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Theorem hicl 28849
Description: Closure of inner product. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
Assertion
Ref Expression
hicl ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih 𝐵) ∈ ℂ)

Proof of Theorem hicl
StepHypRef Expression
1 ax-hfi 28848 . 2 ·ih :( ℋ × ℋ)⟶ℂ
21fovcl 7271 1 ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih 𝐵) ∈ ℂ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398  wcel 2107  (class class class)co 7148  cc 10527  chba 28688   ·ih csp 28691
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-hfi 28848
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-ov 7151
This theorem is referenced by:  hicli  28850  his5  28855  his35  28857  his7  28859  his2sub  28861  his2sub2  28862  hire  28863  hi01  28865  abshicom  28870  hi2eq  28874  hial2eq2  28876  bcs2  28951  pjhthlem1  29160  normcan  29345  pjspansn  29346  adjsym  29602  cnvadj  29661  adj2  29703  brafn  29716  kbop  29722  kbmul  29724  kbpj  29725  eigvalcl  29730  lnopeqi  29777  riesz3i  29831  cnlnadjlem2  29837  cnlnadjlem7  29842  nmopcoadji  29870  kbass2  29886  kbass5  29889  kbass6  29890  hmopidmpji  29921  pjclem4  29968  pj3si  29976
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