Definitiondf-rest 12131* Function returning the subspace topology induced by the topology 𝑦 and the set 𝑥. (Contributed by FL, 20-Sep-2010.) (Revised by Mario Carneiro, 1-May-2015.)
t = (𝑗 ∈ V, 𝑥 ∈ V ↦ ran (𝑦𝑗 ↦ (𝑦𝑥)))

Definitiondf-topn 12132 Define the topology extractor function. This differs from df-tset 12049 when a structure has been restricted using df-ress 11976; in this case the TopSet component will still have a topology over the larger set, and this function fixes this by restricting the topology as well. (Contributed by Mario Carneiro, 13-Aug-2015.)
TopOpen = (𝑤 ∈ V ↦ ((TopSet‘𝑤) ↾t (Base‘𝑤)))

Theoremrestfn 12133 The subspace topology operator is a function on pairs. (Contributed by Mario Carneiro, 1-May-2015.)
t Fn (V × V)

Theoremtopnfn 12134 The topology extractor function is a function on the universe. (Contributed by Mario Carneiro, 13-Aug-2015.)
TopOpen Fn V

Theoremrestval 12135* The subspace topology induced by the topology 𝐽 on the set 𝐴. (Contributed by FL, 20-Sep-2010.) (Revised by Mario Carneiro, 1-May-2015.)
((𝐽𝑉𝐴𝑊) → (𝐽t 𝐴) = ran (𝑥𝐽 ↦ (𝑥𝐴)))

Theoremelrest 12136* The predicate "is an open set of a subspace topology". (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 15-Dec-2013.)
((𝐽𝑉𝐵𝑊) → (𝐴 ∈ (𝐽t 𝐵) ↔ ∃𝑥𝐽 𝐴 = (𝑥𝐵)))

Theoremelrestr 12137 Sufficient condition for being an open set in a subspace. (Contributed by Jeff Hankins, 11-Jul-2009.) (Revised by Mario Carneiro, 15-Dec-2013.)
((𝐽𝑉𝑆𝑊𝐴𝐽) → (𝐴𝑆) ∈ (𝐽t 𝑆))

Theoremrestid2 12138 The subspace topology over a subset of the base set is the original topology. (Contributed by Mario Carneiro, 13-Aug-2015.)
((𝐴𝑉𝐽 ⊆ 𝒫 𝐴) → (𝐽t 𝐴) = 𝐽)

Theoremrestsspw 12139 The subspace topology is a collection of subsets of the restriction set. (Contributed by Mario Carneiro, 13-Aug-2015.)
(𝐽t 𝐴) ⊆ 𝒫 𝐴

Theoremrestid 12140 The subspace topology of the base set is the original topology. (Contributed by Jeff Hankins, 9-Jul-2009.) (Revised by Mario Carneiro, 13-Aug-2015.)
𝑋 = 𝐽       (𝐽𝑉 → (𝐽t 𝑋) = 𝐽)

Theoremtopnvalg 12141 Value of the topology extractor function. (Contributed by Mario Carneiro, 13-Aug-2015.) (Revised by Jim Kingdon, 11-Feb-2023.)
𝐵 = (Base‘𝑊)    &   𝐽 = (TopSet‘𝑊)       (𝑊𝑉 → (𝐽t 𝐵) = (TopOpen‘𝑊))

Theoremtopnidg 12142 Value of the topology extractor function when the topology is defined over the same set as the base. (Contributed by Mario Carneiro, 13-Aug-2015.)
𝐵 = (Base‘𝑊)    &   𝐽 = (TopSet‘𝑊)       ((𝑊𝑉𝐽 ⊆ 𝒫 𝐵) → 𝐽 = (TopOpen‘𝑊))

Theoremtopnpropgd 12143 The topology extractor function depends only on the base and topology components. (Contributed by NM, 18-Jul-2006.) (Revised by Jim Kingdon, 13-Feb-2023.)
(𝜑 → (Base‘𝐾) = (Base‘𝐿))    &   (𝜑 → (TopSet‘𝐾) = (TopSet‘𝐿))    &   (𝜑𝐾𝑉)    &   (𝜑𝐿𝑊)       (𝜑 → (TopOpen‘𝐾) = (TopOpen‘𝐿))

Syntaxctg 12144 Extend class notation with a function that converts a basis to its corresponding topology.
class topGen

Syntaxcpt 12145 Extend class notation with a function whose value is a product topology.
class t

Syntaxc0g 12146 Extend class notation with group identity element.
class 0g

Syntaxcgsu 12147 Extend class notation to include finitely supported group sums.
class Σg

Definitiondf-0g 12148* Define group identity element. Remark: this definition is required here because the symbol 0g is already used in df-gsum 12149. The related theorems will be provided later. (Contributed by NM, 20-Aug-2011.)
0g = (𝑔 ∈ V ↦ (℩𝑒(𝑒 ∈ (Base‘𝑔) ∧ ∀𝑥 ∈ (Base‘𝑔)((𝑒(+g𝑔)𝑥) = 𝑥 ∧ (𝑥(+g𝑔)𝑒) = 𝑥))))

Definitiondf-gsum 12149* Define the group sum for the structure 𝐺 of a finite sequence of elements whose values are defined by the expression 𝐵 and whose set of indices is 𝐴. It may be viewed as a product (if 𝐺 is a multiplication), a sum (if 𝐺 is an addition) or any other operation. The variable 𝑘 is normally a free variable in 𝐵 (i.e., 𝐵 can be thought of as 𝐵(𝑘)). The definition is meaningful in different contexts, depending on the size of the index set 𝐴 and each demanding different properties of 𝐺.

1. If 𝐴 = ∅ and 𝐺 has an identity element, then the sum equals this identity.

2. If 𝐴 = (𝑀...𝑁) and 𝐺 is any magma, then the sum is the sum of the elements, evaluated left-to-right, i.e. (𝐵(1) + 𝐵(2)) + 𝐵(3) etc.

3. If 𝐴 is a finite set (or is nonzero for finitely many indices) and 𝐺 is a commutative monoid, then the sum adds up these elements in some order, which is then uniquely defined.

4. If 𝐴 is an infinite set and 𝐺 is a Hausdorff topological group, then there is a meaningful sum, but Σg cannot handle this case. (Contributed by FL, 5-Sep-2010.) (Revised by FL, 17-Oct-2011.) (Revised by Mario Carneiro, 7-Dec-2014.)

Σg = (𝑤 ∈ V, 𝑓 ∈ V ↦ {𝑥 ∈ (Base‘𝑤) ∣ ∀𝑦 ∈ (Base‘𝑤)((𝑥(+g𝑤)𝑦) = 𝑦 ∧ (𝑦(+g𝑤)𝑥) = 𝑦)} / 𝑜if(ran 𝑓𝑜, (0g𝑤), if(dom 𝑓 ∈ ran ..., (℩𝑥𝑚𝑛 ∈ (ℤ𝑚)(dom 𝑓 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g𝑤), 𝑓)‘𝑛))), (℩𝑥𝑔[(𝑓 “ (V ∖ 𝑜)) / 𝑦](𝑔:(1...(♯‘𝑦))–1-1-onto𝑦𝑥 = (seq1((+g𝑤), (𝑓𝑔))‘(♯‘𝑦)))))))

Definitiondf-topgen 12150* Define a function that converts a basis to its corresponding topology. Equivalent to the definition of a topology generated by a basis in [Munkres] p. 78. (Contributed by NM, 16-Jul-2006.)
topGen = (𝑥 ∈ V ↦ {𝑦𝑦 (𝑥 ∩ 𝒫 𝑦)})

Definitiondf-pt 12151* Define the product topology on a collection of topologies. For convenience, it is defined on arbitrary collections of sets, expressed as a function from some index set to the subbases of each factor space. (Contributed by Mario Carneiro, 3-Feb-2015.)
t = (𝑓 ∈ V ↦ (topGen‘{𝑥 ∣ ∃𝑔((𝑔 Fn dom 𝑓 ∧ ∀𝑦 ∈ dom 𝑓(𝑔𝑦) ∈ (𝑓𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (dom 𝑓𝑧)(𝑔𝑦) = (𝑓𝑦)) ∧ 𝑥 = X𝑦 ∈ dom 𝑓(𝑔𝑦))}))

Syntaxcprds 12152 The function constructing structure products.
class Xs

Syntaxcpws 12153 The function constructing structure powers.
class s

Definitiondf-prds 12154* Define a structure product. This can be a product of groups, rings, modules, or ordered topological fields; any unused components will have garbage in them but this is usually not relevant for the purpose of inheriting the structures present in the factors. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Thierry Arnoux, 15-Jun-2019.)
Xs = (𝑠 ∈ V, 𝑟 ∈ V ↦ X𝑥 ∈ dom 𝑟(Base‘(𝑟𝑥)) / 𝑣(𝑓𝑣, 𝑔𝑣X𝑥 ∈ dom 𝑟((𝑓𝑥)(Hom ‘(𝑟𝑥))(𝑔𝑥))) / (({⟨(Base‘ndx), 𝑣⟩, ⟨(+g‘ndx), (𝑓𝑣, 𝑔𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(+g‘(𝑟𝑥))(𝑔𝑥))))⟩, ⟨(.r‘ndx), (𝑓𝑣, 𝑔𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(.r‘(𝑟𝑥))(𝑔𝑥))))⟩} ∪ {⟨(Scalar‘ndx), 𝑠⟩, ⟨( ·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑠), 𝑔𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ (𝑓( ·𝑠 ‘(𝑟𝑥))(𝑔𝑥))))⟩, ⟨(·𝑖‘ndx), (𝑓𝑣, 𝑔𝑣 ↦ (𝑠 Σg (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(·𝑖‘(𝑟𝑥))(𝑔𝑥)))))⟩}) ∪ ({⟨(TopSet‘ndx), (∏t‘(TopOpen ∘ 𝑟))⟩, ⟨(le‘ndx), {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝑣 ∧ ∀𝑥 ∈ dom 𝑟(𝑓𝑥)(le‘(𝑟𝑥))(𝑔𝑥))}⟩, ⟨(dist‘ndx), (𝑓𝑣, 𝑔𝑣 ↦ sup((ran (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(dist‘(𝑟𝑥))(𝑔𝑥))) ∪ {0}), ℝ*, < ))⟩} ∪ {⟨(Hom ‘ndx), ⟩, ⟨(comp‘ndx), (𝑎 ∈ (𝑣 × 𝑣), 𝑐𝑣 ↦ (𝑑 ∈ (𝑐(2nd𝑎)), 𝑒 ∈ (𝑎) ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑑𝑥)(⟨((1st𝑎)‘𝑥), ((2nd𝑎)‘𝑥)⟩(comp‘(𝑟𝑥))(𝑐𝑥))(𝑒𝑥)))))⟩})))

Theoremreldmprds 12155 The structure product is a well-behaved binary operator. (Contributed by Stefan O'Rear, 7-Jan-2015.) (Revised by Thierry Arnoux, 15-Jun-2019.)
Rel dom Xs

Definitiondf-pws 12156* Define a structure power, which is just a structure product where all the factors are the same. (Contributed by Mario Carneiro, 11-Jan-2015.)
s = (𝑟 ∈ V, 𝑖 ∈ V ↦ ((Scalar‘𝑟)Xs(𝑖 × {𝑟})))

6.2  The complex numbers as an algebraic extensible structure

6.2.1  Definition and basic properties

Syntaxcpsmet 12157 Extend class notation with the class of all pseudometric spaces.
class PsMet

Syntaxcxmet 12158 Extend class notation with the class of all extended metric spaces.
class ∞Met

Syntaxcmet 12159 Extend class notation with the class of all metrics.
class Met

Syntaxcbl 12160 Extend class notation with the metric space ball function.
class ball

Syntaxcfbas 12161 Extend class definition to include the class of filter bases.
class fBas

Syntaxcfg 12162 Extend class definition to include the filter generating function.
class filGen

Syntaxcmopn 12163 Extend class notation with a function mapping each metric space to the family of its open sets.
class MetOpen

Syntaxcmetu 12164 Extend class notation with the function mapping metrics to the uniform structure generated by that metric.
class metUnif

Definitiondf-psmet 12165* Define the set of all pseudometrics on a given base set. In a pseudo metric, two distinct points may have a distance zero. (Contributed by Thierry Arnoux, 7-Feb-2018.)
PsMet = (𝑥 ∈ V ↦ {𝑑 ∈ (ℝ*𝑚 (𝑥 × 𝑥)) ∣ ∀𝑦𝑥 ((𝑦𝑑𝑦) = 0 ∧ ∀𝑧𝑥𝑤𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) +𝑒 (𝑤𝑑𝑧)))})

Definitiondf-xmet 12166* Define the set of all extended metrics on a given base set. The definition is similar to df-met 12167, but we also allow the metric to take on the value +∞. (Contributed by Mario Carneiro, 20-Aug-2015.)
∞Met = (𝑥 ∈ V ↦ {𝑑 ∈ (ℝ*𝑚 (𝑥 × 𝑥)) ∣ ∀𝑦𝑥𝑧𝑥 (((𝑦𝑑𝑧) = 0 ↔ 𝑦 = 𝑧) ∧ ∀𝑤𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) +𝑒 (𝑤𝑑𝑧)))})

Definitiondf-met 12167* Define the (proper) class of all metrics. (A metric space is the metric's base set paired with the metric. However, we will often also call the metric itself a "metric space".) Equivalent to Definition 14-1.1 of [Gleason] p. 223. (Contributed by NM, 25-Aug-2006.)
Met = (𝑥 ∈ V ↦ {𝑑 ∈ (ℝ ↑𝑚 (𝑥 × 𝑥)) ∣ ∀𝑦𝑥𝑧𝑥 (((𝑦𝑑𝑧) = 0 ↔ 𝑦 = 𝑧) ∧ ∀𝑤𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) + (𝑤𝑑𝑧)))})

Definitiondf-bl 12168* Define the metric space ball function. (Contributed by NM, 30-Aug-2006.) (Revised by Thierry Arnoux, 11-Feb-2018.)
ball = (𝑑 ∈ V ↦ (𝑥 ∈ dom dom 𝑑, 𝑧 ∈ ℝ* ↦ {𝑦 ∈ dom dom 𝑑 ∣ (𝑥𝑑𝑦) < 𝑧}))

Definitiondf-mopn 12169 Define a function whose value is the family of open sets of a metric space. (Contributed by NM, 1-Sep-2006.)
MetOpen = (𝑑 ran ∞Met ↦ (topGen‘ran (ball‘𝑑)))

Definitiondf-fbas 12170* Define the class of all filter bases. Note that a filter base on one set is also a filter base for any superset, so there is not a unique base set that can be recovered. (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Stefan O'Rear, 11-Jul-2015.)
fBas = (𝑤 ∈ V ↦ {𝑥 ∈ 𝒫 𝒫 𝑤 ∣ (𝑥 ≠ ∅ ∧ ∅ ∉ 𝑥 ∧ ∀𝑦𝑥𝑧𝑥 (𝑥 ∩ 𝒫 (𝑦𝑧)) ≠ ∅)})

Definitiondf-fg 12171* Define the filter generating function. (Contributed by Jeff Hankins, 3-Sep-2009.) (Revised by Stefan O'Rear, 11-Jul-2015.)
filGen = (𝑤 ∈ V, 𝑥 ∈ (fBas‘𝑤) ↦ {𝑦 ∈ 𝒫 𝑤 ∣ (𝑥 ∩ 𝒫 𝑦) ≠ ∅})

Definitiondf-metu 12172* Define the function mapping metrics to the uniform structure generated by that metric. (Contributed by Thierry Arnoux, 1-Dec-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
metUnif = (𝑑 ran PsMet ↦ ((dom dom 𝑑 × dom dom 𝑑)filGenran (𝑎 ∈ ℝ+ ↦ (𝑑 “ (0[,)𝑎)))))

PART 7  BASIC TOPOLOGY

7.1  Topology

7.1.1  Topological spaces

A topology on a set is a set of subsets of that set, called open sets, which satisfy certain conditions. One condition is that the whole set be an open set. Therefore, a set is recoverable from a topology on it (as its union), and it may sometimes be more convenient to consider topologies without reference to the underlying set.

7.1.1.1  Topologies

Syntaxctop 12173 Syntax for the class of topologies.
class Top

Definitiondf-top 12174* Define the class of topologies. It is a proper class. See istopg 12175 and istopfin 12176 for the corresponding characterizations, using respectively binary intersections like in this definition and nonempty finite intersections.

The final form of the definition is due to Bourbaki (Def. 1 of [BourbakiTop1] p. I.1), while the idea of defining a topology in terms of its open sets is due to Aleksandrov. For the convoluted history of the definitions of these notions, see

Gregory H. Moore, The emergence of open sets, closed sets, and limit points in analysis and topology, Historia Mathematica 35 (2008) 220--241.

(Contributed by NM, 3-Mar-2006.) (Revised by BJ, 20-Oct-2018.)

Top = {𝑥 ∣ (∀𝑦 ∈ 𝒫 𝑥 𝑦𝑥 ∧ ∀𝑦𝑥𝑧𝑥 (𝑦𝑧) ∈ 𝑥)}

Theoremistopg 12175* Express the predicate "𝐽 is a topology". See istopfin 12176 for another characterization using nonempty finite intersections instead of binary intersections.

Note: In the literature, a topology is often represented by a calligraphic letter T, which resembles the letter J. This confusion may have led to J being used by some authors (e.g., K. D. Joshi, Introduction to General Topology (1983), p. 114) and it is convenient for us since we later use 𝑇 to represent linear transformations (operators). (Contributed by Stefan Allan, 3-Mar-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)

(𝐽𝐴 → (𝐽 ∈ Top ↔ (∀𝑥(𝑥𝐽 𝑥𝐽) ∧ ∀𝑥𝐽𝑦𝐽 (𝑥𝑦) ∈ 𝐽)))

Theoremistopfin 12176* Express the predicate "𝐽 is a topology" using nonempty finite intersections instead of binary intersections as in istopg 12175. It is not clear we can prove the converse without adding additional conditions. (Contributed by NM, 19-Jul-2006.) (Revised by Jim Kingdon, 14-Jan-2023.)
(𝐽 ∈ Top → (∀𝑥(𝑥𝐽 𝑥𝐽) ∧ ∀𝑥((𝑥𝐽𝑥 ≠ ∅ ∧ 𝑥 ∈ Fin) → 𝑥𝐽)))

Theoremuniopn 12177 The union of a subset of a topology (that is, the union of any family of open sets of a topology) is an open set. (Contributed by Stefan Allan, 27-Feb-2006.)
((𝐽 ∈ Top ∧ 𝐴𝐽) → 𝐴𝐽)

Theoremiunopn 12178* The indexed union of a subset of a topology is an open set. (Contributed by NM, 5-Oct-2006.)
((𝐽 ∈ Top ∧ ∀𝑥𝐴 𝐵𝐽) → 𝑥𝐴 𝐵𝐽)

Theoreminopn 12179 The intersection of two open sets of a topology is an open set. (Contributed by NM, 17-Jul-2006.)
((𝐽 ∈ Top ∧ 𝐴𝐽𝐵𝐽) → (𝐴𝐵) ∈ 𝐽)

Theoremfiinopn 12180 The intersection of a nonempty finite family of open sets is open. (Contributed by FL, 20-Apr-2012.)
(𝐽 ∈ Top → ((𝐴𝐽𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin) → 𝐴𝐽))

Theoremunopn 12181 The union of two open sets is open. (Contributed by Jeff Madsen, 2-Sep-2009.)
((𝐽 ∈ Top ∧ 𝐴𝐽𝐵𝐽) → (𝐴𝐵) ∈ 𝐽)

Theorem0opn 12182 The empty set is an open subset of any topology. (Contributed by Stefan Allan, 27-Feb-2006.)
(𝐽 ∈ Top → ∅ ∈ 𝐽)

Theorem0ntop 12183 The empty set is not a topology. (Contributed by FL, 1-Jun-2008.)
¬ ∅ ∈ Top

Theoremtopopn 12184 The underlying set of a topology is an open set. (Contributed by NM, 17-Jul-2006.)
𝑋 = 𝐽       (𝐽 ∈ Top → 𝑋𝐽)

Theoremeltopss 12185 A member of a topology is a subset of its underlying set. (Contributed by NM, 12-Sep-2006.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝐴𝐽) → 𝐴𝑋)

7.1.1.2  Topologies on sets

Syntaxctopon 12186 Syntax for the function of topologies on sets.
class TopOn

Definitiondf-topon 12187* Define the function that associates with a set the set of topologies on it. (Contributed by Stefan O'Rear, 31-Jan-2015.)
TopOn = (𝑏 ∈ V ↦ {𝑗 ∈ Top ∣ 𝑏 = 𝑗})

Theoremfuntopon 12188 The class TopOn is a function. (Contributed by BJ, 29-Apr-2021.)
Fun TopOn

Theoremistopon 12189 Property of being a topology with a given base set. (Contributed by Stefan O'Rear, 31-Jan-2015.) (Revised by Mario Carneiro, 13-Aug-2015.)
(𝐽 ∈ (TopOn‘𝐵) ↔ (𝐽 ∈ Top ∧ 𝐵 = 𝐽))

Theoremtopontop 12190 A topology on a given base set is a topology. (Contributed by Mario Carneiro, 13-Aug-2015.)
(𝐽 ∈ (TopOn‘𝐵) → 𝐽 ∈ Top)

Theoremtoponuni 12191 The base set of a topology on a given base set. (Contributed by Mario Carneiro, 13-Aug-2015.)
(𝐽 ∈ (TopOn‘𝐵) → 𝐵 = 𝐽)

Theoremtopontopi 12192 A topology on a given base set is a topology. (Contributed by Mario Carneiro, 13-Aug-2015.)
𝐽 ∈ (TopOn‘𝐵)       𝐽 ∈ Top

Theoremtoponunii 12193 The base set of a topology on a given base set. (Contributed by Mario Carneiro, 13-Aug-2015.)
𝐽 ∈ (TopOn‘𝐵)       𝐵 = 𝐽

Theoremtoptopon 12194 Alternative definition of Top in terms of TopOn. (Contributed by Mario Carneiro, 13-Aug-2015.)
𝑋 = 𝐽       (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))

Theoremtoptopon2 12195 A topology is the same thing as a topology on the union of its open sets. (Contributed by BJ, 27-Apr-2021.)
(𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))

Theoremtopontopon 12196 A topology on a set is a topology on the union of its open sets. (Contributed by BJ, 27-Apr-2021.)
(𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ (TopOn‘ 𝐽))

Theoremtoponrestid 12197 Given a topology on a set, restricting it to that same set has no effect. (Contributed by Jim Kingdon, 6-Jul-2022.)
𝐴 ∈ (TopOn‘𝐵)       𝐴 = (𝐴t 𝐵)

Theoremtoponsspwpwg 12198 The set of topologies on a set is included in the double power set of that set. (Contributed by BJ, 29-Apr-2021.) (Revised by Jim Kingdon, 16-Jan-2023.)
(𝐴𝑉 → (TopOn‘𝐴) ⊆ 𝒫 𝒫 𝐴)

Theoremdmtopon 12199 The domain of TopOn is V. (Contributed by BJ, 29-Apr-2021.)
dom TopOn = V

Theoremfntopon 12200 The class TopOn is a function with domain V. (Contributed by BJ, 29-Apr-2021.)
TopOn Fn V

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