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Theorem tlmlmod 24017
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 2724 . . . 4 ( ·sf𝑊) = ( ·sf𝑊)
2 eqid 2724 . . . 4 (TopOpen‘𝑊) = (TopOpen‘𝑊)
3 eqid 2724 . . . 4 (Scalar‘𝑊) = (Scalar‘𝑊)
4 eqid 2724 . . . 4 (TopOpen‘(Scalar‘𝑊)) = (TopOpen‘(Scalar‘𝑊))
51, 2, 3, 4istlm 24013 . . 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))
76simp2d 1140 1 (𝑊 ∈ TopMod → 𝑊 ∈ LMod)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084  wcel 2098  cfv 6534  (class class class)co 7402  Scalarcsca 17201  TopOpenctopn 17368  LModclmod 20698   ·sf cscaf 20699   Cn ccn 23052   ×t ctx 23388  TopMndctmd 23898  TopRingctrg 23984  TopModctlm 23986
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-iota 6486  df-fv 6542  df-ov 7405  df-tlm 23990
This theorem is referenced by:  tlmtgp  24024  tvclmod  24026
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