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Theorem nmfval2 18643
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
nmfval2  |-  ( W  e.  Grp  ->  N  =  ( x  e.  X  |->  ( x E  .0.  ) ) )
Distinct variable groups:    x, D    x, W    x, X    x,  .0.
Allowed substitution hints:    E( x)    N( x)

Proof of Theorem nmfval2
StepHypRef Expression
1 nmfval.n . . 3  |-  N  =  ( norm `  W
)
2 nmfval.x . . 3  |-  X  =  ( Base `  W
)
3 nmfval.z . . 3  |-  .0.  =  ( 0g `  W )
4 nmfval.d . . 3  |-  D  =  ( dist `  W
)
51, 2, 3, 4nmfval 18641 . 2  |-  N  =  ( x  e.  X  |->  ( x D  .0.  ) )
6 nmfval.e . . . . 5  |-  E  =  ( D  |`  ( X  X.  X ) )
76oveqi 6097 . . . 4  |-  ( x E  .0.  )  =  ( x ( D  |`  ( X  X.  X
) )  .0.  )
8 id 21 . . . . 5  |-  ( x  e.  X  ->  x  e.  X )
92, 3grpidcl 14838 . . . . 5  |-  ( W  e.  Grp  ->  .0.  e.  X )
10 ovres 6216 . . . . 5  |-  ( ( x  e.  X  /\  .0.  e.  X )  -> 
( x ( D  |`  ( X  X.  X
) )  .0.  )  =  ( x D  .0.  ) )
118, 9, 10syl2anr 466 . . . 4  |-  ( ( W  e.  Grp  /\  x  e.  X )  ->  ( x ( D  |`  ( X  X.  X
) )  .0.  )  =  ( x D  .0.  ) )
127, 11syl5req 2483 . . 3  |-  ( ( W  e.  Grp  /\  x  e.  X )  ->  ( x D  .0.  )  =  ( x E  .0.  ) )
1312mpteq2dva 4298 . 2  |-  ( W  e.  Grp  ->  (
x  e.  X  |->  ( x D  .0.  )
)  =  ( x  e.  X  |->  ( x E  .0.  ) ) )
145, 13syl5eq 2482 1  |-  ( W  e.  Grp  ->  N  =  ( x  e.  X  |->  ( x E  .0.  ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726    e. cmpt 4269    X. cxp 4879    |` cres 4883   ` cfv 5457  (class class class)co 6084   Basecbs 13474   distcds 13543   0gc0g 13728   Grpcgrp 14690   normcnm 18629
This theorem is referenced by:  nmf2  18645  nmpropd2  18647
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-13 1728  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-pow 4380  ax-pr 4406  ax-un 4704
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-reu 2714  df-rmo 2715  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-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-fv 5465  df-ov 6087  df-riota 6552  df-0g 13732  df-mnd 14695  df-grp 14817  df-nm 18635
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