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Theorem hicli 28842
 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 28841 . 2 ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih 𝐵) ∈ ℂ)
41, 2, 3mp2an 691 1 (𝐴 ·ih 𝐵) ∈ ℂ
 Colors of variables: wff setvar class Syntax hints:   ∈ wcel 2115  (class class class)co 7130  ℂcc 10512   ℋchba 28680   ·ih csp 28683 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pr 5303  ax-hfi 28840 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-fv 6336  df-ov 7133 This theorem is referenced by:  hisubcomi  28865  normlem0  28870  normlem2  28872  normlem3  28873  normlem7  28877  normlem8  28878  normlem9  28879  bcseqi  28881  norm-ii-i  28898  normpythi  28903  normpari  28915  polid2i  28918  bcsiALT  28940  h1de2i  29314  h1de2bi  29315  h1de2ctlem  29316  eigrei  29595  eigorthi  29598  lnopunilem1  29771  lnopunilem2  29772
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