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Theorem vscacn 21799
 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 21798 . 2 (𝑊 ∈ TopMod ↔ ((𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ TopRing) ∧ · ∈ ((𝐾 ×t 𝐽) Cn 𝐽)))
65simprbi 479 1 (𝑊 ∈ TopMod → · ∈ ((𝐾 ×t 𝐽) Cn 𝐽))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ‘cfv 5804  (class class class)co 6549  Scalarcsca 15771  TopOpenctopn 15905  LModclmod 18686   ·sf cscaf 18687   Cn ccn 20838   ×t ctx 21173  TopMndctmd 21684  TopRingctrg 21769  TopModctlm 21771 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ov 6552  df-tlm 21775 This theorem is referenced by:  cnmpt1vsca  21807  cnmpt2vsca  21808
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