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Definition df-tng 20199
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 20193 . 2  class toNrmGrp
2 vg . . 3  setvar  g
3 vf . . 3  setvar  f
4 cvv 2993 . . 3  class  _V
52cv 1368 . . . . 5  class  g
6 cnx 14192 . . . . . . 7  class  ndx
7 cds 14268 . . . . . . 7  class  dist
86, 7cfv 5439 . . . . . 6  class  ( dist `  ndx )
93cv 1368 . . . . . . 7  class  f
10 csg 15434 . . . . . . . 8  class  -g
115, 10cfv 5439 . . . . . . 7  class  ( -g `  g )
129, 11ccom 4865 . . . . . 6  class  ( f  o.  ( -g `  g
) )
138, 12cop 3904 . . . . 5  class  <. ( dist `  ndx ) ,  ( f  o.  ( -g `  g ) )
>.
14 csts 14193 . . . . 5  class sSet
155, 13, 14co 6112 . . . 4  class  ( g sSet  <. ( dist `  ndx ) ,  ( f  o.  ( -g `  g
) ) >. )
16 cts 14265 . . . . . 6  class TopSet
176, 16cfv 5439 . . . . 5  class  (TopSet `  ndx )
18 cmopn 17828 . . . . . 6  class  MetOpen
1912, 18cfv 5439 . . . . 5  class  ( MetOpen `  ( f  o.  ( -g `  g ) ) )
2017, 19cop 3904 . . . 4  class  <. (TopSet ` 
ndx ) ,  (
MetOpen `  ( f  o.  ( -g `  g
) ) ) >.
2115, 20, 14co 6112 . . 3  class  ( ( g sSet  <. ( dist `  ndx ) ,  ( f  o.  ( -g `  g
) ) >. ) sSet  <.
(TopSet `  ndx ) ,  ( MetOpen `  ( f  o.  ( -g `  g
) ) ) >.
)
222, 3, 4, 4, 21cmpt2 6114 . 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 1369 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 setvar class
This definition is referenced by:  reldmtng  20246  tngval  20247
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