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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 23722 | . . . 4 ⊢ ℂVec = (ℂMod ∩ LVec) | |
3 | 2 | elin2 4173 | . . 3 ⊢ (𝑊 ∈ ℂVec ↔ (𝑊 ∈ ℂMod ∧ 𝑊 ∈ LVec)) |
4 | 3 | simplbi 500 | . 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 2110 LVecclvec 19868 ℂModcclm 23660 ℂVecccvs 23721 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3496 df-in 3942 df-cvs 23722 |
This theorem is referenced by: cvsunit 23729 cvsdiv 23730 cvsmuleqdivd 23732 cvsdiveqd 23733 isncvsngp 23747 ncvsprp 23750 ncvsm1 23752 ncvsdif 23753 ncvspi 23754 ncvspds 23759 cnncvsmulassdemo 23762 ttgcontlem1 26665 |
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