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Theorem nmval2 20840
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 20838 . . 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 6288 . . 3  |-  ( A E  .0.  )  =  ( A ( D  |`  ( X  X.  X
) )  .0.  )
9 id 22 . . . 4  |-  ( A  e.  X  ->  A  e.  X )
102, 3grpidcl 15872 . . . 4  |-  ( W  e.  Grp  ->  .0.  e.  X )
11 ovres 6417 . . . 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 2514 . 2  |-  ( ( W  e.  Grp  /\  A  e.  X )  ->  ( A D  .0.  )  =  ( A E  .0.  ) )
146, 13eqtrd 2501 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 1374    e. wcel 1762    X. cxp 4990    |` cres 4994   ` cfv 5579  (class class class)co 6275   Basecbs 14479   distcds 14553   0gc0g 14684   Grpcgrp 15716   normcnm 20825
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-fv 5587  df-riota 6236  df-ov 6278  df-0g 14686  df-mnd 15721  df-grp 15851  df-nm 20831
This theorem is referenced by:  nmhmcn  21331
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