Theorem cvsclm 25452

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 25450	. . . 4 ⊢ ℂVec = (ℂMod ∩ LVec)
3	2	elin2 4100	. . 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 2144  LVecclvec 21382  ℂModcclm 25388  ℂVecccvs 25449

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-in 4091  df-cvs 25450

This theorem is used by:  cvsunit  25457  cvsdiv  25458  cvsmuleqdivd  25460  cvsdiveqd  25461  isncvsngp  25475  ncvsprp  25478  ncvsm1  25480  ncvsdif  25481  ncvspi  25482  ncvspds  25487  cnncvsmulassdemo  25490  ttgcontlem1  29467