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Mirrors > Home > MPE Home > Th. List > tvctdrg | Structured version Visualization version GIF version |
Description: The scalar field of a topological vector space is a topological division ring. (Contributed by Mario Carneiro, 5-Oct-2015.) |
Ref | Expression |
---|---|
tlmtrg.f | β’ πΉ = (Scalarβπ) |
Ref | Expression |
---|---|
tvctdrg | β’ (π β TopVec β πΉ β TopDRing) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tlmtrg.f | . . 3 β’ πΉ = (Scalarβπ) | |
2 | 1 | istvc 24046 | . 2 β’ (π β TopVec β (π β TopMod β§ πΉ β TopDRing)) |
3 | 2 | simprbi 496 | 1 β’ (π β TopVec β πΉ β TopDRing) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 βcfv 6536 Scalarcsca 17206 TopDRingctdrg 24011 TopModctlm 24012 TopVecctvc 24013 |
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-tvc 24017 |
This theorem is referenced by: tvclvec 24053 |
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