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Theorem imsval 22130
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 5687 . . . 4  |-  ( u  =  U  ->  ( normCV `  u )  =  (
normCV
`  U ) )
2 fveq2 5687 . . . 4  |-  ( u  =  U  ->  ( -v `  u )  =  ( -v `  U
) )
31, 2coeq12d 4996 . . 3  |-  ( u  =  U  ->  (
( normCV `  u )  o.  ( -v `  u
) )  =  ( ( normCV `  U )  o.  ( -v `  U
) ) )
4 df-ims 22033 . . 3  |-  IndMet  =  ( u  e.  NrmCVec  |->  ( (
normCV
`  u )  o.  ( -v `  u
) ) )
5 fvex 5701 . . . 4  |-  ( normCV `  U )  e.  _V
6 fvex 5701 . . . 4  |-  ( -v
`  U )  e. 
_V
75, 6coex 5372 . . 3  |-  ( (
normCV
`  U )  o.  ( -v `  U
) )  e.  _V
83, 4, 7fvmpt 5765 . 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 4995 . 2  |-  ( N  o.  M )  =  ( ( normCV `  U
)  o.  ( -v
`  U ) )
138, 9, 123eqtr4g 2461 1  |-  ( U  e.  NrmCVec  ->  D  =  ( N  o.  M ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1721    o. ccom 4841   ` cfv 5413   NrmCVeccnv 22016   -vcnsb 22021   normCVcnmcv 22022   IndMetcims 22023
This theorem is referenced by:  imsdval  22131  imsdf  22134  cnims  22142  hhims  22627  hhssims  22728
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-iota 5377  df-fun 5415  df-fv 5421  df-ims 22033
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