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Mirrors > Home > MPE Home > Th. List > cvslvec | Structured version Visualization version GIF version |
Description: A complex vector space is a (left) vector space. (Contributed by Thierry Arnoux, 22-May-2019.) |
Ref | Expression |
---|---|
cvslvec.1 | ⊢ (𝜑 → 𝑊 ∈ ℂVec) |
Ref | Expression |
---|---|
cvslvec | ⊢ (𝜑 → 𝑊 ∈ LVec) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cvslvec.1 | . 2 ⊢ (𝜑 → 𝑊 ∈ ℂVec) | |
2 | df-cvs 22732 | . . . 4 ⊢ ℂVec = (ℂMod ∩ LVec) | |
3 | 2 | elin2 3763 | . . 3 ⊢ (𝑊 ∈ ℂVec ↔ (𝑊 ∈ ℂMod ∧ 𝑊 ∈ LVec)) |
4 | 3 | simprbi 479 | . 2 ⊢ (𝑊 ∈ ℂVec → 𝑊 ∈ LVec) |
5 | 1, 4 | syl 17 | 1 ⊢ (𝜑 → 𝑊 ∈ LVec) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1977 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 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-in 3547 df-cvs 22732 |
This theorem is referenced by: cvsunit 22739 cvsdivcl 22741 isncvsngp 22757 |
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