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Theorem hicl 28784
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 28783 . 2 ·ih :( ℋ × ℋ)⟶ℂ
21fovcl 7268 1 ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih 𝐵) ∈ ℂ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wcel 2105  (class class class)co 7145  cc 10523  chba 28623   ·ih csp 28626
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-hfi 28783
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  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 7148
This theorem is referenced by:  hicli  28785  his5  28790  his35  28792  his7  28794  his2sub  28796  his2sub2  28797  hire  28798  hi01  28800  abshicom  28805  hi2eq  28809  hial2eq2  28811  bcs2  28886  pjhthlem1  29095  normcan  29280  pjspansn  29281  adjsym  29537  cnvadj  29596  adj2  29638  brafn  29651  kbop  29657  kbmul  29659  kbpj  29660  eigvalcl  29665  lnopeqi  29712  riesz3i  29766  cnlnadjlem2  29772  cnlnadjlem7  29777  nmopcoadji  29805  kbass2  29821  kbass5  29824  kbass6  29825  hmopidmpji  29856  pjclem4  29903  pj3si  29911
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