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Theorem nmfval2 20843
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 20841 . 2  |-  N  =  ( x  e.  X  |->  ( x D  .0.  ) )
6 nmfval.e . . . . 5  |-  E  =  ( D  |`  ( X  X.  X ) )
76oveqi 6295 . . . 4  |-  ( x E  .0.  )  =  ( x ( D  |`  ( X  X.  X
) )  .0.  )
8 id 22 . . . . 5  |-  ( x  e.  X  ->  x  e.  X )
92, 3grpidcl 15876 . . . . 5  |-  ( W  e.  Grp  ->  .0.  e.  X )
10 ovres 6424 . . . . 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 2521 . . 3  |-  ( ( W  e.  Grp  /\  x  e.  X )  ->  ( x D  .0.  )  =  ( x E  .0.  ) )
1312mpteq2dva 4533 . 2  |-  ( W  e.  Grp  ->  (
x  e.  X  |->  ( x D  .0.  )
)  =  ( x  e.  X  |->  ( x E  .0.  ) ) )
145, 13syl5eq 2520 1  |-  ( W  e.  Grp  ->  N  =  ( x  e.  X  |->  ( x E  .0.  ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    |-> cmpt 4505    X. cxp 4997    |` cres 5001   ` cfv 5586  (class class class)co 6282   Basecbs 14483   distcds 14557   0gc0g 14688   Grpcgrp 15720   normcnm 20829
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-fv 5594  df-riota 6243  df-ov 6285  df-0g 14690  df-mnd 15725  df-grp 15855  df-nm 20835
This theorem is referenced by:  nmf2  20845  nmpropd2  20847
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