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Theorem dfhnorm2 21531
Description: Alternate definition of the norm of a vector of Hilbert space. Definition of norm in [Beran] p. 96. (Contributed by NM, 6-Jun-2008.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
Assertion
Ref Expression
dfhnorm2  |-  normh  =  ( x  e.  ~H  |->  ( sqr `  ( x 
.ih  x ) ) )

Proof of Theorem dfhnorm2
StepHypRef Expression
1 df-hnorm 21378 . 2  |-  normh  =  ( x  e.  dom  dom  .ih  |->  ( sqr `  (
x  .ih  x )
) )
2 ax-hfi 21488 . . . . . 6  |-  .ih  :
( ~H  X.  ~H )
--> CC
32fdmi 5251 . . . . 5  |-  dom  .ih  =  ( ~H  X.  ~H )
43dmeqi 4787 . . . 4  |-  dom  dom  .ih  =  dom  ( ~H 
X.  ~H )
5 dmxpid 4805 . . . 4  |-  dom  ( ~H  X.  ~H )  =  ~H
64, 5eqtr2i 2274 . . 3  |-  ~H  =  dom  dom  .ih
7 eqid 2253 . . 3  |-  ( sqr `  ( x  .ih  x
) )  =  ( sqr `  ( x 
.ih  x ) )
86, 7mpteq12i 4001 . 2  |-  ( x  e.  ~H  |->  ( sqr `  ( x  .ih  x
) ) )  =  ( x  e.  dom  dom 
.ih  |->  ( sqr `  (
x  .ih  x )
) )
91, 8eqtr4i 2276 1  |-  normh  =  ( x  e.  ~H  |->  ( sqr `  ( x 
.ih  x ) ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1619    e. cmpt 3974    X. cxp 4578   dom cdm 4580   ` cfv 4592  (class class class)co 5710   CCcc 8615   sqrcsqr 11595   ~Hchil 21329    .ih csp 21332   normhcno 21333
This theorem is referenced by:  normf  21532  normval  21533  hilnormi  21572
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-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-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-rab 2516  df-v 2729  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-br 3921  df-opab 3975  df-mpt 3976  df-xp 4594  df-dm 4598  df-fn 4603  df-f 4604  df-hnorm 21378
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