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Theorem hicl 27907
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 27906 . 2 ·ih :( ℋ × ℋ)⟶ℂ
21fovcl 6750 1 ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih 𝐵) ∈ ℂ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wcel 1988  (class class class)co 6635  cc 9919  chil 27746   ·ih csp 27749
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897  ax-hfi 27906
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-fv 5884  df-ov 6638
This theorem is referenced by:  hicli  27908  his5  27913  his35  27915  his7  27917  his2sub  27919  his2sub2  27920  hire  27921  hi01  27923  abshicom  27928  hi2eq  27932  hial2eq2  27934  bcs2  28009  pjhthlem1  28220  normcan  28405  pjspansn  28406  adjsym  28662  cnvadj  28721  adj2  28763  brafn  28776  kbop  28782  kbmul  28784  kbpj  28785  eigvalcl  28790  lnopeqi  28837  riesz3i  28891  cnlnadjlem2  28897  cnlnadjlem7  28902  nmopcoadji  28930  kbass2  28946  kbass5  28949  kbass6  28950  hmopidmpji  28981  pjclem4  29028  pj3si  29036
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