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Theorem tvctdrg 18253
Description: The scalar field of a topological vector space is a topological division ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
Hypothesis
Ref Expression
tlmtrg.f  |-  F  =  (Scalar `  W )
Assertion
Ref Expression
tvctdrg  |-  ( W  e.  TopVec  ->  F  e. TopDRing )

Proof of Theorem tvctdrg
StepHypRef Expression
1 tlmtrg.f . . 3  |-  F  =  (Scalar `  W )
21istvc 18252 . 2  |-  ( W  e.  TopVec 
<->  ( W  e. TopMod  /\  F  e. TopDRing ) )
32simprbi 452 1  |-  ( W  e.  TopVec  ->  F  e. TopDRing )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1727   ` cfv 5483  Scalarcsca 13563  TopDRingctdrg 18217  TopModctlm 18218   TopVecctvc 18219
This theorem is referenced by:  tvclvec  18259
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-rex 2717  df-rab 2720  df-v 2964  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-br 4238  df-iota 5447  df-fv 5491  df-tvc 18223
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