MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  imsval Unicode version

Theorem imsval 21179
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
StepHypRef Expression
1 fveq2 5423 . . . 4  |-  ( u  =  U  ->  ( normCV `  u )  =  (
normCV
`  U ) )
2 fveq2 5423 . . . 4  |-  ( u  =  U  ->  ( -v `  u )  =  ( -v `  U
) )
31, 2coeq12d 4801 . . 3  |-  ( u  =  U  ->  (
( normCV `  u )  o.  ( -v `  u
) )  =  ( ( normCV `  U )  o.  ( -v `  U
) ) )
4 df-ims 21082 . . 3  |-  IndMet  =  ( u  e.  NrmCVec  |->  ( (
normCV
`  u )  o.  ( -v `  u
) ) )
5 fvex 5437 . . . 4  |-  ( normCV `  U )  e.  _V
6 fvex 5437 . . . 4  |-  ( -v
`  U )  e. 
_V
75, 6coex 5168 . . 3  |-  ( (
normCV
`  U )  o.  ( -v `  U
) )  e.  _V
83, 4, 7fvmpt 5501 . 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 4800 . 2  |-  ( N  o.  M )  =  ( ( normCV `  U
)  o.  ( -v
`  U ) )
138, 9, 123eqtr4g 2313 1  |-  ( U  e.  NrmCVec  ->  D  =  ( N  o.  M ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    = wceq 1619    e. wcel 1621    o. ccom 4630   ` cfv 4638   NrmCVeccnv 21065   -vcnsb 21070   normCVcnmcv 21071   IndMetcims 21072
This theorem is referenced by:  imsdval  21180  imsdf  21183  cnims  21191  hhims  21676  hhssims  21777
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-rab 2523  df-v 2742  df-sbc 2936  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3769  df-br 3964  df-opab 4018  df-mpt 4019  df-id 4246  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fv 4654  df-ims 21082
  Copyright terms: Public domain W3C validator