Theorem hicli 29443

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

Step	Hyp	Ref	Expression
1		hicl.1	. 2 ⊢ 𝐴 ∈ ℋ
2		hicl.2	. 2 ⊢ 𝐵 ∈ ℋ
3		hicl 29442	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih 𝐵) ∈ ℂ)
4	1, 2, 3	mp2an 689	1 ⊢ (𝐴 ·ih 𝐵) ∈ ℂ

Syntax hints:   ∈ wcel 2106  (class class class)co 7275  ℂcc 10869   ℋchba 29281   ·_ih csp 29284

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-hfi 29441

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fv 6441  df-ov 7278

This theorem is referenced by:  hisubcomi  29466  normlem0  29471  normlem2  29473  normlem3  29474  normlem7  29478  normlem8  29479  normlem9  29480  bcseqi  29482  norm-ii-i  29499  normpythi  29504  normpari  29516  polid2i  29519  bcsiALT  29541  h1de2i  29915  h1de2bi  29916  h1de2ctlem  29917  eigrei  30196  eigorthi  30199  lnopunilem1  30372  lnopunilem2  30373