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Theorem normval 21628
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 5766 . . . 4  |-  ( ( x  =  A  /\  x  =  A )  ->  ( x  .ih  x
)  =  ( A 
.ih  A ) )
21anidms 629 . . 3  |-  ( x  =  A  ->  (
x  .ih  x )  =  ( A  .ih  A ) )
32fveq2d 5427 . 2  |-  ( x  =  A  ->  ( sqr `  ( x  .ih  x ) )  =  ( sqr `  ( A  .ih  A ) ) )
4 dfhnorm2 21626 . 2  |-  normh  =  ( x  e.  ~H  |->  ( sqr `  ( x 
.ih  x ) ) )
5 fvex 5437 . 2  |-  ( sqr `  ( A  .ih  A
) )  e.  _V
63, 4, 5fvmpt 5501 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 4638  (class class class)co 5757   sqrcsqr 11648   ~Hchil 21424    .ih csp 21427   normhcno 21428
This theorem is referenced by:  normge0  21630  normgt0  21631  norm0  21632  normsqi  21636  norm-ii-i  21641  norm-iii-i  21643  bcsiALT  21683
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-pr 4152  ax-un 4449  ax-hfi 21583
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-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-fn 4649  df-f 4650  df-fv 4654  df-ov 5760  df-hnorm 21473
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