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	⊢ · = ( ·𝑠 ‘𝑊)
lmodvscld.k	⊢ 𝐾 = (Base‘𝐹)
lmodvscld.w	⊢ (𝜑 → 𝑊 ∈ LMod)
lmodvscld.r	⊢ (𝜑 → 𝑅 ∈ 𝐾)
lmodvscld.x	⊢ (𝜑 → 𝑋 ∈ 𝑉)

Assertion

Ref	Expression
lmodvscld	⊢ (𝜑 → (𝑅 · 𝑋) ∈ 𝑉)

Proof of Theorem lmodvscld

Step	Hyp	Ref	Expression
1		lmodvscld.w	. 2 ⊢ (𝜑 → 𝑊 ∈ LMod)
2		lmodvscld.r	. 2 ⊢ (𝜑 → 𝑅 ∈ 𝐾)
3		lmodvscld.x	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
4		lmodvscld.v	. . 3 ⊢ 𝑉 = (Base‘𝑊)
5		lmodvscld.f	. . 3 ⊢ 𝐹 = (Scalar‘𝑊)
6		lmodvscld.s	. . 3 ⊢ · = ( ·𝑠 ‘𝑊)
7		lmodvscld.k	. . 3 ⊢ 𝐾 = (Base‘𝐹)
8	4, 5, 6, 7	lmodvscl 20709	. 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → (𝑅 · 𝑋) ∈ 𝑉)
9	1, 2, 3, 8	syl3anc 1368	1 ⊢ (𝜑 → (𝑅 · 𝑋) ∈ 𝑉)

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