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Theorem tlmtmd 23108
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 2738 . . . 4 ( ·sf𝑊) = ( ·sf𝑊)
2 eqid 2738 . . . 4 (TopOpen‘𝑊) = (TopOpen‘𝑊)
3 eqid 2738 . . . 4 (Scalar‘𝑊) = (Scalar‘𝑊)
4 eqid 2738 . . . 4 (TopOpen‘(Scalar‘𝑊)) = (TopOpen‘(Scalar‘𝑊))
51, 2, 3, 4istlm 23106 . . 3 (𝑊 ∈ TopMod ↔ ((𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod ∧ (Scalar‘𝑊) ∈ TopRing) ∧ ( ·sf𝑊) ∈ (((TopOpen‘(Scalar‘𝑊)) ×t (TopOpen‘𝑊)) Cn (TopOpen‘𝑊))))
65simplbi 501 . 2 (𝑊 ∈ TopMod → (𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod ∧ (Scalar‘𝑊) ∈ TopRing))
76simp1d 1144 1 (𝑊 ∈ TopMod → 𝑊 ∈ TopMnd)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1089  wcel 2111  cfv 6397  (class class class)co 7231  Scalarcsca 16829  TopOpenctopn 16950  LModclmod 19923   ·sf cscaf 19924   Cn ccn 22145   ×t ctx 22481  TopMndctmd 22991  TopRingctrg 23077  TopModctlm 23079
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 2113  ax-9 2121  ax-ext 2709
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 2072  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3071  df-v 3422  df-dif 3883  df-un 3885  df-in 3887  df-ss 3897  df-nul 4252  df-if 4454  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4834  df-br 5068  df-iota 6355  df-fv 6405  df-ov 7234  df-tlm 23083
This theorem is referenced by:  tlmtps  23109  tlmtgp  23117
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