MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tngval Unicode version

Theorem tngval 18641
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 2932 . . 3  |-  ( G  e.  V  ->  G  e.  _V )
3 elex 2932 . . 3  |-  ( N  e.  W  ->  N  e.  _V )
4 simpl 444 . . . . . 6  |-  ( ( g  =  G  /\  f  =  N )  ->  g  =  G )
5 simpr 448 . . . . . . . . 9  |-  ( ( g  =  G  /\  f  =  N )  ->  f  =  N )
64fveq2d 5699 . . . . . . . . . 10  |-  ( ( g  =  G  /\  f  =  N )  ->  ( -g `  g
)  =  ( -g `  G ) )
7 tngval.m . . . . . . . . . 10  |-  .-  =  ( -g `  G )
86, 7syl6eqr 2462 . . . . . . . . 9  |-  ( ( g  =  G  /\  f  =  N )  ->  ( -g `  g
)  =  .-  )
95, 8coeq12d 5004 . . . . . . . 8  |-  ( ( g  =  G  /\  f  =  N )  ->  ( f  o.  ( -g `  g ) )  =  ( N  o.  .-  ) )
10 tngval.d . . . . . . . 8  |-  D  =  ( N  o.  .-  )
119, 10syl6eqr 2462 . . . . . . 7  |-  ( ( g  =  G  /\  f  =  N )  ->  ( f  o.  ( -g `  g ) )  =  D )
1211opeq2d 3959 . . . . . 6  |-  ( ( g  =  G  /\  f  =  N )  -> 
<. ( dist `  ndx ) ,  ( f  o.  ( -g `  g
) ) >.  =  <. (
dist `  ndx ) ,  D >. )
134, 12oveq12d 6066 . . . . 5  |-  ( ( g  =  G  /\  f  =  N )  ->  ( g sSet  <. ( dist `  ndx ) ,  ( f  o.  ( -g `  g ) )
>. )  =  ( G sSet  <. ( dist `  ndx ) ,  D >. ) )
1411fveq2d 5699 . . . . . . 7  |-  ( ( g  =  G  /\  f  =  N )  ->  ( MetOpen `  ( f  o.  ( -g `  g
) ) )  =  ( MetOpen `  D )
)
15 tngval.j . . . . . . 7  |-  J  =  ( MetOpen `  D )
1614, 15syl6eqr 2462 . . . . . 6  |-  ( ( g  =  G  /\  f  =  N )  ->  ( MetOpen `  ( f  o.  ( -g `  g
) ) )  =  J )
1716opeq2d 3959 . . . . 5  |-  ( ( g  =  G  /\  f  =  N )  -> 
<. (TopSet `  ndx ) ,  ( MetOpen `  ( f  o.  ( -g `  g
) ) ) >.  =  <. (TopSet `  ndx ) ,  J >. )
1813, 17oveq12d 6066 . . . 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 18593 . . . 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 6073 . . . 4  |-  ( ( G sSet  <. ( dist `  ndx ) ,  D >. ) sSet  <. (TopSet `  ndx ) ,  J >. )  e.  _V
2118, 19, 20ovmpt2a 6171 . . 3  |-  ( ( G  e.  _V  /\  N  e.  _V )  ->  ( G toNrmGrp  N )  =  ( ( G sSet  <. ( dist `  ndx ) ,  D >. ) sSet  <. (TopSet `  ndx ) ,  J >. ) )
222, 3, 21syl2an 464 . 2  |-  ( ( G  e.  V  /\  N  e.  W )  ->  ( G toNrmGrp  N )  =  ( ( G sSet  <. ( dist `  ndx ) ,  D >. ) sSet  <. (TopSet `  ndx ) ,  J >. ) )
231, 22syl5eq 2456 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 359    = wceq 1649    e. wcel 1721   _Vcvv 2924   <.cop 3785    o. ccom 4849   ` cfv 5421  (class class class)co 6048   ndxcnx 13429   sSet csts 13430  TopSetcts 13498   distcds 13501   -gcsg 14651   MetOpencmopn 16654   toNrmGrp ctng 18587
This theorem is referenced by:  tnglem  18642  tngds  18650  tngtset  18651
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pr 4371
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-iota 5385  df-fun 5423  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-tng 18593
  Copyright terms: Public domain W3C validator