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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 | ⊢ 𝐹 = (Scalar‘𝑊) |
Ref | Expression |
---|---|
tvctdrg | ⊢ (𝑊 ∈ TopVec → 𝐹 ∈ TopDRing) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tlmtrg.f | . . 3 ⊢ 𝐹 = (Scalar‘𝑊) | |
2 | 1 | istvc 24216 | . 2 ⊢ (𝑊 ∈ TopVec ↔ (𝑊 ∈ TopMod ∧ 𝐹 ∈ TopDRing)) |
3 | 2 | simprbi 496 | 1 ⊢ (𝑊 ∈ TopVec → 𝐹 ∈ TopDRing) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ‘cfv 6563 Scalarcsca 17301 TopDRingctdrg 24181 TopModctlm 24182 TopVecctvc 24183 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-iota 6516 df-fv 6571 df-tvc 24187 |
This theorem is referenced by: tvclvec 24223 |
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