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Theorem hicli 29008
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 29007 . 2 ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih 𝐵) ∈ ℂ)
41, 2, 3mp2an 692 1 (𝐴 ·ih 𝐵) ∈ ℂ
Colors of variables: wff setvar class
Syntax hints:  wcel 2113  (class class class)co 7164  cc 10606  chba 28846   ·ih csp 28849
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-sep 5164  ax-nul 5171  ax-pr 5293  ax-hfi 29006
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ral 3058  df-rex 3059  df-v 3399  df-sbc 3680  df-csb 3789  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-nul 4210  df-if 4412  df-sn 4514  df-pr 4516  df-op 4520  df-uni 4794  df-iun 4880  df-br 5028  df-opab 5090  df-mpt 5108  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-iota 6291  df-fun 6335  df-fn 6336  df-f 6337  df-fv 6341  df-ov 7167
This theorem is referenced by:  hisubcomi  29031  normlem0  29036  normlem2  29038  normlem3  29039  normlem7  29043  normlem8  29044  normlem9  29045  bcseqi  29047  norm-ii-i  29064  normpythi  29069  normpari  29081  polid2i  29084  bcsiALT  29106  h1de2i  29480  h1de2bi  29481  h1de2ctlem  29482  eigrei  29761  eigorthi  29764  lnopunilem1  29937  lnopunilem2  29938
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