Theorem vscacn 24124

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 24123	. 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 1086   = wceq 1540   ∈ wcel 2108  ‘cfv 6531  (class class class)co 7405  Scalarcsca 17274  TopOpenctopn 17435  LModclmod 20817   ·_sf cscaf 20818   Cn ccn 23162   ×_t ctx 23498  TopMndctmd 24008  TopRingctrg 24094  TopModctlm 24096

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-iota 6484  df-fv 6539  df-ov 7408  df-tlm 24100

This theorem is referenced by:  cnmpt1vsca  24132  cnmpt2vsca  24133