![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > tlmscatps | Structured version Visualization version GIF version |
Description: The scalar ring of a topological module is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.) |
Ref | Expression |
---|---|
tlmtrg.f | ⊢ 𝐹 = (Scalar‘𝑊) |
Ref | Expression |
---|---|
tlmscatps | ⊢ (𝑊 ∈ TopMod → 𝐹 ∈ TopSp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tlmtrg.f | . . 3 ⊢ 𝐹 = (Scalar‘𝑊) | |
2 | 1 | tlmtrg 24214 | . 2 ⊢ (𝑊 ∈ TopMod → 𝐹 ∈ TopRing) |
3 | trgtps 24194 | . 2 ⊢ (𝐹 ∈ TopRing → 𝐹 ∈ TopSp) | |
4 | 2, 3 | syl 17 | 1 ⊢ (𝑊 ∈ TopMod → 𝐹 ∈ TopSp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ‘cfv 6563 Scalarcsca 17301 TopSpctps 22954 TopRingctrg 24180 TopModctlm 24182 |
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 ax-nul 5312 |
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-ne 2939 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-in 3970 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-ov 7434 df-tmd 24096 df-tgp 24097 df-trg 24184 df-tlm 24186 |
This theorem is referenced by: cnmpt1vsca 24218 cnmpt2vsca 24219 tlmtgp 24220 |
Copyright terms: Public domain | W3C validator |