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Theorem hicl 28267
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 28266 . 2 ·ih :( ℋ × ℋ)⟶ℂ
21fovcl 6931 1 ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih 𝐵) ∈ ℂ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wcel 2139  (class class class)co 6814  cc 10146  chil 28106   ·ih csp 28109
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055  ax-hfi 28266
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fv 6057  df-ov 6817
This theorem is referenced by:  hicli  28268  his5  28273  his35  28275  his7  28277  his2sub  28279  his2sub2  28280  hire  28281  hi01  28283  abshicom  28288  hi2eq  28292  hial2eq2  28294  bcs2  28369  pjhthlem1  28580  normcan  28765  pjspansn  28766  adjsym  29022  cnvadj  29081  adj2  29123  brafn  29136  kbop  29142  kbmul  29144  kbpj  29145  eigvalcl  29150  lnopeqi  29197  riesz3i  29251  cnlnadjlem2  29257  cnlnadjlem7  29262  nmopcoadji  29290  kbass2  29306  kbass5  29309  kbass6  29310  hmopidmpji  29341  pjclem4  29388  pj3si  29396
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