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Theorem normval 21649
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
StepHypRef Expression
1 oveq12 5787 . . . 4  |-  ( ( x  =  A  /\  x  =  A )  ->  ( x  .ih  x
)  =  ( A 
.ih  A ) )
21anidms 629 . . 3  |-  ( x  =  A  ->  (
x  .ih  x )  =  ( A  .ih  A ) )
32fveq2d 5448 . 2  |-  ( x  =  A  ->  ( sqr `  ( x  .ih  x ) )  =  ( sqr `  ( A  .ih  A ) ) )
4 dfhnorm2 21647 . 2  |-  normh  =  ( x  e.  ~H  |->  ( sqr `  ( x 
.ih  x ) ) )
5 fvex 5458 . 2  |-  ( sqr `  ( A  .ih  A
) )  e.  _V
63, 4, 5fvmpt 5522 1  |-  ( A  e.  ~H  ->  ( normh `  A )  =  ( sqr `  ( A  .ih  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    = wceq 1619    e. wcel 1621   ` cfv 4659  (class class class)co 5778   sqrcsqr 11669   ~Hchil 21445    .ih csp 21448   normhcno 21449
This theorem is referenced by:  normge0  21651  normgt0  21652  norm0  21653  normsqi  21657  norm-ii-i  21662  norm-iii-i  21664  bcsiALT  21704
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 4101  ax-nul 4109  ax-pr 4172  ax-un 4470  ax-hfi 21604
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 2521  df-rex 2522  df-rab 2525  df-v 2759  df-sbc 2953  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3417  df-if 3526  df-sn 3606  df-pr 3607  df-op 3609  df-uni 3788  df-br 3984  df-opab 4038  df-mpt 4039  df-id 4267  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-fv 4675  df-ov 5781  df-hnorm 21494
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