Theorem iscvs 25083

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 25080	. . 3 ⊢ ℂVec = (ℂMod ∩ LVec)
2	1	elin2 4155	. 2 ⊢ (𝑊 ∈ ℂVec ↔ (𝑊 ∈ ℂMod ∧ 𝑊 ∈ LVec))
3		clmlmod 25023	. . . 4 ⊢ (𝑊 ∈ ℂMod → 𝑊 ∈ LMod)
4		eqid 2736	. . . . . 6 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
5	4	islvec 21056	. . . . 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 707	. . 3 ⊢ (𝑊 ∈ ℂMod → (𝑊 ∈ LVec ↔ (Scalar‘𝑊) ∈ DivRing))
8	7	pm5.32i 574	. 2 ⊢ ((𝑊 ∈ ℂMod ∧ 𝑊 ∈ LVec) ↔ (𝑊 ∈ ℂMod ∧ (Scalar‘𝑊) ∈ DivRing))
9	2, 8	bitri 275	1 ⊢ (𝑊 ∈ ℂVec ↔ (𝑊 ∈ ℂMod ∧ (Scalar‘𝑊) ∈ DivRing))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∈ wcel 2113  ‘cfv 6492  Scalarcsca 17180  DivRingcdr 20662  LModclmod 20811  LVecclvec 21054  ℂModcclm 25018  ℂVecccvs 25079

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708  ax-nul 5251

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-rab 3400  df-v 3442  df-sbc 3741  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-iota 6448  df-fv 6500  df-ov 7361  df-lvec 21055  df-clm 25019  df-cvs 25080

This theorem is referenced by:  iscvsp  25084