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Theorem vscacn 23083
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 23082 . 2 (𝑊 ∈ TopMod ↔ ((𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ TopRing) ∧ · ∈ ((𝐾 ×t 𝐽) Cn 𝐽)))
65simprbi 500 1 (𝑊 ∈ TopMod → · ∈ ((𝐾 ×t 𝐽) Cn 𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1089   = wceq 1543  wcel 2110  cfv 6380  (class class class)co 7213  Scalarcsca 16805  TopOpenctopn 16926  LModclmod 19899   ·sf cscaf 19900   Cn ccn 22121   ×t ctx 22457  TopMndctmd 22967  TopRingctrg 23053  TopModctlm 23055
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 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-iota 6338  df-fv 6388  df-ov 7216  df-tlm 23059
This theorem is referenced by:  cnmpt1vsca  23091  cnmpt2vsca  23092
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