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Theorem tngval 18155
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 2796 . . 3  |-  ( G  e.  V  ->  G  e.  _V )
3 elex 2796 . . 3  |-  ( N  e.  W  ->  N  e.  _V )
4 simpl 443 . . . . . 6  |-  ( ( g  =  G  /\  f  =  N )  ->  g  =  G )
5 simpr 447 . . . . . . . . 9  |-  ( ( g  =  G  /\  f  =  N )  ->  f  =  N )
64fveq2d 5529 . . . . . . . . . 10  |-  ( ( g  =  G  /\  f  =  N )  ->  ( -g `  g
)  =  ( -g `  G ) )
7 tngval.m . . . . . . . . . 10  |-  .-  =  ( -g `  G )
86, 7syl6eqr 2333 . . . . . . . . 9  |-  ( ( g  =  G  /\  f  =  N )  ->  ( -g `  g
)  =  .-  )
95, 8coeq12d 4848 . . . . . . . 8  |-  ( ( g  =  G  /\  f  =  N )  ->  ( f  o.  ( -g `  g ) )  =  ( N  o.  .-  ) )
10 tngval.d . . . . . . . 8  |-  D  =  ( N  o.  .-  )
119, 10syl6eqr 2333 . . . . . . 7  |-  ( ( g  =  G  /\  f  =  N )  ->  ( f  o.  ( -g `  g ) )  =  D )
1211opeq2d 3803 . . . . . 6  |-  ( ( g  =  G  /\  f  =  N )  -> 
<. ( dist `  ndx ) ,  ( f  o.  ( -g `  g
) ) >.  =  <. (
dist `  ndx ) ,  D >. )
134, 12oveq12d 5876 . . . . 5  |-  ( ( g  =  G  /\  f  =  N )  ->  ( g sSet  <. ( dist `  ndx ) ,  ( f  o.  ( -g `  g ) )
>. )  =  ( G sSet  <. ( dist `  ndx ) ,  D >. ) )
1411fveq2d 5529 . . . . . . 7  |-  ( ( g  =  G  /\  f  =  N )  ->  ( MetOpen `  ( f  o.  ( -g `  g
) ) )  =  ( MetOpen `  D )
)
15 tngval.j . . . . . . 7  |-  J  =  ( MetOpen `  D )
1614, 15syl6eqr 2333 . . . . . 6  |-  ( ( g  =  G  /\  f  =  N )  ->  ( MetOpen `  ( f  o.  ( -g `  g
) ) )  =  J )
1716opeq2d 3803 . . . . 5  |-  ( ( g  =  G  /\  f  =  N )  -> 
<. (TopSet `  ndx ) ,  ( MetOpen `  ( f  o.  ( -g `  g
) ) ) >.  =  <. (TopSet `  ndx ) ,  J >. )
1813, 17oveq12d 5876 . . . 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 18107 . . . 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 5883 . . . 4  |-  ( ( G sSet  <. ( dist `  ndx ) ,  D >. ) sSet  <. (TopSet `  ndx ) ,  J >. )  e.  _V
2118, 19, 20ovmpt2a 5978 . . 3  |-  ( ( G  e.  _V  /\  N  e.  _V )  ->  ( G toNrmGrp  N )  =  ( ( G sSet  <. ( dist `  ndx ) ,  D >. ) sSet  <. (TopSet `  ndx ) ,  J >. ) )
222, 3, 21syl2an 463 . 2  |-  ( ( G  e.  V  /\  N  e.  W )  ->  ( G toNrmGrp  N )  =  ( ( G sSet  <. ( dist `  ndx ) ,  D >. ) sSet  <. (TopSet `  ndx ) ,  J >. ) )
231, 22syl5eq 2327 1  |-  ( ( G  e.  V  /\  N  e.  W )  ->  T  =  ( ( G sSet  <. ( dist `  ndx ) ,  D >. ) sSet  <. (TopSet `  ndx ) ,  J >. ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788   <.cop 3643    o. ccom 4693   ` cfv 5255  (class class class)co 5858   ndxcnx 13145   sSet csts 13146  TopSetcts 13214   distcds 13217   -gcsg 14365   MetOpencmopn 16372   toNrmGrp ctng 18101
This theorem is referenced by:  tnglem  18156  tngds  18164  tngtset  18165
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-tng 18107
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