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Theorem h2hsm 22327
Description: The scalar product 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
h2hsm  |-  .h  =  ( .s OLD `  U
)

Proof of Theorem h2hsm
StepHypRef Expression
1 eqid 2388 . . . 4  |-  ( .s
OLD `  <. <.  +h  ,  .h  >. ,  normh >. )  =  ( .s OLD ` 
<. <.  +h  ,  .h  >. ,  normh >. )
21smfval 21933 . . 3  |-  ( .s
OLD `  <. <.  +h  ,  .h  >. ,  normh >. )  =  ( 2nd `  ( 1st `  <. <.  +h  ,  .h  >. ,  normh >. ) )
3 opex 4369 . . . . 5  |-  <.  +h  ,  .h  >.  e.  _V
4 h2h.1 . . . . . . . 8  |-  U  = 
<. <.  +h  ,  .h  >. ,  normh >.
5 h2h.2 . . . . . . . 8  |-  U  e.  NrmCVec
64, 5eqeltrri 2459 . . . . . . 7  |-  <. <.  +h  ,  .h  >. ,  normh >.  e.  NrmCVec
7 nvex 21939 . . . . . . 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 6295 . . . 4  |-  ( 1st `  <. <.  +h  ,  .h  >. ,  normh >. )  =  <.  +h  ,  .h  >.
1110fveq2i 5672 . . 3  |-  ( 2nd `  ( 1st `  <. <.  +h  ,  .h  >. ,  normh >.
) )  =  ( 2nd `  <.  +h  ,  .h  >. )
128simp1i 966 . . . 4  |-  +h  e.  _V
138simp2i 967 . . . 4  |-  .h  e.  _V
1412, 13op2nd 6296 . . 3  |-  ( 2nd `  <.  +h  ,  .h  >. )  =  .h
152, 11, 143eqtrri 2413 . 2  |-  .h  =  ( .s OLD `  <. <.  +h  ,  .h  >. ,  normh >.
)
164fveq2i 5672 . 2  |-  ( .s
OLD `  U )  =  ( .s OLD ` 
<. <.  +h  ,  .h  >. ,  normh >. )
1715, 16eqtr4i 2411 1  |-  .h  =  ( .s OLD `  U
)
Colors of variables: wff set class
Syntax hints:    /\ w3a 936    = wceq 1649    e. wcel 1717   _Vcvv 2900   <.cop 3761   ` cfv 5395   1stc1st 6287   2ndc2nd 6288   NrmCVeccnv 21912   .s OLDcns 21915    +h cva 22272    .h csm 22273   normhcno 22275
This theorem is referenced by:  h2hvs  22329  axhfvmul-zf  22339  axhvmulid-zf  22340  axhvmulass-zf  22341  axhvdistr1-zf  22342  axhvdistr2-zf  22343  axhvmul0-zf  22344  axhis3-zf  22348  hhsm  22520
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-fo 5401  df-fv 5403  df-oprab 6025  df-1st 6289  df-2nd 6290  df-vc 21874  df-nv 21920  df-sm 21925
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