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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 23565 | . 2 β’ (π β TopVec β π β LMod) | |
2 | eqid 2733 | . . . 4 β’ (Scalarβπ) = (Scalarβπ) | |
3 | 2 | tvctdrg 23560 | . . 3 β’ (π β TopVec β (Scalarβπ) β TopDRing) |
4 | tdrgdrng 23541 | . . 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 20580 | . 2 β’ (π β LVec β (π β LMod β§ (Scalarβπ) β DivRing)) |
7 | 1, 5, 6 | sylanbrc 584 | 1 β’ (π β TopVec β π β LVec) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wcel 2107 βcfv 6497 Scalarcsca 17141 DivRingcdr 20197 LModclmod 20336 LVecclvec 20578 TopDRingctdrg 23524 TopVecctvc 23526 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-iota 6449 df-fv 6505 df-ov 7361 df-lvec 20579 df-tdrg 23528 df-tlm 23529 df-tvc 23530 |
This theorem is referenced by: (None) |
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