![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > tlmtrg | Structured version Visualization version GIF version |
Description: The scalar ring of a topological module is a topological ring. (Contributed by Mario Carneiro, 5-Oct-2015.) |
Ref | Expression |
---|---|
tlmtrg.f | β’ πΉ = (Scalarβπ) |
Ref | Expression |
---|---|
tlmtrg | β’ (π β TopMod β πΉ β TopRing) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2728 | . . . 4 β’ ( Β·sf βπ) = ( Β·sf βπ) | |
2 | eqid 2728 | . . . 4 β’ (TopOpenβπ) = (TopOpenβπ) | |
3 | tlmtrg.f | . . . 4 β’ πΉ = (Scalarβπ) | |
4 | eqid 2728 | . . . 4 β’ (TopOpenβπΉ) = (TopOpenβπΉ) | |
5 | 1, 2, 3, 4 | istlm 24102 | . . 3 β’ (π β TopMod β ((π β TopMnd β§ π β LMod β§ πΉ β TopRing) β§ ( Β·sf βπ) β (((TopOpenβπΉ) Γt (TopOpenβπ)) Cn (TopOpenβπ)))) |
6 | 5 | simplbi 497 | . 2 β’ (π β TopMod β (π β TopMnd β§ π β LMod β§ πΉ β TopRing)) |
7 | 6 | simp3d 1142 | 1 β’ (π β TopMod β πΉ β TopRing) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ w3a 1085 = wceq 1534 β wcel 2099 βcfv 6548 (class class class)co 7420 Scalarcsca 17236 TopOpenctopn 17403 LModclmod 20743 Β·sf cscaf 20744 Cn ccn 23141 Γt ctx 23477 TopMndctmd 23987 TopRingctrg 24073 TopModctlm 24075 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-iota 6500 df-fv 6556 df-ov 7423 df-tlm 24079 |
This theorem is referenced by: tlmscatps 24108 |
Copyright terms: Public domain | W3C validator |