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Theorem imsval 22182
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 5731 . . . 4  |-  ( u  =  U  ->  ( normCV `  u )  =  (
normCV
`  U ) )
2 fveq2 5731 . . . 4  |-  ( u  =  U  ->  ( -v `  u )  =  ( -v `  U
) )
31, 2coeq12d 5040 . . 3  |-  ( u  =  U  ->  (
( normCV `  u )  o.  ( -v `  u
) )  =  ( ( normCV `  U )  o.  ( -v `  U
) ) )
4 df-ims 22085 . . 3  |-  IndMet  =  ( u  e.  NrmCVec  |->  ( (
normCV
`  u )  o.  ( -v `  u
) ) )
5 fvex 5745 . . . 4  |-  ( normCV `  U )  e.  _V
6 fvex 5745 . . . 4  |-  ( -v
`  U )  e. 
_V
75, 6coex 5416 . . 3  |-  ( (
normCV
`  U )  o.  ( -v `  U
) )  e.  _V
83, 4, 7fvmpt 5809 . 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 5039 . 2  |-  ( N  o.  M )  =  ( ( normCV `  U
)  o.  ( -v
`  U ) )
138, 9, 123eqtr4g 2495 1  |-  ( U  e.  NrmCVec  ->  D  =  ( N  o.  M ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726    o. ccom 4885   ` cfv 5457   NrmCVeccnv 22068   -vcnsb 22073   normCVcnmcv 22074   IndMetcims 22075
This theorem is referenced by:  imsdval  22183  imsdf  22186  cnims  22194  hhims  22679  hhssims  22780
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-iota 5421  df-fun 5459  df-fv 5465  df-ims 22085
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