Theorem iscvs 22927

Description: A subcomplex vector space is a subcomplex module over a division ring. For example, the subcomplex modules over the rational or real or complex numbers are subcomplex vector spaces. (Contributed by AV, 4-Oct-2021.)

Assertion

Ref	Expression
iscvs	|- ( W e. CVec <-> ( W e. CMod /\ (Scalar ` W ) e. DivRing ) )

Proof of Theorem iscvs

Step	Hyp	Ref	Expression
1		df-cvs 22924	. . 3 |- CVec = (CMod i^i LVec )
2	1	elin2 3801	. 2 |- ( W e. CVec <-> ( W e. CMod /\ W e. LVec ) )
3		clmlmod 22867	. . . 4 |- ( W e. CMod -> W e. LMod )
4		eqid 2622	. . . . . 6 |- (Scalar ` W ) = (Scalar ` W )
5	4	islvec 19104	. . . . 5 |- ( W e. LVec <-> ( W e. LMod /\ (Scalar ` W ) e. DivRing ) )
6	5	a1i 11	. . . 4 |- ( W e. CMod -> ( W e. LVec <-> ( W e. LMod /\ (Scalar ` W ) e. DivRing ) ) )
7	3, 6	mpbirand 530	. . 3 |- ( W e. CMod -> ( W e. LVec <-> (Scalar ` W ) e. DivRing ) )
8	7	pm5.32i 669	. 2 |- ( ( W e. CMod /\ W e. LVec ) <-> ( W e. CMod /\ (Scalar ` W ) e. DivRing ) )
9	2, 8	bitri 264	1 |- ( W e. CVec <-> ( W e. CMod /\ (Scalar ` W ) e. DivRing ) )

Syntax hints:   <-> wb 196   /\ wa 384   e. wcel 1990   ` cfv 5888  Scalarcsca 15944   DivRingcdr 18747   LModclmod 18863   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  ax-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653  df-lvec 19103  df-clm 22863  df-cvs 22924

This theorem is referenced by:  iscvsp  22928