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Theorem hvmulcli 28395
Description: Closure inference for scalar multiplication. (Contributed by NM, 1-Aug-1999.) (New usage is discouraged.)
Hypotheses
Ref Expression
hvmulcl.1 𝐴 ∈ ℂ
hvmulcl.2 𝐵 ∈ ℋ
Assertion
Ref Expression
hvmulcli (𝐴 · 𝐵) ∈ ℋ

Proof of Theorem hvmulcli
StepHypRef Expression
1 hvmulcl.1 . 2 𝐴 ∈ ℂ
2 hvmulcl.2 . 2 𝐵 ∈ ℋ
3 hvmulcl 28394 . 2 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (𝐴 · 𝐵) ∈ ℋ)
41, 2, 3mp2an 684 1 (𝐴 · 𝐵) ∈ ℋ
Colors of variables: wff setvar class
Syntax hints:  wcel 2157  (class class class)co 6879  cc 10223  chba 28300   · csm 28302
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 2378  ax-ext 2778  ax-sep 4976  ax-nul 4984  ax-pr 5098  ax-hfvmul 28386
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 2592  df-eu 2610  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3388  df-sbc 3635  df-csb 3730  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-nul 4117  df-if 4279  df-sn 4370  df-pr 4372  df-op 4376  df-uni 4630  df-iun 4713  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5221  df-xp 5319  df-rel 5320  df-cnv 5321  df-co 5322  df-dm 5323  df-rn 5324  df-iota 6065  df-fun 6104  df-fn 6105  df-f 6106  df-fv 6110  df-ov 6882
This theorem is referenced by:  hvsubsub4i  28440  hvnegdii  28443  hvsubeq0i  28444  hvsubcan2i  28445  hvaddcani  28446  hvsubaddi  28447  normlem0  28490  normlem5  28495  normlem9  28499  bcseqi  28501  norm-iii-i  28520  norm3difi  28528  normpar2i  28537  polid2i  28538  polidi  28539  h1de2i  28936  pjsubii  29061  eigposi  29219  lnop0  29349  lnopunilem1  29393  lnophmlem2  29400  lnfn0i  29425
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