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Theorem istvc 18221
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 5728 . . . 4  |-  ( x  =  W  ->  (Scalar `  x )  =  (Scalar `  W ) )
2 tlmtrg.f . . . 4  |-  F  =  (Scalar `  W )
31, 2syl6eqr 2486 . . 3  |-  ( x  =  W  ->  (Scalar `  x )  =  F )
43eleq1d 2502 . 2  |-  ( x  =  W  ->  (
(Scalar `  x )  e. TopDRing  <-> 
F  e. TopDRing ) )
5 df-tvc 18192 . 2  |-  TopVec  =  {
x  e. TopMod  |  (Scalar `  x )  e. TopDRing }
64, 5elrab2 3094 1  |-  ( W  e.  TopVec 
<->  ( W  e. TopMod  /\  F  e. TopDRing ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   ` cfv 5454  Scalarcsca 13532  TopDRingctdrg 18186  TopModctlm 18187   TopVecctvc 18188
This theorem is referenced by:  tvctdrg  18222  tvctlm  18226  nvctvc  18735
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-iota 5418  df-fv 5462  df-tvc 18192
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