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Theorem tlmlmod 21986
Description: A topological module is a left module. (Contributed by Mario Carneiro, 5-Oct-2015.)
Assertion
Ref Expression
tlmlmod (𝑊 ∈ TopMod → 𝑊 ∈ LMod)

Proof of Theorem tlmlmod
StepHypRef Expression
1 eqid 2621 . . . 4 ( ·sf𝑊) = ( ·sf𝑊)
2 eqid 2621 . . . 4 (TopOpen‘𝑊) = (TopOpen‘𝑊)
3 eqid 2621 . . . 4 (Scalar‘𝑊) = (Scalar‘𝑊)
4 eqid 2621 . . . 4 (TopOpen‘(Scalar‘𝑊)) = (TopOpen‘(Scalar‘𝑊))
51, 2, 3, 4istlm 21982 . . 3 (𝑊 ∈ TopMod ↔ ((𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod ∧ (Scalar‘𝑊) ∈ TopRing) ∧ ( ·sf𝑊) ∈ (((TopOpen‘(Scalar‘𝑊)) ×t (TopOpen‘𝑊)) Cn (TopOpen‘𝑊))))
65simplbi 476 . 2 (𝑊 ∈ TopMod → (𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod ∧ (Scalar‘𝑊) ∈ TopRing))
76simp2d 1073 1 (𝑊 ∈ TopMod → 𝑊 ∈ LMod)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1037  wcel 1989  cfv 5886  (class class class)co 6647  Scalarcsca 15938  TopOpenctopn 16076  LModclmod 18857   ·sf cscaf 18858   Cn ccn 21022   ×t ctx 21357  TopMndctmd 21868  TopRingctrg 21953  TopModctlm 21955
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-rex 2917  df-rab 2920  df-v 3200  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-iota 5849  df-fv 5894  df-ov 6650  df-tlm 21959
This theorem is referenced by:  tlmtgp  21993  tvclmod  21995
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