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Theorem tngval 20230
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 2986 . . 3  |-  ( G  e.  V  ->  G  e.  _V )
3 elex 2986 . . 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 5700 . . . . . . . . . 10  |-  ( ( g  =  G  /\  f  =  N )  ->  ( -g `  g
)  =  ( -g `  G ) )
7 tngval.m . . . . . . . . . 10  |-  .-  =  ( -g `  G )
86, 7syl6eqr 2493 . . . . . . . . 9  |-  ( ( g  =  G  /\  f  =  N )  ->  ( -g `  g
)  =  .-  )
95, 8coeq12d 5009 . . . . . . . 8  |-  ( ( g  =  G  /\  f  =  N )  ->  ( f  o.  ( -g `  g ) )  =  ( N  o.  .-  ) )
10 tngval.d . . . . . . . 8  |-  D  =  ( N  o.  .-  )
119, 10syl6eqr 2493 . . . . . . 7  |-  ( ( g  =  G  /\  f  =  N )  ->  ( f  o.  ( -g `  g ) )  =  D )
1211opeq2d 4071 . . . . . 6  |-  ( ( g  =  G  /\  f  =  N )  -> 
<. ( dist `  ndx ) ,  ( f  o.  ( -g `  g
) ) >.  =  <. (
dist `  ndx ) ,  D >. )
134, 12oveq12d 6114 . . . . 5  |-  ( ( g  =  G  /\  f  =  N )  ->  ( g sSet  <. ( dist `  ndx ) ,  ( f  o.  ( -g `  g ) )
>. )  =  ( G sSet  <. ( dist `  ndx ) ,  D >. ) )
1411fveq2d 5700 . . . . . . 7  |-  ( ( g  =  G  /\  f  =  N )  ->  ( MetOpen `  ( f  o.  ( -g `  g
) ) )  =  ( MetOpen `  D )
)
15 tngval.j . . . . . . 7  |-  J  =  ( MetOpen `  D )
1614, 15syl6eqr 2493 . . . . . 6  |-  ( ( g  =  G  /\  f  =  N )  ->  ( MetOpen `  ( f  o.  ( -g `  g
) ) )  =  J )
1716opeq2d 4071 . . . . 5  |-  ( ( g  =  G  /\  f  =  N )  -> 
<. (TopSet `  ndx ) ,  ( MetOpen `  ( f  o.  ( -g `  g
) ) ) >.  =  <. (TopSet `  ndx ) ,  J >. )
1813, 17oveq12d 6114 . . . 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 20182 . . . 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 6121 . . . 4  |-  ( ( G sSet  <. ( dist `  ndx ) ,  D >. ) sSet  <. (TopSet `  ndx ) ,  J >. )  e.  _V
2118, 19, 20ovmpt2a 6226 . . 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 2487 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 1369    e. wcel 1756   _Vcvv 2977   <.cop 3888    o. ccom 4849   ` cfv 5423  (class class class)co 6096   ndxcnx 14176   sSet csts 14177  TopSetcts 14249   distcds 14252   -gcsg 15418   MetOpencmopn 17811   toNrmGrp ctng 20176
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pr 4536
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-iota 5386  df-fun 5425  df-fv 5431  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-tng 20182
This theorem is referenced by:  tnglem  20231  tngds  20239  tngtset  20240
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