Theorem iscvs 25460

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 25457	. . 3 ⊢ ℂVec = (ℂMod ∩ LVec)
2	1	elin2 4102	. 2 ⊢ (𝑊 ∈ ℂVec ↔ (𝑊 ∈ ℂMod ∧ 𝑊 ∈ LVec))
3		clmlmod 25400	. . . 4 ⊢ (𝑊 ∈ ℂMod → 𝑊 ∈ LMod)
4		eqid 2763	. . . . . 6 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
5	4	islvec 21391	. . . . 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 720	. . 3 ⊢ (𝑊 ∈ ℂMod → (𝑊 ∈ LVec ↔ (Scalar‘𝑊) ∈ DivRing))
8	7	pm5.32i 585	. 2 ⊢ ((𝑊 ∈ ℂMod ∧ 𝑊 ∈ LVec) ↔ (𝑊 ∈ ℂMod ∧ (Scalar‘𝑊) ∈ DivRing))
9	2, 8	bitri 278	1 ⊢ (𝑊 ∈ ℂVec ↔ (𝑊 ∈ ℂMod ∧ (Scalar‘𝑊) ∈ DivRing))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   ∈ wcel 2146  ‘cfv 6546  Scalarcsca 17443  DivRingcdr 20992  LModclmod 21147  LVecclvec 21389  ℂModcclm 25395  ℂVecccvs 25456

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735  ax-nul 5262

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-rab 3416  df-v 3455  df-sbc 3742  df-ss 3900  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-sn 4588  df-pr 4590  df-op 4838  df-uni 4870  df-br 5106  df-iota 6500  df-fv 6554  df-ov 7430  df-lvec 21390  df-clm 25396  df-cvs 25457

This theorem is used by:  iscvsp  25461