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Theorem tvctlm 17879
Description: A topological vector space is a topological module. (Contributed by Mario Carneiro, 5-Oct-2015.)
Assertion
Ref Expression
tvctlm  |-  ( W  e.  TopVec  ->  W  e. TopMod )

Proof of Theorem tvctlm
StepHypRef Expression
1 eqid 2283 . . 3  |-  (Scalar `  W )  =  (Scalar `  W )
21istvc 17874 . 2  |-  ( W  e.  TopVec 
<->  ( W  e. TopMod  /\  (Scalar `  W )  e. TopDRing )
)
32simplbi 446 1  |-  ( W  e.  TopVec  ->  W  e. TopMod )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1684   ` cfv 5255  Scalarcsca 13211  TopDRingctdrg 17839  TopModctlm 17840   TopVecctvc 17841
This theorem is referenced by:  tvclmod  17880
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-tvc 17845
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