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Theorem h2hva 22438
Description: The group (addition) operation of Hilbert space. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
Hypotheses
Ref Expression
h2h.1  |-  U  = 
<. <.  +h  ,  .h  >. ,  normh >.
h2h.2  |-  U  e.  NrmCVec
Assertion
Ref Expression
h2hva  |-  +h  =  ( +v `  U )

Proof of Theorem h2hva
StepHypRef Expression
1 eqid 2412 . . . 4  |-  ( +v
`  <. <.  +h  ,  .h  >. ,  normh >. )  =  ( +v `  <. <.  +h  ,  .h  >. ,  normh >. )
21vafval 22043 . . 3  |-  ( +v
`  <. <.  +h  ,  .h  >. ,  normh >. )  =  ( 1st `  ( 1st `  <. <.  +h  ,  .h  >. ,  normh >. ) )
3 opex 4395 . . . . 5  |-  <.  +h  ,  .h  >.  e.  _V
4 h2h.1 . . . . . . . 8  |-  U  = 
<. <.  +h  ,  .h  >. ,  normh >.
5 h2h.2 . . . . . . . 8  |-  U  e.  NrmCVec
64, 5eqeltrri 2483 . . . . . . 7  |-  <. <.  +h  ,  .h  >. ,  normh >.  e.  NrmCVec
7 nvex 22051 . . . . . . 7  |-  ( <. <.  +h  ,  .h  >. , 
normh >.  e.  NrmCVec  ->  (  +h  e.  _V  /\  .h  e.  _V  /\  normh  e.  _V ) )
86, 7ax-mp 8 . . . . . 6  |-  (  +h  e.  _V  /\  .h  e.  _V  /\  normh  e.  _V )
98simp3i 968 . . . . 5  |-  normh  e.  _V
103, 9op1st 6322 . . . 4  |-  ( 1st `  <. <.  +h  ,  .h  >. ,  normh >. )  =  <.  +h  ,  .h  >.
1110fveq2i 5698 . . 3  |-  ( 1st `  ( 1st `  <. <.  +h  ,  .h  >. ,  normh >.
) )  =  ( 1st `  <.  +h  ,  .h  >. )
128simp1i 966 . . . 4  |-  +h  e.  _V
138simp2i 967 . . . 4  |-  .h  e.  _V
1412, 13op1st 6322 . . 3  |-  ( 1st `  <.  +h  ,  .h  >. )  =  +h
152, 11, 143eqtrri 2437 . 2  |-  +h  =  ( +v `  <. <.  +h  ,  .h  >. ,  normh >. )
164fveq2i 5698 . 2  |-  ( +v
`  U )  =  ( +v `  <. <.  +h  ,  .h  >. ,  normh >.
)
1715, 16eqtr4i 2435 1  |-  +h  =  ( +v `  U )
Colors of variables: wff set class
Syntax hints:    /\ w3a 936    = wceq 1649    e. wcel 1721   _Vcvv 2924   <.cop 3785   ` cfv 5421   1stc1st 6314   NrmCVeccnv 22024   +vcpv 22025    +h cva 22384    .h csm 22385   normhcno 22387
This theorem is referenced by:  h2hvs  22441  axhfvadd-zf  22446  axhvcom-zf  22447  axhvass-zf  22448  axhvaddid-zf  22450  axhvdistr1-zf  22454  axhvdistr2-zf  22455  axhis2-zf  22459  hhva  22629
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-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-fo 5427  df-fv 5429  df-oprab 6052  df-1st 6316  df-vc 21986  df-nv 22032  df-va 22035
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