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Theorem hicli 28066
Description: Closure inference for inner product. (Contributed by NM, 1-Aug-1999.) (New usage is discouraged.)
Hypotheses
Ref Expression
hicl.1 𝐴 ∈ ℋ
hicl.2 𝐵 ∈ ℋ
Assertion
Ref Expression
hicli (𝐴 ·ih 𝐵) ∈ ℂ

Proof of Theorem hicli
StepHypRef Expression
1 hicl.1 . 2 𝐴 ∈ ℋ
2 hicl.2 . 2 𝐵 ∈ ℋ
3 hicl 28065 . 2 ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih 𝐵) ∈ ℂ)
41, 2, 3mp2an 708 1 (𝐴 ·ih 𝐵) ∈ ℂ
Colors of variables: wff setvar class
Syntax hints:  wcel 2030  (class class class)co 6690  cc 9972  chil 27904   ·ih csp 27907
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-hfi 28064
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-ov 6693
This theorem is referenced by:  hisubcomi  28089  normlem0  28094  normlem2  28096  normlem3  28097  normlem7  28101  normlem8  28102  normlem9  28103  bcseqi  28105  norm-ii-i  28122  normpythi  28127  normpari  28139  polid2i  28142  bcsiALT  28164  h1de2i  28540  h1de2bi  28541  h1de2ctlem  28542  eigrei  28821  eigorthi  28824  lnopunilem1  28997  lnopunilem2  28998
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