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Definition df-tng 18123
Description: Define a function that fills in the topology and metric components of a structure given a group and a norm on it. (Contributed by Mario Carneiro, 2-Oct-2015.)
Assertion
Ref Expression
df-tng  |- toNrmGrp  =  ( g  e.  _V , 
f  e.  _V  |->  ( ( g sSet  <. ( dist `  ndx ) ,  ( f  o.  ( -g `  g ) )
>. ) sSet  <. (TopSet `  ndx ) ,  ( MetOpen `  ( f  o.  ( -g `  g ) ) ) >. ) )
Distinct variable group:    f, g

Detailed syntax breakdown of Definition df-tng
StepHypRef Expression
1 ctng 18117 . 2  class toNrmGrp
2 vg . . 3  set  g
3 vf . . 3  set  f
4 cvv 2801 . . 3  class  _V
52cv 1631 . . . . 5  class  g
6 cnx 13161 . . . . . . 7  class  ndx
7 cds 13233 . . . . . . 7  class  dist
86, 7cfv 5271 . . . . . 6  class  ( dist `  ndx )
93cv 1631 . . . . . . 7  class  f
10 csg 14381 . . . . . . . 8  class  -g
115, 10cfv 5271 . . . . . . 7  class  ( -g `  g )
129, 11ccom 4709 . . . . . 6  class  ( f  o.  ( -g `  g
) )
138, 12cop 3656 . . . . 5  class  <. ( dist `  ndx ) ,  ( f  o.  ( -g `  g ) )
>.
14 csts 13162 . . . . 5  class sSet
155, 13, 14co 5874 . . . 4  class  ( g sSet  <. ( dist `  ndx ) ,  ( f  o.  ( -g `  g
) ) >. )
16 cts 13230 . . . . . 6  class TopSet
176, 16cfv 5271 . . . . 5  class  (TopSet `  ndx )
18 cmopn 16388 . . . . . 6  class  MetOpen
1912, 18cfv 5271 . . . . 5  class  ( MetOpen `  ( f  o.  ( -g `  g ) ) )
2017, 19cop 3656 . . . 4  class  <. (TopSet ` 
ndx ) ,  (
MetOpen `  ( f  o.  ( -g `  g
) ) ) >.
2115, 20, 14co 5874 . . 3  class  ( ( g sSet  <. ( dist `  ndx ) ,  ( f  o.  ( -g `  g
) ) >. ) sSet  <.
(TopSet `  ndx ) ,  ( MetOpen `  ( f  o.  ( -g `  g
) ) ) >.
)
222, 3, 4, 4, 21cmpt2 5876 . 2  class  ( g  e.  _V ,  f  e.  _V  |->  ( ( g sSet  <. ( dist `  ndx ) ,  ( f  o.  ( -g `  g
) ) >. ) sSet  <.
(TopSet `  ndx ) ,  ( MetOpen `  ( f  o.  ( -g `  g
) ) ) >.
) )
231, 22wceq 1632 1  wff toNrmGrp  =  ( g  e.  _V , 
f  e.  _V  |->  ( ( g sSet  <. ( dist `  ndx ) ,  ( f  o.  ( -g `  g ) )
>. ) sSet  <. (TopSet `  ndx ) ,  ( MetOpen `  ( f  o.  ( -g `  g ) ) ) >. ) )
Colors of variables: wff set class
This definition is referenced by:  reldmtng  18170  tngval  18171
  Copyright terms: Public domain W3C validator