MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  clmvscl Structured version   Visualization version   GIF version

Theorem clmvscl 24239
Description: Closure of scalar product for a subcomplex module. Analogue of lmodvscl 20128. (Contributed by NM, 3-Nov-2006.) (Revised by AV, 28-Sep-2021.)
Hypotheses
Ref Expression
clmvscl.v 𝑉 = (Base‘𝑊)
clmvscl.f 𝐹 = (Scalar‘𝑊)
clmvscl.s · = ( ·𝑠𝑊)
clmvscl.k 𝐾 = (Base‘𝐹)
Assertion
Ref Expression
clmvscl ((𝑊 ∈ ℂMod ∧ 𝑄𝐾𝑋𝑉) → (𝑄 · 𝑋) ∈ 𝑉)

Proof of Theorem clmvscl
StepHypRef Expression
1 clmlmod 24218 . 2 (𝑊 ∈ ℂMod → 𝑊 ∈ LMod)
2 clmvscl.v . . 3 𝑉 = (Base‘𝑊)
3 clmvscl.f . . 3 𝐹 = (Scalar‘𝑊)
4 clmvscl.s . . 3 · = ( ·𝑠𝑊)
5 clmvscl.k . . 3 𝐾 = (Base‘𝐹)
62, 3, 4, 5lmodvscl 20128 . 2 ((𝑊 ∈ LMod ∧ 𝑄𝐾𝑋𝑉) → (𝑄 · 𝑋) ∈ 𝑉)
71, 6syl3an1 1162 1 ((𝑊 ∈ ℂMod ∧ 𝑄𝐾𝑋𝑉) → (𝑄 · 𝑋) ∈ 𝑉)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1086   = wceq 1539  wcel 2106  cfv 6427  (class class class)co 7268  Basecbs 16900  Scalarcsca 16953   ·𝑠 cvsca 16954  LModclmod 20111  ℂModcclm 24213
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-nul 5229
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-ral 3069  df-rex 3070  df-rab 3073  df-v 3432  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4258  df-if 4461  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4841  df-br 5075  df-iota 6385  df-fv 6435  df-ov 7271  df-lmod 20113  df-clm 24214
This theorem is referenced by:  clmpm1dir  24254  clmnegsubdi2  24256  clmsub4  24257  clmvsubval2  24261  clmvz  24262  nmoleub2lem3  24266  nmoleub3  24270  ncvspi  24308
  Copyright terms: Public domain W3C validator