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Theorem dfhnorm2 22577
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 22424 . 2  |-  normh  =  ( x  e.  dom  dom  .ih  |->  ( sqr `  (
x  .ih  x )
) )
2 ax-hfi 22534 . . . . . 6  |-  .ih  :
( ~H  X.  ~H )
--> CC
32fdmi 5555 . . . . 5  |-  dom  .ih  =  ( ~H  X.  ~H )
43dmeqi 5030 . . . 4  |-  dom  dom  .ih  =  dom  ( ~H 
X.  ~H )
5 dmxpid 5048 . . . 4  |-  dom  ( ~H  X.  ~H )  =  ~H
64, 5eqtr2i 2425 . . 3  |-  ~H  =  dom  dom  .ih
7 eqid 2404 . . 3  |-  ( sqr `  ( x  .ih  x
) )  =  ( sqr `  ( x 
.ih  x ) )
86, 7mpteq12i 4253 . 2  |-  ( x  e.  ~H  |->  ( sqr `  ( x  .ih  x
) ) )  =  ( x  e.  dom  dom 
.ih  |->  ( sqr `  (
x  .ih  x )
) )
91, 8eqtr4i 2427 1  |-  normh  =  ( x  e.  ~H  |->  ( sqr `  ( x 
.ih  x ) ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1649    e. cmpt 4226    X. cxp 4835   dom cdm 4837   ` cfv 5413  (class class class)co 6040   CCcc 8944   sqrcsqr 11993   ~Hchil 22375    .ih csp 22378   normhcno 22379
This theorem is referenced by:  normf  22578  normval  22579  hilnormi  22618
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363  ax-hfi 22534
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-br 4173  df-opab 4227  df-mpt 4228  df-xp 4843  df-dm 4847  df-fn 5416  df-f 5417  df-hnorm 22424
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