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Theorem dfhnorm2 21695
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 21542 . 2  |-  normh  =  ( x  e.  dom  dom  .ih  |->  ( sqr `  (
x  .ih  x )
) )
2 ax-hfi 21652 . . . . . 6  |-  .ih  :
( ~H  X.  ~H )
--> CC
32fdmi 5361 . . . . 5  |-  dom  .ih  =  ( ~H  X.  ~H )
43dmeqi 4881 . . . 4  |-  dom  dom  .ih  =  dom  ( ~H 
X.  ~H )
5 dmxpid 4899 . . . 4  |-  dom  ( ~H  X.  ~H )  =  ~H
64, 5eqtr2i 2307 . . 3  |-  ~H  =  dom  dom  .ih
7 eqid 2286 . . 3  |-  ( sqr `  ( x  .ih  x
) )  =  ( sqr `  ( x 
.ih  x ) )
86, 7mpteq12i 4107 . 2  |-  ( x  e.  ~H  |->  ( sqr `  ( x  .ih  x
) ) )  =  ( x  e.  dom  dom 
.ih  |->  ( sqr `  (
x  .ih  x )
) )
91, 8eqtr4i 2309 1  |-  normh  =  ( x  e.  ~H  |->  ( sqr `  ( x 
.ih  x ) ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1625    e. cmpt 4080    X. cxp 4688   dom cdm 4690   ` cfv 5223  (class class class)co 5821   CCcc 8732   sqrcsqr 11714   ~Hchil 21493    .ih csp 21496   normhcno 21497
This theorem is referenced by:  normf  21696  normval  21697  hilnormi  21736
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1638  ax-8 1646  ax-14 1691  ax-6 1706  ax-7 1711  ax-11 1718  ax-12 1870  ax-ext 2267  ax-sep 4144  ax-nul 4152  ax-pr 4215  ax-hfi 21652
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1531  df-nf 1534  df-sb 1633  df-eu 2150  df-mo 2151  df-clab 2273  df-cleq 2279  df-clel 2282  df-nfc 2411  df-ne 2451  df-ral 2551  df-rab 2555  df-v 2793  df-dif 3158  df-un 3160  df-in 3162  df-ss 3169  df-nul 3459  df-if 3569  df-sn 3649  df-pr 3650  df-op 3652  df-br 4027  df-opab 4081  df-mpt 4082  df-xp 4696  df-dm 4700  df-fn 5226  df-f 5227  df-hnorm 21542
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