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Theorem hicli 28463
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 28462 . 2 ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih 𝐵) ∈ ℂ)
41, 2, 3mp2an 684 1 (𝐴 ·ih 𝐵) ∈ ℂ
Colors of variables: wff setvar class
Syntax hints:  wcel 2157  (class class class)co 6878  cc 10222  chba 28301   ·ih csp 28304
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097  ax-hfi 28461
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-fv 6109  df-ov 6881
This theorem is referenced by:  hisubcomi  28486  normlem0  28491  normlem2  28493  normlem3  28494  normlem7  28498  normlem8  28499  normlem9  28500  bcseqi  28502  norm-ii-i  28519  normpythi  28524  normpari  28536  polid2i  28539  bcsiALT  28561  h1de2i  28937  h1de2bi  28938  h1de2ctlem  28939  eigrei  29218  eigorthi  29221  lnopunilem1  29394  lnopunilem2  29395
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