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Theorem nmval2 18640
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 18638 . . 3  |-  ( A  e.  X  ->  ( N `  A )  =  ( A D  .0.  ) )
65adantl 454 . 2  |-  ( ( W  e.  Grp  /\  A  e.  X )  ->  ( N `  A
)  =  ( A D  .0.  ) )
7 nmfval.e . . . 4  |-  E  =  ( D  |`  ( X  X.  X ) )
87oveqi 6095 . . 3  |-  ( A E  .0.  )  =  ( A ( D  |`  ( X  X.  X
) )  .0.  )
9 id 21 . . . 4  |-  ( A  e.  X  ->  A  e.  X )
102, 3grpidcl 14834 . . . 4  |-  ( W  e.  Grp  ->  .0.  e.  X )
11 ovres 6214 . . . 4  |-  ( ( A  e.  X  /\  .0.  e.  X )  -> 
( A ( D  |`  ( X  X.  X
) )  .0.  )  =  ( A D  .0.  ) )
129, 10, 11syl2anr 466 . . 3  |-  ( ( W  e.  Grp  /\  A  e.  X )  ->  ( A ( D  |`  ( X  X.  X
) )  .0.  )  =  ( A D  .0.  ) )
138, 12syl5req 2482 . 2  |-  ( ( W  e.  Grp  /\  A  e.  X )  ->  ( A D  .0.  )  =  ( A E  .0.  ) )
146, 13eqtrd 2469 1  |-  ( ( W  e.  Grp  /\  A  e.  X )  ->  ( N `  A
)  =  ( A E  .0.  ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726    X. cxp 4877    |` cres 4881   ` cfv 5455  (class class class)co 6082   Basecbs 13470   distcds 13539   0gc0g 13724   Grpcgrp 14686   normcnm 18625
This theorem is referenced by:  nmhmcn  19129
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
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 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-reu 2713  df-rmo 2714  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-fv 5463  df-ov 6085  df-riota 6550  df-0g 13728  df-mnd 14691  df-grp 14813  df-nm 18631
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