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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 | β’ (π β TopVec β π β LVec) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | tvclmod 23693 | . 2 β’ (π β TopVec β π β LMod) | |
| 2 | eqid 2732 | . . . 4 β’ (Scalarβπ) = (Scalarβπ) | |
| 3 | 2 | tvctdrg 23688 | . . 3 β’ (π β TopVec β (Scalarβπ) β TopDRing) |
| 4 | tdrgdrng 23669 | . . 3 β’ ((Scalarβπ) β TopDRing β (Scalarβπ) β DivRing) | |
| 5 | 3, 4 | syl 17 | . 2 β’ (π β TopVec β (Scalarβπ) β DivRing) |
| 6 | 2 | islvec 20707 | . 2 β’ (π β LVec β (π β LMod β§ (Scalarβπ) β DivRing)) |
| 7 | 1, 5, 6 | sylanbrc 583 | 1 β’ (π β TopVec β π β LVec) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β wcel 2106 βcfv 6540 Scalarcsca 17196 DivRingcdr 20307 LModclmod 20463 LVecclvec 20705 TopDRingctdrg 23652 TopVecctvc 23654 |
| 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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 |
| This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-rab 3433 df-v 3476 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-iota 6492 df-fv 6548 df-ov 7408 df-lvec 20706 df-tdrg 23656 df-tlm 23657 df-tvc 23658 |
| This theorem is referenced by: (None) |
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