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Theorem tlmtmd 24042
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 2726 . . . 4 ( ·sf𝑊) = ( ·sf𝑊)
2 eqid 2726 . . . 4 (TopOpen‘𝑊) = (TopOpen‘𝑊)
3 eqid 2726 . . . 4 (Scalar‘𝑊) = (Scalar‘𝑊)
4 eqid 2726 . . . 4 (TopOpen‘(Scalar‘𝑊)) = (TopOpen‘(Scalar‘𝑊))
51, 2, 3, 4istlm 24040 . . 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 1139 1 (𝑊 ∈ TopMod → 𝑊 ∈ TopMnd)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084  wcel 2098  cfv 6536  (class class class)co 7404  Scalarcsca 17207  TopOpenctopn 17374  LModclmod 20704   ·sf cscaf 20705   Cn ccn 23079   ×t ctx 23415  TopMndctmd 23925  TopRingctrg 24011  TopModctlm 24013
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 2697
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 2704  df-cleq 2718  df-clel 2804  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-iota 6488  df-fv 6544  df-ov 7407  df-tlm 24017
This theorem is referenced by:  tlmtps  24043  tlmtgp  24051
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