HomeHome Metamath Proof Explorer
Theorem List (p. 132 of 310)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-21328)
  Hilbert Space Explorer  Hilbert Space Explorer
(21329-22851)
  Users' Mathboxes  Users' Mathboxes
(22852-30955)
 

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  9
 
Definitiondf-ple 13102 Define less-than-or-equal ordering extractor for posets and related structures. We use  10 for the index to avoid conflict with  1 through  9 used for other purposes. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
 |- 
 le  = Slot  10
 
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.)
 |- 
 oc  = Slot ; 1 1
 
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.)
 |- 
 dist  = Slot ; 1 2
 
Definitiondf-unif 13105 Define the uniform structure component of a uniform space. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |- 
 Unif  = Slot ; 1 3
 
Definitiondf-hom 13106 Define the hom-set component of a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |- 
 Hom  = Slot ; 1 4
 
Definitiondf-cco 13107 Define the composition operation of a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |- comp  = Slot ; 1
 5
 
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.)
 |-  ( Fun  `' `' (/)  /\  dom  (/)  C_  ( 1 ... 0 ) )
 
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.)
 |-  ( Fun  `' `' F  /\  dom  F  C_  (
 1 ... I ) )   &    |-  I  e.  NN0   &    |-  I  <  J   &    |-  J  e.  NN   &    |-  A  =  J   &    |-  G  =  ( F  u.  { <. A ,  X >. } )   =>    |-  ( Fun  `' `' G  /\  dom  G  C_  (
 1 ... J ) )
 
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.)
 |-  ( Fun  `' `' F  /\  dom  F  C_  (
 1 ... I ) )   &    |-  I  e.  NN0   &    |-  I  <  J   &    |-  J  e.  NN   &    |-  A  =  J   &    |-  J  <  K   &    |-  K  e.  NN   &    |-  B  =  K   &    |-  G  =  ( F  u.  { <. A ,  X >. ,  <. B ,  Y >. } )   =>    |-  ( Fun  `' `' G  /\  dom  G  C_  ( 1 ... K ) )
 
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.)
 |-  ( Fun  `' `' F  /\  dom  F  C_  (
 1 ... I ) )   &    |-  I  e.  NN0   &    |-  I  <  J   &    |-  J  e.  NN   &    |-  A  =  J   &    |-  J  <  K   &    |-  K  e.  NN   &    |-  B  =  K   &    |-  K  <  L   &    |-  L  e.  NN   &    |-  C  =  L   &    |-  G  =  ( F  u.  { <. A ,  X >. , 
 <. B ,  Y >. , 
 <. C ,  Z >. } )   =>    |-  ( Fun  `' `' G  /\  dom  G  C_  (
 1 ... L ) )
 
Theoremstrleun 13112 Combine two structures into one. (Contributed by Mario Carneiro, 29-Aug-2015.)
 |-  F Struct  <. A ,  B >.   &    |-  G Struct 
 <. C ,  D >.   &    |-  B  <  C   =>    |-  ( F  u.  G ) Struct 
 <. A ,  D >.
 
Theoremstrle1 13113 Make a structure from a singleton. (Contributed by Mario Carneiro, 29-Aug-2015.)
 |-  I  e.  NN   &    |-  A  =  I   =>    |- 
 { <. A ,  X >. } Struct  <. I ,  I >.
 
Theoremstrle2 13114 Make a structure from a pair. (Contributed by Mario Carneiro, 29-Aug-2015.)
 |-  I  e.  NN   &    |-  A  =  I   &    |-  I  <  J   &    |-  J  e.  NN   &    |-  B  =  J   =>    |-  { <. A ,  X >. ,  <. B ,  Y >. } Struct  <. I ,  J >.
 
Theoremstrle3 13115 Make a structure from a triple. (Contributed by Mario Carneiro, 29-Aug-2015.)
 |-  I  e.  NN   &    |-  A  =  I   &    |-  I  <  J   &    |-  J  e.  NN   &    |-  B  =  J   &    |-  J  <  K   &    |-  K  e.  NN   &    |-  C  =  K   =>    |- 
 { <. A ,  X >. ,  <. B ,  Y >. ,  <. C ,  Z >. } Struct  <. I ,  K >.
 
Theoremplusgndx 13116 Index value of the df-plusg 13095 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  ( +g  `  ndx )  =  2
 
Theoremplusgid 13117 Utility theorem: index-independent form of df-plusg 13095. (Contributed by NM, 20-Oct-2012.)
 |- 
 +g  = Slot  ( +g  ` 
 ndx )
 
Theorem2strstr 13118 A constructed two-slot structure. (Contributed by Mario Carneiro, 29-Aug-2015.)
 |-  G  =  { <. (
 Base `  ndx ) ,  B >. ,  <. ( E `
  ndx ) ,  .+  >. }   &    |-  E  = Slot  N   &    |-  1  <  N   &    |-  N  e.  NN   =>    |-  G Struct  <.
 1 ,  N >.
 
Theorem2strbas 13119 The base set of a constructed two-slot structure. (Contributed by Mario Carneiro, 29-Aug-2015.)
 |-  G  =  { <. (
 Base `  ndx ) ,  B >. ,  <. ( E `
  ndx ) ,  .+  >. }   &    |-  E  = Slot  N   &    |-  1  <  N   &    |-  N  e.  NN   =>    |-  ( B  e.  V  ->  B  =  ( Base `  G ) )
 
Theorem2strop 13120 The other slot of a constructed two-slot structure. (Contributed by Mario Carneiro, 29-Aug-2015.)
 |-  G  =  { <. (
 Base `  ndx ) ,  B >. ,  <. ( E `
  ndx ) ,  .+  >. }   &    |-  E  = Slot  N   &    |-  1  <  N   &    |-  N  e.  NN   =>    |-  (  .+  e.  V  ->  .+  =  ( E `  G ) )
 
Theoremgrpstr 13121 A constructed group is a structure on 
1 ... 2. (Contributed by Mario Carneiro, 28-Sep-2013.) (Revised by Mario Carneiro, 30-Apr-2015.)
 |-  G  =  { <. (
 Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. }   =>    |-  G Struct  <. 1 ,  2
 >.
 
Theoremgrpbase 13122 The base set of a constructed group. (Contributed by Mario Carneiro, 2-Aug-2013.) (Revised by Mario Carneiro, 30-Apr-2015.)
 |-  G  =  { <. (
 Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. }   =>    |-  ( B  e.  V  ->  B  =  ( Base `  G ) )
 
Theoremgrpplusg 13123 The operation of a constructed group. (Contributed by Mario Carneiro, 2-Aug-2013.) (Revised by Mario Carneiro, 30-Apr-2015.)
 |-  G  =  { <. (
 Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. }   =>    |-  (  .+  e.  V  ->  .+  =  ( +g  `  G ) )
 
Theoremressplusg 13124  +g is unaffected by restriction. (Contributed by Stefan O'Rear, 27-Nov-2014.)
 |-  H  =  ( Gs  A )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( A  e.  V  ->  .+  =  ( +g  `  H ) )
 
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.)
 |-  B  e.  _V   &    |-  .+  e.  _V   &    |-  G  =  { <. 1 ,  B >. ,  <. 2 ,  .+  >. }   =>    |-  B  =  ( Base `  G )
 
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.)
 |-  B  e.  _V   &    |-  .+  e.  _V   &    |-  G  =  { <. 1 ,  B >. ,  <. 2 ,  .+  >. }   =>    |- 
 .+  =  ( +g  `  G )
 
Theoremmulrndx 13127 Index value of the df-mulr 13096 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  ( .r `  ndx )  =  3
 
Theoremmulrid 13128 Utility theorem: index-independent form of df-mulr 13096. (Contributed by Mario Carneiro, 8-Jun-2013.)
 |- 
 .r  = Slot  ( .r ` 
 ndx )
 
Theoremrngstr 13129 A constructed ring is a structure. (Contributed by Mario Carneiro, 28-Sep-2013.) (Revised by Mario Carneiro, 29-Aug-2015.)
 |-  R  =  { <. (
 Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .x.  >. }   =>    |-  R Struct  <. 1 ,  3 >.
 
Theoremrngbase 13130 The base set of a constructed ring. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Mario Carneiro, 30-Apr-2015.)
 |-  R  =  { <. (
 Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .x.  >. }   =>    |-  ( B  e.  V  ->  B  =  (
 Base `  R ) )
 
Theoremrngplusg 13131 The additive operation of a constructed ring. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Mario Carneiro, 30-Apr-2015.)
 |-  R  =  { <. (
 Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .x.  >. }   =>    |-  (  .+  e.  V  ->  .+  =  ( +g  `  R ) )
 
Theoremrngmulr 13132 The muliplicative operation of a constructed ring. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Mario Carneiro, 30-Apr-2015.)
 |-  R  =  { <. (
 Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .x.  >. }   =>    |-  (  .x.  e.  V  ->  .x.  =  ( .r `  R ) )
 
Theoremstarvndx 13133 Index value of the df-starv 13097 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  ( * r `  ndx )  =  4
 
Theoremstarvid 13134 Utility theorem: index-independent form of df-starv 13097. (Contributed by Mario Carneiro, 6-Oct-2013.)
 |-  * r  = Slot  ( * r `  ndx )
 
Theoremressmulr 13135  .r is unaffected by restriction. (Contributed by Stefan O'Rear, 27-Nov-2014.)
 |-  S  =  ( Rs  A )   &    |-  .x.  =  ( .r `  R )   =>    |-  ( A  e.  V  ->  .x.  =  ( .r `  S ) )
 
Theoremressstarv 13136  * r is unaffected by restriction. (Contributed by Mario Carneiro, 9-Oct-2015.)
 |-  S  =  ( Rs  A )   &    |-  .*  =  ( * r `  R )   =>    |-  ( A  e.  V  ->  .*  =  ( * r `  S ) )
 
Theoremsrngfn 13137 A constructed star ring is a function with domain contained in  1 thru  4. (Contributed by Mario Carneiro, 18-Nov-2013.) (Revised by Mario Carneiro, 14-Aug-2015.)
 |-  R  =  ( { <. ( Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .x.  >. }  u.  {
 <. ( * r `  ndx ) ,  .*  >. } )   =>    |-  R Struct  <. 1 ,  4
 >.
 
Theoremsrngbase 13138 The base set of a constructed star ring. (Contributed by Mario Carneiro, 18-Nov-2013.) (Revised by Mario Carneiro, 6-May-2015.)
 |-  R  =  ( { <. ( Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .x.  >. }  u.  {
 <. ( * r `  ndx ) ,  .*  >. } )   =>    |-  ( B  e.  X  ->  B  =  ( Base `  R ) )
 
Theoremsrngplusg 13139 The addition operation of a constructed star ring. (Contributed by Mario Carneiro, 20-Jun-2015.)
 |-  R  =  ( { <. ( Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .x.  >. }  u.  {
 <. ( * r `  ndx ) ,  .*  >. } )   =>    |-  (  .+  e.  X  ->  .+  =  ( +g  `  R ) )
 
Theoremsrngmulr 13140 The multiplication operation of a constructed star ring. (Contributed by Mario Carneiro, 20-Jun-2015.)
 |-  R  =  ( { <. ( Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .x.  >. }  u.  {
 <. ( * r `  ndx ) ,  .*  >. } )   =>    |-  (  .x.  e.  X  ->  .x.  =  ( .r
 `  R ) )
 
Theoremsrnginvl 13141 The involution function of a constructed star ring. (Contributed by Mario Carneiro, 20-Jun-2015.)
 |-  R  =  ( { <. ( Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .x.  >. }  u.  {
 <. ( * r `  ndx ) ,  .*  >. } )   =>    |-  (  .*  e.  X  ->  .*  =  ( * r `  R ) )
 
Theoremscandx 13142 Index value of the df-sca 13098 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  (Scalar `  ndx )  =  5
 
Theoremscaid 13143 Utility theorem: index-independent form of scalar df-sca 13098. (Contributed by Mario Carneiro, 19-Jun-2014.)
 |- Scalar  = Slot  (Scalar `  ndx )
 
Theoremvscandx 13144 Index value of the df-vsca 13099 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  ( .s `  ndx )  =  6
 
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.)
 |- 
 .s  = Slot  ( .s ` 
 ndx )
 
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.)
 |-  W  =  ( { <. ( Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. ,  <. (Scalar `  ndx ) ,  F >. }  u.  { <. ( .s `  ndx ) ,  .x.  >. } )   =>    |-  W Struct  <.
 1 ,  6 >.
 
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.)
 |-  W  =  ( { <. ( Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. ,  <. (Scalar `  ndx ) ,  F >. }  u.  { <. ( .s `  ndx ) ,  .x.  >. } )   =>    |-  ( B  e.  X  ->  B  =  ( Base `  W ) )
 
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.)
 |-  W  =  ( { <. ( Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. ,  <. (Scalar `  ndx ) ,  F >. }  u.  { <. ( .s `  ndx ) ,  .x.  >. } )   =>    |-  (  .+  e.  X  ->  .+  =  ( +g  `  W )
 )
 
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.)
 |-  W  =  ( { <. ( Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. ,  <. (Scalar `  ndx ) ,  F >. }  u.  { <. ( .s `  ndx ) ,  .x.  >. } )   =>    |-  ( F  e.  X  ->  F  =  (Scalar `  W ) )
 
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.)
 |-  W  =  ( { <. ( Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. ,  <. (Scalar `  ndx ) ,  F >. }  u.  { <. ( .s `  ndx ) ,  .x.  >. } )   =>    |-  (  .x.  e.  X  ->  .x.  =  ( .s `  W ) )
 
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.)
 |-  A  =  ( { <. ( Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .X.  >. }  u.  {
 <. (Scalar `  ndx ) ,  S >. ,  <. ( .s
 `  ndx ) ,  .x.  >. } )   =>    |-  A Struct  <. 1 ,  6
 >.
 
Theoremalgbase 13152 The base set of a constructed algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 29-Aug-2015.)
 |-  A  =  ( { <. ( Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .X.  >. }  u.  {
 <. (Scalar `  ndx ) ,  S >. ,  <. ( .s
 `  ndx ) ,  .x.  >. } )   =>    |-  ( B  e.  V  ->  B  =  ( Base `  A ) )
 
Theoremalgaddg 13153 The additive operation of a constructed algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 29-Aug-2015.)
 |-  A  =  ( { <. ( Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .X.  >. }  u.  {
 <. (Scalar `  ndx ) ,  S >. ,  <. ( .s
 `  ndx ) ,  .x.  >. } )   =>    |-  (  .+  e.  V  ->  .+  =  ( +g  `  A ) )
 
Theoremalgmulr 13154 The multiplicative operation of a constructed algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 29-Aug-2015.)
 |-  A  =  ( { <. ( Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .X.  >. }  u.  {
 <. (Scalar `  ndx ) ,  S >. ,  <. ( .s
 `  ndx ) ,  .x.  >. } )   =>    |-  (  .X.  e.  V  -> 
 .X.  =  ( .r `  A ) )
 
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.)
 |-  A  =  ( { <. ( Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .X.  >. }  u.  {
 <. (Scalar `  ndx ) ,  S >. ,  <. ( .s
 `  ndx ) ,  .x.  >. } )   =>    |-  ( S  e.  V  ->  S  =  (Scalar `  A ) )
 
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.)
 |-  A  =  ( { <. ( Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .X.  >. }  u.  {
 <. (Scalar `  ndx ) ,  S >. ,  <. ( .s
 `  ndx ) ,  .x.  >. } )   =>    |-  (  .x.  e.  V  ->  .x.  =  ( .s
 `  A ) )
 
Theoremresssca 13157 Scalar is unaffected by restriction. (Contributed by Mario Carneiro, 7-Dec-2014.)
 |-  H  =  ( Gs  A )   &    |-  F  =  (Scalar `  G )   =>    |-  ( A  e.  V  ->  F  =  (Scalar `  H ) )
 
Theoremressvsca 13158  .s is unaffected by restriction. (Contributed by Mario Carneiro, 7-Dec-2014.)
 |-  H  =  ( Gs  A )   &    |-  .x.  =  ( .s `  G )   =>    |-  ( A  e.  V  ->  .x.  =  ( .s `  H ) )
 
Theoremipndx 13159 Index value of the df-ip 13100 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  ( .i `  ndx )  =  8
 
Theoremipid 13160 Utility theorem: index-independent form of df-ip 13100. (Contributed by Mario Carneiro, 6-Oct-2013.)
 |- 
 .i  = Slot  ( .i ` 
 ndx )
 
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.)
 |-  H  =  ( { <. ( Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. ,  <. (Scalar `  ndx ) ,  T >. }  u.  { <. ( .s `  ndx ) ,  .x.  >. ,  <. ( .i `  ndx ) ,  .,  >. } )   =>    |-  H Struct  <. 1 ,  8 >.
 
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.)
 |-  H  =  ( { <. ( Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. ,  <. (Scalar `  ndx ) ,  T >. }  u.  { <. ( .s `  ndx ) ,  .x.  >. ,  <. ( .i `  ndx ) ,  .,  >. } )   =>    |-  ( B  e.  X  ->  B  =  (
 Base `  H ) )
 
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.)
 |-  H  =  ( { <. ( Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. ,  <. (Scalar `  ndx ) ,  T >. }  u.  { <. ( .s `  ndx ) ,  .x.  >. ,  <. ( .i `  ndx ) ,  .,  >. } )   =>    |-  (  .+  e.  X  ->  .+  =  ( +g  `  H ) )
 
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.)
 |-  H  =  ( { <. ( Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. ,  <. (Scalar `  ndx ) ,  T >. }  u.  { <. ( .s `  ndx ) ,  .x.  >. ,  <. ( .i `  ndx ) ,  .,  >. } )   =>    |-  ( T  e.  X  ->  T  =  (Scalar `  H ) )
 
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.)
 |-  H  =  ( { <. ( Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. ,  <. (Scalar `  ndx ) ,  T >. }  u.  { <. ( .s `  ndx ) ,  .x.  >. ,  <. ( .i `  ndx ) ,  .,  >. } )   =>    |-  (  .x.  e.  X  ->  .x.  =  ( .s `  H ) )
 
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.)
 |-  H  =  ( { <. ( Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. ,  <. (Scalar `  ndx ) ,  T >. }  u.  { <. ( .s `  ndx ) ,  .x.  >. ,  <. ( .i `  ndx ) ,  .,  >. } )   =>    |-  (  .,  e.  X  ->  .,  =  ( .i `  H ) )
 
Theoremtsetndx 13167 Index value of the df-tset 13101 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  (TopSet `  ndx )  =  9
 
Theoremtsetid 13168 Utility theorem: index-independent form of df-tset 13101. (Contributed by NM, 20-Oct-2012.)
 |- TopSet  = Slot  (TopSet `  ndx )
 
Theoremtopgrpstr 13169 A constructed topological group is a structure. (Contributed by Mario Carneiro, 29-Aug-2015.)
 |-  W  =  { <. (
 Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. ,  <. (TopSet `  ndx ) ,  J >. }   =>    |-  W Struct  <. 1 ,  9
 >.
 
Theoremtopgrpbas 13170 The base set of a constructed topological group. (Contributed by Mario Carneiro, 29-Aug-2015.)
 |-  W  =  { <. (
 Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. ,  <. (TopSet `  ndx ) ,  J >. }   =>    |-  ( B  e.  X  ->  B  =  ( Base `  W ) )
 
Theoremtopgrpplusg 13171 The additive operation of a constructed topological group. (Contributed by Mario Carneiro, 29-Aug-2015.)
 |-  W  =  { <. (
 Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. ,  <. (TopSet `  ndx ) ,  J >. }   =>    |-  (  .+  e.  X  ->  .+  =  ( +g  `  W ) )
 
Theoremtopgrptset 13172 The topology of a constructed topological group. (Contributed by Mario Carneiro, 29-Aug-2015.)
 |-  W  =  { <. (
 Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. ,  <. (TopSet `  ndx ) ,  J >. }   =>    |-  ( J  e.  X  ->  J  =  (TopSet `  W ) )
 
Theoremresstset 13173 TopSet is unaffected by restriction. (Contributed by Mario Carneiro, 13-Aug-2015.)
 |-  H  =  ( Gs  A )   &    |-  J  =  (TopSet `  G )   =>    |-  ( A  e.  V  ->  J  =  (TopSet `  H ) )
 
Theoremplendx 13174 Index value of the df-ple 13102 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  ( le `  ndx )  =  10
 
Theorempleid 13175 Utility theorem: self-referencing, index-independent form of df-ple 13102. (Contributed by NM, 9-Nov-2012.)
 |- 
 le  = Slot  ( le ` 
 ndx )
 
Theoremotpsstr 13176 Functionality of a topological ordered space. (Contributed by Mario Carneiro, 12-Nov-2015.)
 |-  K  =  { <. (
 Base `  ndx ) ,  B >. ,  <. (TopSet `  ndx ) ,  J >. , 
 <. ( le `  ndx ) ,  .<_  >. }   =>    |-  K Struct  <. 1 ,  10 >.
 
Theoremotpsbas 13177 The base set of a topological ordered space. (Contributed by Mario Carneiro, 12-Nov-2015.)
 |-  K  =  { <. (
 Base `  ndx ) ,  B >. ,  <. (TopSet `  ndx ) ,  J >. , 
 <. ( le `  ndx ) ,  .<_  >. }   =>    |-  ( B  e.  V  ->  B  =  (
 Base `  K ) )
 
Theoremotpstset 13178 The open sets of a topological ordered space. (Contributed by Mario Carneiro, 12-Nov-2015.)
 |-  K  =  { <. (
 Base `  ndx ) ,  B >. ,  <. (TopSet `  ndx ) ,  J >. , 
 <. ( le `  ndx ) ,  .<_  >. }   =>    |-  ( J  e.  V  ->  J  =  (TopSet `  K ) )
 
Theoremotpsle 13179 The order of a topological ordered space. (Contributed by Mario Carneiro, 12-Nov-2015.)
 |-  K  =  { <. (
 Base `  ndx ) ,  B >. ,  <. (TopSet `  ndx ) ,  J >. , 
 <. ( le `  ndx ) ,  .<_  >. }   =>    |-  (  .<_  e.  V  -> 
 .<_  =  ( le `  K ) )
 
Theoremressle 13180  le is unaffected by restriction. (Contributed by Mario Carneiro, 3-Nov-2015.)
 |-  W  =  ( Ks  A )   &    |-  .<_  =  ( le `  K )   =>    |-  ( A  e.  V  -> 
 .<_  =  ( le `  W ) )
 
Theoremocndx 13181 Index value of the df-ocomp 13103 slot. (Contributed by Mario Carneiro, 25-Oct-2015.)
 |-  ( oc `  ndx )  = ; 1 1
 
Theoremocid 13182 Utility theorem: index-independent form of df-ocomp 13103. (Contributed by Mario Carneiro, 25-Oct-2015.)
 |- 
 oc  = Slot  ( oc ` 
 ndx )
 
Theoremdsndx 13183 Index value of the df-ds 13104 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  ( dist `  ndx )  = ; 1
 2
 
Theoremdsid 13184 Utility theorem: index-independent form of df-ds 13104. (Contributed by Mario Carneiro, 23-Dec-2013.)
 |- 
 dist  = Slot  ( dist `  ndx )
 
Theoremodrngstr 13185 Functionality of an ordered metric ring. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  W  =  ( { <. ( Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .x.  >. }  u.  {
 <. (TopSet `  ndx ) ,  J >. ,  <. ( le ` 
 ndx ) ,  .<_  >. ,  <. ( dist `  ndx ) ,  D >. } )   =>    |-  W Struct  <. 1 , ; 1 2 >.
 
Theoremodrngbas 13186 The base set of an ordered metric ring. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  W  =  ( { <. ( Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .x.  >. }  u.  {
 <. (TopSet `  ndx ) ,  J >. ,  <. ( le ` 
 ndx ) ,  .<_  >. ,  <. ( dist `  ndx ) ,  D >. } )   =>    |-  ( B  e.  V  ->  B  =  ( Base `  W ) )
 
Theoremodrngplusg 13187 The addition operation of an ordered metric ring. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  W  =  ( { <. ( Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .x.  >. }  u.  {
 <. (TopSet `  ndx ) ,  J >. ,  <. ( le ` 
 ndx ) ,  .<_  >. ,  <. ( dist `  ndx ) ,  D >. } )   =>    |-  (  .+  e.  V  ->  .+  =  ( +g  `  W ) )
 
Theoremodrngmulr 13188 The multiplication operation of an ordered metric ring. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  W  =  ( { <. ( Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .x.  >. }  u.  {
 <. (TopSet `  ndx ) ,  J >. ,  <. ( le ` 
 ndx ) ,  .<_  >. ,  <. ( dist `  ndx ) ,  D >. } )   =>    |-  (  .x.  e.  V  ->  .x.  =  ( .r
 `  W ) )
 
Theoremodrngtset 13189 The open sets of an ordered metric ring. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  W  =  ( { <. ( Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .x.  >. }  u.  {
 <. (TopSet `  ndx ) ,  J >. ,  <. ( le ` 
 ndx ) ,  .<_  >. ,  <. ( dist `  ndx ) ,  D >. } )   =>    |-  ( J  e.  V  ->  J  =  (TopSet `  W ) )
 
Theoremodrngle 13190 The order of an ordered metric ring. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  W  =  ( { <. ( Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .x.  >. }  u.  {
 <. (TopSet `  ndx ) ,  J >. ,  <. ( le ` 
 ndx ) ,  .<_  >. ,  <. ( dist `  ndx ) ,  D >. } )   =>    |-  (  .<_  e.  V  ->  .<_  =  ( le `  W ) )
 
Theoremodrngds 13191 The metric of an ordered metric ring. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  W  =  ( { <. ( Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .x.  >. }  u.  {
 <. (TopSet `  ndx ) ,  J >. ,  <. ( le ` 
 ndx ) ,  .<_  >. ,  <. ( dist `  ndx ) ,  D >. } )   =>    |-  ( D  e.  V  ->  D  =  ( dist `  W ) )
 
Theoremressds 13192  dist is unaffected by restriction. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  H  =  ( Gs  A )   &    |-  D  =  (
 dist `  G )   =>    |-  ( A  e.  V  ->  D  =  (
 dist `  H ) )
 
Theoremhomndx 13193 Index value of the df-hom 13106 slot. (Contributed by Mario Carneiro, 7-Jan-2017.)
 |-  (  Hom  `  ndx )  = ; 1 4
 
Theoremhomid 13194 Utility theorem: index-independent form of df-hom 13106. (Contributed by Mario Carneiro, 7-Jan-2017.)
 |- 
 Hom  = Slot  (  Hom  `  ndx )
 
Theoremccondx 13195 Index value of the df-cco 13107 slot. (Contributed by Mario Carneiro, 7-Jan-2017.)
 |-  (comp `  ndx )  = ; 1
 5
 
Theoremccoid 13196 Utility theorem: index-independent form of df-cco 13107. (Contributed by Mario Carneiro, 7-Jan-2017.)
 |- comp  = Slot  (comp `  ndx )
 
Theoremresshom 13197  Hom is unaffected by restriction. (Contributed by Mario Carneiro, 5-Jan-2017.)
 |-  D  =  ( Cs  A )   &    |-  H  =  ( 
 Hom  `  C )   =>    |-  ( A  e.  V  ->  H  =  (  Hom  `  D ) )
 
Theoremressco 13198 comp is unaffected by restriction. (Contributed by Mario Carneiro, 5-Jan-2017.)
 |-  D  =  ( Cs  A )   &    |-  .x.  =  (comp `  C )   =>    |-  ( A  e.  V  ->  .x.  =  (comp `  D ) )
 
7.1.3  Definition of the structure product
 
Syntaxcrest 13199 Extend class notation with the function returning a subspace topology.
 classt
 
Syntaxctopn 13200 Extend class notation with the topology extractor function.
 class  TopOpen
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-30955
  Copyright terms: Public domain < Previous  Next >