Theorem cvslvec 24194

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 24193	. . . 4 ⊢ ℂVec = (ℂMod ∩ LVec)
3	2	elin2 4127	. . 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 20279  ℂModcclm 24131  ℂVecccvs 24192

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-v 3424  df-in 3890  df-cvs 24193

This theorem is referenced by:  cvsunit  24200  cvsdivcl  24202  isncvsngp  24218