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Theorem lmodvscld 20710
Description: Closure of scalar product for a left module. (Contributed by SN, 15-Mar-2025.)
Hypotheses
Ref Expression
lmodvscld.v 𝑉 = (Base‘𝑊)
lmodvscld.f 𝐹 = (Scalar‘𝑊)
lmodvscld.s · = ( ·𝑠𝑊)
lmodvscld.k 𝐾 = (Base‘𝐹)
lmodvscld.w (𝜑𝑊 ∈ LMod)
lmodvscld.r (𝜑𝑅𝐾)
lmodvscld.x (𝜑𝑋𝑉)
Assertion
Ref Expression
lmodvscld (𝜑 → (𝑅 · 𝑋) ∈ 𝑉)

Proof of Theorem lmodvscld
StepHypRef Expression
1 lmodvscld.w . 2 (𝜑𝑊 ∈ LMod)
2 lmodvscld.r . 2 (𝜑𝑅𝐾)
3 lmodvscld.x . 2 (𝜑𝑋𝑉)
4 lmodvscld.v . . 3 𝑉 = (Base‘𝑊)
5 lmodvscld.f . . 3 𝐹 = (Scalar‘𝑊)
6 lmodvscld.s . . 3 · = ( ·𝑠𝑊)
7 lmodvscld.k . . 3 𝐾 = (Base‘𝐹)
84, 5, 6, 7lmodvscl 20709 . 2 ((𝑊 ∈ LMod ∧ 𝑅𝐾𝑋𝑉) → (𝑅 · 𝑋) ∈ 𝑉)
91, 2, 3, 8syl3anc 1368 1 (𝜑 → (𝑅 · 𝑋) ∈ 𝑉)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  cfv 6533  (class class class)co 7401  Basecbs 17140  Scalarcsca 17196   ·𝑠 cvsca 17197  LModclmod 20691
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-nul 5296
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-ral 3054  df-rab 3425  df-v 3468  df-sbc 3770  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-iota 6485  df-fv 6541  df-ov 7404  df-lmod 20693
This theorem is referenced by:  q1pvsca  33106  r1pvsca  33107  r1plmhm  33112  ply1degltdimlem  33152  algextdeglem8  33226  frlmsnic  41565  selvvvval  41612  prjspvs  41807  prjspeclsp  41809
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