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Theorem hicl 21675
Description: Closure of inner product. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
Assertion
Ref Expression
hicl  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A  .ih  B
)  e.  CC )

Proof of Theorem hicl
StepHypRef Expression
1 ax-hfi 21674 . 2  |-  .ih  :
( ~H  X.  ~H )
--> CC
21fovcl 5965 1  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A  .ih  B
)  e.  CC )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1696  (class class class)co 5874   CCcc 8751   ~Hchil 21515    .ih csp 21518
This theorem is referenced by:  hicli  21676  his5  21681  his35  21683  his7  21685  his2sub  21687  his2sub2  21688  hire  21689  hi01  21691  abshicom  21696  hi2eq  21700  hial2eq2  21702  bcs2  21777  pjhthlem1  21986  normcan  22171  pjspansn  22172  adjsym  22429  cnvadj  22488  adj2  22530  brafn  22543  kbop  22549  kbmul  22551  kbpj  22552  eigvalcl  22557  lnopeqi  22604  riesz3i  22658  cnlnadjlem2  22664  cnlnadjlem7  22669  nmopcoadji  22697  kbass2  22713  kbass5  22716  kbass6  22717  hmopidmpji  22748  pjclem4  22795  pj3si  22803
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-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-hfi 21674
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-csb 3095  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-iun 3923  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-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877
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