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Theorem normval 21817
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 5954 . . . 4  |-  ( ( x  =  A  /\  x  =  A )  ->  ( x  .ih  x
)  =  ( A 
.ih  A ) )
21anidms 626 . . 3  |-  ( x  =  A  ->  (
x  .ih  x )  =  ( A  .ih  A ) )
32fveq2d 5612 . 2  |-  ( x  =  A  ->  ( sqr `  ( x  .ih  x ) )  =  ( sqr `  ( A  .ih  A ) ) )
4 dfhnorm2 21815 . 2  |-  normh  =  ( x  e.  ~H  |->  ( sqr `  ( x 
.ih  x ) ) )
5 fvex 5622 . 2  |-  ( sqr `  ( A  .ih  A
) )  e.  _V
63, 4, 5fvmpt 5685 1  |-  ( A  e.  ~H  ->  ( normh `  A )  =  ( sqr `  ( A  .ih  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1642    e. wcel 1710   ` cfv 5337  (class class class)co 5945   sqrcsqr 11814   ~Hchil 21613    .ih csp 21616   normhcno 21617
This theorem is referenced by:  normge0  21819  normgt0  21820  norm0  21821  normsqi  21825  norm-ii-i  21830  norm-iii-i  21832  bcsiALT  21872
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-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pr 4295  ax-hfi 21772
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-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-fv 5345  df-ov 5948  df-hnorm 21662
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