Theorem iscvs 25255

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 ∈ ℂVec ↔ (𝑊 ∈ ℂMod ∧ (Scalar‘𝑊) ∈ DivRing))

Proof of Theorem iscvs

Step	Hyp	Ref	Expression
1		df-cvs 25252	. . 3 ⊢ ℂVec = (ℂMod ∩ LVec)
2	1	elin2 4164	. 2 ⊢ (𝑊 ∈ ℂVec ↔ (𝑊 ∈ ℂMod ∧ 𝑊 ∈ LVec))
3		clmlmod 25195	. . . 4 ⊢ (𝑊 ∈ ℂMod → 𝑊 ∈ LMod)
4		eqid 2769	. . . . . 6 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
5	4	islvec 21203	. . . . 5 ⊢ (𝑊 ∈ LVec ↔ (𝑊 ∈ LMod ∧ (Scalar‘𝑊) ∈ DivRing))
6	5	a1i 11	. . . 4 ⊢ (𝑊 ∈ ℂMod → (𝑊 ∈ LVec ↔ (𝑊 ∈ LMod ∧ (Scalar‘𝑊) ∈ DivRing)))
7	3, 6	mpbirand 719	. . 3 ⊢ (𝑊 ∈ ℂMod → (𝑊 ∈ LVec ↔ (Scalar‘𝑊) ∈ DivRing))
8	7	pm5.32i 584	. 2 ⊢ ((𝑊 ∈ ℂMod ∧ 𝑊 ∈ LVec) ↔ (𝑊 ∈ ℂMod ∧ (Scalar‘𝑊) ∈ DivRing))
9	2, 8	bitri 278	1 ⊢ (𝑊 ∈ ℂVec ↔ (𝑊 ∈ ℂMod ∧ (Scalar‘𝑊) ∈ DivRing))

Syntax hints:   ↔ wb 209   ∧ wa 400   ∈ wcel 2149  ‘cfv 6537  Scalarcsca 17313  DivRingcdr 20813  LModclmod 20959  LVecclvec 21201  ℂModcclm 25190  ℂVecccvs 25251

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-nul 5271

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-rab 3424  df-v 3465  df-sbc 3754  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-iota 6493  df-fv 6545  df-ov 7414  df-lvec 21202  df-clm 25191  df-cvs 25252

This theorem is referenced by:  iscvsp  25256