Theorem cvslvec 25045

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 25044	. . . 4 ⊢ ℂVec = (ℂMod ∩ LVec)
3	2	elin2 4151	. . 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 2110  LVecclvec 21029  ℂModcclm 24982  ℂVecccvs 25043

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2722  df-clel 2804  df-v 3436  df-in 3907  df-cvs 25044

This theorem is referenced by:  cvsunit  25051  cvsdivcl  25053  isncvsngp  25069