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Mirrors > Home > MPE Home > Th. List > iscvs | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
iscvs | ⊢ (𝑊 ∈ ℂVec ↔ (𝑊 ∈ ℂMod ∧ (Scalar‘𝑊) ∈ DivRing)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-cvs 25064 | . . 3 ⊢ ℂVec = (ℂMod ∩ LVec) | |
2 | 1 | elin2 4197 | . 2 ⊢ (𝑊 ∈ ℂVec ↔ (𝑊 ∈ ℂMod ∧ 𝑊 ∈ LVec)) |
3 | clmlmod 25007 | . . . 4 ⊢ (𝑊 ∈ ℂMod → 𝑊 ∈ LMod) | |
4 | eqid 2728 | . . . . . 6 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
5 | 4 | islvec 20989 | . . . . 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 706 | . . 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)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 395 ∈ wcel 2099 ‘cfv 6548 Scalarcsca 17236 DivRingcdr 20624 LModclmod 20743 LVecclvec 20987 ℂModcclm 25002 ℂVecccvs 25063 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 ax-nul 5306 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-ne 2938 df-rab 3430 df-v 3473 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-iota 6500 df-fv 6556 df-ov 7423 df-lvec 20988 df-clm 25003 df-cvs 25064 |
This theorem is referenced by: iscvsp 25068 |
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