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Theorem h2hsm 21571
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 2296 . . . 4  |-  ( .s
OLD `  <. <.  +h  ,  .h  >. ,  normh >. )  =  ( .s OLD ` 
<. <.  +h  ,  .h  >. ,  normh >. )
21smfval 21177 . . 3  |-  ( .s
OLD `  <. <.  +h  ,  .h  >. ,  normh >. )  =  ( 2nd `  ( 1st `  <. <.  +h  ,  .h  >. ,  normh >. ) )
3 opex 4253 . . . . 5  |-  <.  +h  ,  .h  >.  e.  _V
4 h2h.1 . . . . . . . 8  |-  U  = 
<. <.  +h  ,  .h  >. ,  normh >.
5 h2h.2 . . . . . . . 8  |-  U  e.  NrmCVec
64, 5eqeltrri 2367 . . . . . . 7  |-  <. <.  +h  ,  .h  >. ,  normh >.  e.  NrmCVec
7 nvex 21183 . . . . . . 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 966 . . . . 5  |-  normh  e.  _V
103, 9op1st 6144 . . . 4  |-  ( 1st `  <. <.  +h  ,  .h  >. ,  normh >. )  =  <.  +h  ,  .h  >.
1110fveq2i 5544 . . 3  |-  ( 2nd `  ( 1st `  <. <.  +h  ,  .h  >. ,  normh >.
) )  =  ( 2nd `  <.  +h  ,  .h  >. )
128simp1i 964 . . . 4  |-  +h  e.  _V
138simp2i 965 . . . 4  |-  .h  e.  _V
1412, 13op2nd 6145 . . 3  |-  ( 2nd `  <.  +h  ,  .h  >. )  =  .h
152, 11, 143eqtrri 2321 . 2  |-  .h  =  ( .s OLD `  <. <.  +h  ,  .h  >. ,  normh >.
)
164fveq2i 5544 . 2  |-  ( .s
OLD `  U )  =  ( .s OLD ` 
<. <.  +h  ,  .h  >. ,  normh >. )
1715, 16eqtr4i 2319 1  |-  .h  =  ( .s OLD `  U
)
Colors of variables: wff set class
Syntax hints:    /\ w3a 934    = wceq 1632    e. wcel 1696   _Vcvv 2801   <.cop 3656   ` cfv 5271   1stc1st 6136   2ndc2nd 6137   NrmCVeccnv 21156   .s OLDcns 21159    +h cva 21516    .h csm 21517   normhcno 21519
This theorem is referenced by:  h2hvs  21573  axhfvmul-zf  21583  axhvmulid-zf  21584  axhvmulass-zf  21585  axhvdistr1-zf  21586  axhvdistr2-zf  21587  axhvmul0-zf  21588  axhis3-zf  21592  hhsm  21764
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-13 1698  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-pow 4204  ax-pr 4230  ax-un 4528
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-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  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-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fo 5277  df-fv 5279  df-oprab 5878  df-1st 6138  df-2nd 6139  df-vc 21118  df-nv 21164  df-sm 21169
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