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Theorem tlmtmd 23554
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 2733 . . . 4 ( ·sf𝑊) = ( ·sf𝑊)
2 eqid 2733 . . . 4 (TopOpen‘𝑊) = (TopOpen‘𝑊)
3 eqid 2733 . . . 4 (Scalar‘𝑊) = (Scalar‘𝑊)
4 eqid 2733 . . . 4 (TopOpen‘(Scalar‘𝑊)) = (TopOpen‘(Scalar‘𝑊))
51, 2, 3, 4istlm 23552 . . 3 (𝑊 ∈ TopMod ↔ ((𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod ∧ (Scalar‘𝑊) ∈ TopRing) ∧ ( ·sf𝑊) ∈ (((TopOpen‘(Scalar‘𝑊)) ×t (TopOpen‘𝑊)) Cn (TopOpen‘𝑊))))
65simplbi 499 . 2 (𝑊 ∈ TopMod → (𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod ∧ (Scalar‘𝑊) ∈ TopRing))
76simp1d 1143 1 (𝑊 ∈ TopMod → 𝑊 ∈ TopMnd)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1088  wcel 2107  cfv 6497  (class class class)co 7358  Scalarcsca 17141  TopOpenctopn 17308  LModclmod 20336   ·sf cscaf 20337   Cn ccn 22591   ×t ctx 22927  TopMndctmd 23437  TopRingctrg 23523  TopModctlm 23525
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-iota 6449  df-fv 6505  df-ov 7361  df-tlm 23529
This theorem is referenced by:  tlmtps  23555  tlmtgp  23563
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