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Theorem dfhnorm2 21717
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 21564 . 2  |-  normh  =  ( x  e.  dom  dom  .ih  |->  ( sqr `  (
x  .ih  x )
) )
2 ax-hfi 21674 . . . . . 6  |-  .ih  :
( ~H  X.  ~H )
--> CC
32fdmi 5410 . . . . 5  |-  dom  .ih  =  ( ~H  X.  ~H )
43dmeqi 4896 . . . 4  |-  dom  dom  .ih  =  dom  ( ~H 
X.  ~H )
5 dmxpid 4914 . . . 4  |-  dom  ( ~H  X.  ~H )  =  ~H
64, 5eqtr2i 2317 . . 3  |-  ~H  =  dom  dom  .ih
7 eqid 2296 . . 3  |-  ( sqr `  ( x  .ih  x
) )  =  ( sqr `  ( x 
.ih  x ) )
86, 7mpteq12i 4120 . 2  |-  ( x  e.  ~H  |->  ( sqr `  ( x  .ih  x
) ) )  =  ( x  e.  dom  dom 
.ih  |->  ( sqr `  (
x  .ih  x )
) )
91, 8eqtr4i 2319 1  |-  normh  =  ( x  e.  ~H  |->  ( sqr `  ( x 
.ih  x ) ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1632    e. cmpt 4093    X. cxp 4703   dom cdm 4705   ` cfv 5271  (class class class)co 5874   CCcc 8751   sqrcsqr 11734   ~Hchil 21515    .ih csp 21518   normhcno 21519
This theorem is referenced by:  normf  21718  normval  21719  hilnormi  21758
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-hfi 21674
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-mpt 4095  df-xp 4711  df-dm 4715  df-fn 5274  df-f 5275  df-hnorm 21564
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