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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 23294 | . . 3 ⊢ ℂVec = (ℂMod ∩ LVec) | |
2 | 1 | elin2 4029 | . 2 ⊢ (𝑊 ∈ ℂVec ↔ (𝑊 ∈ ℂMod ∧ 𝑊 ∈ LVec)) |
3 | clmlmod 23237 | . . . 4 ⊢ (𝑊 ∈ ℂMod → 𝑊 ∈ LMod) | |
4 | eqid 2826 | . . . . . 6 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
5 | 4 | islvec 19464 | . . . . 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 700 | . . 3 ⊢ (𝑊 ∈ ℂMod → (𝑊 ∈ LVec ↔ (Scalar‘𝑊) ∈ DivRing)) |
8 | 7 | pm5.32i 572 | . 2 ⊢ ((𝑊 ∈ ℂMod ∧ 𝑊 ∈ LVec) ↔ (𝑊 ∈ ℂMod ∧ (Scalar‘𝑊) ∈ DivRing)) |
9 | 2, 8 | bitri 267 | 1 ⊢ (𝑊 ∈ ℂVec ↔ (𝑊 ∈ ℂMod ∧ (Scalar‘𝑊) ∈ DivRing)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∧ wa 386 ∈ wcel 2166 ‘cfv 6124 Scalarcsca 16309 DivRingcdr 19104 LModclmod 19220 LVecclvec 19462 ℂModcclm 23232 ℂVecccvs 23293 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-nul 5014 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ral 3123 df-rex 3124 df-rab 3127 df-v 3417 df-sbc 3664 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-nul 4146 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4660 df-br 4875 df-iota 6087 df-fv 6132 df-ov 6909 df-lvec 19463 df-clm 23233 df-cvs 23294 |
This theorem is referenced by: iscvsp 23298 |
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