Theorem cvsclm 25033

Description: A subcomplex vector space is a subcomplex module. (Contributed by Thierry Arnoux, 22-May-2019.)

Hypothesis

Ref	Expression
cvslvec.1	⊢ (𝜑 → 𝑊 ∈ ℂVec)

Assertion

Ref	Expression
cvsclm	⊢ (𝜑 → 𝑊 ∈ ℂMod)

Proof of Theorem cvsclm

Step	Hyp	Ref	Expression
1		cvslvec.1	. 2 ⊢ (𝜑 → 𝑊 ∈ ℂVec)
2		df-cvs 25031	. . . 4 ⊢ ℂVec = (ℂMod ∩ LVec)
3	2	elin2 4169	. . 3 ⊢ (𝑊 ∈ ℂVec ↔ (𝑊 ∈ ℂMod ∧ 𝑊 ∈ LVec))
4	3	simplbi 497	. 2 ⊢ (𝑊 ∈ ℂVec → 𝑊 ∈ ℂMod)
5	1, 4	syl 17	1 ⊢ (𝜑 → 𝑊 ∈ ℂMod)

Syntax hints:   → wi 4   ∈ wcel 2109  LVecclvec 21016  ℂModcclm 24969  ℂVecccvs 25030

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 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-v 3452  df-in 3924  df-cvs 25031

This theorem is referenced by:  cvsunit  25038  cvsdiv  25039  cvsmuleqdivd  25041  cvsdiveqd  25042  isncvsngp  25056  ncvsprp  25059  ncvsm1  25061  ncvsdif  25062  ncvspi  25063  ncvspds  25068  cnncvsmulassdemo  25071  ttgcontlem1  28819