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Theorem tvctdrg 18088
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 18087 . 2  |-  ( W  e.  TopVec 
<->  ( W  e. TopMod  /\  F  e. TopDRing ) )
32simprbi 450 1  |-  ( W  e.  TopVec  ->  F  e. TopDRing )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1647    e. wcel 1715   ` cfv 5358  Scalarcsca 13419  TopDRingctdrg 18052  TopModctlm 18053   TopVecctvc 18054
This theorem is referenced by:  tvclvec  18094
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-rex 2634  df-rab 2637  df-v 2875  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-iota 5322  df-fv 5366  df-tvc 18058
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