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Theorem h2hsm 22470
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 2435 . . . 4  |-  ( .s
OLD `  <. <.  +h  ,  .h  >. ,  normh >. )  =  ( .s OLD ` 
<. <.  +h  ,  .h  >. ,  normh >. )
21smfval 22076 . . 3  |-  ( .s
OLD `  <. <.  +h  ,  .h  >. ,  normh >. )  =  ( 2nd `  ( 1st `  <. <.  +h  ,  .h  >. ,  normh >. ) )
3 opex 4419 . . . . 5  |-  <.  +h  ,  .h  >.  e.  _V
4 h2h.1 . . . . . . . 8  |-  U  = 
<. <.  +h  ,  .h  >. ,  normh >.
5 h2h.2 . . . . . . . 8  |-  U  e.  NrmCVec
64, 5eqeltrri 2506 . . . . . . 7  |-  <. <.  +h  ,  .h  >. ,  normh >.  e.  NrmCVec
7 nvex 22082 . . . . . . 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 6347 . . . 4  |-  ( 1st `  <. <.  +h  ,  .h  >. ,  normh >. )  =  <.  +h  ,  .h  >.
1110fveq2i 5723 . . 3  |-  ( 2nd `  ( 1st `  <. <.  +h  ,  .h  >. ,  normh >.
) )  =  ( 2nd `  <.  +h  ,  .h  >. )
128simp1i 966 . . . 4  |-  +h  e.  _V
138simp2i 967 . . . 4  |-  .h  e.  _V
1412, 13op2nd 6348 . . 3  |-  ( 2nd `  <.  +h  ,  .h  >. )  =  .h
152, 11, 143eqtrri 2460 . 2  |-  .h  =  ( .s OLD `  <. <.  +h  ,  .h  >. ,  normh >.
)
164fveq2i 5723 . 2  |-  ( .s
OLD `  U )  =  ( .s OLD ` 
<. <.  +h  ,  .h  >. ,  normh >. )
1715, 16eqtr4i 2458 1  |-  .h  =  ( .s OLD `  U
)
Colors of variables: wff set class
Syntax hints:    /\ w3a 936    = wceq 1652    e. wcel 1725   _Vcvv 2948   <.cop 3809   ` cfv 5446   1stc1st 6339   2ndc2nd 6340   NrmCVeccnv 22055   .s OLDcns 22058    +h cva 22415    .h csm 22416   normhcno 22418
This theorem is referenced by:  h2hvs  22472  axhfvmul-zf  22482  axhvmulid-zf  22483  axhvmulass-zf  22484  axhvdistr1-zf  22485  axhvdistr2-zf  22486  axhvmul0-zf  22487  axhis3-zf  22491  hhsm  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-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fo 5452  df-fv 5454  df-oprab 6077  df-1st 6341  df-2nd 6342  df-vc 22017  df-nv 22063  df-sm 22068
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