Theorem cvslvec 25295

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 25294	. . . 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)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2142  LVecclvec 21234  ℂModcclm 25232  ℂVecccvs 25293

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-tru 1572  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3456  df-in 3911  df-cvs 25294

This theorem is used by:  cvsunit  25301  cvsdivcl  25303  isncvsngp  25319