Theorem clmvscl 24988

Description: Closure of scalar product for a subcomplex module. Analogue of lmodvscl 20784. (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	⊢ · = ( ·𝑠 ‘𝑊)
clmvscl.k	⊢ 𝐾 = (Base‘𝐹)

Assertion

Ref	Expression
clmvscl	⊢ ((𝑊 ∈ ℂMod ∧ 𝑄 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → (𝑄 · 𝑋) ∈ 𝑉)

Proof of Theorem clmvscl

Step	Hyp	Ref	Expression
1		clmlmod 24967	. 2 ⊢ (𝑊 ∈ ℂMod → 𝑊 ∈ LMod)
2		clmvscl.v	. . 3 ⊢ 𝑉 = (Base‘𝑊)
3		clmvscl.f	. . 3 ⊢ 𝐹 = (Scalar‘𝑊)
4		clmvscl.s	. . 3 ⊢ · = ( ·𝑠 ‘𝑊)
5		clmvscl.k	. . 3 ⊢ 𝐾 = (Base‘𝐹)
6	2, 3, 4, 5	lmodvscl 20784	. 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑄 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → (𝑄 · 𝑋) ∈ 𝑉)
7	1, 6	syl3an1 1163	1 ⊢ ((𝑊 ∈ ℂMod ∧ 𝑄 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → (𝑄 · 𝑋) ∈ 𝑉)

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ‘cfv 6511  (class class class)co 7387  Basecbs 17179  Scalarcsca 17223   ·_𝑠 cvsca 17224  LModclmod 20766  ℂModcclm 24962

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-nul 5261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rab 3406  df-v 3449  df-sbc 3754  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-iota 6464  df-fv 6519  df-ov 7390  df-lmod 20768  df-clm 24963

This theorem is referenced by:  clmpm1dir  25003  clmnegsubdi2  25005  clmsub4  25006  clmvsubval2  25010  clmvz  25011  nmoleub2lem3  25015  nmoleub3  25019  ncvspi  25056