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Theorem tngval 21021
Description: Value of the function which augments a given structure  G with a norm  N. (Contributed by Mario Carneiro, 2-Oct-2015.)
Hypotheses
Ref Expression
tngval.t  |-  T  =  ( G toNrmGrp  N )
tngval.m  |-  .-  =  ( -g `  G )
tngval.d  |-  D  =  ( N  o.  .-  )
tngval.j  |-  J  =  ( MetOpen `  D )
Assertion
Ref Expression
tngval  |-  ( ( G  e.  V  /\  N  e.  W )  ->  T  =  ( ( G sSet  <. ( dist `  ndx ) ,  D >. ) sSet  <. (TopSet `  ndx ) ,  J >. ) )

Proof of Theorem tngval
Dummy variables  f 
g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tngval.t . 2  |-  T  =  ( G toNrmGrp  N )
2 elex 3127 . . 3  |-  ( G  e.  V  ->  G  e.  _V )
3 elex 3127 . . 3  |-  ( N  e.  W  ->  N  e.  _V )
4 simpl 457 . . . . . 6  |-  ( ( g  =  G  /\  f  =  N )  ->  g  =  G )
5 simpr 461 . . . . . . . . 9  |-  ( ( g  =  G  /\  f  =  N )  ->  f  =  N )
64fveq2d 5876 . . . . . . . . . 10  |-  ( ( g  =  G  /\  f  =  N )  ->  ( -g `  g
)  =  ( -g `  G ) )
7 tngval.m . . . . . . . . . 10  |-  .-  =  ( -g `  G )
86, 7syl6eqr 2526 . . . . . . . . 9  |-  ( ( g  =  G  /\  f  =  N )  ->  ( -g `  g
)  =  .-  )
95, 8coeq12d 5173 . . . . . . . 8  |-  ( ( g  =  G  /\  f  =  N )  ->  ( f  o.  ( -g `  g ) )  =  ( N  o.  .-  ) )
10 tngval.d . . . . . . . 8  |-  D  =  ( N  o.  .-  )
119, 10syl6eqr 2526 . . . . . . 7  |-  ( ( g  =  G  /\  f  =  N )  ->  ( f  o.  ( -g `  g ) )  =  D )
1211opeq2d 4226 . . . . . 6  |-  ( ( g  =  G  /\  f  =  N )  -> 
<. ( dist `  ndx ) ,  ( f  o.  ( -g `  g
) ) >.  =  <. (
dist `  ndx ) ,  D >. )
134, 12oveq12d 6313 . . . . 5  |-  ( ( g  =  G  /\  f  =  N )  ->  ( g sSet  <. ( dist `  ndx ) ,  ( f  o.  ( -g `  g ) )
>. )  =  ( G sSet  <. ( dist `  ndx ) ,  D >. ) )
1411fveq2d 5876 . . . . . . 7  |-  ( ( g  =  G  /\  f  =  N )  ->  ( MetOpen `  ( f  o.  ( -g `  g
) ) )  =  ( MetOpen `  D )
)
15 tngval.j . . . . . . 7  |-  J  =  ( MetOpen `  D )
1614, 15syl6eqr 2526 . . . . . 6  |-  ( ( g  =  G  /\  f  =  N )  ->  ( MetOpen `  ( f  o.  ( -g `  g
) ) )  =  J )
1716opeq2d 4226 . . . . 5  |-  ( ( g  =  G  /\  f  =  N )  -> 
<. (TopSet `  ndx ) ,  ( MetOpen `  ( f  o.  ( -g `  g
) ) ) >.  =  <. (TopSet `  ndx ) ,  J >. )
1813, 17oveq12d 6313 . . . 4  |-  ( ( g  =  G  /\  f  =  N )  ->  ( ( g sSet  <. (
dist `  ndx ) ,  ( f  o.  ( -g `  g ) )
>. ) sSet  <. (TopSet `  ndx ) ,  ( MetOpen `  ( f  o.  ( -g `  g ) ) ) >. )  =  ( ( G sSet  <. ( dist `  ndx ) ,  D >. ) sSet  <. (TopSet ` 
ndx ) ,  J >. ) )
19 df-tng 20973 . . . 4  |- toNrmGrp  =  ( g  e.  _V , 
f  e.  _V  |->  ( ( g sSet  <. ( dist `  ndx ) ,  ( f  o.  ( -g `  g ) )
>. ) sSet  <. (TopSet `  ndx ) ,  ( MetOpen `  ( f  o.  ( -g `  g ) ) ) >. ) )
20 ovex 6320 . . . 4  |-  ( ( G sSet  <. ( dist `  ndx ) ,  D >. ) sSet  <. (TopSet `  ndx ) ,  J >. )  e.  _V
2118, 19, 20ovmpt2a 6428 . . 3  |-  ( ( G  e.  _V  /\  N  e.  _V )  ->  ( G toNrmGrp  N )  =  ( ( G sSet  <. ( dist `  ndx ) ,  D >. ) sSet  <. (TopSet `  ndx ) ,  J >. ) )
222, 3, 21syl2an 477 . 2  |-  ( ( G  e.  V  /\  N  e.  W )  ->  ( G toNrmGrp  N )  =  ( ( G sSet  <. ( dist `  ndx ) ,  D >. ) sSet  <. (TopSet `  ndx ) ,  J >. ) )
231, 22syl5eq 2520 1  |-  ( ( G  e.  V  /\  N  e.  W )  ->  T  =  ( ( G sSet  <. ( dist `  ndx ) ,  D >. ) sSet  <. (TopSet `  ndx ) ,  J >. ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3118   <.cop 4039    o. ccom 5009   ` cfv 5594  (class class class)co 6295   ndxcnx 14504   sSet csts 14505  TopSetcts 14578   distcds 14581   -gcsg 15927   MetOpencmopn 18278   toNrmGrp ctng 20967
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-iota 5557  df-fun 5596  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-tng 20973
This theorem is referenced by:  tnglem  21022  tngds  21030  tngtset  21031
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