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Theorem List for Metamath Proof Explorer - 13101-13200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Definitiondf-tset 13101 Define the topology component of a topological space (structure). (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
TopSet Slot

Definitiondf-ple 13102 Define less-than-or-equal ordering extractor for posets and related structures. We use for the index to avoid conflict with through used for other purposes. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
Slot

Definitiondf-ocomp 13103 Define the orthocomplementation extractor for posets and related structures. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
Slot ;

Definitiondf-ds 13104 Define the distance function component of a metric space (structure). (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
Slot ;

Definitiondf-unif 13105 Define the uniform structure component of a uniform space. (Contributed by Mario Carneiro, 14-Aug-2015.)
Slot ;

Definitiondf-hom 13106 Define the hom-set component of a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
Slot ;

Definitiondf-cco 13107 Define the composition operation of a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp Slot ;

Theoremstrlemor0 13108 Structure definition utility lemma. To prove that an explicit function is a function using O(n) steps, exploit the order properties of the index set. Zero-pair case. (Contributed by Stefan O'Rear, 3-Jan-2015.)

Theoremstrlemor1 13109 Add one element to the end of a structure. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Mario Carneiro, 30-Apr-2015.)

Theoremstrlemor2 13110 Add two elements to the end of a structure. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Mario Carneiro, 30-Apr-2015.)

Theoremstrlemor3 13111 Add three elements to the end of a structure. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Mario Carneiro, 30-Apr-2015.)

Theoremstrleun 13112 Combine two structures into one. (Contributed by Mario Carneiro, 29-Aug-2015.)
Struct        Struct               Struct

Theoremstrle1 13113 Make a structure from a singleton. (Contributed by Mario Carneiro, 29-Aug-2015.)
Struct

Theoremstrle2 13114 Make a structure from a pair. (Contributed by Mario Carneiro, 29-Aug-2015.)
Struct

Theoremstrle3 13115 Make a structure from a triple. (Contributed by Mario Carneiro, 29-Aug-2015.)
Struct

Theoremplusgndx 13116 Index value of the df-plusg 13095 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)

Theoremplusgid 13117 Utility theorem: index-independent form of df-plusg 13095. (Contributed by NM, 20-Oct-2012.)
Slot

Theorem2strstr 13118 A constructed two-slot structure. (Contributed by Mario Carneiro, 29-Aug-2015.)
Slot                      Struct

Theorem2strbas 13119 The base set of a constructed two-slot structure. (Contributed by Mario Carneiro, 29-Aug-2015.)
Slot

Theorem2strop 13120 The other slot of a constructed two-slot structure. (Contributed by Mario Carneiro, 29-Aug-2015.)
Slot

Theoremgrpstr 13121 A constructed group is a structure on . (Contributed by Mario Carneiro, 28-Sep-2013.) (Revised by Mario Carneiro, 30-Apr-2015.)
Struct

Theoremgrpbase 13122 The base set of a constructed group. (Contributed by Mario Carneiro, 2-Aug-2013.) (Revised by Mario Carneiro, 30-Apr-2015.)

Theoremgrpplusg 13123 The operation of a constructed group. (Contributed by Mario Carneiro, 2-Aug-2013.) (Revised by Mario Carneiro, 30-Apr-2015.)

Theoremressplusg 13124 is unaffected by restriction. (Contributed by Stefan O'Rear, 27-Nov-2014.)
s

Theoremgrpbasex 13125 The base of an explicitly given group. Note: This theorem has hard-coded structure indices for demonstration purposes. It is not intended for general use; use grpbase 13122 instead. (Contributed by NM, 17-Oct-2012.)

Theoremgrpplusgx 13126 The operation of an explicitly given group. Note: This theorem has hard-coded structure indices for demonstration purposes. It is not intended for general use; use grpplusgx 13126 instead. (Contributed by NM, 17-Oct-2012.)

Theoremmulrndx 13127 Index value of the df-mulr 13096 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)

Theoremmulrid 13128 Utility theorem: index-independent form of df-mulr 13096. (Contributed by Mario Carneiro, 8-Jun-2013.)
Slot

Theoremrngstr 13129 A constructed ring is a structure. (Contributed by Mario Carneiro, 28-Sep-2013.) (Revised by Mario Carneiro, 29-Aug-2015.)
Struct

Theoremrngbase 13130 The base set of a constructed ring. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Mario Carneiro, 30-Apr-2015.)

Theoremrngplusg 13131 The additive operation of a constructed ring. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Mario Carneiro, 30-Apr-2015.)

Theoremrngmulr 13132 The muliplicative operation of a constructed ring. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Mario Carneiro, 30-Apr-2015.)

Theoremstarvndx 13133 Index value of the df-starv 13097 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)

Theoremstarvid 13134 Utility theorem: index-independent form of df-starv 13097. (Contributed by Mario Carneiro, 6-Oct-2013.)
Slot

Theoremressmulr 13135 is unaffected by restriction. (Contributed by Stefan O'Rear, 27-Nov-2014.)
s

Theoremressstarv 13136 is unaffected by restriction. (Contributed by Mario Carneiro, 9-Oct-2015.)
s

Theoremsrngfn 13137 A constructed star ring is a function with domain contained in thru . (Contributed by Mario Carneiro, 18-Nov-2013.) (Revised by Mario Carneiro, 14-Aug-2015.)
Struct

Theoremsrngbase 13138 The base set of a constructed star ring. (Contributed by Mario Carneiro, 18-Nov-2013.) (Revised by Mario Carneiro, 6-May-2015.)

Theoremsrngplusg 13139 The addition operation of a constructed star ring. (Contributed by Mario Carneiro, 20-Jun-2015.)

Theoremsrngmulr 13140 The multiplication operation of a constructed star ring. (Contributed by Mario Carneiro, 20-Jun-2015.)

Theoremsrnginvl 13141 The involution function of a constructed star ring. (Contributed by Mario Carneiro, 20-Jun-2015.)

Theoremscandx 13142 Index value of the df-sca 13098 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
Scalar

Theoremscaid 13143 Utility theorem: index-independent form of scalar df-sca 13098. (Contributed by Mario Carneiro, 19-Jun-2014.)
Scalar Slot Scalar

Theoremvscandx 13144 Index value of the df-vsca 13099 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)

Theoremvscaid 13145 Utility theorem: index-independent form of scalar product df-vsca 13099. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
Slot

Theoremlmodstr 13146 A constructed left module or left vector space is a function. (Contributed by Mario Carneiro, 1-Oct-2013.) (Revised by Mario Carneiro, 29-Aug-2015.)
Scalar        Struct

Theoremlmodbase 13147 The base set of a constructed left vector space. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Mario Carneiro, 29-Aug-2015.)
Scalar

Theoremlmodplusg 13148 The additive operation of a constructed left vector space. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Mario Carneiro, 29-Aug-2015.)
Scalar

Theoremlmodsca 13149 The set of scalars of a constructed left vector space. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Mario Carneiro, 29-Aug-2015.)
Scalar        Scalar

Theoremlmodvsca 13150 The scalar product operation of a constructed left vector space. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Mario Carneiro, 29-Aug-2015.)
Scalar

Theoremalgstr 13151 Lemma to shorten proofs of algbase 13152 through algvsca 13156. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 29-Aug-2015.)
Scalar        Struct

Theoremalgbase 13152 The base set of a constructed algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 29-Aug-2015.)
Scalar

Theoremalgaddg 13153 The additive operation of a constructed algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 29-Aug-2015.)
Scalar

Theoremalgmulr 13154 The multiplicative operation of a constructed algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 29-Aug-2015.)
Scalar

Theoremalgsca 13155 The set of scalars of a constructed algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 29-Aug-2015.)
Scalar        Scalar

Theoremalgvsca 13156 The scalar product operation of a constructed algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 29-Aug-2015.)
Scalar

Theoremresssca 13157 Scalar is unaffected by restriction. (Contributed by Mario Carneiro, 7-Dec-2014.)
s        Scalar       Scalar

Theoremressvsca 13158 is unaffected by restriction. (Contributed by Mario Carneiro, 7-Dec-2014.)
s

Theoremipndx 13159 Index value of the df-ip 13100 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)

Theoremipid 13160 Utility theorem: index-independent form of df-ip 13100. (Contributed by Mario Carneiro, 6-Oct-2013.)
Slot

Theoremphlstr 13161 A constructed pre-Hilbert space is a structure. Starting from lmodstr 13146 (which has 4 members), we chain strleun 13112 once more, adding an ordered pair to the function, to get all 5 members. (Contributed by Mario Carneiro, 1-Oct-2013.) (Revised by Mario Carneiro, 29-Aug-2015.)
Scalar        Struct

Theoremphlbase 13162 The base set of a constructed pre-Hilbert space. (Contributed by Mario Carneiro, 6-Oct-2013.) (Revised by Mario Carneiro, 29-Aug-2015.)
Scalar

Theoremphlplusg 13163 The additive operation of a constructed pre-Hilbert space. (Contributed by Mario Carneiro, 6-Oct-2013.) (Revised by Mario Carneiro, 29-Aug-2015.)
Scalar

Theoremphlsca 13164 The ring of scalars of a constructed pre-Hilbert space. (Contributed by Mario Carneiro, 6-Oct-2013.) (Revised by Mario Carneiro, 29-Aug-2015.)
Scalar        Scalar

Theoremphlvsca 13165 The scalar product operation of a constructed pre-Hilbert space. (Contributed by Mario Carneiro, 6-Oct-2013.) (Revised by Mario Carneiro, 29-Aug-2015.)
Scalar

Theoremphlip 13166 The inner product (Hermitian form) operation of a constructed pre-Hilbert space. (Contributed by Mario Carneiro, 6-Oct-2013.) (Revised by Mario Carneiro, 29-Aug-2015.)
Scalar

Theoremtsetndx 13167 Index value of the df-tset 13101 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
TopSet

Theoremtsetid 13168 Utility theorem: index-independent form of df-tset 13101. (Contributed by NM, 20-Oct-2012.)
TopSet Slot TopSet

Theoremtopgrpstr 13169 A constructed topological group is a structure. (Contributed by Mario Carneiro, 29-Aug-2015.)
TopSet        Struct

Theoremtopgrpbas 13170 The base set of a constructed topological group. (Contributed by Mario Carneiro, 29-Aug-2015.)
TopSet

Theoremtopgrpplusg 13171 The additive operation of a constructed topological group. (Contributed by Mario Carneiro, 29-Aug-2015.)
TopSet

Theoremtopgrptset 13172 The topology of a constructed topological group. (Contributed by Mario Carneiro, 29-Aug-2015.)
TopSet        TopSet

Theoremresstset 13173 TopSet is unaffected by restriction. (Contributed by Mario Carneiro, 13-Aug-2015.)
s        TopSet       TopSet

Theoremplendx 13174 Index value of the df-ple 13102 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)

Theorempleid 13175 Utility theorem: self-referencing, index-independent form of df-ple 13102. (Contributed by NM, 9-Nov-2012.)
Slot

Theoremotpsstr 13176 Functionality of a topological ordered space. (Contributed by Mario Carneiro, 12-Nov-2015.)
TopSet        Struct

Theoremotpsbas 13177 The base set of a topological ordered space. (Contributed by Mario Carneiro, 12-Nov-2015.)
TopSet

Theoremotpstset 13178 The open sets of a topological ordered space. (Contributed by Mario Carneiro, 12-Nov-2015.)
TopSet        TopSet

Theoremotpsle 13179 The order of a topological ordered space. (Contributed by Mario Carneiro, 12-Nov-2015.)
TopSet

Theoremressle 13180 is unaffected by restriction. (Contributed by Mario Carneiro, 3-Nov-2015.)
s

Theoremocndx 13181 Index value of the df-ocomp 13103 slot. (Contributed by Mario Carneiro, 25-Oct-2015.)
;

Theoremocid 13182 Utility theorem: index-independent form of df-ocomp 13103. (Contributed by Mario Carneiro, 25-Oct-2015.)
Slot

Theoremdsndx 13183 Index value of the df-ds 13104 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
;

Theoremdsid 13184 Utility theorem: index-independent form of df-ds 13104. (Contributed by Mario Carneiro, 23-Dec-2013.)
Slot

Theoremodrngstr 13185 Functionality of an ordered metric ring. (Contributed by Mario Carneiro, 20-Aug-2015.)
TopSet        Struct ;

Theoremodrngbas 13186 The base set of an ordered metric ring. (Contributed by Mario Carneiro, 20-Aug-2015.)
TopSet

Theoremodrngplusg 13187 The addition operation of an ordered metric ring. (Contributed by Mario Carneiro, 20-Aug-2015.)
TopSet

Theoremodrngmulr 13188 The multiplication operation of an ordered metric ring. (Contributed by Mario Carneiro, 20-Aug-2015.)
TopSet

Theoremodrngtset 13189 The open sets of an ordered metric ring. (Contributed by Mario Carneiro, 20-Aug-2015.)
TopSet        TopSet

Theoremodrngle 13190 The order of an ordered metric ring. (Contributed by Mario Carneiro, 20-Aug-2015.)
TopSet

Theoremodrngds 13191 The metric of an ordered metric ring. (Contributed by Mario Carneiro, 20-Aug-2015.)
TopSet

Theoremressds 13192 is unaffected by restriction. (Contributed by Mario Carneiro, 26-Aug-2015.)
s

Theoremhomndx 13193 Index value of the df-hom 13106 slot. (Contributed by Mario Carneiro, 7-Jan-2017.)
;

Theoremhomid 13194 Utility theorem: index-independent form of df-hom 13106. (Contributed by Mario Carneiro, 7-Jan-2017.)
Slot

Theoremccondx 13195 Index value of the df-cco 13107 slot. (Contributed by Mario Carneiro, 7-Jan-2017.)
comp ;

Theoremccoid 13196 Utility theorem: index-independent form of df-cco 13107. (Contributed by Mario Carneiro, 7-Jan-2017.)
comp Slot comp

Theoremresshom 13197 is unaffected by restriction. (Contributed by Mario Carneiro, 5-Jan-2017.)
s

Theoremressco 13198 comp is unaffected by restriction. (Contributed by Mario Carneiro, 5-Jan-2017.)
s        comp       comp

7.1.3  Definition of the structure product

Syntaxcrest 13199 Extend class notation with the function returning a subspace topology.
t

Syntaxctopn 13200 Extend class notation with the topology extractor function.

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