Theorem vscacn 23245

Description: The scalar multiplication is continuous in a topological module. (Contributed by Mario Carneiro, 5-Oct-2015.)

Hypotheses

Ref	Expression
istlm.s	⊢ · = ( ·sf ‘𝑊)
istlm.j	⊢ 𝐽 = (TopOpen‘𝑊)
istlm.f	⊢ 𝐹 = (Scalar‘𝑊)
istlm.k	⊢ 𝐾 = (TopOpen‘𝐹)

Assertion

Ref	Expression
vscacn	⊢ (𝑊 ∈ TopMod → · ∈ ((𝐾 ×t 𝐽) Cn 𝐽))

Proof of Theorem vscacn

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 23244	. 2 ⊢ (𝑊 ∈ TopMod ↔ ((𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ TopRing) ∧ · ∈ ((𝐾 ×t 𝐽) Cn 𝐽)))
6	5	simprbi 496	1 ⊢ (𝑊 ∈ TopMod → · ∈ ((𝐾 ×t 𝐽) Cn 𝐽))

Syntax hints:   → wi 4   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  ‘cfv 6418  (class class class)co 7255  Scalarcsca 16891  TopOpenctopn 17049  LModclmod 20038   ·_sf cscaf 20039   Cn ccn 22283   ×_t ctx 22619  TopMndctmd 23129  TopRingctrg 23215  TopModctlm 23217

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-iota 6376  df-fv 6426  df-ov 7258  df-tlm 23221

This theorem is referenced by:  cnmpt1vsca  23253  cnmpt2vsca  23254