Theorem vscacn 24326

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 24325	. 2 ⊢ (𝑊 ∈ TopMod ↔ ((𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ TopRing) ∧ · ∈ ((𝐾 ×t 𝐽) Cn 𝐽)))
6	5	simprbi 502	1 ⊢ (𝑊 ∈ TopMod → · ∈ ((𝐾 ×t 𝐽) Cn 𝐽))

Syntax hints:   → wi 4   ∧ w3a 1101   = wceq 1568   ∈ wcel 2150  ‘cfv 6540  (class class class)co 7414  Scalarcsca 17316  TopOpenctopn 17477  LModclmod 20964   ·_sf cscaf 20965   Cn ccn 23364   ×_t ctx 23700  TopMndctmd 24210  TopRingctrg 24296  TopModctlm 24298

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-ext 2742

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2099  df-clab 2749  df-cleq 2762  df-clel 2845  df-rab 3424  df-v 3464  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-iota 6496  df-fv 6548  df-ov 7417  df-tlm 24302

This theorem is referenced by:  cnmpt1vsca  24334  cnmpt2vsca  24335