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Theorem nmval2 20302
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 20300 . . 3  |-  ( A  e.  X  ->  ( N `  A )  =  ( A D  .0.  ) )
65adantl 466 . 2  |-  ( ( W  e.  Grp  /\  A  e.  X )  ->  ( N `  A
)  =  ( A D  .0.  ) )
7 nmfval.e . . . 4  |-  E  =  ( D  |`  ( X  X.  X ) )
87oveqi 6205 . . 3  |-  ( A E  .0.  )  =  ( A ( D  |`  ( X  X.  X
) )  .0.  )
9 id 22 . . . 4  |-  ( A  e.  X  ->  A  e.  X )
102, 3grpidcl 15670 . . . 4  |-  ( W  e.  Grp  ->  .0.  e.  X )
11 ovres 6332 . . . 4  |-  ( ( A  e.  X  /\  .0.  e.  X )  -> 
( A ( D  |`  ( X  X.  X
) )  .0.  )  =  ( A D  .0.  ) )
129, 10, 11syl2anr 478 . . 3  |-  ( ( W  e.  Grp  /\  A  e.  X )  ->  ( A ( D  |`  ( X  X.  X
) )  .0.  )  =  ( A D  .0.  ) )
138, 12syl5req 2505 . 2  |-  ( ( W  e.  Grp  /\  A  e.  X )  ->  ( A D  .0.  )  =  ( A E  .0.  ) )
146, 13eqtrd 2492 1  |-  ( ( W  e.  Grp  /\  A  e.  X )  ->  ( N `  A
)  =  ( A E  .0.  ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758    X. cxp 4938    |` cres 4942   ` cfv 5518  (class class class)co 6192   Basecbs 14278   distcds 14351   0gc0g 14482   Grpcgrp 15514   normcnm 20287
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3072  df-sbc 3287  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-fv 5526  df-riota 6153  df-ov 6195  df-0g 14484  df-mnd 15519  df-grp 15649  df-nm 20293
This theorem is referenced by:  nmhmcn  20793
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