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Theorem vscacn 24210
Description: The scalar multiplication is continuous in a topological module. (Contributed by Mario Carneiro, 5-Oct-2015.)
Hypotheses
Ref Expression
istlm.s · = ( ·sf𝑊)
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
StepHypRef Expression
1 istlm.s . . 3 · = ( ·sf𝑊)
2 istlm.j . . 3 𝐽 = (TopOpen‘𝑊)
3 istlm.f . . 3 𝐹 = (Scalar‘𝑊)
4 istlm.k . . 3 𝐾 = (TopOpen‘𝐹)
51, 2, 3, 4istlm 24209 . 2 (𝑊 ∈ TopMod ↔ ((𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ TopRing) ∧ · ∈ ((𝐾 ×t 𝐽) Cn 𝐽)))
65simprbi 496 1 (𝑊 ∈ TopMod → · ∈ ((𝐾 ×t 𝐽) Cn 𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1086   = wceq 1537  wcel 2106  cfv 6563  (class class class)co 7431  Scalarcsca 17301  TopOpenctopn 17468  LModclmod 20875   ·sf cscaf 20876   Cn ccn 23248   ×t ctx 23584  TopMndctmd 24094  TopRingctrg 24180  TopModctlm 24182
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-ov 7434  df-tlm 24186
This theorem is referenced by:  cnmpt1vsca  24218  cnmpt2vsca  24219
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