Theorem cvsclm 25004

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 25002	. . . 4 ⊢ ℂVec = (ℂMod ∩ LVec)
3	2	elin2 4192	. . 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 2098  LVecclvec 20948  ℂModcclm 24940  ℂVecccvs 25001

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-v 3470  df-in 3950  df-cvs 25002

This theorem is referenced by:  cvsunit  25009  cvsdiv  25010  cvsmuleqdivd  25012  cvsdiveqd  25013  isncvsngp  25028  ncvsprp  25031  ncvsm1  25033  ncvsdif  25034  ncvspi  25035  ncvspds  25040  cnncvsmulassdemo  25043  ttgcontlem1  28646