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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 24120 | . 2 β’ (π β TopVec β π β LMod) | |
| 2 | eqid 2727 | . . . 4 β’ (Scalarβπ) = (Scalarβπ) | |
| 3 | 2 | tvctdrg 24115 | . . 3 β’ (π β TopVec β (Scalarβπ) β TopDRing) |
| 4 | tdrgdrng 24096 | . . 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 20994 | . 2 β’ (π β LVec β (π β LMod β§ (Scalarβπ) β DivRing)) |
| 7 | 1, 5, 6 | sylanbrc 581 | 1 β’ (π β TopVec β π β LVec) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β wcel 2098 βcfv 6551 Scalarcsca 17241 DivRingcdr 20629 LModclmod 20748 LVecclvec 20992 TopDRingctdrg 24079 TopVecctvc 24081 |
| 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 2698 |
| This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2705 df-cleq 2719 df-clel 2805 df-rab 3429 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4325 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-br 5151 df-iota 6503 df-fv 6559 df-ov 7427 df-lvec 20993 df-tdrg 24083 df-tlm 24084 df-tvc 24085 |
| This theorem is referenced by: (None) |
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