Theorem vscacn 24132

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 24131	. 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 1087   = wceq 1542   ∈ wcel 2114  ‘cfv 6491  (class class class)co 7358  Scalarcsca 17182  TopOpenctopn 17343  LModclmod 20813   ·_sf cscaf 20814   Cn ccn 23170   ×_t ctx 23506  TopMndctmd 24016  TopRingctrg 24102  TopModctlm 24104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2714  df-cleq 2727  df-clel 2810  df-rab 3399  df-v 3441  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4285  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-iota 6447  df-fv 6499  df-ov 7361  df-tlm 24108

This theorem is referenced by:  cnmpt1vsca  24140  cnmpt2vsca  24141