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Theorem nmfval2 21280
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 21278 . 2  |-  N  =  ( x  e.  X  |->  ( x D  .0.  ) )
6 nmfval.e . . . . 5  |-  E  =  ( D  |`  ( X  X.  X ) )
76oveqi 6283 . . . 4  |-  ( x E  .0.  )  =  ( x ( D  |`  ( X  X.  X
) )  .0.  )
8 id 22 . . . . 5  |-  ( x  e.  X  ->  x  e.  X )
92, 3grpidcl 16280 . . . . 5  |-  ( W  e.  Grp  ->  .0.  e.  X )
10 ovres 6415 . . . . 5  |-  ( ( x  e.  X  /\  .0.  e.  X )  -> 
( x ( D  |`  ( X  X.  X
) )  .0.  )  =  ( x D  .0.  ) )
118, 9, 10syl2anr 476 . . . 4  |-  ( ( W  e.  Grp  /\  x  e.  X )  ->  ( x ( D  |`  ( X  X.  X
) )  .0.  )  =  ( x D  .0.  ) )
127, 11syl5req 2508 . . 3  |-  ( ( W  e.  Grp  /\  x  e.  X )  ->  ( x D  .0.  )  =  ( x E  .0.  ) )
1312mpteq2dva 4525 . 2  |-  ( W  e.  Grp  ->  (
x  e.  X  |->  ( x D  .0.  )
)  =  ( x  e.  X  |->  ( x E  .0.  ) ) )
145, 13syl5eq 2507 1  |-  ( W  e.  Grp  ->  N  =  ( x  e.  X  |->  ( x E  .0.  ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823    |-> cmpt 4497    X. cxp 4986    |` cres 4990   ` cfv 5570  (class class class)co 6270   Basecbs 14719   distcds 14796   0gc0g 14932   Grpcgrp 16255   normcnm 21266
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-fv 5578  df-riota 6232  df-ov 6273  df-0g 14934  df-mgm 16074  df-sgrp 16113  df-mnd 16123  df-grp 16259  df-nm 21272
This theorem is referenced by:  nmf2  21282  nmpropd2  21284
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