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Theorem tlmtrg 22801
Description: The scalar ring of a topological module is a topological ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
Hypothesis
Ref Expression
tlmtrg.f 𝐹 = (Scalar‘𝑊)
Assertion
Ref Expression
tlmtrg (𝑊 ∈ TopMod → 𝐹 ∈ TopRing)

Proof of Theorem tlmtrg
StepHypRef Expression
1 eqid 2824 . . . 4 ( ·sf𝑊) = ( ·sf𝑊)
2 eqid 2824 . . . 4 (TopOpen‘𝑊) = (TopOpen‘𝑊)
3 tlmtrg.f . . . 4 𝐹 = (Scalar‘𝑊)
4 eqid 2824 . . . 4 (TopOpen‘𝐹) = (TopOpen‘𝐹)
51, 2, 3, 4istlm 22796 . . 3 (𝑊 ∈ TopMod ↔ ((𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ TopRing) ∧ ( ·sf𝑊) ∈ (((TopOpen‘𝐹) ×t (TopOpen‘𝑊)) Cn (TopOpen‘𝑊))))
65simplbi 501 . 2 (𝑊 ∈ TopMod → (𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ TopRing))
76simp3d 1141 1 (𝑊 ∈ TopMod → 𝐹 ∈ TopRing)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084   = wceq 1538  wcel 2115  cfv 6343  (class class class)co 7149  Scalarcsca 16568  TopOpenctopn 16695  LModclmod 19634   ·sf cscaf 19635   Cn ccn 21835   ×t ctx 22171  TopMndctmd 22681  TopRingctrg 22767  TopModctlm 22769
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 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-rab 3142  df-v 3482  df-un 3924  df-in 3926  df-ss 3936  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-iota 6302  df-fv 6351  df-ov 7152  df-tlm 22773
This theorem is referenced by:  tlmscatps  22802
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