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Theorem nmfval2 20296
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 20294 . 2  |-  N  =  ( x  e.  X  |->  ( x D  .0.  ) )
6 nmfval.e . . . . 5  |-  E  =  ( D  |`  ( X  X.  X ) )
76oveqi 6200 . . . 4  |-  ( x E  .0.  )  =  ( x ( D  |`  ( X  X.  X
) )  .0.  )
8 id 22 . . . . 5  |-  ( x  e.  X  ->  x  e.  X )
92, 3grpidcl 15665 . . . . 5  |-  ( W  e.  Grp  ->  .0.  e.  X )
10 ovres 6327 . . . . 5  |-  ( ( x  e.  X  /\  .0.  e.  X )  -> 
( x ( D  |`  ( X  X.  X
) )  .0.  )  =  ( x D  .0.  ) )
118, 9, 10syl2anr 478 . . . 4  |-  ( ( W  e.  Grp  /\  x  e.  X )  ->  ( x ( D  |`  ( X  X.  X
) )  .0.  )  =  ( x D  .0.  ) )
127, 11syl5req 2504 . . 3  |-  ( ( W  e.  Grp  /\  x  e.  X )  ->  ( x D  .0.  )  =  ( x E  .0.  ) )
1312mpteq2dva 4473 . 2  |-  ( W  e.  Grp  ->  (
x  e.  X  |->  ( x D  .0.  )
)  =  ( x  e.  X  |->  ( x E  .0.  ) ) )
145, 13syl5eq 2503 1  |-  ( W  e.  Grp  ->  N  =  ( x  e.  X  |->  ( x E  .0.  ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758    |-> cmpt 4445    X. cxp 4933    |` cres 4937   ` cfv 5513  (class class class)co 6187   Basecbs 14273   distcds 14346   0gc0g 14477   Grpcgrp 15509   normcnm 20282
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 4508  ax-nul 4516  ax-pow 4565  ax-pr 4626  ax-un 6469
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 2599  df-ne 2644  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3067  df-sbc 3282  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-nul 3733  df-if 3887  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4187  df-br 4388  df-opab 4446  df-mpt 4447  df-id 4731  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-res 4947  df-ima 4948  df-iota 5476  df-fun 5515  df-fn 5516  df-f 5517  df-fv 5521  df-riota 6148  df-ov 6190  df-0g 14479  df-mnd 15514  df-grp 15644  df-nm 20288
This theorem is referenced by:  nmf2  20298  nmpropd2  20300
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