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Mirrors > Home > MPE Home > Th. List > iscvs | Structured version Visualization version GIF version |
Description: A complex vector space is a complex left module over a division ring. For example, the (complex) left modules over the rational or real or complex numbers are complex 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 22732 | . . 3 ⊢ ℂVec = (ℂMod ∩ LVec) | |
2 | 1 | elin2 3763 | . 2 ⊢ (𝑊 ∈ ℂVec ↔ (𝑊 ∈ ℂMod ∧ 𝑊 ∈ LVec)) |
3 | clmlmod 22675 | . . . 4 ⊢ (𝑊 ∈ ℂMod → 𝑊 ∈ LMod) | |
4 | eqid 2610 | . . . . . 6 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
5 | 4 | islvec 18925 | . . . . 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 529 | . . 3 ⊢ (𝑊 ∈ ℂMod → (𝑊 ∈ LVec ↔ (Scalar‘𝑊) ∈ DivRing)) |
8 | 7 | pm5.32i 667 | . 2 ⊢ ((𝑊 ∈ ℂMod ∧ 𝑊 ∈ LVec) ↔ (𝑊 ∈ ℂMod ∧ (Scalar‘𝑊) ∈ DivRing)) |
9 | 2, 8 | bitri 263 | 1 ⊢ (𝑊 ∈ ℂVec ↔ (𝑊 ∈ ℂMod ∧ (Scalar‘𝑊) ∈ DivRing)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 ∈ wcel 1977 ‘cfv 5804 Scalarcsca 15771 DivRingcdr 18570 LModclmod 18686 LVecclvec 18923 ℂModcclm 22670 ℂVecccvs 22731 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-nul 4717 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-iota 5768 df-fv 5812 df-ov 6552 df-lvec 18924 df-clm 22671 df-cvs 22732 |
This theorem is referenced by: iscvsp 22736 |
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