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Theorem tvctlm 22812
 Description: A topological vector space is a topological module. (Contributed by Mario Carneiro, 5-Oct-2015.)
Assertion
Ref Expression
tvctlm (𝑊 ∈ TopVec → 𝑊 ∈ TopMod)

Proof of Theorem tvctlm
StepHypRef Expression
1 eqid 2798 . . 3 (Scalar‘𝑊) = (Scalar‘𝑊)
21istvc 22807 . 2 (𝑊 ∈ TopVec ↔ (𝑊 ∈ TopMod ∧ (Scalar‘𝑊) ∈ TopDRing))
32simplbi 501 1 (𝑊 ∈ TopVec → 𝑊 ∈ TopMod)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2111  ‘cfv 6325  Scalarcsca 16563  TopDRingctdrg 22772  TopModctlm 22773  TopVecctvc 22774 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-rab 3115  df-v 3443  df-un 3886  df-in 3888  df-ss 3898  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5032  df-iota 6284  df-fv 6333  df-tvc 22778 This theorem is referenced by:  tvclmod  22813
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