Theorem cvslvec 25158

Description: A subcomplex vector space is a (left) vector space. (Contributed by Thierry Arnoux, 22-May-2019.)

Hypothesis

Ref	Expression
cvslvec.1	⊢ (𝜑 → 𝑊 ∈ ℂVec)

Assertion

Ref	Expression
cvslvec	⊢ (𝜑 → 𝑊 ∈ LVec)

Proof of Theorem cvslvec

Step	Hyp	Ref	Expression
1		cvslvec.1	. 2 ⊢ (𝜑 → 𝑊 ∈ ℂVec)
2		df-cvs 25157	. . . 4 ⊢ ℂVec = (ℂMod ∩ LVec)
3	2	elin2 4203	. . 3 ⊢ (𝑊 ∈ ℂVec ↔ (𝑊 ∈ ℂMod ∧ 𝑊 ∈ LVec))
4	3	simprbi 496	. 2 ⊢ (𝑊 ∈ ℂVec → 𝑊 ∈ LVec)
5	1, 4	syl 17	1 ⊢ (𝜑 → 𝑊 ∈ LVec)

Syntax hints:   → wi 4   ∈ wcel 2108  LVecclvec 21101  ℂModcclm 25095  ℂVecccvs 25156

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 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-in 3958  df-cvs 25157

This theorem is referenced by:  cvsunit  25164  cvsdivcl  25166  isncvsngp  25183