Theorem iscvs 22927

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 22924	. . 3 ⊢ ℂVec = (ℂMod ∩ LVec)
2	1	elin2 3801	. 2 ⊢ (𝑊 ∈ ℂVec ↔ (𝑊 ∈ ℂMod ∧ 𝑊 ∈ LVec))
3		clmlmod 22867	. . . 4 ⊢ (𝑊 ∈ ℂMod → 𝑊 ∈ LMod)
4		eqid 2622	. . . . . 6 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
5	4	islvec 19104	. . . . 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 530	. . 3 ⊢ (𝑊 ∈ ℂMod → (𝑊 ∈ LVec ↔ (Scalar‘𝑊) ∈ DivRing))
8	7	pm5.32i 669	. 2 ⊢ ((𝑊 ∈ ℂMod ∧ 𝑊 ∈ LVec) ↔ (𝑊 ∈ ℂMod ∧ (Scalar‘𝑊) ∈ DivRing))
9	2, 8	bitri 264	1 ⊢ (𝑊 ∈ ℂVec ↔ (𝑊 ∈ ℂMod ∧ (Scalar‘𝑊) ∈ DivRing))

Syntax hints:   ↔ wb 196   ∧ wa 384   ∈ wcel 1990  ‘cfv 5888  Scalarcsca 15944  DivRingcdr 18747  LModclmod 18863  LVecclvec 19102  ℂModcclm 22862  ℂVecccvs 22923

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653  df-lvec 19103  df-clm 22863  df-cvs 22924

This theorem is referenced by:  iscvsp  22928