HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  h2hnm Structured version   Unicode version

Theorem h2hnm 22469
Description: The norm function of Hilbert space. (Contributed by NM, 5-Jun-2008.) (New usage is discouraged.)
Hypotheses
Ref Expression
h2h.1  |-  U  = 
<. <.  +h  ,  .h  >. ,  normh >.
h2h.2  |-  U  e.  NrmCVec
Assertion
Ref Expression
h2hnm  |-  normh  =  (
normCV
`  U )

Proof of Theorem h2hnm
StepHypRef Expression
1 h2h.1 . . 3  |-  U  = 
<. <.  +h  ,  .h  >. ,  normh >.
21fveq2i 5723 . 2  |-  ( normCV `  U )  =  (
normCV
`  <. <.  +h  ,  .h  >. ,  normh >. )
3 eqid 2435 . . 3  |-  ( normCV ` 
<. <.  +h  ,  .h  >. ,  normh >. )  =  (
normCV
`  <. <.  +h  ,  .h  >. ,  normh >. )
43nmcvfval 22076 . 2  |-  ( normCV ` 
<. <.  +h  ,  .h  >. ,  normh >. )  =  ( 2nd `  <. <.  +h  ,  .h  >. ,  normh >. )
5 opex 4419 . . 3  |-  <.  +h  ,  .h  >.  e.  _V
6 h2h.2 . . . . . 6  |-  U  e.  NrmCVec
71, 6eqeltrri 2506 . . . . 5  |-  <. <.  +h  ,  .h  >. ,  normh >.  e.  NrmCVec
8 nvex 22080 . . . . 5  |-  ( <. <.  +h  ,  .h  >. , 
normh >.  e.  NrmCVec  ->  (  +h  e.  _V  /\  .h  e.  _V  /\  normh  e.  _V ) )
97, 8ax-mp 8 . . . 4  |-  (  +h  e.  _V  /\  .h  e.  _V  /\  normh  e.  _V )
109simp3i 968 . . 3  |-  normh  e.  _V
115, 10op2nd 6348 . 2  |-  ( 2nd `  <. <.  +h  ,  .h  >. ,  normh >. )  =  normh
122, 4, 113eqtrri 2460 1  |-  normh  =  (
normCV
`  U )
Colors of variables: wff set class
Syntax hints:    /\ w3a 936    = wceq 1652    e. wcel 1725   _Vcvv 2948   <.cop 3809   ` cfv 5446   2ndc2nd 6340   NrmCVeccnv 22053   normCVcnmcv 22059    +h cva 22413    .h csm 22414   normhcno 22416
This theorem is referenced by:  h2hmetdval  22471  hhnm  22663
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-iota 5410  df-fun 5448  df-fv 5454  df-oprab 6077  df-2nd 6342  df-vc 22015  df-nv 22061  df-nmcv 22069
  Copyright terms: Public domain W3C validator