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Mirrors > Home > MPE Home > Th. List > cvsclm | Structured version Visualization version GIF version |
Description: A subcomplex vector space is a subcomplex module. (Contributed by Thierry Arnoux, 22-May-2019.) |
Ref | Expression |
---|---|
cvslvec.1 | ⊢ (𝜑 → 𝑊 ∈ ℂVec) |
Ref | Expression |
---|---|
cvsclm | ⊢ (𝜑 → 𝑊 ∈ ℂMod) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cvslvec.1 | . 2 ⊢ (𝜑 → 𝑊 ∈ ℂVec) | |
2 | df-cvs 24503 | . . . 4 ⊢ ℂVec = (ℂMod ∩ LVec) | |
3 | 2 | elin2 4158 | . . 3 ⊢ (𝑊 ∈ ℂVec ↔ (𝑊 ∈ ℂMod ∧ 𝑊 ∈ LVec)) |
4 | 3 | simplbi 499 | . 2 ⊢ (𝑊 ∈ ℂVec → 𝑊 ∈ ℂMod) |
5 | 1, 4 | syl 17 | 1 ⊢ (𝜑 → 𝑊 ∈ ℂMod) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 LVecclvec 20578 ℂModcclm 24441 ℂVecccvs 24502 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-v 3446 df-in 3918 df-cvs 24503 |
This theorem is referenced by: cvsunit 24510 cvsdiv 24511 cvsmuleqdivd 24513 cvsdiveqd 24514 isncvsngp 24529 ncvsprp 24532 ncvsm1 24534 ncvsdif 24535 ncvspi 24536 ncvspds 24541 cnncvsmulassdemo 24544 ttgcontlem1 27875 |
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