Theorem cvslvec 22925

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

Hypothesis

Ref	Expression
cvslvec.1	|- ( ph -> W e. CVec )

Assertion

Ref	Expression
cvslvec	|- ( ph -> W e. LVec )

Proof of Theorem cvslvec

Step	Hyp	Ref	Expression
1		cvslvec.1	. 2 |- ( ph -> W e. CVec )
2		df-cvs 22924	. . . 4 |- CVec = (CMod i^i LVec )
3	2	elin2 3801	. . 3 |- ( W e. CVec <-> ( W e. CMod /\ W e. LVec ) )
4	3	simprbi 480	. 2 |- ( W e. CVec -> W e. LVec )
5	1, 4	syl 17	1 |- ( ph -> W e. LVec )

Syntax hints:   -> wi 4   e. wcel 1990   LVecclvec 19102  CModcclm 22862  CVecccvs 22923

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-in 3581  df-cvs 22924

This theorem is referenced by:  cvsunit  22931  cvsdivcl  22933  isncvsngp  22949