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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 25157 | . . 3 ⊢ ℂVec = (ℂMod ∩ LVec) | |
| 2 | 1 | elin2 4203 | . 2 ⊢ (𝑊 ∈ ℂVec ↔ (𝑊 ∈ ℂMod ∧ 𝑊 ∈ LVec)) |
| 3 | clmlmod 25100 | . . . 4 ⊢ (𝑊 ∈ ℂMod → 𝑊 ∈ LMod) | |
| 4 | eqid 2737 | . . . . . 6 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
| 5 | 4 | islvec 21103 | . . . . 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 2108 ‘cfv 6561 Scalarcsca 17300 DivRingcdr 20729 LModclmod 20858 LVecclvec 21101 ℂModcclm 25095 ℂVecccvs 25156 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-nul 5306 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-ov 7434 df-lvec 21102 df-clm 25096 df-cvs 25157 |
| This theorem is referenced by: iscvsp 25161 |
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