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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 25071 | . . . 4 ⊢ ℂVec = (ℂMod ∩ LVec) | |
3 | 2 | elin2 4199 | . . 3 ⊢ (𝑊 ∈ ℂVec ↔ (𝑊 ∈ ℂMod ∧ 𝑊 ∈ LVec)) |
4 | 3 | simplbi 496 | . 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 2098 LVecclvec 20994 ℂModcclm 25009 ℂVecccvs 25070 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 395 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2706 df-cleq 2720 df-clel 2806 df-v 3475 df-in 3956 df-cvs 25071 |
This theorem is referenced by: cvsunit 25078 cvsdiv 25079 cvsmuleqdivd 25081 cvsdiveqd 25082 isncvsngp 25097 ncvsprp 25100 ncvsm1 25102 ncvsdif 25103 ncvspi 25104 ncvspds 25109 cnncvsmulassdemo 25112 ttgcontlem1 28715 |
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