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Theorem istvc 19764
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 5689 . . . 4  |-  ( x  =  W  ->  (Scalar `  x )  =  (Scalar `  W ) )
2 tlmtrg.f . . . 4  |-  F  =  (Scalar `  W )
31, 2syl6eqr 2491 . . 3  |-  ( x  =  W  ->  (Scalar `  x )  =  F )
43eleq1d 2507 . 2  |-  ( x  =  W  ->  (
(Scalar `  x )  e. TopDRing  <-> 
F  e. TopDRing ) )
5 df-tvc 19735 . 2  |-  TopVec  =  {
x  e. TopMod  |  (Scalar `  x )  e. TopDRing }
64, 5elrab2 3117 1  |-  ( W  e.  TopVec 
<->  ( W  e. TopMod  /\  F  e. TopDRing ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   ` cfv 5416  Scalarcsca 14239  TopDRingctdrg 19729  TopModctlm 19730   TopVecctvc 19731
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-rex 2719  df-rab 2722  df-v 2972  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-br 4291  df-iota 5379  df-fv 5424  df-tvc 19735
This theorem is referenced by:  tvctdrg  19765  tvctlm  19769  nvctvc  20278
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