Theorem cvslvec 25263

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 25262	. . . 4 ⊢ ℂVec = (ℂMod ∩ LVec)
3	2	elin2 4155	. . 3 ⊢ (𝑊 ∈ ℂVec ↔ (𝑊 ∈ ℂMod ∧ 𝑊 ∈ LVec))
4	3	simprbi 502	. 2 ⊢ (𝑊 ∈ ℂVec → 𝑊 ∈ LVec)
5	1, 4	syl 18	1 ⊢ (𝜑 → 𝑊 ∈ LVec)

Syntax hints:   → wi 4   ∈ wcel 2141  LVecclvec 21202  ℂModcclm 25200  ℂVecccvs 25261

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1571  df-ex 1808  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3455  df-in 3911  df-cvs 25262

This theorem is referenced by:  cvsunit  25269  cvsdivcl  25271  isncvsngp  25287