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Theorem iscvs 22735
 Description: A complex vector space is a complex left module over a division ring. For example, the (complex) left modules over the rational or real or complex numbers are complex vector spaces. (Contributed by AV, 4-Oct-2021.)
Assertion
Ref Expression
iscvs (𝑊 ∈ ℂVec ↔ (𝑊 ∈ ℂMod ∧ (Scalar‘𝑊) ∈ DivRing))

Proof of Theorem iscvs
StepHypRef Expression
1 df-cvs 22732 . . 3 ℂVec = (ℂMod ∩ LVec)
21elin2 3763 . 2 (𝑊 ∈ ℂVec ↔ (𝑊 ∈ ℂMod ∧ 𝑊 ∈ LVec))
3 clmlmod 22675 . . . 4 (𝑊 ∈ ℂMod → 𝑊 ∈ LMod)
4 eqid 2610 . . . . . 6 (Scalar‘𝑊) = (Scalar‘𝑊)
54islvec 18925 . . . . 5 (𝑊 ∈ LVec ↔ (𝑊 ∈ LMod ∧ (Scalar‘𝑊) ∈ DivRing))
65a1i 11 . . . 4 (𝑊 ∈ ℂMod → (𝑊 ∈ LVec ↔ (𝑊 ∈ LMod ∧ (Scalar‘𝑊) ∈ DivRing)))
73, 6mpbirand 529 . . 3 (𝑊 ∈ ℂMod → (𝑊 ∈ LVec ↔ (Scalar‘𝑊) ∈ DivRing))
87pm5.32i 667 . 2 ((𝑊 ∈ ℂMod ∧ 𝑊 ∈ LVec) ↔ (𝑊 ∈ ℂMod ∧ (Scalar‘𝑊) ∈ DivRing))
92, 8bitri 263 1 (𝑊 ∈ ℂVec ↔ (𝑊 ∈ ℂMod ∧ (Scalar‘𝑊) ∈ DivRing))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383   ∈ wcel 1977  ‘cfv 5804  Scalarcsca 15771  DivRingcdr 18570  LModclmod 18686  LVecclvec 18923  ℂModcclm 22670  ℂVecccvs 22731 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ov 6552  df-lvec 18924  df-clm 22671  df-cvs 22732 This theorem is referenced by:  iscvsp  22736
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