Theorem cvsclm 25082

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 25080	. . . 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 2109  LVecclvec 21065  ℂModcclm 25018  ℂVecccvs 25079

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 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-v 3466  df-in 3938  df-cvs 25080

This theorem is referenced by:  cvsunit  25087  cvsdiv  25088  cvsmuleqdivd  25090  cvsdiveqd  25091  isncvsngp  25106  ncvsprp  25109  ncvsm1  25111  ncvsdif  25112  ncvspi  25113  ncvspds  25118  cnncvsmulassdemo  25121  ttgcontlem1  28869