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| Mirrors > Home > MPE Home > Th. List > tvclvec | Structured version Visualization version GIF version | ||
| Description: A topological vector space is a vector space. (Contributed by Mario Carneiro, 5-Oct-2015.) | 
| Ref | Expression | 
|---|---|
| tvclvec | ⊢ (𝑊 ∈ TopVec → 𝑊 ∈ LVec) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | tvclmod 24207 | . 2 ⊢ (𝑊 ∈ TopVec → 𝑊 ∈ LMod) | |
| 2 | eqid 2736 | . . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
| 3 | 2 | tvctdrg 24202 | . . 3 ⊢ (𝑊 ∈ TopVec → (Scalar‘𝑊) ∈ TopDRing) | 
| 4 | tdrgdrng 24183 | . . 3 ⊢ ((Scalar‘𝑊) ∈ TopDRing → (Scalar‘𝑊) ∈ DivRing) | |
| 5 | 3, 4 | syl 17 | . 2 ⊢ (𝑊 ∈ TopVec → (Scalar‘𝑊) ∈ DivRing) | 
| 6 | 2 | islvec 21104 | . 2 ⊢ (𝑊 ∈ LVec ↔ (𝑊 ∈ LMod ∧ (Scalar‘𝑊) ∈ DivRing)) | 
| 7 | 1, 5, 6 | sylanbrc 583 | 1 ⊢ (𝑊 ∈ TopVec → 𝑊 ∈ LVec) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∈ wcel 2107 ‘cfv 6560 Scalarcsca 17301 DivRingcdr 20730 LModclmod 20859 LVecclvec 21102 TopDRingctdrg 24166 TopVecctvc 24168 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-iota 6513 df-fv 6568 df-ov 7435 df-lvec 21103 df-tdrg 24170 df-tlm 24171 df-tvc 24172 | 
| This theorem is referenced by: (None) | 
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