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Theorem imsval 21362
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 5605 . . . 4  |-  ( u  =  U  ->  ( normCV `  u )  =  (
normCV
`  U ) )
2 fveq2 5605 . . . 4  |-  ( u  =  U  ->  ( -v `  u )  =  ( -v `  U
) )
31, 2coeq12d 4927 . . 3  |-  ( u  =  U  ->  (
( normCV `  u )  o.  ( -v `  u
) )  =  ( ( normCV `  U )  o.  ( -v `  U
) ) )
4 df-ims 21265 . . 3  |-  IndMet  =  ( u  e.  NrmCVec  |->  ( (
normCV
`  u )  o.  ( -v `  u
) ) )
5 fvex 5619 . . . 4  |-  ( normCV `  U )  e.  _V
6 fvex 5619 . . . 4  |-  ( -v
`  U )  e. 
_V
75, 6coex 5295 . . 3  |-  ( (
normCV
`  U )  o.  ( -v `  U
) )  e.  _V
83, 4, 7fvmpt 5682 . 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 4926 . 2  |-  ( N  o.  M )  =  ( ( normCV `  U
)  o.  ( -v
`  U ) )
138, 9, 123eqtr4g 2415 1  |-  ( U  e.  NrmCVec  ->  D  =  ( N  o.  M ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1642    e. wcel 1710    o. ccom 4772   ` cfv 5334   NrmCVeccnv 21248   -vcnsb 21253   normCVcnmcv 21254   IndMetcims 21255
This theorem is referenced by:  imsdval  21363  imsdf  21366  cnims  21374  hhims  21859  hhssims  21960
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-br 4103  df-opab 4157  df-mpt 4158  df-id 4388  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-iota 5298  df-fun 5336  df-fv 5342  df-ims 21265
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