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Theorem iscvs 25174
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
StepHypRef Expression
1 df-cvs 25171 . . 3 ℂVec = (ℂMod ∩ LVec)
21elin2 4213 . 2 (𝑊 ∈ ℂVec ↔ (𝑊 ∈ ℂMod ∧ 𝑊 ∈ LVec))
3 clmlmod 25114 . . . 4 (𝑊 ∈ ℂMod → 𝑊 ∈ LMod)
4 eqid 2735 . . . . . 6 (Scalar‘𝑊) = (Scalar‘𝑊)
54islvec 21121 . . . . 5 (𝑊 ∈ LVec ↔ (𝑊 ∈ LMod ∧ (Scalar‘𝑊) ∈ DivRing))
65a1i 11 . . . 4 (𝑊 ∈ ℂMod → (𝑊 ∈ LVec ↔ (𝑊 ∈ LMod ∧ (Scalar‘𝑊) ∈ DivRing)))
73, 6mpbirand 707 . . 3 (𝑊 ∈ ℂMod → (𝑊 ∈ LVec ↔ (Scalar‘𝑊) ∈ DivRing))
87pm5.32i 574 . 2 ((𝑊 ∈ ℂMod ∧ 𝑊 ∈ LVec) ↔ (𝑊 ∈ ℂMod ∧ (Scalar‘𝑊) ∈ DivRing))
92, 8bitri 275 1 (𝑊 ∈ ℂVec ↔ (𝑊 ∈ ℂMod ∧ (Scalar‘𝑊) ∈ DivRing))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  wcel 2106  cfv 6563  Scalarcsca 17301  DivRingcdr 20746  LModclmod 20875  LVecclvec 21119  ℂModcclm 25109  ℂVecccvs 25170
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-nul 5312
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-ov 7434  df-lvec 21120  df-clm 25110  df-cvs 25171
This theorem is referenced by:  iscvsp  25175
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