Theorem cvsclm 25096

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 25094	. . . 4 ⊢ ℂVec = (ℂMod ∩ LVec)
3	2	elin2 4183	. . 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 2107  LVecclvec 21070  ℂModcclm 25032  ℂVecccvs 25093

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-v 3465  df-in 3938  df-cvs 25094

This theorem is referenced by:  cvsunit  25101  cvsdiv  25102  cvsmuleqdivd  25104  cvsdiveqd  25105  isncvsngp  25120  ncvsprp  25123  ncvsm1  25125  ncvsdif  25126  ncvspi  25127  ncvspds  25132  cnncvsmulassdemo  25135  ttgcontlem1  28831