MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tvctlm Unicode version

Theorem tvctlm 18187
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 2412 . . 3  |-  (Scalar `  W )  =  (Scalar `  W )
21istvc 18182 . 2  |-  ( W  e.  TopVec 
<->  ( W  e. TopMod  /\  (Scalar `  W )  e. TopDRing )
)
32simplbi 447 1  |-  ( W  e.  TopVec  ->  W  e. TopMod )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1721   ` cfv 5421  Scalarcsca 13495  TopDRingctdrg 18147  TopModctlm 18148   TopVecctvc 18149
This theorem is referenced by:  tvclmod  18188
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-rex 2680  df-rab 2683  df-v 2926  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-iota 5385  df-fv 5429  df-tvc 18153
  Copyright terms: Public domain W3C validator