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Mirrors > Home > MPE Home > Th. List > istvc | Structured version Visualization version GIF version |
Description: A topological vector space is a topological module over a topological division ring. (Contributed by Mario Carneiro, 5-Oct-2015.) |
Ref | Expression |
---|---|
tlmtrg.f | β’ πΉ = (Scalarβπ) |
Ref | Expression |
---|---|
istvc | β’ (π β TopVec β (π β TopMod β§ πΉ β TopDRing)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6888 | . . . 4 β’ (π₯ = π β (Scalarβπ₯) = (Scalarβπ)) | |
2 | tlmtrg.f | . . . 4 β’ πΉ = (Scalarβπ) | |
3 | 1, 2 | eqtr4di 2790 | . . 3 β’ (π₯ = π β (Scalarβπ₯) = πΉ) |
4 | 3 | eleq1d 2818 | . 2 β’ (π₯ = π β ((Scalarβπ₯) β TopDRing β πΉ β TopDRing)) |
5 | df-tvc 23658 | . 2 β’ TopVec = {π₯ β TopMod β£ (Scalarβπ₯) β TopDRing} | |
6 | 4, 5 | elrab2 3685 | 1 β’ (π β TopVec β (π β TopMod β§ πΉ β TopDRing)) |
Colors of variables: wff setvar class |
Syntax hints: β wb 205 β§ wa 396 = wceq 1541 β wcel 2106 βcfv 6540 Scalarcsca 17196 TopDRingctdrg 23652 TopModctlm 23653 TopVecctvc 23654 |
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 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-iota 6492 df-fv 6548 df-tvc 23658 |
This theorem is referenced by: tvctdrg 23688 tvctlm 23692 nvctvc 24208 |
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