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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 24222 | . 2 ⊢ (𝑊 ∈ TopVec → 𝑊 ∈ LMod) | |
2 | eqid 2735 | . . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
3 | 2 | tvctdrg 24217 | . . 3 ⊢ (𝑊 ∈ TopVec → (Scalar‘𝑊) ∈ TopDRing) |
4 | tdrgdrng 24198 | . . 3 ⊢ ((Scalar‘𝑊) ∈ TopDRing → (Scalar‘𝑊) ∈ DivRing) | |
5 | 3, 4 | syl 17 | . 2 ⊢ (𝑊 ∈ TopVec → (Scalar‘𝑊) ∈ DivRing) |
6 | 2 | islvec 21121 | . 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 2106 ‘cfv 6563 Scalarcsca 17301 DivRingcdr 20746 LModclmod 20875 LVecclvec 21119 TopDRingctdrg 24181 TopVecctvc 24183 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-iota 6516 df-fv 6571 df-ov 7434 df-lvec 21120 df-tdrg 24185 df-tlm 24186 df-tvc 24187 |
This theorem is referenced by: (None) |
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