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Theorem normval 21703
Description: The value of the norm of a vector in Hilbert space. Definition of norm in [Beran] p. 96. In the literature, the norm of  A is usually written as "||  A ||", but we use function value notation to take advantage of our existing theorems about functions. (Contributed by NM, 29-May-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
Assertion
Ref Expression
normval  |-  ( A  e.  ~H  ->  ( normh `  A )  =  ( sqr `  ( A  .ih  A ) ) )

Proof of Theorem normval
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 oveq12 5867 . . . 4  |-  ( ( x  =  A  /\  x  =  A )  ->  ( x  .ih  x
)  =  ( A 
.ih  A ) )
21anidms 626 . . 3  |-  ( x  =  A  ->  (
x  .ih  x )  =  ( A  .ih  A ) )
32fveq2d 5529 . 2  |-  ( x  =  A  ->  ( sqr `  ( x  .ih  x ) )  =  ( sqr `  ( A  .ih  A ) ) )
4 dfhnorm2 21701 . 2  |-  normh  =  ( x  e.  ~H  |->  ( sqr `  ( x 
.ih  x ) ) )
5 fvex 5539 . 2  |-  ( sqr `  ( A  .ih  A
) )  e.  _V
63, 4, 5fvmpt 5602 1  |-  ( A  e.  ~H  ->  ( normh `  A )  =  ( sqr `  ( A  .ih  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   ` cfv 5255  (class class class)co 5858   sqrcsqr 11718   ~Hchil 21499    .ih csp 21502   normhcno 21503
This theorem is referenced by:  normge0  21705  normgt0  21706  norm0  21707  normsqi  21711  norm-ii-i  21716  norm-iii-i  21718  bcsiALT  21758
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-hfi 21658
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-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-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-hnorm 21548
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