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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 23729 | . . . 4 ⊢ ℂVec = (ℂMod ∩ LVec) | |
3 | 2 | elin2 4124 | . . 3 ⊢ (𝑊 ∈ ℂVec ↔ (𝑊 ∈ ℂMod ∧ 𝑊 ∈ LVec)) |
4 | 3 | simplbi 501 | . 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 2111 LVecclvec 19867 ℂModcclm 23667 ℂVecccvs 23728 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-v 3443 df-in 3888 df-cvs 23729 |
This theorem is referenced by: cvsunit 23736 cvsdiv 23737 cvsmuleqdivd 23739 cvsdiveqd 23740 isncvsngp 23754 ncvsprp 23757 ncvsm1 23759 ncvsdif 23760 ncvspi 23761 ncvspds 23766 cnncvsmulassdemo 23769 ttgcontlem1 26679 |
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