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Theorem istvc 17890
Description: A topological vector space is a topological module over a topological division ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
Hypothesis
Ref Expression
tlmtrg.f  |-  F  =  (Scalar `  W )
Assertion
Ref Expression
istvc  |-  ( W  e.  TopVec 
<->  ( W  e. TopMod  /\  F  e. TopDRing ) )

Proof of Theorem istvc
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 fveq2 5541 . . . 4  |-  ( x  =  W  ->  (Scalar `  x )  =  (Scalar `  W ) )
2 tlmtrg.f . . . 4  |-  F  =  (Scalar `  W )
31, 2syl6eqr 2346 . . 3  |-  ( x  =  W  ->  (Scalar `  x )  =  F )
43eleq1d 2362 . 2  |-  ( x  =  W  ->  (
(Scalar `  x )  e. TopDRing  <-> 
F  e. TopDRing ) )
5 df-tvc 17861 . 2  |-  TopVec  =  {
x  e. TopMod  |  (Scalar `  x )  e. TopDRing }
64, 5elrab2 2938 1  |-  ( W  e.  TopVec 
<->  ( W  e. TopMod  /\  F  e. TopDRing ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   ` cfv 5271  Scalarcsca 13227  TopDRingctdrg 17855  TopModctlm 17856   TopVecctvc 17857
This theorem is referenced by:  tvctdrg  17891  tvctlm  17895  nvctvc  18226
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-tvc 17861
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