Theorem vscacn 24173

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 24172	. 2 ⊢ (𝑊 ∈ TopMod ↔ ((𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ TopRing) ∧ · ∈ ((𝐾 ×t 𝐽) Cn 𝐽)))
6	5	simprbi 499	1 ⊢ (𝑊 ∈ TopMod → · ∈ ((𝐾 ×t 𝐽) Cn 𝐽))

Syntax hints:   → wi 4   ∧ w3a 1093   = wceq 1548   ∈ wcel 2121  ‘cfv 6489  (class class class)co 7360  Scalarcsca 17218  TopOpenctopn 17379  LModclmod 20854   ·_sf cscaf 20855   Cn ccn 23211   ×_t ctx 23547  TopMndctmd 24057  TopRingctrg 24143  TopModctlm 24145

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-rab 3394  df-v 3435  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-br 5076  df-iota 6445  df-fv 6497  df-ov 7363  df-tlm 24149

This theorem is referenced by:  cnmpt1vsca  24181  cnmpt2vsca  24182