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Theorem tngval 18685
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 2966 . . 3  |-  ( G  e.  V  ->  G  e.  _V )
3 elex 2966 . . 3  |-  ( N  e.  W  ->  N  e.  _V )
4 simpl 445 . . . . . 6  |-  ( ( g  =  G  /\  f  =  N )  ->  g  =  G )
5 simpr 449 . . . . . . . . 9  |-  ( ( g  =  G  /\  f  =  N )  ->  f  =  N )
64fveq2d 5735 . . . . . . . . . 10  |-  ( ( g  =  G  /\  f  =  N )  ->  ( -g `  g
)  =  ( -g `  G ) )
7 tngval.m . . . . . . . . . 10  |-  .-  =  ( -g `  G )
86, 7syl6eqr 2488 . . . . . . . . 9  |-  ( ( g  =  G  /\  f  =  N )  ->  ( -g `  g
)  =  .-  )
95, 8coeq12d 5040 . . . . . . . 8  |-  ( ( g  =  G  /\  f  =  N )  ->  ( f  o.  ( -g `  g ) )  =  ( N  o.  .-  ) )
10 tngval.d . . . . . . . 8  |-  D  =  ( N  o.  .-  )
119, 10syl6eqr 2488 . . . . . . 7  |-  ( ( g  =  G  /\  f  =  N )  ->  ( f  o.  ( -g `  g ) )  =  D )
1211opeq2d 3993 . . . . . 6  |-  ( ( g  =  G  /\  f  =  N )  -> 
<. ( dist `  ndx ) ,  ( f  o.  ( -g `  g
) ) >.  =  <. (
dist `  ndx ) ,  D >. )
134, 12oveq12d 6102 . . . . 5  |-  ( ( g  =  G  /\  f  =  N )  ->  ( g sSet  <. ( dist `  ndx ) ,  ( f  o.  ( -g `  g ) )
>. )  =  ( G sSet  <. ( dist `  ndx ) ,  D >. ) )
1411fveq2d 5735 . . . . . . 7  |-  ( ( g  =  G  /\  f  =  N )  ->  ( MetOpen `  ( f  o.  ( -g `  g
) ) )  =  ( MetOpen `  D )
)
15 tngval.j . . . . . . 7  |-  J  =  ( MetOpen `  D )
1614, 15syl6eqr 2488 . . . . . 6  |-  ( ( g  =  G  /\  f  =  N )  ->  ( MetOpen `  ( f  o.  ( -g `  g
) ) )  =  J )
1716opeq2d 3993 . . . . 5  |-  ( ( g  =  G  /\  f  =  N )  -> 
<. (TopSet `  ndx ) ,  ( MetOpen `  ( f  o.  ( -g `  g
) ) ) >.  =  <. (TopSet `  ndx ) ,  J >. )
1813, 17oveq12d 6102 . . . 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 18637 . . . 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 6109 . . . 4  |-  ( ( G sSet  <. ( dist `  ndx ) ,  D >. ) sSet  <. (TopSet `  ndx ) ,  J >. )  e.  _V
2118, 19, 20ovmpt2a 6207 . . 3  |-  ( ( G  e.  _V  /\  N  e.  _V )  ->  ( G toNrmGrp  N )  =  ( ( G sSet  <. ( dist `  ndx ) ,  D >. ) sSet  <. (TopSet `  ndx ) ,  J >. ) )
222, 3, 21syl2an 465 . 2  |-  ( ( G  e.  V  /\  N  e.  W )  ->  ( G toNrmGrp  N )  =  ( ( G sSet  <. ( dist `  ndx ) ,  D >. ) sSet  <. (TopSet `  ndx ) ,  J >. ) )
231, 22syl5eq 2482 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 360    = wceq 1653    e. wcel 1726   _Vcvv 2958   <.cop 3819    o. ccom 4885   ` cfv 5457  (class class class)co 6084   ndxcnx 13471   sSet csts 13472  TopSetcts 13540   distcds 13543   -gcsg 14693   MetOpencmopn 16696   toNrmGrp ctng 18631
This theorem is referenced by:  tnglem  18686  tngds  18694  tngtset  18695
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-iota 5421  df-fun 5459  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-tng 18637
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