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Mirrors > Home > MPE Home > Th. List > cvslvec | Structured version Visualization version GIF version |
Description: A subcomplex 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 22924 | . . . 4 ⊢ ℂVec = (ℂMod ∩ LVec) | |
3 | 2 | elin2 3801 | . . 3 ⊢ (𝑊 ∈ ℂVec ↔ (𝑊 ∈ ℂMod ∧ 𝑊 ∈ LVec)) |
4 | 3 | simprbi 480 | . 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 1990 LVecclvec 19102 ℂModcclm 22862 ℂVecccvs 22923 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-v 3202 df-in 3581 df-cvs 22924 |
This theorem is referenced by: cvsunit 22931 cvsdivcl 22933 isncvsngp 22949 |
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