MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nmval2 Unicode version

Theorem nmval2 18114
Description: The value of the norm function using a restricted metric. (Contributed by Mario Carneiro, 2-Oct-2015.)
Hypotheses
Ref Expression
nmfval.n  |-  N  =  ( norm `  W
)
nmfval.x  |-  X  =  ( Base `  W
)
nmfval.z  |-  .0.  =  ( 0g `  W )
nmfval.d  |-  D  =  ( dist `  W
)
nmfval.e  |-  E  =  ( D  |`  ( X  X.  X ) )
Assertion
Ref Expression
nmval2  |-  ( ( W  e.  Grp  /\  A  e.  X )  ->  ( N `  A
)  =  ( A E  .0.  ) )

Proof of Theorem nmval2
StepHypRef Expression
1 nmfval.n . . . 4  |-  N  =  ( norm `  W
)
2 nmfval.x . . . 4  |-  X  =  ( Base `  W
)
3 nmfval.z . . . 4  |-  .0.  =  ( 0g `  W )
4 nmfval.d . . . 4  |-  D  =  ( dist `  W
)
51, 2, 3, 4nmval 18112 . . 3  |-  ( A  e.  X  ->  ( N `  A )  =  ( A D  .0.  ) )
65adantl 452 . 2  |-  ( ( W  e.  Grp  /\  A  e.  X )  ->  ( N `  A
)  =  ( A D  .0.  ) )
7 nmfval.e . . . 4  |-  E  =  ( D  |`  ( X  X.  X ) )
87oveqi 5871 . . 3  |-  ( A E  .0.  )  =  ( A ( D  |`  ( X  X.  X
) )  .0.  )
9 id 19 . . . 4  |-  ( A  e.  X  ->  A  e.  X )
102, 3grpidcl 14510 . . . 4  |-  ( W  e.  Grp  ->  .0.  e.  X )
11 ovres 5987 . . . 4  |-  ( ( A  e.  X  /\  .0.  e.  X )  -> 
( A ( D  |`  ( X  X.  X
) )  .0.  )  =  ( A D  .0.  ) )
129, 10, 11syl2anr 464 . . 3  |-  ( ( W  e.  Grp  /\  A  e.  X )  ->  ( A ( D  |`  ( X  X.  X
) )  .0.  )  =  ( A D  .0.  ) )
138, 12syl5req 2328 . 2  |-  ( ( W  e.  Grp  /\  A  e.  X )  ->  ( A D  .0.  )  =  ( A E  .0.  ) )
146, 13eqtrd 2315 1  |-  ( ( W  e.  Grp  /\  A  e.  X )  ->  ( N `  A
)  =  ( A E  .0.  ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    X. cxp 4687    |` cres 4691   ` cfv 5255  (class class class)co 5858   Basecbs 13148   distcds 13217   0gc0g 13400   Grpcgrp 14362   normcnm 18099
This theorem is referenced by:  nmhmcn  18601
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-13 1686  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-pow 4188  ax-pr 4214  ax-un 4512
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-reu 2550  df-rmo 2551  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-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-riota 6304  df-0g 13404  df-mnd 14367  df-grp 14489  df-nm 18105
  Copyright terms: Public domain W3C validator