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Theorem tlmtmd 24104
Description: A topological module is a topological monoid. (Contributed by Mario Carneiro, 5-Oct-2015.)
Assertion
Ref Expression
tlmtmd (𝑊 ∈ TopMod → 𝑊 ∈ TopMnd)

Proof of Theorem tlmtmd
StepHypRef Expression
1 eqid 2728 . . . 4 ( ·sf𝑊) = ( ·sf𝑊)
2 eqid 2728 . . . 4 (TopOpen‘𝑊) = (TopOpen‘𝑊)
3 eqid 2728 . . . 4 (Scalar‘𝑊) = (Scalar‘𝑊)
4 eqid 2728 . . . 4 (TopOpen‘(Scalar‘𝑊)) = (TopOpen‘(Scalar‘𝑊))
51, 2, 3, 4istlm 24102 . . 3 (𝑊 ∈ TopMod ↔ ((𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod ∧ (Scalar‘𝑊) ∈ TopRing) ∧ ( ·sf𝑊) ∈ (((TopOpen‘(Scalar‘𝑊)) ×t (TopOpen‘𝑊)) Cn (TopOpen‘𝑊))))
65simplbi 497 . 2 (𝑊 ∈ TopMod → (𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod ∧ (Scalar‘𝑊) ∈ TopRing))
76simp1d 1140 1 (𝑊 ∈ TopMod → 𝑊 ∈ TopMnd)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1085  wcel 2099  cfv 6548  (class class class)co 7420  Scalarcsca 17236  TopOpenctopn 17403  LModclmod 20743   ·sf cscaf 20744   Cn ccn 23141   ×t ctx 23477  TopMndctmd 23987  TopRingctrg 24073  TopModctlm 24075
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-iota 6500  df-fv 6556  df-ov 7423  df-tlm 24079
This theorem is referenced by:  tlmtps  24105  tlmtgp  24113
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