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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 25171 | . . 3 ⊢ ℂVec = (ℂMod ∩ LVec) | |
2 | 1 | elin2 4213 | . 2 ⊢ (𝑊 ∈ ℂVec ↔ (𝑊 ∈ ℂMod ∧ 𝑊 ∈ LVec)) |
3 | clmlmod 25114 | . . . 4 ⊢ (𝑊 ∈ ℂMod → 𝑊 ∈ LMod) | |
4 | eqid 2735 | . . . . . 6 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
5 | 4 | islvec 21121 | . . . . 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 707 | . . 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 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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