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Theorem dfhnorm2 21647
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 21494 . 2  |-  normh  =  ( x  e.  dom  dom  .ih  |->  ( sqr `  (
x  .ih  x )
) )
2 ax-hfi 21604 . . . . . 6  |-  .ih  :
( ~H  X.  ~H )
--> CC
32fdmi 5318 . . . . 5  |-  dom  .ih  =  ( ~H  X.  ~H )
43dmeqi 4854 . . . 4  |-  dom  dom  .ih  =  dom  ( ~H 
X.  ~H )
5 dmxpid 4872 . . . 4  |-  dom  ( ~H  X.  ~H )  =  ~H
64, 5eqtr2i 2277 . . 3  |-  ~H  =  dom  dom  .ih
7 eqid 2256 . . 3  |-  ( sqr `  ( x  .ih  x
) )  =  ( sqr `  ( x 
.ih  x ) )
86, 7mpteq12i 4064 . 2  |-  ( x  e.  ~H  |->  ( sqr `  ( x  .ih  x
) ) )  =  ( x  e.  dom  dom 
.ih  |->  ( sqr `  (
x  .ih  x )
) )
91, 8eqtr4i 2279 1  |-  normh  =  ( x  e.  ~H  |->  ( sqr `  ( x 
.ih  x ) ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1619    e. cmpt 4037    X. cxp 4645   dom cdm 4647   ` cfv 4659  (class class class)co 5778   CCcc 8689   sqrcsqr 11669   ~Hchil 21445    .ih csp 21448   normhcno 21449
This theorem is referenced by:  normf  21648  normval  21649  hilnormi  21688
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 1927  ax-ext 2237  ax-sep 4101  ax-nul 4109  ax-pr 4172  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-rab 2525  df-v 2759  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-br 3984  df-opab 4038  df-mpt 4039  df-xp 4661  df-dm 4665  df-fn 4670  df-f 4671  df-hnorm 21494
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