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Theorem normval 21533
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 5719 . . . 4  |-  ( ( x  =  A  /\  x  =  A )  ->  ( x  .ih  x
)  =  ( A 
.ih  A ) )
21anidms 629 . . 3  |-  ( x  =  A  ->  (
x  .ih  x )  =  ( A  .ih  A ) )
32fveq2d 5381 . 2  |-  ( x  =  A  ->  ( sqr `  ( x  .ih  x ) )  =  ( sqr `  ( A  .ih  A ) ) )
4 dfhnorm2 21531 . 2  |-  normh  =  ( x  e.  ~H  |->  ( sqr `  ( x 
.ih  x ) ) )
5 fvex 5391 . 2  |-  ( sqr `  ( A  .ih  A
) )  e.  _V
63, 4, 5fvmpt 5454 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 4592  (class class class)co 5710   sqrcsqr 11595   ~Hchil 21329    .ih csp 21332   normhcno 21333
This theorem is referenced by:  normge0  21535  normgt0  21536  norm0  21537  normsqi  21541  norm-ii-i  21546  norm-iii-i  21548  bcsiALT  21588
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 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108  ax-un 4403  ax-hfi 21488
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-fv 4608  df-ov 5713  df-hnorm 21378
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