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Theorem istvc 20819
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 5872 . . . 4  |-  ( x  =  W  ->  (Scalar `  x )  =  (Scalar `  W ) )
2 tlmtrg.f . . . 4  |-  F  =  (Scalar `  W )
31, 2syl6eqr 2516 . . 3  |-  ( x  =  W  ->  (Scalar `  x )  =  F )
43eleq1d 2526 . 2  |-  ( x  =  W  ->  (
(Scalar `  x )  e. TopDRing  <-> 
F  e. TopDRing ) )
5 df-tvc 20790 . 2  |-  TopVec  =  {
x  e. TopMod  |  (Scalar `  x )  e. TopDRing }
64, 5elrab2 3259 1  |-  ( W  e.  TopVec 
<->  ( W  e. TopMod  /\  F  e. TopDRing ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819   ` cfv 5594  Scalarcsca 14714  TopDRingctdrg 20784  TopModctlm 20785   TopVecctvc 20786
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-iota 5557  df-fv 5602  df-tvc 20790
This theorem is referenced by:  tvctdrg  20820  tvctlm  20824  nvctvc  21333
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