Metamath Proof Explorer
Description: The scalar multiplication is continuous in a topological module.
(Contributed by Mario Carneiro, 5-Oct-2015)
|
|
Ref |
Expression |
|
Hypotheses |
istlm.s |
⊢ · = ( ·sf ‘ 𝑊 ) |
|
|
istlm.j |
⊢ 𝐽 = ( TopOpen ‘ 𝑊 ) |
|
|
istlm.f |
⊢ 𝐹 = ( Scalar ‘ 𝑊 ) |
|
|
istlm.k |
⊢ 𝐾 = ( TopOpen ‘ 𝐹 ) |
|
Assertion |
vscacn |
⊢ ( 𝑊 ∈ TopMod → · ∈ ( ( 𝐾 ×t 𝐽 ) Cn 𝐽 ) ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
istlm.s |
⊢ · = ( ·sf ‘ 𝑊 ) |
2 |
|
istlm.j |
⊢ 𝐽 = ( TopOpen ‘ 𝑊 ) |
3 |
|
istlm.f |
⊢ 𝐹 = ( Scalar ‘ 𝑊 ) |
4 |
|
istlm.k |
⊢ 𝐾 = ( TopOpen ‘ 𝐹 ) |
5 |
1 2 3 4
|
istlm |
⊢ ( 𝑊 ∈ TopMod ↔ ( ( 𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ TopRing ) ∧ · ∈ ( ( 𝐾 ×t 𝐽 ) Cn 𝐽 ) ) ) |
6 |
5
|
simprbi |
⊢ ( 𝑊 ∈ TopMod → · ∈ ( ( 𝐾 ×t 𝐽 ) Cn 𝐽 ) ) |