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Theorem clmvscl 25135
Description: Closure of scalar product for a subcomplex module. Analogue of lmodvscl 20893. (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 25114 . 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 20893 . 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 1537  wcel 2106  cfv 6563  (class class class)co 7431  Basecbs 17245  Scalarcsca 17301   ·𝑠 cvsca 17302  LModclmod 20875  ℂModcclm 25109
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-nul 5312
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-ov 7434  df-lmod 20877  df-clm 25110
This theorem is referenced by:  clmpm1dir  25150  clmnegsubdi2  25152  clmsub4  25153  clmvsubval2  25157  clmvz  25158  nmoleub2lem3  25162  nmoleub3  25166  ncvspi  25204
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