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Theorem tlmlmod 22400
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 2778 . . . 4 ( ·sf𝑊) = ( ·sf𝑊)
2 eqid 2778 . . . 4 (TopOpen‘𝑊) = (TopOpen‘𝑊)
3 eqid 2778 . . . 4 (Scalar‘𝑊) = (Scalar‘𝑊)
4 eqid 2778 . . . 4 (TopOpen‘(Scalar‘𝑊)) = (TopOpen‘(Scalar‘𝑊))
51, 2, 3, 4istlm 22396 . . 3 (𝑊 ∈ TopMod ↔ ((𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod ∧ (Scalar‘𝑊) ∈ TopRing) ∧ ( ·sf𝑊) ∈ (((TopOpen‘(Scalar‘𝑊)) ×t (TopOpen‘𝑊)) Cn (TopOpen‘𝑊))))
65simplbi 493 . 2 (𝑊 ∈ TopMod → (𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod ∧ (Scalar‘𝑊) ∈ TopRing))
76simp2d 1134 1 (𝑊 ∈ TopMod → 𝑊 ∈ LMod)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1071  wcel 2107  cfv 6135  (class class class)co 6922  Scalarcsca 16341  TopOpenctopn 16468  LModclmod 19255   ·sf cscaf 19256   Cn ccn 21436   ×t ctx 21772  TopMndctmd 22282  TopRingctrg 22367  TopModctlm 22369
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-rex 3096  df-rab 3099  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-br 4887  df-iota 6099  df-fv 6143  df-ov 6925  df-tlm 22373
This theorem is referenced by:  tlmtgp  22407  tvclmod  22409
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