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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 23695 | . 2 β’ (π β TopVec β (π β TopMod β§ πΉ β TopDRing)) |
3 | 2 | simprbi 497 | 1 β’ (π β TopVec β πΉ β TopDRing) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1541 β wcel 2106 βcfv 6543 Scalarcsca 17199 TopDRingctdrg 23660 TopModctlm 23661 TopVecctvc 23662 |
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 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-iota 6495 df-fv 6551 df-tvc 23666 |
This theorem is referenced by: tvclvec 23702 |
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