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Theorem nmval2 20978
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 20976 . . 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 6290 . . 3  |-  ( A E  .0.  )  =  ( A ( D  |`  ( X  X.  X
) )  .0.  )
9 id 22 . . . 4  |-  ( A  e.  X  ->  A  e.  X )
102, 3grpidcl 15947 . . . 4  |-  ( W  e.  Grp  ->  .0.  e.  X )
11 ovres 6423 . . . 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 2495 . 2  |-  ( ( W  e.  Grp  /\  A  e.  X )  ->  ( A D  .0.  )  =  ( A E  .0.  ) )
146, 13eqtrd 2482 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 1381    e. wcel 1802    X. cxp 4983    |` cres 4987   ` cfv 5574  (class class class)co 6277   Basecbs 14504   distcds 14578   0gc0g 14709   Grpcgrp 15922   normcnm 20963
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-br 4434  df-opab 4492  df-mpt 4493  df-id 4781  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-fv 5582  df-riota 6238  df-ov 6280  df-0g 14711  df-mgm 15741  df-sgrp 15780  df-mnd 15790  df-grp 15926  df-nm 20969
This theorem is referenced by:  nmhmcn  21469
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