Theorem cvsclm 25427

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 25425	. . . 4 ⊢ ℂVec = (ℂMod ∩ LVec)
3	2	elin2 4149	. . 3 ⊢ (𝑊 ∈ ℂVec ↔ (𝑊 ∈ ℂMod ∧ 𝑊 ∈ LVec))
4	3	simplbi 502	. 2 ⊢ (𝑊 ∈ ℂVec → 𝑊 ∈ ℂMod)
5	1, 4	syl 18	1 ⊢ (𝜑 → 𝑊 ∈ ℂMod)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145  LVecclvec 21357  ℂModcclm 25363  ℂVecccvs 25424

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3453  df-in 3906  df-cvs 25425

This theorem is used by:  cvsunit  25432  cvsdiv  25433  cvsmuleqdivd  25435  cvsdiveqd  25436  isncvsngp  25450  ncvsprp  25453  ncvsm1  25455  ncvsdif  25456  ncvspi  25457  ncvspds  25462  cnncvsmulassdemo  25465  ttgcontlem1  29444