![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > vscacn | Structured version Visualization version GIF version |
Description: The scalar multiplication is continuous in a topological module. (Contributed by Mario Carneiro, 5-Oct-2015.) |
Ref | Expression |
---|---|
istlm.s | ⊢ · = ( ·sf ‘𝑊) |
istlm.j | ⊢ 𝐽 = (TopOpen‘𝑊) |
istlm.f | ⊢ 𝐹 = (Scalar‘𝑊) |
istlm.k | ⊢ 𝐾 = (TopOpen‘𝐹) |
Ref | Expression |
---|---|
vscacn | ⊢ (𝑊 ∈ TopMod → · ∈ ((𝐾 ×t 𝐽) Cn 𝐽)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | istlm.s | . . 3 ⊢ · = ( ·sf ‘𝑊) | |
2 | istlm.j | . . 3 ⊢ 𝐽 = (TopOpen‘𝑊) | |
3 | istlm.f | . . 3 ⊢ 𝐹 = (Scalar‘𝑊) | |
4 | istlm.k | . . 3 ⊢ 𝐾 = (TopOpen‘𝐹) | |
5 | 1, 2, 3, 4 | istlm 23520 | . 2 ⊢ (𝑊 ∈ TopMod ↔ ((𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ TopRing) ∧ · ∈ ((𝐾 ×t 𝐽) Cn 𝐽))) |
6 | 5 | simprbi 497 | 1 ⊢ (𝑊 ∈ TopMod → · ∈ ((𝐾 ×t 𝐽) Cn 𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ‘cfv 6493 (class class class)co 7353 Scalarcsca 17128 TopOpenctopn 17295 LModclmod 20307 ·sf cscaf 20308 Cn ccn 22559 ×t ctx 22895 TopMndctmd 23405 TopRingctrg 23491 TopModctlm 23493 |
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 2707 |
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 2714 df-cleq 2728 df-clel 2814 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-iota 6445 df-fv 6501 df-ov 7356 df-tlm 23497 |
This theorem is referenced by: cnmpt1vsca 23529 cnmpt2vsca 23530 |
Copyright terms: Public domain | W3C validator |