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Theorem hicli 28218
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 28217 . 2 ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih 𝐵) ∈ ℂ)
41, 2, 3mp2an 710 1 (𝐴 ·ih 𝐵) ∈ ℂ
Colors of variables: wff setvar class
Syntax hints:  wcel 2127  (class class class)co 6801  cc 10097  chil 28056   ·ih csp 28059
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-sep 4921  ax-nul 4929  ax-pr 5043  ax-hfi 28216
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ral 3043  df-rex 3044  df-rab 3047  df-v 3330  df-sbc 3565  df-csb 3663  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-sn 4310  df-pr 4312  df-op 4316  df-uni 4577  df-iun 4662  df-br 4793  df-opab 4853  df-mpt 4870  df-id 5162  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-iota 6000  df-fun 6039  df-fn 6040  df-f 6041  df-fv 6045  df-ov 6804
This theorem is referenced by:  hisubcomi  28241  normlem0  28246  normlem2  28248  normlem3  28249  normlem7  28253  normlem8  28254  normlem9  28255  bcseqi  28257  norm-ii-i  28274  normpythi  28279  normpari  28291  polid2i  28294  bcsiALT  28316  h1de2i  28692  h1de2bi  28693  h1de2ctlem  28694  eigrei  28973  eigorthi  28976  lnopunilem1  29149  lnopunilem2  29150
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