MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  vscacn Structured version   Visualization version   GIF version

Theorem vscacn 24194
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 24193 . 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 1087   = wceq 1540  wcel 2108  cfv 6561  (class class class)co 7431  Scalarcsca 17300  TopOpenctopn 17466  LModclmod 20858   ·sf cscaf 20859   Cn ccn 23232   ×t ctx 23568  TopMndctmd 24078  TopRingctrg 24164  TopModctlm 24166
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-ov 7434  df-tlm 24170
This theorem is referenced by:  cnmpt1vsca  24202  cnmpt2vsca  24203
  Copyright terms: Public domain W3C validator