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Theorem imsval 21254
Description: Value of the induced metric of a normed complex vector space. (Contributed by NM, 11-Sep-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
imsval.3  |-  M  =  ( -v `  U
)
imsval.6  |-  N  =  ( normCV `  U )
imsval.8  |-  D  =  ( IndMet `  U )
Assertion
Ref Expression
imsval  |-  ( U  e.  NrmCVec  ->  D  =  ( N  o.  M ) )

Proof of Theorem imsval
Dummy variable  u is distinct from all other variables.
StepHypRef Expression
1 fveq2 5525 . . . 4  |-  ( u  =  U  ->  ( normCV `  u )  =  (
normCV
`  U ) )
2 fveq2 5525 . . . 4  |-  ( u  =  U  ->  ( -v `  u )  =  ( -v `  U
) )
31, 2coeq12d 4848 . . 3  |-  ( u  =  U  ->  (
( normCV `  u )  o.  ( -v `  u
) )  =  ( ( normCV `  U )  o.  ( -v `  U
) ) )
4 df-ims 21157 . . 3  |-  IndMet  =  ( u  e.  NrmCVec  |->  ( (
normCV
`  u )  o.  ( -v `  u
) ) )
5 fvex 5539 . . . 4  |-  ( normCV `  U )  e.  _V
6 fvex 5539 . . . 4  |-  ( -v
`  U )  e. 
_V
75, 6coex 5216 . . 3  |-  ( (
normCV
`  U )  o.  ( -v `  U
) )  e.  _V
83, 4, 7fvmpt 5602 . 2  |-  ( U  e.  NrmCVec  ->  ( IndMet `  U
)  =  ( (
normCV
`  U )  o.  ( -v `  U
) ) )
9 imsval.8 . 2  |-  D  =  ( IndMet `  U )
10 imsval.6 . . 3  |-  N  =  ( normCV `  U )
11 imsval.3 . . 3  |-  M  =  ( -v `  U
)
1210, 11coeq12i 4847 . 2  |-  ( N  o.  M )  =  ( ( normCV `  U
)  o.  ( -v
`  U ) )
138, 9, 123eqtr4g 2340 1  |-  ( U  e.  NrmCVec  ->  D  =  ( N  o.  M ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684    o. ccom 4693   ` cfv 5255   NrmCVeccnv 21140   -vcnsb 21145   normCVcnmcv 21146   IndMetcims 21147
This theorem is referenced by:  imsdval  21255  imsdf  21258  cnims  21266  hhims  21751  hhssims  21852
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fv 5263  df-ims 21157
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