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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 24053 | . 2 β’ (π β TopVec β π β LMod) | |
| 2 | eqid 2726 | . . . 4 β’ (Scalarβπ) = (Scalarβπ) | |
| 3 | 2 | tvctdrg 24048 | . . 3 β’ (π β TopVec β (Scalarβπ) β TopDRing) |
| 4 | tdrgdrng 24029 | . . 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 20950 | . 2 β’ (π β LVec β (π β LMod β§ (Scalarβπ) β DivRing)) |
| 7 | 1, 5, 6 | sylanbrc 582 | 1 β’ (π β TopVec β π β LVec) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β wcel 2098 βcfv 6536 Scalarcsca 17207 DivRingcdr 20585 LModclmod 20704 LVecclvec 20948 TopDRingctdrg 24012 TopVecctvc 24014 |
| 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 2697 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-iota 6488 df-fv 6544 df-ov 7407 df-lvec 20949 df-tdrg 24016 df-tlm 24017 df-tvc 24018 |
| This theorem is referenced by: (None) |
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